Hochschild homology of reductive $p$-adic groups

Consider a reductive $p$-adic group $G$, its (complex-valued) Hecke algebra $H(G)$ and the Harish-Chandra--Schwartz algebra $S(G)$. We compute the Hochschild homology groups of $H(G)$ and of $S(G)$, and we describe the outcomes in several ways. Our main tools are algebraic families of smooth $G$-representations. With those we construct maps from $HH_n (H(G))$ and $HH_n (S(G))$ to modules of differential $n$-forms on affine varieties. For $n = 0$ this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) $G$-representations. It is known from earlier work that every Bernstein ideal $H(G)^s$ of $H(G)$ is closely related to a crossed product algebra of the from $O(T) \rtimes W$. Here $O(T)$ denotes the regular functions on the variety $T$ of unramified characters of a Levi subgroup $L$ of $G$, and $W$ is a finite group acting on $T$. We make this relation even stronger by establishing an isomorphism between $HH_* (H(G)^s)$ and $HH_* (O(T) \rtimes W)$, although we have to say that in some cases it is necessary to twist $C[W]$ by a 2-cocycle. Similarly we prove that the Hochschild homology of the two-sided ideal $S(G)^s$ of $S(G)$ is isomorphic to $HH_* (C^\infty (T_u) \rtimes W)$, where $T_u$ denotes the Lie group of unitary unramified characters of $L$. In these pictures of $HH_* (H(G))$ and $HH_* (S(G))$ we also show how the Bernstein centre of $H(G)$ acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of $H(G)$ and of $S(G)$ and we relate that to topological K-theory.


Introduction
The Hochschild homology of an algebra A (by default over C) is a fairly subtle invariant. For finitely generated commutative algebras it gives more or less the differential forms on the underlying affine variety -exactly that when the algebra is smooth, and otherwise HH * (A) detects some singularities of the variety. For general algebras Hochschild homology is related to noncommutative versions of differential forms [Lod,Chapter 1].
The vector space HH 0 (A) is particularly interesting, because it equals the cocentre A/[A, A] and via the trace pairing contains information about the set of irreducible representations of A. The higher Hochschild homology groups HH n (A) also have their uses: they say something about higher extensions of A-modules (via Hochschild cohomology) and they interact with further invariants of algebras like (periodic) cyclic homology. When A is the group algebra of a discrete group Γ, HH * (A) computes the group cohomology of the groups Z Γ (γ) with γ ∈ Γ [Bur].
Categories of representations of reductive p-adic groups Let G be a reductive group over a non-archimedean local field, connected as algebraic group. We aim to determine the Hochschild homology of G, by which we mean the Hochschild homology of a suitable group algebra of G. The most natural choice is the Hecke algebra HG), because Mod(H(G)) is naturally equivalent to the category Rep(G) of complex smooth G-representations. By definition HH n (H(G)) = Tor H(G)⊗H(G) op n (H(G), H(G)), so HH n (H(G)) depends only on the category of H(G)-bimodules, which is equivalent to the category of smooth G × G op -representations.
Alternatively we have the Harish-Chandra-Schwartz algebra S(G), for which Mod(S(G)) is the category Rep t (G) of tempered G-representations (by definition). We consider S(G) as a bornological algebra and use the complete bornological tensor product⊗ [Mey]. In that setting On the other hand, the full group C * -algebra C * (G) or its reduced version C * r (G) would not be suitable here, because HH n (C * (G)) = HH n (C * r (G)) = 0 for n > 0 We approach our main goal with representation theory. We start with the Bernstein decomposition Rep(G) = Hochschild homology decomposes accordingly, so we may focus on the algebras H(G) s and S(G) s . We will make ample use of the Morita equivalence between H(G) s and End G (Π s ) op , where Π s is a suitable progenerator of Rep(G) s . In [Sol7] we made a detailed analysis of End G (Π s ) op , which links it to algebras whose Hochschild homology groups have already been determined.
Let σ be a supercuspidal representation of a Levi subgroup L of G, representing s = [L, σ]. To this data one can associate a finite group W (L, s) of transformations of the complex torus of unramified characters X nr (L), satisfying Z(Rep(G) s ) ∼ = Z(End G (Π op s )) ∼ = O(X nr (L)) W (L,s) . There exists a 2-cocycle ♮ s of W (L, s) such that the twisted group algebra C[W (L, s), ♮ s ] acts "almost" on the objects of Rep(G) s by intertwining operators.
Here "almost" means that these intertwining operators depend rationally on χ ∈ X nr (L), and they can have poles. In this setting [Sol7,§5] provides an isomorphism of O(X nr (L)) W (L,s) -algebras This subcategory is stable under tensoring with elements of X unr (G), the group of unitary unramified characters of G. Like above, from [Sol6] one can expect strong similarities between S(G) s and C ∞ (X unr (L)) ⋊ C[W (L, s), ♮ s ]. Let R(A) denote the Grothendieck group of the category of finite length Arepresentations. We abbreviate R(G) s = R(H(G) s ) and R t (G) s = R(S(G) s ).
