Fixed points of coisotropic subgroups of $\Gamma_{k}$ on decomposition spaces

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by  Gregory Arone

FIXED POINTS OF COISOTROPIC SUBGROUPS OF Γ ON
k
DECOMPOSITION SPACES
7
1
GREGORYARONEANDKATHRYNLESH
0
2
n Abstract. We study the equivariant homotopy type of the poset Lpk of or-
a thogonal decompositions of Cpk. The fixed point space of the p-radical sub-
J groupΓk⊂U(cid:0)pk(cid:1)actingonLpk isshowntobehomeomorphictoasymplectic
1 Titsbuilding,awedgeof(k−1)-dimensionalspheres. Oursecondresultcon-
2 cerns∆k=(Z/p)k ⊂U(cid:0)pk(cid:1)actingbytheregularrepresentation. Weidentify
] ahormetortaocptyotfytpheeofifxtehdepuonirnetduspceadcesuosfp∆enksiaocntionfgtohneTLiptks.buTihldisinrgetfroarcGtLhkas(Ftph)e,
T alsoawedgeof(k−1)-dimensionalspheres. Asaconsequenceoftheseresults,
A wefindthatthefixedpointspaceofanycoisotropicsubgroupofΓk contains,
as a retract, a wedge of (k−1)-dimensional spheres. We make a conjecture
.
h about the full homotopy type of the fixed point space of ∆k acting on Lpk,
t basedonamoregeneralbranchingconjecture,andweshowthattheconjecture
a
isconsistent withourresults.
m
[

1
1. Introduction
v
0 A proper orthogonal decomposition of Cn is an unordered collection of nontriv-
7
ial, pairwise orthogonal, proper vector subspaces of Cn whose sum is Cn. These
0
6 decompositionshavea partialorderinggivenby coarseningandaccordinglyforma
0 topologicalposetcategory,denotedL . ThecategoryL hasa(topological)nerve,
n n
1. also denoted Ln, and we trust to context to distinguish whether by Ln we mean
0 the poset (a category) or its nerve (a simplicial space). The action of U(n) on Cn
7 induces a natural action of U(n) on L , and we are interested in the fixed point
n
1
spaces of the action of certain subgroups of U(n) on L .
: n
v ThespaceL wasintroducedin[Aro02],inthecontextoftheorthogonalcalculus
n
i of M. Weiss. It plays an analogous role to that played in Goodwillie’s homotopy
X
calculus by the partition complex P , the poset of proper nontrivial partitions
r n
a of a set of n elements [AM99]. The space Ln made another, related appearance
in [AL07], in the filtration quotients for a filtration of the spectrum bu that is
analogous to the symmetric power filtration of the integral Eilenberg-MacLane
spectrum. The properties of L are particularly of interest in the context of the
n
“bu-Whitehead Conjecture” ([AL10] Conjecture 1.5).
The topologyand some of the equivariantstructure ofL werestudied in detail
n
in [BJL+15], and [BJL+]. In particular, the goal of those papers was to deter-
mine, for a prime p and for all p-toral subgroups H ⊆ U(n), whether (L )H is
n
contractible. This classification question is analogous to questions that had to be
answered in [ADL16], in the course of calculating the Bredon homology of P . In
n
the case of P , for coefficient functors that are Mackey functors taking values in
n
Z -modules, the p-subgroups of Σ with non-contractible fixed point spaces on
(p) n
Date:January24,2017.
1
2 GREGORYARONEANDKATHRYNLESH
P presentobstructionstoP havingthesameBredonhomologyasapoint. Fixed
n n
point spaces of subgroups of Σ acting on P were further studied in [Aro].
n n
Similarly, one expects that p-toral subgroups of U(n) acting on L with non-
n
contractible fixed point spaces will present obstructions to L having the same
n
Bredonhomologyasapoint,forcoefficientsthatareMackeyfunctorstakingvalues
in Z -modules. In this paper, we contribute to the understanding of these fixed
(p)
point spaces by identifying two critical cases of p-toral subgroups of U pk whose
(cid:0) (cid:1)
fixedpointspacesonL arenotonlynon-contractible,butactuallyhavehomology
pk
that is either free abelian or has a free abelian summand. When we put these
together with a join formula from [BJL+], we also obtain a similar result for all
coisotropic subgroups of Γ .
