Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras

We give a detailed calculation of the Hochschild and cyclic homology of the algebra $\CIc(G)$ of locally constant, compactly supported functions on a reductive p-adic group G. We use these calculations to extend to arbitrary elements the definition the higher orbital integrals introduced in \cite{Blanc-Brylinski} for regular semisimple elements. Then we extend to higher orbital integrals some results of Shalika. We also investigate the effect of the ``induction morphism'' on Hochschild homology.


Introduction
Orbital integrals play an important role in the harmonic analysis of a reductive p-adic group G; they are, for instance, one of the main ingredients in the Arthur-Selberg trace formula. Orbital integrals on unimodular groups are a particular case of invariant distribution, which have been used in [2] to prove the irreducibility of certain induced representations of GL n over a p-adic field.
By definition, an invariant distribution on a unimodular group G gives rise to a trace (i.e., a Hochschild cocycle of degree zero) on C ∞ c (G), the Hecke algebra of compactly supported, locally constant, complex valued functions on G. It is interesting then to try to analyse as completely as possible the Hochschild homology and cohomology groups of the algebra C ∞ c (G) (denoted HH * (C ∞ c (G)) and HH * (C ∞ c (G)), respectively).
In this paper, G will be the set of F-rational points of a linear algebraic group G defined over a finite extension F of the field Q p of p-adic numbers, p being a fixed prime number. The group G does not have to be reductive, although this is certainly the most interesting case. When we shall assume G (or G, by abuse of language) to be reductive, we shall state this explicitely. For us, the most important topology to consider on G will be the locally compact topology induced from an embedding of G ⊂ GL n (F). Nevertheless, the Zariski topology on G will also play a role in our study.
To state the main result of this paper, we need to introduce first the concept of standard subgroup. For any set A ⊂ G, we shall denote C(A) := {g ∈ G, ga = ag, ∀a ∈ A} and Z(A) := A ∩ C(A). This latter notation will be used only when A is a subgroup of G. A commutative subgroup S of G is called standard if S = Z(C(s)) for some semi-simple element s ∈ G. Our results will be stated in terms of standard subgroups of G. We shall denote by H u the set of unipotent elements of a subgroup H. Sometimes, the set C(S) u is also denoted by U S , in order to avoid having to many paranthesis in our formulae. One of the main results of this paper (Theorem 1) identifies the groups HH * (C ∞ c (G)) in terms of the following data: the set Σ of (conjugacy classes of) standard subgroups S of G, the subset S reg ⊂ S of S-regular elements, the action of the Weyl group W (S) of S on C ∞ c (S), and the continuous cohomology of the C(S)-module , where ∆ C(S) denotes the modular function of the group C(S). More precisely, if G is a p-adic group defined over a field of characteristic zero, as before, then Theorem 1 states the existence of an isomorphism which can be made natural by using a generalization of the Shalika germs.
It is important to relate this result with the periodic cyclic homology groups of C ∞ c (G). For the Hecke algebra C ∞ c (G), the periodic cyclic homology is related to Hochschild homology by that is, HP * (C ∞ c (G)) is the localization of Hochschild homology to the G-invariant subset of compact elements of G. This relation is implicit in [10]. Consequently, the results of this paper complement the results on the cyclic homology of p-adic groups in [10,19]. It is interesting to remark that HP * (C ∞ c (G)) can also be described in terms of the admissible spectrum of G, see [12], and hence our results have significance for the representation theory of p-adic groups. See also [16] for similar results on (the groups of real points of) algebraic groups defined over R. These periodic cyclic cohomology groups are isomorphic to K * (C * r (G)), by combining results from [1], [13], and [10].
Assume for the moment that G is reductive. Then, in order to better understand the role played by the groups HH * (C ∞ c (G)) and H * (G, C ∞ c (G u )) in the representation theory of G, we relate H * (G, C ∞ c (G u )) to the analogous cohomology groups, H * (P, C ∞ c (P u ) δ ) and H * (M, C ∞ c (M u ) δ ), associated to parabolic subgroups P of G and to their Levi components M . In particular, we define morphisms between these Hochschild homology groups that are analogous to the induction and inflation morphisms that play such a prominent role in the representation theory of p-adic groups. These morphisms are induced by morphisms of algebras.
In [4], Blanc and Brylinski have introduced higher orbital integrals associated to regular semisimple elements by proving first that a result which they dubed "the MacLane isomorphism." (Actually, they did not have to twist with the modular function, because they worked only with unimodular groups G, see Lemma 1 for the slightly more general version needed in this paper), after that we rely more on filtrations of the G-module C ∞ c (G), rather 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 ring of invariant, locally constant functions on the group G. We also use the techniques developed in [16] in the framework of real algebraic groups. It would be interesting to relate the results of this paper to those of [11]. This is the revised version of an preprint that was first circulated in February 1999. I would like to thank Paul Baum, David Kazhdan, Robert Langlands, George Lusztig, Roger Plymen, and Peter Schneider for useful comments and discussions.

Homology of Hecke algebras
In this section we shall use several general results on Hochschild homology of algebras, on algebraic groups, and on the continuous cohomology of totally disconnected groups. Good references are [5,6,15], for the general theory, and [12] for questions related to Hochschild homology.
If G is a group and A ⊂ G is a subset, we denote by C(A) the centralizer of A, that is, the set of elements of G that commute with every element of A, and by N (A) the normalizer of A, that is, the set of elements g ∈ G such that gAg −1 = A. By Z = Z(G) = C(G) we denote the center of G.
If X is a totally disconnected, locally compact, non-discrete space X, we denote by C ∞ c (X) the space of compactly supported, locally constant, complex valued functions on X. Recall that, if U ⊂ X is an open subset of X as above, then restriction defines an isomorphism Let G be a linear algebraic group defined over a totally disconnected locally compact field F. Thus F is a finite algebraic extension of Q p , the field of p-adic numbers. The set G(F) of F-rational points of G is called a p-adic group and will be denoted simply by G. It is known [5] that G = G(F) identifies with a closed subgroup of GL n (F), and hence it has a natural locally compact topology that makes it a totally disconnected space. We fix a Haar measure dg on G.
