© Hindawi Publishing Corp.
HIGHER ORBITAL INTEGRALS, SHALIKA GERMS, AND THE HOCHSCHILD HOMOLOGY
OF HECKE ALGEBRAS
VICTOR NISTOR (Received 1 March 2001)
Abstract.We give a detailed calculation of the Hochschild and cyclic homology of the algebraᏯ∞c(G)of locally constant, compactly supported functions on a reductivep-adic groupG. We use these calculations to extend to arbitrary elements the definition of the higher orbital integrals introduced by Blanc and Brylinski (1992) for regular semi-simple elements. Then we extend to higher orbital integrals some results of Shalika (1972). We also investigate the effect of the “induction morphism” on Hochschild homology.
2000 Mathematics Subject Classification. 19D55, 46L80, 16E40.
1. Introduction. Orbital integrals play a central role in the harmonic analysis of reductive p-adic groups; they are, for instance, one of the main ingredients in the Arthur-Selberg trace formula. An orbital integral on a unimodular groupGis an im- portant particular case of an invariant distribution onG. Invariant distributions have been used in [3] to prove the irreducibility of certain induced representations of GLn
over ap-adic field.
LetGbe a locally compact, totally disconnected topological group. We denote by Ꮿ∞c(G)the space of compactly supported, locally constant, and complex valued func- tions onG. The choice of a Haar measure onGmakesᏯ∞c(G)an algebra with respect to the convolution product. We refer toᏯ∞c(G)with the convolution product as the (full) Hecke algebra ofG. IfG is unimodular, then any invariant distribution on G defines a trace onᏯ∞c(G), and conversely, any trace onᏯ∞c(G)is obtained in this way (this well-known fact follows also fromLemma 3.1). Since the space of traces on an algebraAidentifies naturally with the first (i.e., 0th) Hochschild cohomology group of that algebraA, it is natural to ask what are all the Hochschild cohomology groups ofᏯ∞c(G). The Hochschild cohomology and homology groups of an algebraAare de- noted in this paper by HHq(A)and, respectively, by HHq(A). Since HHq(Ꮿ∞c(G))is the algebraic dual of HHq(Ꮿ∞c(G)), it is enough to determine the Hochschild homology groups ofᏯ∞c(G).
In this paper,Gis typically ap-adic group, which, we recall, means thatGis the set ofF-rational points of a linear algebraic groupGdefined over a finite extensionF of the fieldQpofp-adic numbers,pbeing a fixed prime number. The groupGdoes not have to be reductive, although this is certainly the most interesting case. When we assumeG (orG, by abuse of language) to be reductive, we state this explicitly.
For us, the most important topology to consider onGis the locally compact, totally
disconnected topology induced from an embedding ofG⊂GLn(F). Nevertheless, the Zariski topology onG, which is induced from the Zariski topology ofG, alsoplays a role in our study.
To state the main result of this paper on the Hochschild homology of the algebra Ꮿ∞c(G), we need to introduce first the concepts of a “standard subgroup” and of a
“relatively regular element” of a standard subgroup. For any groupGand any subset A⊂G, we denote
CG(A):=
g∈G, ga=ag,∀a∈A
, NG(A):=
g∈G, gA=Ag
, (1.1) WG(A):=NG(A)/CG(A), and Z(A):=A∩CG(A). This latter notation is used only whenAis a subgroup ofG. The subscriptGis dropped from the notation whenever the groupGis understood. A commutative subgroupSofGis called standard ifSis the group of semi-simple elements of the center ofC(s)for some semi-simple element s∈G. An elements∈Swith this property is called regular relative toS, o rS-regular.
The set ofS-regular elements is denoted bySreg.
We fix from now on ap-adic groupG. Our results are stated in terms of standard subgroups ofG. We denote by Hu the set of unipotent elements of a subgroupH.
Sometimes, the setC(S)uis alsodenoted byᐁS, in order to avoid having too many parentheses in our formulae. Let∆C(S)denote the modular function of the groupC(S) and let
Ꮿ∞cᐁS
δ:=Ꮿ∞cC(S)u
⊗∆C(S) (1.2)
beᏯ∞c(ᐁS)as a vector space, but with the productC(S)-module structure, that is, γ(f )(u)=∆C(S)(γ)f (γ−1uγ), for allγ∈C(S),f∈Ꮿ∞c(ᐁS)δ, andu∈ᐁS.
