Dominant K-theory and Integrable highest weight representations of Kac-Moody groups

0.35 MB·English

by  Nitu Kitchloo

DOMINANT K-THEORY AND INTEGRABLE HIGHEST WEIGHT
REPRESENTATIONS OF KAC-MOODY GROUPS
NITUKITCHLOO
ToHaynesMilleronhis60thbirthday
8
0
0
ABSTRACT. We give a topological interpretation of the highest weight representationsof
2
Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we de-

n fineaversionofequivariantK-theory,KG onthecategoryofproperG-CWcomplexes. We
a
then study Kac-Moody groups of compact type in detail (see Section 2 for definitions).
J
In particular, we show that the Grothendieck group of integrable hightest weight repre-
2
1 sentations of a Kac-Moodygroup G of compacttype, maps isomorphically onto K˜∗G(EG),
whereEG istheclassifyingspaceofproperG-actions. Fortheaffinecase,thisagreesvery

] wellwithrecentresultsofFreed-Hopkins-Teleman.WealsoexplicitlycomputeK∗(EG)for
T G
A Kac-Moodygroupsofextendedcompacttype,whichincludestheKac-MoodygroupE10.
.
h
t
a
m
[

2
v
CONTENTS
7
6
1 1. Introduction 2
0
. Organization of the Paper: 3
0
1 Acknowledgements 3
7
0 2. Background and Statement ofResults 3
:
v Conventions 7
i
X
3. Structure of the space X(A) 8
r
a 4. Dominant K-theory 10
5. The DominantK-theory for the compact type 14
6. The Geometricidentification 15
7. Dominant K-theory for the extended compact type 20
8. Some related remarks 24
9. Dominant K-homology 28
AppendixA. Acompatible familyofmetrics 31
References 32
Date:February2,2008.
NituKitchlooissupportedinpartbyNSFthroughgrantDMS0705159.
1
1. INTRODUCTION
Given a compact Lie group G and a G-space X, the equivariant K-theory of X, K∗(X)
G
can be described geometrically in terms of equivariant vector bundles on X. If one tries
to relax the condition of G being compact, one immediately runs into technical problems
in the definition of equivariant K-theory. One way around this problem is to impose
conditions on the action of the group G on the space X. By a proper action of a topolog-
ical group G on a space X, we shall mean that X has the structure of a G-CW complex
with compact isotropy subgroups. There exists a universal space with a proper G-action,
known as the classifying space for proper G-actions, which is a terminal object (up to G-
equivariant homotopy) in the category of proper G-spaces [LM, tD]. One may give an
alternate identification of thisclassifying space as:
Definition 1.1. For a topological group G, the classifying space for proper actions EG is a G-
CW complex with the property that all the isotropy subgroups are compact, and given a compact
subgroup H ⊆ G, thefixedpointspaceEGH isweakly contractible.
Notice that if G is a compact Lie group, then EG is simply equivalent to a point, and so
K∗(EG) is isomorphic to the representation ring of G. For a general non-compact group
G
G, the definition orgeometric meaningof K∗(EG) remainsunclear.
G
In this paper, we deal with a class of topological groups known as Kac-Moody groups
[K1, K2]. By a Kac-Moody group, we shall mean the unitary form of a split Kac-Moody
C
group over . We refer the reader to [Ku] for a beautiful treatment of the subject. These
groups form a natural extension of the class of compact Lie groups, and share many of
theirproperties. Theyare known to contain the class of(polynomial) loop groups, which
go by the name of affine Kac-Moody groups. With the exception of compact Lie groups,
C
Kac-Moody groups over are not even locally compact (local compactness holds for
Kac-Moody groups defined over finite fields). However, the theory of integrable highest
weightrepresentationsdoesextendtotheworldofKac-Moodygroups. Foraloopgroup,
these integrable highest weight representations form the well studied class of positive
energy representations [PS].