k
Our results have a similar flavor to results of [AD01] and [ADL16] in that they
involve Tits buildings. We also show that the results obtained are consistent with
amoregeneralconjectureaboutthe equivarianthomotopytypeofL analogousto
n
the branching rule of [Aro] for P .
n
The results of the current work are used in [BJL+] to give a complete classi-
fication of p-toral subgroups of U(n) with contractible fixed point spaces on L .
n
Unlike the case for P , where many elementary abelian p-subgroups of Σ have
n n
non-contractible fixed point sets [Aro], it turns out that the fixed point spaces of
most p-toral subgroups of U(n) are actually contractible. [BJL+] shows that the
only exceptions occur when n = qipj, where q is a prime different from p. Theo-
rems 1.2 and 1.3 below are used in [BJL+] to settle these cases.
To state our results explicitly, we need some notation for the two p-toral sub-
groups that we study. First, let ∆ denote the subgroup (Z/p)k ⊂ U pk where
k
(cid:0) (cid:1)
(Z/p)k acts on Cpk by the regular representation. Associated to ∆ is the Tits
k
building for GL (F ), denoted T GL (F ), which is the poset of proper, nontrivial
k p k p
subgroups of ∆ and has the homotopy type of a wedge of spheres. Second, let Γ
k k
be the irreducible projective elementary abelian p-subgroup of U pk (unique up
(cid:0) (cid:1)
to conjugacy), which is given by an extension
(1.1) 1→S1 →Γ →(Z/p)2k →1.
k
Here S1 denotes the center of U pk . (See Section 2 for a brief discussion, or
(cid:0) (cid:1)
[Oli94] or [BJL+] for a detailed discussion from basic principles.) The extension
(1.1) induces a symplectic form on (Z/p)2k by lifting to Γ and looking at the
k
commutator, which lies in S1 and has order p. Hence associated with Γ we have
k
the Tits building for the symplectic group, denoted T Sp (F ), which is the poset
k p
of proper coisotropic subgroups of (Z/p)2k, and like T GL (F ) has the homotopy
k p
type of a wedge of spheres.
Given a space X, let X⋄ denote the unreduced suspension of X. The following
are our main results.
Theorem 1.2. The fixed point space L Γk is homeomorphic to T Sp (F ).
(cid:0) pk(cid:1) k p
Theorem 1.3. The fixed point space L ∆k has T GL (F )⋄ as a retract.
(cid:0) pk(cid:1) k p
Wecanuseajoinformulafrom[BJL+]toidentifyawedgeofspheresasaretract
of the fixed point space of any coisotropic subgroup of Γ , where a coisotropic
k
subgroup means a subgroup of Γ that is the preimage in (1.1) of a coisotropic
k
subspace of (Z/p)2k.
FIXED POINTS OF COISOTROPIC SUBGROUPS 3
H
Corollary 1.4. If H ⊆Γ is coisotropic, then L has a retract that is homo-
k (cid:0) pk(cid:1)
topy equivalent to a wedge of spheres of dimension k−1.
Proof. Because H is coisotropic, it has the form Γ × ∆ for some s + t = k
s t
(Lemma 2.9). By [BJL+] Theorem 9.2, we find that
(cid:0)Lpk(cid:1)H ∼=(Lpt)∆t ∗(Lps)Γs.
Hence L H has T GL (F )⋄ ∗T Sp (F ) as a retract. But the Tits buildings
TGL (F(cid:0) )pka(cid:1)nd T Sp (F )t eapch have thse hpomotopy type of a wedge of spheres, of
t p s p
dimension t−2 and s−1, respectively, and the result follows. (cid:3)
Theorem 1.3 is good enough to complete the classification of [BJL+], for which
allthatisneededisthattheintegralhomologyof L ∆k hasasummandthatisa
(cid:0) pk(cid:1)
free abelian group. However, we actually have a conjectural description of the full
homotopytypeofthefixedpointspace L ∆k,basedonamoregeneralconjecture
(cid:0) pk(cid:1)
regardingtheequivarianthomotopytypeofL . WecanembedU(n−1)⊆U(n)(in
n
anonstandardway)asthesymmetriesoftheorthogonalcomplementofthediagonal
C ⊂ Cn, since that complement is an (n − 1)-dimensional vector space over C.