One of the main results of this paper, namelyTheorem 3.6, identifies the groups HH∗(Ꮿ∞c(G))in terms of the following data:
(1) the setΣof conjugacy classes of standard subgroupsSofG;
(2) the subsetsSreg⊂SofS-regular elements;
(3) the actions of the Weyl groupsW (S)onᏯ∞c(S); and (4) the continuous cohomology of theC(S)-modulesᏯ∞c(ᐁS)δ;
whereSranges through a set of representatives ofΣ. More precisely, ifGis ap-adic group defined over a field of characteristic zero, as before, thenTheorem 3.6states the existence of an isomorphism
HHq Ꮿ∞c(G)
S∈Σ
Ꮿ∞cSregW (S)
⊗Hq
C(S),Ꮿ∞cᐁS
δ
. (1.3)
This isomorphism is obtained by identifying theE∞-term of an implicit, convergent spectral sequence, and hence it is not natural. This isomorphism can be made natural by using a generalization of the Shalika germs. The isomorphism (1.3) is in the spirit of the results of Karoubi [13] and Burghelea [8]. See also[10].
It is important to relate the determination of the Hochschild homology in (1.3) with the periodic cyclic homology groups ofᏯ∞c(G). Let HH[q]:= ⊕k∈ZHHq+2k. Recall that an elementγ∈Gis called compact, by definition, if it belongs to a compact subgroup ofG. The set of compact elements ofGis open and closed and is clearlyG-invariant for the action ofGon itself by conjugation. Also, we denote by HH[q](Ꮿ∞c(G))compthe
localization of the homology group HH[q](Ꮿ∞c(G))to the set of compact elements of G(see [5] o r [18]). Then, the periodic cyclic homology of the Hecke algebraᏯ∞c(G)is related to its Hochschild homology by
HPq Ꮿ∞c(G)
HH[q]
Ꮿ∞c(G)
comp. (1.4)
This relation is implicit in [12]. Consequently, the results of this paper complement the results on the cyclic homology ofp-adic groups in [12,23]. More precisely, letScomp
be the set of compact elements of a standard subgroupS and let H[q]:= ⊕k∈ZHq+2k, then
HPq Ꮿ∞c(G)
S∈Σ
Ꮿ∞cScompreg W (S)
⊗H[q]
C(S),Ꮿ∞cᐁS
δ
. (1.5)
It is interesting toremark thatHP∗(Ꮿ∞c(G))can alsobe related tothe admissible dual (or spectrum) of G, see [15], and hence our results have significance for the representation theory ofp-adic groups. (See also [18] for similar results on the groups of real points of algebraic groups defined overR.) These periodic cyclic cohomology groups are seen to be isomorphic toK∗(Cr∗(G))⊗C, by combining results from [1,16]
to those of [12]. See also[15].
Assume for the moment thatGis reductive. Then, in order to better understand the role played by the groups HH∗(Ꮿ∞c(G)) and H∗(G,Ꮿ∞c(Gu)) in the representa- tion theory of G, we relate H∗(G,Ꮿ∞c(Gu)) to the analogous cohomology groups, H∗(P,Ꮿ∞c(Pu)δ)and H∗(M,Ꮿ∞c(Mu)δ), associated to parabolic subgroupsPofGand to their Levi components M. In particular, we define morphisms between these Hochschild homology groups that are analogous to the induction and inflation mor- phisms that play such a prominent role in the representation theory ofp-adic groups.
These morphisms are induced by morphisms of algebras.
In [5], Blanc and Brylinski have introduced higher orbital integrals associated to regular semi-simple elements by proving first that
HHq Ꮿ∞c(G)
Hq
G,Ꮿ∞c(G)δ
, (1.6)
a result which they called “the MacLane isomorphism.” (Actually, they did not have to twist with the modular function, because they worked only with unimodular groups G, see Lemma 3.1for the slightly more general version needed in this paper.) Our approach also starts from the MacLane isomorphism, but after that we rely more on filtrations of theG-moduleᏯ∞c(G)than on localization. This allows us to define higher orbital integrals at arbitrary elements. Then, we study the properties of these orbital integrals and we obtain, in particular, a proof of the existence of abstract Shalika germs for the higher orbital integrals. Actually, the existence of Shalika germs turns out to be a consequence of some general homological properties of the ringR∞(G) of (conjugacy) invariant, locally constant functions on the groupG. We alsouse the techniques developed in [18] in the framework of real algebraic groups. It would be interesting to relate the results of this paper to those of [2] on the periodic cyclic homology of Iwahori-Hecke algebras and those of [14] on invariant distributions.
This paper is the revised version of a preprint that was first circulated in February 1999.
2. Standard subgroups. Our description of the Hochschild homology of Hecke al- gebras is in terms of “standard” subgroups, a class of commutative groups that we define and study below. The main role of the standard subgroups is to define a strat- ification ofGby sets invariant with respect to inner automorphisms. This section is devoted to establishing the basic facts about standard subgroups. We begin by fixing notation.
IfGis a group andA⊂Gis a subset, we denote byCG(A)the centralizer ofA, that is, the set of elements ofGthat commute with every element ofA. WhenGis understood, we omit it from notation. Also, we denote byNG(A)the normalizer ofAinG, that is, the set of elementsg∈Gsuch thatgAg−1=A. We then setWG(A):=NG(A)/CG(A) andZ(A):=A∩CG(A). ByZ=Z(G)=CG(G), we denote the center ofG. Again, we omitGif the group is understood.