It is therefore natural to ask if some version of equivariant K-theory for Kac-Moody
groups encodes the integrable highest weight representations. The object of this paper
K
is two fold. Firstly, we define a version of equivariant K-theory as a functor on the
G
category of proper G-CW complexes, where G is a Kac-Moody group. For reasons that
will become clear later, we shall call this functor ”Dominant K-theory”. For a compact
Lie group G, this functor is usual equivariant K-theory. Next, we build an explicit model
for the classifying space of proper G-actions, EG, and calculate the groups K∗(EG) for
G
Kac-Moody groupsG ofcompact, andextendedcompacttype(seeSection 2). Indeed,we
construct a map from the Grothendieck group of highest weight representations of G, to
K∗(EG) for a Kac-Moody group of compact type, and show that this map is an isomor-
G
phism.
This document is directly inspired by the following recent result of Freed, Hopkins and
˜
Teleman [FHT] (see also [M]): Let G denote a compact Lie group and let LG denote the
universal central extension ofthe group ofsmooth free loops on G. In [FHT], the authors
calculatethetwisted equivariantK-theory oftheconjugation actionofGonitselfandde-
˜
scribeitastheGrothendieckgroupofpositiveenergyrepresentationsofLG. Thepositive
energy representations form an important class of (infinite dimensional) representations
2
˜
ofLG[PS]thatare indexed byan integerknown as’level’. IfGissimplyconnected, then
˜
theloopgroup LGadmitsaKac-Moody form knownastheaffineKac-Moodygroup. We
will show that the Dominant K-theory of the classifying space of proper actions of the
corresponding affine Kac-Moody group is simply the graded sum (under the action of
the central circle) of the twisted equivariant K-theory groups of G. Hence, we recover
the theorem of Freed-Hopkins-Teleman in the special case of the loop group of a simply
connected, simple, compact Liegroup.
Organization of the Paper:
WebegininSection2withbackgroundonKac-Moodygroupsanddescribethemaindefi-
nitions,constructionsandresultsofthispaper. Section3describesthefinitetypetopolog-
ical Tits building as a model for the classifying space for proper actions of a Kac-Moody
group, and in Section 4 we study the properties of Dominant K-theory as an equivariant
cohomologytheory. InSections5wecomputetheDominantK-theoryoftheTitsbuilding
foraKac-Moodygroupofcompacttype,andinsection6wegiveourcomputation ageo-
metricinterpretation intermsofanequivariantfamilyofcubicDiracoperators. Section 7
explorestheDominantK-theory ofthebuildingfortheextendedcompacttype. Section 8
is a discussion on various related topics, including the relationship of our work with the
work of Freed-Hopkins-Teleman. Included also in this section are remarks concerning
realformsofKac-Moody groups, andthegroup E . Finally,insection 9,wedefineacor-
10
respondingDominantK-homology theory, byintroducingtheequivariantdualofproper
complexes. We also compute the Dominant K-homology of the building in the compact
type. The Appendix is devoted to the construction of a compatible family of metrics on
the TitsBuilding, which is required in section 6.
Acknowledgements.
The author acknowledges his debt to the ideas described in [FHT] and [AS]. He would
also like to thank P. E. Caprace, L. Carbone, John Greenlees and N. Wallach for helpful
feedback.
2. BACKGROUND AND STATEMENT OF RESULTS
Kac-Moody groups have been around for more than twenty years. They have been ex-
tensivelystudiedandmuchisknownabouttheirgeneralstructure, representationtheory
andtopology[K2,K3,K4,Ki,Ku,KW,T](see[Ku]foramodernprespective). Onebegins
with a finite integral matrix A = (a ) with the properties that a = 2 and a ≤ 0 for
i,j i,j∈I i,i i,j
i 6= j. Moreover, we demand that a = 0 if and only if a = 0. These conditions define
i,j j,i
a Generalized Cartan Matrix. A generalized Cartan matrix is said to be symmetrizable if it
becomessymmetric aftermultiplication with asuitable rational diagonal matrix.