Theorem A. (see Theorem 2.5) There exists a group isomorphism which restricts to a bijection ζ ∨ t : R t (G) s → R C ∞ (X unr (L)) ⋊ C[W (L, s), ♮ s ] . These bijections are compatible with parabolic induction and with twists by unramified characters. When an isomorphism (1) has been fixed, ζ ∨ and ζ ∨ t are canonical. Hochschild homology and twisted extended quotients In a sense that we will make precise later, Theorem A induces isomorphisms on Hochschild homology.
Let Irr cusp (L) be the set of supercuspidal irreducible L-representations (up to isomorphism), so that Irr(L) s is one X nr (L)-orbit in Irr cusp (L). The group W (G, L) = N G (L)/L acts naturally on Irr(L). Let W (G, L) s be the stabilizer of Irr(L) s in W (G, L). The covering map X nr (L) → Irr(L) s : χ → σ ⊗ χ induces a bijection Let Lev(G) be a set of representatives for the conjugacy classes of Levi subgroups of G. Theorem B and the above entail that HH n (H(G)) can be regarded as the Z(Rep(G))-module of algebraic differential n-forms on L∈Lev(G) Irr cusp (L)//W (G, L) ♮ L .
Similarly we may interpret HH n (S(G)) as the Z(Rep t (G))-module of smooth differential n-forms on Notice that these descriptions mainly involve data that are much easier than Rep(G) s , only the 2-cocycles ♮ s contain information about non-supercuspidal representations. Fortunately ♮ s is known to be trivial in many cases, and we expect that it is trivial whenever G is quasi-split. We find it remarkable that such a simple description of a strong invariant of very complicated algebras is possible.
Hochschild homology via families of representations For more precise statements we employ algebraic families of G-representations. The families relevant for us come from a parabolic subgroup P = M U of G and a tempered representation η of a Levi factor M of P . All the representations I G P (η ⊗χ) with χ ∈ X nr (M ) can be realized on the same vector space V P,η , and their matrix coefficients depend algebraically on χ. The family of representations For χ ∈ X unr (M ) the members of F(M, η) are tempered. Then Harish-Chandra's Plancherel isomorphism (Theorem 1.2) shows that for f ∈ S(G) s the matrix coefficients of I G P (η ⊗ χ)(f ) are smooth functions on X unr (M ). We obtain a map HH n (F t M,η ) : HH n (S(G) s ) → Ω n sm (X unr (M )), where the subscript sm means smooth differential forms on a real manifold. With [Sol8,§1.2] this setup can be generalized to algebraic families of virtual representations, then we may speak of algebraic families in C ⊗ Z R(G) s or in C ⊗ Z R t (G) s .
For each w ∈ W (L, s) and each connected component X nr (L) w c of X nr (L) w , we will construct a particular algebraic family F(w, c) = ν 1 w,χ : χ ∈ X nr (L) w c in C ⊗ Z R(G) s . .
The canonicity of Theorem B can be formulated in similar terms. Namely, for each algebraic family F(M, η) in Rep(G) s there are equalities HH n (F M,η ) • HH n (ζ ∨ ) = HH n (F M,ζ ∨ (η) ), HH n (F t M,η ) • HH n (ζ ∨ t ) = HH n (F t M,ζ ∨ (η) ). Hochschild homology groups in degree 0 Theorem C admits a nice alternative description in degree n = 0. Let us say that a linear function on C ⊗ Z R(G) s is regular if it transforms every algebraic family F(M, η) into a regular function on X nr (M ). Similarly we call a linear function on C ⊗ Z R t (G) s smooth if it transforms F t (M, η) into a smooth function on X unr (M ).
Theorem D. (see Propositions 2.9 and 3.11) (i) The trace pairing H(G) s × R(G) s → C induce a natural isomorphism of Z(Rep(G) s )-modules We note that Theorem D.(i) was already shown in [BDK], with much more elementary methods. Theorem D.(ii) implies that the traces of irreducible tempered representations in Rep(G) s space a dense subspace of the space of trace functions on S(G) s .
The action of the Bernstein centre Theorems B and C do not yet reveal how the Bernstein centre Z(Rep(G) s ) ∼ = O(X nr (L)) W (L,s) acts on HH n (H(G) s ). That action is more tricky than it could seem, because in Theorem A it is changed. We are aided by the finer decomposition of R t (G) and S(G) in "Harish-Chandra blocks". Namely, to each square-integrable (modulo centre) representation δ of a Levi subgroup M of G one canonically associates a direct factor R t (G) d of R t (G), and a two-sided ideal S(G) d of S(G). If the supercuspidal support of δ is (L, σ), for a suitable finite set ∆ s G of square-integrable (modulo centre) representations of Levi subgroups of G. This gives rise to a decomposition For H(G) s no decomposition like (2) exists. Nevertheless something similar can be achieved with Hochschild homology groups, see below.
