THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD KASPER K. S. ANDERSEN, JESPER GRODAL, JESPER M. MØLLER, AND ANTONIO VIRUEL Contents 1. Introduction 1 Relationship to the Lie case and the conjectural picture for p = 2 * * 7 Organization of the paper and notation * * 7 Acknowledgments 8 2. The map Aut(BX) ! Aut(BNX ) 8 3. Automorphisms of maximal torus normalizers 9 4. Reduction to connected, center-free simple p-compact groups * * 12 5. An integral version of a theorem of Nakajima and realization of p-compact groups 18 6. Proof of the main theorem using Sections 8, 9, 10, 11, and 12 * * 21 7. Consequences of the main theorem 32 8. Elementary abelian subgroups of the exceptional groups * * 38 8.1. Recollection of some results on linear algebraic groups * * 39 8.2. The groups E6(C) and 3E6(C), p = 3 42 8.3. The group E8(C), p = 3 52 8.4. The group 2E7(C), p = 3 59 9. Non-toral elementary abelian p-subgroups of projective unitary groups * * 62 10. Calculation of the obstruction groups * * 63 10.1. The toral part 64 10.2. The non-toral part for the exceptional groups * * 66 10.3. The non-toral part for the projective unitary groups * * 69 11. Appendix: The classification of finite Zp-reflection groups * * 71 12. Appendix: Invariant rings of finite Zp-reflection groups, p odd (followi* *ng Notbohm) 76 13. Appendix: Outer automorphisms of exotic finite Zp-reflection groups * * 79 References 82 1. Introduction It has been a central goal in homotopy theory for about half a century to sin* *gle out the homotopy theoretic properties characterizing compact Lie groups, and obtain a c* *orrespond- ing classification, starting with the work of Hopf [68] and Serre [113, x4] on * *H-spaces and loop spaces. Materializing old dreams of Sullivan [123, p. 5.96] and Rector [1* *11], Dwyer ___________ The first named author was supported by EU grant EEC HPRN-CT-1999-00119. The second named author was supported by NSF grant DMS-0104318, a Clay liftoff * *fellowship, and the Institute for Advanced Study for different parts of the time this research was * *carried out. 1 2 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL and Wilkerson, in their seminal paper [48], introduced the notion of a p-compac* *t group, as a suitably "finite" p-complete loop space, and proved that p-compact groups * *have many Lie-like properties. Even before their introduction it has been the hope [110]* *, and later the conjecture [51, 80, 41], that these objects should admit a classification m* *uch like the classification of compact connected Lie groups, and the work toward this has be* *en carried out by many authors. The goal of this paper is to complete the proof of the cla* *ssification theorem for p an odd prime, showing that there is a one-to-one correspondence b* *etween connected p-compact groups and finite reflection groups over the p-adic integer* *s Zp. We do this by providing the last_and rather intricate_piece, namely that the p-com* *pletion of the exceptional compact Lie groups are uniquely determined as p-compact groups * *by their Weyl groups, seen as Zp-reflection groups. In fact we will give an essentially * *self-contained proof of the entire classification theorem. We start by very briefly introducing p-compact groups and some objects associ* *ated to them necessary to state the classification theorem_we will later in the introdu* *ction return to the history behind the various steps of the proof. We refer the reader to [4* *8] for more details on p-compact groups and also recommend the overview articles [41, 80, 8* *8]; we likewise point out that it is the technical advances on homotopy fixed points b* *y Miller [84], Lannes [79], and others which makes this theory possible. A space X with a loop space structure, for short a loop space, is a triple (X* *, BX, e) where BX is a pointed connected space, called the classifying space of X, and e : X !* * BX is a homotopy equivalence. A p-compact group is a loop space with the two additional* * properties that H*(X; Fp) is finite dimensional over Fp (to be thought of as öc mpactness"* *) and that BX is Fp-local [16][48, x11] (or, in this connection, equivalently Fp-complete * *[17]). Often we refer to a loop space simply as X. When working with a loop space we shall o* *nly be concerned with its classifying space BX, since this determines the rest of the * *structure_ indeed, we could instead have defined a p-compact group to be a space BX with t* *he above properties. The loop space (G^p, BG^p, e), corresponding to a pair (G, p) (whe* *re p is a prime, G a compact Lie group with component group a finite p-group, and (.)^pde* *notes Fp-completion [17][48, x11]) is a p-compact group. (Note however that the Lie g* *roup G is not uniquely determined by BG^p, since we are only focusing on the structure "v* *isible at the prime p"; e.g., B SO(2n + 1)^p' B Sp(n)^pif p 6= 2, as originally proven by* * Friedlander [59]; see Theorem 11.4 for a complete analysis.) A morphism X ! Y between loop spaces is a pointed map of spaces BX ! BY . We * *say that two morphisms are conjugate if the corresponding maps of classifying space* *s are freely homotopic. A morphism is called an equivalence if it has an inverse up to conju* *gation, or in other words if BX ! BY is a homotopy equivalence. If X and Y are p-compact grou* *ps, we call a morphism a monomorphism if the homotopy fiber of the map BX ! BY is Fp-f* *inite. The loop space corresponding to the p-completed classifying space BT = (BU(1)* *r)^pis called a p-compact torus of rank r. A maximal torus in X is a monomorphism T !* * X such that the homotopy fiber of BT ! BX has non-zero Euler characteristic. (We * *define the Euler characteristic as the alternating sum of the Fp-dimensions of the Fp-* *homology groups.) Fundamental to the theory of p-compact groups is the theorem of Dwyer-* *Wilkerson [48] that, analogously to the classical situation, any p-compact group admits a* * maximal torus. It is unique in the sense that for any other maximal torus i0 : T 0! X,* * there exists an equivalence ' : T ! T 0such that i0' and i are conjugate. (Note that * *there is a subtle difference between this statement and the classical statement of being ü* * nique up THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 3 to conjugation" due to the fact that a maximal torus is defined to be a map and* * not a subgroup.) Fix a p-compact group X with maximal torus i : T ! X of rank r. Replace the map Bi : BT ! BX by an equivalent fibration, and define the Weyl space WX (T ) * *as the topological monoid of self-maps BT ! BT over BX. The Weyl group is defined* * as WX (T ) = ß0(WX (T )) [48, Def. 9.2]. By [48, Prop. 9.5] WX (T ) is a finite gr* *oup of order Ø(X=T ). Furthermore, by [48, Thm. 9.7], if X is connected then WX (T ) identi* *fies with the set of conjugacy classes of self-equivalences ' of T such that i and i' are* * conjugate. In other words, the canonical homomorphism WX (T ) ! Aut(ß1(T )) is injective, * *so we can view WX (T ) as a subgroup of GL r(Zp), and this subgroup is independent of* * T up to conjugation in GL r(Zp). Now, by [48, Thm. 9.7] this exhibits (WX , ß1(T )) as a finite reflection gro* *up over Zp. Finite reflection groups over Zp have been classified for p odd by Notbohm [95]* * extending the classification over Qp by Clark-Ewing [29] (which again builds on the class* *ification over C of Shephard-Todd [115]); we recall this classification in Section 11 and exte* *nd Notbohm's result to all primes. (Recall that a finite Zp-reflection group is a pair (W, * *L) where L is a finitely generated free Zp-module, and W is a finite subgroup of Aut(L) gener* *ated by elements ff such that 1-ff has rank one viewed as a matrix over Qp; two finite * *Zp-reflection groups (W, L) and (W 0, L0) are called equivalent, if we can find a Zp-linear i* *somorphism ' : L ! L0such that the group 'W '-1 equals W 0.) The main classification theorem which we complete in this paper, is the follo* *wing. Theorem 1.1. Let (W, L) be the assignment which to each connected p-compact gro* *up X assigns the Weyl group WX , seen as a Zp-reflection group via the action of WX * * on its Zp-lattice LX = ß1(T ). If p is an odd prime, then the assignment (W, L) gives * *a bijection between the set of connected p-compact groups up to equivalence and the set of * *finite Zp- reflection groups up to equivalence. Furthermore the group of outer automorphisms of X (that is the group of unpoi* *nted homotopy classes of self-homotopy equivalences of BX) is naturally isomorphic t* *o the outer automorphism group of (WX , LX ), that is NGL(LX)(WX )=WX . (See Lemma 2.1 for * *the map giving this equivalence of automorphism groups, and Section 13 for a calculatio* *n of these.) In particular this proves, for p odd, Conjecture 5.3 in [41] (see Theorem 1.4* *). The self- map part of the statement can be viewed as an extension to p-compact groups, p * *odd, of the main result in [75], which again extends [74]. (Our method of proof is via * *centralizers and so üd al", but logically independent, of the one in [74, 75] (see e.g. [40,* * 65]).) By inspection of the classification of finite Zp-reflection groups one sees t* *hat the theorem has as a corollary that the theory of p-compact groups on the level of objects * *splits in two parts, as has been conjectured (Conjecture 5.1 and 5.2 in [41]). Theorem 1.2. Let X be a connected p-compact group, p odd. Then X can be written* * as a product of p-compact groups X ~=G^px X0 where G is a compact connected Lie group, and X0 is a direct product of exotic * *p-compact groups. (A p-compact group is called exotic if (W, L Q) is irreducible and does not* * come from a Z-reflection group; see Section 11 for more information.) Theorem 1.1 has both * *an existence and a uniqueness part to it, the existence part being to realize all possible f* *inite Zp-reflection 4 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL groups as Weyl groups of connected p-compact groups. The finite Zp-reflection * *groups which come from compact Lie groups are of course realizable, and the finite Zp-* *reflection groups where p does not divide the order of the group can also relatively easil* *y be dealt with, as done by Sullivan [123, p. 5.96] and Clark-Ewing [29] long before p-com* *pact groups were officially defined. The remaining cases were realized by Quillen [108, x10* *], Zabrodsky [134], Aguad'e [3] and Notbohm-Oliver [102] [97, Thm. 1.4]. (The classification* * of finite Zp- reflection groups guarantees that the construction of these examples actually e* *nables you to construct all finite Zp-reflection groups as Weyl groups of connected p-comp* *act groups.) The work toward the uniqueness part, to show that a p-compact group is unique* *ly de- termined by its Weyl group, also predates the introduction of p-compact groups.* * The quest was initiated by Dwyer-Miller-Wilkerson [56, 44] (building on [1]) who proved t* *he state- ment (using slightly different language) in the case where p is prime to the or* *der of WX as well as for SU (2)^2. Notbohm [100] and Møller-Notbohm [87, Thm. 1.9] extend* *ed this to a uniqueness statement for all p-compact groups X where Zp[LX ]WX (the ring* * of invari- ant polynomial functions on LX ) is a polynomial algebra and (W, L) comes from * *a finite Z-reflection group. Notbohm [102, 97] subsequently also handled the cases wher* *e (W, L) does not come from a finite Z-reflection group. (Beware that in [102, 97] Notbo* *hm used a apparently weaker uniqueness statement, from which the above statement however * *can be deduced; see Remark 7.11.) To get general statements beyond the case where Zp[LX ]WX is a polynomial al* *gebra, i.e., to attack the cases where there exists p-torsion in the cohomology ring, the fi* *rst step is to reduce the classification to the case of simple, center-free p-compact groups. * *(A p-compact group X is called simple if LX Q is an irreducible W -representation and that* * it is center- free is for p odd equivalent to that (LX Z=p1 )W = 0.) The results necessary t* *o obtain this reduction was achieved by the splitting theorem of Dwyer-Wilkerson [50] and Not* *bohm [99] along with properties of the center of a p-compact group established by Dwyer-W* *ilkerson [49] and Møller-Notbohm [86]. (We carry out this reduction in Section 4; see al* *so [89].) An analysis of the classification of finite Zp-reflection groups together wit* *h explicit calcu- lations (see [96] and Theorem 12.2) shows that, for p odd, Zp[LX ]WX is a polyn* *omial algebra for all irreducible center-free finite Zp-reflection groups except the reflecti* *on groups coming p-compact groups P U(n)^p, E8^5, F4^3, E6^3, E7^3, and E8^3. (For exceptional c* *ompact Lie groups the notation E6 etc. denotes their adjoint form.) The case P U(n)^pwas handled by Broto-Viruel [19] (using a Bockstein spectral* * sequence argument to deduce it from the result for SU (n)), generalizing earlier partial* * results of Broto-Viruel [20] and Møller [85]. The remaining step in the classification is* * hence to handle the exceptional compact Lie groups, in particular the problematic E-fami* *ly at the prime 3, and this is what is carried out in this paper. (The fourth named autho* *r has also given alternative proofs for F4^3and E8^5in [126] and [125].) Theorem 1.3. Let X be a connected p-compact group, for p odd, with Weyl group e* *qual to (WG , LG Zp) for one of the pairs (F4, p = 3), (E8, p = 5), (E6, p = 3), (* *E7, p = 3), or (E8, p = 3). Then X is equivalent, as a p-compact group, to the p-completio* *n of the corresponding exceptional group G. We will in fact give an essentially self-contained proof of the entire classi* *fication The- orem 1.1, since most of the work is necessary anyway for setting up the proof f* *or the exceptional cases. We however still rely on the classification of finite Zp-ref* *lection groups THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 5 (see [95, 96] and Section 11 and 12) as well as the above mentioned structural * *results from [48, 49, 86, 50, 99, 90]. The main ingredient in handling the exceptional groups is to get sufficiently* * detailed information about their many conjugacy classes of elementary abelian p-subgroup* *s (carried out in Section 8, expanding on the work of Griess [63]), and then to use this i* *nformation to show the relevant obstruction groups are trivial (carried out in Section 10), u* *sing formulas of Oliver [103] (see also [65]). It is possible to formulate a more topological version of the uniqueness part* * of Theorem 1.1 which holds for all p-compact groups, not necessarily connected, which is howev* *er easily seen to be equivalent to the first one using [5, Thm. 1.2]. It should be viewed* * as a topological analog of Chevalley's isomorphism theorem for linear algebraic groups (see [70,* * x32] [122, Thm. 1.5] and [35, 106, 101]). To state it, we define the maximal torus normali* *zer NX (T ) to be the loop space such that BNX (T ) is the Borel construction of the canoni* *cal action of WX (T ) on BT . Note that by construction we have a morphism NX (T ) ! X, an* *d that this map is independent of the choice of T , up to conjugacy. By [48, Prop. 9.5* *], WX (T ) is a discrete space so BNX (X) has only two non-trivial homotopy groups and fit* * into a fibration sequence BT ! BNX (T ) ! BWX . (Beware that in general NX (T ) will n* *ot be a p-compact group since its group of components WX need not be a p-group.) Theorem 1.4 (Topological isomorphism theorem for p-compact groups, p odd). Let * *p be an odd prime and let X and X0 be p-compact groups with maximal torus normalizer* *s NX and NX0. Then X ~=X0 if and only if BNX ' BNX0. Furthermore the spaces of self-homotopy equivalences Aut(BX) and Aut(BNX ) ar* *e equiv- alent as topological monoids. Explicitly, turn f : BNX ! BX into a fibration wh* *ich we will again denote by f, and let Aut(f) denote the topological monoid of self-homotop* *y equiv- alences of f. Then the following canonical zig-zag, given by restrictions, is * *a zig-zag of equivalences: ~= ~= Aut(BX) Aut (f) ! Aut (BNX ). (In the above theorem, the fact that the evaluation map Aut(f) ! Aut (BX) is * *an equivalence follows by a short general argument (Lemma 2.1), whereas the equiva* *lence Aut(f) ! Aut(BNX ) requires a detailed case-by-case analysis.) We point out that the classification of course gives easy, although somewhat * *unsatisfac- tory, proofs that many theorems from Lie theory extends to p-compact groups, si* *mply using the models provided via Theorem 1.2 and using that the theorem is known to be t* *rue in the Lie type case, and a case checking in the remaining cases. Since all the exotic* * p-compact groups have cohomology ring a polynomial algebra, this can turn out to be rathe* *r straight- forward. In this way one for instance sees that Bott's celebrated result about * *the structure of G=T [12] still holds true for p-compact groups, at least on cohomology. Theorem 1.5 (Bott's theorem for p-compact groups). Let X be a connected p-compa* *ct group, p odd, with maximal torus T . Then H*(X=T ; Zp) is a free Zp-module of d* *imension |WX |, concentrated in even degrees. Likewise combining the classification with a case-by-case verification for th* *e exotic groups by Castellana [24, 25], we get that the Peter-Weyl theorem holds for connected * *p-compact groups, p odd: Theorem 1.6 (Peter-Weyl theorem for connected p-compact groups). Let X be a con* *nected p-compact group, p odd. Then there exists a monomorphism X ! U(n)^pfor some n. 6 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL We also still have the `standard' formula for the fundamental group (the subs* *cript denotes coinvariants). Theorem 1.7. Let X be a connected p-compact group, p odd. Then ß1(X) = (LX )WX The classification (but in fact only the part pertaining to finite Zp-reflect* *ion groups (W, L) with Zp[L]W a polynomial algebra) also gives a verification that results of Bo* *rel, Steinberg, Demazure, and Notbohm [97, Prop. 1.11] extends to p-compact groups, p odd. Theorem 1.8. Let X be a connected p-compact group, p odd. The following conditi* *ons are equivalent: (1)X has torsion free Zp-cohomology. (2)Zp[L]W is a polynomial algebra over Zp. (3)All elementary abelian p-subgroups of X lie inside a maximal torus. Even in the Lie case, the proof of the above theorem is still not entirely sa* *tisfactory despite much effort_see the comments surrounding our proof as well as Borel's c* *omments [8, p. 775] and the references [7, 36, 121]. Likewise the following related re* *sult from Lie theory is true. Theorem 1.9. Let X be a connected p-compact group, p odd. Then the following co* *nditions are equivalent: (1)ß1(X) is torsion free. (2)Every rank one elementary abelian p-subgroup : Z=p ! X has connected ce* *ntral- izer CX ( ) (= map(BZ=p, BX)B ; see Section 6). (3)Every rank two elementary abelian p-subgroup is contained in a torus. Results about p-compact groups can in general, via Sullivan's arithmetic squa* *re, be translated into results about finite loop spaces, and the last theorem in this * *introduction is an example of such a translation. Recall that a finite loop space is a loop spa* *ce (X, BX, e), where X is a finite CW-complex. A maximal torus of a finite loop space is simpl* *y a map BU(1)r ! BX for some r, such that the homotopy fiber is homotopy equivalent to a finite CW -complex of non-zero Euler characteristic. The classical maximal toru* *s conjecture (stated in 1973 by Wilkerson [129, Conj. 1] as ä popular conjecture toward whi* *ch the author is biased"), asserts that connected compact Lie groups are the only conn* *ected finite loop spaces which admit maximal tori. (A slightly more elaborate version state* *s that the classifying space functor should set up a one to one correspondence between* * compact connected Lie groups, up to isomorphism, and connected finite loop spaces admit* *ting a maximal torus, up to isomorphism, and furthermore the outer isomorphisms of G e* *qual homotopy classes of self-homotopy equivalences of BG, the last part being true * *by [75].) It is well known that a proof of the conjectured classification of p-compact * *groups for all primes p would imply the maximal torus conjecture. Our results at least imply * *that the conjecture is true after inverting the single prime 2. Theorem 1.10. Let X be a connected finite loop space with a maximal torus. Then* * there exists a compact connected Lie group G such that BX[1_2] and BG[1_2] are homoto* *py equivalent spaces, where [1_2] indicates Z[1_2]-localization. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 7 Relationship to the Lie case and the conjectural picture for p = 2. We now stat* *e a common formulation of both the classification of compact Lie groups and the cla* *ssification of p-compact groups for p odd, which conjecturally should also hold for p = 2. * *We have not encountered this_in our opinion quite natural_description before in the lit* *erature (compare [41] and [80]). Let R be an integral domain and W a finite R-reflection group. For an RW -la* *ttice L (i.e., an RW -module which is finitely generated and free as an R-module) defin* *e SL to be the sublattice of L generated by (1 - w)x where w 2 W and x 2 L. Define an R-re* *flection datum to be a triple (W, L, L0) where (W, L) is a finite R-reflection group and* * L0 is an RW - lattice such that SL L0 L and L0 is of the form SL0 for some other RW -latt* *ice L0. (If R = Zp, p odd, then "S" is idempotent (since H1(W ; LW ) = 0 for all (W, L)* * by a case- by-case computation given in [5, Thm. 3.2]) so L0 = SL in this case.) Two refle* *ction data (W, L, L0) and (W 0, L0, L00) are said to be isomorphic if there exists an R-li* *near isomorphism ' : L ! L0such that 'W '-1 = W 0and '(L0) = L00. If D is either the category of connected compact Lie groups or connected p-co* *mpact groups, then we can consider the assignment which to each object X in D assigns* * the triple (W, L, L0), where W is the Weyl group, L = ß1(T ) is the dual weight lat* *tice, and L0 = ker(ß1(T ) ! ß1(X)) is the coroot lattice. Theorem 1.1 and 1.7 as well as the classification of compact connected Lie gr* *oups (see e.g. [15]) can now be reformulated as follows: Theorem 1.11. If D is the category of connected compact Lie groups or p-compact* * groups for p odd, then (W, L, L0) is an R-reflection datum (R = Z for compact Lie grou* *ps and Zp for p-compact groups) and this assignment sets up a bijection between the el* *ements of D up to isomorphism and R-reflection datum, up to isomorphism. Furthermore the * *outer automorphisms of X equal outer automorphisms of the corresponding R-reflection * *datum. Conjecture 1.12. Theorem 1.11 is also true if D is the category of connected 2-* *compact groups. One can check that the conjecture on objects is equivalent to the conjecture * *given in [41] and [80], and the self-map statement would then follow from [75] and [98].* * (The role of the coroot lattice L0 in the above theorem and conjecture is in fact in prac* *tice only to be able to distinguish direct summands isomorphic to SO (2n + 1) from direct su* *mmands isomorphic to Sp(n) (see Remark 7.3 and Proposition 7.4); alternatively on can * *use the extension class fl 2 H3(W ; LX ) of the maximal torus normalizer (see Section 3* *) rather than L0 but in that picture it is not a priori clear which triples (W, L, fl) are re* *alizable.) It would be desirable to give a öt pological" version of Theorem 1.11 and Conjecture 1.1* *2 which were statements on the level of automorphism spaces like Theorem 1.4 but we do * *not know a general formulation which incorporates this feature. Organization of the paper and notation. The sections of this paper can be read * *in an almost arbitrary order. The short Section 2 sets up the map from space of autom* *orphism of X to the space of automorphisms of NX , and in Section 3 we give an algebraic d* *escription of the automorphism group of NX (which we expand on in Section 13). In Sectio* *n 4 we reduce the classification Theorem 1.1 to the case of simple, center-free, co* *nnected p- compact groups. In Section 5 we prove a theorem about invariant rings and show* * how this leads to an easy construction of the exotic p-compact groups. In Section * *6 we give proofs of the main theorem, modulo obstruction group calculations which are car* *ried out in 8 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Section 10. The applications listed in the introduction are proved in Section 7* *. The purely algebraic Section 8 contains complete information about all non-toral elementar* *y abelian p- subgroups of the exceptional Lie groups, along with their Weyl groups and centr* *alizers, and Section 9 gives the analogous (but much easier) results for the projective unit* *ary groups. (This information is used in a crucial way for the calculations in Section 10 a* *s well as, in a milder way, directly in Section 6 for information about rank two non-toral subg* *roups.) In the appendix Section 11 we give a classification of finite Zp-reflection groups* * generalizing Notbohm's classification to all primes and in the appendix Section 12 we recall* * Notbohm's results on invariant rings of Zp-reflection groups. Finally in the appendix Se* *ction 13 we briefly calculate the outer automorphism group of the exotic finite Zp-reflecti* *on groups to make the result of Theorem 1.1 more explicit. We have tried to introduce all definitions relating to p-compact group as the* *y are used, but it is probably nevertheless helpful for the reader unfamiliar with p-compac* *t groups to keep copies of the excellent papers [48] and [49] of Dwyer-Wilkerson (whose ter* *minology we follow) within reach. As a technical term we say that a p-compact group X is de* *termined by N if it is true that any other p-compact group X0with the same maximal torus no* *rmalizer is isomorphic isomorphic to X (which will be true for all p-compact groups, p o* *dd, by Theorem 1.4). Acknowledgments. We would like to thank H. H. Andersen, D. Benson, G. Kemper, A. Kleschev, G. Malle, and J.-P. Serre for helpful correspondence. We would in * *particular like to thank W. Dwyer, D. Notbohm, and C. Wilkerson for several useful tutoria* *ls on their beautiful work, which this paper builds upon. 2. The map Aut(BX) ! Aut(BNX ) The purpose of this very short section is to construct the map Aut(BX) ! Aut(* *BNX ) which we will later prove is an equivalence. We have been unable to find this d* *escription in the literature. For a fibration f : E ! B we let Aut(f) denote the space of commutative diagr* *ams E_____//E |f| |f| fflffl|fflffl| B ____//_B such that the horizontal maps are homotopy equivalences. (This is a subspace of* * Aut(E) x Aut(B).) Lemma 2.1 (Adams-Mahmud lifting). Let X be a p-compact group with maximal torus normalizer NX . Turn the inclusion of the maximal torus normalizer into a fibr* *ation f : BNX ! BX. Then the restriction map Aut(f) ! Aut(BX) is an equivalence of topolo* *gical monoids. In particular any self-homotopy equivalence of BX lifts to a self-homotopy eq* *uivalence of BNX , which is unique in the strong sense that the space of lifts is contrac* *tible. Choos- ing an inverse to the equivalence Aut(f) ! Aut(BX), we get by restriction a can* *onical homomorphism of topological monoids ~= Aut(BX) ! Aut (f) ! Aut(BNX ). THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 9 Proof.For any ' 2 Aut(BX), there exists, e.g. by [90, Thm. 1.2(3)], a _ 2 Aut(B* *NX ) such that 'f is homotopic to f_. Since f is assumed to be a fibration, _ can further* *more be modi- fied such that the equality is strict. This shows that the evaluation map Aut(f* *) ! Aut(BX) is surjective on components. This map of group-like topological monoids is furt* *hermore eas- ily seen to have the homotopy lifting property. To see that it is a homotopy eq* *uivalence we hence just have to verify that the fiber over the identity map AutBX (BNX ) is * *contractible. We have the following diagram with rows and columns fibrations WX ____//_X=T____//X=NX || | | || | | || fflffl| fflffl| WX _____//BT_____//_BNX | | | | | | fflffl| fflffl| fflffl| * ______//BX_______BX. Taking homotopy T -fixed points of the top row produces a fibration sequence WX* * ! (X=T )hT ! (X=NX )hT, where, by the definition of WX , the inclusion of the fib* *er in the total space is a homotopy equivalence. (We refer to [48, 3.3, x10] for basic fa* *cts and defi- nitions about homotopy actions.) Hence (X=NX )hT is contractible. But then (X=N* *X )hNX is contractible as well, since (X=NX )hN~X' ((X=NX )hT~)hWX , where N~X is a di* *screte ap- proximation to NX (cf. [48, Prop. 3.13]). Since the space of maps BNX ! BNX ove* *r BX identifies with (X=NX )hNX , we see that any self-map of BNX over BX is an equi* *valence, and that AutBX (BNX ) is contractible as wanted. 