Consider now the space C ∞ c (G) of compactly supported, locally constant functions on G. Fix a Haar measure dh on G. Then the convolution product, denoted * , is defined by makes C ∞ c (G) an algebra, the Hecke algebra of G. It is important in representation theory to determine the (Ad G -)invariant linear functionals on C ∞ c (G). If G is unimodular, the space of invariant linear functionals on C ∞ c (G) coincides with the space of traces on C ∞ c (G); moreover, since the space of traces of C ∞ c (G) identifies with HH 0 (C ∞ c (G)), the first Hochschild cohomology group of C ∞ c (G), it is reasonable to ask what are the groups HH q (C ∞ c (G)), the Hochschild homology groups of C ∞ c (G), in general. Since HH q (C ∞ c (G)) is the algebraic dual of HH q (C ∞ c (G)), it is enough to concentrate on Hochschild homology. The computation of the groups HH q (C ∞ c (G)) is the main purpose of this paper. We first recall the definition of HH * (C ∞ c (G)). Let (q + 1)-times, be the usual (algebraic) tensor product of vector spaces. The Hoch- is an inductive limit of unital algebras, this definition coincides with the usual definition of Hochschild homology for non-unital algebras (using algebras with adjoint unit).
The group G acts by conjugation on C ∞ c (G), and we denote by C ∞ c (G) ad the G-module defined by this action. Also, let ∆ G denote the modular function of G, which we recall, is defined by the relation We shall be especially interested in the G-module C ∞ c (G) δ obtained from C ∞ c (G) ad by twisting it with the modular function. More precisely, let C ∞ c (G) δ = C ∞ c (G) as vector spaces, and let the action of G on functions be given by the formula The reason for this twisting is that, for G non-unimodular, the traces of C ∞ c (G) are the G-invariant functionals on C ∞ c (G) δ , not on C ∞ c (G) (this is an immediate consequence of Lemma 1). More generally, our approach to the Hochschild homology of C ∞ c (G) is based on Lemma 1. Before stating and proving Lemma 1, we need to introduce some notation. First, if M is an arbitrary G-module, we denote by M ⊗ ∆ G the tensor product of the G-modules M and C, where the action on C is given by the multiplication with the modular function of G. (We sometimes write this as where G × X is the quotient (G × X)/H for the action h(g, x) = (gh −1 , hx). This isomorphism is obtained by observing that the natural map passes to the quotient to give the desired isomorphism. Sometimes it will be convenient to regard a left G-module as a right G-module by replacing g with g −1 .
Also, recall that a G-module M is smooth if, and only if, the stabilizer of each element of M is open in G. Then one can define the continuous homology groups of G with coefficients in the smooth module M , denoted H k (G, M ), using tensor products as follows. Let B q (G) = C ∞ c (G q+1 ), q = 0, 1, . . . , be the Bar complex of the group G, with differential Then the complex (B q , d) gives a resolution of C with projective C ∞ c (G)-modules, and the complex computes H q (G, M ). See [3,6].
As in equation (7), the map h G descends to the quotient to induce an isomorphismh , as desired. To better justify the twisting of the module C ∞ c (G) by the modular function in the above lemma, note that the trivial representation of G gives rise to an obvious morphism π 0 : C ∞ c (G) → C, by π 0 (f ) = G f (g)dg, which hence defines a trace on C ∞ c (G). However π 0 is not G-invariant for the usual action of G, but is invariant if we twist the adjoint action of G by the modular function, as indicated.
We proceed now to a detailed study of the G-module C ∞ c (G) δ . First we define a natural Ad G -invariant stratification of G, called the standard stratification of G.
Let g be the Lie algebra of G in the sense of linear algebraic groups. Denote by a i (g) the coefficients of the polynomial det(t + 1 − Ad g ), Let a r be the first non-zero coefficient a i , and define Thus V 0 = G, by convention, and G \ V 1 = G ′ , the set of regular elements of G if G is reductive. Also, V m+1 = ∅ because a m = 1. We observe that the functions a i (g) are G-invariant polynomial functions on G, and that they depend only on the semisimple part of g.
The description of the Hochschild homology of C ∞ c (G) that we shall obtain is formulated in terms of certain commutative subgroups of G, called standard, that we now define. Definition 1. A commutative subgroup S ⊂ G is called standard if, and only if, there exists a semisimple element s 0 ∈ G such that S is the center of C(s 0 ), the centralizer of s 0 in G.
If s 0 and S are as in the definition above, then S is the set of F-rational points of a subgroup S of G and C(S) = C(s 0 ). The element s 0 in the above definition is not unique in general, and, for a standard subgroup S ⊂ G, we denote by S reg ⊂ S the set of semisimple elements s ∈ S such that C(S) = C(s). An element s ∈ S reg will be called S-regular. This set is not empty by the definition of a standard subgroup. Note that S reg depends on G also, and not only on S.
Standard subgroups exist. Indeed, if γ ∈ G is semisimple, then S(γ) := Z(C(γ)), the center of the centralizer of γ, is a standard subgroup of G. In particular, every semisimple element of G belongs to S reg , for some standard subgroup S of G. For all standard subgroups S, the set S reg is an open subset of S in the Zariski topology.
For any p-adic group H, we denote by H u the set of unipotent elements of H, and call it the unipotent variety of H. In the particular case of H = C(S), where S ⊂ G is a standard subgroup, we also denote C(S) u = U S .
In order to proceed further, recall that the Jordan decomposition of an element g ∈ G is g = g s g u , where g s is semisimple, g u is unipotent, and g s g u = g u g s . This decomposition is unique [5]. If g = g s g u is the Jordan decomposition of g ∈ G and if g s ∈ S reg , then g u ∈ U S , by definition, and hence g ∈ S reg U S .