Observe that the standard inclusion Σ ֒→U(n) by permutation matrices actually
n
factors through this inclusion U(n−1) ⊂ U(n). Finally, let Sn−1 denote the one-
pointcompactificationofthe reducedstandardrepresentationofΣ onRn−1. The
n
general conjecture is as follows.
Conjecture 1.5. There is a U(n−1)-equivariant homotopy equivalence
L ≃U(n−1) ∧ P⋄∧Sn−1 .
n + Σn (cid:0) n (cid:1)
Remark 1.6. Conjecture 1.5 is motivated by the role of L in orthogonal calcu-
n
lus. On the one hand, L is closely related to the n-th derivative of the functor
n
V 7→ BU(V). This, together with the fibration sequence S1 ∧SV → BU(V) →
BU(V ⊕C) implies that the restriction of L to U(n−1) is closely related to the
n
n-th derivative of the functor V 7→ S1 ∧SV. On the other hand, by connection
with Goodwillie’s homotopy calculus, the n-th derivative of this functor is closely
relatedtoP⋄∧Sn−1. Infact, onecanusethis connectionto provethatthe equiva-
n
lenceinConjecture1.5istrueaftertakingsuspensionspectrumandsmashproduct
withEU(n) . Formoredetails see[Aro02], especiallyTheorem3,whichis equiva-
+
lenttothisassertion,modulostandardmanipulationsinvolvingSpanier-Whitehead
duality.
In the final section of this paper, we show what Conjecture 1.5 would imply
about the actual homotopy type of L ∆k. After some calculation, we find that
(cid:0) pk(cid:1)
Conjecture 1.5 implies the following conjecture.
Conjecture 1.7. Let C˜ = C (∆ )/ ∆ ×S1 . There is a homotopy equiv-
U(pk) k (cid:0) k (cid:1)
alance
(1.8) L ∆k ≃C˜ ∧T GL (F )⋄.
(cid:0) pk(cid:1) + k p
We observe that Theorem 1.3 is consistent with Conjecture 1.7.
Organization of the paper
In Section 2, we collect some background information about L , the p-toral
n
group Γ , and the symplectic Tits building. Section 3 proves Theorem 1.2, and
k
4 GREGORYARONEANDKATHRYNLESH
Section 4 proves Theorem 1.3. Finally, in Section 5 we show how to deduce Con-
jecture 1.7 from Conjecture 1.5, and we compute an example.
Throughout the paper, we assume that we have fixed a prime p. By a subgroup
of a Lie group, we always mean a closed subgroup.
2. Background on L and Γ
pk k
In this section, we give background results on decomposition spaces L , the
n
group Γ , and the symplectic Tits building.
k
As explained in Section 1, L is a poset category internal to topologicalspaces:
n
the objects and morphisms have an action of U(n) and are topologized as disjoint
unions of U(n)-orbits. If λ is an object of L , then we write cl(λ) for the set of
n
subspaces that make up λ, which are called the classes or components of λ. If a
decomposition λ is stabilized by the action of a subgroup H ⊆U(n), then there is
an action of H on cl(λ), which may be nontrivial.
In analyzing (L )H, there are two operations that are particularly helpful in
n
constructing deformation retractions to subcategories.
Definition 2.1. Suppose that H ⊆ U(n) is a closed subgroup, and λ is a decom-
position in (L )H.
n
(1) We define λ/H as the decomposition of Cn obtained by summing compo-
nents of cl(λ) that are in the same orbit of the action of H on cl(λ).
(2) If µis adecompositionofCn suchthatH actstriviallyoncl(µ) (i.e., every
component of µ is a representation of H), then we define µ as the
iso(H)
refinement of µ obtained by taking the canonical decomposition of each
component of µ into its H-isotypical summands.
Example 2.2. Let {e ,e ,e ,e } denote the standard basis for C4, and let Σ ⊂
1 2 3 4 4
U(4)actbypermutingthebasisvectors. LetǫdenotethedecompositionofC4 into
the four lines determined by the standard basis. Let H ∼= Z/2 ⊂ Σ be generated
4
by (1,2)(3,4). Then µ := ǫ/H consists of two components v = Span{e ,e } and
1 1 2
v =Span{e ,e }.
2 3 4
SinceeachcomponentofµisarepresentationofH,wecanrefineµas(ǫ/H) .
iso(H)
Eachof the components v andv decompose into one-dimensionaleigenspacesfor
1 2
the action of H, one for the eigenvalue +1 and one for the eigenvalue −1. Hence
(ǫ/H) is a decomposition of C4 into four lines, each of which is fixed by H,
iso(H)
whereH actsontwoofthembytheidentityandontheothertwobymultiplication
by −1.