3.Automorphisms of maximal torus normalizers The purpose of this short section is establish some easy facts about automorp* *hisms of maximal torus normalizers which are needed to carry out the reduction to connec* *ted, center- free simple p-compact groups in Section 4. At the same time the section serves * *to make the automorphism statement of Theorem 1.1 more explicit. Proposition 3.1. Suppose that N is an extended p-compact torus, i.e., a loop sp* *ace such that W = ß0(N ) is a finite group and the identity component of N is a p-compac* *t torus T . Let N~be the unique discrete approximation to N (see [48, 3.12]), and recal* *l that N~will have a unique largest p-divisible subgroup ~T, which will be the discrete appro* *ximation of T . The obvious map associating to a self-homotopy equivalence of BN~ a self-homo* *topy equiv- alence of BN (via fiberwise Fp-completion; cf. [17, Ch. I x8]) induces a homoto* *py equivalence of aspherical group-like topological monoids ~= Aut (BN~)^p! Aut (BN ). If ß0(N ) acts faithfully on ß1(N1) then Aut1(BN~), the component of Aut(BN~)* * of the identity map, has the homotopy type of B(T~W) where ~Tis a discrete approximati* *on to T . Sketch of proof.The statement on the level of component groups follows directly* * from [48, 3.12]. (The point is that the homotopy fiber of BN~ ! BN will have homotopy ty* *pe K(V, 1) for a Qp vector space V , and hence the existence and uniqueness to lif* *ting a map BN~ ! BN~ to BN lie in Hn(N~; V ) where n = 2, 1 which are easily seen to be ze* *ro.) It is 10 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL likewise easy to see that both spaces are aspherical and that we get a homotopy* * equivalence on component groups. The last statement is likewise obvious. Let L be a finitely generated free Zp-module and suppose that W GL (T~), wh* *ere we set ~T= L Z=p1 . Consider the second cohomology group H2(W ; ~T) which classifies e* *xtensions of W by ~Twith the fixed action of W on ~T. Given an isomorphism ff : L ! L0sen* *ding W GL (L) to W 0 GL (L0) we get an isomorphism of cohomology groups H2(W ; ~T* *) ! -1cffOp H2(W 0; ~T)0by sending an extension ~T!i N~ p!W to the extension ~T 0iOff!~N! * *W 0, where cffdenotes conjugation by ff. An isomorphism between two triples (W, L, * *fl) and (W 0, L0, fl0) is an isomorphism L ! L0sending W to W 0and fl to fl0. The autom* *orphism group of a triple (W, L, fl) thus identifies with flN 2 ~ GL(T~)(W ) = {ff 2 NGL(T~)(W ) | ff(fl) = fl 2 H (W ; T)}. It follows directly from the definition (and using that ~Tis characteristic in * *N~) that two triples as above are isomorphic if and only if the associated groups ~Nand ~N 0* *are isomorphic, where ~Nis obtained from the extension 1 ! ~T! ~N! W ! 1 given by fl, and analo* *gously for fl0. However, N~ and (W, L, fl) in general have slightly different automor* *phisms, as is described in the following lemma (see also [128]): Proposition 3.2. In the notation above, for any exact sequence 1 ! ~T! N~!i W !* * 1 with extension class fl we have a canonical exact sequence (3.1) 1 ! Der(W, ~T) ! Aut(N~) ! flNGL(T~)(W ) ! 1 where we embed the derivations Der(W, ~T) in Aut(N~) by sending a derivation s * *to the automorphism given by x 7! s(ß(x))x, and the map Aut(N~) ! flNGL(T~)(W ) is giv* *en by restricting an automorphism ' 2 Aut(N~) to ~T. This exact sequence has a exact subsequence 1 ! ~T=T~W ! N~=ZN~ ! W ! 1 and t* *he quotient exact sequence is 1 ! H1(W, ~T) ! Out(N~) ! flNGL(T~)(W )=W ! 1. In particular if (W, L) is a finite Zp-reflection group and p is odd then by * *[5],[74, proof ~= of Prop. 3.5] H1(W ; ~T) = 0, so we get an isomorphism Out(N~) ! flNGL(T~)(W )=* *W . Proof.Let ' 2 Aut(N~), and consider the restriction map ' 7! '|T~2 Aut(T~). Not* *e that for all x 2 ~N, l 2 ~Twe have (' O cx)(l) = '(xlx-1) = '(x)'(l)'(x)-1 = (c'(x)O ')(l) so '|T~2 NGL(T~)(W ). That the image is the elements which fixes the extension * *class follows easily from the definitions: The diagram iO' c'Oi ~T____//_~N__//_W || | || || '| || || i fflffl|||i ~T____//_~N__//_W shows that ' leaves fl invariant. Likewise, to see that the right map in (3.1)i* *s surjective let _ 2 flNGL(T~)(W ) and let ~T! ~N! W be the extension obtained by first pushing * *forward THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 11 along _ : ~T! ~Tand then pulling back along _-1(-)_ : W ! W . Since _ fixes fl * *there exists an isomorphism ~N! ~Nmaking the following diagram commute _ ~T_____//~T____~T | | | | | | fflffl|fflffl|fflffl| ~N_____//~N___//~N | | | | | | fflffl|fflffl|fflffl|_(-)_-1 W ____//_W______W which shows that Aut(N~) ! flNGL(T~)(W ) is surjective. Now suppose ' 2 Aut(N~) restricts to the identity on ~T. For a given x 2 ~Nwe* * have '(x)l'(x-1) = '(x)'(l)'(x-1) = '(xlx-1) = xlx-1 so the induced map ' : W ! W is the identity since W acts faithfully on ~T. Thi* *s means that we can define a map s : W ! ~Tby s(w) = ~w-1'(w~) where ~wis a lift of w, * *and this is easily seen to be a derivation. Furthermore taking the automorphism of N~associ* *ated to s gives back ', which establishes exactness in the middle, and we have proven tha* *t we have the first exact sequence. The existence of the short exact subsequence is clear, noting that ZN~ = ~T W* *(since W acts faithfully on ~T) and that ~T=T~W embeds in Der(W ; ~T) as the principal d* *erivations by sending l to the derivation w 7! cw(l)l-1. The last exact sequence is now obvio* *us. Remark 3.3. See [66] for a related exact sequence for compact Lie groups, fitti* *ng with the conjectured classification of p-compact groups for p = 2. Proposition 3.4. Suppose {(Wi, Li, fli)}ki=1is a collection of non-isomorphic t* *riples where Liis a finitely generated free Zp-module, Wiis a finite subgroup of GL (Li) suc* *h that Li Q is irreducible, and fli2 H2(Wi; ~Ti). Let (W, L, fl) = xki=1(Wi, Li, fli)mi den* *ote the product. Then ~ xki=1((fliNGL(Li)(Wi)=Wi) o mi) =!flNGL(L)(W )=W. Proof.Assume for ease of notation that L = L1 L2; the general case follows fr* *om this by induction. Consider ' 2 NGL(L1 L2)(W1 x W2). For every w 2 W1 x W2 there exi* *sts a unique ~w2 W1 x W2 such that '(wx) = ~w'(x) for allx 2 L. Let ff denote the element in Aut(W1 x W2) given by w 7! ~w. By the definition of ff the canonical map 'ji: Li! L1 L2 '!ff(L1 L2) ! ffLj is W1xW2-equivariant, where the superscript ff means that we are acting through* * ff. Hence this map, after tensoring with Q has to be either an isomorphism or zero, since* * Li Q and ffLj Q are irreducible. But combined with the fact that ' is an isomorphism, t* *his means that ` ' ' = '11' '12 21'22 12 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL has to consist of either just "diagonal" elements or just ö ff diagonal" elemen* *ts. If we furthermore require ' to respect the extension classes it is clear that we can * *only have the "diagonal" case if (W1, L1, fl1) is not isomorphic to (W2, L2, fl2) whereas* * if they are isomorphic both cases can really occur. This proves the lemma. 4. Reduction to connected, center-free simple p-compact groups In this section we prove some lemmas, which, together with the splitting theo* *rems of Dwyer-Wilkerson [50] and Notbohm [99], reduce the proof of Theorem 1.4 to the c* *ase of connected, center-free simple p-compact groups. (Recall that a connected p-comp* *act group is called simple if (WX , LX Q) is an irreducible Qp-reflection group; it is c* *alled center-free if Aut1(BX) is contractible_equivalent definitions are given in [49].) This re* *duction is known and most of it appears in [89] (relying on earlier work of that author). * * We here provide a self-contained and a bit more direct proof using [49]. Lemma 4.1 (Behavior with respect to products). Let X and X0 be p-compact groups* * with maximal torus normalizers N and N 0. Then N x N 0is a maximal torus normalizer* * for X x X0 and the following statements hold: ~= ~= (1)Aut1(BX) x Aut1(BX0) ! Aut 1(BX x BX0) and Aut 1(BN ) x Aut1(BN 0) ! Aut1(BN x BN 0), where Aut1 denotes homotopy equivalences homotopic to the identity. (2)If Aut(BX) ! Aut(BN ) and Aut(BX0) ! Aut(BN 0) are injective on ß0, then * *so is Aut(B(X x X0)) ! Aut(B(N x N 0)). (3)Suppose p is odd and that X and X0 are connected and center-free. If Aut(* *BX) ! Aut(BN ) and Aut(BX) ! Aut(BN 0) are surjective on ß0, then so is Aut(B(X* * x X0)) ! Aut(B(N x N 0)). Proof.Recall that the map Aut(BX) ! Aut(BN ) was described in Section 2. To see* * (1) first note that (4.1) map (BX x BX0, BX x BX0) ' map(BX, map(BX0, BX)) x map(BX0, map(BX, BX0)). The evaluation map map(BX0, BX)const! BX is an equivalence by the Sullivan conj* *ecture for p-compact groups [49, Thm 9.3 and Prop. 10.1], (and likewise with X and X0s* *witched). Since the component of the identity map on the left hand side of (4.1)lands in * *the component of the constant map in map (BX0, BX) this shows that map (BX x BX0, BX x BX0)1 ' map(BX, BX)1x map(BX0, BX0)1 as wanted. (The statement just says that the cente* *r of a product of p-compact groups is the product of the centers, which of course al* *so follows from the equivalence of the different definitions of the center from [49].) To see (2)suppose that ' is a self-equivalence of BX x BX0such that its restr* *iction to a self-equivalence of B(N x N 0) becomes homotopic to the identity. The restricti* *on '|BXx* composed with the projection onto BX0 becomes null homotopic upon restriction t* *o BN , which by e.g, [91, Thm. 6.1] implies that it is null homotopic. Likewise the p* *rojection of '|*xBX0 onto BX0 becomes homotopic to the identity map upon restriction to B* *N , which by assumption means that the projection of '*xBX0 onto BX0 is the identit* *y. But by adjointness, repeating the argument of the first claim, this implies that ' * *composed with the projection onto BX0 is homotopic to the projection map onto BX0 (this * *is [49, Lem. 5.3]). By symmetry this holds for the projection onto BX as well, and we c* *onclude that ' is homotopic to the identity as wanted. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 13 Finally, combining Proposition 3.2 and 3.4 gives (3), since ß0(BN ) = Out(N~)* * by Propo- sition 3.1. Remark 4.2. Claim (3)of the above lemma is in general false if p = 2. For insta* *nce if X = SO(3)^2then both for Y = X and Y = X x X we have ß0(Aut (BY )) ~=NGL(LY)(WY )=W* *Y . But for Y = X x X we have H1(WY ; ~TY) ~= Z=2 x Z=2, so BNY (T ) has non-trivi* *al automorphisms which restrict to the identity on BT (see Proposition 3.2). Recall the observation that for p odd the component group of X is determined * *by WX : Lemma 4.3. Let X be a p-compact group for p odd, with maximal torus normalizer * *j : N ! X. By definition WX = ß0(N ). The map ß0(j) : WX = ß0(N ) ! ß0(X) is surjec* *tive. The kernel is Op(WX ), the subgroup generated by elements of order prime to p. * *It can also be identified with the Weyl group of the connected component X1 of X, and is th* *e largest Zp-reflection subgroup of WX . Proof.By [49, Rem. 2.11] ß0(j) is surjective with kernel the Weyl group of the * *connected component of X. Since ß0(X) is a p-group, Op(ß0(N )) is contained in the kernel* *. On the other hand, since p is odd, the Weyl group of X1 is generated by elements of or* *der prime to p, since it is a Zp-reflection group, so equality has to hold. Remark 4.4. For p = 2 the component group of X cannot be read off from NX , and* * one would have to remember ß0(X) as part of the data. The 2-compact groups SO (3)^* *2and O(2)^2have for example the same maximal torus normalizer, namely O(2)^2. (Note * *however that if X is the centralizer of a toral abelian subgroup A of a connected p-com* *pact group Y , then the component group of X can be read off from A and NY (see [49, Thm. * *7.6]); a case of frequent interest.) Before proceeding recall that by [43] (see also [49, Prop. 11.9]) we have, fo* *r a fibration F ! E f!B, a fibration sequence map (B, B Aut(F))C(f)! B Aut(f) ! B Aut(B). Here C(f) denotes the components corresponding to the orbit of the ß0(Aut (B))-* *action on the class in [B, B Aut(F)] classifying the fibration. We are interested in when the map of monoids Aut(f) ! Aut(E) is a homotopy eq* *uiva- lence. This will follow if we can see that Aut1(f) ! Aut1(E) and ß0(Aut (f)) ! * *ß0(Aut (E)) are equivalences. By an easy general argument given as [49, Prop. 11.10] the s* *tatement about identity components follows if B ! map(F, B)0 is an equivalence. Lemma 4.5 (Behavior with respect to components). Let X be a p-compact group with maximal torus normalizer N , and assume that p is odd (so that ß0(X) can be rea* *d off from N ). Let N1 denote the kernel of the map N ! ß0(X), and note that N1 is a maxim* *al torus normalizer for X1. ~= ~= If Aut(BX1) ! Aut (BN1), then Aut(BX) ! Aut (BN ). If furthermore BX1 is dete* *r- mined by BN1 then BX is determined by BN . Proof.First note that by an inspection of Euler characteristics and using [90, * *Thm. 1.2(3)], N1 is really a maximal torus normalizer in X1 as claimed. Set ß = ß0(X) for sho* *rt. We want to apply the setup described before the lemma to the fibrations BX1 ! BX !* * Bß and BN1 ! BN ! Bß and to see that in both cases the map of monoids Aut(f) ! Aut* *(E) are homotopy equivalences. By the remarks above this follows if it is an isomor* *phism on 14 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL ß0 and that B ! map (F, B)0 is an equivalence. The statement about ß0 is true i* *n both cases since a self-map of E determines a unique self-map of Bß. Likewise it is * *easy to see that Bß '! map(BX1, Bß)0 and that Bß '! map(BN1, Bß)0. This means that our map B Aut(BX) ! B Aut(BN ) (from Lemma 2.1) fits in a map of fibration sequences map (Bß, B Aut(BX1))C(f)____//_B Aut(BX)____//B Aut(Bß) | | | | | | fflffl| fflffl| fflffl| map (Bß, B Aut(BN1))C(f)____//_B Aut(BN_)__//_B Aut(Bß). Here the maps between the fibers and bases are homotopy equivalences by assumpt* *ion, and we conclude that we get homotopy equivalences between the total spaces as well. Now assume furthermore that X1 is determined by N1, and let X0be another p-co* *mpact group with maximal torus normalizer N . By Lemma 4.3 we get that ß = ß0(X) ~=ß0* *(X0) and that N1 is also a maximal torus normalizer in X01. We want to show that the two fibrations BX ! Bß and BX0 ! Bß are equivalent as fibrations over Bß, or equivalently that the ß-spaces BX1 and BX01are hß-equiva* *lent, i.e., that we can find a zig-zag of ß-maps which are non-equivariant equivalences con* *necting the two (see e.g., [38] where this equivalence relation is called equivariant w* *eak homotopy equivalence). By the assumptions on X1 we can choose a homotopy equivalence Bf : BX1 ! BX01 such that BN1 F Bj xxxx FFFBj0 xx FF --xxx Bf F##F BX1 ______________//BX01 commutes up to homotopy, and Bf is unique up to homotopy. We now want to see that we can change Bf so that it becomes a ß-map. For this* *, consider the restriction ß-map map (BX1, BX01) ! map(BN1, BX01). By the assumptions on Aut(BX1) this map sends distinct components of map (BX1, * *BX01) corresponding to homotopy equivalences to distinct components of map(BN1, BX01)* *. More- over, by the proof of Lemma 2.1, we have a homotopy equivalence map (BX1, BX01)* *Bf ' map(BN1, BX01)BfOBj. In particular the component map (BX1, BX01)Bf is preserve* *d un- der the ß-action, since this obviously is so for map (BN1, BX01)Bj0. Furthermo* *re since map(BN1, BX01)iBj0contains Bj0 we see that map (BX1, BX01)hiBf' map (BN1, BX01)* *hiBj0is non-empty, and so there exists a ß-map EßxBX1 ! BX01which is a homotopy equival* *ence. This shows that BX1 and BX01are hß-homotopy equivalent as wanted. Remark 4.6. If X is a connected p-compact group, and p is odd, then it follows * *from [49, Thm. 7.5] that Z(N~) is a discrete approximation to the center of X. The p* *roof of the above lemma extends this to X non-connected provided we know that the self-maps* * of X1 are detected by their restriction to N1, which will be a consequence of Theorem* * 1.4. Having to appeal to this is a bit unfortunate but seems unavoidable. The point is that* * if there for a connected p-compact group X, existed a self-equivalence oe of X of finite p-pow* *er order and not detected by N , then we could form Xo , where oe would be central in th* *e normalizer but not in the whole group. (See also Lemma 10.2.) THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 15 Lemma 4.7 (Behavior with respect to centers). Let X be a connected p-compact gr* *oup with center Z. (1)If ß0(Aut (BX=Z)) ! ß0(Aut (BN =Z)) is surjective and X=Z is determined by N =Z then X is determined by N . (2)If p is odd and Aut(BX=Z) ! Aut(BN =Z) is a homotopy equivalence then Aut* *(BX) ! Aut(BN ) is as well. Proof.Suppose that X and X0have the same maximal torus normalizer N and choose * *fixed inclusions j : N ! X and j0: N ! X0. By [49, Thm. 7.6] X0 and X have the same c* *enter Z. Suppose that X=Z is isomorphic to X0=Z. If ß0(Aut (BX=Z)) ! ß0(Aut (BN =Z)) * *is surjective we furthermore have that we can choose the homotopy equivalence BX=Z* * ! BX=Z in such a way that BN =ZK j=Ztttt KKj0=ZKK tt KKK yyttt K%% BX=Z _______________//_BX0=Z commutes up to homotopy. We have canonical maps BX ! B2Z and BX0 ! B2Z classifying the extensions, and we claim that in fact the bottom triangle in the diagram BN =ZK tttt KKK ttt KKK yytt KK%% BX=Z K_______________//_BX0=Z KKK ssss KKK ssss KK%% yyss B2Z commutes up to homotopy. By construction the outer square commutes up to homoto* *py (since both composites agree with the map classifying map BN =Z ! B2Z since we * *kept j and j0 fixed). Since the top triangle also commutes up to homotopy, an appli* *cation of the transfer [48, 9.12], using that B2Z is a product of Eilenberg-Mac Lane spac* *es and that Ø((X=Z)=(N =Z)) = 1, shows that the bottom triangle commutes up to homotopy as * *well. Since we have constructed a map BX=Z ! BX0=Z over B2Z we get an induced homotopy equivalence BX ! BX0. (Note that this construction does not a priori give this * *map as a map under BN .) We now want to get the second statement about automorphism groups. Consider * *the homotopy commutative diagram f0 BN ____//_BN =Z | | | | | fflffl|f fflffl| BX ____//_BX=Z where we can suppose that the two horizontal maps f0 and f are fibrations. We first claim that we can replace B Aut(f) with B Aut(BX) and B Aut(f0) with B Aut(BN ). As in the case of the component group (see the proof of Lemma 4.5)* * we 16 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL just have to justify that in the appropriate fibration sequences we have equiva* *lences B ! map(F, B)0 and ß0(Aut (f)) ! ß0(Aut (E)). The map BX=Z ! map (BZ, BX=Z)0 is a homotopy equivalence since the trivial map is central [49, Prop. 10.1]. That B* *N =Z ! map(BZ, BN =Z)0 is an equivalence is a similar (but easier) argument. By Lemma 2.1 a self-equivalence of BX induces a unique self-equivalence of BN* * , and hence a canonical self-equivalence of BZ. Now, by the description of X=Z as a * *Borel construction (given in [48, proof of Prop. 8.3]) we get a canonical self-equiva* *lence of BX=Z. This self-equivalence is furthermore unique, in the sense that given a diagram g BX ______//_BX | | | | fflffl|g0 fflffl| BX=Z ____//_BX=Z the homotopy type of g0is uniquely given by that of g. To see this note that by* * Lemma 2.1 the diagram restricts to a unique diagram g~ BN _______//_BN | | | | fflffl|~g0 fflffl| BN =Z ____//_BN =Z. By looking at discrete approximations we see that the homotopy class of ~g0is d* *etermined by ~g. Since by assumption the homotopy class of g0is determined by ~g0, we con* *clude that a self-equivalence of BX induces a unique self-equivalence of BX=Z, and so ß0(A* *ut (f)) ~= ß0(Aut (BX)). The last part of the argument furthermore shows that also ß0(Aut * *(f0)) ~= ß0(Aut (BN )). We hence have the following diagram where horizontal maps are fibration seque* *nces map (BX=Z, B Aut(BZ))C(f) ____//_B Aut(BX)____//_B Aut(BX=Z) | | | | | | fflffl| fflffl| fflffl| map (BN =Z, B Aut(BZ))C(f0)___//_B Aut(BN_)___//B Aut(BN =Z)). Examining when the middle vertical arrow is a homotopy equivalence reduces to f* *inding out when the restriction map map(BX=Z, B Aut(BZ))C(f)! map(BN =Z, B Aut(BZ))C(f0)is a homotopy equivalence, which we now proceed to analyze. Note that since BZ is an Eilenberg-Mac Lane space we have a fibration sequence B2Z ! B Aut(BZ) ! B Aut(Z~) where ~Zis the discrete approximation to Z and Aut(Z~) is the discrete group of* * automor- phisms. Since our extensions are central this gives a diagram of fibration sequ* *ences map(BX=Z, B2Z)C(f) _____//map(BX=Z, B Aut(BZ))C(f)_____//_map(BX=Z, B Aut(Z~))0 | | | | | | fflffl| fflffl| fflffl| map (BN =Z, B2Z)C(f0)____//_map(BN =Z, B Aut(BZ))C(f0)__//map(BN =Z, B Aut(Z~))* *0. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 17 Again, in this diagram the map between the base spaces is obviously an equivale* *nce, so we are reduced to studying (4.2) map (BX=Z, B2Z)C(f)! map(BN =Z, B2Z)C(f0). Since B2Z is a product of Eilenberg-Mac Lane spaces a transfer argument (cf. [4* *8, 9.12]) shows that this gives an embedding as a retract. Since we assume ß0(Aut (BX=Z)* *) ~= ß0(B Aut(BN =Z)) we furthermore get that this is an isomorphism on ß0 by the de* *finition of C(f) and C(f0). Now set X0 = X=Z and N 0= N =Z and let (W, L0) denote the We* *yl group of X0. Write BZ ' B2A x BA0, where A is torsion free and A0 is finite (s* *ee [49, Thm. 1.1]). On ß1 the map (4.2)identifies with H1(BX0; A0) H2(BX0; A) ! H1(BN 0; A0) H2(BN 0; A). The group H1(BN 0; A0) is zero since ß1(BN 0) = WX is generated by elements of* * order prime to p, since p is assumed to be odd. Furthermore, H2(BN ; A) is related via the Serre spectral sequence to the gro* *ups H2(BW ; H0(B2L0; A)), H1(BW ; H1(B2L0; A)), and H0(BW ; H2(B2L0; A)). The first of these groups is zero since W is generated by elements prime to p b* *y the as- sumption that p is odd. The second is obviously zero, and the last group is ze* *ro since H0(W ; Hom (L0, Zp)) = Hom ((L0)W , Zp) = 0 because (L0)W is finite. Hence we get an isomorphism on ß1, since we already know that the map is inje* *ctive. On ß2 and ß3 the map identifies with H0(BX0; A0) H1(BX0; A) ! H0(BN 0; A0) H1(BN 0* *; A) and H0(BX0; A) ! H0(BN 0; A) respectively, and these maps are obviously isomorp* *hisms. Hence map (BX=Z, B2Z)C(f)! map(BN =Z, B2Z)C(f0)is a homotopy equivalence, which via the fibration sequences above imply that B Aut(BX) ! B Aut(BN ) is a homoto* *py equivalence as wanted. Remark 4.8. Consider BX = B(SO(3) x S1)^2. This has center Z = S1^2and X=Z = SO(3)^2. It is easy to calculate directly that B Aut(BX=Z) '! B Aut(BN =Z) (or * *appeal to [75]). However ß0(Aut (BX)) ! ß0(Aut (BN )) is not onto by Proposition 3.2,* * since Hom (WSO(3), Z=21 ) = Z=2. This shows that the p odd assumption is necessary in* * the last part of the above lemma. (Compare also Remark 4.2.) Remark 4.9. Suppose that X is a connected p-compact group. Fibration sequences * *with base B2ß1(X) and fiber B(X<1>) are one-to-one correspondence with the set of ma* *ps [B2ß1(X), B Aut(B(X<1>))]. Likewise self-equivalences of BX can be expressed in* * terms of self-equivalences of B(X<1>) and ß1(X), analogously to the lemmas above. Hence * *if we a priori knew that Theorem 1.7 held true, i.e., if we could read off ß1(X) from N* *X then the above methods would reduce the proof of the main theorems to the simply connect* *ed case, which could be used advantageously in the proofs. (See also Remark 7.3.) Remark 4.10. The assumption in Lemma 4.7(1)that ß0(Aut (BX=Z)) ! ß0(Aut (BN =Z)) is surjective has the following origin. We have a canonical restriction map H2(* *BX=Z; ~Z) ! H2(BN =Z; ~Z), which is injective by a transfer argument. Two extension classes* * in H2(BX=Z; ~Z) give arise to isomorphic total spaces if the extension classes are conjugate vi* *a the actions of Aut(BX=Z) and Aut(Z~) on this H2. The total spaces have isomorphic maximal t* *orus normalizer if the extension classes have images in H2(BN =Z; ~Z) which are conj* *ugate under the action of Aut(BN =Z) and Aut(Z~), which could a priori be a weaker notion. 18 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL 5.An integral version of a theorem of Nakajima and realization of p-compact groups The goal of this section is to prove an integral version of an algebraic resu* *lt of Naka- jima (Theorem 5.1) and use this to prove a theorem (Theorem 5.3) which, as part* * of our induction proof of Theorem 1.1, will allow us to construct the center-free p-co* *mpact groups corresponding to Zp-reflection groups (W, L) such that Zp[L]W is a polynomial * *algebra. This will provide the existence part of Theorem 1.1. We feel that this way of * *showing existence, is perhaps more straightforward that previous approaches; compare fo* *r instance [97]. (We refer to the introduction for the history behind this result.) Theorem 5.1. Let p be an odd prime and let (W, L) be a finite Zp-reflection gro* *up. For a subgroup V of L Fp we let WV denote the pointwise stabilizer of V in W . Th* *en the following conditions are equivalent: (1)Zp[L]W is a polynomial algebra. (2)Zp[L]WV is a polynomial algebra for all non-trivial subspaces V L Fp. (3)(WV , L) is a Zp-reflection group for all non-trivial subspaces V L F* *p. Remark 5.2. An analog of the implication (1)) (2)where the ring Zp is replaced * *by a field was proven by Nakajima [92, Lem. 1.4] (in the case of finite fields see a* *lso [53] and [94]). For fields of positive characteristic the implication (3)) (1)does not h* *old; see [78] for more information about this case. Our proof unfortunately involves the cla* *ssification of finite Zp-reflection groups and some case-by-case checking. (See the discuss* *ion following the proof of Theorem 1.8 for related information.) Proof of Theorem 5.1.To start, note that the implication (2)) (3)follows from t* *he fact that if Zp[L]WV is a polynomial algebra then Qp[L Q]WV is as well, so (WV ,* * L) is a Zp-reflection group by the Shephard-Todd-Chevalley theorem ([6, Thm. 7.2.1] or * *[116, Thm. 7.4.1]). To go further we want to see that the theorem is well behaved under products,* * i.e., that if (W, L) = (W 0, L0) x (W 00, L00), then the theorem holds for (W, L) if it holds* * for (W 0, L0) and (W 00, L00). This follow from the fact that the stabilizer in W 0xW 00of an arb* *itrary subgroup in (L0 Fp) (L00 Fp) equals the stabilizer of the smallest product subgroup * *containing it, combined with the fact that the tensor product of two algebras is a polynom* *ial algebra if and only if each of the factors are. Hence to prove the remaining implicatio* *ns it follows from Theorem 11.1 that it suffices to consider separately the cases where (W, L* *) comes from a compact connected Lie group and the cases where (W, L) is one of the exotic Z* *p-reflection groups. Assume first that (W, L) = (WG , LG Zp) for a compact connected Lie group G* *. If Zp[L]W is a polynomial algebra then by Theorem 12.2 (which involves case-by-ca* *se con- siderations and p odd) BX = BG^phas the property H*(BX; Zp) ~=H*(B2L; Zp)W . We can identify V L Fp with a toral elementary abelian p-subgroup in X and by* * [53, Rem. 1.3] H*(BCX (V ); Zp) is again a polynomial algebra concentrated in even d* *egrees. In particular CX (V ) is connected and by [49, Thm. 7.6] WCX(V )= WV . Hence, by * *Theo- rem 12.1, H*(BCX (V ); Zp) ~=H*(B2L; Zp)WV , so Zp[L]WV is a polynomial algebr* *a. This shows that (1) ) (2) when (W, L) comes from a compact Lie group. To prove (3) ) (1)for Lie groups suppose that (W, L) is a finite Zp-reflection group correspon* *ding to a p-compact group X = G^psuch that (WV , L) is a Zp-reflection group for all non-* *trivial V L Fp. Since p is odd it follows by [49, Thm. 7.6] that CX (V ) is connect* *ed for all THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 19 non-trivial V L Fp. Hence, since X is assumed to come from a compact Lie gr* *oup [7, Thm. B] (or [121, Thm. 2.28]) implies that H*(BX; Zp) does not have p-torsion a* *nd hence H*(BX; Zp) ~=H*(B2L; Zp)W (cf. Theorem 12.1). So Zp[L]W is a polynomial algeb* *ra as wanted. Next we assume that (W, L) is one of the exotic Zp-reflection groups. By Theo* *rem 12.2, Zp[L]W is a polynomial algebra, so we only need to prove that Zp[L]WV is a po* *lynomial algebra for any non-trivial V L Fp. Furthermore, by Theorem 12.2(2), Fp[L * * Fp]W is a polynomial algebra. Nakajima's result [92, Lem. 1.4] shows that Fp[L Fp* *]WV is a polynomial algebra as well. Thus we are done if p - |WV | by Lemma 12.6, whi* *ch in particular covers the cases where p - |W |. If (W, L) belongs to family number 2 on the Clark-Ewing list, then since p is* * odd, it is easily seen from the form of the representing matrices (see Section 11 for a* * concrete description) that reduction mod p gives a bijection between reflections in (W, * *L) and (W, L Fp). As Fp[L Fp]WV is a polynomial algebra it follows by the Shephard-Todd-C* *hevalley theorem [6, Thm. 7.2.1] that WV GL (L Fp) is a reflection group. Thus (WV , L* *) is a Zp- reflection group. Since the representing matrices are monomial, it follows by [* *92, Thm. 2.4] that Zp[L]WV is a polynomial algebra. By Theorem 11.1 only four cases remain, namely the Zabrodsky-Aguad'e cases (W* *12, p = 3), (W29, p = 5), (W31, p = 5) and (W34, p = 7). For each of these a direct co* *mputation (for instance easily done with the aid of a computer) shows that if S is a Sylo* *w p-subgroup of W , then U = (L Fp)S is 1-dimensional and (WU , L) is equivalent to ( p, L* *SU(p) Zp) (the construction of such a subgroup can also be found in Aguad'e [3]). Hence w* *e see that if V L Fp is non-trivial then either p - |WV | or V is W -conjugate to U. B* *ut in these cases we already know that Zp[L]WV is a polynomial algebra. Theorem 5.3. Let p be an odd prime and let (W, L) be a finite Zp-reflection gro* *up with the property that Zp[L]W is a polynomial algebra over Zp. By Theorem 5.1 (WV , L) * *is again a Zp-reflection group and Zp[L]WV is a polynomial algebra, for all non-trivial* * elementary abelian p-subgroups V ~T= L Z=p1 . Assume that we know that for all such V there exists a p-compact group F (V )* * with discrete approximation to its maximal torus normalizer given by ~ToWV such that F (V ) i* *s deter- ~= mined by NF(V ), Aut(BF (V )) ~=Aut(BNF(V )), and H*(BF (V ); Zp) ! H*(B2L; Zp* *)WV . Then there exists a p-compact group X with discrete approximation to its maxima* *l torus normalizer given by ~ToW satisfying the same properties as listed for F (V ). Proof.Set N~ = T~oW . We want to construct a candidate "centralizer decomposit* *ion" diagram. Let A be the category with objects the non-trivial elementary abelian * *p-subgroup of ~Tand morphisms the homomorphisms between them induced by inclusion of subgr* *oups and conjugation by elements in W . If V is a non-trivial elementary abelian p-s* *ubgroup of ~T, then by assumption there exists an, up to isomorphism unique, p-compact group F* * (V ) with discrete approximation to the maximal torus normalizer jV : CN~(V ) ! F (V ) sa* *tisfying the assumptions listed in the theorem. 