LetGbe a linear algebraic group defined over a totally disconnected, locally compact fieldFof characteristic zero. ThusFis a finite algebraic extension ofQp, the field of p-adic numbers. The setG(F)ofF-rational points ofGis called ap-adic group and is denoted simply byG. It is known [6] thatG=G(F)identifies with a closed subgroup o f GLn(F), and hence it has a natural locally compact topology that makes it a totally disconnected space.
Definition2.1. A commutative subgroupS⊂Gis called standard if and only if, there exists a semi-simple element s0∈G such thatS is the group of semi-simple elements of the center of C(s0), the centralizer ofs0 in G. A semi-simple element s0∈S with this property will be called regular relative toS or, simply,S-regular. The set ofS-regular elementss∈Sis denoted bySreg.
Clearly, every standard subgroup is commutative. More properties of standard sub- groups are summarized inProposition 2.2.
We denote byHss the subset of semi-simple elements of a groupH.
Proposition2.2. LetS be a standard subgroup ands0∈Sreg.
(i) The groupSis the set ofF-rational points of a subgroupS⊂Gdefined over the fieldF.
(ii) We have thatC(S)=C(s0), soS=Z(C(s0))ss andN(C(S))=N(S).
(iii) Every semi-simple elementγ∈Gis S-regular for one, and only one, standard subgroupS.
(iv) The setSregis a Zariski open subset ofS.
Proof. (i) We identify the groupGwith its set of ¯F-rational points, for some al- gebraically closed extension ofF. LetΓ be the Galois group of ¯FoverF. ThenΓ acts onGandGcan be identified with the set of fixed points of this action becauseFis a perfect field.
Lets0be a semi-simple element ofG. From the above identification, we easily obtain thatCG(s0), the centralizer ofs0inG, is invariant with respect toΓ. From this it follows that CG(s0)is defined overFand C(s0):=CG(s0) is the set ofF-rational points of CG(s0). LetSbe the center ofCG(s0). Then we see, using the same reasoning, thatS is defined overFand thatSis the set of itsF-rational points.
(ii) We haveC(s0)⊃C(S)becauses0∈S. But,S⊂Z(C(s0)), soC(S)⊂C(s0), too.
By definition,S =Z(C(s0)), soS=Z(C(S)). The last part follows becauseN(H)⊂ N(Z(H))andN(H)⊂N(C(H)), for any groupH.
(iii) Let γ∈Gbe a semi-simple element. Define S(γ):=Z(C(γ))ss. ThenS(γ)is a standard subgroup, by definition, andγ is regular relative toS(γ). Clearly, ifγ is S-regular, thenS=S(γ).
(iv) Let S ⊂G. We may assume that G⊂GLn(¯F) and that G =G∩GLn(F). The statement is obvious ifG=GLn(F). In general, the result follows because CG(s)= CGLn(F)(s)∩G.
For anyp-adic groupH, we denote byHuthe set of unipotent elements ofH, and call it the unipotent variety ofH. In the particular case ofH=C(S), whereS⊂Gis a standard subgroup, we also denoteC(S)u=ᐁS.
We now define a natural AdG-invariant stratification ofG, called the standard strat- ification ofG.
Letgbe the Lie algebra ofGin the sense of linear algebraic groups. Denote byai(g) the coefficients of the polynomial det(t+1−Adg),
det
t+1−Adg
= m i=0
ai(g)ti∈F[t]. (2.1) Letar be the first nonzero coefficientai, and define
Vk=
g∈G, ar(g)=ar+1(g)= ··· =ar+k−1(g)=0
. (2.2)
ThusV0=G, by convention, andG&V1=G, the set of regular elements ofGifGis reductive. Also,Vm+1= ∅becauseam=1. We observe that the functionsai(g)are G-invariant polynomial functions onG, and that they depend only on the semi-simple part ofg.
In order to proceed further, recall that the Jordan decomposition of an element g∈Gisg=gsgu, wheregs is semi-simple,guis unipotent, andgsgu=gugs. This decomposition is unique [6]. LetS ⊂G be a standard subgroup. Ifg=gsgu is the Jordan decomposition ofg∈Gand ifgs∈Sreg, thengu∈ᐁS:=C(S)u, by definition, and henceg∈SregᐁS.
Fix now a standard subgroupS⊂G, and let FS=AdG
Sreg
, FSu=AdG SregᐁS
(2.3) be the set of semi-simple elements ofGconjugated to an element ofSregand, respec- tively, the set of elementsg∈Gconjugated to an element ofSregᐁS (i.e., the set of elementsg∈Ssuch whose semi-simple part is inFS).
Also, letN(S):= {g∈G, gSg−1=S}be the normalizer ofSandW (S)=N(S)/C(S).