Since L has a topology, it is necessary that the operations of Definition 2.1 be
n
continuous, which is proved in [BJL+] using the following lemma.
Lemma 2.3. The path components of the object and morphism spaces of (L )H
n
are orbits of the identity component of the centralizer of H in U(n).
TheproofofcontinuityoftheoperationsofDefinition2.1thengoesbyobserving
thattheoperationscommutewiththeactionofthecentralizerofH inU(n),which
defines the topology of (L )H, since the orbits of U(n) determine the topology
n
of L . See [BJL+] Section 4.
n
Our next job is to identify a smaller subcomplex of (L )H that is sometimes
n
good enough to compute the homotopy type of (L )H.
n
FIXED POINTS OF COISOTROPIC SUBGROUPS 5
Definition 2.4. Let H ⊆U(n) be a subgroup and suppose that λ is a decomposi-
tion in (L )H.
n
(1) For v ∈ cl(λ), we define the H-isotropy group of v, denoted I , as I =
v v
{h∈H :hv =v}.
(2) Wesaythatλhasuniform H-isotropy ifallelementsofcl(λ)havethesame
H-isotropygroup. Inthiscase,wewriteI fortheH-isotropygroupofany
λ
v ∈cl(λ), provided that the group H is understood from context.
Example 2.5. Suppose that λ ∈ Obj(L )H, and that H acts transitively on the
n
set cl(λ). If there exists v ∈cl(λ) such that I ⊳H, then λ necessarily has uniform
v
H-isotropy. This is because the transitive action of H means that the H-isotropy
groups of all components of λ are conjugate in H. Since I is normal, all the
v
isotropy groups are actually the same.
More specifically, suppose that H ⊂ U(n) has the property that H/(H ∩S1) is
elementary abelian, where S1 denotes the center of U(n). In this case we say that
H is “projective elementary abelian.” By the discussion above, if λ ∈ Obj(L )H
n
hasa transitive actionofH on cl(λ), thenλ has uniform H-isotropybecause every
subgroup of H containing H ∩S1 is normal.
ForH ⊂U(n),letUnif(L )H denotethesubposetof(L )H consistingofobjects
n n
withuniformH-isotropy. Asin[BJL+],wehavethefollowinglemma,statedslightly
more generally here.
Lemma 2.6. If H ⊂ U(n) is a projective abelian subgroup, then the inclusion
Unif(L )H →(L )H induces a homotopy equivalence on nerves.
n n
Proof. Exactly the same proof as in [BJL+] works here. If cl(λ) = {v ,...,v },
1 j
then because H is projective abelian, each I is normal in H, and the product
vi
J =I ...I is a normal subgroup of H. If λ/J were not proper, we would have
λ v1 vj λ
J (and hence also H) acting transitively on cl(λ). This would imply that J =
λ λ
I = ... = I acts transitively on cl(λ), which could only have one component, a
v1 vj
contradiction.
From this point, the proof is precisely as in [BJL+], by doing the routine checks
that λ 7→ λ/J is a continuous deformation retraction from (L )H to Unif(L )H.
λ n n
(cid:3)
Ournextorderofbusinessis toprovidealittle backgroundonthe groupswhose
fixed points we study in this paper. As in the introduction, we write ∆ for the
k
group(Z/p)k ⊂U pk acting onthe standardbasisof Cpk by the regularrepresen-
(cid:0) (cid:1)
tation. One of the goals of this paper is to understand the fixed point space of ∆
k
acting on L (Theorem 1.3 and Conjecture 1.7).
pk
The other important group in our results is Γ ⊂ U pk , which denotes a sub-
k
(cid:0) (cid:1)
group of U pk given by an extension
(cid:0) (cid:1)
1→S1 →Γ →(Z/p)k×(Z/p)k →1.
k
The groupΓ is discussedextensively anddescribed explicitly in terms ofmatrices
k
in [Oli94]. (See also [BJL+] for a discussion from first principles.) Each factor
of (Z/p)k has a splitting back into Γ , though the splittings of the two factors
k
do not commute in Γ . As a subgroup of Γ ⊆ U pk , the first factor of (Z/p)k
k k
(cid:0) (cid:1)
can be regarded as ∆ itself, acting on the standard basis of Cpk by the regular
k
6 GREGORYARONEANDKATHRYNLESH
representation. The second factor of (Z/p)k acts via the regular representation on
the pk one-dimensionalirreducible representationsof ∆ , whichare nonisomorphic
k
and span Cpk.
