is stubborn in
U(n) iff P = P 0\ SU(n) for some p-stubborn subgroup P 0 U(n).
Proof. Everything is proved in [O], but the third statement. That follows from *
*the
first two properties.
Remark. This theorem describes standard models for every conjugacy class of
p-stubborn subgroups in U(n) and SU(n).
6.3 Proposition ([J-M-O 1, proposition 1.6]).
(1) For connected Lie groups G and H, Rp(G x H) ~=Rp(G) x Rp(H).
(2) For a finite covering A- ! eG-! G of compact connected Lie groups, we
have Rp(Ge) ~=Rp(G). In particular, eP eGis p-stubborn iff
P = eP=(Pe\ A) G is p-stubborn.
(3) If P G is p-stubborn then CG (P ) = Z(P ).
Information about centralizers of p-toral subgroups of mod-p subgroups is pr*
*o-
vided by the next lemma.
24
__
6.4 Lemma. Let ae : G- ! G be a mod-p isomorphism of compact connected Lie
groups.
__ __
(1) For any p-toral subgroup P G_ there exists a p-toral subgroup P G,
which is mod-p isomorphic_to P .
(2) CG (P )- ! C__G(P_) is a mod-p isomorphism.
(3) Z(P )- ! Z(P ) is a mod-p isomorphism.
__
Proof. Q := ae-1 (P ) is toral,_i.e a finite extension of a torus. The p-toral*
* Sylow
subgroup P of Q maps_onto P .
CG (P )- ! C__G(P ) is a mod-p subgroup._ Let l be the order of the kernel o*
*f ae,
let x 2 P , and let g 2 ae-1 (C__G(P )). Then gxlg-1 = (gxg-1 )l = (xa)l = xl, *
*where
a 2 ker(ae). The map P -!_P : x 7! xl is a surjection because l is coprime to *
*p.
That is CG (P )- ! C__G(P ) is a surjection which proves (2).
(3) follows analogously by replacing G by P .
The following lemma is the key to get control over the p-stubborn subgroups *
*of
all p-convenient simple simply connected Lie groups, in particular of the
p-convenient exeptional Lie groups.
6.5 Lemma. Let G be a compact connected Lie group and P G be p-stubborn.
Then P is p-stubborn in CG (Z(P )).
Proof. P is contained in K := CG (Z(P )). The normalizer NG (P ) of P in G nor-
malizes Z(P ) and K. The intersection NG (P ) \ K = NK (P ) is a normal subgroup
in NG (P ). Let Q be the intersection of all p-toral Sylow subgroups of NK (P *
*).
A conjugation by an element of NG (P ) permutes the p-toral Sylow subgroups of
NK (P ). Thus Q is normal in NG (P ). If Q=P is not trivial, i.e P is not stubb*
*orn
in K, then P is also not stubborn in G.
Now we are able to describe the p-stubborn subgroups of the simple simply
connected Lie groups. By im(ae) we denote the image of a homomorphism ae :
H- ! G between groups.
6.6 Proposition. Let P G be a p-stubborn group of a p-convenient simple
simply connected Lie group G.
(1) If G 6= SU(n); G2, there exist a mod-p subgroup
ae : H ~=U(n1)x; :::; xU(nk)- ! G
and a p-stubborn subgroup P 0= P1x; :::; xPk, such that Pi is irreducib*
*le
in U(ni), im(ae) = CG (Z(P )), and im(ae|P0) = P .
(2) For G = G2 and p = 3, P is stubborn in SU(3), and Z(P ) = Z(SU(3)).
For G = G2 and p 5 we have P = TG2 .
Proof. Let ae : K-_!_G be a mod-p subgroup of G satisfying the conditions of le*
*mma
5.2 (2), and let K =_im(ae)._ Since the Weylgroup index [WG : WK ] is coprime*
* to
p, the groups G and K have isomorphic and G and K mod-p isomorphic p-toral
25
___
Sylow subgroups. Therefore any p-stubborn subgroup P G sits in K up to
conjugation. By lemma 6.4 there exists a p-toral subgroup P 0 K with ae(P 0) = P
and ae(Z(P 0) = Z(P ). In the sequence
Z(P ) = CG (P ) C__K(P ) Z(P )
all_the_groups are equal. The first identity follows from lemma 6.3. We get Z(P*
* )
Z(K ) and CG (Z(P )) = C__K(Z(P )). By lemma 6.5, P is p-stubborn in C__K(Z(P )*
*),
and by proposition 6.3, P 0is p-stubborn in CK (Z(P 0)).
For G 6= SU(n); G2, the group K is a product of unitary groups. The centrali*
*zer
of any subgroup of such a group always is of the same type. Thus CK (Z(P 0)) ~=
U(n1) x ::: x U(nr). By theorem 6.2 we have P 0= P1x; :::; xPr and Pi U(ni) is
irreducible p-stubborn because CU(ni)(Z(Pi)) = U(ni).
For G = G2 and p 5_we_have P = TG2 because (p; |WG2 |) = 1. For G = G2
and p = 3 we have K = K = SU(3). If CG2 (Z(P )) = CSU(3)(Z(P )) 6= SU(3) this
centralizer is isomorphic to U(2) or TG2 = TSU(3). In both cases P = TG = TSU(3*
*),
because |WCG2(Z(P))| 2. This is a contradiction because WG2 has a normal
subgroup of order 3. Hence P is p-stubborn in SU(3).
Theorem 6.2 finishes the proof.
For the proof of theorem 1.2, we need certain properties of p-stubborn group*
*s.
With the help of the last proposition we can do this for a special class of com*
*pact
connected Lie group, namely pseudo simply connected Lie groups.
6.7 Proposition. Let G be a p-convenient pseudo simply connected Lie groupand
TG ,!G a fixed maximal torus. In the conjugacy class (P ) of any p-stubborn sub-
group of G exists a representative i : P ,!N(TG ),! G satisfying the following *
*con-
ditions:
(1) PT := P \ TG = S x V , where S is a torus and V an elementary abelian
p-group, and Z(P ) PT .
(2) CG (PT ) = TG .
(3) For any extension ff : P -! G of i : PT -! G, we have CG (ff) = i(Z(P )*
*).
(4) The canonical map
ss0(map(BP; BG^p)Bff|BPT=Bi ) -! Hom(H*(BG; Z=p); H*(BP ; Z=p))
is an injection.
Remark. By map(BP; BG^p)Bff|BPT =Biwe denote the components of
map(BP; BG^p) given by maps Bff : BP -! BG^p, such that Bff|BPT ' Bi. The
homotopy classes of these extensions are classified by obstructions in the grou*
*ps
H*(BP=PT ; ss*(map(BPT ; BG^p)B i) (lemma 3.13). By (1) of the above theorem
follows that there is only one obstruction group, namely
H2(BP=PT ; ss2(map(BPT ; BG^p)B i)) ~=H*(BP=PT ; ss2(BTG ^p)).
In the rest of this chapter we will prove this proposition in several steps.
26
Step 1. Let G = U(pk) and P = Upk.
Proof of (1) and (2). For abbreviation we set := Upk. The condition (1) follo*
*ws
from remark 6.1, and T ~= (Z=p)k x S1. The j-th coordinate is generated by Aj-*
*1,
and the last coordinate is the the center of U(pk). The inclusion i : T -! TU(*
*pk) is
a splitting in 1-dimensional representations of the pk-dimensional representati*
*on
-! U(pk). The j-th summand is given by the homomorphism ij : T -! S1 defined
by Y s
ij(a1; :::; ak; ) = i[(j-1)=p ].as. ; i = exp(2ssi=p) :
s
All these 1-dimensional representation are pairwise nonisomorphic (see the next
lemma), which implies that CU(pk)( T) = TU(pk).
6.8 Lemma. The two homorphism ij; il : T -! S1 are equal iff j = l.
Proof. If ij = il we have for all (a1; :::; ak) 2 (Z=p)k
Yk s-1 Yk s-1
i[(j-1)=p ].as= i[(l-1)=p ].as
s=1 s=1
respectively
Xk Xk
[(j - 1)=ps-1] . as [(l - 1)=ps-1] . as mod p :
s=1 s=1
This condition is satisfied iff [(j - 1)=ps-1] [(l - 1)=ps-1] mod p for 1 s *
*k.
This implies j = l, because j; l k.
Proof of (3) and (4). Extensions Bff : B -! BU(pk)^pof Bi : B T- ! BU(pk) are
classified by obstructions in the groups Hr( = T; ssr(map(B T; BU(pk)^p)) (s*
*ee
lemma 3.12). Because of theorem 3.9 and (1), there is only one obstruction group
H2( = T; ss2(BTU(pk))) ~= H2( = T; (Z^p)pk). Here = T ~= Epk ~= (Z=p)pk*
* acts
on Z^ppkvia the inclusion Epk,! pk and the permutation representation. In par-
ticular = T acts transitively on the standard basis of Z^ppk. The canonical *
*map
Z^p[ = T]- ! Z^ppk: 1 7! e1 is an isomorphism, where e1 is the first basis ve*
*ctor of
the standard basis. Hence the obstruction group H2( = T; (Z^p)pk) vanishes a*
*nd
the two extensions ff and fi are conjugate. This completes the proof of (3) and*
* (4)
as well as of step 1.
Step 2. Let P G = SU(p) be a standard p-stubborn subgroup.
Proof. By theorem 6.2 the only p-stubborn subgroups of SU(p) are TSU(p) (for
p 5), P1 = Up \ SU(p) and P2 = (S1 o Z=p) \ SU(p). We have P1T =
(Z=p)2 because UpT = S1 x Z=p and because CSU(p)(P1T ) = TSU(p). For P2
we get P2T = TSU(p) and CSU(p)(P2T ) = TSU(p). To calculate the extensions of
27
BPiT- ! BSU(p) we have only to consider the obstruction group H2(Z=p; Z^pp-1),
where Z=p acts on Z^pp-1 via the action of p on ss2(BTSU(p)). The exact sequence
Z^pp-1-! Z^pp-! Z^pof Z=p-modules induces a long exact sequence
::: -! H1(Z=p; Z^p) -! H2(Z=p; Z^pp-1) -! H2(Z=p; Z^pp) -! ::: :
The left and the right term vanish. Hence there exists only one extension and
CSU(p)(Pi) = Z(Pi). For the p-stubborn subgroup TSU(p) all the statements are
obvious.
For the next step, namely the case of an arbitrary p-stubborn subgroup of U(*
*pk),
we need some technical lemmas.
6.9 Lemma. Let P U(pk) be a standard irreducible p-stubborn subgroup. Then
PT is normal in P , the qoutient P=PT is an iterated wreath product of elementa*
*ry
abelian p-groups, and the homomorphism P -! P=PT splits.
Proof. By theorem 6.2, P = Upr0o Epr1o :::: o Eprs and
P=PT = Upr0=Upr0To Epr1o :::: o Eprs which is an iterated wreath product of ele*
*men-
tary abelian p-groups. The homomorphism Upr0-! Upr0=Upr0Tsplits by remark
6.1. Thus P -! P=PT splits.
6.10 Lemma. Let ff : Q- ! G be a homomorphism of an iterated wreath product
of elementary abelian p-groups into a compact Lie group. If H*(Bff; Z=p) = 0,
then ff is constant.
Proof. By theorem 3.1 the statement is true for elementary abelian p-groups. Now
let Q = Q0 o (Z=p)k. By induction hypothesis the homomorphism
ff|Q0x:::xQ0 : Q0 x ::: x Q0- ! G is constant restricted to one factor, and hen*
*ce
constant on the product. ff splits over __ff: (Z=p)k- ! G which is trivial too,*
* because
H*(B__ff; Z=p) = 0.
Step 3. P U(pk) is a standard irreducible p-stubborn subgroup.