20 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Let ' : V ! V 0be a morphism in A, induced by conjugation by an element x 2 W* * . We F(') now want to define a map F (V 0) ! F (V ). Consider the diagram cx-1 V 0//__//CN~(V_0)_//_CN~(V ) | | | | fflffl| fflffl| F (V 0) F (V ) Taking centralizer of the composite map x-1 : V 0! F (V ) we get a space BCF(V * *)(x-1) = map(BV 0, BF (V ))Bx-1, which has discrete approximation to its maximal torus n* *ormalizer equal to CN~(V 0). By assumption we get a unique (up to conjugacy) isomorphism * *F (V 0) ! CF(V )(x-1) under CN~(V 0). By composing with the evaluation CF(V )(x-1) ! F (* *V ), we get a map F (') : F (V 0) ! F (V ). We need to check that this gives us a well* * defined functor from Aop to the homotopy category of spaces, i.e., that for V !' V 0_!V* * 00, we have F (_') ~=F (')F (_). To see this consider the following diagram with obvious ma* *ps ~= ev -1~= F (V )oevoCF(V_)(x-1)oo______ F (V 0)oo______ CF(V 0)(y o)o__F (V 00) ffNNNN hhQQQQ mm ppp NNNN QQevQQ mmmm pppp NNNN QQQQ vvmmm~=mmm ppppp NN NNNN ^-1-1 pppp evNNNN CCF(V()x-1)(x y ) pppp~=pp NNNNN | ppp NNNN |~= pppp N fflfwwppppfl| CF(V )(x-1y-1) (Here ~(.)denotes the adjoint map which is explained in Remark 6.4.) Note that * *the bottom composite from F (V 00) to F (V ) is F (_') and the top composite is F (')F (_)* *. The top tri- angle is commutative, since the lower isomorphism in that triangle is just the * *map obtained by taking centralizers of the upper one. The rightmost square is homotopy commu* *tative, since the corresponding square of isomorphisms between centralizers in ~Nis com* *mutative, using our assumptions that maps are detected here. Finally, the leftmost square* * is homotopy commutative, by definition of the adjoint construction. Hence we have constructed a well defined functor A ! Ho(Spaces), where Ho(Spa* *ces) is the homotopy category of spaces. By construction the functor obtained when * *taking cohomology of this diagram, can be identified with the canonical functor which * *on objects is given by V 7! H*(BT~; Zp)WV . We want to lift this to a diagram in the category of topological spaces. The * *obstruction theory for doing this is described in [42, Thm. 1.1], noting that by [49, Lem. * *11.15] our diagram is a so-called centric diagram so the assumptions of that theorem are s* *atisfied. By looking at their cohomology we see that all the spaces F (V ) are connecte* *d and hence by [49, Thm. 7.5] have center given by ~T WV, since p is odd. In particul* *ar (see e.g. Lemma 10.2) the homotopy groups of ZF (V ) are given by ß0(ZF (V )) = H1(WV ; L* *) and ß1(ZF (V )) = LWV . By [47, x8] (for details see Section 10) lim*V 2Aß*(F (-)) * *= 0, so by [42, Thm. 1.1] there exists a (unique) lift of our functor BF to a functor gBF landi* *ng in Spaces. Set BX = (hocolimA gBF)^p. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 21 The spectral sequence for the calculating the cohomology of a homotopy colimi* *t [17, Ch. XII 4.5] has E2-term given as Ei,j2= limiV 2AHj(BT~; Zp)WV . But again by [* *47, x8] these groups vanish for i > 0 and for i = 0 give lim0AH*(BT~; Zp)WV ~=H*(BT~; Zp)W . * *Hence the spectral sequence collapses onto the vertical axis, and we get H*(BX; Zp) ~=H*(* *BT~; Zp)W . Since H*(BX; Zp) is a polynomial algebra H*(X; Zp) will be an exterior algebr* *a on odd generators (cf. Theorem 12.1), so X is indeed a p-compact group. The fact that* * BX is determined by N and satisfies Aut(BX) ~=Aut(BN ), also follows easily from the * *above_ the details are given in the proof of Lemma 6.3. Remark 5.4. Note that Theorem 5.3 in itself does not quite give a stand-alone p* *roof of the realization and uniqueness of all center-free p-compact groups with Weyl group * *satisfying that Zp[L]W is a polynomial algebra, since (WV , L) need not be center-free wh* *ich prevents the obvious induction from working, the problem being the unitary groups. 6. Proof of the main theorem using Sections 8, 9, 10, 11, and 12 The purpose of this section is to prove the main Theorems 1.1 and 1.4, but in* * the proofs referring forward to Section 8 for information about elementary abelian p-subgr* *oups of the Lie groups and to Section 10 for the obstruction group calculations. We start by explaining the strategy in general terms. Recall that the central* *izer of an elementary abelian p-subgroup : E ! X of a p-compact group X is defined as th* *e p- compact group CX ( ) with classifying space BCX ( ) = map (BE, BX)B . It is a t* *heorem of Dwyer-Wilkerson [48, Prop. 5.1 and 5.2] that this actually is a p-compact gr* *oup and that the evaluation map to X is a monomorphism. A theorem of Dwyer-Zabrodsky [39] [* *74, Thm. 3.2] says that if G is a compact Lie group with component group a p-group,* * then the map BCG ( (E))^p! BCG^p( ) = map(BE, BG^p)B induced by the adjoint of the canonical homomorphism E x CG ( (E)) ! G is a hom* *otopy equivalence. Note however that CX ( ) is not in any natural way a subobject of * *X, and that the map to X is defined in terms of . For a p-compact group X, let A(X) denote the Quillen category of X. The objec* *ts of A(X) are conjugacy classes of monomorphisms :E ! X of non-trivial elementary * *abelian p-subgroups E into X. The morphisms ( : E ! X) ! ( 0: E0! X) of A(X) consists * *of all group homomorphisms ': E ! E0 such that and 0' are conjugate. The centralizer construction gives a functor (6.1) BCX : A(X)op! Spaces that takes the monomorphism ( : E ! X) 2 Ob (A(X)) to its centralizer BCX ( ) = map(BE, BX)B and a morphism ' to composition with B' : BE ! BE0. By a theorem of Dwyer-Wilkerson [49, x8], generalizing a theorem for compact * *Lie groups by Jackowski-McClure [73], the evaluation map hocolimA(X)BCX ! BX induces an isomorphism on mod p homology. If X is connected center-free, then f* *or all , the centralizer CX ( ) is a p-compact group with smaller cohomological dimensio* *n setting the stage for a proof by induction. (The cohomological dimension of a p-compact* * group Y is defined as cd(Y ) = max{n|Hn(Y ; Fp) 6= 0}; see [48, 6.13] and [50, 3.8].) T* *o make use of this we need a way to construct a map from the elementary abelian p-subgroups a* *nd their 22 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL centralizers in X, to another any other p-compact group X0 with the same maxima* *l torus normalizer N . Suppose that N is embedded in connected p-compact groups X and X0 via homomor- phisms j and j0 respectively. If : E ! X can be conjugated into T , i.e., if * *there exists ~ : E ! T such that ~ composed with inclusion T ! X equals , then ~ is unique * *up to conjugation as a map to N by [50, Prop. 3.4], and furthermore by [49, Thm. 7.6]* *, CN (~) is a maximal torus normalizer in CX ( ). In this case j0~ will be an elementar* *y abelian p-subgroup of X0, which we have assigned without making any choices, and CX0(j0* *~) will have maximal torus normalizer CN (~). Elementary abelian p-subgroups which can * *be con- jugated into T are called toral subgroups, and elementary abelian p-subgroups w* *hich do not have this property are called non-toral subgroups. The problem hence arises how* * to com- pare the centralizers in the case of non-toral elementary abelian p-subgroups. * *This problem was addressed by the third-named author in [90]: Theorem 6.1. [90] Let X be a p-compact group with maximal torus normalizer N . If : E ! X is an elementary abelian p-subgroup of X, then there exists a li* *ft ~ : E ! N with = j~ such that CN (~) ! CX ( ) is a maximal torus normalizer. Furthermo* *re, if E0 E and ~0is a lift of |E0with the above property, then ~0can be extended to* * a lift ~ of E which also has that property. Assume X is connected. If : E ! X has rank one then is toral [48, Prop. * *5.5.] and the above lift ~ is unique up to conjugation in N . If has rank two, then* * the lift is unique if is toral and if is non-toral then there are precisely p + 1 diffe* *rent lifts with the further property that ß0(~) : E ! ß0(N ) is not injective, corresponding to the* * p + 1 rank one subgroups of E. Remark 6.2. The analogous theorem of the above in the classical case of compact* * Lie groups does not seem to appear in the literature. However, as was suggested to * *the third- named author by H. R. Miller and subsequently explained in detail by J.-P. Serr* *e [112], this can be obtained by a modification of the proof of [119, Thm. 5.16]_strictly spe* *aking, we shall only need this theorem in the cases where either X is the p-completion of* * a compact Lie group, or where the mod p cohomology of BX is a polynomial algebra, the lat* *ter case being trivial. If X is assumed connected then we can always arrange that the lift in Theorem* * 6.1 furthermore satisfies that the kernel of the map ß0(~) : E ! ß0(N ) is non-triv* *ial, and such a map will be called a preferred lift. Note that if : E ! X can be factored t* *hrough T then the corresponding map ~ : E ! N is a preferred lift and it is furthermore * *unique up to conjugacy in N , i.e., the preferred lift is exactly the factorization throu* *gh the maximal torus described earlier (see also [90, Prop. 4.10]). We now want to show how to use Theorem 6.1 to construct a map between the cen* *tralizer diagrams of connected p-compact groups X and X0. Suppose that : V ! X has rank one and that CX ( ) is determined by NCX( )a* *nd satisfies Aut(BCX ( )) ~=Aut(BNCX( )). Let ~ be a preferred lift of . Since * *is rank one and X is connected the unique preferred lift ~ will in fact factor through T by* * [48, Prop. 5.5], and ~ is a preferred lift of j0~ by [49, Thm. 7.6]. Hence CN (~) is a maximal t* *orus normalizer for both CX ( ) and CX0(j0~), so by assumption there exists a homomorphism h , * *unique THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 23 up to conjugacy, making the diagram (6.2) CN (~)K juuuu KKKj0K uuu KK zzuu h K%%K CX ( )_______~=_____//_CX0(j0~) commute. For an arbitrary elementary abelian p-subgroup : E ! X we can restri* *ct to a rank one subgroup V , and all the corresponding maps in the general case by v* *iewing things as going on inside the centralizer of the rank one subgroup. However it * *is not a priori clear that this construction does not depend on the choice of the rank one subg* *roup V . Furthermore one has the problem that this procedure only gives a map of diagram* *s in the homotopy category, and one then needs to prove that this diagram can be rigidif* *ied to a diagram in the category of spaces. The next lemma and its proof makes this cons* *truction precise and states what needs to be checked_the calculations to verify that the* *se conditions are indeed verified for all p-compact groups is essentially the contents of the* * rest of the paper. Lemma 6.3. Let X and X0be two connected p-compact groups with the same maximal * *torus normalizer N embedded via j and j0 respectively. Assume that for all rank one e* *lementary abelian p-subgroups : E ! X of X the centralizer CX ( ) is determine by NCX( * *)and that Aut(BCX ( )) ~=Aut(BNCX( )) when is of rank one or two. (1)Assume that for every rank two non-toral object : E ! X with a preferre* *d lift ~, neither the conjugacy class of j0~ : E ! X0 nor the conjugacy class of th* *e induced map h ,V: CX ( ) ! CX0(j0~) induced by taking ä djoints relative to the r* *ank one subgroup |V : V ! X" (see diagram (6.3)) depends on the p + 1 choices of* * a rank one subgroup V in E. Then there exists a map in the homotopy category of * *spaces from the centralizer diagram of BX to BX0 (seen as a constant diagram), i* *.e., an element in lim0 2A(X)[BCX ( ), BX0], given via maps h ,Vas hinted above. (2)Assume furthermore limi 2A(X)ßj(BZCX ( )) = 0 for j = 1, 2 and i = j, j +* * 1, then there is a unique lift of this to a map in the (diagram) category of spac* *es, i.e., there exists a unique equivalence f :X ! X0 under N and Aut(BX) ~=Aut(BN ). Before the proof, let us recall the construction of adjoint maps, since these* * play a central role in the proof: Remark 6.4 (Adjoint maps). Let A be an abelian p-compact group, X be a p-compact group, and : A ! X be a homomorphism. Suppose that E is a subgroup of A and n* *ote that we have a canonical map B_ : BA x BE mult!BA ! BX which only depends on the conjugacy class of . Since furthermore a ß0(map (BA x BE, BX)) = ß0(map (BA, map(BE, BX),)) ,2[BE,BX] 24 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL we get that every element : A ! X gives rise to an element ~ : A ! CX ( |E) m* *aking the diagram CX ( |E) w;;w w~www |ev| www fflffl| A _______//X commutative. Here ~ is well defined up to conjugacy, from the conjugacy class o* *f . We will always use the notation f(.)for this construction. Proof of Lemma 6.3.Let ~ be a preferred lift of . Since X is connected we can * *by Theorem 6.1 choose a rank one subgroup V in the kernel of E ! ß0(N ). Since ~|V factors* * through T it follows from [49, Thm. 7.6] that ~|V is a preferred lift of both |V and j* *0~|V. By assumption CX ( |V) is determined by NCX( |V)and Aut(BCX ( |V)) ~=Aut (BNCX( |V* *)) so we get a commutative diagram CN (~|V) s MMM sss MMM sss MMM yysss h |V M&& CX ( |V)_______~=______//_CX0(j0~|V) The adjoint maps of , ~, and j0~ (see Remark 6.4) map into the diagram in a co* *herent way, which expresses j0~ in terms of and the rank one subgroup V , so it only depe* *nds on those parameters. Taking further adjoints with respect to , ~ and j~ produces a comm* *utative diagram (6.3) CNO(~)OO qqq | OOOO qqq |~= OOOO qqq | OOO jqqqqq C (~~) OOOj0OO qqq CN (~|V)O OOOO qqq qqq OOO OOO qqq qqq OOOO OOOO qqq qqq OOO OOO xxqqq xxqq gh |V '' O'' CX ( )oo~=CCX(_|V)(~)_________~=_______//_CCX0(j0~|V)(jf0~)~=//_CX0(j0~) where ~h |Vis the map induced from h |Von the centralizers. But this means that* * CN (~) is a maximal torus normalizer also in CX (j0~), e.g., by the characterizing property* * for maximal torus normalizers give in [90, Thm. 1.2(3)], so in particular ~ is a preferred * *lift of j0~. Call the bottom left-to-right composite in the above diagram for h ,V. We wan* *t to see that this map does not depend on V (and we will then drop the V from the notati* *on). h ,V * * 0 Note that by the above diagram (6.3)the composition E !~ CX ( ) ! CX0(j~) !* * X equals j0~. By a slight abuse of notation we denote this map hV ( ). If has rank two and is toral then h ,Vdoes not depend on the choice of V by* * diagram (6.3)since ~ is uniquely determined from in this case, and we are assuming th* *at centralizers of rank two subgroups have the same automorphisms as their maximal torus normal* *izer. If has rank two and is non-toral, then we are simply assuming that h ,Vdoes not * *depend on V . THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 25 We want to see that the rank two condition forces this to hold in general. Le* *t V1 and V2 be two different rank one subgroups of E, and set U = V1 V2. Since h |U,Vd* *oes not depend on the choice of V the following diagram commutes up to conjugation h |V1 CX ( |V1)___//_CX0(h( |V1)) xx< 1 then Theorem 9.1 shows that the assumptions of Lemma 6.8 are * *satisfied which implies that the assumptions of Lemma 6.3(1)are satisfied. By Theorem 10.* *1 the assumptions of Lemma 6.3(2)are also satisfied, so Lemma 6.3 produces an equival* *ence of p-compact groups P U(n)^p! X, and shows that X satisfies the conclusion of Theo* *rem 1.4. If n = p then we argue as above but replace the reference to Lemma 6.8 with a r* *eference to Lemma 6.7, noting that the statement about h ,Vin this case follows for free fr* *om that of j0~, since E ~=CPU(p)(E). If (W, L) = (WE8, LE8 Z5) then by Theorem 8.2(3) E8 does not have any non-t* *oral rank 2 elementary abelian 5-subgroups. Furthermore the higher limits obstructio* *ns which feature in the assumptions for Lemma 6.3(2)vanish by Theorem 10.1. Hence Lemma * *6.3 implies that there exists an equivalence of p-compact groups E8^5! X, and that * *X satisfies the conclusion of Theorem 1.4. Likewise if (W, L) equals (WF4, LF4 Z3), (WE7, LE7 Z3), or (WE8, LE8 Z3* *), then Theorem 8.2(3)shows that there are no rank two non-toral elementary abelian 3-s* *ubgroups and the obstruction groups vanish by Theorem 10.1, so Lemma 6.3 again implies t* *hat X is homotopy equivalent to the p-completion of the appropriate Lie group and sat* *isfies the conclusion of Theorem 1.4. Finally, if (W, L) = (WE6, LE6 Z3) then the rank 2 non-toral subgroup E2bE6* *of The- orem 8.9 satisfies the conditions of Lemma 6.8 so the assumptions of Lemma 6.3(* *1) are satisfied for this subgroup. Likewise for the subgroup E2aE6the custom made Le* *mma 6.9 shows that the assumptions of Lemma 6.3(1)are also satisfied for this subgroup.* * Since the assumptions of Lemma 6.3(2) are satisfied by Theorem 10.1 we conclude by Lemma * *6.3 that X is homotopy equivalent to E6^3and satisfies the conclusion of Theorem 1.* *4 also in this case. This concludes the proof of the main theorem. Remark 6.10. Note that taking the case (WE6, LE6 Z3) last in the above theore* *m is a bit misleading, since groups with adjoint form E6 appear as centralizers in E7 * *and E8, so a separate inductive proof of uniqueness in those case would require knowledge of* * uniqueness of E6. Remark 6.11. The very careful reader might have noticed that the splitting resu* *lt in [5], which we use in the above proof, to conclude the splitting for (WPU(3), LPU(3))* * refers to a uniqueness result in [20]. We now quickly sketch a more direct way to see th* *is, which we were told by Dwyer-Wilkerson: We need to see that a 3-compact group with We* *yl group (WPU(3), LPU(3) Z3) has to have split maximal torus normalizer N . So, s* *uppose that X is a hypothetical 3-compact group like above but with non-split maximal * *torus normalizer. By a transfer argument (cf. [48, 9.12]) N3 has to be non-split as w* *ell. Since every elementary abelian 3-subgroup in X can be conjugated into N3 (since Ø(X=N* *p) is prime to p), this means that all elementary abelian 3-subgroups in X are toral.* * Furthermore by [50, Prop. 3.4] conjugation between toral elementary abelian p-subgroups is * *controlled by the Weyl group, so the Quillen category of X in fact agrees with the Quillen* * category of 32 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL N . The category has up to isomorphism one element of rank two and two of rank * *one. The centralizers CN (V ) of these are respectively T , T : Z=2, and T . Z=3. The un* *ique 3-compact groups corresponding to these centralizers are in fact given by BCN (V )^3. Hen* *ce the map BN ! BX is an equivalence by the centralizer cohomology decomposition theorem [* *49, x8]. But since N is non-split, we can find a map Z=p2 ! N , which is not conjug* *ate in N to a map into T . Hence the corresponding map Z=p2 ! X cannot be conjugated int* *o T either, contradicting [48, Prop. 5.5]. 7. Consequences of the main theorem In this section we prove the theorems listed in the introduction which are co* *nsequences of the main theorem. We also give a couple of remarks relating our method to other* * possible approaches. Proof of Theorem 1.2.The theorem follows directly from Theorem 1.1 together wit* *h the classification of finite Zp-reflection groups (Theorem 11.1). Proof of Theorem 1.5.By [86, Thm. 1.4] X is equivalent to a p-compact group of * *the form (X0x T 00)=A, where X0 is a simply connected p-compact group, T 00is a p-compac* *t torus, and A is a finite central subgroup of the product. Hence we have X=T ~=X0=T 0, * *where T and T 0are maximal tori of X and X0 respectively. So we can without restriction* * assume that X is simply connected. For compact connected Lie groups the statement of this theorem is the celebra* *ted result of Bott [12, Thm. A]. Hence by Theorem 1.2 it is enough to prove the theorem w* *hen X is an exotic p-compact group. In that case H*(BX; Zp) is a polynomial algebr* *a with generators in even degrees, and number of generators equal the rank of X (by th* *e proof of Theorem 1.4). The same is true over Fp, and since H*(BT ; Fp) is finitely g* *enerated over H*(BX; Fp) by [48, Prop. 9.11], we have that H*(BX; Fp) ! H*(BT ; Fp) has * *to be injective by a Krull dimension consideration. But since they are both polynomia* *l algebras it follows by e.g., [52, x11] that H*(BT ; Fp) is in fact free over H*(BX; Fp).* * Hence the Eilenberg-Moore spectral sequence of the fibration X=T ! BT ! BX collapses and H*(X=T ; Fp) ~=Fp H*(BX;Fp)H*(BT ; Fp). In particular H*(X=T ; Fp) is concentrated in even degrees so the rank equals t* *he Euler characteristic Ø(X=T ) which again equals |WX | by [48, Prop. 9.5]. By the lon* *g exact sequence in cohomology and Nakayama's lemma we get that H*(X=T ; Zp) is a free * *Zp- module of rank |WX | as wanted. Remark 7.1. For any connected p-compact group X the natural map X=T ! BT induces an isomorphism H*Qp(X=T ) ~=Qp H*Qp(BX)H*Qp(BT ) since the Eilenberg-Moore spectral sequence of the fibration X=T ! BT ! BX coll* *apses by [48, Prop. 9.7] and [52, x11]. It then follows from [27] that the natural WX* * -action on H*(X=T ; Zp) Qp = H*Qp(X=T ) is isomorphic to the regular representation of W* *X when ignoring the grading. Just as for Lie groups this is usually not true over Zp * *as SU (2)^2 shows. Proof of Theorem 1.6.By Theorem 1.2 it is enough to prove the statement for the* * case where X is the p-completion of a compact connected Lie group and the case where* * X is an exotic p-compact group separately. The case where X is the p-completion of the * *compact THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 33 Lie group of course follows directly from the classical Peter-Weyl theorem (cf.* * e.g. [18, Thm. III.4.1], so we can concentrate on the case where X is exotic. If p does n* *ot divide the order of the Weyl group the statement is also obvious: The inclusion ~T! U(* *r) induces a map ~ToW ! U(r|W |) whose p-completion is a faithful representation. The rema* *ining cases have been shown to have faithful representations by Castellana: If (W, L)* * is in the 2a family then this is carried out in [25] and if (W, L) is one of the pairs (G* *12, p = 3), (G29, p = 3), (G31, p = 5), or (G34, p = 7) this is carried out in [24]. We now turn to Theorem 1.7 which in fact follows easily from the classificati* *on. But let us first state the part which one can see by elementary means. (See [52, Lem. 9* *.3] for the Lie group version.) Proposition 7.2. Let X be a connected p-compact group then the natural composit* *e map (LX )W ~= H0(W ; H2(BT ; Zp)) ! H2(BX; Zp) ~=ß1(X) induced by the inclusion T ! X is surjective with finite kernel. In particular if (LX )W is torsion free then it is an isomorphism. Proof.By [86, Thm. 1.4] X is equivalent to a p-compact group of the form (X0x T* * 00)=A, where X0is a simply connected p-compact group, T 00is a p-compact torus, and A * *is a finite central subgroup of the product. Since the center of a p-compact group is conta* *ined in a maximal torus by [49, Thm. 7.5] we can assume A is a subgroup of T 0x T 00, whe* *re T 0is maximal torus for X0, and hence (T 0x T 00)=A is a maximal torus for X. Therefo* *re we get the following diagram of fibration sequences BA ____//_BT 0x BT_00__//B((T 0x T 00)=A) || | | || | | || fflffl| fflffl| BA ____//_BX0x BT 00________//BX The long exact sequence of homotopy groups and the five-lemma now shows that ß2* *(B((T x T 00)=A)) ! ß2(BX) is surjective which is the first statement in the propositio* *n. To see that the kernel is finite note that by [48, Thm. 9.7(3)] H2Qp(BX) ! H2Qp(BT )W * * is an isomorphism, which by dualizing to homology shows the claim. That we get an isomorphism when (LX )W is torsion free is obvious from the g* *eneral statement. Remark 7.3. One easily shows that the image of the differential d3 : H3(W ; Zp)* * ! H0(W ; H2(BT ; Zp)) in the Serre spectral sequence for the fibration BT ! BNX !* * BW is always in the kernel of the surjective map of Proposition 7.2. By standard g* *roup coho- mology (cf. [26]) the image of this differential identifies with the image of t* *he map given by capping with the k-invariant fl 2 H3(W ; H2(BT ; Zp)) of the extension. If one * *knew that * * * the double coset formula held for p-compact groups (more precisely that H*(BN ;* * Zp) tr! H*(BX; Zp) res!H*(BT ; Zp) is the identity, cf. [57, Ex. VI.4]) then it would e* *asily follow that this image was in fact equal to the kernel of the map in Proposition 7.2, * *getting a good proof of the formula for the fundamental group. Note that by a result of Tits [* *124] (see also [54, 93, 5]) this extension class fl is always of order 2 for compact connected* * Lie groups. The next proposition gives the complete answer in the Lie case. 34 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Proposition 7.4. Let G be a connected compact Lie group. Then the map ß1(T )W !* * ß1(G) is surjective with kernel (Z=2)s, where s is the number of direct factors of G * *isomorphic to a symplectic group Sp(n), n 1. Proof.That the map is surjective follows as in the p-compact case, so we just h* *ave to identify the kernel. By [83, Thm. 1.6], for any compact connected Lie group G, T W = Z(G* *) (Z=2)s, where s is the number of direct factors of G isomorphic to an odd special ortho* *gonal group SO(2n + 1), n 1. Consider the dual group G_ of G obtained as the taking the dual root diagram * *(see [15, x4 8]), then G_ has fundamental group isomorphic to [Z(G), where the hat d* *enotes the Poincare dual group (see [15, x4 9]). Likewise ("LG~)Wis canonically isomor* *phic to T W. Since duality is an involution on the set of compact Lie groups which sends dir* *ect summands to direct summands and SO(2n+1) to Sp(n) the claim about the fundamental group * *follows directly from the dual result about the center. Proof of Theorem 1.7.By Theorem 11.1 (LX )W = 0 for all the exotic p-compact g* *roups, so Proposition 7.2 shows the formula in this case. By Theorem 1.1 we are hence * *reduced to showing the formula for X of the form G^pfor some connected compact Lie grou* *p G. In this case the formula is well known and easy. Namely it follows from Remark * *7.3 that the kernel of (LX )W ! ß1(X) is an elementary abelian 2-group. Alternatively * *the same conclusion follows from the formula for the fundamental group of a compact Lie * *group (see [15, x4.6. Prop. 11] or [2, Thm. 5.47], noting that in the notation of [15] (1-* *s`)v` = 2v`). We now start to prove Theorem 1.8 and 1.9. Lemma 7.5. Suppose X and X0 are two connected p-compact groups both with maximal torus normalizer N . Then all elementary abelian p-subgroups of X are toral if * *and only if all elementary abelian p-subgroups of X0 are toral. Furthermore, if for all toral elementary abelian p-subgroups V ! X the centra* *lizer CX (V ) is connected (something which can be calculated from N by [49, Thm. 7.6]) then * *all elemen- tary abelian p-subgroups in X are toral. Proof.Suppose that X has a non-toral elementary abelian p-subgroup V ! X. We c* *an assume that it is minimal, in the sense that any elementary abelian p-subgroup * *of smaller rank is toral. Write V = V 0 V 00, where V 0has rank one. We can assume that* * V 00! X factors through T (by the minimality) and that V ! X factors through N (by [* *49, Prop. 2.14]). Let CN (V 00)1 denote the maximal torus normalizer of CX (V 00)1* *, which by [49, Thm. 7.6] can be described in terms of V 00and N . The adjoint map V 0! CN* * (V 00) cannot factor through CN (V 00)1 since otherwise V 0! CX (V 00) would be conjug* *ate to T in CX (V 00) (by [48, Prop. 5.5]), contradicting that V is assumed not to be toral* *. Note that CN~(V 00)1 is normal in CN~(V 00) and CN~(V )=CN~(V )1 ~=ß0(CX (V 00)) ~=ß0(CX0* *(V 00)) (see [49, Rem. 2.11]). Hence V 0! ß0(CX0(V 00)) is non-trivial so V ! N ! X0 cannot * *be toral in X0. The last part of the lemma is clear from the proof of the first part. Remark 7.6. Despite the above lemma, it is a priori not clear how to determine * *whether a p-compact group X has the property that all elementary abelian p-subgroups ar* *e toral just from looking at NX (but see [121, Thm. 2.28] for the Lie case). However, b* *y a case-by- case analysis (Theorem 1.8), the property is equivalent to that all toral eleme* *ntary abelian p-subgroups have connected centralizers. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 35 Remark 7.7. Note that by Lannes theory [79, Thm. 0.4] the property that every e* *lementary abelian p-subgroup of X lie inside a maximal torus is equivalent to that H*(BX;* * Fp) ! H*(BT ; Fp)WX is an F -isomorphism. (See also Theorem 12.1 and Remark 12.3.) We state the following well known lemma for easy reference: Lemma 7.8. Suppose that X is a connected p-compact group such that H*(BX; Zp) is a polynomial algebra with generators concentrated in even degrees. Then all el* *ementary abelian p-subgroups of X are toral. Proof.For every elementary abelian p-subgroup : E ! X, H*(BCX ( ); Zp) is a p* *oly- nomial algebra with generators concentrated in even degrees by [53, Thm. 1.3] (* *note that Lannes T -functor preserves objects concentrated in even degrees by [79, Prop. * *2.1.3]). In particular CX ( ) is connected so by Lemma 7.5 all elementary abelian p-subgrou* *ps are toral. (Alternatively one can use Remark 12.3.) Proof of Theorem 1.8.First note that the implication (1)) (2)follows from Theor* *em 12.1. The implication (2)) (3)follows easily from Theorem 5.1. Namely for all toral e* *lementary abelian p-subgroups V ! X, Theorem 5.1 implies that WCX(V )is a reflection grou* *p, so by [49, Thm. 7.6] CX (V ) is connected, using that p is odd. But this implies that* * all elementary abelian p-subgroups are toral by Lemma 7.5. We now prove the implication (3)) (1). First note that by Theorem 11.1 and [* *50, Thm. 1.4] we can write X ~=X0x X00where X0 has Weyl group (WG , LG Zp), for s* *ome compact connected Lie group G, and (W 00, L00) is a product of exotic finite Zp* *-reflection groups. Furthermore, since the normalizer of a connected p-compact group is sp* *lit for p odd by [5, Thm. 1.2] we can in fact choose the compact Lie group such that NG^p* *~=NX0. Since by Lemma 7.5 the property of having all elementary abelian p-subgroups to* *ral is a property which only depends on N we conclude that G has to have this property a* *s well. But this implies that G has torsion free Zp-cohomology by [7, Thm. B] (see also* * [121, Thm. 2.28]). In the exotic case, we know by the proof of Theorem 1.4 that we ca* *n find a p-compact group X~00which has the same maximal torus normalizer as X00and which* * has torsion free Zp-cohomology. Hence we have found a p-compact group G^px ~X00which has the same maximal tor* *us normalizer as X and has torsion free Zp-cohomology. Since by Theorem 1.4 a p-co* *mpact group is determined by its normalizer we conclude that X in fact has torsion fr* *ee Zp- cohomology. (Alternatively, one can appeal to Remark 7.11 which shows that the * *property of having torsion free Zp-cohomology only depends on N .) Remark 7.9. Since the implication (1)) (3)follows from Lemma 7.8, we see that i* *n the above theorem the implications (1)) (2)and (1)) (3)follow by general arguments.* * In the case of compact Lie groups the implication (3)) (2)likewise has a general p* *roof, by combining [121, Thm. 2.28] with [36], but at the moment we do not know such a p* *roof for p-compact groups. (See [121, x4].) The remaining implications do not seem to ha* *ve general proofs even for compact Lie groups. Proof of Theorem 1.9.As in the proof of Theorem 1.8 we can by Theorem 11.1 and * *[50, Thm. 1.4] write X ~=X0x X00where X0 has Weyl group (WG , LG Zp), for some com* *pact connected Lie group G, and the Weyl group (W 00, L00) of X00is a product of exo* *tic finite Zp-reflection groups. By Theorem 1.1 and its proof we know that X00is uniquely determined0by0its We* *yl group00 (W 00, L00) and has cohomology isomorphic to H*(B2L00; Zp)W . Since H2(B2L00; * *Zp)W = 36 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL (L00*)W00= 0 we conclude by the universal coefficient theorem that ß1(X00) = H2* *(BX00; Zp) = 0. Hence by the T -functor, as in the proof of Theorem 1.8, we conclude that X0* *0satisfies all three equivalent conditions of the theorem. Furthermore we can, again by [5* *, Thm. 1.2] since p is odd, choose the compact Lie group such that NG^p~=NX0. Hence by Lemm* *a 7.5 and [49, Thm. 7.6] the theorem holds for X0 if and only if it holds for G. But* * by [121, Thm. 2.27] the theorem is true for G, which finishes the proof of the theorem. Remark 7.10. The same arguments as above shows that the conjectural classificat* *ion for p = 2 implies that Theorem 1.9 and Theorem 1.8 (1), (3)) (2)holds true for p = * *2. However, in Theorem 1.8, (2)is not equivalent to the other conditions since Zp[* *LSO(2n+1)]W is a polynomial algebra, since this is true for Sp(n), despite SO(2n + 1) havin* *g 2-torsion. Remark 7.11. Notbohm states his classification of connected p-compact groups wi* *th Zp[L]W a polynomial algebra in the setup of spaces BX with polynomial cohomolo* *gy (cf. [97]). This means that his uniqueness statement is a priori only uniqueness am* *ongst p- compact groups with torsion free Zp-cohomology (cf. Theorem 12.1). We will here* * briefly sketch a direct but case-by-case way (following a line of argument given in a s* *pecial case in [87, Proof of Thm. 5.3]) to show that for a p-compact the property of having to* *rsion free Zp-cohomology depends only on (W, L), which allows us to remove the extra assum* *ption. Assume that X is a connected p-compact group, p odd, such that Zp[LX ]W is a* * poly- nomial algebra. We want to show that H*(BX; Zp) is a polynomial algebra as wel* *l. By Theorem 12.2(1), (LX )W is torsion free, so ß1(X) = (LX )W by Proposition 7.2. * *By the Serre and Eilenberg-Moore spectral sequences H*(BX; Zp) is a polynomial algebra if an* *d only if H*(B(X<1>); Zp) is a polynomial algebra. Furthermore by construction LX<1>=* * SLX and by Theorem 12.2(1)Zp[LX<1>]WX is also a polynomial algebra, so we can with* *out loss of generality assume that X is simply connected. By [50, Thm. 1.4 and Rem. 1.6]* * we can furthermore assume that X is a simple p-compact group. By [5, Thm. 1.2] N~X = ~TXoW . Using the classification of finite Zp-reflect* *ion groups (Theorem 11.1) we see that the cohomology of ~Nis always detected on elementary* * abelian p-subgroups: If p - |W | then the statement is obvious; if (W, L) is in the fam* *ily 2a then N~ contains N~U(n)of coprime index; if (W, L) is either (G12, p = 3), (G29, p = 3)* *, (G31, p = 5), or (G34, p = 7) then it contains N~SU(n)of coprime index (cf. the proof of Theo* *rem 5.1); if (W, L) is of Lie type then it contains either ~NU(n)^por ~NSU(n)^pof coprime in* *dex. Both ~NU(n) and ~NSU(n)both have cohomology which is detected on elementary abelian p-subgr* *oups by [107, Prop. 3.4] (for NU(n)^p; NSU(n)^pfollows from this). Hence, by a transfer* * argument, the mod p cohomology of BX is detected on elementary abelian p-subgroups. Next, we want to show that all elementary abelian p-subgroups of X can be con* *jugated into a maximal torus. By Lemma 7.5 we just have to show that we can find some p* *-compact group X0 with the same maximal torus normalizer which has that property. If (W,* * L) is of Lie type this follows from Borel's theorem [7, Thm. B]. If (W, L) is exotic thi* *s is also true since we know (by Theorem 5.3 or Notbohm's work [97]) that there exist a p-comp* *act group with Weyl group (W, L) and classifying space having polynomial Zp-cohomology al* *gebra. The fact that all elementary abelian p-subgroups of X are toral combined with* * the fact that the cohomology is detected on elementary abelian p-subgroups implies that * *the mod p cohomology of BX is concentrated in even degrees. Hence H*(BX; Zp) is torsion* * free as wanted. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 37 Proof of Theorem 1.10.Let X be a connected finite loop space with maximal torus* * i : T ! X. Note that (X=T )^p' X^p=T ^pby the fiber lemma [17, II 5.1], and consequentl* *y, by the definition of Euler characteristic, Ø(X=T ) = Ø(X^p=T ^p). Hence T ^p! X^pwill * *be a maximal torus normalizer for the p-compact group X^p, for all primes p. For our connected finite loop space X, define WX (T ) to be the set of conjug* *acy classes of self-equivalences ' of T such that i and i' are conjugate. We obviously have* * an injective homomorphism WX ! WX^p for all primes p and we now want to see that this map is surjective as well, so that we can naturally identify (WX , ß1(T ) Zp) with (* *WX^p, LX^p). First note that by [48, Proof of Thm. 9.7] we can view WX^pas the Galois grou* *p of the extension of polynomial algebras H*Qp(BX) ! H*Qp(BT ). But, since BX has finite* *ly many cells in each dimension and since BX is nilpotent, we can identify H*(BX; Q) Q Qp ____//_H*(BT ; Q) Q Qp |~=| ~=|| | | H*Qp(BX) ____________//H*Qp(BT ) so the extensions H*(BX; Q) ! H*(BT ; Q) and H*Qp(BX) ! H*Qp(BT ) have canonica* *lly isomorphic Galois groups. Hence any element in WX^plifts to a canonical element* * in the Galois group of the extension H*(BX; Q) ! H*(BT ; Q). However, since BXQ and BTQ are Eilenberg-Mac Lane spaces (as follows easily by looking at their cohomology* *), this Galois group just identifies with the self-equivalences BTQ ! BTQ which commute* * with the evaluation map BiQ : BTQ ! BXQ up to homotopy. Hence any element in WX^pgiv* *es rise to a compatible family of self-equivalences of BT ^land BT ^l, for all pri* *mes l. So by the arithmetic square [17, VI.8.1], we get a self-equivalence of BT which commu* *te with evaluation map to BX up to homotopy, i.e., an element in WX . The constructed e* *lement is a lift of the element in WX^pwe started with, so the map WX ! WX^pis surjective* * as well. Likewise, the argument above showed that WX is the Galois group of the exten* *sion H*(BX; Q) ! H*(BT ; Q), so we have an isomorphism ~= * W H*(BX; Q) ! H (BT ; Q) X . Since H*(BX; Q) is a polynomial algebra we get by the Shephard-Todd-Chevalley t* *heorem (see [6, Thm. 7.2.1]) that (WX , ß1(T )) is a Z-reflection group. Hence, by th* *e proof of Theorem 11.1, (WX , ß1(T )) is the Weyl group of some compact connected Lie gro* *up G. For each p we have an extension class xp 2 H3(WX ; ß1(T ) Zp) corresponding* * to the fibration sequence BT ^p! BNX^p! BWX . Since H3(WX ; ß1(T )) is a finite abelia* *n group, and hence given as a sum of its p-primary parts, these extension classes identi* *fy with a unique extension class x 2 H3(WX ; ß1(T )). We define the loop space NX to be t* *he loop space of the total space in the fibration sequence BT ! BNX ! BWX with the cano* *nical action of WX on BT and extension class x. Since, the partial Fp-completion of* * BNX identifies with BNX^p, the arithmetic square produces a canonical morphism NX * *! X. (NX is, quite naturally, called the maximal torus normalizer of the finite loop* * space X [87, Def. 1.3].) By [5] (see also [93], [54]) the extension classes defining T ! NX ! WX and* * T ! NG (T ) ! WG (T ) are both 2-torsion. Let gBN denote the fiber-wise Z[1_2]-loca* *lization of the total space of the fibration BT ! BNG (T ) ! BWG (T ) or equivalently the corre* *sponding 38 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL fibration with BNX . We hence have embeddings BgN xxx FFF xxx FFF --xx FF## BX[1_2] BG[1_2] By the arithmetic square [17, VI.8.1], the following square is a pullback Q ^ BX[1_2]_____//_p6=2BXp | | | | fflffl| Q fflffl| BXQ _____//( p6=2BX^p)Q and similarly for BG. By Theorem 1.4 we can construct unique maps between p-com* *pletions under gBN, and we obviously also have a unique map between the rationalizations* *. By con- struction (as maps under gBN) these maps agree on the rationalization of the p-* *completions, so by the arithmetic square we get an induced map BX[1_2] ! BG[1_2], which by c* *onstruction is an Fp-equivalence for all primes p. Since both spaces are one-connected this* * implies that the map is a homotopy equivalence. 8. Elementary abelian subgroups of the exceptional groups In this section we find all conjugacy classes of non-toral elementary abelian* * p-subgroups E, p odd, of any exceptional compact Lie group G, as well as their centralizers* * CG (E) and Weyl groups W (E) = NG (E)=CG (E). (Recall that a subgroup of G is called toral* * if it is contained in a torus in G and non-toral otherwise.) We do this by expanding on * *the work of Griess [63], who found the maximal non-toral elementary abelian p-subgroups. Our strategy is as follows. Using the work of Griess [63], we first find rep* *resentatives of the conjugacy classes of maximal non-toral elementary abelian p-subgroups. * *We then get lower bounds for their Weyl groups by producing explicit elements in these.* * From this we are able to identify the non-maximal non-toral elementary abelian p-subgroup* *s and get lower bounds for their Weyl groups. Finally we get exact results on the Weyl g* *roups by computing centralizers. To be compatible with the standard literature we will in this section state a* *nd prove all theorems in the context of linear algebraic groups over the complex numbers C_w* *e state in Proposition 8.4 why this is equivalent to considering compact Lie groups. (T* *he results for G(C) can furthermore be translated into results for G(F ) for any algebraic* *ally closed field F of characteristic prime to p, see [64, Thm. 1.22] and [60].) This section is divided into four subsections. The first recalls some results* * from the theory of linear algebraic groups and discusses the relation to compact Lie groups. Th* *e remaining subsections deal with the elementary abelian 3-subgroups of the groups of type * *E6, E7 and E8 respectively. (The remaining non-trivial cases E8(C), p = 5 and F4(C), p = 3* * are treated completely in [63, Lem. 10.3 and Thm. 7.4].) For some of our computations for the groups 3E6(C) and E8(C) we have used the* * com- puter algebra system Magma [11], although this reliance on computers could if n* *eeded be replaced by some rather tedious hand calculations. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 39 Notation 8.1. We use standard names for the linear algebraic groups we consider* *, i.e. 3E6(C) and E6(C) are respectively the simply connected and the adjoint versions* * of the group of type E6 over C. We let Tn denote an n-dimensional torus, i.e. Tn = (Cx* * )n. To describe centralizers we follow standard notation for extensions of groups* *, cf. the ATLAS [32, p. xx]. Thus A : B denotes a group which is the semidirect product o* *f the normal subgroup A with the subgroup B, and A . B denotes a non-split extension * *of A with B. If p is a prime number, we let pn denote an elementary abelian p-group of ra* *nk n. Whenever E is a concrete elementary abelian p-subgroup of rank r we will alwa* *ys fix an ordered basis of E, so that W (E) may can be considered as a subgroup of GL r(F* *p). We make the standing convention that all matrices acts on columns. We identify a permutation oe in the symmetric group n with its permutation m* *atrix A = [aij] given by aij= ffii,ff(j)where ffi is the Kronecker delta. If K is a field, we let Mn(K) denote set of n x n-matrices over K. For a1, . * *.,.an 2 K we let diag(a1, . .,.an) 2 Mn(K) denote the diagonal matrix with the ai's in th* *e diagonal. For 1 i, j n, we let eij2 Mn(K) denotes the matrix whose only non-zero entr* *y is a 1 in position (i, j). Given matrices A1 2 Mn1(K), . . . , Am 2 Mnm(K) we let A1 * * . . .Am denote the n x n-block matrix with the Ai's in the diagonal, n = n1 + . .+.nm .* * We also need the "blowup" homomorphism n,m : Mn(K) -! Mmn (K) defined by replacing each entry aijby aijIm , where Im 2 Mm (K) is the identity matrix. As p = 3 in all the cases we consider, we use some special notation. An arbit* *rary element of F3 is denoted by *, and " denotes an element of the multiplicative group Fx3* *. We let ! = e2ii=3and j = e2ii=9and define elements fi, fl, ø1, ø2 2 SL3(C) by fi = dia* *g(1, !, !2), fl = (1, 2, 3), 2 3 2 1 e-ii=18 1 ! 2 ø1 = ______p_41 1 ! 5 3 !2 1 1 and ø2 = diag(j, j-2, j). Note that fifi1= fifl, flfi1= fl, fifi2= fi and flfi2* *= fifl. 8.1. Recollection of some results on linear algebraic groups. Recall that a (not necessarily connected) linear algebraic group G is called reductive if the unip* *otent radical, i.e., the largest normal connected unipotent subgroup of G, is trivial. Theorem 8.2. Let G be a linear algebraic group over an algebraically closed fie* *ld K. (1)If A is a subgroup of G and S is some subset of A, then A is toral in G i* *f and only if A is toral in CG (S). (2)If H is a maximal torus of G, then two subsets of H are conjugate in G if* * and only if they are conjugate in NG (H). If A is a toral subgroup of G, then* * W (A) = NG (A)=CG (A) is isomorphic to a subquotient of the Weyl group W = NG (H)* *=H of G. (3)Assume that G is a connected reductive group such that G0 is simply conne* *cted. Then the centralizer of any semisimple element in G is connected. In part* *icular, if A is an abelian subgroup G consisting of semisimple elements generated by* * at most two elements, then A is toral. (4)If G is reductive and oe is a semisimple automorphism of G, then Gffis re* *ductive and contains a regular element of G. (5)Assume that G is a connected reductive group, let Z G be a central subg* *roup, and let ß : G ! G=Z be the quotient homomorphism. If A is a subgroup of G, th* *en A is toral in G if and only if ß(A) is toral in G=Z. 40 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL (6)Assume charK = 0 and let g be the Lie algebra of G. If S G is a finite * *subset of G, then the Lie algebra of CG (S) is given by cg(S) = {x 2 g| Ad(s)(x) = x for all s 2}S. In particular, if S G is a finite subgroup, then 1 X dim CG (S) = ___ trAd (s)|g. |S|s2S Proof.(1): Obviously, if A is toral in CG (S) then A is toral in G. Conversely,* * if A is toral in G, then A H for a torus H in G. Since S A we get H CG (S) and thus A i* *s toral in CG (S). (2): The first part follows by a Frattini argument. Assume that A, Ag H are* * conjugate subsets of H. Then H and Hg-1 are maximal tori of CG (A) and thus conjugate in * *CG (A) (cf. [70, Cor. 21.3.A]). Thus we may write H = Hg-1cfor some c 2 CG (A) and we * *conclude that n = g-1c 2 NG (H). Then An-1 = Ac-1g= Ag, which proves the first part. The* * second part follows similarly, cf. [81, Prop. 1.1(i)]. (3): The first part which is due to Steinberg is proved in [23, Thm. 3.5.6]. * *The second part follows from the first, cf. [119, II.5.1, p. 206]. (4): We can assume G to be connected. In case G is semisimple and simply conn* *ected the first claim is proved in [120, Thm. 8.1] and the general case reduces to th* *is one. Indeed we can find a finite cover ~Gof G which is a direct product of a semisimple sim* *ply connected group and a torus, and oe lifts to a semisimple automorphism of ~Gby [120, 9.16* *]. For the second claim see [130, Thms. 2 and 3] or [119, Pf. of Thm. 5.16,p. E-45] in c* *ase G is semisimple; the general case clearly reduces to this one. (5): By [70, Cor. 21.3.C] we know that if H is a maximal torus of G, then ß(H* *) is a maximal torus of G=Z, and all maximal tori of G=Z are of this form. Thus if A i* *s toral in G, then ß(A) is toral in G=Z. Conversely, if H0 is a maximal torus of G=Z co* *ntaining ß(A), then by the above we have H0 = ß(H) for some maximal torus H of G. Thus * *we get A . However since G is connected and reductive, we get Z H by [7* *0, Cor. 26.2.A(b)]. Thus A H and we are done. (6): In case S consists of a single element, the first part follows from [70,* * Thm. 13.4(a)] (note that the connectivity assumption in [70, Thm. 13.4] is only used in part * *(b)). The general case follows from this by applying [70, Thm. 12.5(b)] to the centralize* *rs CG (s), s 2 S. Now assume that S G is a finite subgroup, and let Ø denote the character of* * the adjoint representation of G restricted to S. Then the dimension of cg(S) = {x 2 g| Ad(s)(x) = x for all s 2}S. equals the multiplicity of the trivial character in Ø. By the orthogonality rel* *ations this is given by 1 X (Ø|1)= ___ Ø(s), |S|s2S and we are done. We also need the following result whose proof is extracted from [112]. Theorem 8.3. Let G be a reductive linear algebraic group, H a maximal torus of * *G and let N = NG (H). Let U N be a subgroup consisting of semisimple elements such* * that THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 41 U=(U \ H) is cyclic. Let S be the identity component of HU (the subgroup of H f* *ixed by U), and assume that S is a maximal torus of CG (U). Then CN (U) = NCG(U)(S) and* * in particular CN (U) is a maximal torus normalizer in CG (U). Proof.As any element of CN (U) normalizes HU and hence also its identity compon* *ent S, the inclusion CN (U) NCG(U)(S) is clear. Suppose conversely that x 2 NCG(U)(S* *). Let C = U \ H. From [10, 2.15(d)] it follows that GC is reductive. By assumption th* *e cyclic group U=C acts by semisimple automorphisms on GC . It now follows from Theorem * *8.2(4) that GU = (GC )U=C is reductive and that every maximal torus of GU is contained* * in a unique maximal torus of GC . Since C H, we see that H is the maximal torus o* *f GC containing S. As Hx is also a maximal torus of GC and Hx Sx = S we conclude t* *hat Hx = H. Thus x 2 CN (U) proving the result. We now explain the relationship between reductive complex linear algebraic gr* *oups and compact Lie groups. If G is a complex linear algebraic group then the underlyin* *g variety of G is an affine complex variety. By endowing this variety with the usual Euclide* *an topology instead of the Zariski topology we may view G as a complex Lie group since the * *group operations are given by polynomial maps. Proposition 8.4. Let G be a complex linear algebraic group. (1)Viewed as a Lie group, G contains a maximal compact subgroup which is uni* *que up to conjugacy, and for any such subgroup K we have a diffeomorphism G ~=K * *x Rs for some s. (2)Let K be a maximal compact subgroup of G, and let S, S0 K be two subset* *s. If S0= Sg for some g 2 G, then there exists k 2 K such that xk = xg for all * *x 2 S. (3)Assume that G is reductive. If S is a finite subgroup of G, then CG (S) * *is also reductive. If K is a maximal compact subgroup of G containing S, then CK * *(S) is a maximal compact subgroup of CG (S). (4)If G is reductive and K is a maximal compact subgroup of G, then we have a diffeomorphism Z(G) ~=Z(K) x Rs for some s. Proof.Note first that the identity component G1 of G seen as a Lie group coinci* *des with the identity component of G seen as a linear algebraic group [105, Ch. 3, x3,no* *. 1]. Thus G=G1 is finite by [70, Prop. 7.3(a)]. The first claim is now part of the Cartan* *-Chevalley- Iwasawa-Malcev-Mostow theorem [67, Ch. XV, Thm. 3.1] and the second claim also * *follows from this, cf. [9, Ch. V, x24,Prop. 2]. In case G is reductive it is possible to give a more explicit form of the dec* *omposition above. By [70, Thm. 8.6] we may assume that G is a closed subgroup of GL (V ) f* *or some complex vector space V . From [105, Thm. 5.2.8] it follows that G has a compac* *t real form K and we may thus choose a non-degenerate Hermitian inner product on V whi* *ch is invariant under K (see for instance [105, Thm. 3.4.2]). Let U(V ) denote the se* *t of operators in GL (V ) which are unitary with respect to the chosen inner product. Using [1* *05, Problems 5.2.3 and 5.2.4] we see that G GL (V ) is self-adjoint and that K = G \ U(V )* *. The last part now follows by combining [105, Cor. 2 of Thm. 5.2.2] with [105, Cor. 2 of * *Thm. 5.2.1]. If S is a subgroup of K, then S is selfadjoint since K consists of unitary op* *erators. In particular CG (S) is also a selfadjoint subgroup of GL (V ), so by [105, Cor. 3* * of Thm. 5.2.1] it follows that CK (S) = CG (S) \ U(V ) is a maximal compact subgroup of CG (S). It only remains to prove that CG (S) is reductive for a finite subgroup S of * *G. However by [105, Problem 6.11] and [105, Ch. 4, x1,no. 2] we see that a complex linear alg* *ebraic group 42 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL is reductive if and only if its Lie algebra is reductive. Thus it suffices to p* *rove that the Lie algebra of CG (S) is reductive. However by Theorem 8.2(6) this Lie algebra equa* *ls cg(S) = {x 2 g| Ad(s)(x) = x for all s 2}S. where g denotes the Lie algebra of G. The claim now follows from [28, Ch. V, x* *2,no. 2, Prop. 8]. 8.2. The groups E6(C) and 3E6(C), p = 3. In this subsection we consider the ele* *mentary abelian 3-subgroups of the groups of type E6 over C. The group 3E6(C) has two * *non- isomorphic faithful irreducible 27-dimensional representations. These have high* *est weight ~1 and ~6 respectively and are dual to each other. An explicit construction of * *3E6(C) based on one of these representations was originally given by Freudenthal [58]. This * *construction is described in more detail in [31, x2] from which we take most of our notation* *. In particular we let K be the 27-dimensional complex vector space consisting of triples m = (* *m1, m2, m3) of complex 3 x 3-matrices mi, 1 i 3 where addition and scalar multiplicatio* *n is defined coordinatewise. We define a cubic form <.>on K by = det(m1) + det(m2) + det(m3) - tr(m1m2m3). Then 3E6(C) is the subgroup of GL (K ) preserving the form <.>. Moreover the st* *abilizer in 3E6(C) of the element (I3, 0, 0) 2 K is the group F4(C). For g1, g2, g3 2 SL3(C* *) we have the element sg1,g2,g3of 3E6(C) given by -1 -1 -1 sg1,g2,g3(m1, m2, m3)= g1m1g2 , g2m2g3 , g3m3g1 for m = (m1, m2, m3) 2 K . This gives a representation of SL3(C)3 which has ke* *rnel C3 generated by (!I3, !I3, !I3), and we thus get an embedding of SL3(C)3=C3 in 3E6* *(C). We will denote the element sg1,g2,g3by [g1, g2,.g3] We let {eij,k}, 1 i, j, k 3 be the natural basis of K consisting of the e* *lements eij,kwhose entries are all 0 except for the (j, k)-entry of the i'th matrix which equals 1* *. The elements of 3E6(C) which acts diagonally with respect to this basis of K form a maximal * *torus H in 3E6(C). Let mj,kidenote the (j, k)-entry of the matrix mi. We then have H-in* *variant subgroups uff1(t) = [I3, I3 + te1,3,,I3] u-ff1(t) = [I3, I3 + te3,1,,I3] uff2(t) = [I3 + te2,1, I3,,I3] u-ff2(t) = [I3 + te1,2, I3,,I3] uff3(t) = [I3, I3 + te2,1,,I3] u-ff3(t) = [I3, I3 + te1,2,,I3] 0 2 2,3 31 0 -mi+2 0 uff4(t) : (mi)i=1,2,37! @mi+ t . 4 0 0 0 5A 0 m2,1i+20 i=1,2,3 0 2 31 0 0 0 u-ff4(t) : (mi)i=1,2,37! @mi+ t . 4 m3,2i+10 -m1,2i+15A 0 0 0 i=1,2,3 uff5(t) = [I3, I3, I3 + te2,1], u-ff5(t) = [I3, I3, I3 + te1,2], uff6(t) = [I3, I3, I3 + te1,3], u-ff6(t) = [I3, I3, I3 + te3,1]. Here, in the description of u ff4(t), the mi's should be counted cyclicly mod 3* *, e.g. mi+2= m1 for i = 2. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 43 The associated roots ffi, 1 i 6, of these root subgroups form a base for * *the root system (E6) of 3E6(C) (our numbering agrees with [13, p. 260-262]). For this * *base of (E6), the highest weight of K is ~1. Furthermore the root subgroups u ffi, 1* * i 6, have been chosen so that they satisfy [118, 8.1.1(i) and 8.1.4(i)], i.e. they f* *orm part of a realization ([118, p. 133]) of (E6) in 3E6(C). For ff = ffi, 1 i 6, and t* * 2 Cx , we may then define the elements nff(t) = uff(t)u-ff(-1=t)uff(t), hff(t) = nff(t)nff(1)-1. Q 6 Then the maximal torus consists of the elements h(t1, t2, t3, t4, t5, t6) = i* *=1hffi(ti) and the normalizer N(H) of the maximal torus is generated by H and the elements ni=* * nffi(1), 1 i 6. It should be noted that this notation differs from the one used in [* *31]. More precisely, the element h(ff, fi, fl, ffi, ffl, i) in [31] is h(ffi, ff-1, fl-1,* * fi, ffl-1, i) in our notation, and the elements n1, n2, n3, n4, n5 and n6 in [31] equals respectively n1hff1(-* *1)hff3(-1), n2h(-1, 1, 1, -1, 1, -1), n3hff1(-1), n4, n5hff6(-1) and n6hff5(-1)hff6(-1) in * *our notation. From the description of the root system of type E6 in [13, p. 260-262] we see* * that the center Z of 3E6(C) is cyclic of order 3 and is generated by the element z = I3* *, !2I3, !I3. We consider also the element a = [!I3, I3,.I3]A straightforward computation sho* *ws that the roots of the centralizer C3E6(C)(a) are { ff1, ff2, ff3, ff5, ff6, eff, (ff1 + ff3), (ff5 + ff6), (ff2* * - eff)}, where effis the longest root. The Dynkin diagram for this centralizer is the s* *ame as the extended Dynkin diagram for E6 with the node ff4 removed. In particular it* * has type A2A2A2 and basis {ff1, ff3, ff5, ff6, ff2, -eff}. Since 3E6(C) is simply * *connected, The- orem 8.2(3) implies that the centralizer C3E6(C)(a) is connected, and thus it i* *s generated by the maximal torus H and the root subgroups u ff(t) where ff runs through the* * simple roots in the basis {ff1, ff3, ff5, ff6, ff2, -eff} of the root system of centra* *lizer. Now note that ueff(t) = [I3 + te3,1, I3,aI3]nd u-eff(t) = [I3 + te1,3, I3,aI3]re root subgrou* *ps with associ- ated roots effand -effrespectively. Since these along with H and the root subgr* *oups u ff1, u ff2, u ff3, u ff5and u ff6generate the subgroup SL3(C)3=C3 of 3E6(C) from abo* *ve, we conclude that C3E6(C)(a) = SL3(C)3=C3. To describe the conjugacy classes of elementary abelian 3-subgroups we need t* *o introduce some more elements. Consider the following elements in SL3(C)3=C3 3E6(C): 2 x1 = [I3, fi,,fi] x2 = [fi, fi,,fi] y1 = I3, fl,,fl y2 = [fl, fl,.fl] We also need the following elements in N(H): s1 =n1n3n4n2n5n4n3n1n6n5n4n2n3n4n5n6, s2 =n1n2n3n1n4n2n3n1n4n3n5n4n2n3n1n4n3n5n4n2n6n5n4n2n3n1n4. n3n5n4n2n6n5n4n3n1 The action of these elements are as follows: s1(m1, m2, m3) = (m3, m1, m2), s2(m1, m2, m3) = (mT3, mT2, mT1), where mTidenotes the transpose of mi. Thus these elements acts by conjugation * *on the subgroup SL3(C)3=C3 as follows h T T Ti [g1, g2,sg3]1= [g2, g3,,g1] [g1, g2,sg3]2= g-11 , g-13 , g-12 . 