Consider now a standard subgroup S ⊂ G and let F S = Ad G (S reg ), and F u S = Ad G (S reg U S ) be the set of semisimple elements of G conjugated to an element of S reg and, respectively, the set of elements g ∈ G conjugated to an element of S reg U S (i.e., such that the semisimple part of g is in F S ). Also, let N (S) be the normalizer of S and W (S) = N (S)/C(S). Since N (S) leaves S reg invariant and is actually the normalizer of this set, it follows that the quotient W (S) can be identified with a set of automorphisms of S ss , the subgroup of semisimple elements of S. Since N (S) is the set of F-rational points of an algebraic group, the rigidity of tori ( [5], page 117) shows that W (S) is finite.
The natural map (g, s) → gsg −1 descends to a map Similarly, we obtain a map In the following proposition we consider the locally compact (and Hausdorff) topology of G, and not the Zariski topology. Denote by G ss the set of semisimple elements of G. Proposition 1. Let S be a standard subgroup of G. Using the above notation, we have: (i) The set F S is an analytic submanifold of G, and the maps φ S and φ u S are homeomorphisms. ( The same argument shows that if F S and F S ′ have a point in common, then the standard subgroups S and S ′ are conjugated in G. The injectivity of φ u S follows from the injectivity of φ S , indeed, if g 1 (s 1 u 1 )g −1 1 = g 2 (s 2 u 2 )g −1 2 , let g = g −1 2 g 1 as above, and conclude that gs 1 g −1 = s 2 , by the uniqueness of the Jordan decomposition. As above, this implies that g ∈ N (S).
Since the differential dφ S is a linear isomorphism onto its image (i.e. it is injective) and φ S is injective, it follows that φ S is a local homeomorphism onto its image (for the locally compact topologies), and that its image is an analytic submanifold ( [21], p. 38, Theorem (2.3)). The set G ss ∩ (V k \ V k+1 ) is an algebraic variety on which G acts with orbits of the same dimension, and hence φ S is proper. This proves that φ S is a homeomorphism. Using an inverse for φ S , we obtain that φ u S is also a homeomorphism.
To prove now (ii), consider a standard subgroup S ⊂ G, and let d be the dimension of C(S).
This must then be a disjoint union because the sets F S are either equal or disjoint, as proved above.
See also [24]. Let R ∞ := R ∞ (G) be the ring of locally constant Ad G -invariant functions on G with the pointwise product, which we regard as a subset of the set of endomorphisms where φ is a locally constant function φ : F k → C such that φ(0, 0, . . . , 0) = 0. (Recall that each of the polynomials a 0 , . . . , a r−1 is the 0 polynomial.) By convention, we set I 0 = (0); also, it follows that I m+1 = R ∞ .
Fix now k, and let p n ∈ I k be the function φ n (a r , a r+1 , . . . , a r+k−1 ), where φ n : F k → C is 1 on the set and vanishes outside this set. (Here q is the number of elements of the residual field of F, and the non-archimedean norm "| |" is normalized such that its range is {0}∪{q n , n ∈ Z}.) Then p n = p 2 n = p n p n+1 and I k = ∪p n R ∞ . For further reference, we state as a lemma a basic property of the constructions we have introduced.
As a consequence of this above lemma, we obtain the following result.
of vector spaces.
Proof. There exists a (not natural) isomorphism of vector spaces. By the above lemma, the inclusion of because the functor H q is compatible with inductive limits and with direct sums. The naturality of these isomorphisms and the Five Lemma show that . This is enough to complete the proof.
We now study the homology of the subquotients M k = I k+1 C ∞ c (G) δ /I k C ∞ c (G) δ by identifying them with induced modules. Let Σ k be a set of representative of conjugacy classes of standard subgroups S such that Lemma 3. Using the above notation, we have Proof. It follows from the definition of , then we can find some polynomial a i , with i ≤ r + k − 1, such that |a i | is bounded from below on the support of f by, say, q −n , then p n f = f . The second isomorphism follows from the first isomorphism using (4) and Lemma 2.
. Then Shapiro's lemma, see [7], states that (A proof of Shapiro's Lemma in our setting is contained in the proof of Theorem 4.) The basic examples of induced modules are obtained from H-spaces. If X is an H-space (we agree that H acts on X from the left), then . Shapiro's lemma is an easy consequence of the Serre-Hochschild spectral sequence, see [7], which states the following. Let M be a smooth G-module and H ⊂ G be a normal subgroup. Then the action of G on H q (H, M ) descends to an action of G/H, and there exists a spectral sequence with E 2 as before. Proposition 2. Using the above notation, we have a natural Proof. Let S be a standard subgroup of G. Recall first that W (S) = N (S)/C(S) is a finite group that acts freely on S reg , which gives a N (S)-equivariant isomorphism Combining these two isomorphisms, we obtain . The result then follows from Lemma 3, which implies directly that Combining this proposition with Corollary 1, we obtain the main result of this section. Recall that a p-adic group G = G(F) is the set of F-rational points of a linear algebraic group G defined over a non-archemedean, non-discrete, locally compact field F of characteristic zero. Also, recall that U S is the set of unipotent elements commuting with the standard subgroup S, and that the action of C(S) on Theorem 1. Let G be a p-adic group. Let Σ be 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 Remark. The isomorphism of the above theorem is not natural. A more natural description of HH q (C ∞ c (G)) will be obtained in one of the following sections by considering higher orbital integrals and their Shalika germs

Higher orbital integrals and their Shalika germs
Proposition 2 of the previous section allows us to determine the structure of the localized cohomology groups HH * (C ∞ c (G)) m , where m is a maximal ideal of R ∞ (G). This will lead to an extension of the higher orbital integrals introduced by Blanc and Brylinski in [4], and to a generalization of some results of Shalika [22] to higher orbital integrals. In this way, we shall also obtain a more natural description of HH q (C ∞ c (G)). First recall the following result.
Proposition 3. Let G be a reductive p-adic group over a field of characteristic 0, S ⊂ G be a standard subgroup, and γ ∈ S reg (that is, γ is a semisimple element such that C(S) = C(γ)). Then there exists a N (S)-invariant closed and open neighborhood of γ in C(S) such that Proof. The result follows from Luna's Lemma. For p-adic groups, Luna's Lemma is proved in [17], page 109, Properties "C" and "D." From this proposition we obtain the following consequences for the ring R ∞ (G).

