ADJOINT SPACES AND FLAG VARIETIES OF p-COMPACT GROUPS TILMAN BAUER AND NATA`LIA CASTELLANA Abstract.For a compact Lie group G with maximal torus T, Pittie and Smith showed that the flag variety G/T is always a stably framed boundary. We * *gener- alize this to the category of p-compact groups, where the geometric argu* *ment is replaced by a homotopy theoretic argument showing that the class in the * *stable homotopy groups of spheres represented by G/T is trivial, even G-equivar* *iantly. As an application, we consider an unstable construction of a G-space mim* *icking the adjoint representation sphere of G inspired by work of the second au* *thor and Kitchloo. This construction stably and G-equivariantly splits off its to* *p cell, which is then shown to be a dualizing spectrum for G. 1. Introduction Let G be a compact, connected Lie group of dimension d and rank r with max- imal torus T. Left translation by elements of G on the tangent space g = TeG induces a framing of G. By the Pontryagin-Thom construction, G with this fram- ing represents an element [G] in the stable homotopy groups of spheres; this has been extensively studied for example in [Smi74, Woo76 , Kna78, Oss82]. The following classical argument shows that the flag variety G/T, while not necessarily framed, is still a stably framed manifold: since every element in a* * com- pact Lie group is conjugate to an element in the maximal torus, the conjugation map G T ! G, (g, t) 7! gtg 1 is surjective, and furthermore, it factors throu* *gh c: G/T T ! G. An element s 2 T is called regular if the centralizer CG(s) ' T equals T, or, equivalently, if cjG/T fsgis an embedding; it is a fact from Lie * *the- ory that the set of irregular elements has positive codimension in T. Thus there is a regular element s such that the derivative of c has full rank along G/T * *fsg, and by the tubular neighborhood theorem, it induces an embedding of G/T U, where U is a contractible neighborhood of s in T. Thus the framing of G can be pulled back to a stable framing of G/T. Pittie and Smith showed in [Pit75, PS75] that G/T is always the boundary of another framed manifold M, and moreover, that M has a G-action which agrees with the standard G-action on G/T on the boundary. In terms of homotopy the- ory, this is saying that the class [G/T] 2 sssd rinduced by the Pontryagin-Thom construction is trivial. The first main result of this paper generalizes this fact to p-compact groups. Theorem 1.1. Let G be a Z/p-local, p-finite group with maximal torus T such that dim(G) > dim(T). Then the Pontryagin-Thom construction [G/T]: SG ! ST is G- equivariantly null-homotopic, with G acting trivially on ST. ____________ Date: February 20, 2006. N. Castellana is partially supported by MCYT grant MTM2004-06686. Support by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully ack* *nowledged. 1 2 TILMAN BAUER AND NATA`LIA CASTELLANA The statement of this theorem requires some explanation. A p-compact group [DW94 ] is a triple (G, BG, e) such that fflG is p-finite, i. e., H (G; Fp) is finite; fflBG is Z/p-local, i. e. whenever f :X ! Y is a mod-p homology equiva- lence of CW-complexes, then f :[Y, BG] ! [X, BG] is an isomorphism; ffle: G ! BG is a homotopy equivalence. Clearly, G and e are determined by BG up to homotopy, making this definition somewhat redundant. Although a priori G is only a loop space, we will henceforth assume we have chosen a rigidification such that G is actually a topological gr* *oup. This is always possible, for example by using the geometric realization of Kan's group model of the loops on a simplicial set [Kan56 ]. A Z/p-local, p-finite loop space is only slightly more general than a p-compa* *ct group in that the latter also requires ss0(G) to