GivenageneralizedCartanmatrixA,onemayconstructacomplexLiealgebrag(A)using
the Harishchandra-Serre relations. This Lie algebra contains a finite dimensional Cartan
subalgebra h that admits an integral form hZ and a real form hR = hZ ⊗ R. The lattice hZ
contains a finite set of primitive elements h ,i ∈ I called ”simple coroots”. Similarly, the
i
dual lattice h∗ contains a special set of elements called ”simple roots” α ,i ∈ I. One may
Z i
decomposeg(A)undertheadjointactionofhtoobtainatriangularformasintheclassical
theory of semisimple Lie algebras. Let η denote the positive and negative ”nilpotent”
±
subalgebrasrespectively,andletb = h⊕η denotethecorresponding”Borel”subalgebas.
± ±
3
Thestructuretheoryforthehighestweightrepresentationsofg(A)leadstoaconstruction
(in much the same way that Chevalley groups are constructed), of a topological group
G(A)calledthe(minimal,split)Kac-Moodygroupoverthecomplexnumbers. Thegroup
G(A)supportsacanonicalanti-linearinvolutionω,andonedefinestheunitaryformK(A)
as the fixed group G(A)ω. It is the group K(A) that we study in this article. We refer the
readerto [Ku]for detailson the subject.
Given a subset J ⊆ I, one may define a parabolic subalgebra g (A) ⊆ g(A) generated by
J
b+ and the root spaces corresponding to the set J. For example, g∅(A) = b+. One may
exponentiate these subalgebras to parabolic subgroups G (A) ⊂ G(A). We then define
J
the unitary Levi factors KJ(A) to be the groups K(A) ∩ GJ(A). Hence K∅(A) = T is a
torus of rank 2|I|− rk(A), called the maximal torus of K(A). The normalizer N(T) of T
in K(A), is an extension of a discrete group W(A) by T. The Weyl group W(A) has the
structure ofacrystallographic Coxetergroup generated byreflectionsr ,i ∈ I. ForJ ⊆ I,
i
let W (A) denote the subgroup generated by the corresponding reflections r ,j ∈ J. The
J j
group W (A) is a crystallographic Coxeter group in its own right that can be identified
J
with the Weylgroup of K (A).
J
We will identify the type of a Kac-Moody group K(A), by that of its generalized Cartan
matrix. For example, a generalizedCartan matrix Aiscalled of Finite Typeif the Lie alge-
bra g(A) is a finite dimensional semisimple Lie algebra. In this case, the groups G(A) are
the corresponding simply connected semisimple complex Lie groups. Another sub-class
ofgroups G(A)correspond toCartanmatricesthatareofAffineType. Thesearenon-finite
type matrices A which are diagonalizable with nonnegative real eigenvalues. The group
of polynomial loops on a complex simply-connected semisimple Lie group can be seen
as Kac-Moody group of affine type. Kac-Moody groups that are not of finite type or of
affine type, are said to be of Indefinite Type. We will say that a generalized Cartan matrix
A is of Compact Type, if for every proper subset J ⊂ I, the sub matrix (a ) is of finite
i,j i,j∈J
type (see [D3] (Section 6.9) for a classification). It is known that indecomposable gener-
alized Cartan matrices of affine type are automatically of compact type. Finally, for the
purposes of this paper, we introduce the Extended Compact Type defined as a generalized
CartanmatrixA = (a ) ,forwhichthereisa(unique)decompositonI = I J ,with
i,j i,j∈I 0` 0
thepropertythatthesubCartanmatrix(a ) isofnon-finitetypeifandonlyifI ⊆ J.
i,j i,j∈J 0
Given a generalized Cartan matrix A = (a ) , define a category S(A) to be the poset
i,j i,j∈I
category (under inclusion) of subsets J ⊆ I such that K (A) is a compact Lie group.
J
This is equivalent to demanding that W (A) is a finite group. Notice that S(A) contains
J
all subsets of I of cardinality less than two. In particular, S(A) is nonempty and has an
initial object given by the empty set. However, S(A) need not have a terminal object. In
fact, it hasaterminal object exactly ifAisoffinite type. Thecategory S(A) isalsoknown
asthe poset ofspherical subsets [D4].