Proof. By proposition 6.2, P = P 0o Epr, where P 0 U(pk-r ) isralso a stan-
dard irreducible p-stubborn subgroup. Therefore PT = (PT0)p is a product of a
torus and an elementary abelian p-group. Sincerthe splitting of PT into the pro*
*d-
uct comes from the homomorphism U(pk-r )p - ! U(pk), we have CU(pk)(PT ) =
(CU(pk-r)(PT0))pr = (TU(pk-r))pr = TU(pk) by induction hypothesis. This proves
the first two parts.
Now let ff; fi : P -! U(pk) be two homomorphisms, such that H*(Bff; Z=p) =
H*(Bfi; Z=p) and Bff|BPT = Bfi|BPT = Bi|BPT , where i : P -! U(pk) is the sta*
*n-
dard inclusion. Let r
ff0; fi0 : (P 0)p - ! U(pk)
be the restrictions of ff and fi. r
We get Z(P 0) = CU(pk-r)(P 0) = Z(U(pk-r )) = S1 and (U(pk-r ))p = CU(pk)(V*
* ),
28
r r
where V (S1)p = (Z(P 0))p PT is the maximal elementary abelian p-
subgroup. The homorphism ff0 and fi0 split over
r k-r pr
ff00; fi00: P 0p-! (U(p )) :
The induced maps in cohomology are determined by the application of the Lannes
functor TBVi*(remark 3.4), in particular H*(Bff00) = H*(Bfi00).
The homomorphism ff00can be described by a (pr x pr)-matrix with homo-
morphisms ffij : Pi0-! U(pk-r )j as entries. The indices i; j denote the compo-
nents in the products. Analogously fi00is described by a matrix B with entries
fii;j: Pi0-! U(pk-r )j. By induction hypothesis Bffii ' Bfiii, i.e ffii and fi*
*ii are
conjugate (theorem 3.6), and
CU(pk-r)i(ffii) = CU(pk-r)i(fiii) = Z(Pi0) = Z(U(pk-r )i) = S1i:
We can assume that ffii= fiii. Because Pi0and Pj0commute, the homomorphisms
ffij; fiij split over the inclusion S1j= Z(U(pk-r )j)- ! U(pk-r )j. Since ff00*
*|PT =
fi00|PT , the homomorphisms ffij. fiij-1 : Pi0-! S1jfactor over homomorphisms a*
*eij:
(P 0=PT0)i-! S1j. and fit together to a map ae : (P 0=PT0)pr- ! (S1)pr. fi00*
*can be
described by the composition
r rff00xae r r r
(P 0)pr-! (P 0)p x (P 0=PT0)p---! U(pk-r )p x (S1)p -! U(pk-r )p ;
where is the diagonal composed with taking the quotient, and where is given
by the canonical map S1 x U(pk-r )- ! U(pk-r ). Because H*(Bff00) = H*(Bfi00)
and because aeiiis constant, the composition
Bae* r r
H*(BU(pk-r )pr) ----! H*((BP 0=PT0)p ) ----! H*(BP 0p)
is trivial. By lemma 6.9 and lemma 6.10 the homorphism ae is constant.r Thatris
that ff00= fi00, and CU(pk)(ff0) = CU(pk-r)pr(ff00) = (Z(U(pk-r ))p = (S1)p .
The extensions Bff : BP = BP 0o Epr- ! BU(pk)^p of Bff0: BP 0pr-! BU(pk)^p
are classified by the obstructions in the groups
r 0pr k ^
H*(P=(P 0p); ss*(map(BP ; BU(p )p )Bff0^p)
(lemma 3.13). By theorem 3.8
r k ^ 0 ^
map(BP 0p; BU(p )p )Bff0 = BCU(pk)(ff )p
= BCU(pk-r)pr(ff00)^p
r
= (BZ(U(pk-r )^p))p
r
= (BS1^p)p :
The only obstruction group H2(Epr; Z^ppr) vanishes because Z^ppris a free
Epr-module (see the proof of step 1). There exists one extension ff of ff0, and*
* the
centralizer CU(pk)(P ) is given by the fixed-point set
r E r 1pr E r 1
CU(pk)(P ) = (CU(pk-r)pr(P 0p)) p = (S ) p = S = Z(P ) :
29
Step 4. Let P G be a p-stubborn subgroup of a pseudo simply connected Lie
group.
Proof. According to proposition 6.6 we choose a mod-p subgroup
ae : H = H1 x ::: x Hs- ! G
and a p-stubborn subgroup P 0= P1 x ::: x Ps, such that Hi is isomorphic to a
unitary group or to SU(p), Pi Hi is p-stubborn,_Pi Hi is irreducible if Hi is
a unitary group, ae(P 0) = P_,_and H- ! H := CG (ae|V ) is a mod-p isomorphism*
* of
groups, where V Z(P ) = Z(H ) is the maximal elementary abelian p-subgroup.
Up to conjugacy Pi Hi is the standard model of theorem 6.2. PT is a product of
a torus and an elementary abelian p-group, which is the statement of (1).
Because PT Z(P ) V , step 3 induces the sequence
Y Y
TH = THi = CHi (Pi) = CH (PT0) -! C__H(PT ) = CG (PT )
i i
which is a mod-p isomorphism by lemma 6.4. Thus, CG (PT ) = TG as required in
(2).
Let ffG ; fiG : P -! G be two homomorphisms such that
H*(BffG ; Z=p) = H*(BfiG ; Z=p) and ffG |PT = fiG |PT = i|PT :
___
Both, ffG and fiG , split over homomorphisms ff__H; fi__H: P -! H = CG (V ). A*
*pplying
the Lannes functor TBVi*we get H*(Bff__H) = H*(Bfi__H) (remark 3.4). By lemma 6*
*.4
there exist lifts ffH ; fiH : P 0-! H of ff__Hand fi__H. Of course H*(Bff__H) *
*= H*(BffH )
and H*(Bfi__H) = H*(BfiH ). Now we are in a position to apply the methods of st*
*ep
3. This implies BffH ' BfiH , Bff__H' Bfi__H, and hence BffG ' BfiG . This pro*
*ves
(4). The condition (3) is satisfied, because
Z(P 0) = CH (ffH ) -! C__H(ff__H) = CG (ffG )
and
Z(P 0) -! Z(P )
are mod-p isomorphisms by lemma 6.4 again and step 3. This finishes the proof of
step 4 as well as the proof of proposition 6.7.
7. Maximal tori and Weyl groups.
In [R] the concept of maximal tori and Weyl groups of a finite loop space is*
* intro-
duced (see also [NS 1,2,3], [R-S]). An algebraic version of this homotopy theor*
*etical
definition is given in [D-W]. For our purpose the following definition is conve*
*nient.
30
7.1 Definition. A maximal torus of a space X with the mod-p type of BG is a
map
fT : BTX ----! X or fT ^p: BTX ^p----! X ;
where TX is a torus, such that
(1) rank TX = rank G
(2) H*(BTX ; Z=p) is a finite generated module over H*(X; Z=p).
The Weylgroup of X is the group
^ ^ ^ ^
WX := [w] : BTX p- ! BTX p : fT pO w ' fT p
of homotopy classes of self maps of BTX .
The definition of the Weyl group might depend on the chosen maximal torus
fT : BTX -! X. If p is odd or G is 2-convenient, similar methods as in the pro*
*of
of [N-S 1; proposition 2.4] can be applied to show that any two maximal tori
f0T; f00T: BTX----! X
fit into a commutative diagram
g
BTX ^p ----! BTX ^p
? ?
f0T?y ?yf00T
X ________ X ;
where g is an equivalence. Therefore WX is well defined up to `conjugation'.
Following ideas of Dwyer, Miller, and Wilkerson [D-M-W 2] we will construct
a maximal torus of X. This construction we recall in detail because later we u*
*se
similar methods in more complicated situations.
Now we assume that G satisfies the condition H*(BG; Z=p) ~=H*(BTG ; Z=p)WG .
Let X be a p-complete space with the mod-p type of BG. By theorem 3.1 the
composition
H*(X; Z=p)- OE!H*(BG; Z=p)- ! H*(BTG ; Z=p)- ! H*(BVG ; Z=p) :
has a topological realization fV : BVG -! X. Applying the Lannes-functor we get
the diagram
~=
TfVG*VH*(X; Z=p) -! TBVGi*VH*(BG; Z=p)
?? ?
y ?y
~=
H*(map(BVG ; X)fV ; Z=p)) H*(map(BVG ; BG)BiV ) -! H*(BTG ; Z=p) :
The last isomorphism comes from theorem 3.5 and lemma 4.5. The upper*horizontal
arrow is an isomorphism. The space BTG is 1-connected. Thus, TVfVGH*(X; Z=p)
31
vanishes in degree 1, and the left vertical arrow is also an isomorphism (theor*
*em
3.2). The mapping space map(BVG ; X)fV is p-complete by theorem 3.3. Among
the p-complete spaces, BTG ^pis determined up to homotopy by its mod-p cohomol-
ogy. The evaluation map e : BTX ^p:= map(BVG ; X)fV -! X, defined by choosing
a suitable base point in BVG , induces a map fT : BTX ^p-! X which plays the ro*
*le
of a maximal torus of X. Here TX is a torus of the same rank as TG .
The composition BVG x BVG -! BVG -! X of the multiplication of the H-space
BVG and fV has as adjoint the map
Bj : BVG -! map(BVG ; X)fV = BTX :
There exists an equivalence BTG ^p-! BTX ^pwhich fits into the diagram
BVG == BVG
# #
BTG -! BTX ^p
# #
BG 99K X ;
where the upper square commutes up to homotopy and the lower square in mod-p
cohomology. The dotted arrow means a map which only exists in cohomology, in
this case in mod-p cohomology.
The Weyl group WG acts on BVG and hence on map(BVG ; X). The action on
the components of map(BVG ; X) is determined by pure cohomological properties,
the action on HomAp (H*(X; Z=p); H*(BVG ; Z=p)) (theorem 3.1). Up to homotopy
WG fixes the map Bi : BVG -! BG and therefore the component map(BVG ; X)fV
of map(BVG ; X). We get an action of WG on BTX ^psuch that fT O w ' fT for
every w 2 WG . All the spaces in the above diagram carry an WG -action ( BG and
X the trivial action), and the maps induce equivariant maps in mod-p cohomology.
We collect the results in the following proposition
7.2 Proposition. Let G be a compact connected Lie group, and let X be a p-
complete space with the mod-p type of BG. If H*(BG; Z=p) ~=H*(BTG ; Z=p)WG
then there exist a maximal torus fT : BTX ^p-! X, a Weyl group action of WG on
BTX , and a mod-p equivalence BTG -! BTX ^p, equivariant in mod-p cohomology,
such that the diagram
BTG -! BTX ^p
# #
BG 99K X
commutes in mod-p cohomology. Moreover, H*(X; Z^p) ~=H*(BTX ; Z^p)WG .
Proof. Only the last statement needs a comment. It follows from the isomorphism
H*(X; Z=p) ~=H*(BTX ; Z=p)WG and the Nakayama lemma.
32
Remark. All but the last statement is due to Dwyer, Miller and Wilkerson and
is true under the weaker assumption CG (VG ) = TG , in particular, for all comp*
*act
connected Lie groups at odd primes [D-M-W 2].
To prove theorem 1.1 we have to study the WG -modules
L*TX ^p= H2(BTX ; Z^p) and L*TG ^p= H2(BTG ; Z^p) :
7.3 Definition. A Z^p[WG ]-module L is called p-reducible if
P (L)WG Z=p ~=P (L Z=p)WG ;
where P (-) denotes the symmetric part of the tensor algebra.
Proof of theorem 1.1.