SinceN(S)leavesSreginvariant and is actually the normalizer of this set, it follows that the quotientW (S)can be identified with a set of automorphisms ofS. SinceN(S) is the set ofF-rational points of an algebraic group, the rigidity of tori (see [6, page 117]) shows thatW (S)is finite.
The natural map(g,s)→gsg−1descends toa map φS:
G×Sreg
N(S) (g,s) →gsg−1∈FS. (2.4)
Similarly, we obtain a map φuS :
G×SregᐁS
N(S) (g,su) →gsug−1∈FSu. (2.5) InProposition 2.3consider the locally compact (and Hausdorff) topology ofG, and not the Zariski topology. Recall that we denote byGssthe set of semi-simple elements ofG.
Proposition2.3. LetSbe a standard subgroup ofG. Using the above notation, we have
(i) The setFSis an analyticsubmanifold ofG, and the mapsφSandφuS are home- omorphisms.
(ii) For each k, the set Vk& Vk+1 is the disjoint union of the sets FSu that have a nonempty intersection withVk& Vk+1, and eachFSu⊂Vk& Vk+1 is an open subset of Vk&Vk+1.
(iii) Similarly, the set Gss∩(Vk& Vk+1)is a disjoint union of the sets FS that have a nonempty intersection withVk& Vk+1, and eachFS ⊂Vk& Vk+1is an open subset of Gss∩(Vk&Vk+1).
Proof. (i) First we check that φS and φuS are injective. Indeed, assume that g1s1g1−1=g2s2g2−1, fo r so mes1,s2∈Sreg. Then, ifg=g2−1g1, we have
gC s1
g−1=C s2
⇒gC(S)g−1=C(S) ⇒gSg−1=S, (2.6) and henceg∈N(S). Consequently, we have(g1,s1)=(g2g,g−1s2g)=g−1(g2,s2), withg∈N(S), as desired. The same argument shows that ifFS andFS have a point in common, then the standard subgroupsSandSare conjugated inG.
The injectivity ofφuS follows from the injectivity ofφS, indeed, ifg1(s1u1)g1−1= g2(s2u2)g2−1, letg=g2−1g1as above, and conclude thatgs1g−1=s2, by the uniqueness of the Jordan decomposition. As above, this implies thatg∈N(S).
Since the differentialdφSis a linear isomorphism onto its image (i.e., it is injective) andφS is injective, it follows thatφS is a local homeomorphism onto its image (for the locally compact topologies), and that its image is an analytic submanifold (see [22, Theorem 2.3, page 38]). The setGss∩(Vk&Vk+1)is an algebraic variety on which G acts with orbits of the same dimension, and henceφS is proper [6]. This proves thatφS is a homeomorphism. Using an inverse forφS, we obtain thatφuS is alsoa homeomorphism.
Now to prove (ii), consider a standard subgroupS⊂G, and letdbe the dimension of C(S). Then a0=a1 = ··· =ad−1=0 o nS, andSreg is an open component of S∩ {ad=0}. It follows that, ifs∈(Vk& Vk+1)∩Sreg, thenFS⊂Vk& Vk+1. This shows thatGss∩(Vk& Vk+1)is a union of sets of the formFS. This must then be a disjoint union because the setsFS are either equal or disjoint, as proved above.
Now, ifg∈Vk& Vk+1has semi-simple part s, thens∈FS⊂Gss∩(Vk& Vk+1), fo r some standard subgroupS, and hence g∈FSu⊂Vk& Vk+1. The setsFSu are open in the induced topology because the mapVk&Vk+1→Gss∩(Vk&Vk+1)is continuous. See also[27].
3. Homology of Hecke algebras. In this section, we obtain a first identification of the Hochschild homology groups of Hecke algebras ofp-adic groups. To this end, we use several general results on Hochschild homology of algebras, on algebraic groups, and on the continuous cohomology of totally disconnected groups. Good references are [6, 7, 17], for the general theory, and [15] for questions related to Hochschild homology.
LetGbe ap-adic group on which we fix a Haar measuredg. Consider now the space Ꮿ∞c(G)of compactly supported, locally constant functions onG. Fix a Haar measure dhonG. Then the convolution product, denoted∗, is defined by
f1∗f2(g)=
Gf1(h)f2 h−1g
dh. (3.1)
The convolution product makesᏯ∞c(G)an algebra, called the Hecke algebra ofG. It is important in representation theory to determine the (AdG-)invariant linear functionals on Ꮿ∞c(G). If G is unimodular, the space of invariant linear functionals onᏯ∞c(G) coincides with the space of traces onᏯ∞c(G). The space of traces ofᏯ∞c(G)identifies with HH0(Ꮿ∞c(G)), the first Hochschild cohomology group ofᏯ∞c(G). It is reasonable then to ask, what are all Hochschild cohomology groups ofᏯ∞c(G)? Since Hochschild cohomology is the algebraic dual of Hochschild homology, it is enough to concentrate on the latter.