Corollary 2.
Let U and V be as in Proposition 3 above. To prove (iii), observe that the maximal ideal m is generated by a sequence of projections p n , that is, m = ∪p n R ∞ (G), with p 2 n = p n . We know from [17], Proposition 2.5, that R ∞ (G) is isomorphic to C ∞ , for some locally compact, totally disconnected topological space X. Moreover, if M is a C ∞ (X)-module and m is the maximal ideal of functions vanishing at x 0 , for some fixed point x 0 ∈ X, then C ∞ (X) m ≃ C ∞ (X)/mC ∞ (X), and hence Since X is metrizable, we can choose a basis V n of compact open neighborhoods of x 0 in X. Then, if we let p n to be the characteristic function of V c n , then p n are projections generating m. By choosing V n to be decreasing, we obtain a decreasing sequence p n .
We now consider for each maximal ideal m ⊂ R ∞ = R ∞ (G) the localization HH q (C ∞ c (G)) m . Proposition 4. Let m be a maximal ideal of R ∞ (G). If m consists of the functions that vanish at the semisimple element γ ∈ G and S ⊂ G is a standard subgroup such that γ ∈ S reg , then Since all the isomorphisms of Proposition 2 are compatible with this the localization functor, we obtain that The only quotient C ∞ c (S reg ) W (S) /mC ∞ c (S reg ) W (S) that does not vanish is the one containing (a conjugate of) γ, and then it is isomorphic to C. This completes the proof.
An alternative proof can be obtained by writing , by Corollary 2 (iii). However the above proof is more convenient when dealing with orbital integrals. See also [16], first circulated in 1990 as a preprint of the Mathematical Institute of the Romanian Academy (INCREST) Nr. 18, March 1990, and where the localization techniques were first introduced.
We now extend the definition of higher orbital integrals introduced by Blanc and Brylinski to cover non-regular semisimple elements also. Fix a standard subgroup S ⊂ G, and let k be such that S reg ⊂ V k \ V k+1 . As in the above proof, Proposition 2 gives a natural R ∞ (G)-linear, degree preserving, surjective morphism , and hence a linear map Fix c ∈ H q (C(S), C ∞ c (U S ) δ ) and γ ∈ S reg , and let O γ,c = O S γ,c : I k+1 HH q (C ∞ c (G)) −→ C be the evaluation of the map at γ and c in (13). We obtain, in particular, that for any f ∈ I k+1 HH q (C ∞ c (G)), the function γ → O γ,c (f ) is a locally constant, compactly supported function on S reg . The function O γ,c can then be extended to the whole group HH q (C ∞ c (G)) using a simple observation. For any γ ∈ S reg there exists a locally constant function φ ∈ I k+1 such that φ(γ) = 1. Then let which is independent of φ. It follows from definition of O γ,c that, for any f ∈ HH q (C ∞ c (G)), the function γ → O γ,c (f ), obtained as above, is a locally constant function on S reg , but not necessarily compactly supported. We thus obtain the following result.
, which is an isomorphism when localized at each maximal ideal m ⊂ R ∞ (G) consisting of functions vanishing at an element γ ∈ S reg .
We call the maps O S and O γ,c = O S γ,c "higher orbital integrals" because they generalize the usual notion of orbital integral. (If c is a cocycle of dimension q, we call O γ,c a q-higher orbital integral.) Indeed, assume that G and C(S) are unimodular, following thorough the identifications in the previous section, we obtain, for c 0 = 1 ∈ H 0 (C(S), C ∞ c (U S ) δ ) the evaluation at the identity element e ∈ G, and f ∈ C ∞ c (G) = HH 0 (C ∞ c (G)), that where dg is the induced measure on G/C(S).
If γ ∈ G is a semisimple element and S is a standard subgroup of G such that C(γ) = C(S), (i.e., γ ∈ S reg ), then restriction at γ defines a map . A word on notation, whenever we write O S γ,c or O S γ , we assume that γ ∈ S reg , which actually determines S. This means that we can omit S from notation. However, if we want to write that O γ,c = O S γ,c is obtained by specializing to a point γ ∈ S reg and then by evaluating at c, that is that , c , then it is obviously more convenient to include S in the notation.
Let γ ∈ G be a semisimple element. We want now to investigate the behavior of the orbital integrals O g,c with g in a small neighborhood of γ. Fix a standard subgroup S ⊂ G such that γ is in the closure of Ad G (S reg ), but is not in Ad G (S reg ), and a class c ∈ H q (C(S), C ∞ c (U S )). More precisely, we want to study the germ of the function g → O g,c (f ) at an element γ, where f ∈ HH q (C ∞ c (G)) is arbitrary. The germ of a function h at γ will be denoted h γ .
The following theorem extends one of the basic properties of Shalika germs from usual orbital integrals to higher orbital integrals.
Theorem 2. Let S ∈ G be a standard subgroup γ ∈ S an element in the closure of S reg , but γ ∈ S reg . Then there exists a degree preserving linear map , for all f ∈ HH * (C ∞ c (G)). Note that, in the notation for the maps σ S γ , the standard subgroup S is no longer determined by γ.

Proof. By the definition of the localization of a module, the map
is an isomorphism by Proposition 4, we may define σ S γ = F • O −1 γ , and all desired properties for σ S γ will be satisfied.