The proof is based on results of the sections 8 and 9. By proposition 7.2 the m*
*odules
L*TX ^pand L*TG ^pare p-reducible, and there mod-p reduction are isomorphic as
Z=p[WG ]-modules. Let G be p-convenient.
For p 6= 2 and G simply connected or pseudo simply connected, for p = 2 and
G simply connected, and for (p; |WG |) = 1 we can apply proposition 8.1. If G *
*is
pseudo projective we can use proposition 9.5, and if G is a product of unitary *
*groups
we can apply proposition 9.9. This proves that in these cases L*TG ^p~=L*TX ^pas
Z^p[WG ]-modules.
In the general case there exists a compact connected Lie group H, such that *
*BH
has the mod-p type of BG and X, and such that L*TH ^p~=L*TX ^p. This follows
from proposition 9.2.
We can identify BTG ^p, respectively BTH ^p, and BTX ^pvia this map. This pr*
*oves
that X has the p-adic type of BG respectively BH. By the Nakayama lemma
follows that H*(X; Z^p) ~=H*(BTH ; Z^p)WG . .
8. p-adic Weyl group representations, for simply connected Lie groups.
The action of the Weyl group WG on the maximal torus BTG of a compact
connected Lie group G gives a Z^p[WG ]-module L*TG ^p:= H2(BTG ; Z^p). In this
chapter we will study the p-adic liftings of the associated mod-p representation
H2(BTG ; Z=p) for a simply connected Lie group G. The general case is discussed
in the next chapter.
Definitions and Notation. A Z^p[WG ]-module L is called simply connected, if
LWG = 0. The module LZ=pk is denoted by L=pk, and L=pk* = Hom(L=pk; Z=pk)
is the dual module of L=pk as a Z=pk[WG ]-module. Gl(L=pk) denotes the isomor-
phisms and M(L=pk) the endomorphisms of L=pk. The symmetric part of the
tensor algebra of L is given by P (L).
33
8.1 Proposition. Let G be a compact connected Lie groupand L be a torsion free
Z^p[WG ]-module. Let
__ff: L*T
G =p----! L=p
be a Z^p[WG ]-isomorphism. Then there exists a Z^p[WG ]-isomorphism
ff : L*TG ^p----! L ;
which is a lift of __ff, if one of the following conditions is satisfied:
(1) p is odd, and G is p-convenient and simply connected or pseudo simply
connected.
(2) p = 2, L is p-reducible, and G is p-convenient and simply connected.
(3) (p; |WG |) = 1.
For the proof we need several lemmas.
A Z^p[WG ]-module L, which is free as Z^p-module of rank n, induces a homo-
morphism
aeL : WG ----! Gl(L) ~=Gl(n; Z^p):
We denote by aeL (k) the homorphism associated to the Z=pk[WG ]-module L=pk.
The kernel of the restriction Gl(n; Z=pk+r )- ! Gl(n; Z=pk) is the group
ker := {id + pkA | A 2 M(n; Z=pr)} ;
where M(n; Z=pr) ~=M(L=pr) ~=Hom(L=pr; L=pr) ~=L=pr* L=pr is the group of
n x n matrices over Z=pr. The multiplication in ker is given by
(id + pkA)(id + pkB) = (id + pk(A + B + pkAB)) :
Thus, if r k, the map A 7! id+pk .A induces an isomorphism M(n; Z=pk) ~=ker.
The lifts in the diagram
Gl(n; Z=pk+r )
??
y
aeL(k)
WG ----! Gl(n; Z=pk)
are classified up to conjugation by the obstruction group
H1(WG ; Hom(L=pr; L=pr)) ;
where WG acts on Hom(L=pr; L=pr) via conjugation and the homomorphism
WG aeL(r)-!Gl(n; Z=pr). To prove lemma 8.1 we have to calculate this obstruct*
*ion
group.
Let G and H be two compact connected Lie groups. Then L*TG ^px L*TH ^pas
well as L*TG ^pand L*TH ^pare WG x WH -modules, where WH acts trivially on
L*TG ^pand WG trivially on L*TH ^p.
34
8.2 Lemma. Let G and H be two connected Lie groups.
(1) H1(WG x WH ; Hom(L*TG =pk; L*TG =pk))
~=H1(WG ; Hom(L*TG =pk; L*TG =pk))
H1(WH ; Z=pk) Hom(L*TG =pk; L*TG =pk)WG
If p is odd the second summand vanishes.
(2) H1(WG x WH ; Hom(L*TG =pk; L*TH =pk))
~=(LTG =pk)WG H1(WH ; L*TH =pk)
H1(WG ; LTG =pk) (L*TH =pk)WH :
If G and H are p-convenient and simply connected or if p is odd and G is p-
convenient and pseudo simply connected then both summands vanish.
(3) H1(n; Hom(L*TU(n)=pk; L*TU(n)=pk))
~=H1(n-1 ; Z=pk) H1(n-2 ; Z=pk)
(4) For n 3 we have
H1(n; Hom(L*TSU(n)=pk; L*TSU(n)=pk)) ~=H1(n-2 ; Z=pk) :
(5) If G is p-convenient and pseudo simply connected or simply connected, and
if p is odd,
H1(WG ; Hom(L*TG =pk; L*TG =pk)) = 0 :
(6) If (p; |WG |) = 1, then H1(WG ; Hom(L*TG =pk; L*TG =pk) = 0.
Proof. The Hochschild-Serre spectral sequence and some easy calculations with t*
*he
coefficients establish the two isomorphism of (1) and (2) . If p is odd H1(WH ;*
* Z=pk)
as well as H1(WH ; L*TH =pk) vanish, and H1(WG ; LTG =pk) vanishes if G is pseu*
*do
simply connected. All this follows from lemma 5.3. Because W H is generaterd
by elements of order 2 the first group H1(WH ; Z=pk) vanishes for every compact
connected Lie group. If G and H are simply connected then (L*TH =pk)WH = 0 =
(L*TH =pk)WH . These facts prove the vanishing of the summands which finishes t*
*he
proof of (1) and (2) .
The representation L*TU(n)=pk ~=indnn-1 Z=pk is induced by the 1-dimensional
trivial n-1 -module Z=pk. Therefore
Hom(L*TU(n)=pk; L*TU(n)=pk) ~=indnn-1 Hom(Z=pk; L*TU(n)=pk)
~=indnn-1 (L*TU(n-1) =pk Z=pk) ;
and
H1(n; Hom(L*TU(n)=pk; L*TU(n)=pk))
~=H1(n-1 ; L*TU(n-1) =pk) H1(n-1 ; Z=pk)
~=H1(n-2 ; Z=pk) H1(n-1 ; Z=pk) :
35
The exact sequence
0 --! Z=pk --! L*TU(n)=pk --! L*TSU(n)=pk --! 0
splits as sequence of Z=pk-modules and induces an exact sequence
0 -! Hom(L*TSU(n)=pk; L*TSU(n)=pk) -! Hom(L*TU(n)=pk; L*TSU(n)=pk)
-! Hom(Z=pk; L*TSU(n)=pk) ~=L*TSU(n)=pk -! 0 :
Because of lemma 5.3, the associated long exact sequence in cohomology shows th*
*at
H1(n; Hom(L*TSU(n)=pk; L*TSU(n)=pk))
~=H1(n; Hom(L*TU(n)=pk; L*TSU(n)=pk))
~=H1(n-1 ; Hom(Z=pk; L*TSU(n)=pk))
~=H1(n-1 ; L*TU(n-1) =pk)
~=H1(n-2 ; Z=pk) :
This proves (3) and (4).
To prove (5) we choose by remark 5.2 a mod-p subgroup H- ! G of G, such
that the Weyl group index is coprime to p and such that H is a product of unita*
*ry
groups and special unitary groups. The restriction
H1(WG ; Hom(L*TG =pk; L*TG =pk))!- H1(WH ; Hom(L*TH =pk; L*TH =pk))
is an injection. Let H = H1 x ::: x Hr be the splitting into the factors; i.e *
*Hi ~=
SU(ni), ni 3, or Hi ~=U(ni). If p is odd H1(WHi ; Z=pk) = 0, and, by (1), (2),
(3), and (4),
H1(WH ; Hom(L*TH =pk; L*TH =pk))
~=M H1(WHi ; Hom(L*THi =pk; L*THi =pk))
i
= 0
which finishes the proof of (5).
The statement (6) follows because Hom(L*TG =pk; L*TG =pk) is a Z^p-modul and
(p; |WG |) = 1.
8.3 Lemma. Let G and H be 2-convenient compact connected Lie groups. If G
is simply connected then
H1(WG x WH ; Hom (L*TG =4; L*TG =4))
-! H1(WG x WH ; Hom(L*TG =2; L*TG =2)
is the trivial map.
Proof. The only 2-convenientcompact connected Lie groups are qoutients of a pro*
*d-
uct of SU(n)'s, n 3 and a torus. Hence WG and WH are products of symmetric
groups. Because of the last lemma we have only to show that
H1(n; Z=4)- ! H1(n; Z=2)
is the trivial map. This is obvious, because H1(n; Z) = Z=2.
Now we are in the position to prove proposition 8.1.
36
Proof of proposition 8.1.
Proof. The two Z^p[WG ]-modules L*TG ^pand L induce two homomorphisms
aeL*TG ^p; aeL : WG----!Gl(n; Z^p) :
By assumption
aeL*TG ^p(1); aeL (1) :-WG---!Gl(n; Z=p)
are conjugate. We can assume they are equal. Because the obstruction groups
H1(WG ; Hom(L*TG =p; L*TG =p)) vanish for p odd and for (p; |WG |) = 1 (lemma
8.2), all the lifts
aeL*TG ^p(k); aeL (k) :-WG---!Gl(n; Z^pk)
are conjugate. Thus, aeL*TG ^pand aeL are conjugate. This proves 8.1 under t*
*he
assumption (1) and (3).
Next we consider the case G = SU(n); n 3, and p = 2. We denote by
det : n- ! Z^2the composition of aeL*TSU(n)^pand the determinant. For n 4 the
group H1(n-2 ; Z=2) ~=Z=2, and for n = 3 the group vanishes; i.e. for n = 3 the*
*re
is no obstruction and the lift aeL*TG ^p(2) is unique up to conjugation, and fo*
*r n 4
there are two lifts of aeL*TG ^p(1), given by aeL*TG ^p(2) and aeL*TG ^p(2) de*
*t. But these
two lifts can be distinguished by looking at the invariants
P (L*TSU(n)=4)aeL*TSU(n)^p(2)(n)and P (L*T SU(n)=4)(aeL*TSU(n)^p(2)det)(n);
as a short calculation of the invariants of degree 6 shows. We assumed that L=4*
* is
p-reducible, and hence, L*TSU(n)=4 ~=L=4.
If G is simply connected and 2-convenient G splits_into_factors G = G1x:::xG*
*r,
where_Gi = SU(ni) and ni 3. In this case, L=4 ~=L 1x:::xL rby lemma_8.2, where
L iis a Z=4[WG1 x ::: x WGr ]-module. Because L is 2-reducible, L 1is 2-reducib*
*le
as WG1 x WH -module, where H = G2 x :::Gr. In the commutative diagram
__ ff __
P (L 1)WG1xWH ----! P (L 1)WG
? ?
(*) ?yfi ?yfl
__ ffi __
P (L 1=2)WG1xWH ----! P (L 1=2)WG
ffi and fi Z=2 are isomorphisms,_and fl Z=2 is an injection, hence also an is*
*o-
morphism. This shows that_L_1 is 2-reducible as WG -module. Therefore, by the
above argument, L*TG1 ^p~=L 1 as WG -modules, and ff is an isomorphism by the
Nakayama lemma. __
The Weyl group WH must act on L 1via a homorphism
WH ----! CGL(__L1)(WG ) ~=CGL(L*TG1=4) (WG ) ~=Z=4 :
37
That_is to say that WH acts via scalar multiplication. A non trivial action of*
* WH
on L 1contradicts_the fact that the map ff in the diagram (*) is an isomorphism.