By the Plancherel isomorphism (Theorem 1.2) . Theorem E. (see Theorem 3.13.b, Lemma 3.8 and Lemma 2.10) (i) There exists a canonical decomposition where HH n (H(G) s ) d is the inverse image of HH n (S(G) d ) under the natural map HH n (H(G) s ) → HH n (S(G) s ). (ii) Suppose that Z(Rep t (G) d ) does not annihilate the contribution (via Theorem C) of Ω n sm (X unr (L) w c ) to HH n (S(G) s ) . Then we can arrange that X unr (L) w c is contained in χ δ X unr (M ). For χ ∈ X unr (L) w c , Z(Rep t (G) d ) acts on the fibre of HH n (S(G) s ) over W (L, s)(w, χ) via the character W (M, d)χ −1 δ χ. (iii) In the setting of part (ii), for χ ∈ X nr (L) w c , Z(Rep(G) s ) acts on the fibre of HH n (H(G) s ) over W (L, s)(w, χ) via the character W (L, s)t + δ χ. Other homology theories There are standard techniques to derive the cyclic homology HC * (A) and the periodic cyclic homology HP * (A) from the Hochschild homology of a C-algebra A [Lod]. In our cases A = H(G) s and A = S(G) s , we can get them as the homology of HH * (A) with respect to the usual exterior differential on forms.
Theorem F. (see (4.7), (4.8) and Corollary 4.3) Theorem C induces isomorphisms The periodic cyclic homology of a Fréchet algebra relates to its topological Ktheory via a Chern character. When can pass from S(G) s to its C * -completion via suitable Morita equivalent Fréchet subalgebras. In this way we compute the topological K-theory of any Bernstein block in the reduced C * -algebra of G: There is an isomorphism of vector spaces Here K * W,♮ denotes W -equivariant K-theory, twisted by a 2-cocycle ♮. Theorem G confirms [ABPS2,Conjecture 5], modulo torsion elements in the K-groups.

Relation with previous work and outlook
The Hochschild homology of H(G) has been determined earlier in [Nis]. The methods of Nistor are completely different from ours, he obtains a description of HH n (H(G)) in terms of several algebraic subgroups of G and of the continuous group cohomology of certain modules. This arises from a generalization of the standard techniques for discrete groups, a filtration of H(G) as bimodule, and spectral sequences. In [Nis,§6] a "parabolic induction map" HH n (H(G)) → HH n (H(M )) is constructed, for a Levi subgroup M of G. It would be interesting to relate this to our methods and results, maybe that could provide some information about supercuspidal representations.
A technique prominent in Nistor's work is localization of HH * (H(G)) at conjugacy classes in G. That can be regarded as a higher order version of taking the trace of a representation at a conjugacy class. Of particular interest is the localization of HH * (H(G)) at the set of compact elements of G, for that yields the periodic cyclic homology HP * (H(G)) [HiNi]. While localization at one conjugacy class in G appears to be intractable in our setup, localization at all compact elements is within reach. Since every compact element lies in the kernel of every unramified character, such localization removes all differential forms that are not locally constant on (subvarieties of) X nr (L). Moreover, in the description from Theorem C the locally constant differential forms constitute a set of representatives for HP * (H(G)), that follows from Lemma 4.4 (and with Theorem F it also works for S(G)). Hence the localization of HH * (H(G)) at the compact elements of G is given precisely by the subspace of locally constant differential forms.
At the same time HP * (H(G)) is naturally isomorphic to the equivariant homology of the Bruhat-Tits building of G, which yields yet another, more geometric, picture of HH * (H(G)) and HH * (S(G)). It would be nice if the Bernstein decomposition of H(G) and of S(G) could be expressed in such geometric terms, as suggested in [BHP].
Structure of the paper This paper is part of a larger project that includes [Sol8] and [KaSo]. Initially those two and the current text were conceived as one paper. When that grew too big, two parts were split off and transformed into independent papers. Although neither [Sol8] nor [KaSo] deals with p-adic groups, both prepare for this paper. Many results in Section 2 rely on the study of the Hochschild homology of slightly simpler algebras in [Sol8]. In Section 3 we need several nontrivial results about topological algebras and modules involving smooth functions. These are formulated and proven in larger generality in [KaSo].
Section 1 is preparatory, its main purpose is to describe precisely what kind of families of representations we will use. Already there we see that it is convenient to replace H(G) s by its subalgebra of functions that are biinvariant under a well-chosen compact open subgroup K.