MovingontoTitsbuildings,recallthatasymplecticformonanF -vectorspace
p
V is a nondegenerate alternating bilinear form. It necessarily has even dimension.
LiftingelementsofΓ /S1toΓ andcomputingthecommutatordefinesasymplectic
k k
formon(Z/p)k×(Z/p)k. Olivershowsin[Oli94]thattheWeylgroupofΓ inU pk
k
(cid:0) (cid:1)
is the full group of automorphisms of this form. Therefore is is not surprising that
the fixed point space of Γ acting on L should be related to the symplectic Tits
k pk
building, which we describe next.
Definition 2.7.
(1) AsubspaceW ofasymplecticvectorspaceiscalledcoisotropic ifW⊥ ⊆W.
(2) We say that J ⊆Γ is a coisotropic subgroup if J is the inverse image of a
k
coisotropic subspace of (Z/p)2k.
(3) The symplectic Tits building, T Sp (F ), is the poset of proper coisotropic
k p
subgroups of Γ .
k
Example 2.8. To compute T Sp (F ), consider
1 p
1→S1 →Γ →(Z/p)2 →1.
1
Coisotropic subspaces have dimension at least half the dimension of the ambient
vector space, so here a proper coisotropic subspace of (Z/p)2 has dimension one.
Further, every one-dimensional subspace of a two-dimensional symplectic vector
space is coisotropic. The vector space (Z/p)2 has p+1 one-dimensionalsubspaces.
Since there are no possible inclusions between the subspaces, there are no mor-
phisms in the poset, and therefore the nerve of T Sp (F ) consists ofp+1 isolated
1 p
points.
In general, T Sp (F ) has the homotopy type of a wedge of spheres of dimen-
k p
sion k−1.
Finally, we need a couple of concrete lemmas about coisotropic subgroups. Let
H denote an s-dimensional vector space over Z/p with a symplectic form, and let
s
T denote a t-dimensional vector space with trivial form.
t
Lemma2.9. IfH ⊆Γ iscoisotropic, thenH hastheformΓ ×∆ wheres+t=k.
k s t
Proof. Acoisotropicsubspaceof(Z/p)2khasanalternatingformisomorphictoH ⊕
s
T where s+t=k. Further, H is classified up to isomorphism by its commutator
t
form, with H corresponding to Γ and T corresponding to ∆ . (A proof is given
s s t t
in [BJL+].) The result follows. (cid:3)
Lemma 2.10. If H ⊆Γ is coisotropic, then H has irreducibles of dimension ps,
k
iff H ∼=Γ ×∆ where s+t=k.
s t
Proof. We already know from Lemma 2.9 that H is isomorphic to H ∼= Γ ×∆
s t
where s+t = k. The lemma follows from the fact that Γ is acting on Cpk by
s
a multiple of the standard representation, and the irreducible representations of
Γ ×∆ are products of irreducible representations of Γ and (one-dimensional)
s t s
irreducible representations of ∆ . (cid:3)
t
FIXED POINTS OF COISOTROPIC SUBGROUPS 7
3. Fixed points of Γ acting on L
k pk
In this section, we prove the first theorem announced in the introduction.
Theorem 1.2. The fixed point space L Γk is homeomorphic to T Sp (F ).
(cid:0) pk(cid:1) k p
Thestrategyfortheproofisstraightforward: toestablishfunctorsfromT Sp (F )
k p
to L Γk and back, and to show that their compositions are identity functors.
(cid:0) pk(cid:1)
Defining the functions on objects is not difficult. To show that the maps are func-
torial and compose to identity functors requires some representation theory.
First, we observe that while T Sp (F ) is a discrete poset, it is not initially
k p
clear that L Γk is discrete, because L itself is a topological poset. While it is
(cid:0) pk(cid:1) pk
not logically necessary to verify discreteness up front, we begin this section with a
freestanding proof that L Γk is a discrete poset.
(cid:0) pk(cid:1)
Lemma 3.1. The object and morphism spaces of L Γk are discrete.