Let γ ∈ S \ S reg be such that γ is in the closure of S reg , as above, and also let c ∈ H q (C(S), C ∞ c (U S ) δ ). Then a consequence of the above theorem, Theorem 2, is that the germ at γ of the higher orbital integrals O S g,c depends only on O γ . More precisely, if g ∈ S reg , f ∈ HH q (C ∞ c (G)), and we regard O S g,c (f ) as a function of g, then its germ at γ, denoted O S g,c (f ) γ , is given by This observation allows us to relate Theorem 2 with results of Shalika [22] and Vigneras [24]. So assume now that G is reductive and let ξ i ∈ H 0 (C(γ), C ∞ c (C(γ) u ) δ ) be the basis dual to the basis of H 0 (C(γ), C ∞ c (C(γ) u ) δ ) given by the orbital integrals over the orbits of γu, for u nilpotent in C(γ). If we let F i = σ γ (ξ i ), then we recover the usual definition of Shalika germs. Due to this fact, we shall call the maps σ S γ introduced in Theorem 2 the higher Shalika germs. We can now characterize the range of the higher orbital integrals. Combining all higher orbital integrals for S ⊂ G ranging through a set Σ of representatives of standard subgroups of G, we obtain a map Theorem 3. Let Σ be a set of representatives of standard subgroups of G and σ S γ be the maps introduced in Theorem 2 for γ ∈ S reg \ S reg . Also, let be the space of sections ξ satisfying ξ γ = σ S γ (ξ(γ)). Then O establishes a R ∞ (G)linear isomorphism O : HH * (C ∞ c (G)) −→ F . Proof. Note first that the map O is well defined, that is, that its range is contained in F , by Theorem 2.
To prove that O is an isomorphism, filter both HH * (C ∞ c (G)) and F by the subgroups I k HH * (C ∞ c (G)) and, respectively, by I k F , using the ideals I k introduced in Section 1. Since O is R ∞ (G)-linear, it preserves this filtration and induces maps I k+1 HH * (C ∞ c (G))/I k HH * (C ∞ c (G)) → I k+1 F /I k F . These maps are, by construction, exactly the isomorphisms of Proposition 2. Standard homological algebra then implies that O itself is an isomorphism, as desired.
A consequence of the above result the following "density" corollary.
Corollary 3. Let a ∈ HH q (C ∞ c (G)). If all q-higher orbital integrals of a vanish, then a = 0.
We shall also need certain specific cocycles below. Let τ 0 be the trace τ 0 (f ) = f (e) on C ∞ c (G), G unimodular, obtained by evaluating f at the identity e of G. Let G 0 be the kernel of all unramified characters of G. Then G/G 0 ≃ Z r , where r is the rank of a split component of G. Let p j : G → Z be the morphisms obtained by considering the jth component of Z r . Then δ j (f )(g) = p j (g)f (g) defines a derivation of C ∞ c (G). Moreover, we can identify H * (G) with Λ * C r , the exterior algebra with generators δ 1 , . . . , δ r . Fix c ∈ H * (G). We can assume that c = δ 1 ∧ . . . ∧ δ q , and then we define the map D c : Then τ c = τ 0 • D c (f 0 , . . . , f q ) defines a Hochschild q cocycle on C ∞ c (G), and if we naturally identify c with an element of the cohomology group H * (G, C ∞ c (G u )).

The cohomology of the unipotent variety
It follows from the main result of the first section, Theorem 1, that, in order to obtain a more precise description of the Hochschild homology of C ∞ c (G), we need to understand the continuous cohomology of the H-module C ∞ c (H u ) δ , where H ranges through the set of centralizers of standard subgroups of G and H u is the variety of unipotent elements in H. (We call the variety H u the unipotent variety of H.) Since the cohomology groups H k (H, C ∞ c (H u ) δ ) depend only on H, it is enough to consider the case H = G. In this section we gather some results on the groups H q (G, C ∞ c (G u ) δ ). We first need to recall the computation of the groups H * (G) = H * (G, C), [6]. More generally, we also need to compute H * (G, C χ ), where χ : G → C * is a character of G and C χ = C as a vector space, but with G-action given by the character χ.
Assume first that G = S is a commutative p-adic group, and let S 0 be the union of all compact-open subgroups of S. Then S 0 is a subgroup of S and S/S 0 is a free abelian subgroup, whose rank we denote by rk(S). For this group we then have For an arbitrary p-adic group G, we may identify the cohomology groups H q (G) with those of a commutative p-adic group. Indeed, if G 0 is the connected component of G (in the sense of algebraic groups) then G/G 0 is finite, and hence H q (G) ≃ H q (G 0 ), by the Hochschild-Serre spectral sequence. This tells us that we may assume G to be connected as an algebraic group. Choose then a Levi decomposition G = M N , where N is the unipotent radical of G, M is a reductive subgroup, uniquely determined up to conjugation, and the product M N is a semidirect product. Since H q (N ) = 0 for q > 0, it follows that H q (G) ≃ H q (M ). Let M 1 ⊂ M be the commutator subgroup of M , which is also a p-adic group, see [5]. The cohomology groups H q (M 1 ) were computed in [4], Proposition 6.1, page 316, and [6] and they also vanish for q > 0 (because the fundamental domain of the building of G is a simplex). All in all, we obtain that where M ab = M/M 1 is the abelianization of M .
We summarize the above discussion in the following well known statement.
and H q (G, C χ ) = 0, if χ is a nontrivial character of G.
We continue with a few elementary remarks on H k (G, C ∞ c (G u ) δ ). Remark 1. If G 1 → G is a surjective morphism with finite kernel F , then there exists a natural homeomorphism G 1u ≃ G u of the unipotent varieties of the two groups. Since the kernel F acts trivially on G 1u , using the Hochschild-Serre spectral sequence we obtain an isomorphism Remark 2. If G ⊂ G 1 is a normal p-adic subgroup with F ≃ G 1 /G finite, then we again have a natural homeomorphism G 1u = G u . This gives using once again the Hochschild-Serre spectral sequence. In particular, if the characteristic morphism F → Aut(G)/ Inn(G) is trivial, then we get a natural isomor- Remark 4. If Z is a commutative p-adic group of split rank r, then Remark 5. The above isomorphisms reduce the computation of H k (G, C ∞ c (G u ) δ ) for G reductive, to the computation of the cohomology groups corresponding to its semisimple quotient H := G/Z(G): where r is the rank of a split component of G.