We start our investigations of the Hecke algebra in earnest by transforming it into simpler algebras via Morita equivalences, in Paragraph 2.1. This relies largely on [Sol7], but we go a little further and establish Theorem A. In Paragraph 2.2 we set up a good array of algebraic families of G-representations, and we approach HH n (H(G) s ) via formal completions at central characters. That yields a rough description in terms of differential forms on varieties like X nr (M ), not yet indexed by W (L, s) as desired, but already sufficient for Theorem D.i. The local results thus obtained are glued together in Paragraph 2.3. When that is done, Theorems B, C and E for H(G) s follow quickly.
For the Schwartz algebra S(G) no such simplifying Morita equivalences are available, but Harish-Chandra's Plancherel isomorphism from Theorem 1.2 works better than for H(G). Our main technique to determine HH n (S(G) s ) is to derive it from HH n (H(G) s ) via a comparison of formal completions with respect to central characters. To carry out that strategy completely, we need to check that the relevant modules are Fréchet spaces, which is done in Paragraph 3.1. In Paragraph 3.2 we first show Theorem E.ii, so that we can work with C ∞ (X unr (M )) W (M,d) -modules. That plays a role in the proof of Theorem C.ii, from which Theorem D.ii follows readily. Then we establish Theorem B.ii and we compare HH n (S(G) d ) with HH n (H(G) s ) d .
Section 4 contains the derivation of the (periodic) cyclic homology of H(G) s and S(G) s . We also draw conclusions for the topological K-theory of G. In the final section we work out the examples G = SL 2 (F ) and G = GL n (F ).

Acknowledgements.
We thank David Kazhdan, Roman Bezrukavnikov and Alexander Braverman for interesting email discussions about group algebras of reductive p-adic groups. We are grateful to Roger Plymen for his feedback on an earlier version, which inspired us to add the last two sections.

Algebraic families of G-representations
Let G be a connected reductive group defined over a non-archimedean local field F , and consider the group of rational points G = G(F ). Let Rep(G) be the category of smooth G-representations and let Rep f (G) be the subcategory of finite length representations. Let R(G) be the Grothendieck group of Rep f (G). By imposing temperedness we obtain the category Rep t f (G) and the Grothendieck group R t (G). We fix a Haar measure on G we let H(G) be the algebra of locally constant compactly supported complex-valued functions on G, endowed with the convolution product. Recall that the Schwartz algebra S(G) [Wal,§III.6] satisfies Irr(S(G)) = Irr t (G), where the latter denotes the space of irreducible tempered G-representations. We fix a compact open subgroup K of G and we consider the algebras H(G, K) and S(G, K) of K-biinvariant functions in, respectively, H(G) and S(G). By definition where the inductive limit runs over the set of all compact subgroups K of G, partially ordered by reverse inclusion.
Let X nr (G) be the group of unramified characters of G and let X unr (G) be the subgroup of unitary unramified characters. The first is a complex algebraic torus and the second is a compact real torus of the same dimension.
Let P be a parabolic subgroup of G with a Levi factor M , and let I G P : Rep(M ) → Rep(G) be the normalized parabolic induction functor. Let σ ∈ Rep t f (M ) and suppose that the space I G P (V σ ) K , which has finite dimension by the admissibility of I G P (σ), is nonzero. Since I G P (V σ ) can be realized as a space of functions on a good maximal compact subgroup of G, we may identify the vector spaces I G P (V σ ) K and I G P (V σ ⊗ χ) K for χ ∈ X nr (M ). Every f ∈ S(G, K) gives a family of operators I G P (σ⊗χ)(f ) on I G P (V σ ) K , parametrized by χ ∈ X unr (M ). It turns out [Wal,Proposition VII.1.3] that I G P (σ ⊗ χ)(f ) depends smoothly on χ. When f ∈ H(G, K), this even works for all χ ∈ X nr (M ), and the outcome depends algebraically on χ. More precisely, this enables us to define algebra homomorphisms .
Recall the natural pairing .
This and its analogue for S(G, K) induce bilinear maps We say that a linear function f on C ⊗ Z R(G) is regular if ) is a regular function, for all (M, σ) as above. Similarly we call f ∈ (C ⊗ Z R t (G)) * smooth if X unr (M ) → C : χ → f (I G P (σ ⊗ χ)) is a smooth function, for all (M, σ) as above. We write ) * : f is smooth}. With these notations, (1.1) and (1.3) induces maps It is easy to see that the former is a homomorphism of Z(H(G, K))-modules and that the latter is is a homomorphism of Z(S(G, K))-modules.