Thus L*TG1 =4 ~=L 1and L*TG =4 ~=L=4 as WG -module.
Now, for an induction argument, we can assume that
L*TG =2k ~=L=2k; aeL*TG ^p(k) = aeL (k) ; and k 2 :
The two lifts aeL*TG ^p(k + 2) and aeL (k + 2) differ by an obstruction in
H1(WG ; Hom(L*TG =4; L*TG =4)). Because the map
H1(WG ; Hom(L*TG =4; L*TG =4)) ----! H1(WG ; Hom(L*TG =2; L*TG =2))
is trivial (lemma 8.3) the two lifts aeL*TG ^p(k + 1) and aeL (k + 1) are conju*
*gate,
i.e. L*TG =2k+1 ~= L=2k+1 and L*TG ^p~=L as Z^p[WG ]-modules. This proves part
(2).
9. p-adic Weyl group representations, for general compact connected
Lie groups.
In this section we use the same notation as in the last one. G denotes in t*
*his
chapter a compact connected Lie group. Uniqueness results about the p-adic lift*
*ings
of the Weyl group representation L*TG =p are not true in general. In remark 9.6
we consrtruct a counterexample. Nevertheless we can describe all torsion free *
*p-
reducible p-adic representations L of WG which are lifts of L*TG =p. The result*
*s to
be applied in section 7 are the propositions 9.5 and 9.9.
__ Any compact connected Lie group G fits into an exact sequence
G s-! G- ! T , where Gs is simply connected and_where T is a torus. If G is p-
convenient Gs is also p-convenient, and Gs- ! G sis a mod-p isomorphism of grou*
*ps
(see section 4). Moreover, L*TGs ^p~=L*T__Gs^pand L*TGs =p ~=L*T__Gs=p.
9.1 Lemma. Let G be a p-convenient compact connected Lie group. Let L be
a torsion free p-reducible Z^p[WG ]-module. If L=p ~= L*TG =p as Z^p[WG ]-modu*
*le
then there is an exact sequence
1 --! L*Tp^ --! L --! L*TGs ^p--! 1
Proof. The qoutient L=LWG is torsion free because L is. Thus, applying the fun*
*ctor
Z=p to the exact sequence
0 --! LWG --! L --! L=LWG --! 0 ;
establishes the diagram
0 --! LWG =p --! L=p --! (L=LWG )=p --! 0
?? ? ?
y ?y ?y
0 --! (L=p)WG --! L=p --! (L=p)=(L=p)WG --! 0
?? ? ?
y~= ?y~= ?y~=
0 --! (L*TG =p)WG --! L*TG =p --! (L*TG =p)=(L*TG =pWG ) --! 0
38
__
of exact rows. The exact sequence G s-! G- ! T establishes the isomorphism
(L*TG =p)WG ~=L*T=p. Because G is p-convenient we have
(L*TG ^p=L*TG ^pWG)=p ~=(L*TG =p)=(L*TG =pWG ) ~=L*T__Gs=p ~=L*TGs =p :
The two lower rows in the diagram are isomorphic by assumption. The two upper
rows also are isomorphic, because L is p-reducible. We have LWG ~=L*Tp^ and, by
proposition 8.1, we get L=LWG ~=L*TGs ^pas Z^p[WG ]-module. That is there exis*
*ts
an exact sequence
0 --! L*Tp^ --! L --! L*TGs ^p--! 0
of Z^p[WG ]-modules as desired.
Remark. If
ExtWG (L*TGs ^p; L*Tp^) ~=H1(WG ; Hom(L*TGs ^p; L*Tp^))
~=H1(WG ; LTGs ^p) L*Tp^= 0 ;
then L ~=L*TG ^pas Z^p[WG ]-modules. Thus, proposition 8.1 is true if (p; |WG |*
*) = 1,
as already shown, or if Gs is a p-convenient simply connected Lie group , which
contains no factor isomorphic to some SU(n), i.e. Gs is also pseudo simply con-
nected.
Next we will study the group of extensions ExtWG (L*TGs ^p; L*Tp^) or, more *
*gen-
erally, the extensions ExtWG (L*TG ^p; L*Tp^) for a compact connected Lie group*
* G
and a torus T . We denote by GrExt(T; G) the set of exact sequences G- ! G0-! T
of compact connected Lie groups dividing out the usual equivalence relations for
extensions. There is a canonical map
L : GrExt(T; G) -! ExtWG (L*TG ^p; L*Tp^)
(G! G0! T ) 7! (L*Tp^! L*TG 0^p!L*TG ^p)
9.2 Proposition. Let G be a compact connected Lie group. If the inclusion
Z(G)- ! TG WG has a cokernel of order coprime to p the map L is a surjection.
Remark. By lemma 4.4 the assumption of 9.2 are satisfied if G is p-convenient.
Proof. Let E : L*Tp^-! L- ! L*TG ^pbe an exact sequence. Because
ExtWG (L*TG ^p; L*Tp^) ~= H1(WG ; Hom(L*TG ^p; L*Tp^)) is a finite group we can
choose an injective self map ff : L*Tp^-! L*Tp^ with finite cokernel K such that
ff*(E) splits. Therefore, there exists a commutative diagram
L*Tp^ --! L --! L*TG ^p
? ? fl
ff?y fi?y flfl
L*Tp^ --! L*Tp^ L*TG ^p --! L*TG ^p
?? ?
y ?y
K ______ K :
39
The equivariant maps L*Tp^-! K ~= H2(BBK; Z^p) and L*Tp^ L*TG ^p-! K ~=
H2(BBK; Z^p) are induced by homomorphisms K- ! T and K- ! T x TG . Because
K is a finite abelian p-group and because H*(BK; Z=p) is a finitely generated
H*(BT ; Z=p)-module, both homomorphisms are injections [Q]. We have K T x
TG WG , and hence K maps into Z(G) T WG . The compact connected Lie group
G0:= (T x G)=K fits into an exact sequence G- ! G0-! T . By construction we get
an isomorphism of extensions, namely
L*Tp^ --! L --! L*TG ^p
flfl ? fl
fl ?y flfl
L*Tp^ --! L*TG0^p --! L*TG ^p:
Because L*TG ^pis torsionfree the functor Z=p induces maps
ff : ExtWG (L*TG ^p; L*Tp^)--! ExtWG (L*TG =p; L*T=p)
fi : ExtWG (L*TG ^p; L*T=p)--! ExtWG (L*TG =p; L*T=p) :
The exact sequence
m qT
L*Tp^ --! L*Tp^ --! L*T=p ;
where m is the multiplication with p, and the projection qG : L*TG ^p-! L*TG =p
establish the diagram
ExtWG (L*TG ^p; L*Tp^)
|m*|
|fflffl
ExtWG (L*TG ^p; L*Tp^)U
UU UU
|qT*| UUUffUUUUU
|fflffl ___fi_//UUUU**
ExtWG (L*TG ^p; L*T=p) oo____ExtWG (L*TG =p; L*T=p) :
q*G
The map m* is given by multiplication with p in the Ext-group, and fi is a left
inverse of q*G, i.e. fiq*G= id. The commutativity of
L*Tp^ --! L --! L*TG ^p
? ? fl
qT?y ?y flfl
L*T=p --! L0 --! L*TG ^p
flfl ? ?
fl ?y qG?y
L*T=p --! L=p --! L*TG =p
shows that fiqT* = ff and q*Gff = qT *. This implies the following lemma:
40
9.3 Lemma. ker(ff) = ker(qT *) = im(m*).
Next we will show that the p-adic lifting of the Weyl group representation
L*TG =p is unique for another class of Lie groups, namely for pseudo projective
p-convenient Lie groups. We introduce the following definition.
Definition. An extension
E : L*Tp^-! L- ! L*TG ^p2 ExtWG (L*TG ^p; L*Tp^)
is called initial if, for every other extension ^E2 ExtWG (L*TG ^p; L*Tp^), the*
*re exists
a self map g : L*Tp^-! L*Tp^ such that E^= g*E.
The next lemma produces initial extensions.
9.4 Lemma. Let Gs be simply connected. If G 2 GrExt(T; Gs) is pseudo projec-
tive the extension
E : L*Tp^ -! L*TG ^p-! L*TGs ^p
is initial.
Proof. By proposition 9.2 every extension E0 : L*Tp^-! L- ! L*TGs ^pis induced *
*by
a group extension Gs- ! G0-! T or alternatively by an exact sequence
K - ! Gs x T - ! G0 ;
where the composition K- ! GsxT -! T is an injection. The inclusion K- ! Z(Gs )
can be extended to
K --! Z(Gs )
?? ?
y ?y
T --! T :
This establishes a finite covering G0-! G. By construction we get
L*Tp^ --! L*TG ^p --! L*TGs ^p
?? ? fl
y ?y flfl
L*Tp^ --! L*TG0^p= L --! L*TGs ^p
which shows that E is initial.
Remember that a compact connected Lie group is pseudo projective if Z(G) is
connected.
9.5 Proposition. Let Gs be a p-convenient simply connected Lie group, and let
G 2 GrExt(T; Gs) be pseudo projective. Let L be a torsion free p-reducible
Z^p[WG ]-module, and let __ff: L=p- ! L*TG =p be a Z=p[WG ]-isomor-
phism. Then there exists a Z^p[WG ]-isomorphism
ff : L -! L*TG ^p;
41
__
which is a lift of ff.
Proof. Let EG : L*Tp^-! L*TG ^p-! L*TGs ^pbe the sequence associated to G, and
E : L*Tp^-! L- ! L*TGs ^pthe sequence associated to L (lemma 9.1). The differen*
*ce
E1 := E -EG is mod-p trivial. Thus, by lemmma 9.3, there is an extension E2 such
that E1 = m*(E2). The map m : L*Tp^-! L*Tp^ is the multiplication with p. The
exact sequence EG is initial (lemma 9.4). Hence E2 = a*(EG ) and E1 = m*a*(EG )
for a suitable homomorphism a : L*Tp^-! L*Tp^. This yields
EG : L*Tp^ --! L*TG ^p --! L*TGs ^p
? ? ?
(id;am)?y ?y ?y
EG + E1 : L*Tp^ L*Tp^ --! L*TG ^p L1 --! L*TGs ^p L*TGs ^p
?? ? fl
y ?y flfl
__
L*Tp^ --! L --! L*TGs ^p L*TGs ^p
flfl x x
fl ?? ??
E : L*Tp^ --! L --! L*TGs ^p;
where the three lower rows represent the sum EG + E1 = E and where is the
diagonal map. Therefore E = (id + am)*(EG ), and, because id + am id mod p,
the map L*TG ^p-! L is an isomorphism.
9.6 Remark. Let Z=p ,! Z=p2 = Z(SU(p2) ) and Z=p,! S1 be the standard in-
clusions, and let G = (SU(p2) x S1)=Z=p. G is a p-fold cover of U(p2). This fits
into
BZ=p ______BZ=p
?? ?
y ?y
BSU(p2) --! BG --! BS1
flfl ? ?
fl ?y ?yf
BSU(p2) --! BU(p2) --! BS1
and establishes
L*S1^p --! L*TG ^p --! L*TSU(p2)^p
x x fl
f*?? ?? flfl
L*S1^p --! L*TU(p2)^p --! L*TSU(p2)^p;
where f* is multiplication by p. Therefore, by lemma 9.3, the upper row splits *
*after
applying the functor Z=p. That is to say that L*TG =p ~=L*TSU(p2)=pL*S1=p as
p2-modules. Because G is p-convenient BG and BSU(p2) x BS1 have the same
42
mod-p type, but not the same p-adic type. This is the generic example of compact
Lie groups having the same mod-p type but different p-adic type, i.e. all examp*
*les
come from p-fold coverings.