Let τ 0 be the trace obtained by evaluating at the identity. Using τ 0 , we obtain an injection H j (G) ∋ c → τ 0 ⊗ c ∈ H j (G, C ∞ c (G u )). In order to obtain more precise results on H * (G, C ∞ c (G u ) δ ), we need to take a closer look at the structure of C ∞ c (G u ) as a G-module. For a G-space X, we denote by X the quotient space X/G with the induced topology, which may be non-Hausdorff. Thus G u is the set of unipotent conjugacy classes of G.
Assume now that G u is a finite set. (This happens for example if G is reductive, because the ground field F has characteristic zero.) Then the space G u can be written as an increasing union of open G-invariant sets U l ⊂ G u , U −1 = ∅, such that each difference set U l \ U l−1 is a disjoint union of open and closed G-orbits, A filtration U l with these properties will be called "nice." There may be several nice filtrations of G u .
A nice filtration of G u , as above, gives rise, by standard arguments, to a spectral sequence converging to H k (G, C ∞ c (G u ) δ ), as follows. First, let g ∈ G u be the orbit through an element g ∈ G u . Also, let C(g) denote the centralizer of g ∈ G u and r g denote the rank of a split component of C(g) if C(g) is unimodular, r g = 0 otherwise. This definition of r g is justified by H k (C(g), ∆ C(g) ) ≃ Λ k C rg .
Proposition 6. Let G be a p-adic group with finitely many unipotent orbits (i.e., G u is finite). Then, for any nice filtration (U l ) of G u by open G-invariant subsets, there exists a natural spectral sequence with The argument is standard and goes as follows. Recall first that any filtration . Now, associated to the open sets U l of a nice filtration, there exists an increasing where each X l,j is the orbit of a unipotent element, and U l \ U l−1 has the topology given by the disjoint union of the orbits X l,j . Fix l and j, and let u be a unipotent element in X l,j (so that then X l,j is the orbit through u), which implies that ). Finally, from Shapiro's lemma we obtain that H k (G, C ∞ c (X l,j ) δ ) ≃ H k (C(u), ∆ C(u) ) ≃ Λ k C ru , and this completes the proof.
We expect this spectral sequence to converge for G reductive. This is the case, for example for G = GL n (F) and for SL n (F). See Section 5. The convergence of the spectral sequence implies, in particular, the convergence of the orbital integrals of unipotent elements in reductive groups (which is a well known fact due to Deligne and Rao [18]). In general, the convergence of the spectral sequence of the above proposition can be interpreted to represent the convergence of "higher orbital integrals."

Induction and the unipotent variety
We assume from now on that G is reductive, and we fix a parabolic subgroup P ⊂ G, P = G, and a Levi subgroup M ⊂ P , so that P = M N , where N is the unipotent radical of P , and the product is a semidirect product. In this section, we relate H * (G, C ∞ c (G u ) δ ) to the groups, H * (P, C ∞ c (P u ) δ ) and H * (M, C ∞ c (M u ) δ ). He have considered non-unimodular subgroups in the previous sections in order to be able to handle subgroups like P .
Let K be a "good" maximal compact subgroup of G (see [9], Theorem 5), so that G = KP . This decomposition shows that the map is proper, and hence the map G × P P := (G × P )/P ∋ (g, p) → gpg −1 ∈ G is also proper. This gives a map of G-modules. This map of G-modules and the standard identification of Hochschild homology with continuous cohomology, equation (9), then give a morphism ind G P : HH * (C ∞ c (G)) −→ HH * (C ∞ c (P )), (22) defined as the composition of the following sequence of morphisms of Hochschild homology groups. The main result of this section states that ind G P is induced by a morphism of algebras, which we now proceed to define.
Let dk be the normalized Haar measure on the maximal compact subgroup K, normalized such that K has volume 1. The composition of kernels be defined by φ G P (f )(k 1 , k 2 , p) = f (k 1 pk −1 2 ). Recall [9] that the push-forward of the product dpdk of Haar measure on P × K, via the multiplication map P × K ∋ (p, k) → pk ∈ G, is a left invariant measure on G, and hence a multiple λdg of the Haar measure dg on G. Suppose that the measure dk of K is the restriction of dg to K, and has total mass 1. Then the Haar measures on G and P will be called compatible if λ = 1. We shall need the following result of Harish Chandra (implicitly stated in [23]): Lemma 5. Suppose the Haar measures on G and P are compatible. Then the linear map φ G P , defined above in equation (23), is a morphism of algebras. Proof. The product on C ∞ (K × K) ⊗ C ∞ c (P ) = C ∞ c (K × K × P ) is given by the formula Let * denote the multiplication (i.e., convolution product) on C ∞ c (G). Thus, we need to prove that . Consider the map P × K ∋ (q, k) → g := qk −1 ∈ G, and let dµ be the push-forward of the measure dqdk. Then the right-hand side of the above formula becomes We know that dµ = dg, by assumptions (see the discussion before the statement of this lemma), and then by the invariance of the Haar measure. The lemma is proved.
Theorem 4. Let P be a parabolic subgroup of a reductive p-adic group G. Consider the morphisms (φ G P ) * and ind G P : HH * (C ∞ c (G)) −→ HH * (C ∞ c (P )), defined above (Equations (24) and (22)). Then (φ G P ) * = ind G P . Proof. Let M 1 and M 2 be two left G-modules. We can regard M 1 as a right module, and then the tensor product M 1 ⊗ G M 2 is the quotient of M 1 ⊗ M 2 by the group generated by the elements gm 1 ⊗ gm 2 − m 1 ⊗ m 2 , as before. Alternatively, we can think of We shall prove the theorem by an explicit computation. To this end, we shall use the results and notation (h G andh G = h G ⊗ G 1) of Lemma 1.
By direct computation, we see that the morphism between Hochschild complexes, is given by the formula We now want to realize the map ind G P : HH * (C ∞ c (G)) −→ HH * (C ∞ c (P )), at the level of complexes. In the process, it will be convenient to identify the smooth Gmodule C ∞ c ((G × P )/P ) ≃ ind G P (C ∞ c (P ) δ ) with a subspace of the space of functions on G × P , using the projection G × P → (G × P )/P .