44 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Lemma 8.5. We have z = h(!, 1, !2, 1, !, !2), a = h(!, 1, !2, 1, !2, !), x1 = h(!, 1, * *!, 1, !, !), x2 = h(1, !2, !2, 1, !2, 1), y1 = n1n3n5n6hff5(-1), y2 = n1n2n3n4n3n1n5n4n2n3n4n5n6n5n4n2n3n1n4n3n5n4n6n5hff2(-1). Moreover conjugation by the element n1n4n2n3n1n4n5n4n6n5n4n2n3n1n4 . hff2(-1)hff4(-1) acts as follows: a 7! x2, x2 7! a, y17! s1, y27! y22, x2x-117! hff4(!) = [ø2, ø2,.ø2] Proof.Both parts of the lemma may be checked by direct computation. The second * *part also follows from the first by using the following relations in N(H): The eleme* *nt ni has image sffiin W ([118, 8.1.4(i)]), we have n2i= hffi(-1) ([118, 8.1.4(ii)]) and ninjni. .=.njninj. . . for 1 i, j 6, where the number of factors on both sides equals the order of* * sffisffjin W ([118, 9.3.2]). Notation 8.6. For our calculations, we need some information on the conjugacy c* *lasses of elements of order 3 in 3E6(C). These are given in [31, Table 2]: There are 7 su* *ch conjugacy classes, which we label 3A , 3B , 3B0, 3C , 3D , 3E and 3E0, where 3B0 and 3E0d* *enotes the inverses of the classes 3B and 3E. This notation is almost identical to the not* *ation in [31], but differs from [63]. We will need the following, which follows quickly from [* *31, Table 2] using the action of W on H: We have z 2 3E, a, x2, y2 2 3C , x1, y1 2 3D and x2* *x-112 3A . Multiplication by z acts as follows on the conjugacy classes: 3A 7! 3B,3B 7! 3B0,3B07! 3A ,3C 7! 3C ,3D7! 3D ,3E 7! 3E0,3E07! 1, where 1 denotes the conjugacy class consisting of the identity element. Theorem 8.7. The conjugacy classes of non-toral elementary abelian 3-subgroups * *of 3E6(C) are given by the following table. ___________________________________________________________ |_rank_|name|ordered_basis|3E6(C)-class_distribution|C3E6(C)(E)_|_ 3 3C26 E4 |__3__|E3E6_|______2___2|_______________________|___3E6___|__ | 4 |E4 | 3C783E13E01 | E4 | |_____|_3E6_|________2__|2______________________|___3E6___|_ Their Weyl groups with respect to the given ordered bases are as follows: 2 3 __1__|*__*__*__ 6 0 | 7 W (E33E6) = SL3(F3), W (E43E6) = 64 0 | 7 |SL 3(F3) 5 0 || Proof.Non-toral subgroups: By [63, Thm. 11.13], there are two conjugacy classes* * of non- toral elementary abelian 3-subgroups in 3E6(C), one non-maximal of rank three a* *nd one maximal of rank 4. We may concretely realize these as follows. Consider the sub* *groups E33E6= andE43E6= which are readily seen to be elementary abelian 3-subgroups of rank 3 and 4 res* *pectively. In particular both groups are subsets of C3E6(C)(a) = SL3(C)3=C3, and since fi,* * fl 2 SL3(C) THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 45 does not commute, we see that the preimages of E33E6and E43E6under the projecti* *on SL3(C)3 ! SL3(C)3=C3 are non-abelian. Thus by Theorem 8.2(5) E33E6and E43E6are * *non- toral in SL3(C)3=C3 = C3E6(C)(a) and hence also non-toral in 3E6(C) by Theorem * *8.2(1). Thus by the above these two groups represent the conjugacy classes of non-toral* * elementary abelian 3-subgroups in 3E6(C). Lower bounds for Weyl groups: By [63, Thm. 7.4] there is a unique non-toral e* *lementary abelian 3-subgroup of F4(C) of rank 3. Since we have an inclusion F4(C) 3E6(C* *) this subgroup may also be considered as a subgroup of 3E6(C). As its Weyl group in * *F4(C) is SL3(F3), its Weyl group in 3E6(C) must contain SL3(F3). In particular it ha* *s order divisible by 13 and since the order of W (E6) is 27 . 34 . 5 which is not divis* *ible by 13, we conclude by Theorem 8.2(2) that E is non-toral in 3E6(C) as well. Thus by the * *above E must be conjugate to E33E6, and we get that W (E33E6) contains SL3(F3). From * *this we immediately see that W (E43E6) contains the group 2 3 __1__|0__0__0__ 66 0 | 77 4 0 ||SL3(F3)5 0 || Note that the element I3, fi, fi2commutes with z, a and x2 and conjugates y2 t* *o y2z. Thus it normalizes E43E6and produces the element I4+ e1,4in W (E43E6). As a result w* *e see that W (E43E6) contains the group 2 3 __1__|*__*__*__ 66 0 | 77 4 0 ||SL3(F3)5 0 || Class distributions: Since a 2 3C by 8.6 and W (E33E6) contains SL3(F3) whi* *ch acts transitively on E33E6\ {1}, the class distribution of E33E6follows immediately.* * Using this and the information given in 8.6 about the action of z on the conjugacy classes* *, the class distribution of E43E6is easily found. Centralizers: We have already seen that C3E6(C)(a) = SL3(C)3=C3. From this we* * directly get C3E6(C)(a, x2)= CSL3(C)3=C3(x2) = , C3E6(C)(a, x2, y2)= x x )=C3>= E43E6, proving that C3E6(C)(E33E6) = C3E6(C)(E43E6) = E43E6. Exact Weyl groups: From the lower bounds above and the fact that z is central* * we get SL3(F3) W (E33E6) GL 3(F3) and 2 3 2 3 __1__|*__*__*__ __1__|*__*__*__ 66 0 | 77 4 66 0 | 77 4 0 ||SL3(F3)5 W (E3E6) 4 0 ||GL3(F3) 5 0 || 0 || As C3E6(C)(a, x2) = , we see that no element in C3E6(C)(* *a, x2) conjugates y2 to y-12. Hence diag(1, 1, 2) =2W (E33E6) and diag(1, 1, 1, 2) =2W* * (E43E6) which shows that Weyl groups are the ones given in the Theorem. 46 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL We now turn to the group E6(C). As above we let Z be the center of 3E6(C) and* * we let ß : 3E6(C) ! E6(C) = 3E6(C)=Z denote_the projection. For g 2 3E6(C) we writ* *e _g instead of ß(g) and similarly we let S = ß(S) for a subset S 3E6(C). Lemma 8.8. Let E be a rank 2 non-toral elementary abelian 3-subgroup of E6(C). * *Then the Weyl group W (E) is a subgroup of SL2(F3). Proof.Let E = <__g1,.__g2>By Theorem 8.2 part (5) and (3) the group 3* *E6(C) is non-abelian. Thus setting~z0 = [g1,~g2]2 Z we have z0 6= 1. Assume that oe 2 * *W (E) is represented by the matrix a11aa12 , i.e. we have oe(__g1) = (__g1)a11(__g2)* *a21and oe(__g2) = __ __ 21a22 (g1)a12(g2)a22. Since oe is given by a conjugation in E6(C), it lifts to a conj* *ugation in 3E6(C). Now the relation [g1, g2]= z0 2 Z shows (z0)a11.a22-a12.a21= z0, so since z0 6=* * 1 we have oe 2 SL2(F3). Theorem 8.9. The conjugacy classes of non-toral elementary abelian 3-subgroups * *of E6(C) are given by the following table: _______________________________________________________________________________* *______ |_rank|name|_ordered_basis|__3E6(C)-class_distribution|__CE6(C)(E)_____|Z(CE6(C* *)(E))__ | 2a __ __ 18 6 1 01 2a 2a |__2__|EE6_|_____|______3C__3D_3E_3E________|__EE6x_PSL3(C)___|____EE6_* *_____|_2b__241012b2b |__2__|EE6_|_____|_______3D__3E_3E__________|___EE6x_G2(C)____|____EE6_* *_____|__ 3a _ __ __ 60 18 1 01 3a _ __ 3a |__3__|EE6_|__|_____3C__3D__3E_3E________|_EE6O(T2:_)__|_EE6_* *_____|_3b___781013b33b |__3__|EE6_|__|_______3C__3E_3E__________|_____EE6._3_______|___EE6_* *_____|_3c___6721013c3c |__3__|EE6_|__|______3C_3D__3E_3E________|__EE6O<_a>SL3(C)___|__EE6_* *_____|_D_E | | 3d | -1 __ __| 2 2 02 48 24 1 01| 3d ø____Æ | 3d ø__* *__Æ | | 3 |EE6 | x2x1 , y1, x1|3A 3B 3B 3C 3D 3E 3E |EE6O x2x-1GL 2(C) |EE6O x2* *x-1T1 | |_____|____|_____________|__________________________|________1_________|_______* *_1____|_ 4a _ __ __ __ 186 54 1 01 4a 4a |__4__|EE6_|_|__3C___3D__3E_3E_______|______EE6_______|_____EE6_* *_____|_D_E | | 4b | _ -1 __ |__ 6 6 06 150 72 1 01|4b ø ____Æ |4b ø * *____Æ | | 4 |EE6 | a, x2x1, y1,|x13A 3B 3B 3C 3D 3E 3E| EE6O _a,x2x-1T2 E|E6O _a,* *x2x-1T2 | |_____|____|_____________|__________________________|____________1_____|_______* *___1__ | In particular we have 3Z(CE6(C)(E)) = E for any non-toral elementary abelian 3-* *subgroup of E6(C). (In the table the 3E6(C)-class distribution of E E6(C) denotes the* * class distribution of ß-1(E) 3E6(C).) The Weyl groups of these groups with respect to the given ordered bases are g* *iven as follows: ~ ~ 2 "1 * * 3 W (E2aE6) = "0*" , W (E2bE6) = SL2(F3), W (E3aE6) = 4 0 "2 * 5 0 0 "2 2 3 2 * * 3 __"__|_*__*___ __"__|_0__0_* *__ W (E3bE6) = SL3(F3), W (E3cE6) = 4 0 | 5 , W (E3dE6) = 4 0 | * * 5 0 ||SL2(F3) 0 ||SL2(F* *3) 2 3 2 3 |* * "1 * | * * 6 GL 2(F3) |* * 7 6 0 "2 0 0 7 W (E4aE6) = 64_________|________0075, W (E4bE6) = 64________|________75 |det * 0 0 |SL2(F3) 0 0 | 0 det 0 0 || where detdenotes the determinant of the matrix from GL2(F3) in the description * *of W (E4aE6). Proof.Maximal non-toral subgroups: By [63, Them. 11.14], there are two conjugac* *y classes of maximal non-toral elementary abelian 3-subgroups in E6(C), both of which hav* *e rank 4. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 47 We may concretely realize these as follows. Consider the subgroups -1 ff Ea = ndEb = z, a, x2x1 , y1, x1 of C3E6(C)(a) = SL3(C)3=C3. Since the commutator subgroup of both of these is Z* *, we see that E4aE6= ß(Ea) and E4bE6= ß(Eb) are elementary abelian subgroups of rank 4 i* *n E6(C). It follows from Theorem 8.2(5) that both E4aE6and E4bE6are non-toral in E6(C). * *We will see below that their class distributions are as given in the table. From this it fo* *llows that they are not conjugate and thus represents the two conjugacy classes of maximal elem* *entary abelian 3-subgroups in E6(C). Lower bounds for Weyl groups of maximal non-toral subgroups: We can find lowe* *r bounds for the Weyl groups of the maximal non-toral elementary abelian 3-subgroups by * *conjugating with elements coming from the centralizer C3E6(C)(a) = SL3(C)3=C3 and the norma* *lizer N(H) of the maximal_torus._________ The elements [fi2, I3,,I3][I3, ø1,,ø21]_s1and __s2normalize E4aE6and conjugat* *ion by these elements induce the automorphisms given by the matrices I4 + e1,2, I4 + e3,4, I* *4 + e2,3and diag(2, 1, 2, 2) on E4aE6.DMoreover, by Lemma 8.5 we may conjugate the ordered * *basis of E4aE6 __ __ __E ________ * * __ into the ordered basis __x2, y22, s1,.aNoting that the element [ø1, ø1,cø1]omm* *utes with y2, __s __ __ ____ 4a a1nd a and conjugates x2 into x2y2, we see that W (EE6) contains the element * *I4 + e2,1. The above matrices are easily seen to generate the group 2 3 |* * 6 GL 2(F3) |* * 7 W 0(E4aE6) = 64_________|________0075 |det * 0 0 | 0 det and thus WE4a6contains this group. __Now_consider_E4bE6and_let oe = -(2, 3) 2 SL 3(C). We then see that the eleme* *nts ________ ________ _________ __ [I3, ø1,,ø21][I3, ø2fi,,ø22][oe,,I3,[I3]fl,,I3,[I3]I3,afi2,nI3]d s2normalize E4* *bE6, and conjuga- tion by these elements induce the automorphisms given by the matrices I4+ e3,4,* * I4+ e4,3, diag(1, 2, 1, 1), I4 + e1,2, I4 + e1,3and -I4 on E4bE6. These matrices generate* * the group 2 3 "1 * | * * 6 0 "2 0 0 7 W 0(E4bE6) = 64________|________007 |SL (F ) 5 0 0 || 2 3 and thus WE4b6contains this group. Orbit computation: Any elementary abelian 3-subgroup of rank 1 is toral since* * E6(C) is connected. As we already know that E4aE6and E4bE6are representatives of the* * maximal non-toral elementary abelian 3-subgroups, we may find the conjugacy classes of * *non-toral elementary abelian 3-subgroups of rank 2 and 3 by studying subgroups of these. Under the action of W 0(E4aE6), the set of rank 2 subgroups of E4aE6has orbit* * representatives E2aE6= <__y1,,__x2><_a,,__x2><_a,a__y1>nd<_a,,__y2> and under the action of W 0(E4bE6), the set of rank 2 subgroups of E4bE6has orb* *it representatives D ______ED___* *_ E E2aE6= <__y1,,__x2>E2bE6= <__y1,,__x1><_a,,__x2><_a,,__y1>_a, x2x-11andx2* *x-11,.__x1 48 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Similarly we find that under the action of W 0(E4aE6), the set of rank 3 subgro* *ups of E4aE6has orbit representatives E3aE6= <__a, __y1,,__x2>E3bE6= <__a, __y2,a__x2>nd<_a,,__y1, _* *_y2> and that under the action of W 0(E4bE6), the set of rank 3 subgroups of E4bE6ha* *s orbit repre- sentatives D______ DE * * ______ E E3aE6= <__a, __y1,,__x2>E3cE6= <__a, __y1,,__x1>E3dE6= x2x-11, __y1,a__x1* *nd_a, x2x-11,.__x1 D * * ______E D______ E Other non-toral subgroups: We see directly that the subgroups <_a, __x2>, __* *a, x2x-11, x2x-11, __x1 D ______ E and __a, x2x-11,a__x1re toral. Noting that the elements fi and fl are conjugat* *e in SL3(C) we D ________ E see that the group <__a, __y1,i__y2>s conjugate to the group __a, [I3, fi,,fi2* *]_x2which is obviously toral. Thus we see that the groups <__a, __y1,,__y2><_a,a__y1>nd <__a, __y2>are* * also toral. Using the fact that [y1, x1]= [y1, x2]= z we see from Theorem 8.2(5) that both E2aE6and E* *2bE6are non-toral in E6(C). Since the groups E3aE6, E3cE6and E3dE6all contain either E2* *aE6or E2bE6they are also non-toral. Using Theorem 8.2(5) we see that the group E3bE6is non-tora* *l in E6(C), since we know that ß-1(E3bE6) = E43E6is non-toral in 3E6(C) by Theorem 8.7. Class distributions: Using 8.6 and the action of the groups W 0(E4aE6) and W * *0(E4bE6) it is not hard to verify the class distributions in the table. As an example consider* * the group E4bE6. From the action of W 0(E4bE6) we see that E4bE6\ {1} contains 2 elements* * conjugate to __a, 6 elements conjugate to ______x -1 __ __ , 2x124 elements conjugate to x1 and 48 eleme* *nts conjugate to x2. Thus by 8.6, the set ß-1(E4bE6\ {1}) contains 6 elements from each of th* *e classes 3A , 3B and 3B0, 3 . (2 + 48) = 150 elements from the class 3C and 3 . 24 = 72 eleme* *nts from the class 3D . Including the elements z and z2 from the classes 3E and 3E0respe* *ctively, we get the class distribution of ß-1(E4bE6) \ {1} given in the table. Similar comp* *utations give the remaining entries in the table. Since these distributions are different we * *see that the groups in the table are not conjugate and thus they provide a set of representa* *tives for the conjugacy classes of non-toral elementary abelian 3-subgroups of E6(C). Lower bounds for other Weyl groups: We now show that the other matrix groups * *in the table are all lower bounds for the remaining Weyl groups. To do this consider o* *ne of the non-maximal groups E from the table. We then have E E4aE6or E E4bE6, and we* * get a lower bound on W (E) by considering the action on E of the subgroup of W 0(E4* *aE6) or W 0(E4bE6) fixing E. As an example we see that E2aE6 E4aE6and that the stabili* *zer of E2aE6 inside W 0(E4aE6) is 2 3 |0 0 66GL_2(F3)_||0__0___77 4 0 0 |det x 5 0 0 | 0 det where detis the determinant of the matrix from GL 2(F3). The action of such a m* *atrix on E2aE6is given by __y __ det __ __ x __ det 17! (y1) , x27! (y1) (x2) . THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 49 Thus W (E2aE6) contains the group ~ ~ W 0(E2aE6) = "0*" as claimed. Similar computations show that for the groups E = E2bE6, E3aE6, E3c* *E6and E3dE6, the group W 0(E) occurring in the Theorem is a lower bound for the Weyl group W* * (E). For the group E3bE6= <__a, __x2,w__y2>e know the structure of W (ß-1(E3bE6)) * *= W (E43E6) by Theorem 8.7. From this we immediately get W (E3bE6) = SL3(F3). Exact Weyl groups: We now prove that the lower bounds on the Weyl groups esta* *blished above are in fact equalities. By Lemma 8.8 the Weyl groups W (E2aE6) and W (E2* *bE6) are subgroups of SL2(F3). From this we see that W (E2bE6) = SL2(F3) and that W (E2a* *E6) is equal to either W 0(E2aE6) or SL2(F3), since these are the only subgroups of SL2(F3) * *containing W 0(E2aE6). We have E2aE6= <__y1,,__x2>and by 8.6 we see that the elements __y* *1and __x2are not conjugate in E6(C). In particular we see that W (E2aE6) cannot act transiti* *vely on the non-trivial elements of E2aE6, and we conclude that W (E2aE6) = W 0(E2aE6) is t* *he group from above. For each of the remaining non-toral subgroups we now show that a strictly lar* *ger Weyl group contradicts the Weyl group results already established. The groups E = E3* *aE6, E3dE6, and E4bE6all contain E2aE6. A direct computation shows that any proper overgrou* *p of W 0(E) in GL (E) contains an element which normalizes the subgroup E2aE6and induces an* * auto- morphism which does not lie in W (E2aE6). Hence W (E) = W 0(E). If E = E3cE6a* * similar argument, using the subgroup E2bE6, again shows that W (E) = W 0(E). Consider * *finally E = E4aE6. Each proper overgroup of W 0(E) contains an element which normalizes* * one of the subgroups E2aE6or E3bE6and induces an automorphism on it not contained in i* *ts Weyl group. Hence W (E) = W 0(E). This concludes the proof that the Weyl groups list* *ed in the theorem are the correct ones. Centralizers: Let : SL3(C) -! SL3(C)3=C3 3E6(C) denote the homomorphism given by (g) = [g, g, g]for g 2 SL 3(C). By Lemma 8.5 the group E2aE6= <__x2,* *___y1>is_ conjugate to the group <__a,.__s1>Since as1= az2 we obtain CE6(C)(__a) = , and hence __________________ ___________________ __ __ CE6(C)(__a, __s1) = = x (SL 3(C))= x* * PSL3(C), proving the claims for E2aE6. By abusing the notation slightly, we let _gdenot* *e the image of g 2 SL3(C)_in_the_quotient PSL 3(C). From Lemma 8.5 we then see that the ele* *ments __a, __y -1 2a __ __2 __ 2and x2x1 in CE6(C)(EE6) correspond to the elements fi, fl and ø2in th* *e PSL 3(C) component of CE6(C)(E2aE6). Thus we immediately get __ 3d 2a __ CE6(C)(E3aE6)= E2aE6x CPSL3(C)(fi), CE6(C)(EE6) = EE6 x CPSL3(C)(ø2), __ __ 4b 2a __ __ CE6(C)(E4aE6)= E2aE6x CPSL3(C)(fi, fl2),CE6(C)(EE6)= EE6 x CPSL3(C)(fi, ø2). __ _______ __ __ __ __ff * * __ __ Note_that CPSL3(C)(fi) = T2 : , giving CPSL3(C)(fi, fl2) = fi, fland CPSL3* *(C)(fi, ø2) = T2. From this the results on E3aE6, E4aE6and E4bE6follow directly. Note also th* *at CPSL3(C)(__ø2) ~= GL2(C) from which we deduce the claims about E3dE6. 50 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL _______________ Now consider the group E3bE6. Since CE6(C)(__a) = we get ___________________________ CE6(C)(__a, __x2)= , _____________________________________________ CE6(C)(__a, __x2,=__y2)x x )=C3> D ________E and thus CE6(C)(E3bE6) = E3bE6, __s1, __y1,.[I3,Ifi,tfi2]is now easy to check * *that CE6(C)(E3bE6) has the structure E3bE6. 33 and that Z(CE6(C)(E3bE6)) = E3bE6. For the group E3* *cE6we obtain _________________________ CE6(C)(__a, __x1)= , ________________________________ CE6(C)(__a, __x1,=__y1)x )=C3>. Thus the centralizer CE6(C)(E3cE6) equals the central product E3cE6O<_a>SL3(C) * *and we obtain the claims about E3cE6. Finally we consider the group E2bE6= <__y1,.__x1>If g 2 ß-1(CE6(C)(E2bE6)) th* *en [g, y1], [g, x2]2 Z, and since [y1, x2]= z_it_follows_that_g 2 ß-1(E2bE6) OZ C3E6(C)(ß-1(E2bE6)).* * Thus we have CE6(C)(E2bE6) = E2bE6x C3E6(C)(ß-1(E2bE6)). A direct computation shows that C3E6(C)(x1) has type T2D4 and a basis for the* * root system of the centralizer is given by {ff1 + ff3 + ff4, ff2, ff4 + ff5 + ff6, f* *f3 + ff4 + ff5}. From this we see that the 2-dimensional torus consists of the elements h(ff, 1, fl, * *1, ff, fl) where ff, fl 2 Cx . Moreover we see that C3E6(C)(x1) = T2 OC Spin(8, C), where the c* *entral product is over the group C = Z(Spin(8, C)) = C2 x C2 which consist of the elem* *ents h(ff, 1, fl, 1, ff, fl), ff, fl = 1. Let oe denote the automorphism of C3E6(C)(x1) given by conjugation with y1. A* * direct check shows that the map from C to C given by x 7! x-1xffis surjective. It then* * follows that C3E6(C)(ß-1(E2bE6)) = (T2 OC Spin(8, C))ff= T2ffOCoeSpin(8, C)ff. We have T2ff= , so C3E6(C)(ß-1(E2bE6)) = x Spin(8, C)ff. Using the class * *distribution of ß-1(E2bE6) found above together with [31, Table 2] and Theorem 8.2(6) we find 1 2 dim C3E6(C)(ß-1(E2bE6)) = __. 3 . 78 + 24 . (30 + 24! + 24!=)14. 33 Thus Spin(8, C)ffhas dimension 14 and since Z(Spin(8, C))ff= 1 we also see that* * Spin(8, C)ff has rank less than 4. From this it follows that the identity component of Spin* *(8, C)ff must have type G2. By [120, Thm. 8.1] we know that Spin(8, C)ffis connected, so* * we get Spin(8, C)ff= G2(C) and hence C3E6(C)(ß-1(E2bE6)) = x G2(C). Combining this * *with the computation from above we conclude CE6(C)(E2bE6) = E2bE6x G2(C). For the proof of our main results we need the following auxiliary results abo* *ut the two non-toral elementary abelian 3-groups of rank 2. Proposition 8.10. Let E be an elementary abelian_3-group_of rank 2 with basis (* *e1, e2) and consider the 4 homomorphisms ~i: E ! NE6(C)(H ) (the maximal torus normaliz* *er in THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 51 E6(C)), 1 i 4 defined as follows: ~1 : e17! __y1,e27! __x2, ~2 : e17! __y1,e27! ____x2y1, _____ ~3 : e17! __y1,e27! x2y-11, _____ ~4 : e17! __x1,e27! x2y-11. __ Then ~i(E) is conjugate to E2aE6for all i and ~-1i(H ) equals the 4 distinct * *rank 1 subgroups __~i(E) of E for i = 1, . .,.4. Moreover TE = (H )1 is independent of i and CNE (* *__H)(~i(E)) 6(C) is a maximal torus normalizer in CE6(C)(~i(E)) for all i. The canonical homomo* *rphism E x TE ~ix1-!N(H) ! E6(C) does not depend on i up to conjugacy in E6(C). __ -1 __ -1 __ -* *1 __ Proof.Obviously_~-11(H ) = , ~2 (H ) = , ~3 (H ) = and ~4* * (H ) = , so ~-1i(H ) equals the 4 distinct rank 1 subgroups of E for i = 1, . .,.4. __ __~i(E) Since __x1,___x22_H_, it is clear_that the identity component of H equals* * the identity component of H y1. Hence TE = (H ~i(E))1 is independent of i and as y1_=__I3,_* *fl,,fl2a direct computation shows that TE consist of the elements of the form [g, I3,,I3* *]g 2 T2, where T2 SL3(C) denotes the maximal torus consisting of diagonal matrices. We now prove that the homomorphisms E -~i!CE6(C)(TE ) are conjugate_in CE6(C)* *(TE ). As __a2 TE we find that CE6(C)(TE ) consists of the elements [g1, g2,,g3]where_* *g1_2_T2_and g2, g3 2 SL3(C) are arbitrary. Note first that the conjugation by the element * *I3, ø1, ø212 CE6(C)(TE ) sends __y1to itself and __x2to ____x2y1. Hence the homomorphism E -* *~i!CE6(C)(TE ) ~i+1 is conjugate to the homomorphism E -! CE6(C)(TE ) for i = 1, 2. Letting 2 2 3 eii=18 1 ! 12 ø3 = -_____p_41 1 ! 5 2 SL3(C), 3 1 ! ! _______* *_____ we get fifi3= fifl and flfi3= fi2. Thus_the_conjugation by the element I3, ø2* *ø-11,2ø3 CE6(C)(TE ) sends __y1to __x1and __x2to x2y-11and hence the homomorphism E -~1!* *CE6(C)(TE ) is conjugate to the homomorphism E -~4!CE6(C)(TE ). This proves the claim. We c* *onclude that E x TE ~ix1-!N(H) ! E6(C) is independent of i up to conjugacy in E6(C) and* * also that ~i(E) is conjugate to ~1(E) = E2aE6for all i. Since CE6(C)(E2aE6) has rank 2 it follows that TE is a maximal_torus in CE6(C* *)(~i(E)) for all i. Since ~i(E) is elementary abelian of rank 2 and ~i(E) \ H 6= 1 it now fo* *llows from Theorem 8.3 that CNE (__H)(~i(E)) is a maximal torus normalizer in CE6(C)(~i(* *E)) for all 6(C) i. Proposition 8.11. An element ff 2 W (E2bE6) acts up to conjugacy by ffx1 on CE6* *(C)(E2bE6) = E2bE6x G2(C). Proof.Since G2(C) is connected and has trivial center it is clear that Out(E2bE* *6x G2(C)) = Out(E2bE6)xOut (G2(C)). The result now follows as Out(G2(C)) = 1 (i.e. by [70, * *Thm. 27.4]). 52 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL 8.3. The group E8(C), p = 3. In this section we consider the elementary abelian* * 3- subgroups of the group E8(C). By using [14, Table 2, p. 214] we see that the sm* *allest faithful representation of E8(C) is the adjoint representation, i.e. the representation * *given by the action of E8(C) on its Lie algebra e8, which has dimension 248. For our computa* *tions, we explicitly construct this representation on a computer by following the recipe * *in [22, Ch. 4]. (As explained in [22, Ch. 4] there is some ambiguity in choosing a Chevalley ba* *sis of e8 and we fix a certain such choice; a different choice affects our formulas at only o* *ne point_see Remark 8.12.) Letting (E8) denote the root system of type E8 (we use the notation of [13, * *p. 268- 270]), we have in particular a maximal torus H generated by the elements hffi(t* *), 1 i 8, t 2 Cx ([22, p. 92, p. 97]) and root subgroups uff(t), ff 2 (E8), t 2 C. The n* *ormalizer N(H) of the maximal torus, is generated by H and the elements ni= nffi, 1 i * * 8 ([22, p. 93, p. 101]). We let Y8 h(t1, t2, t3, t4, t5, t6, t7, t8) = hffi(ti). i=1 Note that by [22, p. 100 and Lem. 6.4.4] the root subgroups uffform a realiza* *tion ([118, p. 133]) of (E8) in E8(C). In particular we have the following relations. The * *element ni has image sffiin W = W (E8) ([118, 8.1.4(i)]), we have n2i= hffi(-1) ([118, 8.1* *.4(ii)]) and ninjni. .=.njninj. . . for 1 i, j 8, where the number of factors on both sides equals the order of* * sffisffjin W ([118, 9.3.2]). Now let __a= hff1(!)hff2(!)hff3(!2) 2 E8(C). Direct computation shows that fo* *r any root ff 2 (E8) we have ff(__a) = !2. From this we see that Dynkin diagram of* * the centralizer CE8(C)(__a) is the same as the extended Dynkin diagram of E8 with the node ff2 * *removed. Thus it has type A8 and basis {ff1, ff3, ff4, ff5, ff6, ff7, ff8, -eff}, where effis the longest root. As in [13, p. 250-251] we identify (A8) with the* * set of elements in R9 of the form ei-ej with i 6= j and 1 i, j 9, where eidenotes the i'th * *canonical basis vector in R9. We now consider SL9(C), which is the simply connected group of ty* *pe A8 over C. For any 1 i, j 9 we let ei,jbe the 9 x 9-matrix over C whose only non-ze* *ro element is 1 occurring at the (i, j)-entry. Given a root ff0= ei- ej 2 (A8) we let u0f* *f0(t) = I9+ tei,j for t 2 C. With respect to the maximal torus consisting of the diagonal matrice* *s, this is a root subgroup of SL9(C) corresponding to the root ff0. The roots ff0i= ei- ei+1* *, 1 i 8, form a basis of (A8). From the above we then see that u0 ff01(t)7! u ff1(t),u0 ff02(t)7! u ff3(t),u0 ff03(t)7! u ff4(t),u07ff04(t)! u* * ff5(t), u0 ff05(t)7! u ff6(t),u0 ff06(t)7! u ff7(t),u0 ff07(t)7! u ff8(t),u07ff08(t)! u* * eff(t) defines a homomorphism SL9(C) ! E8(C) onto the centralizer CE8(C)(__a). It is e* *asy to check that this map has kernel C3 = and thus we may make the identification CE8(* *C)(__a) = SL9(C)=C3. For any g 2 SL9(C) we denote by _gits image in SL9(C)=C3 = CE8(C)(__* *a) E8(C). In particular we see that a = jI9 corresponds to the element __afrom abo* *ve. We also THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 53 define the following elements in SL9(C) : x1 = diag(1, !, !2, 1, !, !2, 1, !,x!2),2=diag(1, 1, 1, !, !, !, !2, !2, * *!2), x3 = diag(1, 1, 1, 1, 1, 1, !, !, !),y1(=1, 2, 3)(4, 5, 6)(7, 8, 9), y2 =(1, 4, 7)(2, 5, 8)(3, 6, 9). From the explicit homomorphism above we easily find __a=h 2 __ ff1(!)hff2(!)hff3(! ), x1= hff1(!)hff5(!)hff8(!), __x 2 2 __ 2 2 =2hff1(!)hff3(! )hff5(! )hff6(!),x3=hff1(! )hff3(!)hff5(! )hff6(!), and a direct computation in E8(C) shows that n-eff=n8n7n6n5n4n2n3n1n4n3n5n4n2n6n5n4n3n1n7n6n5n4n2n3n4n5n6n7n8. n7n6n5n4n2n3n1n4n3n5n4n2n6n5n4n3n1n7n6n5n4n2n3n4n5n6n7n8. Remark 8.12. A different choice of Chevalley basis for e8 may effect the expres* *sion for n-effby a order two element in H. If a Chevalley basis is chosen such that the * *above formula holds then all further formulas will be independent of the choice. From this and the explicit homomorphism above we find, either by direct compu* *tation or by using the relations in N(H), that y1 =n1n3n5n6n7n6n5n4n2n3n1n4n3n5n4n2n6n5n4n3n1n7n6n5. n4n2n3n4n5n6n7n8n7n6n5n4n2n3n1n4n3n5n4n2n6n5n4n3. n1n7n6n5n4n2n3n4n5n6n7n8 . hff1(-1)hff2(-1)hff7(-1), y2 =n2n3n1n4n2n3n4n5n4n2n3n4n6n5n4n2n3n1n4n7n6n5n4. n2n3n1n4n3n5n6n7n8n7n6n5n4n2n3n1n4n3n5n4n2n6n5. n4n3n1n7n6n5n4n2n3n4n5n8n7n6 . hff2(-1)hff5(-1). Notation 8.13. To distinguish subgroups of E8(C), we need some information on t* *he conjugacy classes of elements of order 3. These are given in [63, Table VI] (wh* *ich is taken from [30, Table 4]): There are 4 such conjugacy classes, which we label 3A , 3B* * , 3C and 3D . Moreover these classes may bePdistinguished by their traces on e8. Since t* *he trace of the element h 2 H is given by 8 + ff2 (E8)ff(h) we get __a2 3A , __x1, __x2, * *__x3, __y1, __y22 3B and _____ x3a-12 3D . Notation 8.14. If K is a field and n is a natural number, we define the group o* *f symplectic similitudes as CSp2n(K) = {X 2 GL 2n(K)|XtBX = cB, c 2 Kx }, where ~ ~ ~ ~ B = 01-10 . . . 01-10 ____________-z___________" n times We define the homomorphism Ø : CSp2n(K) ! Kx by Ø(X) = c, where XtBX = cB. The kernel of Ø is the symplectic group Sp2n(K). (The notation CSp is taken from [8* *2].) 54 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Theorem 8.15. The conjugacy classes of non-toral elementary abelian 3-subgroups* * of E8(C) are given by the following table. _______________________________________________________________________________* *______ |_rank_|name|____ordered_basis__|E8(C)-class_dist.