We first recall the definition of the Hochschild homology groups of the algebra Ꮿ∞c(G). Let
Ꮿ∞cGq+1
=Ꮿ∞c(G)⊗Ꮿ∞c(G)⊗···⊗Ꮿ∞c(G), (3.2) (q+1)-times, be the usual (algebraic) tensor product of vector spaces. The Hochschild differentialb:Ꮿ∞c(Gq+2)→Ꮿ∞c(Gq+1)is given by
(bf )
g0,g1,...,gq
= q j=0
(−1)j
Gf
g0,...,gj−1,γ,γ−1gj,gj+1,...,gq dγ
+(−1)q+1
Gf
γ−1g0,g1,...,gq,γ dγ.
(3.3)
By definition, theqth Hochschild homology group ofᏯ∞c(G), denoted by HHq(Ꮿ∞c(G)), is theqth homology group of the complex(Ꮿ∞c(Gq+1),b). Hochschild homology can be defined for any algebra. Our definition takes into account the particular structure ofᏯ∞c(G), in particular, that it is an inductive limit of unital algebras, so there is no need to first adjoin a unit in order to define Hochschild homology. The computation of the groups HHq(Ꮿ∞c(G))is the main purpose of this paper.
The groupGacts by conjugation onᏯ∞c(G), and we denote byᏯ∞c(G)adtheG-module defined by this action. Also, let∆Gdenote the modular function ofG, which, we recall, is defined by the relation
∆G(h)
Gf (gh)dg=
Gf (g)dg. (3.4)
We are especially interested in theG-moduleᏯ∞c(G)δobtained fromᏯ∞c(G)adby twist- ing it with the modular function. More precisely, letᏯ∞c(G)δ=Ꮿ∞c(G)as vector spaces,
and let the action ofGon functions be given by the formula (γ·f )(g)=∆G(γ)f
γ−1gγ
, f∈Ꮿ∞c(G)δ. (3.5) The reason for this twisting is that, forGnonunimodular, the traces ofᏯ∞c(G)are the G-invariant functionals onᏯ∞c(G)δ, no t o nᏯ∞c(G)(this is an immediate consequence ofLemma 3.1). More generally, our approach to the Hochschild homology ofᏯ∞c(G) is based onLemma 3.1.
Before stating and provingLemma 3.1, we need to introduce some notation. First, if Mis an arbitraryG-module, we denote byM⊗∆Gthe tensor product of theG-modules MandC, where the action onCis given by the multiplication with the modular func- tion ofG. (In particular,Ꮿ∞c(G)δ=Ꮿ∞c(G)⊗∆G.)
IfMis a rightG-module andMis a leftG-module, thenM⊗GMis defined as the quotient ofM⊗Mby the submodule generated bymg⊗m−m⊗gm. For example, ifH⊂Gis a closed subgroup and ifXis a leftH-space, then we have an isomorphism ofG-spaces
Ꮿ∞c(G)⊗H
Ꮿ∞c(X)⊗∆H
Ꮿ∞cG×HX
, (3.6)
whereG×Xis the quotient(G×X)/Hfor the actionh(g,x)=(gh−1,hx). This iso- morphism is obtained by observing that the natural map
tX:Ꮿ∞c(G)⊗Ꮿ∞c(X)=Ꮿ∞c(G×X)→Ꮿ∞cG×HX , tX(f )
(g,x)
=
Hf
gh,h−1x
dh, (3.7)
passes to the quotient to give the desired isomorphism. Sometimes it will be conve- nient toregard a leftG-module as a rightG-module by replacinggwithg−1. Equation (3.6) is one of the main reasons why we need to consider the modular function.
Also, recall that aG-moduleMis smooth if and only if the stabilizer of each element of M is open in G. The continuous homology groups of G with coefficients in the smooth moduleM, denoted Hk(G,M), can be defined using tensor products as follows.
LetᏮq(G)=Ꮿ∞c(Gq+1),q=0,1,...,be the Bar complex of the groupG, with differential
(df )
g0,g1,...,gq
=
q+1
j=0
(−1)j
Gf
g0,...,gj−1,γ,gj,...,gq
dγ. (3.8)
Then the complex(Ꮾq,d)gives a resolution ofCwith projectiveᏯ∞c(G)-modules, and the complex
Ꮾq(G)⊗GM (3.9)
computes Hq(G,M). See [4,7].
We need the following extension of a result from [5].