For p = 2, the last proposition only covers the case of products of unitary *
*groups
U(n), n 3, which are all pseudo projective, but not the group U(2) which is not
2-convenient. Nevertheless, with one extra assumption, the same statement is st*
*ill
true for products of all unitary groups at p = 2 as we will see. First we cons*
*ider
the case of a product of U(2)'s.
9.7 Proposition. Let G = U(2)r and let L be a torsion free 2-reducible Z^2[WG *
*]-
module. Let __
ff: L*TG =2!- L=2
be a Z^2[WG ]-isomorphism. Then there exists a Z^2[WG ]-isomorphism
ff : L*TG!- L
which is a lift of __ff.
Proof. The two representations L*TG and L establish homomorphisms aeL ; aeL*TG :
WG!- Gl(2r; Z^2). Because L is 2-reducible and because P (L)WG =2 ~=P (L=2)WG
is a polynomial algebra, the 2-adic invariants P (L)WG are also a polynomial a*
*lgebra
as well as P (L)WG Q ~= P (L Q)WG . Therefore the rational representation
aeL : WG!- Gl(2r; Q^2) represents WG as a pseudo reflection group. We can choo*
*se
s1; :::; sr 2 (WG ) ~= (Z=2)r which generate WG and which are pseudo reflectio*
*ns
with respect to aeL . The only pseudo reflections in the image of WG in Gl(2r*
*; Z^2
are given by the generators of the Weyl groups of the factors of G. Therefore, *
*after
reordering, the element si generates the Weyl group of the i-th factor.
Let W 0 W be the subgroup generated by s2; :::; sr. The fixed-points of a
pseudo reflection acting on a0Z^2-module or Q^2-module has codimension01 or cod*
*i-
mension 0. Therefore, rk(LW ) 2r - r + 1 = r + 1 and rk(L=LW ) r - 1. 0
Here rk( ) denotes the rank of free modules. We notice that the qoutient L=LW *
* is
torsion free.
In the commutative diadram of exact rows
0 --! LW0 --! L --! L=LW0 --! 0
?? ? ?
y ?y ?y
0 --! LW0 =2 --! L=2 --! (L=LW0 )=2 --! 0
?? fl ?
y flfl ?y
0 --! L=2W0 --! L=2 --! (L=2)=(L=2W0 ) --! 0
? ? ?
~=?y ~=?y ~=?y
0 --! L*TG =2W0 --! L*TG =2 --! (L*TG =2)=(L*TG =2W0 ) --! 0
43
0
the right vertical composition of maps is an epimorphism. Hence, rkZ^2L=LW =
rkZ=2(L*TG =2)=((L*TG =2)W0 ) = r - 1. This implies that
(L=LW0 )=2!- (L=2)=((L=2)W0 ) is an isomorphism as well as_LW0 =2!- (L=2)W0 .
According to the splitting G = U(2)r we choose a basis B = {__x1; __y1; :::*
*; __xr; __yr}
of L=2 characterized by the properties si(__xj) = __xjand si(__yj) = __yjfor i *
*6= j and
the property si(__xi) = __yi. By the above considerations there exist lifts x1;*
* y1 2 LW0
of __x1and __y1, such that s1(x1) = y1. Analogously we can choose lifts xi; yi*
* of __xi
and __yifor all i such that B = {x1; y1; :::; xr; yr} is a basis of L, such tha*
*tsi(xi) = yi
and such that, for i 6= j, si(xj) = xj and si(yj) = yj. This implies that there*
* exist
a Z^2[WG ]-isomorphism L*TG!- L which is a lift of __ff.
Let X be a space with the mod-p type of BG x BH. The assumption that G
and H are p-convenient is too strong for what follows. We only assume that
(*) H*(BG; Z=p) ~=H*(BTG ; Z=p)WG and H*(BH; Z=p) ~=H*(BTH ; Z=p)WH :
Let BTX -! X be the maximal torus of X, let VG TG and VH TH be the
maximal elementary abelian p-subgroups of the maximal tori TG and TH , and let
g : BVH -! BVH x BVG -! X be the obvious map (see section 7). The isotropy
group of g is WG . Hence, for Y := map(BVH ; X)g,
H*(Y ; Z=p) ~=H*(BTX ; Z=p)WG ~=H*(BTG x BTH ; Z=p)WG
(theorem 10.1). In particular, H*(Y ; Z=p) is concentrated in even degrees, and
therefore, H*(Y ; Z^p)- ! H*(Y ; Z=p) is a surjection.
9.8 Lemma. If G and H are two compact connected Lie groups satifying (*) then
L*TX ^pis p-reducible as WG -module.
Proof. We have H*(BTX ;^p) ~=P (L*TX ^p) which is the symmetric part of the ten*
*sor
algbra of L*TX ^p. This induces maps
H*(Y ; Z^p)=p -! P (L*TX ^p)WG =p ,! P (L*TX =p)WG ~=H*(Y ; Z=p) :
The composition is an isomorphism. This implies the statement.
Now we are in the position to prove the following result.
9.9 Proposition. Let G be a product of unitary groups. If X is a space with the
mod-2 type of BG, then L*TX ^p~=L*TG ^pas WG -modules.
Proof. The group G = G0x G00splits into a product, where G0 contains all factors
isomorphic to U(2) and G00all the other factors U(n); n 3. Then L*TX ^2is
2-reducible as WG0-module (lemma 9.7), and L*TX ^2WG0 has as WG00-module the
mod-2 type of L*Z(G0) L*TG00^2. Moreover, Z(G0) x G00is pseudo projective, and
44
therefore, L*TX ^2WG0 ~=L*Z(G0)^2 L*TG00^2as WG00-modules (proposition 9.5).
Now we consider the diagram of exact sequences
0 --! L*TG00^2 --! L*TX ^2 --! L*TX ^2=L*TG00^2 --! 0
?? ? ?
y ?y ?y
0 --! L*TG00=2 --! L*TX =2 --! (L*TX ^2=L*TG00^2)=2 --! 0
flfl ? ?
fl ~=?y ~=?y
0 --! L*TG00=2 --! L*TG0=2 L*TG00=2 --! L*TG0=2 --! 0 :
The middle arrow is also an exact sequence because L*TX ^2=L*TG00^2is torsion f*
*ree.
Moreover, the analogous diagram for the polynomial algebras of the modules shows
that L*TX ^2=L*TG00^2is 2-reducible as WG0-module. By proposition 9.7 this impl*
*ies
that L*TX ^2=L*TG00^2~=L*TG0^2as WG0-modules.
L*TX can be considered as an element in
ExtWG0xWG00 (L*TG0^2; L*TG00^2) ~=H1(WG0 x WG00; Hom(L*TG0^2; L*TG00^2))
~=LTG0^2WG0 H1(WG00; L*TG00^2)
H1(WG0; LTG0^2) L*TG00^2WG00
= 0 :
This follows by lemma 8.2 and because H1(WG0; LTG0^2) = H1(WG00; L*TG00^2) = 0.
Finally we get L*TX ^2~=L*TG0^2 L*TG00^2= L*TG ^2as W G-modules.
10. Mapping spaces.
In this section G is a compact connected Lie group satifying the condition
H*(BG; Z=p) ~= H*(BTG ; Z=p)WG . Let X be a space with the mod-p type of
BG. In section 7 we constructed a maximal torus fT : BTX -! X, respectively
fT : BTX ^p-! X, with Weyl group WG such that the diagram
BTG ^p -'! BTX ^p
# #
BG^p 99K X
commutes in mod-p cohomology. In this chapter we always work with mod-p
cohomology and define H*( ) := H*( ; Z=p).
Let A be an abelian p-toral group. Then WG acts on map(BA; BTX ). This
action induces a map
[BA; BTX ^p]=WG ----! [BA; X] :
For a map g : BA- ! BTX ^pwe define
Iso(g) := {w 2 WG |w O g ' g}
to be the isotropy group of g. The composition of g with the maximal tori of X *
*is
also denoted by g.
45
10.1 Theorem. Let G be a compact connected Lie group such that H*(BG; Z=p) ~=
H*(BTG ; Z=p)WG . Let X be a space with the mod-p type of BG, and let A be an
abelian p-toral group. Then the following hold:
(1) [BA; BTX ^p] -! [BA; X] is a surjection.
(2) [BA; BTX ^p]=WG - ! [BA; X] is a bijection.
(3) For any g : BA- ! BTX ^p,
H*(map(BA; X)g) ~=H*(map(BA; BTX ^p)g)Iso(g)
and map(BA; X)g is p-complete.
10.2 Remark.
(1) By 10.1 (1) there exists for any map g : BA- ! X a lift g0 : BA- ! BTX ^*
*p,
which, by 10.1 (2), is unique up to conjugation by elements of the Weyl group.
Thus Iso(g0) WG is uniquely determined up to conjugation by g.
(2) The isomorphism of 10.1 (3) is induced by the maximal torus BTX ^p-! X.
Of course, BG^psatisfies the assumptions of the theorem. Let X have the p-adic
type of BG. Part (3) and theorem 3.8 induce canonical isomorphisms
H*(map(BA; X)g) ~=H*(map(BA; BG^p)g) ~=H*(map(BA; BG)g) :
(3) Let g : BA- ! BTG be a map and g ' Bff for a suitable homomorphism
ff : A- ! TG (theorem 3.6). ss0(CG (ff)) is a p-group [J-M-O; A.4]. Therefore, *
*by 10.1
(3), the centralizer CG (ff) is connected and p-convenient, and WCG (ff)= Iso(g*
*).
(4) Theorem 10.1 is true under some weaker assumptions as the proof will sho*
*w.
We only have to assume that X is a complete space with maximal torus f : BTX ^p*
*-!
X and Weyl group WX such that H*(X; Z=p) ~= H*(BTX ^p; Z=p)WX ; i.e. it is
not necessary that X is of the mod p-type of the classifying space of a compact
connected Lie group.
We can reformulate theorem 10.1, using the identity
Y
H*(map(BA; BTX ^p))WG ~= H*(map(BA; BTX ^p)g)Iso(g) :
g2[BA;BTX ]=WG
This identity is in anology to theorem 3.14.
10.3 Theorem. Under the assumption of 10.1, the maximal torus BTX ^p-! X
induces an isomorphism
H*(map(BA; X)) ~=H*(map(BA; BTX ^p))WG ;
and map(BA; X) is p-complete, i.e. every component is p-complete.
For the next statement, we assume that X has the p-adic type of BG and that
there exists an extension
BN(TG ) ----! X ;
46
of the maximal torus BTG -! X, such that the diagram
BN(TG )H
tt HHH
ttt HHH
zzttt H$$
BG _ _ _ _ _ _ _ _ _+3X
commutes in mod-p cohomology. For any abelian abelian p-toral group A and any
map g : BA!- BTG , theorem 10.3 establishes the following sequence of isomor-
phisms
H*(map(BA; X)g) ~=H*(map(BA; BTX ^p)g)WG
~=H*(map(BA; BTG ^p)g)WG
~=H*(map(BA; BG^p)g) ;
and therefore, a dotted arrow map(BA; BG)g 999K map(BA; X)g
10.4 Proposition. For any map g : BA- ! BTG , the diagram
map(BA; BN(TG ))gS
kkk SSSS
kkkk SSSS
kkkk SSS S
uukkk ))
map(BA; BG)g _ _ _ _ _ _ _ _ _ _ _ _ _ _ +map(BA;3X)g
commutes in mod-p cohomology.