(cid:0) pk(cid:1)
Proof. By Lemma 2.3, the path components of Obj L Γk are orbits of the cen-
(cid:0) pk(cid:1)
tralizer of Γ in U pk . However, Γ is centralized in U pk only by the center S1
k k
(cid:0) (cid:1) (cid:0) (cid:1)
ofU pk [Oli94, Prop.4]). Since S1 actually fixes everyobjectofL , the S1-orbit
(cid:0) (cid:1) pk
of an object of L is just a point. Hence the path components of the object space
pk
of L Γk are single points, andthe object space of L Γk is discrete. The same
(cid:0) pk(cid:1) (cid:0) pk(cid:1)
isthennecessarilytrueofthemorphismspace,sincethereisatmostonemorphism
between any two objects and the source and target maps are continuous on the
morphism space. (cid:3)
We will define functions in both directions between the proper coisotropic sub-
groups of Γ and the objects of L Γk. If H is a subgroup of Γ , let λ de-
k (cid:0) pk(cid:1) k H
note the canonical decomposition of Cpk by H-isotypicalsummands. On the other
hand, recall that if λ is an object of L Γk, then λ necessarily has uniform Γ -
(cid:0) pk(cid:1) k
isotropy(Example2.5,becauseΓ actsirreduciblyonCpk). Wedenotethisisotropy
k
by I ⊂ U pk . Then we define the required correspondences between subgroups
λ
(cid:0) (cid:1)
and decompositions as follows: if H is a coisotropic subgroup of Γ , then
k
F(H)=λ
H
and if λ is a decomposition in L Γk, then
(cid:0) pk(cid:1)
G(λ)=I .
λ
We need to check that the image of F consists of proper decompositions of Cpk,
thatthe image ofG consistsof coisotropicsubgroups,thatF andG arefunctorial,
and that F and G are inverses of each other when F is restricted to coisotropic
groups.
To show that F and G are functors, we need a representation-theoretic lemma.
Lemma3.2. IfH is acoisotropic subgroupof Γ , thenthestandardrepresentation
k
of Γ on Cpk breaks into the sum of [Γ :H] irreducible representations of H, all
k k
of equal dimension, and pairwise non-isomorphic.
8 GREGORYARONEANDKATHRYNLESH
Proof. By Lemma 2.9, we know H ∼= Γ × ∆ with s + t = k, and the action
s t
of Γ ×∆ on Cpk ∼= Cps ⊗Cpt is conjugate to the action where Γ acts on the
s t s
first factor by the standard representation and ∆ acts on the second factor by
t
the regularrepresentation. Since H is a product, irreducible H-representationsare
obtained as tensor products of irreducible representations of Γ and of ∆ . There
s t
are pt = [Γ :H] irreducibles of ∆ acting on Cpt, all non-isomorphic, and the
k t
tensor products of these irreducibles with the standard representation of Γ are
s
again irreducible, span Cpk, and pairwise non-isomorphic (for example, since they
have different characters). (cid:3)
We obtain the following corollary to Lemma 3.2.
Corollary 3.3. If J ⊆Γ is coisotropic, then λ is the only J-isotypical decompo-
k J
sition of Cpk.
Proof. A decomposition of Cpk is J-isotypical if and only if each one of its com-
ponents is an isotypical representation of J. Every J-isotypical decomposition of
Cpk is a refinement of λ . By Lemma 3.2, each component of λ is irreducible.
J J
Hence λ has no J-isotypical refinements, and therefore it is the only J-isotypical
J
decomposition of Cpk. (cid:3)
With Corollary3.3in hand,we canestablishthatF is functorialfromthe poset
of coisotropic subgroups of Γ .
k
Proposition 3.4. F is a functor from TSp (F ) to L Γk.
k p (cid:0) pk(cid:1)
Proof. SupposeH isanobjectofT Sp (F ),thatis,apropercoisotropicsubgroup
k p
ofΓ . SinceH⊳Γ ,theactionofΓ onCpk permutestheirreduciblerepresentations
k k k
of H and hence stabilizes λ (while possibly permuting its components). Further,
H
by Lemma 3.2, λ has [Γ :H]>1 components, so λ is a proper decomposition
H k H
of Cpk.