|__CE8(C)(E)_____|__Z(CE8(C)(* *E))___|_ 3a <__x, __y, __a> 3A183B8 E3a x PSL (C) E3a |__3__|EE8__|_________1____1____|_______________|___E8______3______|_______E8__* *_____|_||3b|___|26|3b|3b| |__3__|EE8__|_______|_____3B________|___EE8_x_G2(C)____|______EE8__* *_____|__D_E | 4 |E4aE | __x1, __y1, __x3,|x3a-13A523B263D2|E4aEO _____-1GL2(C)E|4aEO ___* *__-1T1 | |_____|____8|___________________|_______________|_____8_________|___8____|||4b|____|5426|4b_|4b| |__4__|EE8__|____|___3A__3B______|EE8_O<__x2>(T2_:_)_|_EE8__* *_____|_4c____804c4c |__4__|EE8__|___|_____3B________|__EE8_O<__x2>SL3(C)___|__EE8__* *_____|__D_E | 5 |E5aE | __x2, __x1, __y1, __x3,|x3a-13A1563B803D6|E5aEO __ _____-1T2E5|aE* *O __ _____-1T2 | |_____|____8|___________________|_______________|___________8______|__* *8||5b||_____|16280|5b|5b| |__5__|EE8__|__|3A___3B______|_______EE8________|______EE8__* *_____|_ In particular we have 3Z(CE8(C)(E)) = E for any non-toral elementary abelian 3-* *subgroup of E8(C). The Weyl groups of these groups with respect to the given ordered bases are g* *iven as follows: 2 3 2 |0* * 3 |* 6 SL (F ) |0* * 7 W (E3aE8) = 4_GL_2(F3)_||*_5, W (E3bE8) = SL3(F3), W (E4aE8) = 64 3 3 ||* *0 75 0 0 |det ___________|_* *__000 | " 2 3 2 3 __"__|_*__*___|*___ __"__|*__*__*__ 6 0 | | * 7 6 0 | 7 W (E4bE8) = 64 0 |GL2(F3) | 75, W (E4cE8) = 64 | 75 _____|________|_*__ 0 |SL 3(F3) 0 | 0 0 |det 0 || 2 3 2 3 _"1_|_*__*__*_|_*__ ||* 66 0 | |0 77 66 |* 77 W (E5aE8) = 66 0 ||SL3(F3) ||077, W (E5bE8) = 66 CSp 4(F3) ||*77 4__0__||_________||0_5 4______________||*_5 0 | 0 0 0 |"2 0 0 0 0 | Ø where det is the determinant of the matrix from GL 2(F3) in the description of * *W (E3aE8) and W (E4bE8). In the description of W (E5bE8), Ø denotes the value of the hom* *omorphism Ø : CSp4(F3) ! Fx3defined in 8.14 evaluated on the matrix from CSp4(F3). Remark 8.16. Note that our information on the rank five subgroup E5aE8corrects * *[63]. Proof of Theorem 8.15.Maximal non-toral subgroups: By [63, Lems. 11.7 and 11.9]* *, any maximal non-toral elementary abelian 3-subgroup of E8(C) contains an element of* * type 3A . We may thus find representatives in CE8(C)(__a) = SL9(C)=C3. From [63, Cor* *. 11.10], it follows that there are two conjugacy classes of these maximal non-toral element* *ary abelian 3-subgroups: E5aE8and E5bE8, both of which have rank 5. Moreover, by [63, Lem. * *11.5], their preimages in SL9(C) may be chosen to have the shape 31+2OC3 9 x 3 x 3 and 31+4O* *C3 9. Using the representationDtheory of extraspecial 3-groups ([62, Ch. 5.5]) we fin* *d that E5aE8is _____E __* * __ __ __ __ represented by __x2, __x1, __y1, __x3,ax3a-1nd that E5bE8is represented by THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 55 Lower bounds for Weyl groups of maximal non-toral subgroups: We can find lowe* *r bounds for the Weyl groups of E5aE8and E5bE8by conjugating with elements in the centra* *lizer CE8(C)(__a) = SL9(C)=C3 and the normalizer N(H) of the maximal torus. Note that a = jI3 jI3 jI3, x1 = fi fi , fi x2 = I3 !I3 !2I3, x3= I3 I3 !I3, y1 = fl fl fl and (A B C)y2= B C A. Conjugation by ø1 ø1 ,ø1ø2 ø2 aø2nd I3 fi* *2 fi gives (8.1) ø1 ø1 ø1:7a! a,x17! x1y1, x27! x2, x37! x3, y1 7! y1, y2 7! y2. (8.2) ø2 ø2 ø2:7a! a,x17! x1, x27! x2, x37! x3, y1 7! x1y1,y2 7! y2. (8.3) I3 fi2 :fia7! a,x17! x1, x27! x2, x37! x3, y1 7! x2y1,y2 7! x1y2. _______* *___ Now consider the group E5aE8. From (8.1)_(8.3)we see that the elements ø1 ø1* * ø1, __________ __________ 5a ø2 ø2 ø2and I3 fi2 nfiormalize EE8 and that conjugation by these element* *s induces the automorphisms on E5aE8given by the matrices I5 + e3,2, I5 + e2,3and I5 + e1* *,3. Letting oe = -(1, 4)(2, 5)(3, 6) 2 SL9(C) we see_that_(A_ B C)ff= B A * *C. Using this and the above we obtain that __oe, __y2and I3 I3 fi2normalize E5aE8and* * that conjugation by these elements induces the automorphisms on E5aE8given by the matrices diag(* *2, 1, 1, 1, 1), I5 + e1,4+ e1,5and I5 + e4,3. By using the relations in N(H) given above or by* * direct computation, it may be checked that conjugation by the element n = n1n2n4n2n3n5n4n2n3n1n4n3n5n4n6n5n4n2n3n4n7n6n5n4n8n7n6. n5n4n2n3n1n4n3n5n4n2n6n5n4n7 . h(1, 1, -1, -1, -1, 1, -1, -1) induces the automorphism on E5aE8represented by the matrix diag(1, 1, 1, 1, 2) * *+ e2,4. It is easy to see that the above matrices generate the group 2 3 _"1_|_*__*__*_|_*__ 66 0 | |0 77 W 0(E5aE8) = 66 0 ||SL3(F3) ||077 4__0__||_________||0_5 0 | 0 0 0 |"2 and thus W (E5aE8) contains this group. _____* *_____ Next consider the group E5bE8. From (8.1)_(8.3)we see that the elements ø1 * *ø1 ø1, __________ __________ 5b ø2 ø2 ø2and I3 fi2 nfiormalize EE8 and that conjugation by these element* *s induces the automorphisms on E5bE8given by the matrices I5 + e2,1, I5 + e1,2and I5 + e1* *,4+ e3,2. Now note that a = 3,3(jI3), x2 = 3,3(fi)and y2 = 3,3(fl). Noting also that__* *3,3(M1)_ commutes_with_M2 M2 M2 for any M1, M2 2 M3(C) we see that the elements 3,3(ø* *1) and 3,3(ø2)normalize E5bE8. The automorphisms induced on E5bE8by conjugation * *with these elements have the matrices I5+ e4,3and I5+ e3,4. The upper left 4 x 4-cor* *ner of these matrices are easily seen to generate the group Sp4(F3) from 8.14. By using the * *relations in N(H) given above or by direct computation, we get that conjugation by the eleme* *nt n = n2n8n7n6n5n4n2n3n1n4n3n5n4n2n6n5n4n3n1n7n6. n5n4n2n3n4n5n6n7n8 . h(1, -1, -1, -1, -1, 1, 1, 1) 56 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL of N(H) induces the automorphism on E5bE8represented by the matrix diag(1, 2, 1* *, 2, 2)+e3,5. It now follows that W (E5bE8) contains the group 2 | 3 * 66 ||* 77 W 0(E5bE8) = 66 CSp 4(F3) ||* 77 4_____________||*__5 0 0 0 0 | Ø Lower bounds for other Weyl groups: We now show that the other Weyl groups in* * the table are all lower bounds. To do this consider one of the non-maximal groups E* * from the table. We then have E E5aE8, and we get a lower bound on W (E) by considerin* *g the action on E of the subgroup of W 0(E5aE8) fixing E. As an example we find that * *the stabilizer of E3aE8inside W 0(E5aE8) is2 3 "1 0 0 x x 66 0 a11 a12 a13 0 7 66 0 a21 a22 a23 0 777 4 0 0 0 det 0 5 0 0 0 0 det where det= a11. a22- a12. a216= 0. The action of such a matrix on E3aE8is given* * by __x __ a11 __ a21 __ __ a12 __ a22 __ __ a13 __ a23 __det 17! (x1) (y1) , y17! (x1) (y1) , a7! (x1) (y1) (a) . Thus W (E3aE8) contains the group 2 3 |* W 0(E3aE8) = 4_GL_2(F3)_||*_5 0 0 |det as claimed. Similar computations show that for the remaining groups E = E3bE8, * *E4aE8, E4bE8 and E4cE8, the group W 0(E) occurring in the Theorem is a lower bound for the W* *eyl group W (E). Orbit computation: Note first that all elementary abelian subgroups of rank a* *t most two are toral by Theorem 8.2(3). By using the lower bounds on the Weyl groups of E5* *aE8and E5bE8established above, we may find a set of representatives for the conjugacy * *classes of subgroups of E5aE8and E5bE8of rank 3 and 4. Under the action of W 0(E5aE8), the set of rank 3 subgroups of E5aE8has orbit* * representatives E3aE8= <__x1, __y1,,__a>E3bE8= <__x1, __y1,,__x3><_x1,,__x2, * *__y1> <__a, __x1,,__x2><_a,a__x1,n__x3>d<_a,,__x2, __x3> and under the action of W 0(E5bE8), the set of rank 3 subgroups of E5bE8has orb* *it representatives E3aE8= <__x1, __y1,,__a><_x1, __x2,a__y1>nd<_a,.__x1, __x2> Similarly we find that under the action of W 0(E5aE8), the set of rank 4 subg* *roups of E5aE8 has orbit representatives D _____E E4aE8= __x1, __y1, __x3,,x3a-1E4bE8= <__x2, __x1,,__y1, __a> E4cE8= <__x2, __x1,a__y1,n__x3>d<_a,,__x1, __x2, __x3> THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 57 and that under the action of W 0(E5bE8), the set of rank 4 subgroups of E5bE8ha* *s orbit repre- sentatives E4bE8= <__x2, __x1,a__y1,n__a>dE0 = <__x1, __x2,.__y1, __y2> Class distributions:_Recall that by 8.13, __ais in the conjugacy class 3A , _* *_x1and __x2are in the class 3B and x3a-1belongs to the class 3D . Using the actions of W 0(E5aE8)* * and W 0(E5bE8) it is then straightforward to verify the class distributions given in the table* *. As an example consider the group E5aE8. Under the action of W 0(E5aE8) it contains 156 elemen* *ts conjugate to___a,_78 elements conjugate to __x1, 2 elements conjugate to __x2and 6 elemen* *ts conjugate to x3a-1, which gives the class distribution in the table. Similar computations gi* *ve the results for the remaining groups. We also see that the group E0 = <__x1, __x2,h__y1,a__y2>s class distribution * *3B80 and from the class distribution of E5bE8we get E0 = (E5bE8\ 3B) [ {1}. It then follows from * *[63, Lem. 11.5] that E0 is toral. Other non-toral subgroups: We see directly that the groups <_a, __x1, __x2,,__x3><_a,,__x1,<__x2>_a,a__x1,n__x3>d<_a, __x* *2, __x3> are toral. Since the group <__x1, __x2,i__y1>s a subgroup of E0 it is also tora* *l. Alternatively, from the action of W 0(E5aE8) we see that it is conjugate to the group <__x1, __x2,,* *__x3>which is visibly toral. Thus any non-toral elementary abelian 3-subgroup of E8(C) is conjugate t* *o a group in the table. Moreover, since their class distributions differ, none of the gro* *ups occurring in the table are conjugate. To see that the groups in the table are actually non-toral we may proceed as * *follows. The group E3aE8contains the element __a, so by Theorem 8.2(1) it is toral if an* *d only if it is toral in CE8(C)(__a) = SL9(C)=C3. However this is not the case by Theorem 8.2(5* *), since its lift to SL9(C) is non-abelian. The groups E4aE8and E4bE8are thus also non-toral* * since they contain E3aE8. We saw above that the Weyl group of E3bE8contains SL3(F3), which* * has order 24. 33. 13. Since the Weyl group of E8(C) has order 214. 35. 52. 7 which is not* * divisible by 13 it follows from Theorem 8.2(2) that E3bE8is non-toral. Since E4cE8contains E* *3bE8it is also non-toral. Centralizers: The subgroups E = E3aE8, E4aE8, E4bE8, E5aE8and E5bE8are easy t* *o deal with since they all contain __a, and hence we have CE8(C)(E) = CSL9(C)=C3(E) for the* *se. It is however notationally convenient first to change the representatives as follo* *ws. Define x4 = ø-12 ø-12 ø-122 SL9(C), and note that conjugation by (2, 7, 3, 4)(5, 8, 9,* * 6) 2 SL9(C) acts as follows: a7! a, x1 7! x2, x27! x21, x3a-17! x4, y17! y2, y27! y21. In particular we see that E3aE8is conjugate to <__x2, __y2,.__a>Moreover we have _____________________________ CE8(C)(__a, __x2) = . From this we directly get _______________________________D E CE8(C)(__a, __x2,x__y2)2=, y2, {A A A | (detA)3= 1} ______________________________ = ~=<__x2, __y2,x__a>PSL3(C). 58 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Thus CE8(C)(E3aE8) = E3aE8x PSL3(C) and Z(CE8(C)(E3aE8)) = E3aE8. From the abov* *e we see _____ __ * *___ ____ that_the elements __x2, x3a-1and y2in CE8(C)(E3aE8) correspond to the elements * *fi2, ø-12and fl2in the PSL 3(C) component of CE8(C)(E3aE8). From this we easily compute the * *structure of C(E) for the representatives E which contain E3aE8, cf. the proof of Theorem* * 8.9. For the computation of the centralizers of E3bE8and E4cE8we consider the elem* *ent g = hff1(!)hff3(!2) 2 E8(C). By using [30, Tables 4 and 6] we get that g belongs to* * the conjugacy class 3B and that the centralizer CE8(C)(g) has type E6A2. The precise structur* *e of this centralizer may be found as follows. Since E8(C) is simply connected, Theorem * *8.2(3) implies that CE8(C)(g) is connected. Setting ff0= ff1 + ff2 + 2ff3 + 3ff4 + 2ff5 + ff6, we see that {ff5, ff8, ff6, ff7, ff0, ff2} [ {ff1, ff3} is a basis of the root * *system of CE8(C)(g) (the bases of the components of type E6 and A2 have been ordered so that the nu* *mbering is consistent with [13, p. 250-251, 260-262]). From this we get an explicit ho* *momor- phism 3E6(C)fxfSL3(C) ! E8(C) onto the centralizer CE8(C)(g). The kernel is giv* *en by (z, !2I3), where z 2 3E6(C) denotes the central element defined in Section 8.2* *. Thus CE8(C)(g) = 3E6(C) OC3 SL3(C), and we denote elements in this central product b* *y A . B where A 2 3E6(C) and B 2 SL3(C). In particular we have g = z . I3 = 1 . !I3. Now consider the subgroup E = ch is seen to be an e* *lementary abelian 3-group of rank 3 (here the elements x1, y1 2 3E6(C) from Section 8.2 s* *hould not be confused with the elements x1, y1 2 SL9(C) from above). We have * * ff CE8(C)(z . I3, x1 . fi) = C3E6(C)OC3SL3(C)(x1 . fi) = y1 . fl, C3E6(C)(x1) O* *C3 CSL3(C)(fi). We note that y1 . fl is not conjugate to its inverse in CE8(C)(z . I3, x1 . fi)* * since fl is not conjugate to fl-1 times a power of !I3 in SL3(C). Thus we have diag(1, 1, -1) =* *2W (E) and in particular W (E) 6= GL 3(F3). From the above we also get ff CE8(C)(E) = y1 . fl, x1 . fi, C3E6(C)(x1, y1) OC3 CSL3(C)(fi, fl) D E = y1 . fl, x1 . fi, C3E6(C)(ß-1(E2bE6)) OC3 Z(SL 3(C)) = x G2(C)) OC3 Z(SL 3(C))> = E x G2(C), using the computation of C3E6(C)(ß-1(E2bE6)) from the last part of the proof of* * Theorem 8.9. Since the preimage of E in 3E6(C) x SL3(C) is non-abelian it follows from Theor* *em 8.2(5) that E is non-toral in 3E6(C) OC3 SL3(C). Now Theorem 8.2(1) shows that E is no* *n-toral in E8(C) (alternatively one could also just observe that CE8(C)(E) has rank les* *s than 8). From what we have already proved we then see that E is conjugate to either E3aE* *8or E3bE8in E8(C). Since we already know CE8(C)(E3aE8) we conclude that E must be conjugate* * to E3bE8 (alternatively one could also compute the class distribution of E directly). In* * particular we have CE8(C)(E3bE8) = E3bE8x G2(C) and W (E3bE8) 6= GL 3(F3). Using the inclusion 3E6(C) 3E6(C) OC3 SL3(C) E8(C) we may also consider t* *he subgroup E43E6 3E6(C) from Theorem 8.7 as a subgroup of E8(C). Since E43E6is n* *on- toral in 3E6(C), it is also non-toral in 3E6(C) OC3 SL3(C), and hence also in E* *8(C) by Theorem 8.2(1). Thus E43E6must be conjugate in E8(C) to one of the groups E4aE* *8, E4bE8 or E4cE8. Comparing with the class distributions we can rule out E4aE8and E4bE* *8, so we THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 59 conclude that E43E6is conjugate to E4cE8. From Theorem 8.7 we have C3E6(C)(E43E* *6) = E43E6. Hence CE8(C)(E43E6) = E43E6OC3SL 3(C) from which we get CE8(C)(E4cE8) = E4cE8OC* *3SL 3(C). We determine the precise structure of the central product below after the compu* *tation of W (E4cE8). Exact Weyl groups: Recall from above that E3aE8is conjugate to <__x2, __y2,._* *_a>If W (E3aE8) was larger than the group W 0(E3aE8) from above, we then see that W (E3aE8) wou* *ld have to contain one of the groups 2 3 |* 4_GL_2(F3)_||*_5 orSL 3(F3) 0 0 | " which are the minimal overgroups of W 0(E3aE8) inside GL 3(F3). Thus W (E3aE8) * *would have to contain one of the matrices diag(1, 2, 1) or I3 + e3,2. This would mean tha* *t inside CE8(C)(__x2, __a) there would be an element which conjugates __y2into either __* *y22or ___y2a. However from above we have _____________________________ CE8(C)(__x2, __a) = , and from this it is easily seen that no such element exists. Thus W (E3aE8) = W* * 0(E3aE8) as claimed. For the group E3bE8we have SL3(F3) W (E3bE8) 6= GL 3(F3) and hence W* * (E3bE8) = SL3(F3). As in the proof of Theorem 8.9 we show that the remaining Weyl groups equal t* *he lower bounds already established, by looking at what this would imply for the subgrou* *ps E3aE8and E3bE8. For E = E4aE8, E4bE8, E5aE8, and E5bE8, any proper overgroup of W (E) co* *ntains an element which normalizes E3aE8but induces an automorphism on it not contained in its We* *yl group. For E4cE8the result follows by considering the subgroup E3bE8. It remains only to determine the precise structure of the central product CE8* *(C)(E4cE8) = E4cE8OC3 SL3(C). From the structure of W (E4cE8) we see that the subgroup <__x2* *>is invariant under the action of W (E4cE8). Thus a conjugates which sends to E4cE8to E43E6we* * see that <_x2> must be send to a W (E43E6)-invariant subgroup of E43E6. From the structure of * *W (E43E6) we see that there is only one such subgroup, namely = <1 . !I3>. As thi* *s is exactly the center of the SL3(C)-component of CE8(C)(E43E6) = E43E6OC3 SL3(C), we see t* *hat CE8(C)(E4cE8) = E4cE8O<__x2>SL3(C). 8.4. The group 2E7(C), p = 3. In this section we consider the elementary abelia* *n 3- subgroups of 2E7(C). We let H be a maximal torus of 2E7(C), (E7) be the root s* *ystem relative to H, and choose a realization ([118, p. 133]) (uff)ff2 (E7)of (E7) i* *n 2E7(C). By [118, 8.1.4(iv)] we may suppose that the root subgroups (u0ff)ff2 (E6)in 3E6(C)* * 2E7(C) coming from the choice of root subgroups for 3E6(C) from Section 8.2 satisfy uf* *f= u0fffor ff 2 (E6). For ff = ffi, 1 i 7, and t 2 Cx we define the elements nff(t) = uff(t)u-ff(-1=t)uff(t), hff(t) = nff(t)nff(1)-1. Q 7 Then the maximal torus consists of the elements i=1hffi(ti) and the normalize* *r N(H) of the maximal torus is generated by H and the elements ni= nffi(1), 1 i 7. 60 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL As in Section 8.2 we define the following elements in 3E6(C) 2E7(C): z = hff1(!)hff3(!2)hff5(!)hff6(!2), a = hff1(!)hff3(!2)hff5(!2)hff6* *(!), x2 = hff2(!2)hff3(!2)hff5(!2), y2 = n1n2n3n4n3n1n5n4n2n3n4n5n6n5n4n2n3n1n4n3n5n4n6n5hff2(-1). Notation 8.17. The conjugacy classes of elements of order 3 in 2E7(C) are given* * in [63, Table VI] and [30, Table 6] from which we take our notation. In particular, the* *re are 5 such conjugacy classes, which we label 3A , 3B , 3C , 3D and 3E. Moreover these clas* *ses may be distinguished by their traces on e7, except for the classes 3APand 3D which hav* *e the same trace. Since the trace of the element h 2 H is given by 7 + ff2 (E7)ff(h) we * *easily obtain the inclusions 3C [3E6] 3C [2E7], 3E [3E6] 3B[2E7], 3E0[3E6] 3B[2E7], corresponding to the inclusion 3E6(C) 2E7(C). Theorem 8.18. The conjugacy classes of non-toral elementary abelian 3-subgroups* * of 2E7(C) are given by the following table. __________________________________________________________________________ |_rank_|name|ordered_basis|2E7(C)-class_dist.|C2E7(C)(E)__Z|(C2E7(C)(E))___| 3 3C26 E3 x SL (C) E3 x Z(2E (C)) |__3__|E2E7_|______2___2|________________|_2E7____2____|__2E7______7_____|__ 4 3B23C78 E4 O T E4 O T |__4__|E2E7_|________2__|2_______________|___2E7__1__|____2E7__1____| In particular we have 3Z(C2E7(C)(E)) = E for any non-toral elementary abelian 3* *-subgroup of 2E7(C). The Weyl groups of these groups with respect to the given ordered bases are g* *iven as follows: 2 3 __"__|*__*__*__ 6 0 | 7 W (E32E7) = SL3(F3), W (E42E7) = 64 0 | 7 |SL 3(F3) 5 0 || Remark 8.19. Note that our information on the rank 3 subgroup E32E7corrects [63* *]. Proof of Theorem 8.18.Non-toral subgroups: From the way the realization (uff)ff* *2 (E7)is chosen above, it follows from Theorem 8.7 that E32E7and E42E7are elementary abe* *lian 3-subgroups of 2E7(C) and that we have 2 3 __1__|*__*__*__ 6 0 | 7 W (E32E7) SL3(F3), W (E42E7) 64 0 | 7 |SL 3(F3) 5 0 || In particular we see that both W (E32E7) and W (E42E7) have orders divisible by* * 13 and since 13 - |W (E7)|= 210. 34 . 5 . 7, we conclude by Theorem 8.2(2) that E32E7and E42* *E7are non-toral in 2E7(C). By [63, Thm. 11.16] we know that there are precisely two c* *onjugacy classes of non-toral elementary abelian 3-subgroups in 2E7(C), and thus E32E7an* *d E42E7 represent these two conjugacy classes. Class distributions: The class distributions follows directly from the class * *distributions of the groups E33E6and E43E6given in Theorem 8.7 and the information in 8.17 ab* *out the behavior of conjugacy classes in 3E6(C) under the inclusion 3E6(C) 2E7(C). THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 61 Weyl groups: Using our realization (uff)ff2 (E7)we may define a canonical map* * OE : W ! N(H) as follows ([118, 9.3.3]): If w = sffi1.s.f.firis a reduced expression for* * w 2 W we let OE(w) = ni1. .n.ir(by [118, 8.3.3 and 9.3.2] this does not depend on the reduce* *d expression for w). Note that the element OE(w) is a representative in N(H) for w 2 W . N* *ow let w0 2 W be the longest element in W , and let n0 = OE(w0). From [13, p. 264-266]* * it follows that w0 equals the scalar transformation -1, and so conjugation by n0 acts as i* *nversion on H. Now let w 2 W and define w0 by ww0 = w0. Since w0 is central in our cas* *e, we have (ww0)w-1 = w-1 (ww0)= w0so we conclude that w0w = ww0= w0. Now let ` be the length function on W . By [71, p. 16] we have `(w) + `(w0) = `(w0). In general * *the map OE is not a homomorphism, but we do have OE(w1w2) = OE(w1)OE(w2) if `(w1w2) = `(w1* *) + `(w2) by [118, 9.3.4(i)]. From this it follows that OE(w)OE(w0) = OE(w0)OE(w) = OE(w0* *) = n0, and we conclude that n0 commutes with OE(w) for all w 2 W . Now consider the element w = s1s2s3s4s3s1s5s4s2s3s4s5s6s5s4s2s3s1s4s3s5s4s6s5. Using the fact that the length of an element is given by the number of positive* * roots it sends to negative roots ([71, Cor. 1.7]), we see that the above product is a reduced * *expression for w. Thus we have y2 = OE(w)hff2(-1). From the above we then conclude that conjug* *ation by n0 acts as follows: z7! z2, a7! a2, x2 7! x22, y2 7! y2. Thus n0 normalizes E42E7and gives the element diag(2, 2, 2, 1) in W (E42E7). Co* *mbined with the above we conclude that 2 3 __"__|*__*__*__ 6 0 | 7 W (E42E7) 64 0 | 7 |SL 3(F3) 5 0 || From the inclusion (E7) (E8) we get the inclusion 2E7(C) E8(C), so we may consider E32E7and E42E7as subgroups of E8(C) as well. Since the orders of thei* *r Weyl groups in 2E7(C) are divisible by 13 and 13 - |W (E8)| = 214. 35 . 52 . 7 we se* *e from Theorem 8.2(2) that E32E7and E42E7remain non-toral in E8(C). Using Theorem 8.15* * and the class distributions from above we conclude that E32E7and E42E7are conjugate* * to E3bE8 and E4cE8respectively in E8(C). Theorem 8.15 now shows that the lower bounds f* *ound above are indeed the Weyl groups of E32E7and E42E7in 2E7(C). Centralizers: For the computation of the centralizer of E32E7we consider the* * element g = hff1(!)hff3(!2) 2 2E7(C). By using [30, Table 6] we see that g belongs to t* *he conjugacy class 3B and that the centralizer C2E7(C)(g) has type A5A2. The precise structu* *re of this centralizer may be found as follows. Since 2E7(C) is simply connected, Theorem* * 8.2(3) implies that C2E7(C)(g) is connected. Setting ff0= ff1 + ff2 + 2ff3 + 3ff4 + 2ff5 + ff6, we see that {ff5, ff6, ff7, ff0, ff2} [ {ff1, ff3} is a basis of the root syste* *m of C2E7(C)(g) (the bases of the components of type A5 and A2 have been ordered so that the numberi* *ng is consistent with [13, p. 250-251]). From this we get an explicit homomorphism SL* *6(C) xff SL3(C) ! 2E7(C) onto the centralizer C2E7(C)(g). The kernel is given by (!I6,* * !2I3). 62 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Thus C2E7(C)(g) = SL6(C) OC3 SL3(C), and we denote elements in this central pro* *duct by A . B where A 2 SL6(C) and B 2 SL3(C). In particular we have g = !I6 . I3 =fI6f* *. !I3. Now consider the subgroup E = !I6 . I3, (fi fi) . fi, (flwhifl)c.hfl2is se* *en to be an elementary abelian 3-group of rank 3. We have C2E7(C)(!I6 . I3, (fi fi)=.Cfi)SL6(C)OC3SL3(C)((fi fi) . fi) 2 * * ff = (fl fl) . fl , CSL6(C)(fi fi) OC3.CSL3(C)(* *fi) From this we get 2 * * ff C2E7(C)(E)= (fl fl) . fl , (fi fi) . fi, CSL6(C)(fi fi, fl fl) OC* *3 CSL3(C)(fi, fl) 2 * * ff = (fl fl) . fl , (fi fi) . fi, CSL6(C)(fi fi, fl. fl) OC* *3 Z(SL 3(C)) Here CSL6(C)(fi fi, fl fl) = 2,3({A 2 GL 2(C)|(detA)3 = 1}) is generated by 2* *,3(!2I2) = !2I6 and 2,3(SL 2(C)). From this we get C2E7(C)(E) = ~=E x SL2(C). Since the preimage of E in SL6(C) x SL3(C) is non-abelian it follows from Theor* *em 8.2(5) that E is non-toral in SL6(C) OC3 SL3(C). Now Theorem 8.2(1) shows that E is n* *on- toral in 2E7(C) (alternatively one could also just observe that C2E7(C)(E) has * *rank less than 7). It then follows that E is conjugate to E32E7in 2E7(C). In particular* * we have C2E7(C)(E32E7) = E32E7x SL2(C). Hence Z(C2E7(C)(E32E7)) = E32E7x Z(2E7(C)) sinc* *e the center of 2E7(C) has order 2. To compute the centralizer of E42E7we note that C2E7(C)(z) has centralizer ty* *pe E6T1, and that the E6 component corresponds to the subgroup 3E6(C) 2E7(C). A comput* *ation shows that the T1 component is given by T1 = {h(t2, t3, t4, t6, t5, t4, t3)|t 2* * Cx }, and thus we get C2E7(C)(z) = 3E6(C) OT1. Theorem 8.7 now shows that C2E7(C)(E42E* *7) = C3E6(C)(E42E7) OT1 = E42E7OT1. 9.Non-toral elementary abelian p-subgroups of projective unitary groups The purpose of this short section is to describe the non-toral elementary abe* *lian sub- groups of PGL n(C), which by Theorem 8.4 is equivalent to finding them for its * *compact form P U(n), as well as give information about centralizers and Weyl groups. T* *he sub- groups are easily determined and are described in [63, x3]_we here just add som* *e extra information centralizers and Weyl groups which we need in our proof of Theorem * *1.1. We first introduce a useful subgroup. Suppose n = prk and consider the extra* *special group p1+2r+embedded in GL n(C) by taking k copies of one of the p - 1 faithful* * irreducible pr-dimensional representations. (They all have the same image; see [72, Satz 16* *.14].) Note that this embedding maps the center of p1+2r+to the elements of order p in the * *center of GLn(C). Let r denote the subgroup of GL n(C) given by the subgroup generated b* *y the image of p1+2r+and the center of GL n(C). Note that as an abstract group r fit* *s into an extension sequence 1 ! Cx ! r ! ~ r! 1 where Cx identifies with the center of Cx and ~ r~=(Z=p)2ridentifies with the i* *mage of r in PGL n(C). (The matrices for r are written explicitly for k = 1 in [104, p. * *56] where it is called Upr.) THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 63 Theorem 9.1. Suppose E is a non-toral elementary p-subgroup of PGL n(C) (any pr* *ime p). Then, up to conjugacy E can be written E = ~ rx ~A, for some r 1 and some* * abelian subgroup A of CGLn(C)( r) ~=GL k(C). For a given r the number of conjugacy classes of such subgroups E are in one-* *to-one cor- respondence with conjugacy classes of toral elementary abelian p-subgroups ~Aof* * PGL k(C) ~= CPGLn(C)(~ r)1 (allowing the trivial subgroup), and the centralizer of E is giv* *en by CPGLn(C)(E) ~= ~ rx CPGLk(C)(A~). The Weyl group equals ~ ~ ~ WPGLn(C)(E) = Sp(*r) W 0 ~ PGLk(C)(A ) Here Sp(~ r) is the symplectic group relative to the symplectic product coming * *from the commutator product [., .] : ~ rx ~ r! Z=p Cx and the symbol * denotes a rank(* *A~) x 2r matrix with arbitrary entries. An element ff 2 Sp(~ r) WPGLn(C)(E) acts as up to conjugacy as ffx1 on CPGL* *n(C)(E) ~= ~ rx CPGLk(C)(A~). Sketch of proof of Theorem 9.1.The existence of the decomposition E = ~ rx ~Afo* *llows from Griess [63, Thm. 3.1] and the statements about uniqueness follows by repre* *sentation theory of the extraspecial p-groups. Since the image of p1+2r+is the sum of k identical irreducible representation* *s we have by Schur's lemma that CGLn(C)( r) ~=GL k(C) (see also [104, Prop. 4]). From th* *is the centralizer in PGL n(C) can easily be worked out. In the case where ~Ais trivial the statement about Weyl groups is given in [1* *04, Thm. 6] (and just uses elementary character theory). The general case follows similarly* *, again using character theory. For the statement about the Weyl group action, first note that Out(~ rx PGL k* *(C)) ~= Aut(~ r) x Out(PGL k(C)). An element ff 2 Sp( r) = WPGLn(C)(~ r) acts as an in* *ner automorphism on PGL k(C) since this is true for the action on CGLn(C)( r) ~= GL* *k(C) by character theory. Hence we can choose a representative g 2 NPGLn(C)(~ r) of * *ff which acts as ff x 1 on CPGLn(C)(~ r) ~= ~ rx PGL k(C). Hence g is also a representa* *tive of ff 2 Sp(~ r) WPGLn(C)(~ rx ~A). The claim now follows. 10.Calculation of the obstruction groups In this section we show that the existence and uniqueness obstructions to lif* *ting our diagram in the homotopy category to a diagram in the category of spaces identic* *ally vanish. More precisely, we will show the following theorem. Theorem 10.1. Suppose that X is any of the following p-compact groups F4^3, E6^* *3, E7^3, E8^3, E8^5or P U(n)^p(any p), or suppose that p is odd and X is connected and c* *enter-free with H*(BX; Zp) a polynomial algebra. Then limißj(BZ(CX (-))) = 0, for all i,.j A(X) (See Theorem 12.2 for an explanation of why exactly these p-compact groups ne* *ed atten- tion.) Note that for the purpose of Theorem 1.4 we only need to calculate the a* *bove groups for j = 1, 2 and i = j or i = j + 1. 64 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL We prove the theorem by filtering the functor Fj = ßj(BZ(CX (-))), and showin* *g that all filtration quotients vanish (with a small twist for P U(2)^2). First we show th* *at the quotient functor of Fj concentrated on all the toral elementary abelian p-subgroups has * *vanishing limits, using a Mackey functor argument which first appeared in [47]. This take* *s care of the case where H*(BX; Zp) is a polynomial algebra since in this case all subgroups * *are toral by Lemma 7.8. For the exceptional compact Lie groups we then continue and filt* *er the non-toral part of the functor by functors concentrated on only one non-toral su* *bgroup, and use a formula of Oliver [103] to show that the higher limits of these subquotie* *nt functors all vanish. For P U(n) we use a variant of this technique by suitably grouping * *the non-toral subgroups and using a combination of Oliver's formula and the Mackey functor ar* *gument we used for the toral part. We use the notation StG to denote the Steinberg module over Zp of a finite gr* *oup of Lie type G of characteristic p, defined as the top homology group with Zp coefficie* *nts of the Tits building of G (see e.g., [69]). In the special case of GL (E) we also writ* *e St(E) for the Steinberg module. 