Consider the G-morphism . . , g q , g, p) = f (g 0 , g 1 , . . . , g q , gpg −1 ). Then the resulting morphism The G-morphism is well defined and induces an isomorphism in homology, because the only nonzero homology groups are in dimension 0, and they are both isomorphic to ind G P (C ∞ c (P ) δ ). We have an isomorphism , of complexes. This shows that the homology of the second complex in (26) is isomorphic to H q (P, C ∞ c (P ) δ ), and that the map induced on homology, that is is induced by the morphism of complexesh q defined in Lemma 1, Equation (9). From the definition of the morphism ind G P : HH * (C ∞ c (G)) → HH * (C ∞ c (P )) and the above discussion, we obtain the equality of the morphisms H Thus, in order to complete the proof, it would be enough to check thath P • χ(rl ⊗ G 1) = τ • φ G P •h G at the level of complexes. Let where dk = dk 0 . . . dk q , as before. Then r ′ induces a morphism This completes the proof.
For simplicity, we have stated and proved the above result only for G reductive, however, it extends to general G and P such that G/P is compact, by including the modular function of G, where appropriate.
In order to better understand the effect of the morphism ind G P = (φ G P ) * : HH * (C ∞ c (G)) −→ HH * (C ∞ c (P )), it is sometimes useful to look at its action on the geometric fibers of the group HH * (C ∞ c (G)). This is especially useful because the action on the geometric fibers also recovers the classical results on the characters of induced representations.
First we observe that restriction defines a morphism ρ G P : R ∞ (G) → R ∞ (P ). In case the group G is reductive and M is a Levi component of the parabolic subgroup P , we also have R ∞ (P ) ≃ R ∞ (M ). Lemma 6. Let P be a parabolic subgroup of a reductive p-adic group G, and let ρ G P : R ∞ (G) → R ∞ (P ) be the morphism induced by restriction, used to define a R ∞ (G)-module structure on HH * (C ∞ c (P )). Then ind G P : , for all f ∈ R ∞ (G) and all ξ ∈ HH * (C ∞ c (G)). Proof. The result of the lemma follows from the fact that the map Alternatively, one can use the explicit formula of equation (25).
is the maximal ideal of functions vanishing at a semisimple element γ ∈ G, then its image (ρ G P ) * (m) := ρ G P (m)R ∞ (P ) ⊂ R ∞ (P ) = R ∞ (M ) is the ideal of functions vanishing at all g ∈ M that are conjugated to γ in G. If γ is elliptic, then m = R ∞ (P ). If γ ∈ M , then m need not, in general, be maximal. Nevertheless, we obtain a morphism where # γ = l is the set of conjugacy classes in M that consist of elements that are conjugated to γ in the bigger group G. Let γ 1 , γ 2 , . . . , γ l ∈ M be representatives of the conjugacy classes of element in M that are conjugated to γ in G.
We are ready now to study the morphisms HH q (C ∞ c (P )) γj .
Let C P (γ j ) be the centralizer of γ j in P and C G (γ j ) ≃ C G (γ) be the centralizer of γ j in G. Then C P (γ j ) u identifies with a subspace of C G (γ) u , which gives rise to a continuous proper map C G (γ) × CP (γ) C P (γ) u → C G (γ) u , and hence to a morphism Passing to cohomology, we obtain using Shapiro's Lemma a morphism

Recall that Proposition 4 gives isomorphisms
HH q (C ∞ c (P )) γj ≃ H q (C P (γ j )), C ∞ c (C P (γ j ) u ) δ ).  − 1). Then, using localization at the maximal ideal defined by γ in R ∞ (G) = R ∞ (P ) and the above notation, Proof. Fix γ ∈ G, not necessarily semisimple and let N γ be the subgroup of elements of N commuting with γ. We choose a complement V γ of Lie(N γ ) in Lie(N ) and we use the exponential map to identify V γ with a subset of N . Then the Jacobian of the map , and from this the result follows.
This result is compatible with the results of van Dijk on characters of induced representations, see [8].

Examples
Our results can be used to obtain some very explicit results in certain particular cases.
Example 1. Let Z be a commutative p-adic group of split rank r (so that H q (Z) ≃ Λ q C r , for all q ≥ 0). Then Example 2. Let P be the (parabolic) subgroup of upper triangular matrices in SL 2 (F), and A ⊂ P be the subgroup of diagonal matrices. Then inflation defines a morphism inf P A : , with I the identity matrix of SL 2 (F). (We see this by localizing at each γ.) To describe the kernel of inf P A , let Then, if we choose b to range through Σ u , a set of representative of F * /F * 2 , the set of elements u b forms a set of representatives of the set of nontrivial conjugacy classes of unipotent elements of P . Recall that F has characteristic zero, so Σ u is a discrete set. Let O b,+ be the orbital integral associated to u b , and let O b,− be the orbital integral associated to −u b , then A as follows. The map F + ⊕ F − : ker(inf P A ) → C ±Σu is injective, and the range of each of F ± is the set of elements with zero sum.
Note that evaluation at ±I does not define a trace on C ∞ c (P ). Actually, in the spectral sequence of Proposition 6, the obstruction to extend the evaluation at I to a trace is responsible for "killing" the 1 cohomology supported at I (which explains our claim on the range of inf P A above). Example 3. Consider now the group G = SL 2 (F), where F is a p-adic field of characteristic zero such that the characteristic of the residual field is not 2, for simplicity. Let F q be the residual field of F (thus q = p n , for some n ∈ N and some prime p, denotes the number of elements of F q . We choose ǫ in the valuation ring of F, whose image in F q is not a square. Also, let τ be a generator of the (unique) maximal ideal of the valuation ring of F. We shall use the notation of [20] and thus let θ range through the set {ǫ, τ, ǫτ } and let T θ T # θ be the elliptic tori defined there. (Recall that T θ = {[a ij ], a 11 = a 22 , a 21 = θa 12 } and T # θ = {[a ij ], a 11 = a 22 , a 21 = θ # a 12 }, where θ # = θa 2 , for some a ∈ F 2 not in the image of the norm map N : F[θ] * → F * .) We distinguish two cases, first the case where −1 is a square and then the case where it is not a square. If −1 is a square, then the Weyl group of each of the tori T = T θ , T # θ has order 2. Otherwise W (T ) = {1}, for each T = T θ or T = T # θ , but T θ and T # θ are conjugate for each fixed θ. Let X = ∪ θ T θ /S 2 ∪ ∪ θ T # θ /S 2 , if −1 is a square, and X = ∪ θ T θ otherwise, with the induced topology. Then X \ {±} identifies with the set of elliptic conjugacy classes of SL 2 (F).