be a p-group. We will have no * *need to assume this in Theorem 1.1. By [DW94 ], every p-compact group has a maximal torus T; that is, there is a monomorphism T ! G with T ' Lp(S1)r and r is maximal with this property. By definition, a monomorphism of p-compact groups is a group monomorphism H ! G such that G/H is p-finite (see [Bau04] for this slightly nonstandard point of view). Dwyer and Wilkerson show that T is essentially unique. Since a maximal torus is always contained in the identity component of a p-compact group, the same works for Z/p-local, p-finite groups. Denote by S0[X] the suspension spectrum of a space X with a disjoint base poi* *nt added. Definition ([Kle01]). Let G be a topological group. Define SG, the dualizing sp* *ec- trum of G, to be the spectrum of homotopy fixed points of the right action of G* * on its own suspension spectrum. That is, SG = (S0[G])hGopas left G-spectra. In [Bau04], the first author showed that for a connected, d-dimensional p-com- pact group G, SG is always homotopy equivalent to a Z/p-local sphere of dimen- sion d. Furthermore, there is a G-equivariant logarithm map S0[G] ! SG, where G acts on the left by conjugation. If G is the Z/p-localization of a connected * *Lie group, then SG is canonically identified with the suspension spectrum of the on* *e- point compactification of the Lie algebra of G. Thus we may call SG the adjoint (stable) sphere of G. In a spectacular case of shortsightedness, [Bau04] restricts its scope to con* *nected p-compact groups where everything would have worked for Z/p-local, p-finite groups G as well. In this case, SG has the mod-p homology of a d-dimensional sphere. Similarly, the proof of the following was given in [Bau04, Cor. 24] f* *or connected groups, but immediately generalizes. Let DM be the Spanier-Whitehead dual of a finite CW-spectrum M. Lemma 1.2. Let H < G be a monomorphism of Z/p-local, p-finite groups. Then ther* *e is a relative G-equivariant duality weak equivalence i j G+ ^H SH ' D S0[G/H] ^ SG. For any space X, there is a canonical map ffl: S0[X] ! S0 given by applying t* *he functor S0[ ] to X ! . If T < G is a sub-torus in a Z/p-local, p-finite group * *then ADJOINT SPACES AND FLAG VARIETIES OF p-COMPACT GROUPS 3 there is a stable G-equivariant map i j (1.3)[G/T]: SG id^Dffl!SG ^ D S0[G/T] ' G+ ^T ST ' S0[G/T] ^ ST ffl!ST Lemma1.2 where the homotopy equivalence on the right hand side holds because ST has a homotopy trivial T-action as T is homotopy abelian. The first map is studied in [Bau04]. This is the map referred to in Theorem 1.1; it generalizes the Pontrya* *gin- Thom construction. In the second part of this paper, as an application of Theorem 1.1, we study * *the relationship between two notions of adjoint objects of p-compact groups. It is * *an interesting question to ask whether the action of G on SG actually comes from an unstable action of G on Sd. We will not be able to answer this question in this paper. However, there is an alternative, unstable construction of an adjoint ob* *ject for a connected p-compact group G inspired by the following: Theorem 1.4 ([CK02 , Mit88]). Let G be a semisimple, connected Lie group of ran* *k r. There exist subgroups GI < G for every I ( f1, : :,:rg and a homeomorphism of G- spaces AG := hocolimG/GI ! g [ f1g I(f1,:::,rg to the one-point compactification of the Lie algebra g of G. In the second part of this paper, we define a G-space AG for every connected p-compact group G and show: Theorem 1.5. For any connected p-compact group G, there is a G-equivariant spli* *tting S0[AG] ' SG ` R for some finite G-spectrum R. This result links the two notions of adjoint objects together. Thus stably, t* *he adjoint sphere is a wedge summand of the adjoint space. Unfortunately, AG is in general not a sphere. Acknowledgements. We would like to thank the Institut Mittag-Leffler for its support while finishing this work, and Nitu Kitchloo for helpful discussions. 