Remark 2.1. The topology on the group K(A) is the strong topology generated by the compact
subgroups K (A) for J ∈ S(A) [K2, Ku]. More precisely, K(A) is the amalgamated product
J
of the compact Lie groups K (A), in the category of topological groups. For a arbitrary subset
J
L ⊆ I, the topology induced on homogeneous space of the form K(A)/K (A) makes it into a
L
CW-complex,with only evencells,indexedby the setofcosetsW(A)/W (A).
L
4
We now introduce the topological Tits building, which is a space on which most of our
constructionsrest. AssumethatthesetI hascardinalityn+1,andletusfixanorderingof
the elements of I. Notice that the geometric n simplex ∆(n) has faces that can be canon-
ically identified with proper subsets of I. Hence, the faces of codimension k correspond
to subset of cardinality k. Let ∆ (n) be the face of ∆(n) corresponding to the subset J.
J
If we let B∆(n) denote the Barycentric subdivision of ∆(n), then it follows that the faces
of dimension k in B∆(n) are indexed on chains of length k consisting of proper inclu-
sions ∅ ⊆ J ⊂ J ⊂ ...J ⊂ I. Let |S(A)| denote the subcomplex of B∆(n) consisting
1 2 k
of those faces for which the corresponding chain is contained entirely in S(A). Hence-
forth, we identify |S(A)| as a subspace of ∆(n). The terminology is suggestive of the fact
that |S(A)| is canonically homeomorphic to the geometric realization of the nerve of the
category S(A).
Definition 2.2. Definethe (finite-type)TopologicalTitsbuilding X(A)as the K(A)-space:
K(A)/T ×|S(A)|
X(A) = ,
∼
wherewe identify(gT,x)with (hT,y)iffx = y ∈ ∆ (n),and g = hmodK (A).
J J
Remark2.3. Analternate definitionofX(A) isas the homotopy colimit[BK]:
X(A) = hocolim F(J),
J∈S(A)
whereF isthe functor fromS(A) toK(A)-spaces,such thatF(J) = K(A)/K (A).
J
Notice that by construction, X(A) is a K(A)-CW complex such that all the isotropy sub-
groups are compact Lie groups. In fact, we prove the following theorem in Section 3.
Theorem 2.4. The space X(A) is equivalent to the classifying space EK(A) for proper K(A)-
actions.
The WeylchamberC, itsfacesC , andthe space Y, are defined assubspacesof h∗:
J R
C = {λ ∈ h∗ |λ(h ) ≥ 0, i ∈ I}, C = Interior{λ ∈ C|λ(h ) = 0,∀j ∈ J}, Y = W(A)C.
R i J j
ThespaceYhasthestructureofacone. IndeeditiscalledtheTitscone. TheWeylchamber
isthe fundamental domain for the W(A)-action on the Titscone. Moreover, the stabilizer
ofany pointin C isW (A).
J J
The subset of h∗ contained in C is called the set of dominant weights D, and those in the
Z
interior of Care called the regular dominantweights D :
+
D = {λ ∈ h∗Z | λ(hi) ≥ 0, i ∈ I} D+ = {λ ∈ h∗Z | λ(hi) > 0, i ∈ I}.
Asisthecaseclassically,thereisabijectivecorrespondance betweenthesetofirreducible
highest weight representations, and the setD. Thisbijection identifies an irreducible rep-
resentation L of hightest weight µ, with the element µ ∈ D. It is more convenient for
µ
our purposes to introduce a Weyl element ρ ∈ h∗ with the property that ρ(h ) = 1 for all
Z i
i ∈ I. The Weyl element may not be unique, but we fix a choice. We may then index
the set of irreducible highest weight representations on D by identifying an irreducible
+
representation L ofhightestweightµ,with theelementµ+ρ ∈ D . Therelevanceofthe
µ +
elementρ will bemade clearin Section 5.
For the sake of simplicity, assume that A is a symmetrizable generalized Cartan matrix.