Lemma3.1. LetᏯ∞c(G)δ=Ꮿ∞c(G)⊗∆G be the G-module obtained by twisting the adjoint action ofGonᏯ∞c(G)by the modular function. Then we have a natural isomor- phism
HHq Ꮿ∞c(G)
Hq
G,Ꮿ∞c(G)δ
. (3.10)
Proof. Consider the complex (3.9), which computes the continuous cohomology ofM=Ꮿ∞c(G)δ, and let
hG:Ꮾq(G)⊗Ꮿ∞c(G)δᏯ∞c(G)⊗Ꮿ∞cGq+1
=Ꮿ∞cGq+2
→Ꮿ∞c(G)⊗q+1 (3.11) be the map
hG(f )
g0,g1,...,gq
=
Gf
g−1hg,g−1g0,g−1g0g1,...,g−1g0g1···gq dg,
h=g0g1···gq. (3.12)
As in (3.6), the maphG descends to the quotient to induce an isomorphism
˜hG:Ꮾq(G)⊗GᏯ∞c(G)δᏯ∞cGq+2
⊗GCᏯ∞c(G)⊗q+1 (3.13) of complexes, that is, ˜hG◦(d⊗G1)=b◦˜hG, which establishes the isomorphism Hq(G,Ꮿ∞c(G)δ)HHq(Ꮿ∞c(G)), as desired.
To better justify the twisting of the module Ꮿ∞c(G) by the modular function in Lemma 3.1, note that the trivial representation ofG gives rise to an obvious mor- phismπ0:Ꮿ∞c(G)→C, byπ0(f )=
Gf (g)dg, which hence defines a trace onᏯ∞c(G).
However,π0is notG-invariant for the usual action ofG, but is invariant if we twist the adjoint action ofGby the modular function, as indicated.
We proceed now to a detailed study of theG-moduleᏯ∞c(G)δ using the standard stratification introduced in the previous section.
LetR∞(G) be the ring of locally constant AdG-invariant functions on G with the pointwise product, which we regard as a subset of the set of endomorphisms of the G-moduleᏯ∞c(G)δ=Ꮿ∞c(G)⊗∆G. Let det(t+1−Adg)= mi=0ai(g)ti, as before. For eachk≥1, denote byIk⊂R∞(G)the ideal generated by functionsf:G→Cof the form
f=φ
ar,ar+1,...,ar+k−1
, (3.14)
whereφis a locally constant functionφ:Fk→Csuch thatφ(0,0,...,0)=0. (Recall that each of the polynomialsa0,...,ar−1is the zero polynomial.) By convention, we setI0=(0); also, it follows thatIm+1=R∞(G).
Fix nowk, and letφn:Fk→Cbe 1 on the set ξ=
ξ0,...,ξk−1
∈Fk,max|ξi| ≥q−n
, (3.15)
and vanishes outside this set. (Hereqis the number of elements of the residual field ofF, and the non-Archimedean norm “| |” is normalized such that its range is{0} ∪ {qn, n∈Z}.) Also, letpn=φn(ar,ar+1,...,ar+k−1)∈Ik. Thenpn=pn2=pnpn+1and Ik= ∪pnR∞(G).
For further reference, we state as a lemma a basic property of the constructions we have introduced.
Lemma3.2. IfMis anR∞(G)-module, thenIkM= ∪pnM.
As a consequence of the above lemma, we obtain the following result.
Corollary3.3. Consider theG-modules, Mk=
Ik+1
Ik
⊗R∞(G)Ꮿ∞c(G)δIk+1Ꮿ∞c(G)δ
IkᏯ∞c(G)δ . (3.16) Then, for eachq≥0, we have an isomorphism
Hq
G,Ꮿ∞c(G)δ
m k=0
Hq G,Mk
(3.17)
of vector spaces.
Proof. There exists a (not natural) isomorphism Hq
G,Ꮿ∞c(G)δ
m
k=0Ik+1Hq
G,Ꮿ∞c(G)δ IkHq
G,Ꮿ∞c(G)δ (3.18)
of vector spaces.
ByLemma 3.2, the inclusion ofIkᏯ∞c(G)δ→Ꮿ∞c(G)δofG-modules induces natural isomorphisms
Hq
G,IkᏯ∞c(G)δ Hq
G,lim
→ pnᏯ∞c(G)δ lim→ pnHq
G,Ꮿ∞c(G)δ IkHq
G,Ꮿ∞c(G)δ ,
(3.19)
because the functor Hqis compatible with inductive limits and with direct sums.
The naturality of these isomorphisms andLemma 6.1shows that HqG,Ik+1Ꮿ∞c(G)δ
IkᏯ∞c(G)δ Ik+1Hq
G,Ꮿ∞c(G)δ IkHq
G,Ꮿ∞c(G)δ . (3.20) This is enough to complete the proof.
If Xis a totally disconnected, locally compact space X, we denote byᏯ∞c(X)the space of compactly supported, locally constant, complex valued functions onX. Recall that, ifU⊂Xis an open subset ofXas above, then restriction defines an isomorphism
Ꮿ∞c(X)
Ꮿ∞c(U)Ꮿ∞c(X &U). (3.21) We now study the homology of the subquotients
Mk=Ik+1Ꮿ∞c(G)δ
IkᏯ∞c(G)δ Ꮿ∞cVk&Vk+1
(3.22) by identifying them with induced modules. LetΣkbe a set of representatives of con- jugacy classes of standard subgroups S such that FS ⊂Vk& Vk+1 (or, equivalently, FSu⊂Vk&Vk+1).