2.The stable p-complete splitting of complex projective space 2.1. Stable splittings from homotopy idempotents. Let p be a prime. We denote by Lp the localization functor on topological spaces with respect to mod-p homo* *l- ogy, which coincides with p-completion on nilpotent spaces [BK72 ]. Let S = LpS1 be the p-complete 1-sphere, and set P = S0[BS]. It is a classical result that p`2 (2.1) P ' Ps s=0 for certain (2i 1)-connected spectra Pi. In this section, we will investigate* * this splitting and its compatibility with certain transfer maps. Let X be a spectrum, e 2 [X, X] and define eX = hocolimfX !e X !e g. 4 TILMAN BAUER AND NATA`LIA CASTELLANA If e is idempotent, this is a homotopy theoretic analog of the image of e. Any such idempotent e yields a stable splitting X ' eX ` (1 e)X. If fe1, : :,:eng* * are a complete set of orthogonal idempotents (this means that each eiis idempotent, eiej' , and idX' e1+ + en), then they induce a splitting X ' e1X ` ` enX. Example 2.2. Let p be an odd prime. Denote by _ :P ! P the map induced by multiplication with a (p 1)st root of unity i. Define es:P ! P by _ l 1 ! es= __1__p X 1i is_i. i=0 It is straightforward to check that fe0, : :,:ep 2g are a complete set of ortho* *gonal idempotents in [P, P]. They induce the splitting (2.1)by defining Ps= esP. Setting H (P) = Zpfxjg with jxjj = 2j, we have that (ei) : H (P) ! H (P) is given by ( (2.3) (ei) (xj) = xj; j j i (mod p 1) 0; otherwise. 2.2. Transfers as splittings. Let 1 ! H i! G ! W ! 1 be an extension of compact Lie groups. Then associated to the fibration W ! BH ! BG there are two versions of functorial stable transfer maps [BG75 , BG76]: (1)The Becker-Gottlieb transfer __o:S0[BG] ! S0[BH] (2)The stable Umkehr map o :BGg ! BHh of Thom spaces of the adjoint representation of the Lie groups. Both versions can be generalized to a setting where the groups involved are not Lie groups but only Z/p-local and p-finite [Dwy96 , Bau04]. For such a group G, BGg is defined to be the homotopy orbit spectrum of G acting on the dualizing spectrum SG; since H (SG) = H (Sd; Zp), we have a (possibly twisted) Thom isomorphism between_H (BG) and H (BGg). Note that ofactors through o in the following way: 0 comult: (2.4)S0[BG] o! BH ! BH ^BG S0[BH] id^! BH ^ 0 0 eval^id 0 BG S [BH] ^BG S [BH] ! S [BH] where = h i g is the normal fibration along the fibers of BH ! BG, o0 is o twisted by g, and the right hand side evaluation map is defined by identifying BH with the fiberwise Spanier-Whitehead dual of BH over BG. Proposition 2.5. Let W = Clbe a finite cyclic group acting freely on S, with l * *j p 1. Denote by N = S o W the semidirect product with respect to this action. Then th* *e Becker- Gottlieb transfer map __o:S0[BN] ! P factors through fP ! P for some idempotent f :P ! P which induces the same map in homology as e0+ el+ + ep 1 l, and the induced map S0[BN] ! fP is a mod-p homology equivalence. Proof.Since p - jWj, the Serre spectral sequence associated to the group extens* *ion S !i N ! W is concentrated on the vertical axis and shows that H (BN; Zp) ,=H (BS; Zp)W ,=Zp[zl] ,! Zp[z] ,=H (BS; Zp). ADJOINT SPACES AND FLAG VARIETIES OF p-COMPACT GROUPS 5 In this case, the Becker-Gottlieb transfer is nothing but the usual transfer fo* *r finite coverings, therefore i ffi __ois multiplication by jWj = l 2 Zp . Setting I = l* * 1i: P ! S0[LpBN], we thus get orthogonal idempotents in [P, P]: f = __offi I and e = idP f. Clearly, e ffi __o' , thus __ofactors through fP and induces an isomorphism S0[LpBN] ! fP, in particular a mod-p