In this case, any irreducible representation L admits a K(A)-invariant Hermitian inner
µ
product. Letusnow definea dominantrepresentation ofK(A) in a Hilbertspace:
5
Definition 2.5. We say that a K(A)-representation in a seperable Hilbert space is dominant if
it decomposes as a sum of irreducible highest weight representations. In particular, we obtain a
maximaldominantK(A)-representationH(inthesensethatanyotherdominantrepresentationis
asummand)by takingacompletedsumofcountably manycopiesof allirreduciblehighestweight
representations.
The DominantK-theoryspectrum:
We are now ready to define Dominant K-theory as a cohomology theory on the category
of proper K(A)-CW complexes. As is standard, we will do so by defining a representing
object for Dominant K-theory. Recall that usual 2-periodic K-theory is represented by
homotopy classes ofmapsinto the infinite grassmannian Z×BU in evenparity, and into
the infinite unitary group U in odd parity. The theorem of Bott periodicity relates these
spacesvia ΩU = Z×BU, ensuringthat thisdefinesa cohomology theory.
This structure described above can be formalized using the notion of a spectrum. In par-
E
ticular, a spectrum consists of a family of pointed spaces indexed over the integers,
n
endowed with homeomorphisms E → ΩE . For a topological group G, the objects
n−1 n
that represent a G-equivariant cohomology theory are known as G-equivariant spectra.
E
In analogy to a spectrum, an equivariant spectrum consists of a collection of pointed
G-spacesE(V),indexedonfinitedimensionalsub-representationsV ofaninfinitedimen-
sional unitary representation of G in a separable Hilbert space (known as a ”universe”).
In addition, these spacesare related on taking suitable loop spaces. For a comprehensive
reference on equivariantspectra, see [LMS].
Henceforth, all our K(A)-equivariant spectra are to be understood as naive equivariant
spectra,i.e. indexedonatrivialK(A)-universe,orequivalently,indexedovertheintegers.
Indeed, there are no interesting finite dimensional representations of K(A), and so it is
even unclearwhata nontrivial universe would mean.
Let KU denote the K(A)-equivariant periodic K-theory spectrum represented by a suit-
able model F(H), for the space of Fredholm operators on H [AS, S]. The space F(H) is
chosensothattheprojective unitarygroup PU(H)(withthecompactopentopology) acts
continuouslyonF(H)[AS](Prop3.1). Bymaximality,noticethatwehaveanequivalence:
H⊗H = H. Hence KU isnaturally a twoperiodic, equivariantringspectrum.
Definition 2.6. Givena properK(A)-CW complexY, the DominantK-theory of Y is definedas
the group of equivarianthomotopy classesof maps:
K2k (Y) = [Y,F(H)] , and K2k+1(Y) = [Y,ΩF(H)] .
K(A) K(A) K(A) K(A)
There is nothing specialabout spaces. In fact, we may define the Dominant K-cohomology groups
of aproperK(A)-CW spectrumY as thegroup of equivarianthomotopy classesofstable maps:
Kk (Y) = [Y,ΣkKU] .
K(A) K(A)
Remark 2.7. Given a closed subgroup G ⊆ K(A), we introduce the notation AK∗(X), when
G
necessary,tomeanthe restrictionof DominantK-theory fromK(A)-spectrato G-spectra. Hence
AKk(X) = Kk (K(A) ∧ X) = [X,ΣkKU] .
G K(A) + G G
6
Conventions.
To avoid redundancy, we will prove our results in this paper under the assumption that
the Kac-Moody group beingconsidered isnot offinite type.
LetAut(A)denotetheautomorphismsofK(A)inducedfromautomorphismsofg(A). The
outer automorphism group of K(A) is essentially a finite dimensional torus, extended
by a finite group of automorphisms of the Dynkin diagram of A [KW]. Given a torus
T′ ⊆ Aut(A), we call groups of the form T′ ⋉ K(A), Reductive Kac-Moody Groups. The
unitary Levi factors K (A) are examplesofsuch groups.
J
AllargumentsinthispaperarebasedontwogeneralnotionsinthetheoryofKac-Moody
groups: Theory of Highest Weight representations [Ku], and the theory of BN-pairs and
Buildings[D4,K2,T]. BoththesenotionsextendnaturallytoreductiveKac-Moodygroups,
andconsequentlytheresultsofourpaperalsoextendinanobviouswaytoreductiveKac-
Moody groups. Thiswill beassumed implicitly throughout.