The normalized parabolic induction functor I G P induces a Z-linear map I G M : R(M ) → R(G). It may be denoted this way, because given a Levi subgroup M of G it does not depend on the choice of the parabolic subgroup P with Levi factor M . We define where the sum runs over all proper Levi subgroups M of G. We say that a finite dimensional G-representation is elliptic if it admits a central character and does not belong to R I (G). By [BDK,Proposition 3.1] every Bernstein component of Irr(G) contains only a finite number of X nr (G)-orbits of irreducible elliptic representations. It follows from the Langlands classification that every such X nr (G)-orbit contains a tempered G-representation.
Definition 1.1. Let η ∈ Irr(M ) be elliptic and tempered. Then : χ ∈ X nr (M )} is an algebraic family of G-representations. Its dimension is dim C (X nr (M )), that is, the dimension of the maximal split torus in Z(M ). The subset We fix a minimal parabolic subgroup P 0 of G and a maximal split torus S 0 of P 0 . A parabolic (resp. Levi) subgroup of G is standard if it contains P 0 (resp. S 0 ). In the above definition it suffices to consider standard parabolic and standard Levi subgroups of G, because every pair (P, M ) is G-conjugate to a standard such pair.
Consider a Bernstein block Rep(G) s of Rep(G), determined by a tempered supercuspidal representation of a standard Levi subgroup L of G. Let R(G) s be the Grothendieck group of Rep f (G) s . Similarly we define R t (G) s as the Grothendieck group of the category Rep t f (G) s of tempered modules in Rep f (G) s . If we restrict to standard parabolic/Levi subgroups of G (as we will often do tacitly), Rep(G) s contains only finitely algebraic families of G-representations as in Definition 1.1. Moreover, by [BDK,Corollary 3.1] these families span Q ⊗ Z R(G) s .
We want to minimize the redundancy, by choosing a smaller collection of algebraic families of G-representations. One step in that direction is to determine which members of an algebraic family are equivalent in R(G). To that end we briefly recall Harish-Chandra's Plancherel isomorphism for G [Wal].
We denote such an equivalence class by d = [M, δ]. (When δ is supercuspidal, we have the equivalence class s = [M, δ], which determines a Bernstein component of Irr(G).) Let P be a parabolic subgroup of G with Levi factor M and let χ ∈ X unr (M ). To (M, δ, χ) we associate the tempered G-representation I G P (δ ⊗ χ) (which under these conditions does not really depend on the choice of P ). Then the connected component of Irr t (G) associated to (M, δ) consists of the irreducible summands (or equivalently subquotients) of the representations I G P (δ ⊗ χ) with χ ∈ X unr (M ). The Plancherel isomorphism describes the image of F t M,δ , as the invariants for an action of a certain finite group. The group X nr (M, δ) = {χ ∈ X nr (M ) : δ ⊗ χ ∼ = δ} is finite and contained in X unr (M ), because it consists of characters that are trivial on Z(M ). Consider the subset It is not canonical, because it depends on the choice of δ in Irr(M ) d t . For each χ ′ ∈ X nr (M, δ) we fix a unitary M -isomorphism δ ∼ = δ ⊗ χ ′ , and we induce it to a family of G-isomorphisms (1.7) I(χ ′ , P, δ, χ) : By [Sol6,Lemma 3.3] the action of an element w ∈ W d on Irr(M ) d can be lifted (non-canonically) along (1.6), to an automorphism of the complex algebraic variety X nr (M ) such that By [Wal,Lemme V.3.1] there exists a unitary G-isomorphism , depending smoothly and rationally on χ ∈ X unr (M ). Let W (M, d) be the group of transformations of X nr (M ) generated by X nr (M, δ) and the actions of elements of W d . By [Sol6,Lemma 3.3] it fits in a short exact sequence The intertwining operators (1.7) and (1.8) give rise to analogous families of G-isomorphisms for any element of W (M, d). These are far from unique, but for any fixed χ ∈ X unr (M ) they are unique up to scalars. The group W (M, d) acts on C ∞ (X unr (M )) ⊗ End C (I G P (V δ ) K ) by (w · (f ⊗ A))(χ) = f (w −1 χ) ⊗ I(w, P, δ, w −1 χ)AI(w, P, δ, w −1 χ) −1 .
Let ∆ G,K be a set of representatives for the (M, δ) with I G P (V δ ) K = 0, modulo the equivalence relation (1.5). We assume that every M occurring here is a Levi factor of a standard parabolic subgroup P .
There is an isomorphism of Fréchet algebras .
Consider an algebraic family F(M ′ , η ′ ) contained in Rep(G) s . We may assume that if an element w ∈ W (M, s) fixes the point or the intertwining operator associated to w has a singularity at the cuspidal support of the point, then then w has that property for all members of F(M ′ , η ′ ).