Lemma3.4. Using the above notation, we haveIkᏯ∞c(G)=Ꮿ∞c(G &Vk)and Ik+1Ꮿ∞c(G)δ
IkᏯ∞c(G)δ
S∈Σk
Ꮿ∞cFSu
. (3.23)
Proof. It follows from the definition ofIkthat, iff∈IkᏯ∞c(G)δ, thenfvanishes in a neighborhood ofVk. Conversely, iff is inᏯ∞c(G & Vk), then we can find some polynomialai, withi≤r+k−1, such that|ai|is bounded from below on the sup- port offby, say,q−n, thenpnf=f. The second isomorphism follows from the first isomorphism using (3.21) andLemma 3.2.
IfH⊂Gis a closed subgroup andMis a smooth (left)H-module (i.e., the stabilizer of eachm∈Mis an open subgroup ofM), we denote
indGH(M)=Ꮿ∞c(G)⊗HM= Ꮿ∞c(G)⊗M
f h⊗m−f⊗hm, h∈H, (3.24) where the rightH-module structure onᏯ∞c(G)is(f h)(g)=f (gh−1). Then Shapiro’s lemma, see [9], states that
Hk
G,indGH(M)⊗∆G
Hk(H,M). (3.25)
(A proof of Shapiro’s lemma for nonunimodular groups is contained in the proof of Theorem 6.2.)
The basic examples of induced modules are obtained fromH-spaces. If X is an H-space (we agree thatHacts onXfrom the left), then
Ꮿ∞cG×X H
indGH
Ꮿ∞c(X)⊗∆H
indGH
Ꮿ∞c(X)δ
(3.26) asG-modules, whereHacts onG×Xbyh(g,x)=(gh−1,hx). For example,Proposition 2.3identifiesᏯ∞c(FSu)with an induced module:
Ꮿ∞cFSu
indGN(S)
Ꮿ∞cSregᐁS
⊗∆N(S)
=indGN(S)
Ꮿ∞cSregᐁS
δ
. (3.27)
Shapiro’s lemma is an easy consequence of the Serre-Hochschild spectral sequence, see [9], which states the following. LetMbe a smoothG-module andH⊂Gbe a normal subgroup. Then the action ofGo n Hq(H,M)descends to an action ofG/H, and there exists a spectral sequence withEp,q2 =Hp(G/H,Hq(H,M)), convergent to Hp+q(G,M).
LetMk=Ik+1Ꮿ∞c(G)δ/IkᏯ∞c(G)δ, as before.
Proposition3.5. Using the above notation, we have Hq
G,Mk
S∈Σk
Ꮿ∞cSregW (S)
⊗Hq
C(S),Ꮿ∞cᐁS
δ
, (3.28)
a natural isomorphism ofR∞(G)-modules.
Proof. LetS be a standard subgroup ofG. Recall first thatW (S)=N(S)/C(S)is a finite group that acts freely onSreg, which gives anN(S)-equivariant isomorphism
Ꮿ∞cSregᐁS
δ=Ꮿ∞cᐁS
δ⊗Ꮿ∞cSreg
. (3.29)
LetM be a smoothN(S)-module. The Hochschild-Serre spectral sequence applied tothe moduleMand the normal subgroupC(S)⊂N(S)gives natural isomorphisms
Hq
N(S),M H0
W (S),Hq
C(S),M Hq
C(S),MW (S). (3.30)
Combining these two isomorphisms, we obtain Hk
G,Ꮿ∞cFSu
δ Hk
G,indGN(S)
Ꮿ∞cSregᐁS
δ
δ Hk
N(S),Ꮿ∞cSregᐁS
δ
Hk
C(S),Ꮿ∞cᐁS
δ
⊗Ꮿ∞cSregW (S)
Ꮿ∞cSregW (S)
⊗Hk
C(S),Ꮿ∞cᐁS
δ
.
(3.31)
The result then follows fromLemma 3.4, which implies directly that Mk ⊕S∈ΣkᏯ∞cFSu
δ. (3.32)
The proof is now complete.
Combining Proposition 3.5 with Corollary 3.3, we obtain the main result of this section. Recall that ap-adic groupG=G(F)is the set ofF-rational points of a linear algebraic groupGdefined over a non-Archimedean, nondiscrete, locally compact field Fof characteristic zero. Also, recall thatᐁSis the set of unipotent elements commut- ing with the standard subgroupS, and that the action ofC(S)onᏯ∞c(ᐁS)is twisted by the modular function ofC(S), yielding the moduleᏯ∞c(ᐁS)δ=Ꮿ∞c(ᐁS)⊗∆C(S).