Insection 5 we planto prove the following theorem:
Theorem 2.8. Assume that the generalized Cartan matrix A has size n+1, and that K(A) is of
∅
compact type, but not of finite type. Let T ⊂ K(A) be the maximal torus, and let R denote the
T
regular dominantcharacterringof K(A) (i.e. charactersof T generated bythe weightsinD ).
+
For the space X(A), define the reduced Dominant K-theory K˜∗ (X(A)), to be the kernel of the
K(A)
restrictionmap along any orbitK(A)/T inX(A). We havean isomorphismofgradedgroups:
K˜∗ (X(A)) = R∅[β±1],
K(A) T
∅
whereβ isthe Bottclassindegree2,and R isgradedsoas to belongentirely indegreen.
T
∅
Remark 2.9. Note that we may identify R with the Grothendieck group of irreducible highest
T
weight representations of K(A), as described earlier. Under this identification, we will goemetri-
callyinterpretthe above theoremusing an equivariantfamilyof cubicDiracoperators.
InSection 7, we extend the above theorem to showthat:
Theorem2.10. LetK(A)beaKac-Moodygroupofextendedcompacttype,with|I | = n+1and
0
n > 1. Let A = (a ) denote the sub Cartan matrix of compact type. Then X(A ) is also a
0 i,j i,j∈I0 0
classifyingspaceof properK (A)-actions, and thefollowing restrictionmapis aninjection:
I0
r : K˜∗ (X(A)) −→ AK˜∗ (X(A )).
K(A) KI0(A) 0
AdominantK(A)-representationrestrictstoadominantK (A)-representation,inducingamap:
I0
St : AK˜∗ (X(A )) −→ K˜∗ (X(A )).
KI0(A) 0 KI0(A) 0
Since K (A) is a (reductive) Kac-Moody group of compact type, the previous theorem identifies
I0
K˜∗ (X(A ))withtheidealgeneratedbyregulardominantcharactersofK (A). Furthermore,
KI0(A) 0 I0
the map St is injective, and under the above identification, has image generated by those regular
dominantcharactersofK (A)whichareintheW(A)-orbitofdominantcharactersofK(A). The
I0
imageofK˜∗ (X(A))insideK˜∗ (X(A ))maythenbeidentifiedwithcharactersintheimage
K(A) KI0(A) 0
of Stwhichare alsoantidominant charactersof K (A).
J0
7
3. STRUCTURE OF THE SPACE X(A)
Before we begin a detailed study of the space X(A) that will allow us prove that X(A)
is equivalent to the classifying space of proper K(A)- actions, let us review the structure
of the Coxeter group W(A). Details can be found in [H]. The group W(A) is defined as
follows:
W(A) = hr , i ∈ I|(r r )mi,j = 1i,
i i j
where the integers m depend on the product of the entries a a in the generalized
i,j i,j j,i
Cartan matrix A = (a ). In particular, m = 1. The word length with respect to the
i,j i,i
generators r defines a length function l(w) on W(A). Moreover, we may define a partial
i
order on W(A) known as the Bruhat order. Under the Bruhat order, we say v ≤ w if v
may be obtained from some (in fact any!) reduced expression for w by deleting some
generators.
Given any two subsets J,K ⊆ I, let W (A) and W (A) denote the subgroups generated
J K
by the elements r for j ∈ J (resp. K). Let w ∈ W(A) be an arbitrary element. Consider
j
the double coset W (A)wW (A) ⊆ W(A). This double coset contains a unique element
J K
w of minimal length. Moreover, any other element u ∈ W (A)wW (A) can be written
0 J K
in reduced form: u = αw β, where α ∈ W (A) and β ∈ W (A). It should be pointed
0 J K
out that the expression αw β above, is not unique. We have αw = w β if and only if
0 0 0
α ∈ W (A),whereW (A) = W (A) ∩ w W (A)w−1 (anysuchintersection canbeshown
L L J 0 K 0
to begenerated by a subsetL ⊆ J).