Proof. The action of w ∈ W (M, d) on the collection of direct summands of the I G P (δ ⊗ χ) comes from an algebraic action on X nr (M ) and conjugation by some operator. Hence, for a generic π = I G P 0 M ′ (η ′ ⊗ χ ′ ) ∈ F(M ′ , η ′ ), the representation wπ lies in F(M ′ , η ′ ) if and only if w ∈ W (M ′ , M, η ′ ). In combination with (1.10), that implies the second claim for generic tempered members of F(M ′ , η ′ ).
In fact Harish-Chandra's commuting algebra theorem (1.10) also holds for generic χ 1 , χ 2 ∈ X nr (M ), one only needs to avoid the poles of the intertwining operators I(w, P, δ, χ). This follows for instance from [ABPS1, Theorem 1.6]. Then the above argument can be applied to all generic members, and yields the first claim.
For any f ∈ H(G) and w ∈ W (M ′ , M, η ′ ), and tr f, I G P 0 M ′ (η ′ ⊗ wχ ′ ) are algebraic functions of χ ′ ∈ X nr (M ′ ). These two functions agree for generic tempered χ ′ ∈ X unr (M ′ ), so they agree on the whole of X nr (M ′ ). Now we can finally describe how to choose a minimal set of algebraic families of G-representations in Rep(G) s .
We start with the family F(L, σ) and proceed recursively. Suppose that for every dimension D > d we have chosen a set of D-dimensional algebraic families F(M i , ω i ), where i runs through some index set I D , with the following property: for generic χ i ∈ X nr (M i ) the set Here we regard all χ j in one W (M j , M, ω j )orbit as the same, because by Lemma 1.3 they yield the same element , were we still regard χ j as an element of X nr (M j )/W (M j , M, ω j ). Then we add F(M ′ i , ω ′ i ) to our collection of algebraic families. Consider the remaining d-dimensional algebraic families. For F(M ′ j , ω ′ j ) we look at the same condition as for If that condition is fulfilled, we add F(M ′ j , ω ′ j ) to our set of algebraic families. We continue this process until none of the remaining d-dimensional algebraic families is (over generic points of that family) Q-linearly independent from the algebraic families that we chose already. At that point our set of d-dimensional algebraic families is complete, and we move on to families of dimension d − 1.
In the end, this algorithm yields a collection We note that these conditions do not imply that (1.11) is a basis of Q ⊗ Z R(G) s . Some linear dependence is still possible for representations with a specific cuspidal support (L, σ ⊗ χ), namely when the algebraic R-group of σ ⊗ χ acts on I G P 0 L (σ ⊗ χ) via a projective, non-linear representation. That does not happen often though.
The formula (1.1) for the partial Fourier transform F M,δ also applies with any elliptic M -representation instead of δ (which is square-integrable modulo centre). .
These induce maps on Hochschild homology . We added a subscript sm to emphasize that we consider smooth differential forms on a real manifold. We will describe HH n (H(G, K)) and HH n (S(G, K)) in terms of the maps (1.13).

The Hecke algebra of
At this point we need the following continuity property of the functors HH n from [Lod,E.1.13]. Namely, let A = lim − →i A i be an inductive limit of algebras. Then In particular We fix a Bernstein block Rep(G) s in Rep(G), where s = [L, σ] G . According to [BeDe] there exist arbitrarily small compact open subgroups K of G such that The centre of H(G, K) s is isomorphic to the centre of the category Rep(G) s . The latter can be made more explicit with the notations from Section 1. Namely, by . By [BeDe] and (1.9) there are isomorphisms It is also known from [BeDe] that 2.1. Structure of the module category. We aim to describe H(G, K) s and its modules locally on X nr (L). We write X + nr (L) = Hom(L, R >0 ) and we fix u ∈ X unr (L). We let U u ⊂ X nr (L) be a connected neighborhood of u in X nr (L) (for the analytic topology) satisfying [Sol7, Condition 6.3]: • U u is stable under W (L, s) u and under X + nr (L), • a technical condition to ensure that u is the "most singular" point of U u . The tangent space at 1 of the complex torus X nr (L) is This means that we regard t also as the tangent space of X nr (L) at u. Let log u be the branch of exp −1 u with log u (u) = 0. By [Sol7,Condition 6.3] exp u restricts to a diffeomorphism, log u (U u ) → U u . From [Sol7,§7] we get a root system Φ u in t, whose Weyl group is a subgroup of W (L, s) u , a basis ∆ u of Φ u , a parameter function k u : ∆ u → R ≥0 and a 2-cocyle ♮ u of W (L, s)/W (Φ u ). In [Sol7] some of these objects have a subscript σ ⊗ u instead of u, but since W (L, s) u is naturally isomorphic with (W s ) σ⊗u , we may omit σ⊗. To these data one can associate a twisted graded Hecke algebra H(t, W (L, s) u , k u , ♮ u ). For any Levi subgroup M of G containing L, there is a parabolic subalgebra H(t, W (M, L, s) u , k u , ♮ u ), constructed in the same way.