In the rest of this chapter we will prove theorem 10.3 and proposition 10.4.
10.5 Lemma. It is sufficient to prove 10.3 and 10.4 for finite abelian p-group*
*s.
Proof. Let A be a p-toralSabelian group. We denote by Ak A the elements of or-
der pk and define A1 := Ak. Then Ak is a finite p-group. The map BA1 -! BA
is a mod-p equivalence, which implies that map(BA; X) ' holim-map(BAk; X) and
map(BA; BTX ) ' holim- map(BAk; BTX ). The inclusions BAk-1 -! BAk induce
a commutative diagram
:::----! map(BAk; BTX ^p) ----! map(BAk-1 ; BTX ^p) ----! :::
?? ?
y ?y
:::----! map(BAk; X) ----! map(BAk-1 ; X) ----! ::::
By theorem 10.3 for finite abelian p-groups we have H1(map(BAk; BTX ^p)) =
H1(map(BAk; X)) = 0.
47
The two function spaces map(BAr; BTX ^p) and map(BAr; X) are p-complete.
Hence, both mapping spaces are also 1-connected [B-K; I.6]. The Milnor sequence
for calculating components of homotopy inverse limits therefore reduces to
~=
[BA; BTX ^p] ----! lim-[BAk; BTX ^p]
?? ?
y ?y
~=
[BA; X] ----! lim-[BAk; X] :
The obvious map
[BA; BTX ^p]=WG - ! lim-([BAk; BTX ^p]=WG )
is a bijection. That is to say that
H0(map(BA; X)) ~=H0(map(BA; BTX ^p))WG :
Let g : BA- ! BTX ^pbe a map and denote by gk the restriction g|BAk . The seque*
*nce
::: Iso(gk-1 ) Iso(gk) :::
stabilizes, since WG is a finite group. We can choose k0 (big enough), such t*
*hat
Iso(g) = Iso(gk) for k k0. Thus
map(BAk+1 ; X)gk+1 ----! map(BAk; X)gk
are mod-p equivalences and therefore equivalences for k k0 because all the spa*
*ces
are p-complete. This implies
H*(map(BA; X)g) ~= H*(map(BAk0; X)gk0)
~= H*(map(BAk0; BTX ^p)gk0)Iso(gk0)
~= H*(map(BA; BTX ^p)g)Iso(g);
which finishes the proof of first part of the statement.
To prove 10.4, we consider a map g : BA- ! BTG . According toTremark
10.2 (3), we can speak of the centralizers CG (gk) and CG (g) = kCG (gk), whi*
*ch
are compact Lie groups. The sequence CG (gk+1 ) CG (gk) stabilizes. Again we c*
*an
choose k0 big enough such that CG (g) = CG (gk0) and, analogously, CN(TG )(g) =
CN(TG )(gk0). Theorems 10.1 and 3.6 and the above considerations show that the
maps
map(BA; BN(TG ))g ----! map(BAk0; BN(TG ))gk0
map(BA; BG)g ----! map(BAk0; BG)gk0
map(BA; X)g ----! map(BAk0; X)gk0
48
are mod-p equivalences which reduces 10.4 to the case of a finite group.
Let A be a finite abelian p-group. We can choose a subgroup A0 A of index
p and get an exact sequence
1 --! A0 --! A --! Z=p --! 1 :
We will prove 10.3 and 10.4 for a finite group A by an induction over the order*
* of A.
In remark 3.12 we defined gBA0 := EA=A0 ' BA0 which carries a free Z=p-action.
To this end we introduce the following notation and abbreviations:
MX := map(BA; X)
MX0 := map(gBA 0; X)
MT := map(BA; BTX ^p)
MT0 := map(gBA 0; BTX ^p)
W := WG :
Then MX ~= MX0hZ=p and MT ~= MT0hZ=p. Because the actions of W and Z=p
on MT0 commute the group W acts on the Borel product EZ=p xZ=p MT0.
Now let us assume that 10.3 and 10.4 is true for A0.
10.6 Lemma.
(1) The fibration
MT0 ----! EZ=p xZ=p MT0 ----! BZ=p
is fiber homotopic trivial.
(2) H*(EZ=p xZ=p MX0) ~=H*(EZ=p xZ=p MT0)W .
Proof. (1) Every map g0 : BA0- ! BTX ^pis, up to homotopy, induced by a ho-
momorphism. For every g0 we can choose an_extension_gS: BA- ! BTX ^psuch
that g|BA0 = g0. Z=p acts trivially on MT := g map(BA; BTX ^p)g and on
_____ _____
EZ=p xZ=p MT = BZ=p x MT . By theorem 3.9 the restriction from BA to gBA 0
induces a homotopy equivalence
_____
MT ----! MT0 :
Moreover, this map is Z=p-equivariant, and fits into a commutative diagram
_____ _____
MT ----! BZ=p x MT ----! BZ=p
?? ? fl
y ?y flfl
MT0 ----! EZ=p xZ=p MT0 ----! BZ=p
49
of homotopy equivalent fibrations, which proves (1).
(2) By the assumptions H*(MX0) ~= H*(MT0)W . Moreover, this is a direct
summand in H*(MT0) considered as vector spaces. The diagram
MT0 ----! EZ=p xZ=p MT0 ----! BZ=p
?? ? fl
y ?y flfl
MX0 ----! EZ=p xZ=p MX0 ----! BZ=p
and (1) show that both fibrations are oriented and that all differentials in the
associated Serre spectral sequences are trivial. For every n we have
Hn (BZ=p; H*(MX0)) ~=H*(MX0)
~=H*(MT0)W
~=Hn (BZ=p; H*(MT0))W :
The isomorphism between the first and last group is compatible with the inclusi*
*on
Hn (BZ=p; H*(MX0))- ! Hn (BZ=p; H*(MT0)) because W acts on the space level.
Applying the 5-lemma to the extension problems yields the desired isomorphism
H*(EZ=p xZ=p MX0) ~=H*(EZ=p xZ=p MT0)W :
Proof of 10.3 for A a finite abelian p-group.
By induction hypothesis H*(MX0) ~=H*(MT0)W . To calculate MX = MX0hZ=p
we apply the Lannes functor for homotopy fixed-point sets (see section 3). Beca*
*use
MT is p-complete and H1(MT ) = 0, the W -equivariant map
HF Z=p(MT0) -! H*(MT )
is an isomorphism (theorem 3.10). Using this fact and theorem 3.14 we get
HF Z=p(MX0) ~=TsZ=pec*H*(EZ=p xZ=p MX0) H*(BZ=p) Z=p
~=TsZ=pec*(H*(EZ=p xZ=p MT0)W ) H*(BZ=p) Z=p
~=TsZ=pec*((H*(BZ=p) H*(MT0))W ) H*(BZ=p) Z=p
~=(TiZ=pd(H*(BZ=p)) T Z=p(H*(MT0)W ) H*(BZ=p) Z=p
~=H*(BZ=p) T Z=p(H*(MT0))W H*(BZ=p) Z=p
~=H*(MT )W ;
where sec* denotes the cohomological maps induced by sections in the bundle
EZ=p xZ=p MT0. The third isomorphism follows from lemma 10.6 which says that
the bundle EZ=p x MT0!- BZ=p is fiber homotopic trivial. The fourth isomor-
phism follows because T commutes with tensor products and because every section
50
in the trivial bundle BZ=p x MTO!- BZ=p is given by a pair (idBZ=p ; g) of ma*
*ps
where g : BZ=p!- MT0. Hence HF Z=p(MX0) vanishes in degree 1. By theorem
3.10 again
H*(MT )W ~=HF Z=p(MX0) ----! H*(MX)
is an isomorphism, and by theorem 3.11, MX is p-complete which finishes the
proof of 10.3.
Proof of 10.4 for A a finite abelian p-group.
Let g ' Bff : BA! BTG be a map and denote by SpN(TG ) N(TG ) a p-toral
Sylow subgroup. Then CG (ff) is connected (remark 10.2) and
CSpN(TG )(ff) CN(TG )(ff) = N(TCG (ff))
is an inclusion of index coprime to p and therefore, also a p-toral Sylow subgr*
*oup.
In particular,
H*(BCN(TG )(ff)) -! H*(BCSpN(TG )(ff))
and, by theorem 3.6,
H*(map(BA; BN(TG ))) -! H*(map(BA; BSpN(TG )))
are injections. In order to prove 10.4 it is sufficient to replace N(TG ) by Sp*
*N(TG ).
BCSpN(TG )(ff)^p' (map(BA; BSpN(TG ))g)^pare p-complete (theorem 3.9), and
map(BA; BSpN(TG ))g^p ' map(BA; BSpN(TG )^p)g^p
' (map(BA0; BSpN(TG )^p)g0^p)hZ=p
' (map(BA0; BSpN(TG ))g0^p)hZ=pg^p
is the homotopy fixed-point set of a p-complete space. Therefore and because of
theorem 3.11
HFgZ=p*(map(BA0; BSpN(TG ))g0) ~=HFgZ=p*(map(BA0; BSpN(TG )^p)g0^p)
~=H*(map(BA; BSpN(TG )^p)g^p)
~=H*(map(BA; BSpN(TG ))g) :
The diagram
(*)
H*(map(BA; BSpN(TG ))g)
ii44 jjVVVVV
iiiii VVVVV
iiiii VVVVV
iiiii VV
H*(map(BA; X)g) _______________________________________H*(map(BA;/BG)g)/
51
comes up by repeated applications of the functor HF Z=pto the commutative dia-
gram
H*(BSpN(TG ))
o77 ggPPP
ooo PPPP
ooo PPP
oo o P
H*(X) ____________________________H*(BG)// :
That is that the diagram (*) also commutes (remark 3.4) which finishes the proof
of 10.4.
11. The normalizer of the maximal torus I.
Let G be a p-convenient compact connected Lie group, and let X be a space of
the p-adic type of BG. In this section we construct an extension of the maximal
torus
fT : BTG - ! X
to a map
fN : BN(TG ) -! X
of the classifying space of the the normalizer N(TG ) of TG .
The space BN(TG ) is a two stage Posnikov system given by the fibration
BTG - ! BN(TG ) -! BWG :
We have to find a suitable element in map(BN(TG ); X) ' map(BTG ; X)hWG (re-
mark 3.12); i.e. a section in the bundle
EWG xWG map(]BTG ; X)fT - ! BWG :
The obstructions are lying in
H*+1 (BWG ; ss*(map(]BTG ; X)fT) ~=H3(BWG ; LTG ^p)
(twisted coefficients). The isomorphism follows from theorem 10.1 and the ident*
*ity
ss2(BTG ^p) ~=H2(BTG ; Z^p) = LTG ^p. By lemma 5.3 the obstruction group vanish*
*es
for odd primes which is sufficient to prove the following statement for odd pri*
*mes.
11.1 Proposition. Let G be a compact connected Lie group, and let X be a space
with the p-adic type of BG. If p is odd and G is p-convenient or if p = 2 and G*
* is
a product of unitary groups, there is an extension
BTG __fT____X//v;;
| vvv
| vvf
|fflfflNvv
BN(TG )
52
of the maximal torus BTG -! X to BN(TG )- ! X.
Because the obstruction group does not vanish for p = 2, we need a different
approach. The proof in this case is postponed to the end of the section.