homology isomorphism between S0[BN] and fP. The computation of the homology of BN together with (2.3)implies that f = (e0+ el+ + ep 1 l) . Corollary 2.6. Let S, N, W be as above. Then the stable Umkehr map BNn ! BSs ' P factors through fP ! P for some f :P ! P which induces the same morphism p_1_ in homology as Pi=l0e(i+1)l.1The induced map BNn ! fP is a mod-p homology equivalence. Proof.This follows from a similarly simple homological consideration. The S- fibration n is not orientable, thus we have a twisted Thom isomorphism H"n+1(BNn) ,=Hn(BN; H1(S; Zp)) where ss1(BN) = Z/l acts on H1(S; Zp) ,=Zp by multiplication by an lth root of unity. Thus ( Hi(BNn; Zp) = Zp; i j 1 (mod l) 0; otherwise. The factorization (2.4)of __othrough o 0 comult: S0[BN] o! BS ! BS ^BN S0[BS] id^!BS ^ 0 0 eval^id 0 BN S [BS] ^BN S [BS] ! S [BS] simplifies considerably since i is the trivial 1-dimensional fibration over B* *S, and the composition of the three right hand_side_maps is an equivalence. In Prop. 2.5 it was shown that I ffi o = idS0[LpBN], thus the same holds after twisting with n: nIn n idBNn:LpBNn o! BSs ! BSi ! LpBN . n If we denote the composition BSs ! BSi nI! LpBNn by I, overriding its previous meaning, the argument now proceeds as in Prop. 2.5. Using the computation of H (BNn; Zp), we find that LpBNn ' (o ffi I)P, and p_1_ l (o ffi I) = X (e(i+1)l)1 i=0 6 TILMAN BAUER AND NATA`LIA CASTELLANA 3.Framing p-compact flag varieties Before proving Theorem 1.1, we need an alternative description of the Pontrya- gin-Thom construction (1.3)on G/T. Lemma 3.1. The map [G/T] is G-equivariantly homotopic to the map rffl SG incl!BGg o! BTt ' rS0[BT] ! Sr, where BGg, BTt, and o are as in Section 2.2, and all spectra except SG have a t* *rivial G-action. Proof.Applying homotopy G-orbits to (1.3), we get a G-equivariant diagram SG ______//SG ^ D(S0[G/T]),__//_S0[G/T] ^ ST___//ST |incl| || |||| fflffl| o , fflffl| || BGg __________//_BTt__________//_S0[BT] ^_ST___//ST which is commutative by the definition of o [Bau04, Def. 25]. In the proof of Theorem 1.1, certain special subgroups will play an important role. In order to define them we need to recall certain facts about the Weyl gr* *oup of a p-compact group. Dwyer and Wilkerson showed in their ground-breaking paper [DW94 ] that given any connected p-compact group G with maximal torus T, there is an as- sociated Weyl group W(G), which is defined as the group of components of the homotopy discrete space of automorphisms of the fibration BT ! BG. This gen- eralizes the notion of Weyl groups of compact Lie groups; they are canonically subgroups of GL(H1(T)) = GLr(Zp), and they are so-called finite complex reflec- tion groups. This means that they are generated by elements (called reflections or, more classically, pseudo-reflections) that fix hyperplanes in Zrp. The comp* *lete classification of complex reflection groups over C is classical and due to Shep* *hard and Todd [ST54], the refinement to the p-adics is due to Clark and Ewing [CE74 * *]. Call a reflection s 2 W primitive if there is no reflection s02 W of strictly* * larger order such that s = (s0)k for some k. Denote by s 2 W a primitive reflection of minimal order l > 1. Let Ts < T be the fixed point subtorus under s. Since s is primitive, hsi = fw 2 W j wjTs= idTsg. Definition. Given connected p-compact group G and a primitive reflection s 2 W(G) of minimal order l > 1, define Csto be the centralizer of Ts in G. Since G is connected, so is the subgroup Cs[DW95 , Lemma 7.8]. Furthermore, Cs has maximal rank because T < Cs by definition, and the inclusion Cs < G induces the inclusion of Weyl groups hsi < W [DW95 , Thm. 7.6]. Since the Weyl group of Csis Z/l, the quotient of Csby its p-compact center, Cs/Z(Cs), can