Theorem 2.1. [Sol7,Corollary 8.1 and its proof, Proposition 9.5.a] There is an equivalence between the following categories: • finite length G-representations, all whose irreducible subquotients have cus- . This equivalence of categories commutes with parabolic induction and preserves temperedness.
Hence we may replace right In particular the centres of the algebras H(G, K) s and End G (Π s ) op are canonically isomorphic.
(v) Check that the above localizations do not change the categories of finite dimensional modules with O(X nr L))-weights (respectively O(t)-weights) in the set on which one localizes. (vi) Show that the isomorphism To make full use of Theorem 2.1, we also need a variation on step (iii) above. We will construct an algebra H G W (L,s)u which is Morita equivalent with H G u and closer to End G (Π s ) op than H G u . We start with wu∈W (L,s)u H G wu . In this algebra the unit element of H G wu is denoted e wu . For every element wu ∈ W (L, s)u we fix a w which has minimal length in wW (L, s) u (see [Sol7,end of §3] for the definition of the length function). From [Sol7,Lemma 8.3] we get an isomorphism and hence also an isomorphism between their opposite algebras: The advantage of this particular isomorphism comes from [Sol7,Lemma 8.3.b]: intertwines Theorem 2.1 for wu with Theorem 2.1 for u. Ad(T w ) really is conjugation by an element T w in a larger algebra, and satisfies: The multiplication of H G W (L,s)u is given by where h i ∈ H G w i u and all the w i are as chosen above. The elements e wu T w T −1 w ew u of H G W (L,s)u multiply like matrices with just one nonzero entry. It follows readily that We note that this algebra is of the form H(V, G, k, ♮), as in [Sol8,§2.3]. The centre of this algebra is This algebra contains H G Uu as a Morita equivalent subalgebra, analogous to (2.9).
In this diagram the vertical arrows are inclusions of Morita equivalent subalgebras and each of the two horizontal arrows induces an equivalence between the categories of finite length modules all whose weights for (2.10) belong to ⊔ wu∈W (L,s)u log wu (wU u ).
Proof. (a) The elements T w involved in H G W (L,s)u stem from [Sol7,§5]. It was shown in the proof of [Sol7,Lemma 8.3] that We define the parabolic subalgebras of H G u to be the analogous algebras The translation from right to left modules via (2.6) commutes with parabolic induction. Namely, for a right fulfill the conditions from [Sol8,. Indeed, that follows from (2.11), Theorem 2.1 and the properties of elliptic G-representations discussed at the start of Section 1.
Let F(M, η) be an algebraic family in Rep(G) s , with η irreducible and elliptic. We may and will assume that M is standard and we let P be the unique standard parabolic subgroup of G with Levi factor M . All the representations I G P (η ⊗ χ) with χ ∈ X nr (M ) admit a central character, so Z(H(G, K) s ) acts by a character on is a homomorphism of Z(H(G, K) s )-algebras and that HH n (F M,δ ) is a homomorphism of Z(H(G, K) s )-modules. Assume that some members of F(M, η) have cuspidal support in (L, σ⊗W (L, s)U u ). Then the image of F(M, η) under Theorem 2.1 is an algebraic family F(M,η) of H G umodules, whereη ∈ Irr(H M u ) is elliptic and tempered. More precisely Theorem 2.1 only applies to an open part of F(M, η), and the image of that is the part of F(M,η) with O(t) W (L,s)u -weights in log u (U u ). By the Langlands classification (for graded Hecke algebras in [Eve], generalized to our setting with the method from [Sol3, §2.2]) every such family of H G u -modules arises from an elliptic representation of (2.12) u acts on C λ ⊗η by evaluation at 0. We may replaceη by C λ ⊗η without changing F(M,η). Thenη has In general the full structure of the algebra End G (Π s ) (or its opposite) seems to be rather complicated. Fortunately, it can be approximated with simpler algebras. The normalized parabolic induction functor I G P 0 L gives an embedding O(X nr (L)) → End G (Π s ).
From (2.7) and (2.4) we know that Let C(X nr (L)) be the quotient field of O(X nr (L)), i.e. the field of rational functions on the complex affine variety X nr (L). It is easy to see that the multiplication map (2.14) is a field isomorphism. According to [Sol7,Corollary 5.8], (2.14) extends to an algebra isomorphism With (2.6) we also obtain the opposite version Unfortunately the isomorphisms (2.15) and (2.16) are not canonical, they depend on the choice of a suitable σ ∈ Irr(L) s and on the normalization of certain intertwining operators. In the remainder of this paragraph we fix those choices. We emphasize that (except in very special cases) Remarkably, it turns out that nevertheless there is a canonical bijection We describe step-by-step how it is obtained.