Further, if J ⊆ H are two coisotropic subgroups of Γ , then every component
k
of λ is a representation of H, and hence also of J. Consider the decomposi-
H
tion (λ ) . It is J-isotypical, by definition, and so by Corollary 3.3, we know
H iso(J)
that (λ ) =λ . It followsthat λ is a refinementof λ , so F is a functor on
H iso(J) J J H
the poset of coisotropic subgroups of Γ . (cid:3)
k
Next we turn our attention to the function G from objects of L Γk to sub-
(cid:0) pk(cid:1)
groupsof Γ . By way ofpreparation,we need a key representation-theoreticresult
k
similar to Lemma 3.2. Given an irreducible representation σ of a group G and
another representation τ of G, let [τ :σ] denote the multiplicity of σ in τ.
Lemma 3.5. Let λ be an object of L Γk, and let I denote the (uniform) Γ -
(cid:0) pk(cid:1) λ k
isotropy subgroup of its components. Then the representations of I afforded by the
λ
components of λ are pairwise non-isomorphic irreducible representations of I .
λ
Corollary 3.6. If λ∈Obj L Γk, then FG(λ)=λ.
(cid:0) pk(cid:1)
Proof. By definition, G(λ)=I , so the question is to find the canonical isotypical
λ
decompositionof I . Lemma 3.5 says that all components of λ are non-isomorphic
λ
irreducible representations of I , so in fact F(I )=λ. (cid:3)
λ λ
FIXED POINTS OF COISOTROPIC SUBGROUPS 9
Proof of Lemma 3.5. Let ρ denote the standard representation of Γ on Cpk. The
k
action of Γ /I on cl(λ) is free and transitive (the latter because Γ acts irre-
k λ k
ducibly), so if we choose v ∈cl(λ), then ρ is induced from the representation of I
λ
given by v. We conclude that v is an irreducible representation of I , since it in-
λ
ducesthe irreduciblerepresentationρ. The sameistrue foreveryothercomponent
of λ, so the components of λ are a decomposition of Cpk into I -irreducibles.
λ
We canapply Frobenius reciprocity (see, for example,[Kna96, Theorem9.9]) to
conclude that:
IndΓk(v):ρ =[ρ| :v].
h Iλ i Iλ
Because IndΓk(v) ∼= ρ, we conclude that [ρ| :v] = 1. However, ρ| is a direct
Iλ Iλ Iλ
sum of the irreducible I -modules given by the components of λ. If any other
λ
componentofλwereisomorphictov asarepresentationofI ,thenwewouldhave
λ
[ρ| :v]≥2, contrary to the calculation above. (cid:3)
Iλ
In addition to showing that F is a left inverse for G, Lemma 3.5 also allows us
tocheckthatsubgroupsintheimageofGareactuallycoisotropicsubgroupsofΓ .
k
Lemma 3.7. If λ is an object of L Γk, then I is a coisotropic subgroup of Γ .
(cid:0) pk(cid:1) λ k
Proof. We have the following ladder of short exact sequences:
1 −−−−→ S1 −−−−→ I −−−−→ W −−−−→ 1
λ
=  
  
1 −−−−→ Sy1 −−−−→ Γy −−−−→ (Z/yp)2k −−−−→ 1.
k
We must show that if z ∈ W⊥ ⊆ (Z/p)2k, then in fact z ∈ W. Recall that the
symplectic form on (Z/p)2k is given by the commutator pairing: if we denote lifts
of z and w by z˜ and w˜, then the symplectic form evaluated on the pair (z,w) is
givenby the commutator [z˜,w˜]∈S1. Hence if z pairs to 0 with all elements of W,
it means that z˜is actually in the centralizer of I in Γ . Thus is it sufficient for us
λ k
to show that if z˜∈Γ centralizes I , then z˜∈I .
k λ λ
However,ifz˜centralizesI andv ∈cl(λ),thenz˜givesanontrivialI -equivariant
λ λ
mapbetweentheI -representationsv andz˜v. ByLemma3.5,ifv 6=z˜v,thenv and
λ
z˜v are non-isomorphic irreducible representations of I , so Schur’s Lemma tells us
λ
thatthereisnonontrivialI -equivariantmap. We concludethatz˜v =v,soz˜∈I ,
λ λ
as required. (cid:3)
Finally, the last step is to show that the functors F and G are inverses of each
other.
Proof of Theorem 1.2.