We denote the set of minimal W (A)-W (A) double coset representatives by JWK. If
J K
K = ∅, then we denote the set of minimal left W (A)-coset representatives by JW. The
J
following claim follows from the above discussion:
Claim 3.1. Let J,K be any subsets of I. Let w be any element of JW. Then w belongs to the set
JWK ifand only ifl(wr ) > l(w)for allk ∈ K.
k
Let us now recall the generalized Bruhat decomposition for the split Kac-Moody group
G(A). Given any two subsets J,K ⊆ I. Let G (A) and G (A) denote the corresponding
J K
parabolic subgroups. Then the group G(A) admits a Bruhat decomposition into double
cosets:
G(A) = a GJ(A)w˜GK(A),
w∈JWK
where w˜ denotes any element of N(T) lifting w ∈ W(A). The closure of a double coset
G (A)w˜G (A) isgiven by:
J K
GJ(A)w˜GK(A) = a GJ(A)v˜GK(A).
v∈JWK,v≤w
One maydecompose the subspace G (A)w˜G (A) further as
J K
GJ(A)w˜GK(A) = a Bv˜B,
v∈WJ(A)wWK(A)
where B = G∅(A) denotes the positive Borel subgroup. The subspace Bv˜B has the
structure of the right B space Cl(v) ×B and one hasanisomorphism:
s
Bv˜B = YBr˜sB,
i=1
8
where v = r ...r is a reduced decomposition. The above structure in fact gives the
1 s
homogeneous space G(A)/G (A) thestructure ofa CWcomplex, andassuch itishome-
K
omorphic to the corresponding homogeneous space for the unitary forms: K(A)/K (A).
K
Nowrecall the proper K(A)-CW complexX(A) definedin the previous section:
K(A)/T ×|S(A)|
X(A) = ,
∼
whereweidentify(gT,x)with(hT,y)iffx = y ∈ ∆ (n),andg = hmodK (A). Thespace
J J
|S(A)| was the subcomplex of the barycentric subdivision of the n-simplex consisting of
those facesindexedon chains∅ ⊆ J ⊂ J ... ⊂ J ⊆ I forwhich J ∈ S(A) for allk.
0 1 m k
Theorem 3.2. The space X(A) is equivalent to the classifying space EK(A) for proper K(A)-
actions.
Proof. ToprovethetheoremitissufficienttoshowthatX(A)isequivariantlycontractible
with respect to any compact subgroup of K(A). By the general theory of Buildings [D4]
[K3], any compact subgroup of K(A) is conjugate to a subgroup of K (A) for some J ∈
J
S(A). Hence,itissufficienttoshowthatX(A)isK (A)-equivariantlycontractibleforany
J
J ∈ S(A). We proceed asfollows:
Define aK (A) equivariantfiltration of G(A)/B by finite complexes:
J
(G(A)/B)k = a GJ(A)w˜B/B, k ≥ 0.
w∈JW,l(w)≤k
Identifying K(A)/T with G(A)/B, we get a filtration of X(A) by K (A) invariant finite
J
dimensional subcomplexes:
(G(A)/B) ×|S(A)| G (A)/B ×|S (A)|
k J J
X (A) = , k > 0, X (A) = ,
k 0
∼ ∼
where |S (A)| denotes the subspace |S(A)|∩∆ (n), which is the realization of the nerve
J J
of the subcategory under the object J ∈ S(A) (and is hence contractible). Notice that
X (A)isinfactatrivial G (A)-spacehomeomorphicto|S (A)|. Wenowproceedtoshow
0 J J
thatX (A)equivariantlydeformationretractsontoX (A)therebyshowingequivariant
k k−1
contractibility.