Denote by A ⊂ SL 2 (F) the set of diagonal matrices in SL 2 (F). Let W (A) = S 2 act on C ∞ c (A) ⊗ Λ * C by conjugation on C ∞ c (A) and act by the nontrivial character on C. Then we have the following. Recall that there are 10 conjugacy classes of unipotent elements in SL 2 (F), if p = 2.
Proposition 9. The composition φ := inf P A • ind G P : HH * (C ∞ c (SL 2 (F))) → HH * (A) = C ∞ c (A) ⊗ Λ * C has range consisting of W (A) invariant elements, and the kernel of φ is isomorphic to C ∞ c (X \ {±I}) ⊕ C 10 , via orbital integrals with respect to elliptic and unipotent elements.
Proof. First of all, it is clear that the composition φ = inf P A • ind G P is invariant with respect to the Weyl group W (A), and hence its range consists of W (A)-invariant elements.
The localization of φ at a regular, diagonal conjugacy class γ is onto by Proposition 4. Next, we know that every orbital integral extends to C ∞ c (SL 2 (F)), and this implies directly that the spectral sequence of Proposition 6 collapses at the E 2 term. This proves that the localization of φ at γ = 1 is also onto, and hence φ is onto. The rest of the proposition follows also from Proposition 6 by localization.
This example is also discussed in [4], but from a different perspective.
Example 4. We end this section with a description of the ingredients entering in the formula (1) for the of the Hochschild homology of C ∞ c (G), if G = GL n (F). Let γ ∈ G be a semisimple element. The minimal polynomial Q γ of γ decomposes as Q γ = p 1 p 2 . . . p r into irreducible polynomials with coefficients in F. (We assume, for simplicity, that each polynomial p j is a monic polynomial.) Also, let P γ = p l1 1 p l2 2 . . . p lr r be the characteristic polynomial of γ. Then the algebra generated by γ is F[γ] ≃ K 1 ⊕ . . . ⊕ K r , where K i = F[t]/(p i (t)) are not necessarily distinct fields. The commutant {γ} ′ of γ in M n (F) is the commutant of this algebra, and hence {γ} ′ ≃ M l1 (K 1 ) ⊕ M l2 (K 2 ) ⊕ · · · ⊕ M lr (K r ), and S reg = {(x i ) ∈ S, x i generates K i and the minimal polynomials of x i are distinct}.
By the Skolem-Noether theorem, the Weyl group W (S) = N (S)/C(S) coincides with the group of algebra automorphisms of {γ} ′ . This group has as quotient a group isomorphic to the subgroup Π ⊂ N (S) which permutes the algebras M li (K i ) Then Π ∼ = S m1 × . . . × S mt , that is, Π is a product of symmetric groups. We denote the kernel of this morphism by W 0 (S). It is isomorphic to r i=1 Aut F (K i ) (again by the Skolem-Noether theorem). The group W (S) is then the semidirect product of W 0 (S) by Π. We hence obtain exact sequences  (1), the only other ingredients necessary to compute HH * (C ∞ c (G)) are the groups H * (C(S), C ∞ c (U S )). Now, the unipotent variety of C(S) is the product of the unipotent varieties of GL li (K i ), i = 1, r, and the subgroup C(S) preserves this product decomposition. We see then that in order to prove that the spectral sequence of Proposition 6 collapses (for any choice of open subsets U i ), it is enough to check this for the spectral sequence converging to the cohomology of C ∞ c (GL n (K) u ), for an arbitrary characteristic zero p-adic field K.
Fix a unipotent element γ ∈ GL n (K). Define then V 0 = 0, V l = ker(γ −1) l ⊂ K n , if l > 0. Also, choose W l such that V l = V l−1 ⊕ W l , and define P = {γ ∈ GL n (K), γV l ⊂ V l }, and M = {γ ∈ GL n (K), γW l = W l }. Then P is a parabolic subgroup with unipotent radical N = {γ ∈ GL n (K), (γ − 1)V l ⊂ V l−1 }, and M is a Levi component of P . It is easy to check, from definition, that the P -orbit of u in N is dense. The centralizer of u is then contained in P and has split rank ≤ the split rank of P . Fix a maximal split torus A in the centralizer of u. We can assume that this split torus is contained in M . Fix now a cohomology class c 0 ∈ H q (C(γ)) ≃ H q (A) and choose a cohomology class c ∈ H q (M ) that maps to c 0 under the above restriction map. Also, let τ be the trace on τ 0 (f ) = f (e) on C ∞ c (M ) (obtained by evaluation at the identity e). Then the formula φ 0 (f 0 , . . . , f q ) = τ 0 (D c (f 0 , . . . , f q )) (30) defines a Hochschild cyclic cocycle on C ∞ c (M ). Consequently, φ = φ 0 • inf P A • • ind G P defines a Hochschild cocycle on C ∞ c (G). For any filtration U i of G u by open, invariant open sets, such that each U l \ U l−1 consists of a single orbit. Suppose that the orbit U l \ U l−1 is the orbit of γ ∈ GL n (F) considered above. Then the cocycle φ will vanish on C ∞ c (U l ) and represent the cohomology class c ∈ H q (C(γ)) ≃ H q (G, C ∞ c (U l \ U l−1 )). From this it follows that the spectral sequence of Proposition 6 degenerates at E 2 .
It is very likely that the above argument extends to arbitrary reductive G by choosing M and P as in [18].