Choosing a fixed-point of the WG -action on BTG the evaluation at this fixpo*
*int
map(BTG ^p; X)fT - ! X
is equivariant with respect to the trivial action of WG on X. The group WG ac*
*ts
freely on the product BTG ^p' EWG x map(BTG ^p; X)fT. We get a well defined
map on the orbit space Y := (EWG x map(BTG ^p; X)fT)=WG and an extension
BTG ^p' map(BTG ^p; X)fTfT _____X_//l66
ll
| llll l
| llll
|fflfflllll
Y
We will show that Y is nothing but the fiberwise completion BN(TG )Opof BN(TG )
of the fibration BTG -! BN(TG )- ! BWG (see [B-K; I.8]). By construction Y fi*
*ts
into a fibration
BTG ^p-! Y - ! BWG :
We will consider a more general situation. Let W be a finite group, acting o*
*n a
torus T via homomorphisms. Fibrations of the form
BTp^ -! Y - ! BW
can be classified by homotopy classes of maps
BW -! BHE(BT ^p) ;
where HE(BT ^p) is the monoid of self equivalences of BT ^p[St]. Denote by
SHE(BT ^p) the component of the identity. Then Gl(LT ^p) ~=ss0(HE(BT ^p)) is the
group of the components. The Dwyer-Zabrodsky map
BT ^p-! SHE(BT ^p)
is a homotopy equivalence. Considering BT as a group the map is given by mapping
each element to the associated left translation. Therefore, it is a homomorphis*
*m of
monoids. Moreover, it is W -equivariant because W acts via homomorphisms. We
get an W -equivariant equivalence
BBT ^p-! BSHE(BT ^p) :
Up to homotopy, the composition BW -! BHE(BT ^p) - ! BGl(LT ^p) is in-
duced by an homomorphism which is given by the action of W on BT ^passociated
53
to the fibration. The difference between two clasifying maps
OE; : BW -! BHE(BT ^p) with the same action on BT ^pis measured by an ob-
struction class
d(OE; ) 2 H3(BW ; ss3(BSHE(BT ^p) ~=H3(BW ; ss2(BT ^p) ~=H3(W ; LT ^p)
for lifting homotopies in the fibration
BSHE(BT ^p) -! BHE(BT ^p) -! BGl(LT ^p) :
Notice that H3(W ; LT ^p) ~=H2(W ; T )^pwhich, forgetting the completion, descr*
*ibes
the equivalence classes of group extensions T!- H!- W .
Let s : BW -! BHE(BT ^p) be the classifying map of the fibration
BT ^p-!B(T o W )Op-! BW .
Definition. By F ib(BW; BT ^p) we denote the set of fibrations divided out the
equivalence relations given by
BT ^p ----! Y1 ----! BW
flfl ? fl
fl ?y flfl
BT ^p ----! Y2 ----! BW :
11.2 Lemma. The map
F ib(BW; BT ^p) -! H3(W ; LT ^p)
(OE : BW ! BHE(BT ^p)) 7! d(s; OE)
is a bijection.
Proof. Every cohomology class can be realized as an obstruction.
11.3 Corollary. The canonical map
B : GrExt(W; T ) -! F ib(BW; BT ^p)
T ! N! W 7! BT ^p!BNOp! BW
is a surjection.
Proof. Group extensions are described by cohomology classes in
H2(W ; T ) ~=H3(W ; LT ). The canonical map H2(W ; T )- ! H3(W ; LT ^p) is a su*
*r-
jection and reflects the map B on the cohomological level.
54
11.4 Remark. Obviously there is an integral version of corollary 11.3. That is
that
GrExt(W; T ) -! F ib(BW; BT )
is a bijection.
Now let us assume that G = G1 x G2 is a product of unitary groups. WG acts
trivially on the centers Z(G) and Zi := Z(Gi), which are all tori. The canonic*
*al
map
F ib(BWG ; BZ1^p) -! H3(WG ; LZ1^p)
is a bijection and establishes a map
F ib(BWG ; BZ1^p) -! F ib(BWG ; BTG1 ^p)
given by the inclusion Z1- ! TG1 .
The inclusion S1 = Z(U(n)),! TU(n) induces
H3(n; Z^p) ~=H3( n; LS1^p)
-! H3(n; LTU(n)^p) ~=H3(n-1 ; LS1^p) ~=H3(n-1 ; Z^p)
This composition is given by the restriction and therefore an isomorphism for n*
* 6= 4
and a surjection for n = 4. In dimensions 2 this restriction induces an isomo*
*r-
phism for n 6= 2; 4 and a surjection in all cases.
Now a spectral sequence argument shows that
H3(WG ; LZ1^p) -! H3(WG ; LTG1 ^p)
and
F ib(BWG ; BZ1^p) -! F ib(BWG ; BTG1 ^p)
are surjective. They are isomorphisms, if G1 has no factor isomorphic to U(4) or
U(2).
11.5 Lemma. Let G be a product of unitary groups. Let
BTG ^2-! Y - ! BWG 2 F ib(BWG ; BTG ^2)
be a fibration such that H2(Y ; Z=2)- ! H2(BTG ^2; Z=2)WG is a surjection. Th*
*en
there exists an equivalence of fibrations
BTG ^2 --! Y --! BWG
flfl ? fl
fl ?y flfl
BTG ^2 --! B(TG o WG )O2 --! BWG :
Proof. Let G = U(n1)x:::xU(nk). By corollary 11.3 there exists a group extension
E : TG - ! N - ! WG
55
such that
BTG ^2 --! BNO2 --! BWG
flfl ? ?
fl ?y ?y
BTG ^2 --! Y --! BWG
is an equivalence of fibrations. We have to show that BNO2' B(TG o WG )O2.
The isomorphism
M
H2(WG ; TG ) ~= H2(WG ; TU(ni))
i
establishes group extensions
Ei : TU(ni) -! Ni -! WG :
E is given by the semi direct product, i.e N ~=TG o WG , if and only if Ei is
isomorphic to the semi direct product for all i. Moreover, we have a map
BTG ^2 --! BNO2 --! BWG
?? ? fl
y ?y flfl
BTU(ni)^2 --! BNiO2 --! BWG
between the two fibrations.
H2(Y ; Z=2) ~= H2(BNO2; Z=2)- ! H2(BTG ; Z=2)WG is an epimorphism. That is
to say that the differential
d2 : H0(BWG ; H2(BTG ; Z=2)) ~=H2(BTG ; Z=2)WG -! H3(BWG ; Z=2)
of the Leray-Serre spectral sequence is trivial. Because
H0(WG ; H2(BTU(ni); Z=2)) -! H0(WG ; H2(BTG ; Z=2))
is an injection the differential
d2 : H2(BTU(ni); Z=2)WG -! H3(BWG ; Z=2)
is also trivial.
By the above remark there is a commutative diagram
BS1^2 --! Yi --! BWG
?? ? fl
y ?y flfl
BTU(ni)^2 --! BNiO2 --! BWG :
56
Because the action of WG on BS1 is trivial the classifying map of the top fibra*
*tion
lifts to a map BWG -! BSHE(BS1^2) ' BBS1^2. This map is totally determined
by the first transgression
d2 : H2(BS1^2; Z^2) -! H3(BWG ; Z^2)
in the Serre spectral sequence of the 2-adic cohomology and, because H3(Bn; Z^2)
and H3(WG ; Z^2) are annihilated by 2, also by the differential
d2 : H2(BS1^2; Z=2) -! H3(Bn; Z=2) :
Now we first assume that n is odd. Then H2(BTU(n); Z=2)WG ~= H2(BS1; Z=2),
and the above transgression is given by the differential
d2 : H2(BTU(n)^2; Z^2)WG -! H3(BWG ; Z^2)
which, by assumption, vanishes. Thus, the fibration BS1^2-! Yi-! BWG is trivia*
*l,
and the fibration BTU(ni)^2-! BNiO2-! BWG is given by the fiber wise completion
B(TU(ni)o WG )O2.
Now let ni be even, and let G = U(ni) x Gi. Let S1- ! TU(ni) be the inclusion
in the last factor and Z=2- ! S1 the standard inclusion. Then
CNi(Z=2)=CTU(ni)(Z=2) = CNi(Z=2)=TU(ni) = ni-1 x Gi ;
___
and S1- ! CN (Z=2) is central. We define N i:= CNi(Z=2)=S1 and get a commuta-
tive diagram of fibrations
___
BTU(ni-1)^2 --! BN iO2 --! Bni-1 x BWGi
x? x fl
? ?? flfl
BTU(ni)^2 --! BCNi(Z=2)O2 --! Bni-1 x BWGi
flfl ? ?
fl ?y ?y
BTU(ni)^2 --! BNiO2 --! Bni x BWGi :
A comparison of the two lower rows shows that the transgression
d2 : H2(BTU(ni); Z=2)ni-1xWGi - ! H3(Bni-1 x BWGi; Z=2)
is trivial, and a comparison between the two upper rows implies that the differ*
*ential
of the top fibration
d2 : H2(BTU(ni-1); Z=2)ni-1xWGi - ! H3(Bni-1 x BWGi; Z=2)
is also trivial. Because ni - 1 is odd the top fibration is given by the semi *
*direct
product.
57
On the level of the cohomology groups, describing the group extensions, the
above construction goes along with the maps
H2(nixWGi; TU(ni)) -! H2(ni-1 xWGi; TU(ni)) -! H2(nixWGi; TU(ni-1)) :
The composition
~= 2 n 2 n
H2(n-1 ; R) -! H (n; R ) -! H (n-1 ; R )
~= 2
-! H2(n-1 ; Rn-1 ) -! H (n-2 ; R)
is given by the restriction. This is an isomorphism for n = 4; 2; n 6, and R =*
* S1
or R = Z=2 [Na]. Hence, by a spectral sequence argument follows that the above
composition is an isomorphism for even ni's.
Passing to the obstruction groups, describing the associated fibrations, we *
*see
that
BTU(ni)^2-! BNiO2-! Bni x WGi
is given by the semi direct product if and only if the fibration
___O
BTU(ni-1)^2- ! BN i2 -! Bni-1 x WGi
is given by the semi direct product which we already proved.
Now we are in the position to finish the proof of proposition 11.1.
Proof of proposition 11.1. It is only left to consider one case, namely G is a
product of unitary groups and p = 2. The composition of maps
BTG ^2-! Y - ! X
shows that H*(Y ; Z=2)- ! H*(BTG ^2; Z=2)WG is surjective. By lemma 11.5 the
fibration
BTG ^2-! Y - ! BWG
is given by the fiberwise completion of the classifying spaces of the exact seq*
*uence
TG - ! TG o WG - ! WG :
The observation TG o WG = N(TG ) completes the proof.
58
12. The normalizer of the maximal torus II.
In this section G is a compact connected Lie group which is p-convenient for*
* p
odd and a product of unitary groups for p = 2. Let X be a space with the p-adic
type of BG.
In section 11 we constructed an extension fN : BN(TG )- ! X of the maximal
torus fT : BTG -! X, which fits into the diagram
* Bi*
BTG __Bi__BN(TG/)/UUUU__BG_//O
UUU UU JJJ O
UUU JfNJ O
fT UUUUUUJJ ff
U**J$$
X ;
where the outer triangle commutes in p-adic cohomology. We consider the questio*
*n,
whether the inner triangle commutes in mod-p cohomology.
The extension BN(TG )- ! X induces a lift in
H*(BN(TG5);5Z=p)
llll
(*) lllalll Bi*||
lll * |fflffl
H*(BG; Z=p) ___Bi__H*(BTG/;/Z=p) :
Here we work in the category of unstable algebras over the Steenrod algebra. The
lift a is given by the composition
__OE f*
N *
H*(BG; Z=p) ---- H*(X; Z=p) ----! H (BN(TG ); Z=p) ;
__
where OE is an isomorphism. The map Bi* : H*(BG; Z=p)- ! H*(BTG ; Z=p) is
induced by the standard inclusion. Another lift comes from the standard inclusi*
*on
Bi* : BN(TG )- ! BG.