have rank at most 1. By the (almost trivial) classification of rank-1 p-compact grou* *ps, we find that its rank is equal to 1 and i j (3.2) Cs,= Lp(S1)r 1 LpS2l 1 / , ADJOINT SPACES AND FLAG VARIETIES OF p-COMPACT GROUPS 7 where LpS2l 1is simply Lp SU(2) for l = 2, and the Sullivan group given by i j LpS2l 1= Lp Lp(BS1)hZ/l for p odd, and is a finite central subgroup. Proof of Thm. 1.1.By Lemma 3.1, showing equivariant null-homotopy is equiva- lent to showing that the map proj r h(G/T): BGg o! BTt ! S is null. Note that for any given subgroup H < G of maximal rank, there is a factorization of o through BHh. In particular, we may assume that G is connecte* *d. By the dimension hypothesis of the theorem, W(G) is nontrivial. If H = Cs is the subgroup associated to a primitive reflection s 2 W(G) of minimal order l > 1, then the map h(Cs/T) is the (r 1)-fold suspension of h(LpS2l 1/S) by (3.2). Therefore, it is enough to prove the theorem for those p-compact groups Cs. We distinguish two cases. First suppose that l = 2. By the classification of complex reflection groups [ST54], and with the terminology of that paper, this is always the case except * *when W is a product of any number of groups from the list fG4, G5, G16, G18, G20, G25, G32g. This comment is only meant to intimidate the reader and is insubstantial for wh* *at follows. In this case, the map h(Lp SU(2)/S) is null by Pittie [Pit75, PS75] since the spaces involved are Lie groups, thus h(G/T) ' . Now suppose that l > 2. This forces p > 2 as well, and since hsi acts faithfu* *lly on some line in H1(T; Zp) while fixing the complementary hyperplane, we must have that it acts by an lth root of unity, and thus l j p 1. The proof is fin* *ished if we can show that h(LpS2l 1/S) = 0 where S is the 1-dimensional maximal torus in the Sullivan group G0:= LpS2l 1. To see this, note that the inclusion S ! G0 factors through the maximal torus normalizer NG0(S) ,=S o Z/l, and thus 0 o1 o2 h(G0/S): BG0g ! BNn ! S0[BS] ! S1. Wp 2 If P ' i=0 eiP is any stable splitting of the p-completed complex projective s* *pace P = S0[BS] induced by idempotents eias in the previous section, then the right- most projection map clearly factors through e0P, which is the part containing t* *he bottom cell. Since p > 2, Corollary 2.6 shows that there is an idempotent f 2 [* *P, P] such that o ' f ffi o and o ffi e0 = e0ffi o = 0, proving the theorem. 4.The adjoint representation Let G be a d-dimensional connected p-compact group with maximal torus T of rank r. Choose a set fs1, : :,:sr0g of generating reflections of W = W(G) with * *r0 minimal. The classification of pseudo-reflection groups [ST54, CE74] implies th* *at for G semisimple, most of the time r = r0, but there are cases where r0= r + 1. 8 TILMAN BAUER AND NATA`LIA CASTELLANA Example 4.1 (The group no. 7). Let p j 1 (mod 12). Let G7 be the finite group generated by the reflection s of order 2 and the two reflections t, u of order* * 3, where s, t, u 2 GL2(Zp) are given by ` ' ` 7 ' ` 7 7 ' s = 10 01 , t = _1_p_ i i7 , u = _1_p_ i i . 2 i i 2 i i Here i is a 24th primitive root of unity. Note that although possibly i 62 Zp, _1_p_i 2 Z . In Shephard and Todd's classification, this is the restriction to * *Z of the 2 p * * p complex pseudo-reflection group no. 7. They show that even over C, G7 cannot be generated by two reflections. The associated p-compact group is given by Lp((BT2)hG7). If G is not semisimple (i. e. it contains a nontrivial normal torus subgroup)* *, then r0may be smaller than r. Set ~ = r + 1 r0~ 0. Let Ir0be the set of proper subsets of f1, : :,:r0g, and for I ` f1, : :,:r0g* *, let TI be the fixed point subtorus Thsiji2Iiand CI = CG(TI) be the centralizer in G, w* *hich is connected by [DW95 , Lemma 7.8]. Definition. Let G be a connected p-compact group. Define the adjoint space AG by the homotopy colimit AG = ~hocolim G/CI I2Ir with the induced left G-action, and the trivial G-action on the suspension coor* *di- nates. Theorem 1.4 shows that if G is a the p-completion of a connected, semisimple Lie group (in this case r = r0and ~ = 1), then AG is a d-dimensional sphere G- equivariantly homotopy equivalent to g [ f1g. This holds more generally: if G is a connected, compact Lie group with maximal normal torus Tk then AG ,= kAG/Tk = (t [ f1g) ^ (g/t [ f1g) = g [ f1g. Lemma 4.2. Let Irbe the poset category of proper subsets of f1, : :,:rg and F, G :Ik ! ffinite CW-complexes or finite CW-spectrag. be two functors. Then (1)If F is has the property that dimF(;) > dim F(I) for every I 6= ;, then dimhocolim F = dim F(;) + k 1. (2)If f :F ! G is a natural transformation of two such functors such that ,= f (;): HdimF(;)(F(;)) ! HdimG(;)(G(;)), then f induces an isomorphism hocolim f : HdimhocolimF(hocolimF) ! HdimhocolimG(hocolimG). (3)Let F :Ir! Topbe the functor given by F(;) = Sn, F(I) = for I 6= ;. Then hocolimIrF ' Sn+r 1. ADJOINT SPACES AND FLAG VARIETIES OF p-COMPACT GROUPS 9 Proof.The first two assertions follow from the Mayer-Vietoris spectral sequence [BK72 , Chapter XII.5], M E1p,q= Hq(F(I)) =) Hp+q(hocolimF), I2Ik, jIj=k 1 p along with the observation that under the dimension assumptions of (1), E1p,q= 0 for q ~ dim F(;) except for E1k 1,dimF(;)= HdimF(;)(F(;)). In particular, this group cannot support a nonzero differential and thus Hi(F(;)) ,=Hi+k 1(hocolimF) fori ~ dim F(;). The third one is an immediate consequence of the Mayer-Vietoris spectral se- quence. Corollary 4.3. For any connected p-compact group G, AG is a d-dimensional G-spa* *ce. Proof.This follows from Lemma 4.2. Indeed, since any CI (I 6= ;) is connected and has the nontrivial Weyl group WI, its dimension is greater than dim T. So t* *he condition dim F(;) = dim G/T > dim F(I) is satisfied, and dimhocolim F = d r + r0 1 = d ~. As mentioned at the end of the introduction, for p-compact groups G, AG is not usually a sphere, as the following example illustrates. Example 4.4. Let p ~ 5 be a prime, and letiG = S2pj3 be the Sullivan sphere, whose group structure is given by BG = Lp BShCp 1, where Cp 1 ` Zp acts on BS = K(Zp, 2) by multiplication on Zp. Clearly, G has rank 1, and I1 consists o* *nly of a point, thus AG = G/T ' Lp CPp 2. Since p ~ 5, this is not a sphere. For the proof of Theorem 1.5 we need a preparatory result. Proposition 4.5. Let P be a p-compact subgroup of maximal rank in a p-compact g* *roup G. Denote by T a maximal torus of P (and thus also of G). Then the following co* *mposition is G-equivariantly null-homotopic: fG,P:SG ^ DST ! S0[G/T] ! S0[G/P]. The second map is the canonical projection, whereas the first map is given by u* *sing the duality isomorphism SG ^ DST ! SG ^ D(S0[G/T]) ^ DST ,! G+ ^T ST ^ DST ' S0[G/T] ^ ST ^ DST id^ev!S0[G/T]. Proof.In [Bau04, Cor. 24] it was shown that the relative duality isomorphism fr* *om Lemma 1.2 is natural in the sense that the following diagram commutes: (S0[G])hPop_,__//G+ ^P SP___,//_D(S0[G/P]) ^ SG |res| |D(proj)^id| fflffl|op, , fflffl| (S0[G])hT _____//G+ ^T ST____//D(S0[G/T]) ^ SG 10 TILMAN BAUER AND NATA`LIA CASTELLANA Taking duals and smashing with DSG, we find that the map of the proposition is the left hand column in the diagram SG ^ D(G+O^POSP)___,____//S0[G/P]OO | | | | | , | SG ^ D(G+O^TOST) _______//_S0[G/T] , || | SG ^ D(S0[G/T])O^ODST | | | SG ^ DST Thus we need to show that the composition G+ ^P SP ! G+ ^T ST ' S0[G/T] ^ ST ! ST is G-equivariantly trivial, or equivalently, that SP ! P+ ^T ST ! ST is P-equivariantly trivial. But this map