An element (gB,x) ∈ X (A) belongs to X (A) if either gT ∈ (G(A)/B) or if x ∈
k k−1 k−1
∆ (n) and gG (A) = hG (A) for some h such that hB ∈ (G(A)/B) . This observation
K K K k−1
maybe expressed as:
Xk(A)/Xk−1(A) = _ (GJ(A)w˜B/B)+ ∧(|S(A)|/|Sw(A)|),
w∈JW,l(w)=k
where(G (A)w˜B/B) indicatestheonepointcompactification ofthespace,and|S (A)|
J + w
is a subcomplex of |S(A)| consisting of faces indexed on chains ∅ ⊆ J ⊂ J ... ⊂ J ⊆ I
0 1 m
for which w does not belong to JWJ0. Hence, to show that X (A) equivariantly retracts
k
onto X (A) it is sufficient to show that there is a deformation retraction from |S(A)| to
k−1
|S (A)|for all w ∈ JW. Itisenough to showthat the subspace |S (A)|iscontractible.
w w
Let I ⊆ I be the subset defined as I = {i ∈ I|l(wr ) < l(w)}. By studying the coset
w w i
wW (A), it is easy to see that I ∈ S(A) (see [D3](4.7.2)). Using claim 3.1 we see that
Iw w
9
|S (A)|isthe geometric realization ofthe nerve ofthe subcategory:
w
S (A) = {J ∈ S(A)|J ∩I 6= ∅}.
w w
Note that S (A) is clearly equivalent to the subcategory consisting of subsets that are
w
contained in I , which has a terminal object (namely I ). Hence the nerve of S (A) is
w w w
(cid:3)
contractible.
Remark3.3. OnemayconsidertheT-fixedpointsubspaceofX(A),whereT isthemaximaltorus
of K(A). It follows that the X(A)T is a proper W(A)-space. Moreover, it is the classifying space
forproperW(A)-actions. ThisspacehasbeenstudiedingreatdetailbyM.W.Davisandcoauthors
[D1, D2, D3], and is sometimes called the Davis complex Σ, for the Coxeter group W(A). It is
not hardto seethat
W(A)×|S(A)|
Σ = X(A)T = ,
∼
where we identify (w,x) with (v,y) iff x = y ∈ ∆ (n), and w = vmodW (A). As before, this
J J
may be expressed as a homotopy colimitof a suitable functor defined on the category S(A) taking
values inthecategoryof W(A)-spaces.
4. DOMINANT K-THEORY
In this section we study Dominant K-theory in detail. For the sake of simplicity, we as-
sumethroughout thissectionthatthegeneralizedCartanmatrixAissymmetrizable. Un-
derthisassumption,anyirreduciblehighestweightrepresentationofK(A)isunitary. We
believe that this assumption is not strictly necessary, and that most of our constructions
maybe extendedto arbitrary generalizedCartan matrices.
NowrecallsomedefinitionsintroductedinSection2. Wesaythataunitaryrepresentation
of K(A) is dominant if it decomposes as a completed sum of irreducible highest weight
representations. HenceonehasamaximaldominantrepresentationHobtainedbytaking
a completed sum of countably many copies of all highest weight representations. Since
K(A) is the amalgamated product (in the category of topological groups) of compact Lie
groups, we obtain a continuous map from K(A) to the group U(H) of unitary operators
on H, with the compact open topology.
Now given any highest weight representation L , one may consider the set of weights
µ
in h∗ for L . It is known [Ku] that this set is contained in the convex hull of the orbit:
Z µ
W(A)µ. Henceweseethatthesetofweightsofanyproperhighestweightrepresentation
L belongs to the setof weights contained in the Tits cone Y. Conversely, any weightin Y
µ
isthe W(A) translate ofsome weight in the WeylchamberC, andso we have:
Claim 4.1. The set of distinct weights that belong to the maximal dominant representation H is
exactlythe setof weightsintheTitscone Y. Moreover,this setis closedunder addition.
Definition 4.2. For G ⊆ K(A), a compact Lie subgroup, let the dominant representation ring
of G denoted by DR , be the free abelian group on the set of isomorphism classes of irreducible G
G
representationsbelongingtoH. Thisgroupadmitsthestructureofasubringoftherepresentation
ring R of G.
G
KU
Recall from Section 2 that denotes the periodic K-theory spectrum represented by a
model F(H)for the space ofFredholm operators on H.
10