For any homomorphism : WG -! TG WG , we define j : N(TG )- ! N(TG ) to be
the automorphism
idx
N(TG ) ----! N(TG ) x WG ----! N(TG ) x TG WG ----! N(TG ) ;
where is the diagonal composed with the projection on WG and the multipli-
cation. Obviously, j |TG = id|TG , and N(TG )-j! N(TG )- i!G induces another l*
*ift
in the diagram (*). We will prove the following statement:
12.1 Proposition. Let G be a p-convenient pseudo simply connected Lie group
or a product of unitary groups, and let
a : H*(BG; Z=p) -! H*(BN(TG ); Z=p)
be a lift of H*(BG; Z=p)- ! H*(BTG ; Z=p).
(1) If p is odd then a = Bi*; i.e. there is only one algebraic lift, given *
*by the
standard inclusion.
(2) If p = 2 there exists a homomorphism : WG -! TG WG such that
a = Bj* Bi*; i.e a can be realized by an self automorphism of N(TG )
composed with the standard inclusion.
59
12.2 Corollary. Let G be a p-convenient pseudo simply connected Lie group or
a product of unitary groups. Then there exists an extension fN : N(TG )- ! X *
*of
fT : BTG -! X such that
*
BTG Bi* ______BN(TG/)/UUUBi_BG//O
UUUUU JJJ O
U UU JfNJ O
fT UUUUUUUJJ ff
U**J$$
X ;
commutes in mod-p cohomology.
Proof. By proposition 11.1 we choose an extension f0N : BN(TG )- ! X of fT :
BTG -! X. If p is odd we_can_directly apply the above theorem.
If p = 2 we have f0N*OE-1= Bj* Bi* for a suitable homomorphism
: WG -! TG WG . By construction j is an isomorphism. Hence fN := f0NB(j-1 )
satisfies the statemant.
For the following we assume that G = U(n1) x ::: x U(nr) is a product of
unitary groups. We fix a lift a : H*(BG; Z=p)- ! H*(BN(TG ); Z=p) of the standa*
*rd
inclusion into H*(BTG ; Z=p).
The next lemma says that, for the proof of proposition 12.1, it suffices to *
*look
at compositions
a Bj*
H*(BG; Z=p) ----! H*(BN(TG ); Z=p) ----! H*(BV ; Z=p)
for every elementary abelian p-subgroup j : V -! N(TG ).
12.3 Lemma. Let G be a product of unitary groups. Then H*(BN(TG ); Z=p) is
detected by elementary abelian subgroups.
Proof. [G-L-Z].
Every element x 2 N(TU(p)) can be written in the form x = (1; :::; p; o ), i*
* 2
S1 and o 2 p. Let oe 2 p be the permutation represented by the cycle (1; 2; :::*
*; p),
let V2 N(TU(p)) be the group generated by (1; :::; 1; oe), and let V1 N(TU(p)*
*) be
the subgroup generated by diag(!; :::; !), ! = exp(2ssi=p). Here diag( ) denote*
*s the
canonical element in TU(p) N(TU(p)). Then V1 x V2- ! N(TU(p)) is a subgroup.
12.4 Lemma. The elements
(1; :::; p; oe); (!1; :::; !p; oe); (1; :::; p; oel); (; :::; ; oe)
Q
are conjugate in N(TU(p)) if (l; p) = 1 and p = ii.
Proof. Choose ff1 = 1 and ffi = ffi-1i-1-1 . Then (ff1; :::; ffp; 1) conjugate*
*s the
first element into the last one. oe and oel have the same cycle type. They are *
*conju-
gate in p. With these operation we can construct all necessary conjugations.
For abbreviation, we set
Hn(k) := U(p)k x (S1)n-pk U(n) ;
Wn(k) := (V1 x V2)k x (Z=p)n-pk Hn(k) U(n) ;
Vn(k) := (V1)k x (Z=p)n-pk = Wn(k) \ TU(n) Hn(k) U(n) :
60
12.5 Lemma. Every element x 2 N(TG ) of order p is conjugate to an element in
a suitable subgroup
Wn1(k1) x ::: x Wnr(kr) N(TU(n1)) x ::: x N(TU(nr)) = N(TG ) :
Q
Proof. x can be written in the form i(i; oi), i 2 TU(ni) and oi 2 ni. oi
is represented by a product of cycles of length p. An application of lemma 12.4
finishes the proof.
Let j : V -! N(TG ) be an elementary abelian p-subgroup of N(TG ). By the
theorems 3.1 and 3.6 the map Bj*a can be realized by a homomorphism aeV : V -! G
which is unique up to conjugation. The character OaeV(x) depends on the chosen
element x 2 N(TG ) and on a, but not on the subgroup V containing x, nor on
the homomorphism aeV . Therefore O(x) := Oa(x) := OaeV(x) is well defined for a*
*ll
elements x 2 N(TG ) of order p. By T r(x) we denote the trace of the associated
matrix, given by N(TG )- ! G = U(n1) x ::: x U(nr). Now we are prepared for the
proof of theorem 12.1.
Proof of proposition 12.1 for G a product of unitary groups.
Let G = U(n1) x ::: x U(nr). Let j : V 0-! N(TG ) be an elementary abelian
p-subgroup of N(TG ). Because G is a product of unitary groups, it is a questi*
*on
of characters to determine aeV 0. Thus we have only to consider cyclic subgroup*
*s of
order p.
Let x 2 N(TG ) be an element of order p. Up to conjugation
j
x 2 WN;K := Wn1(k1) x ::: x Wnr(kr) ----! N(TG )
for suitable k1; :::; kr. We choose k1; :::; kr minimal; i.e x is not containe*
*d in a
group Wn1(l1) x ::: x Wnr(lr) of the same form such that li ki for all i and at
least for one i there is strict inequality. Because the cohomological restrict*
*ion of
a : H*(BG; Z=p)- ! H*(BN(TG ); Z=p) to H*(BTG ; Z=p) is the standard inclusion,
we can assume that aeWN;K |VN;K is the standard inclusion, where VN;K := Vn1(k*
*1) x
::: x Vnr(kr).
We set
HN;K := Hn1(k1) x ::: x Hnr(kr) = CG (V ) :
Applying the Lannes functor TBVi*to
a
H*(BG; Z=p) -! H*(BN(TG ); Z=p) -! H*(BTG ; Z=p)
gives
a
H*(BHN;K ; Z=p) -! H*(BN(TH ); Z=p) -! H*(BTH ; Z=p) = H*(BTG ; Z=p)
61
(theorem 3.5) which shows that aeWN;K can be taken to have image in HN;K .
x 2 WN;K can be thought of having the form
x = (x1; :::; xr)
xi= ((1;_:::;_1;_oe);_:::;-(1;z:::;_1;_oe)_"; 1(i); :::; ni-pki(i)) 2*
* Wni(ki) :
ki times
We also consider y = (y1; :::; yr) with
yi = ((1;_:::;_1;_oe);_:::;-(1;z:::;_1;_oe)_"; 1; :::; 1) 2 Wn*
*i(ki) ;
ki times
and write aeWN;K (y) = (B1; :::; Br) 2 HN;K with
Bi = (A1(i); :::; Aki(i); fi1(i); :::; fini-pki(i)) 2 Hni(ki) ;
where Aj(i) 2 U(p) and fil(i) 2 S1. This implies
aeWN;K (x)= (D1; :::; Dr)
Di = (A1(i); :::; Aki(i); fi1(i)1(i); :::; fini-pki(i)ni-pki(i))
= Bidiag(1; :::; 1; 1(i); :::; ni-pki(i)) :
Since yiand y0i:= yidiag(!; :::; !; 1; :::; 1) (! pki-times) are conjugate in N*
*(THni(ki))
(lemma 12.4), O(y) = O(y1; :::; y0i; :::; yr). This shows that T r(A1(i)) = 0.
Analogously follows that T r(Aj(i)) = 0 for all j and i.
We first assume p is odd. Since yi is conjugate to (yi)2Pin N(THni(ki))Pand
yidiag(1; :::; 1; !) to (yi)2diag(1; :::; 1; !), we get j fij(i) = j fij(i)*
*2 and
fini-pki(i)(! - 1) = fini-pki(i)2(! - 1). This implies fini-pki(i) = 1. An anal*
*ogous
argument shows that fij(i) = 1 for all j and i.
Let ssl : G- ! U(nl) be the projection onto the l-th factor. Because aeWN;K *
*|VN;K
is the standard inclusion, it follows immediately that
Xnl
O(x) = OsslaeWN;K(x) = j(l) ;
j=1
which is the character of the inclusion
i ssl
V (x),! WN;K --! G --! U(nl) ;
where V (x) is the cyclic group of order p generated by x. Thus, sslaeWN;K j is
conjugate to ssli|V (x)and aeWN;K j = aeV (x)is conjugatePto the standard inclu*
*sion.
Now we assume p = 2. We have O(x) = T r(aeWN;K (x)) = i;jfij(i)j(i). If we
permute the j(i)'s for a fixed i the new element is conjugate to x in N(TG ). W*
*e can
vary x to realize different values for the j(i)'s. Therefore, fij(i) = fil(i) f*
*or all j and
62
l. As elements of order 2, fij(i) = 1. This implies that aeWN;K (x) = (D1; :::;*
* Dr)
with
Di = (A1(i); :::; Aki(i); "i(x)1(i); :::; "i(x)ni-pki(i)) :
The function "i : WG -! {1} is constant on conjugate elements and multiplica-
tive if restricted to an 2-elementary abelian subgroup. We choose for every i a
transposition oi 2 ni and set
z(oi):= (z1; :::zr) ; zi 2 N(TU(ni))
(
diag(1; :::; 1) if j 6= i
zj :=
(1; :::; 1; oi) if j = i :
We define a homomorphism
Yr Yr
= ( j;k) : WG = nj -! TG WG = S1
j=1 k=1
by setting
(
sign(oe) if "j(z(ok)) = -1
j;k: nj -! S1 : oe 7!
+1 if "j(z(ok)) = +1 :
Again we denote by ssl : G- ! U(nl) the projection on the l-th factor, and by
V (x) W the subgroup generated by x. Then, by construction, sslaeV (x)and
the map V (x),! N(TG )-j! N(TG )- i!G-ssl!U(nl) have the same character for all
x 2 N(TG ). That is to say that both homomorphisms are conjugate. To finish the
proof we can proceed as for p odd.
In order to prove theorem 12.1 in the other case, namely for a p-convenient
pseudo simply connected Lie group G, we will use the mod-p subgroup H- ! G of
G given by proposition 5.2. Unfortunately, for p = 3, H might have a factor SU(*
*p)
besides unitary groups. We have to prove the theorem for products of unitary
groups and SU(p)'s. For odd primes the p-toral Sylow subgroup of N(TSU(p)) is
TSU(p) o Z=p. Thus, lemma 12.4 is still true.
If we define
H0p(0) := TSU(p) SU(p)
H0p(1) := SU(p)
Wp0(1) := V1 x V2 SU(p)
Wp0(0) := TSU(p) =: VP0(0)
Vp0(1):= V1 Wp0(1) \ TSU(p) H0p(1) SU(p) ;
few minor changes make the proof work. Only the replacement of lemma 12.3 needs
some comments.
63
12.6 Lemma. For p odd, H*(BN(TSU(p)); Z=p) is detected by elementary abelian
subgroups.
Proof. We consider the Gysin sequence
::: --! H*-2 (BN(TSU(p)); Z=p) --! H*(BN(TU(p)); Z=p)
Bi* * d *-1
--! H (BN(TSU(p)); Z=p) --! H (BN(TU(p)); Z=p) --! :::
of the fibration Bi
S1!- BN(TSU(p)) -! BN(TU(p)) :
Let x 2 H*(BN(TU(p)); Z=p), such that d(x) 6= 0. We choose an elementary abelian
p-subgroup V -! BN(TU(p)), detecting d(x). Let