Theorem3.6. LetGbe ap-adicgroup. LetΣbe a set of representative of conjugacy classes of standard subgroups of S ⊂ G and W (S)= N(S)/C(S), then we have an isomorphism
HHq Ꮿ∞c(G)
S∈Σ
Ꮿ∞cSregW (S)
⊗Hq
C(S),Ꮿ∞cᐁS
δ
. (3.33)
Remark3.7. The isomorphism ofTheorem 3.6is not natural. A natural description o f HHq(Ꮿ∞c(G))will be obtained in one of the following sections by considering higher orbital integrals and their Shalika germs.
4. Higher orbital integrals and their Shalika germs. Proposition 3.5allows us to determine the structure of the localized cohomology groups HH∗(Ꮿ∞c(G))m, where m is a maximal ideal ofR∞(G). This will lead to an extension of the higher orbital integrals introduced by Blanc and Brylinski in [5] and to a generalization of some results of Shalika [24] to higher orbital integrals, all discussed in this section. In this way, we also obtain a new, more natural description of the groups HHq(Ꮿ∞c(G)).
First recall the following result.
Proposition4.1. LetGbe a reductivep-adic group over a field of characteristic0, and letS⊂Gbe a standard subgroup, andγ∈Sreg(i.e.,γis a semi-simple element such thatC(S)=C(γ)). Then there exists anN(S)-invariant closed and open neighborhood UofγinC(S)such that
G×U(g,h) →ghg−1∈G (4.1)
defines a homeomorphism ofG×N(S)U:=(G×U)/N(S)onto aG-invariant, closed, and open subsetV⊂Gcontainingγ.
Proof. The result follows from Luna’s lemma. Forp-adic groups, Luna’s lemma is proved in [19, page 109, Properties “C” and “D”].
For any maximal idealm ⊂R∞(G) and any R∞(G)-moduleM, we denote by Mm
the localization ofM atm, that is,Mm S−1M, whereS is the multiplicative subset R∞(G)&m.
FromProposition 4.1we obtain the following consequences for the ringR∞(G).
Corollary4.2. Letγ∈S,U, andV be as inProposition 4.1.
(i) The ringR∞(G)decomposes as the direct sumᏯ∞(V )G⊕Ꮿ∞(Vc)G, andᏯ∞(V )G Ꮿ∞(U)N(S)⊂Ꮿ∞(C(S))N(S)=R∞(C(S))W (S). (HereVcis the complement ofVinG.)
(ii) For any two semi-simple elements γ,γ∈G, ifφ(γ)=φ(γ)for all functions φ∈R∞(G), thenγandγare conjugated inG.
(iii) Letm∈R∞(G)be the maximal ideal consisting of functions that vanish at a semi- simple elementγ∈G. Thenpis generated by an increasing sequence of projections, andMmM/mM, for anyR∞(G)-moduleM.
Proof. (i) is an immediate consequence ofProposition 4.1.
(ii) follows from [19, Proposition 2.5].
To prove (iii), observe that the maximal idealm is generated by an increasing se- quence of projectionspn, that is,m= ∪pnR∞(G), withpn2=pnandpn+1pn=pn.
We know from Proposition 2.5 of [19] thatR∞(G)is isomorphic toC∞(X), for some locally compact, totally disconnected topological spaceX. Moreover, ifMis aC∞(X)- module andmis the maximal ideal of functions vanishing atx0, for some fixed point x0∈X, thenᏯ∞(X)mᏯ∞(X)/mᏯ∞(X), and hence
Mm=M⊗Ꮿ∞(X)Ꮿ∞(X)mM⊗Ꮿ∞(X)Ꮿ∞(X) mᏯ∞(X) M
mM. (4.2)
SinceXis metrizable, we can choose a basisVnof compact open neighborhoods of x0inX. If we letpn to be the characteristic function ofVnc, thenpnare projections generatingm. By choosingVn to be decreasing, we obtain an increasing sequence of projectionspn.
We now consider for each maximal ideal m ⊂ R∞ = R∞(G) the localization HHq(Ꮿ∞c(G))m.
Proposition4.3. Letmbe a maximal ideal ofR∞(G). Ifmconsists of the functions that vanish at the semi-simple elementγ∈GandS⊂Gis a standard subgroup such thatγ∈Sreg, then
HHq Ꮿ∞c(G)
mHq
C(S),Ꮿ∞cᐁS
δ
. (4.3)
For all other maximal idealsm⊂R∞(G), we haveHHq(Ꮿ∞c(G))m=0.
Note that Hq(C(S),Ꮿ∞c(ᐁS))Hq(C(γ),Ꮿ∞c(C(γ)u)).
Proof. Letmγ:= {f∈R∞(G), f (γ)=0}. The vanishing of HH∗(Ꮿ∞c(G))m in the last part ofProposition 4.4becauseᏯ∞c(G)m=0 for all maximal idealsmthat are not of the formmγ, for some semi-simple elementγ∈G.
Assume now thatm=mγ. The localization functorV→Vm is exact by standard homological algebra. Let(0)=I0⊂I1⊂ ··· ⊂Im⊂Im+1=R∞(G)be the sequence of