10.1. The toral part. Define a quotient functor Fjtorof Fj by setting Fjtor(V )* * = Fj(V ) if V is toral and Fjtor(V ) = 0 if V is non-toral. Let Ator(X) denote the full * *subcategory of A(X) consisting of toral subgroups. From the chain complex defining higher l* *imits (see e.g., [61, 3.3]) it follows that lim*Fjtor~= lim*Fjtor A(X) Ator(X) In order to use a Mackey functor argument on the right-hand side we need a more* * explicit description of the functor Fjtor. Lemma 10.2. Fix a connected p-compact group X and let ~Tbe a discrete approxima* *tion to a maximal torus T in X. For a non-trivial elementary abelian p-subgroup V * *~T, let WX (V ) denote the Weyl group of CX (V ) and let WX (V )1 denote the Weyl group* * of CX (V )1 (see [49, Thm. 7.6]). If T~WX(V )1is a discrete approximation to Z(Y1) then T~WX(V )is a discrete a* *pproxi- mation to Z(CX (V )). In particular in this case ß1(BZ(CX (V ))) = H1(WX (V );* * LX ) and ß2(BZ(CX (V )) = (LX )WX(V ), where LX = ß1(T ). Remark 10.3. For a connected p-compact group X and p odd, the fixed points T~WX always equals a discrete approximation to the center of X by [49, Thm. 7.6]. If* * X is the p-completion of a connected compact Lie group then this is likewise the case fo* *r p = 2 unless X contains a direct factor isomorphic to SO(2n + 1)^2, by [83, Thm. 1.6]. Proof of Lemma 10.2.Set Y = CX (V ) and ß = ß0(Y ) for short. Since V is toral,* * ~Tis in a canonical way a discrete approximation to a maximal torus in Y . First observe that the center of Y has discrete approximation in ~T. Indeed, * *otherwise there would by [49, Thm. 6.4] exist a central homomorphism f : Z=pn ! ~Np,Ywith* * image not in ~T, which would produce a homomorphism f0: Z=pn ! ~Np,Xcommuting with ~T* *but not in ~T, which contradicts the fact that T is self-centralizing in X by [48, * *Thm. 9.1], since X is connected. Suppose that ~T WX(V )1is a discrete approximation to Z(Y1) and set C = ~T WX* *(V.)We want to show that C is central in Y . Let f : BC ! BY1 be the natural inclusion* *. We have THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 65 an obvious diagram with horizontal maps fibrations map (BC, BY1){f}____//_map(BC, BY )f___//map(BC, Bß)0 | | || | | || fflffl| fflffl| || BY1 _______________//BY______________//_Bß where {f} denotes the set of homotopy classes of maps BC ! BY1 generated by f u* *nder the ß-action on BY1. If we can show that {f} consists of just f then it follows* * from the five-lemma that the middle vertical map is a homotopy equivalence, since our as* *sumption implies that C is central in Y1. To see that the action is trivial consider the following diagram: ~g BNY1;____//_BNY1; f~wwww| | www | | ww f fflffl|g fflffl| BC _____//BY1_____//BY1 where ~fis the natural inclusion of BC in BNY1, g is an element in Aut(BY1) ind* *uced by an element in ß and ~gis the corresponding self-map of BNY1 defined via Lemma 2.1.* * However by the definition of C, the composite ~g~fhomotopic to ~ffor all g induced by a* *n element in ß, so f is homotopic to gf as well. Hence we have shown that C is central in Y * *and since the center of Y has discrete approximation in ~Tit is obviously the largest sub* *group with this property. So C is a discrete approximation to ZY as wanted. The last statement about the homotopy groups now follows easily using the lon* *g exact sequence in group cohomology. Remark 10.4. The above lemma should be compared to Lemma 4.5 and Remark 4.6 whi* *ch have slightly different assumptions and conclusions. The following lemma is essentially contained in [47, x8]. Lemma 10.5. Let X be a connected p-compact group, and assume that for each non-* *trivial toral elementary abelian p-subgroup V 2 A(X) the fixed point set ~T WX(V )1is * *a discrete approximation to Z(CX (V )1). Then æ 2-j lim iFjtor= H (WX ; LX ) ifi = 0 and j = 1, 2 A(X) 0 otherwise. where H2-j(WX ; LX ) ~=ßj(BZ(X)) if ~T Wis a discrete approximation to Z(X). In particular, if p is odd, or more generally if for all reflections s 2 WX * *the singu- lar set oe(s) equals the fixed point set T~then the assumptions above are sa* *tisfied and limA(X)iFjtor= ßj(BZ(X)) if i = 0 and zero otherwise. (See [49, Def. 7.3] and R* *emark 10.6 for the definition of oe(s).) Proof.The first part of the proof consists of a translation of [47, x8] into th* *e current notation. By [50, Prop. 3.4] all morphism in A(X) between toral subgroups V ! X and V 0! * *X are induced by inclusion and action by WX . Hence we can identify Ator(X), up to eq* *uivalence of categories with a category which has objects non-trivial subgroups of p~T~=(Z=p* *)r (where r is the rank of T ) and morphisms the homomorphism between subgroups induced by inclusions and action by WX . Also, by [49, Thm. 7.6], WX (V ) consists of the * *elements in 66 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL WX which pointwise fixes V . Hence Lemma 10.2 shows that the functor F2toron At* *or(X) is isomorphic to the functor ff0 ,Mon A from [47, x8], where = WX and M = LX . * *Likewise F1toris isomorphic to ff1 ,M. (Note that there is the slight difference from [4* *7, x8] that M is a Zp -module rather than an Fp -module, but this makes no difference.) Therefor* *e [47, x8] (which is a Mackey functor argument, which can also be deduced from [46] or [73* *]) implies the first part of the lemma about obstruction groups. To see the last part about the singular set recall that for an abelian subgro* *up A ~Twe have by [49, Thm. 7.6] that " oe(s) reflections s 2 WX such that A oe(s) is a discrete approximation to Z(CX (A)1). Hence if oe(s) equals ~T then the* * assumptions of the first part are obviously satisfied since (again by [49, Thm. 7.6]) WX (A* *)1 is generated by reflections s 2 WX with A oe(s). Remark 10.6. Let G be a compact connected Lie group with maximal torus T , and * *let ff be a root of G relative to T with corresponding reflection sff. In this case th* *e singular set oe(sff) is just the discrete approximation of the kernel Uffof ff on T . (To se* *e this note that by [15, x4, no. 5] the reflection sfflift to an element nff(denoted by (`) in* * [15]) which satisfies n2ff= exp(ff_=2); the statement now follows_cf. [83, Pf. of Prop. 3.1* *(ii)].) Explicit calculations [83, Prop. 3.1(ii)] (see also [55], [75, Prop. 3.2(vi)]* *, and [106, x4]) show that for a compact connected Lie group G, oe(s) in fact always equal to ~T* * except when G contains a direct factor isomorphic to SO(2n+1). Combining this with Lem* *ma 10.5, now gives the following calculation of the toral part of the obstruction groups* *, whose full strength at p = 2 we will however not use here. Corollary 10.7. Let G be a compact connected Lie group with no direct factors i* *somorphic to SO (2n + 1) and p a prime. Set X = G^p. Then æ lim iFjtor= ßj(BZ(X)) ifi = 0 A(X) 0 otherwise. Proof of Theorem 10.1 when H*(BX; Zp) is polynomial, pIodd.f H*(BX; Zp) is a po* *lyno- mial algebra concentrated in even degrees then all elementary abelian p-subgrou* *ps are toral by Lemma 7.8, so F = F tor. Since p is odd the assumptions of Lemma 10.5 are sa* *tisfied and Theorem 10.1 follows. 10.2. The non-toral part for the exceptional groups. In this subsection we prove Theorem 10.1 when X is the p-completion of one of the exceptional groups and p * *odd. Let FjE denote the subquotient functor of Fj concentrated on a non-toral elementary* * abelian p-group E. By Oliver's formula [103, Prop. 4] æ limiFjE= Hom W (St(E), Fj(E)) ifi = rkE - 1 A(X) 0 otherwise. We now embark on proving some lemmas which will be used to show that these obst* *ructions groups identically vanish. (For the use in Theorem 1.4 we actually only need th* *is when E has rank at most four.) THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 67 Since ZCX (E) is the p-completion of an abelian compact Lie group (by [49, Th* *m. 1.1]), Fj = 0 unless j = 1, 2. The following lemma reduces the problem of showing tha* *t the obstruction groups vanish to showing that Hom W(E)(St(E), pZCX E) = 0, where pZ* *CX E is the finite group of elements of order p. Lemma 10.8. Let A be an abelian compact Lie group, and let pA and Ap denote the* * kernel and the image of the pth power map on A (using multiplicative notation). Let P * *be a finitely generated projective ZpW -module for a finite group W , and assume that A has a* * module action of W . Then Hom W(P, pA) = 0 if and only if Hom W(P, ß1(A) Zp) = Hom W (P, ß0(A) Zp)* * = 0. Proof.The long exact sequence of homotopy groups associated to the exact sequen* *ce of groups 1 ! Ap ! A ! A=Ap ! 1 shows that the inclusion Ap ,! A induces an isomor* *phism ß1(Ap) '!ß1(A) and an injection ß0(Ap) ,! ß0(A). Hence the exact sequence 1 ! pA ! A p!Ap ! 1 produces the following diagram, * *where the row, as well as the sequence going through ßi(A) instead of ßi(Ap), is exac* *t. p p p p ß1(A)I____//_ß1(A_)__//_ß0(pA)__//_ß0(A)__//_Iß0(A")` IIIpI | IIIpI | II |' III | I$$Ifflffl| $$Ifflffl| ß1(A) ß0(A) Apply the exact functor Hom W (P, - Zp) to this diagram. The lemma now follow* *s from Nakayama's lemma, using that ß0(A) is finite and ß1(A) is finitely generated. The following elementary observation is so useful that it is worth stating ex* *plicitly. Lemma 10.9. Suppose that W is a subgroup of GL (E), p odd, such that -1 2 W . T* *hen Hom W (St(E), E) = 0. Proof.Set Z = <-1>. Since Z acts trivially on St(E) we have Hom W(St(E), E) Hom Z(St(E), E) = Hom Z(St(E), EZ ) = 0. We also need the following lemma, which is a special case of a theorem of Ste* *ve Smith [117]. Lemma 10.10. Let G be a finite group of Lie type of characteristic p, and let P* * be a parabolic subgroup of G with corresponding unipotent radical U and Levi subgrou* *p L ~=P=U. Suppose that W is a subgroup U W P , and let M be an FpW -module. (1)If U acts trivially on M, then Hom W (StG, M) = Hom W=U(StL, M). (2)If StL Fp is irreducible as an FpW=U-module and if M has a finite filtrat* *ion as a FpW -module, with filtration quotients of Fp-dimension strictly less than* * dimZpStL then Hom W (StG, M) = 0. Proof.Since U acts trivially on M, Hom W (StG, M) = Hom W=U ((StG)U , M) where * *(-)U denotes coinvariants. But since the Steinberg module is self-dual, as is clear * *from its def- inition as a homology module, (StG)U ~= (StG)U . Now Smith's theorem [117] say* *s that (StG)U ~=StL, which proves the first part of the lemma. 68 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL For the second part, we can assume that the filtration quotients are simple F* *pW -modules. Since U Op(W ), U acts trivially on any irreducible FpW -module, by elementar* *y repre- sentation theory. Hence the second part follows from the first together with a * *dimension consideration. The above lemma is usually used in conjunction with the following obvious obs* *ervation. Lemma 10.11. Let E be a non-toral elementary abelian p-subgroup of a compact Li* *e group G. Then the Fp-dimension of pZCE is no more than the dimension of one of the ma* *ximal non-toral elementary abelian p-subgroups of G, and E is a W (E)-submodule of pZ* *CE. The last lemma we shall need is a concrete calculation. Lemma 10.12. Let E be a rank 4 elementary abelian 3-group, and let W = SL3(F3) * *x 1 GL(E). Then Hom W (St(E), E) = 0. Proof.This most easily checked by computer, e.g., using magma [11], but is inde* *ed a sufficiently small calculation so that the computer's algorithm with a bit of e* *ffort can be redone by hand. Alternatively one can use some ad hoc Lie theoretic arguments. * *(We are grateful to A. Kleschev and H. H. Andersen for sketching a couple of such argum* *ents to us_ however, since these arguments are rather involved compared to the size of the * *calculation at hand we will not provide them here.) Before we start going through the exceptional groups, we need to introduce a * *bit of notation. For an Fp-vector space E = ,we let Eij...denote the subs* *pace generated by ei, ej, . ... Likewise we let Pij...(resp. Uij...) denote the parabolic sub* *group (resp. unipotent radical) of GL (E) corresponding to the simple roots ffi, ffj, . .i.n* * the standard basis and notation. For example in GL 3(Fp), U2 is the subgroup 2 3 1 * * 4 0 1 0 5. 0 0 1 Proof of Theorem 10.1 when X = E8^5, F4^3, E7^3, E6^3,BoryE8^3.Lemma 10.8 it is* * enough to see that Hom W(E)(St(E), p(Z(CG (V ))) = 0. We now show this for all the E i* *n the list. (E8, 5): By [63, Lem. 10.3] E8 has, up to conjugacy, one non-toral elementary * *abelian 5-subgroup E, which has rank 3, Weyl group SL3(F5), and (since E is necessarily* * maxi- mal) E = pZCG E. Since St(E) is an irreducible module of dimension 125 we have* * that Hom W (St(E), E) = 0. (F4, 3): By [63, Thm. 7.4] there is, up to conjugacy, one non-toral elementary * *abelian 3- group E, which has rank 3, Weyl group W = SL(E), and E = pZCG (E). Since St(E) * *is an irreducible SL(E)-module of dimension 27, Hom W (St(E), pZCE) = 0. (E7, 3): By Theorem 8.18 E7 has, up to conjugacy, two non-toral elementary abel* *ian 3- subgroups E32E7and E42E7of rank 3 and 4 respectively. Since W (E32E7) = SL 3(F* *3) a dimension consideration as above gives Hom W(E32E)(St(E32E), pZCG E32E) = 0. Fo* *r E42E 7 7 7 * * 7 (whose Weyl group is listed in Theorem 8.18) we use Lemma 10.10(2), taking U = * *U23, which immediately gives that also Hom W(E42E)(St(E42E), pZCG E42E) = 0. 7 7 7 (E6, 3): By Theorem 8.9 E6 has eight non-toral elementary abelian 3-groups all * *of rank less than or equal to four. We follow the notation of this theorem. By Lemma 10* *.10(2), Hom W (St(E), pZCG E) = 0 when E = E2bE6, E3bE6, E3cE6or E4aE6(taking U = 1, 1,* * U2, and U1 THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 69 respectively). For E = E2aE6, E3aE6, E3dE6, and E4bE6we use that by Theorem 8.9* * E = pZCG E in these cases (a fact that we did not need above), and also -1 2 W (E) so Lemm* *a 10.9 applies to show that Hom W (St(E), pZCG E) = 0. (E8, 3): By Theorem 8.15 E8 has seven conjugacy classes of non-toral subgroups* *. If E = E3aE8, E3bE8, E4bE8, or E4cE8then Lemma 10.10 shows that Hom W (St(E), pZC(* *E)) = 0 (taking U = U1, 1, U2, and U23respectively). (Note that we do not need to know * *pZCG (E) exactly since the rough bound from Lemma 10.11 will do.) We now consider E = E4* *aE8. By Theorem 8.15 we have that pZ(CG (E)) = E, and by Lemma 10.12 Hom W(E)(St(E), E)* * = 0. Suppose that E = E5aE8. Then E has an invariant subspace E1 upon which U = U234 acts trivially. Now Hom W (St(E), E1) = Hom W=U (St(E1) St(E2345), E1) = 0, * *where we use Lemma 10.10 (to see that St(E)U ~= St(E1) St(E2345)) and Lemma 10.9 (u* *s- ing that "1 act trivially on St(E1) but fixed point free on E1). Now Hom W (St(* *E), E=E1) = Hom W=U(St(E1) St(E2345), E=E1) = 0 by Lemma 10.12. Suppose that E = E5bE8. T* *ake U = U123and note that E1234is an invariant subspace under W . Then Hom W(St(E),* * E1234) = Hom W=U(St(E1234), E1234) = 0 since -1 2 Sp(E1234) W=U. By [4] (or a direct c* *alcula- tion) Hom Sp(E1234)(St(E1234), E=E1234) = 0, which shows Hom W (St(E), E) = 0 as well. This exhausts the list. 10.3. The non-toral part for the projective unitary groups. We now embark in proving Theorem 10.1 for X = P U(n)^p. We will throughout this subsection use t* *he notation for elementary abelian p-subgroups of X introduced in Section 9. We first state the toral case. Lemma 10.13. Let X = P U(n)^p. Then æ limiFjtor= Z=2 ifn = p = 2, i = 0 and j = 1 A(X) 0 otherwise. Proof.If n 6= 2 then it is immediate to check that for an arbitrary reflection * *s 2 WX then ~T is connected so oe(s) = ~T by definition of oe(s). Hence if n 6= 2 or * *p odd the lemma follows by Lemma 10.5. Now suppose that X = P U(2)^2. Since for the non-trivial V ~Twe have that W* *X (V )1 is trivial and CX (V )1 ~=T the first part of Lemma 10.5 still apply to finish * *the proof also in this case. We next record the following general lemma, which is obvious from the Künneth* * formula. Lemma 10.14. Suppose D1 and D2 are two categories with only finitely many morph* *isms. Let CDi be "the cone on Di" i.e., the category constructed from Di by adding an* * initial object e to Di, and let D1 ? D2 = CD1 x CD2 \ (e, e), "the join of D1 and D2" (* *see [109, x1]). If Fi: CDi! Zp-mod , i = 1, 2 are functors then C*(CD1 x CD2, D1 ? D2; F1 F2) ~=C*(CD1, D1; F1) C*(CD2; D2; F2). In particular if one of the chain complexes has torsion free homology or if eve* *rything is defined over Fp then H*(CD1 x CD2, D1 ? D2; F1 F2) ~=H*(CD1, D1; F1) H*(CD2; D2; F2). 70 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL The following result gives that certain filtration quotients have (almost) va* *nishing coho- mology. Theorem 10.15. Set X = P U(n)^pand fix an integer r > 0. Let F r: A(X) ! Zp-mod denote the functor on objects given by Fjr(E) = ßj(BZCX (E)) if E is of the for* *m ~ rx ~A (in the notation of Section 9) and zero otherwise. Writing n = prk we have that æ lim iFjr= Z=2 ifj = i = k = r = 1 and p = 2 A(X) 0 otherwise. Proof.Define a functor ~Fjrby ~Fj(E) = ßj(BZCPU(k)(A~)^p) if E = ~ rx ~Afor a f* *ixed r and zero otherwise. This is a subfunctor of Fjrvia the identification P U(k) ~=CPU(* *n)(~ r)1. Set ~~F r r ~r j = Fj=Fj and observe that this is the trivial functor unless j = 1 where it* * is given by ~~F r ~ ~ ~ j(E) = rif E is of the form rx A and zero otherwise. Consider the category D = Ae(X) ~ rx Ae(P U(k)^p) \ (e, e) where the superscript e means that we do not exclude the trivial subgroup. We * *have a natural inclusion of categories ' : D ! A(X) on objects given by (V, ~A) 7! V x* * ~A. Step 1: We claim that this map induces an equivalence lim *Fjr! lim*Fjr. A(X) D By filtering the functor and using Nakayama's lemma it is enough to show this f* *or ~Fjr Fp and ~~Fjr Fp. We can furthermore replace these functors by the subquotient fun* *ctors which are only concentrated on one subgroup ~ rx ~A. Consider first such a subquotient of ~Fjr Fp. In this case the formula of O* *liver [103, Prop. 4] together with Lemma 10.14 shows that the higher limits on both sides a* *re only non-zero in a single degree, where the map identifies with the restriction map Hom WX(~ rxA~)(St(~ rx ~A), ßj(BZCPU(k)(A~)) Fp) ! Hom Sp(~ r)xWPU(k)(A~)(St(~ r) St(A~), ßj(BZCPU(k)(A~)) Fp). Now note that the elements U in WPU(n)(~ rx ~A) which sends ~ rto ~Aact trivial* *ly on the target by Theorem 9.1. Furthermore we have by the theorem of Steve Smith [117]* * (and self-duality of the Steinberg module) that St(~ rx ~A)U ~=St(~ rx ~A)U ~=St(~ r* *) St(A~), where the (-)U and (-)U denotes coinvariants and invariants respectively. Hence* * this map is an isomorphism. The case of a subquotient of ~~Fjr Fp is completely analog* *ous. This shows the isomorphism. Step 2: We now proceed to calculate the higher limits over D, which we do by * *calculating the limits of F~rjand F~~rj. We first consider F~~rj. We have already remarke* *d that only ~~F r * tor tor 1 6= 0. Furthermore if k 6= 1 then H (CA (P U(k)), A (P U(k)); Zp) = 0 by * *[47, x8] so lim*D~~Fjr= 0 by Lemma 10.14. If k = 1 we get limiD~~F r~=Hom Sp(~ r)(St(~ r), * *~ r) if i = 2r-1 and zero otherwise, by [103, Prop. 4]. Now consider ~Fjr. By Lemma 10.14 limi~Fjr= Hom Sp(~()St(~ r), Zp) limi-2rßj(BZCPU(k)^(A~)). D r ~A2Ator(PU(k)^p) p THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 71 By Lemma 10.13 limi-2rAtor(PU(k)^p)ßj(BZCPU(k)^p(A~)) = 0 unless p = k = 2, j* * = 1 and i - 2r = 1 where we get lim1Ator(PU(2)^2)ßj(BZCPU(2)^2(A~)) = Z=2. By an argument of H. H. Andersen and C. Stroppel [4] we have that Hom Sp(~ r)(St(~ r), Zp) = 0 for all r and p. To sum up we get that lim iFjr~=limi~~Fjr~=Hom Sp(~ r)(St(~ r), ~ r) A(PU(n)) D if j = k = 1 and i = 2r - 1 and zero otherwise. However the same argument of H. H. Andersen and C. Stroppel [4] shows that Hom Sp(~ r)(St(~ r), ~ r) = 0 unless r = 1 and p = 2 where it equals F2. (Note that this is obvious if p is * *odd by Lemma 10.9.) This shows the wanted formula. Remark 10.16. Note that slightly non-trivial statements from [4] are only used * *above for p = 2 and furthermore become trivial when r = 1, where Sp(~ 1) ~=SL(~ 1), and t* *his is in fact the only case which involve obstruction groups in the range needed for the* * proof of (the p = 2 version of) Theorem 1.4. Remark 10.17. It is in fact possible to give a short proof of Smith's theorem i* *n the special case used above, using the geometric definition of the Steinberg module St(E) v* *ia flags. Proof of Theorem 10.1 for X = P U(n)^p.Lemma 10.15 and 10.13 directly shows the* * con- clusion unless n = 2 and p = 2, so assume we are in this case. By writing down* * the definition it follows that lim0A(X)F1 = 0. Now, consider the exact sequence of* * functors 0 ! F11! F1 ! F1tor! 0. The long exact sequence of higher limits starts out as 0 ! 0 ! 0 ! lim 0F1tor! lim 1F11! lim 1F1 ! 0 ! . . . A(X) A(X) A(X) So, since limA(X)0F1tor~=Z=2 ~=limA(X)1F11we get that limiA(X)F1 = 0 for i > 0 * *as well. This concludes the proof of this last case of Theorem 10.1. 11. Appendix: The classification of finite Zp-reflection groups The purpose of this appendix is to give a classification of finite Zp-reflect* *ion groups extending and simplifying work of Notbohm [95, 96] who gave a classification fo* *r odd primes p. We start by recalling some definitions. Let R be an integral domain with fiel* *d of fractions K. An R-reflection group is a pair (W, L) where L is a finitely generated free * *R-module, and W is a subgroup of Aut(L) generated by elements ff such that 1 - ff has ran* *k one viewed as a matrix over K. Two finite R-reflection groups (W, L) and (W 0, L0) * *are called equivalent, if we can find a R-linear isomorphism ' : L ! L0such that the group* * 'W '-1 equals W 0. A finite R-reflection group (W, L) is said to be irreducible if the* * corresponding representation of W on L R K is irreducible. If R has characteristic zero we d* *efine the character field of an R-reflection group (W, L) as the field extension of Q gen* *erated by the values of the character of the representation W ,! Aut(L). For R = Zp or Qp we * *define an 72 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL exotic R-reflection group to be a finite irreducible R-reflection group with ch* *aracter field larger than Q. The classification of finite Zp-reflection groups is based on the work of Cla* *rk and Ewing [29], which is again based on the classification of finite C-reflection groups * *by Shephard and Todd [115]. Clark and Ewing showed that there is a bijection between finite Qp-* *reflection groups and finite C-reflection groups whose character field may be embedded in * *Qp. The classification of finite complex reflection groups [115] is as follows: The irr* *educible ones fall into 3 infinite families (in the following called family 1, 2 and 3) and 34 spo* *radic cases (in the following labeled Gi, 4 i 37). Moreover any finite complex reflection g* *roup can be written as a direct product of irreducible finite complex reflection groups, cf* *. [55, Rem. 2.3] (in fact this holds over any field of characteristic 0). It is convenient to split family 2 further depending on the character field. * *The associated complex reflection group is the group G(m, r, n) (where m, r and n are integers* * with m, n 2, r 1, r | m and (m, r, n) 6= (2, 2, 2)) from [115, p. 277] which consists o* *f monomial nxn- matrices such that the non-zero entries are m'th roots of unity and the product* * of the non-zero entries is an (m=r)'th root of unity. Thus G(m, r, n) is the semidirec* *t product of its subgroup A(m, r, n) of diagonal matrices with the subgroup of permutation m* *atrices. Let im = e2ii=m. For n 3 or n = 2 and r 6= m the character field of G(m, r, n* *) equals Q(im ), and for n = 2 and r = m it equals Q(im + i-1m) (see [29, p. 432-433]). * *We label the two cases family 2a and 2b respectively. A complete list of the irreducible finite complex reflection groups, their ch* *aracter fields and the primes for which these embed in Qp can be found in [77, p. 165] or [5, * *Table 1]. If (W, V ) is a finite Qp-reflection group, then by [34, Prop. 23.16] we can * *find a finitely generated ZpW -submodule L V with L Q = V . Thus any finite Qp-reflection m* *ay be obtained from a finite Zp-reflection group by extension of scalars, but this ma* *y give several inequivalent Zp-reflection groups. The following result extends [96, Thm. 1.5 a* *nd Prop. 1.6] to all primes. Theorem 11.1 (The classification of finite Zp-reflection groups). Let (W, L) be* * a finite Zp-reflection group. Then there exists a decomposition (W, L) = (W1 x W2, L1 L2) where (W1, L1) ~=(WG , LG Zp), for some (non-unique) compact connected Lie gr* *oup G with integral lattice LG , and (W2, L2) is a (up to permutation unique) direct * *product of exotic Zp-reflection groups. The exotic Zp-reflection groups are in natural one-to-one correspondence with* * the exotic Qp-reflection groups, i.e. the finite irreducible Qp-reflection groups on Clar* *k-Ewing's list whose character field is not contained in Q. If (W, L) is any exotic Zp-reflection group, then L Fp is an irreducible Fp* *W -module, and in particular we have (L Z=p1 )W = 0 and H0(W ; L) = 0. Remark 11.2. For odd primes p the last two results says by definition that any * *exotic Zp-reflection group is respectively center-free and simply connected, cf. [95]. Note also that [96, Thm. 1.5] imposes the unnecessarily strong condition that* * the invariant ring Zp[L]W is a polynomial algebra, but this condition is not actually used i* *n [96]. In the proof we shall use the following result concerning the mod-p reduction* * GL n(Zp) ! GLn(Fp). THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 73 Lemma 11.3. Let G GL n(Zp) be a finite subgroup. Then the composition G ,! GLn(Zp) ! GL n(Fp) of the inclusion with the mod-p reduction is injective if p * *is odd. For p = 2 the kernel of the composition is an elementary abelian 2-subgroup of * *rank at most n2. In particular the kernel is contained in O2(G), the largest normal 2-subgro* *up of G. Proof.For p odd this follows from [116, Pf. of Lem. 10.7.1]. For p = 2 the same* * argument shows that the composition G ,! GL n(Z2) ! GL n(Z=4) is injective. As the kern* *el of GLn(Z=4) ! GL n(F2) is an elementary abelian 2-group of rank n2 the result foll* *ows. Proof of Theorem 11.1.We start by showing that for any exotic Qp-reflection gro* *up (W, V ) we can find a finitely generated ZpW -submodule L V with L Q = V , such tha* *t L Fp is an irreducible FpW -module. Assume first that p - |W |. By [34, Prop. 23.16] we can find a finitely gener* *ated ZpW - submodule L V with L Q = V . It follows from [33, 75.6 and 76.15] that L * * Fp is automatically an irreducible FpW -module. Assume now that W has order divisible by p. From Clark-Ewing's list we see th* *at the only exotic Qp-reflection groups satisfying this condition are the groups G(m, * *r, n) from family 2a or one of the groups G12 for p = 3, G24 for p = 2, G29 and G31 for p * *= 5 or G34 for p = 7. In case W = G(m, r, n) from family 2a we get the extra conditions m 3, p * * 1 (mod m) and p n. Note in particular that n 3. The description above direct* *ly gives a representation with entries in Zp since the multiplicative group of Zp contains* * the (p - 1)'st roots of unity. Let L = (Zp)n be the natural ZpW -module, i.e. the set of col* *umns with entries in Zp. Assume that 0 6= M L Fp is a FpW -submodule of L Fp. Cho* *ose x 2 M with x 6= 0 and let ` 2 Fp be a primitive m'th root of unity. Since W con* *tains the permutation matrices and the diagonal matrix diag(`, `-1, 1, . .,.1) we see tha* *t M contains an element of the form x0= (x1, x2, 0, . .,.0)T with x1 6= 0. Since n 3, W al* *so contains the diagonal matrix diag(`, 1, `-1, 1, . .,.1) and hence M contains ((1 - `)x1,* * 0, . .,.0)T. As ` 6= 1 and W contains all permutation matrices we conclude that M = L Fp, pro* *ving the claim for the groups from family 2a. Next consider W = G12at p = 3. Since W is isomorphic to GL 2(F3), Lemma 11.3 * *shows that for any finitely generated Z3W -submodule L (Q3)2 of rank 2, we may iden* *tify L F3 as the natural F3W -module. In particular L F3 is an irreducible F3W -module. For W = G24 at p = 2 we have W ~=Z=2 x GL3(F2). Hence Lemma 11.3 shows that for any finitely generated Z2W -submodule L (Q2)3 of rank 3, we may identify * *L F2 as the F2(Z=2 x GL3(F2))-module where Z=2 acts trivially and GL 3(F2) acts natu* *rally. In particular L F2 is an irreducible F2W -module. Next consider the groups G29and G31at p = 5. Since G29is contained in G31it s* *uffices to show the result for W = G29. The representation in [115, p. 298] is defined * *over Z[1_2, i] and hence we get a representation over Z5 by mapping i to a primitive 4'th root* * of unity in Z5. Let L = (Z5)4 be the natural Z5W -module. There are 40 reflections in G2* *9: The 24 reflections in the hyperplanesPof the form xj- iffxkP= 0, j 6= k and the 16 * *reflections in the hyperplanes of the form 4j=1iffjxj = 0 with 4j=1ffj 0 (mod 4). In pa* *rticular G29 contains the reflections in the hyperplanes xj-xk = 0 and thus G29contains all * *permutation matrices. The product of the reflections in the hyperplanes x1 - ix2 = 0 and x1* * - x2 = 0 equals the diagonal matrix diag(i, -i, 1, 1) and thus this element is also cont* *ained in G29. Now the same argument used in the case of the groups from family 2a shows that * *L F5 is an irreducible F5W -module. 