The functors F : H 7→ λ and G : λ 7→I induce the desired homeomorphism,
H λ
onceweshowthattheyareinversesofeachother. Corollary3.6alreadytellsusthat
FG(λ) = λ. To finish the proof of the theorem, we must show if H is coisotropic,
then GF(H)=H, that is, the Γ -isotropy subgroup of λ is H itself.
k H
By definition of λ , the components of λ are H-representations, so certainly
H H
H ⊆ I . Both H and I are coisotropic, by assumption and by Lemma 3.7,
λH λH
respectively. However, a coisotropic subgroup of Γ is determined up to isomor-
k
phismbythe dimensionofitsirreduciblesummandsinthestandardrepresentation
ofΓ (Lemma2.10). Further,thecomponentsofλ areirreduciblerepresentations
k H
10 GREGORYARONEANDKATHRYNLESH
for both H (Lemma 3.2) and I (Lemma 3.5). Hence H ⊆ I have the same
λH λH
irreducible summands on Cpk and must be isomorphic, and therefore equal. (cid:3)
4. Fixed points of ∆ acting on L
k pk
LetT GL (F )denotetheTitsbuildingforGL (F ),thatis,theposetofproper
k p k p
nontrivial subgroups of ∆ . In this section, we prove the following result.
k
Theorem 1.3. The fixed point space L ∆k has T GL (F )⋄ as a retract.
(cid:0) pk(cid:1) k p
To setup the proof, we follow a similar strategyto [BJL+]. RecallUnif L ∆k
(cid:0) pk(cid:1)
denotes the subposet of L ∆k consisting of objects with uniform ∆ -isotropy,
(cid:0) pk(cid:1) k
and that Unif L ∆k ֒→ L ∆k is a homotopy equivalence (Lemma 2.6). We
(cid:0) pk(cid:1) (cid:0) pk(cid:1)
analyze Unif L ∆k in terms of two subposets.
(cid:0) pk(cid:1)
Definition 4.1.
(1) Let L ∆k ⊆ Unif L ∆k consist of objects λ such that ∆ does not
(cid:0) pk(cid:1)Ntr (cid:0) pk(cid:1) k
act transitively on cl(λ).
(2) Let L ∆k ⊆Unif L ∆k consist of objects λ such that ∆ acts non-
(cid:0) pk(cid:1)move (cid:0) pk(cid:1) k
trivially on cl(λ).
Example 4.2. ChooseanorthonormalbasisE ofCpk onwhich∆ actsfreelyand
k
transitively. (Recall that ∆ is acting on Cpk by the regular representation.) Let
k
ǫ be the corresponding decomposition of Cpk into the lines, each line generated by
an element of E. Then ǫ is an object of L ∆k but not of L ∆k, and the
(cid:0) pk(cid:1)move (cid:0) pk(cid:1)Ntr
same is true for ǫ/K for any proper subgroup K ⊆∆ .
k
Conversely, let H be any subgroup of ∆ . Then λ is an element of L ∆k
k H (cid:0) pk(cid:1)Ntr
but not of L ∆k .
(cid:0) pk(cid:1)move
We observe that refinements of objects in L ∆k are still in L ∆k, and
(cid:0) pk(cid:1)Ntr (cid:0) pk(cid:1)Ntr
refinements ofobjects in L ∆k arestill in L ∆k . Further, everyobject of
(cid:0) pk(cid:1)move (cid:0) pk(cid:1)move
Unif L ∆k is in one of these two subposets. Hence we have a pushout diagram
(cid:0) pk(cid:1)
of nerves
L ∆k ∩ L ∆k −−−−→ L ∆k
(cid:0) pk(cid:1)Ntr (cid:0) pk(cid:1)move (cid:0) pk(cid:1)Ntr
(4.3)
 
 
L y∆k −−−−→ Unif Ly ∆k
(cid:0) pk(cid:1)move (cid:0) pk(cid:1)
whichisinfactahomotopypushoutbecausethemapsoriginatinginthe upperleft
corner are cofibrations on the level of nerves.
To prove Theorem 1.3, we will use the expected steps to show that the nerve of
Unif L ∆k has T GL (F )⋄ as a retract: finding a retraction map, exhibiting a
(cid:0) pk(cid:1) k p
correspondinginclusion, and showing that the inclusion andretractioncompose to
a self-equivalence of T GL (F )⋄.
k p
Our first step is to use diagram (4.3) to produce a map from the nerve of
Unif L ∆k to the double cone on T GL (F ). Unlike the rest of the arguments
(cid:0) pk(cid:1) k p