is exactly the homotopy class represen* *ted by [P/T], thus the assertion follows from Theorem 1.1. Proof of Thm. 1.5.Let G be a connected p-compact group whose Weyl group is gen- erated by a minimal set of r0reflections. Let F, A: Ir0! Topbe the functors giv* *en by F(;) = SG ^ DST, F(I) = for I 6= ;, and A(I) = G/CI. Note that, since G is connected, CG(T) = T [DW94 , Proposition 9.1] and A(;) = G/T. There is a map : F ! A of Ir0-diagrams in the homotopy category of G-spectra which is fully described by defining (;) = fG,T:F(;) = SG ^ DST ! S0[G/T] as the map given in Prop. 4.5. The strategy of the proof is to obtain a functor F :Ir0! Top such that F(;) = SG ^ DST, F(I) ' for I 6= ;, and a map of Ir0-diagrams : F ! A in the category of G-spectra such that (;) = fG,T. From this we get a G-equivariant map 0 1 ~ ~ 0 0 SG ' S~^ r SG ^ DST ' S ^ hocolimF ! hocolim S [G/CI] ' S [AG], Ir0 Ir0 which will give us the splitting. We will proceed by induction on the number of generating reflections r0. If r0= 1 then AG is S~^ G/T and (;) = S~^ fG,T. We can construct the functor F and the natural transformation step by step. Fix a subset I of cardinality k,* * and assume that F and have been defined for all vertices in the diagram correspon* *d- ing to I0with jI0j < k. Let P(I) be the poset category of all proper subsets of I. Since F and are defined over P(I) by induction hypothesis, we can consider hocolimP(I)F ' k 1SG ^ DST ! S0[G/CI]. It is enough to show that this map is G-equivariantly nullhomotopic. Then, we can fix a null-homotopy and extend the map to the cone ADJOINT SPACES AND FLAG VARIETIES OF p-COMPACT GROUPS 11 of hocolimP(I)F. Finally, we define F(I) = C(hocolimP(I)F) and (I) is the cor- responding extension. Note that SG ^ DST ! S0[G/T] factors through G+ ^CISCI^ DST. By induc- tion, we know there is a map k 1SCI^ DST ! S0[hocolimCI /CJ], J2Ik which splits the top cell. We get a factorization (4.6) k 1SG ^ DST ! k 1G+ ^CISCI^ DST ! G+ ^CIS0[hocolimCI /CJ] ! G+ ^CIS0. J2Ik It thus suffices to show that in the Ik-diagram SCI^ DST _________////_//_f gJ2Ikhf;gocolim//_ k 1SCI^ DST | | | | | | fflffl| // fflffl| hocolim fflffl| S0[CI/T] _____//__//_fS0[CI/CJ]gJ2Ik_f;g//_S0[hocolimJ2IkCI/CJ] | | | | | | fflffl|______//_ fflffl| hocolim fflffl| S0 ___________//_//_fS0gI2Ik_f;g___________//_S0 the right hand side composition k 1SCI^ DST ! S0 is CI-equivariantly null- homotopic. In the latter diagram, it makes no difference whether the centralize* *rs are taken in CIor in G. But by Theorem 1.1, the left hand column is already nul* *l- homotopic, thus, as a colimit of null-homotopic maps over a contractible diagra* *m, so is the right hand column. Conclusion and questions. In this paper, we have compared two imperfect no- tions of adjoint representations of a p-compact group G. One (SG) is a sphere, * *but has a G-action only stably; the other (AG) is an unstable G-space, but fails to* * be a sphere. 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Pittie and Larry Smith. Generalized flag manifolds as framed bo* *undaries. Math. Z., 142:191-193, 1975. [Smi74]Larry Smith. Framings of sphere bundles over spheres, the plumbing pairi* *ng, and the framed bordism classes of rank two simple Lie groups. Topology, 13:401-4* *15, 1974. [ST54] G. C. Shephard and J. A. Todd. Finite unitary reflection groups. Canadia* *n J. Math., 6:274-304, 1954. [Woo76]R. M. W. Wood. Framing the exceptional Lie group G2. Topology, 15(4):303* *-320, 1976. Institut f"ur Mathematik Westf"alische Wilhelms-Universit"at M"unster Einsteinstr. 62 48149 M"unster, Germany E-mail address: tbauer@math.uni-muenster.de Departament de Matem`atiques Universitat Aut`onoma de Barcelona 08193 Bellaterra, Spain E-mail address: natalia@mat.uab.es