74 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL The argument for the group W = G34 at p = 7 is similar. The representation gi* *ven in [115, p. 298] is defined over Z[1_3, !], ! = i3 and hence we get a representati* *on over Z7 by mapping ! to a primitive 3'rd root of unity in Z7. Let L = (Z7)6 be the natura* *l Z7W - module. There are 126 reflections in G34: The 45 reflections in the hyperplanes* *Pof the form xj- !ffxk = 0, j 6= k and the 81 reflections in the hyperplanes of the form 6* *j=1!ffjxj = 0 P 6 with j=1ffj 0 (mod 3). In particular G34 contains all permutation matrice* *s. The product of the reflections in the hyperplanes x1 - !x2 = 0 and x1 - x2 = 0 equa* *ls the diagonal matrix diag(!, !2, 1, 1, 1, 1) and thus this element is also contained* * in G34. As above we then see that L F7 is an irreducible F7W -module. This proves the claim made above that for any exotic Qp-reflection group (W, * *V ) we can find a finitely generated ZpW -submodule L V with L Q = V , such that L F* *p is an irreducible FpW -module. It now follows from [114, Ex. 15.3] that, up to scalin* *g, L is the only such ZpW -submodule. This gives the natural bijection between exotic Zp-r* *eflection groups and exotic Qp-reflection groups. Since L Fp is an irreducible FpW -module we also conclude that (L Fp)W =* * 0 and H0(W ; L Fp) = 0 since the W -action on L Fp is non-trivial. Hence we get (L Z=* *p1 )W = 0 as claimed. We also see that multiplication by p is surjective on H0(W ; L) a* *nd from this we obtain H0(W ; L) = 0 by Nakayama's lemma. This proves the part of the theor* *em pertaining to exotic Zp-reflection groups. Now consider a finite Zp-reflection group (W, L) such that we have a direct s* *um decom- position L Q = V1 V2 as QpW -modules. Let W1 (resp. W2) be the subgroup of* * W which fixes V2 (resp. V1) pointwise. It it easy to see (cf. [50, Prop. 6.3]) th* *at (Wi, Vi) is a Qp-reflection group and that we get the decomposition (W, L Q) = (W1 x W2, V1* * V2). We now claim that if (W2, V2) is an exotic Qp-reflection group, then we have * *the de- composition (W, L) = (W1 x W2, L1 L2) with Li = L \ Vi. Let ff : L1 L2 -! L be the addition map. As in [50, Pf. of Thm. 1.5] it suffices to prove that ff* * Z=p1 : (L1 Z=p1 ) (L2 Z=p1 ) -! L Z=p1 is injective. Assume that (x1, x2) is* * in the kernel of ff Z=p1 , xi 2 Li Z=p1 . Thus x1 + x2 = 0. If s 2 W2 is a reflec* *tion we have by definition s . x1 = x1 and hence s also fixes x2 = -x1. Since W2 is gen* *erated by reflections we get x2 2 (L2 Z=p1 )W and hence x2 = 0 by the results already * *proved for exotic Zp-reflection groups. Hence x1 = 0 as well, and thus ff Z=p1 is injec* *tive proving the claim. Since any finite Qp-reflection group may be decomposed into a (up to permutat* *ion unique) product of finite irreducible ones, we see by using the claim repeatedly that a* *ny finite Zp- reflection group (W, L) may be decomposed as a product (W, L) ~=(W1 x W2, L1 * *L2) where (W1, L1) is a Zp-reflection group with character field equal to Q and (W2* *, L2) is as in the theorem. To finish the proof we thus need to show that for any finite Zp-reflection gr* *oup (W, L) with character field equal to Q we may find a compact Lie group G such that (W,* * L) is equivalent to (WG , LG Zp). We start by reducing the problem to finite Z-refle* *ction groups. The representation W ! GL (L Q) is a reflection representation and hence has Sc* *hur index 1 by [29, Cor. p. 429]. Thus this representation is equivalent to a representa* *tion defined over Q. Hence [34, Cor. 30.10] applied to R = Z(p)shows that there exist a (uni* *que) finitely generated Z(p)W -submodule L0 L with L0 Zp = L. Now [34, Cor. 23.14] applie* *d to R = Z shows that L0 contains a (non-unique) finitely generated ZW -submodule L0* *0with L0= L00 Z(p). We conclude in particular that (W, L) ~=(W, L00 Zp). THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 75 We finish the proof by showing that there exists a (non-unique) compact conne* *cted Lie group G whose Weyl group (WG , LG ) is equivalent to (W, L00). For each reflect* *ion s 2 W the group {x 2 L00| s(x) = -x} is an infinite cyclic group with two generators * *which we label ffs. Let = { ffs | s is a reflectionWin} and L000= (L00)W . It then * *follows (cf. [106, p. 85]) that (L00, L000, ) is a reduced root diagram whose associated Z-* *reflection group equals (W, L00) (see [15] for definitions). From the classification of compact * *connected Lie groups ([15, x4,no. 9, Prop. 16]) it then follows that there exists a compact c* *onnected Lie group G whose root diagram equals (L00, L000, ). In particular (WG , LG ) is * *equivalent to (W, L00) and we are done. The following result answers the question of when two connected compact Lie g* *roups give rise to the same p-compact group. For a compact connected Lie group G, le* *t H be the direct product of Z(G)1 with the universal cover of the derived group of G.* * We have a canonical finite covering homomorphism ' : H ! G with finite kernel (cf. [15, P* *rop. 1.4.4]). 0 If p is a prime number, we let Covp(G) denote the covering of G corresponding t* *o subgroup of ß1(G) given as the preimage of the Sylow p-subgroup of ß1(G)='(ß1(H)). Theorem 11.4 (Addendum to Theorem 1.1 and 11.1). Let G and G0 be two compact connected Lie groups and p a prime number. Then (1)(WG , LG ) and (WG0, LG0) are equivalent if and only if G is isomorphic t* *o G0up to the substitution of direct factors isomorphic to Sp(n) with SO (2n + 1). 0 (2)(WG , LG Z2) and (WG0, LG0 Z2) are equivalent if and only if Cov 2(G)* * and 0 0 Cov 2(G ) are isomorphic up to the substitution of direct factors isomorp* *hic to Sp(n) with SO (2n + 1). Moreover the following conditions are equivalent: (a)(WG , LG Z2, LG<1> Z2) and (WG0, LG0 Z2, LG0<1> Z2) are equivale* *nt. 0 20 0 (b)Cov 2(G) is isomorphic to Cov (G ). (c)(BG)^2' (BG0)^2. (3)For p odd the following conditions are equivalent: (a)(WG , LG Zp) and (WG0, LG0 Zp) are equivalent. 0 p0 0 (b)Cov p(G) and Cov (G ) are isomorphic up to the substitution of direc* *t factors isomorphic to Sp(n) with Spin(2n + 1). (c)(BG)^p' (BG0)^p. Sketch of proof:By [106, x4] or [75, Prop. 3.2(vi)] we can recover the root dat* *um of a compact connected Lie group from its integral lattice up to substitution of dir* *ect factors isomorphic to Sp(n) with direct factors isomorphic to SO(2n + 1). Part (1)now f* *ollows. Now, suppose that G is a compact connected Lie group of the form H=K where H * *is a direct product of a torus with a simply connected compact Lie group and K is a * *finite central p-group. Suppose moreover that G does not contain any direct factors isomorphic* * to Sp(n). By Proposition 7.4 the fundamental group of G equals the coinvariants (LG )W a* *nd hence LH = (LG )W SLG . This shows that (W, LH Zp) can be reconstructed from (W, LG* * Zp). By the classification of simply connected compact Lie groups we can for p = 2 r* *econstruct H from (W, LH Zp). For p odd the only ambiguity arises from direct factors iso* *morphic to Sp(n) or Spin(2n+1). However, if p is odd then by the assumption on G, H cannot* * contain any direct factors isomorphic to Sp(n), and we conclude that in all cases we ca* *n reconstruct H from (W, LG Zp). Since K is the cokernel of the inclusion LH Zp ! LG Zp* *, we can also recover the inclusion K LH Z=p1 = ~Z(H) Z(H) using [55, Thm. 1.4] f* *or the middle equality. This shows that we can recover G from (W, LG Zp). 76 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL The above analysis directly shows the first claim in (2)as well as (3a), (3b)* *. From the first claim in (2), (2a), (2b)follows, since Sp(n) and SO(2n+1) have different * *Z2-reflection data (WG , LG Z2, LG<1> Z2). The implications (2b)) (2c)) (2a)are clear. To finish off we remark that (3c)) (3a)is clear and (3b)) (3c)follows since B* *SO(2n+ 1)^pis homotopy equivalent to BSp(n)^peither by a very special case of our main* * Theorem 1.1 or by the original proof due to Friedlander [59]. 12.Appendix: Invariant rings of finite Zp-reflection groups, p odd (following Notbohm) The purpose of this appendix is to recall Notbohm's determination [96] of fin* *ite Zp- reflection groups (W, L), p odd, such that invariant ring Zp[L]W is a polynomi* *al algebra. Before stating it let us however for easy reference recall the following `cla* *ssical' character- izations of being a `p-torsion free' p-compact group, which has a proof by non-* *case-by-case arguments which we will sketch. Theorem 12.1. Let X be a connected p-compact group with maximal torus normalize* *r T . The following statements are equivalent: (1)H*(BX; Zp) is torsion free. (2)H*(X; Zp) is torsion free. (3)H*(X; Zp) is an exterior algebra over Zp with generators in odd degrees (* *or equiv- alently with Fp instead of Zp). (4)H*(BX; Zp) is a polynomial algebra over Zp with generators in even degree* * (or equivalently with Fp instead of Zp). ~= (5)H*(BX; Zp) ! H*(BT ; Zp)WX . We now give Notbohm's classification. The first part (which is a general arg* *ument reducing to the simply connected case) is [96, Thm. 1.3] and the second (which * *is a case-by- case argument of the simply connected cases) is a slight extension of [96, Thm.* * 1.4]. For the benefit of the reader we give a streamlined proof of the second part. Recall th* *at for a finite Zp-reflection group (W, L) we define SL to be the submodule of L generated by e* *lements of the form (1 - w)x with w 2 W and x 2 L. We call (W, L) simply connected if L = * *SL0for some ZpW -lattice L0(note that for p odd this is equivalent to SL = L since S2L* *0= SL0, cf. the discussion of Zp-reflection data in the introduction). Theorem 12.2 (Finite Zp-reflection groups with polynomial invariants, p odd). L* *et p be an odd prime and (W, L) a finite Zp-reflection group. Then we have the following s* *tatements: (1)Zp[L]W is a polynomial algebra if and only if Zp[SL]W is a polynomial a* *lgebra and the group of coinvariants LW is torsion free. (2)Suppose (W, L) is simply connected. The following conditions are equivale* *nt: (a)Zp[L]W is a polynomial algebra. (b)Fp[L Fp]W is a polynomial algebra. (c)(W, L) is not equivalent to (WG , LG Zp) for the following pairs (G,* * p): (F4, 3), (3E6, 3), (2E7, 3), (E8, 3) and (E8, 5). In particular, if X is an exotic p-compact group then Zp[LX ]WX is a polynomia* *l algebra and if (W, L) = (WG , LG Zp) for a connected compact Lie group G then Zp[LG Zp]* *WG is a polynomial algebra if and only if H*(G; Zp) is torsion free. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 77 Sketch of proof of Theorem 12.1.The equivalence of (1), (2), (3), and (4)are ol* *d H-space and loop space arguments which we first very briefly sketch. By a Bockstein sp* *ectral sequence argument (cf. e.g., [77, x11-2]) H*(X; Zp) torsion free if and only if* * H*(X; Zp) is an exterior algebra on odd dimensional generators so (2)is equivalent to (3). T* *his is again equivalent to that H*(BX; Zp) is a polynomial algebra on even dimensional gener* *ators (using the Eilenberg-Moore and the cobar spectral sequence; see e.g., [77, x7-4* *]), so (3) is equivalent to (4). Obviously (4)implies (1). Furthermore (1)implies (2), s* *ince if we had a torsion class in H*(X; Zp) of lowest degree then it would correspond to a* * torsion class in H*(BX; Zp) by the path-loop fibration. Also (4)implies (1)since if H*(* *BX; Zp) is concentrated in even degrees then it is torsion free by the universal coeffi* *cient theorem. That (5)implies (1)is obvious since (5)implies that H*(BX; Zp) is concentrate* *d in even degrees. The fact that (1)also imply (5)requires more machinery and is probabl* *y first found in [44, Thm. 2.11] (see also [87, Thm. 4.2(1)])_we quickly sketch an argu* *ment. We want to show that the map r : H*(BX; Zp) ! H*(BT ; Zp)W is an isomorphism. By * *[48, Thm. 9.7(3)] ~= * W (12.1) H*(BX; Zp) Q ! H (BT ; Zp) Q. This implies by comparing Krull dimensions that the number of polynomial genera* *tors equals the rank of T . Since H*(BT ; Fp) is finitely generated over H*(BX; Fp)* * by [48, Prop. 9.11] it follows by comparing Krull dimensions again that H*(BX; Fp) ! H** *(BT ; Fp) is injective. Hence H*(BX; Zp) ! H*(BT ; Zp) has to be injective by Nakayama's * *lemma. Likewise r has to be surjective: By (12.1)the cokernel of r has to be p-torsion* *. Since the reduction mod p of r is still injective (as seen above) the cokernel of r has t* *o be p-torsion free as well (since Tor(coker(r), Fp) = 0). Remark 12.3. If p is odd then Fp-coefficients can also be used in Theorem 12.1(* *5)by a Galois theory argument using Lemma 11.3. For p = 2, this is not true as can be * *seen by taking X = SU(2)^2. See [45] for a p = 2 version. Remark 12.4. If (W, L) is a finite Zp-reflection group then Zp[L]W is a polyno* *mial alge- bra if and only if Fp[L Fp]W is a polynomial algebra and the canonical monom* *orphism Zp[L]W Fp -! Fp[L Fp]W is an isomorphism as shown in [96, Lem. 2.3]. Note * *that this can be reformulated as saying that Zp[L]W is a polynomial algebra if and only i* *f Fp[L Fp]W is a polynomial algebra with generators in the same degrees as the generators o* *f Qp[L Q]W , since dimQp(Qp[L Q]W )n = dimFp(Zp[L]W Fp)n dimFp(Fp[L Fp]W )n for any * *n. Remark 12.5. The finite Z3-reflection group (W, L) = (WPU(3), LPU(3) Z3) does * *not have invariant ring a polynomial ring (e.g., since LW ~= Z=3 is not torsion fre* *e). However a short calculation shows that F3[L F3]W is a polynomial ring with generators * *in degrees 1 and 6 (as opposed to the degrees over Q3 which are 2 and 3). (See also [44, R* *em. 5.3].) It turns out that this example is essentially the only example since one can pr* *ove that if (W, L) is a finite Zp-reflection group, p odd, such that Fp[L]W is a polynomia* *l algebra, then Zp[L]W is also a polynomial algebra unless p = 3 and (W, L) contains (WPU(3), L* *PU(3) Z3) a direct summand. We omit the proof which is an extension of the technique used* * in the examples in Section 7 in a preprint version of [53], which can at the time of w* *riting be found on Wilkerson's homepage. 78 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Lemma 12.6. Assume that L is a finitely generated free Zp-module and that W is * *a finite subgroup of GL (L). If p - |W | and Fp[L Fp]W is a polynomial algebra, then * *Zp[L]W is also a polynomial algebra. Proof.By assumption we have the averagingPhomomorphisms Zp[L] -! Zp[L]W and Fp[L Fp] -! Fp[L Fp]W given by f 7! _1_|W|w2Ww . f. These are obviously surjectiv* *e and hence the commutative diagram Zp[L]_________//Zp[L]W | | | | fflffl| fflffl| Fp[L Fp]____//_Fp[L Fp]W shows that the reduction homomorphism Zp[L]W ! Fp[L Fp]W is surjective. The* * result now follows easily from Nakayama's lemma (cf. [96, Lem. 2.3]). Proof of Theorem 12.2.Part (1) is contained in [96, Thm. 1.3]. To prove part (2* *) note that by Notbohm [95] (see also [96, Thm. 1.2(iii)]), there is a unique finite irredu* *cible simply connected Zp-reflection group for each group on the Clark-Ewing list. We now go* * through the list, verifying the result in each case. If p - |W | the invariant ring Fp[L Fp]W is a polynomial algebra by the Sh* *ephard-Todd- Chevalley theorem ([6, Thm. 7.2.1] or [116, Thm. 7.4.1]), and thus Lemma 12.6 s* *hows that Zp[L]W is a polynomial algebra. Next, assume that (W, L) is an exotic Zp-reflection group. If (W, L) belongs * *to family number 2 on the Clark-Ewing list, the representing matrices with respect to the* * standard basis are monomial and so Zp[L]W is a polynomial algebra by [92, Thm. 2.4]. An inspection of the Clark-Ewing list now shows that only 4 exotic cases rema* *in, namely (G12, p = 3), (G29, p = 5), (G31, p = 5) and (G34, p = 7). In the first case w* *e have G12~=GL 2(F3) and Lemma 11.3 shows that the action on L F3 = (F3)2 is the can* *onical one. The invariant ring F3[L F3]GL2(F3)was computed by Dickson [37]. In the r* *emaining 3 cases the mod-p invariant ring was calculated by Xu [132, 133] using computer* *, see also Kemper and Malle [78, Prop. 6.1]. The conclusion of these computations is that * *in all 4 cases the invariant ring Fp[L Fp]W is a polynomial algebra with generators i* *n the same degrees as the generators of Qp[L Q]W . By Remark 12.4 we then see that Zp[L]* *W is a polynomial algebra in these cases. The only remaining cases are the finite simply connected Zp-reflection groups* * which are not exotic. Since p is odd and ß1(G) and (LG )WG only differ by an elementary * *abelian 2- group (cf. proof of Theorem 1.7 and Remark 7.4), we may assume that (W, L) = (W* *G , LG Zp) for some simple connected Lie group G. In this case Demazure [36] shows tha* *t if p is not a torsion prime for the root system associated to G, then the invariant rings Z* *p[LG Zp]WG and Fp[LG Fp]WG are polynomial algebras. By the calculation of torsion primes for the simple root systems, [36, Prop. * *8], the excluded pairs (G, p) in the last part of the theorem are exactly the cases where the ro* *ot system of G has p-torsion. In these cases Kemper and Malle [78, Prop. 6.1 and pf. of Thm. 8* *.5] shows that Fp[LG Fp]WG is not a polynomial algebra. Hence in these cases Zp[LG Z* *p]WG is not a polynomial algebra by [96, Lem. 2.3(i)]. This proves the second claim. Finally, let G be a compact connected Lie group with Weyl group W and integra* *l lattice L = LG . We now prove that Zp[L Zp]W is a polynomial algebra if and only if * *H*(G; Zp) THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 79 is torsion free. (See also [97, Prop. 1.11]). One direction follows from Theo* *rem 12.1, so assume now that Zp[L Zp]W is a polynomial algebra. From Theorem 12.2 we see* * that Zp[S(L Zp)]W is a polynomial algebra and that (L Zp)W is torsion free. Si* *nce p is odd, we have (L Zp)W = ß1(G) Zp and S(L Zp) = LG<1> Zp, cf. proof of Theorem 1.7 and Remark 7.4. From what we have proved above we conclude that H*(G<1>; Zp) is tor* *sion free. Since ß1(G) does not contain p-torsion, it now follows easily from the Se* *rre spectral sequence that H*(G; Zp) is torsion free. Remark 12.7. Let p be an odd prime and (W, L) a finite Zp-reflection group. We * *claim that the following conditions are equivalent: (1)Zp[L]W is a polynomial algebra. (2)Fp[L Fp]W is a polynomial algebra and LW is torsion free. (3)Fp[SL Fp]W is a polynomial algebra and LW is torsion free. Indeed we have (1), (3)by Theorem 12.2 since (W, SL) can be decomposed as a dir* *ect product of irreducible finite simply connected Zp-reflection groups by [95, Thm* *. 1.4]. The implication (1)) (2)follows from [96, Thm. 1.3 and Lem. 2.3]. Finally (2)) (3)f* *ollows from [92, Prop. 4.1] as LW torsion free implies that SL Fp ! L Fp is injec* *tive. 13. Appendix: Outer automorphisms of exotic finite Zp-reflection groups Theorem 1.1 states that the outer automorphism group of a p-compact group, p * *odd, equals NGL(L)(W )=W , which makes it useful to have a complete case-by-case cal* *culation of this object. Theorem 11.1 and Proposition 3.4 reduces the calculation to the* * case where (W, L) = (WG , LG Zp) for some connected compact Lie group G and the case wher* *e (W, L) is exotic. The purpose of this appendix section is to provide such a calculation when (W* *, L) is exotic based on calculations of Brou'e, Malle and Michel [21, Prop. 3.13] over * *the complex numbers. (Information about the, perhaps more familiar, Lie case can be obtaine* *d similarly, or by a very close reading of [75].) For the statement of the result (which will take place in the theorem below a* *s well as in the following elaborations), we fix the realizations G(m, r, n) of the groups f* *rom family 2 as described in Section 11. Moreover we also fix the realization of the complex* * reflection groups Gi(4 i 37) to be the one described in [115]. Finally we let ~n denot* *e the group of n'th roots of unity. Theorem 13.1 (Outer automorphisms of the exotic Zp-reflection groups). Let (W, * *L) be an exotic Zp-reflection group and let (W, V ) be the associated complex reflection* * group. Then NGL(V )(W ) = and hence NGL(L)(W )=W = Zxp=Z(W ) and NGL(L)(W )=ZxpW = 1 except in the following cases: (1)W = G(m, r, n) from family 2 with (m, r, n) 6= (4, 2, 2), (3, 3, 3): NG* *L(V )(W ) = nd NGL(L)(W )=ZxpW = Cgcd(r,n), cf. 13.4. (2)W = G(4, 2, 2): NGL(V )(W ) = and NGL(L)(W )=ZxpW = 3, cf. 13.5. (3)W = G(3, 3, 3): NGL(V )(W ) = and NGL(L)(W )=ZxpW = A4, cf. 13.* *6. (4)W = G5: NGL(V )(W ) = and NGL(L)(W )=ZxpW = C2, cf. 13.7. (5)W = G7: NGL(V )(W ) = and NGL(L)(W )=ZxpW = C2, cf. 13.8. Lemma 13.2. Let K K0be fields of characteristicDzero,Eand W GL n(K) an irre* *ducible reflection group. Then NGLn(K0)(W ) = NGLn(K)(W ), K0x. 80 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL Proof.The inclusion " " is clear, so suppose g 2 NGLn(K0)(W ). Consider the sy* *stem of equations Xw = gwg-1X, w 2 W where X is an n x n-matrix. Over K0 this has the solution X = g. By [55, Lem. 2.10], the representation W ! GL n(K0) is irreduci* *ble, so the solution space is the 1-dimensional space spanned by g. Since the coefficients * *lie in K, the solution space over K is 1-dimensional as well, so we can write g = ~g1 with ~ * *2 K0 and g1 2 Mn(K). As g 6= 0 we get ~ 6= 0 and g1 2 NGLn(K)(W ). We can now start the proof of Theorem 13.1. By [21, Prop. 3.13] we easily get* * the results on NGL(V )(W ) claimed above (note that and G(3, * *1, 3) G26). Now assume that W does not belong to family 2 and W 6= G5, G7. Let n d* *e- note the rank of W and K the field extension of Q generated by the entries of t* *he ma- trices representing W . Our assumption ensures that NGL(V )(W ) = andf* *hence NGLn(K)(W ) = . Lemmaf13.2fnow shows that NGLn(Qp)(W ) = W, Qxp. Hence we get NGL(L)(W ) = W, Zxpand since W is irreducible we have W \ Zxp= Z(W ), c* *f. [55, Lem. 2.9]. This proves Theorem 13.1 in case W does not belong to family 2 and W 6= G5, G* *7. In the remaining cases we have also proved the statements concerning NGL(V )(W ), * *and we thus only need to find the structure of NGL(L)(W ) in these cases. This is don* *e in the Examples 13.4, 13.5, 13.6, 13.7 and 13.8 below. To treat the dihedral group G(m, m, 2) from family 2 we need the following au* *xiliary result. Note also that the exotic groups from family 2a are also handled in [10* *2, x6] (here the non-standard notation G(q, r; n) for G(q, q=r, n)). Lemma 13.3. Assume that m 3 and p 1 (mod m) so that im + i-1m2 Zp. Then 2 + im + i-1mis a unit in Zp. Proof.It suffices to prove that the norm NQ(``m+``-1m)=Q(2 + im + i-1m) is not * *divisible by p. Since its square equals the norm NQ(``m)=Q(2 + im + i-1m) it is enough to see t* *hat this norm is not divisible by p. In Q(im ) we have 2 + im + i-1m= (1 + im )2=im and since* * im is a unit it is enough to see that NQ(``m)=Q(1 + im ) is not divisible by p. By definition Y Y NQ(``m)=Q(1 + im ) = (1 + ikm) = (-1)ffi(m) (-1 - ikm) = m (-1* *). 0 k m 0 k m gcd(k,m)=1 gcd(k,m)=1 The first claim now follows from [127, Lem. 2.9]. Elaboration 13.4 (Family 2, generic case). Let W = G(m, r, n) from family 2 and* * let p be a prime number such that W is an exotic Zp-reflection group. Thus if n 3 or n* * = 2 and r < m we have m 3 and p 1 (mod m), and for n = 2 and m = r we have m 5, * *m 6= 6 and p 1 (mod m). Assume moreover that (m, r, n) 6= (4, 2, 2), (3, 3, 3) (th* *ese two cases will be dealt with below). Assume first that p 1 (mod m). The realizations of the groups G(m, r, n) a* *nd G(m, 1, n) from above are both defined overfthefring Z[im ] which embeds in Zp. Lemma 13.2* * shows that NGLn(Zp)(W ) = G(m, 1, n), Zxpwhence the natural homomorphism (A(m, 1, n)=A(m,* * r, n))x Zxp-! NGLn(Zp)(W )=W is surjective. The kernel is the cyclic group generated by* * the el- ement ([im In], i-1m) (here [im In] 2 A(m, 1, n)=A(m, r, n) denotes the coset o* *f im In) and thus NGLn(Zp)(W )=W = (A(m, 1, n)=A(m, r, n)) OCm Zxp. Note that A(m, 1, n)=A(m* *, r, n) is cyclic of order r generated by the element x = [diag(1, . .,.1, im )] and that * *[im In] = xn. THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD 81 If the assumption p 1 (mod m) is not satisfied, then W = G(m, m, 2) is the* * dihedral group of order 2m with m 5, m 6= 6 and p -1 (mod m). Conjugating the reali* *zation of G(m, m, 2) from above with the element ~ -1 ~ g = 11-im-i m gives a realization G(m, m, 2)g defined over the character field Q(im + i-1m). * * Note that if m is odd, then NGL2(C)(G(m, m, 2) = nd hen* *ce NGL2(Zp)(G(m, m, 2)g)=G(m, m, 2)g = Zxp, so we may assume m to be even. Since G* *(m, 1, 2) is generated by G(m, m, 2) and diag(1, im ) we find ø ~ ~ Æ NGL2(Zp)(G(m, m, 2)g) = G(m, m, 2)g, -11 1 + i1 -1 , Qxp\ GL2(Zp) m + im using Lemma 13.2. From Lemma 13.3 we see that the above matrix is invertible ov* *er Zp and hence ø ~ ~ Æ NGL2(Zp)(G(m, m, 2)g) = G(m, m, 2)g, -11 1 + i1 -1 , Zxp m + im Thus the homomorphism Z x (Zxp=~2) -! NGL2(Zp)(G(m, m, 2)g)=G(m, m, 2)g which m* *aps ~ ~k (k, [~]) to the coset of ~ -11 1 + i1 -1 is surjective. The kernel is eas* *ily seen to m + im be the infinite cyclic group generated by the element (-2, [1 + im + i-1m]) and* * thus we get NGL2(Zp)(G(m, m, 2)g)=G(m, m, 2)g ~=ZOZ(Zxp=~2). It easily checked that [2+im +* *i-1m] has a square root in Zxp=~2 if and only if either m 0 (mod 4) or m 2 (mod 4) * *and p -1 (mod 2m). In this case we have NGL2(Zp)(G(m, m, 2)g)=G(m, m, 2)g ~=C2 x (Zxp=~* *2). Elaboration 13.5 (G(4, 2, 2)). The realization of the group G(4, 2, 2) from abo* *ve and the realization of the group G8 from [115, p. 280-281] are both defined over their * *common character field Q(i). Thus the relevant primes p are the ones satisfying p 1* * (mod 4). More precisely the representations are defined over Z[1_2, i] andfasfthis ring * *embeds in Zp for all p as above, we see that NGL2(Zp)(G(4, 2, 2)) = G8, Zxp. It is easily * *checked that G8 = where H is the group of order 24 generated by the elements ~ ~ ~ ~ 0 i 1_+_i 1 1 1 0 , 2 i -i ff Since G(4, 2, 2) \ H, Zxp= Z(H) = ~4 we conclude that NGL2(Zp)(G(4, 2, 2))=G(4* *, 2, 2) ~= (H=Z(H)) x (Zxp=~4) ~= 3 x (Zxp=~4). Elaboration 13.6 (G(3, 3, 3)). The realization of the group G(3, 3, 3) from abo* *ve and the realization of the group G26 from [115, p. 296-297] are both defined over their* * common character field Q(!) where ! = e2ii=3. Thus the relevant primes p are the ones * *satisfying p 1 (mod 3). More precisely the representations are defined over Z[1_3, !] * *andfasfthis ring embeds in Zp for all p as above, we see that NGL3(Zp)(G(3, 3, 3)) = G26, * *Zxp. It is easily checked that G26is the semidirect product of G(3, 3, 3) with the group H* * ~=SL2(F3) generated by the elements 2 3 2 3 1 0 0 1 ! !2 !2 R1 = 4 0 1 0 5 , R2 = _____p_4!2 ! !2 5 0 0 !2 -3 !2 !2 ! 82 K. ANDERSEN, J. GRODAL, J. MfflLLER, AND A. VIRUEL The center of H is generated by the element 2 3 0 -1 0 z = 4 -1 0 0 5 0 0 -1 ff and G(3, 3, 3) \ H, Zxp= <-z, ~3>. Thus NGL3(Zp)(G(3, 3, 3))=G(3, 3, 3) ~=H OC2 (Zxp=~3) ~=SL2(F3) OC2 (Zxp=~3) where the central product is given by identifying z 2 H with the element in Zxp* *=~3 repre- sented by -1. Elaboration 13.7 (G5). The realization of the group G5 from [115, p. 280-281] i* *s defined over the field Q(i12), but the group has character field Q(!) and thus the rele* *vant primes p are the one satisfying p 1 (mod 3). Conjugation by the matrix ~ p _ ~ g = p _ 2 3- 1 ( 3- 1)(1 - i)i - 1 gives a realization defined over Z[1_3, !] which embeds in Zp for all p as abov* *e. Its easily checked that G14is generated by G5 and the reflection ~ ~ 1 -1 i S = ___p_ 2 -i 1 From this we get ø ~ ~ Æ NGL2(Zp)(Gg5) = Gg5, -20! 10 , Zxp and thus the homomorphism Z x (Zxp=~6) -! NGL2(Zp)(Gg5)=Gg5which maps (k, [~]) * *to the ~ ~k coset of ~ -20! 10 is surjective. The kernel is easily seen to be the infini* *te cyclic group generated by the element (-2, [2]) and we get NGL2(Zp)(Gg5)=Gg5~=Z OZ (Zxp=~6).* * It is easy to check that [2] has a square root in Zxp=~6 if and only if p 1, 7, 19 (mod * * 24) (that is unless p 13 (mod 24)). In this case we get the simpler description NGL2(Zp)(* *Gg5)=Gg5~= C2 x (Zxp=~6). Elaboration 13.8 (G7). The realizations of the groups G7 and G10given in [115, * *p. 280- 281] are both defined over their common character field Q(i12). 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Department of Mathematics, University of Copenhagen, 2100 Copenhagen, Denmark E-mail address: kksa@math.ku.dk Department of Mathematics, University of Chicago, Chicago, IL 60637, USA E-mail address: jg@math.uchicago.edu Department of Mathematics, University of Copenhagen, 2100 Copenhagen, Denmark E-mail address: moller@math.ku.dk Department of Mathematics, University of Malaga, 29080 Malaga, Spain E-mail address: viruel@agt.cie.uma.es