Construction 2.4. (i) With the equivalences of categories (2.7) we go from R(G) s to R(End G (Π s ) op ).
(ii) By decomposing finite length End G (Π s ) op -modules along their O(X nr (L)) W (L,s)weights, it suffices to consider G-representations π as in Theorem 2.1. (iii) Via Theorem 2.1 and (2.6) we obtain the H G u -module 1 Uu Hom G (Π s , π). (iv) There is a canonical Z-linear bijection The construction is given in [Sol4,Theorem 2.4], while the bijectivity follows from [Sol5,Theorem 1.9 Theorem 2.5. The map ζ ∨ from (2.18) has the following properties.
In the setting of (c), suppose that π is tempered. Then (e) ζ ∨ commutes with parabolic induction and unramified twists, in the sense that Proof. (a) Since each step in Construction 2.4 is Z-linear and bijective, so is ζ ∨ . The bijectivity of (vi) comes from the Morita equivalence between (b) By Theorem 2.1, steps (i)-(iii) respect temperedness. It is known from [Sol4,Theorem 2.4] that ζ ∨ u in (iv) respects temperedness, and for (v) that is obvious because the O(X nr (L))-weights are not changed in that step. Step (i) translates "cuspidal support (L, σ ⊗ W (L, s)χ)" into "all O(X nr (L))weights in W (L, s)χ". After that the only step that changes the O(X nr (L))-weights is (iv), and by [Sol4,Theorem 2.4.(3)] it only adjusts O(X nr (L))-weights by elements of X + nr (L). (d) The only tricky point is to see that step (iv) of Construction 2.4 sends where C 0 means that O(t) acts via evaluation at 0 ∈ t. That is the content of [Sol4,Theorem 2.4.(4)].
(e) First we check that ζ ∨ respects parabolic induction, at least when the input is a tempered (virtual) representation tensored with an unramified character. Steps (i) and (ii) commute with parabolic induction by [Sol6,Condition 4.1 and Lemma 6.1]. For step (iii) that follows from [Sol7, Lemma 6.6 and Proposition 7.3]. In step (iv), property (e) is an important part of the construction of ζ ∨ u in [Sol4,Theorem 2.4], that is where we need the shape of the input. That step (v) respects parabolic induction follows from the properties of the isomorphism between the analytically localized versions of the involved algebras, as in [Sol7,Proposition 7.3]. For step (vi) we obtain the desired behaviour from [Sol7,Lemma 6.6].
The compatibility with unramified twists requires an explicit computation. By the above it suffices to check that In steps (i) The properties listed in Theorem 2.5 imply for instance that ζ ∨ maps algebraic families in R(G) s (or equivalently in R(H(G, K) s ) to algebraic families in

Local descriptions of Hochschild homology.
In this section we will determine The Hochschild homology of H(G) with a method based on the families of G-representations from Section 1. With the procedure from page 13 we pick a finite number (say n s ) of algebraic families of G-representations F(M i , η i ), such that the η i are irreducible and the members of these families span Q ⊗ Z R(G) s in a minimal way. We may assume that each M i is the standard Levi factor of a standard parabolic subgroup P i of G. Writing we obtain a homomorphism of Z(H(G, K) s )-modules where Z(H(G, K) s ) acts on the right hand side via the central characters of the involved representations π(M i , η i , χ i ). We aim to establish an analogue of [Sol8, Theorems 1.13 and 2.8] for HH n (F s ). The families that have no cuspidal supports in σ ⊗ W (L, s)U u can be ignored for the current purposes (we may call them U u -irrelevant). For the remaining families, as explained above we may assume without loss of generality that (2.13) holds. Select . Thus we are in the setting of [Sol8,.
For g ∈ W (L, s) u and v ∈ t g , in [Sol8,(1.19)] an element which also occurs in [Sol8,(2.10)]. Here the U u -irrelevant indices i are left out of the sum, but we may still include by setting λ g,i = 0 for those i. From [Sol8, Lemma 1.10] we know that each φ g,i : t g → t M i is given by an element of W (L, s) u . Hence φ g,i induces regular maps Since the right hand side is well-defined for any u ′ ∈ uX nr (L) g,• , we may extend the definition of ν 1 g,u ′ to such u ′ . The map (2.24) induces a homomorphism of O(X nr (L)) W (L,s) -algebras (2.25) Here O(X nr (L)) W (L,s) acts on the domain via the central characters of the members of F(M i , η i ), whereas the O(X nr (L)) W (L,s) -module structure on the range is given at χ ∈ uX nr (L) g,• ) by W (L, s)χt + η i , where the central character of η i is represented by In other words, the natural module structure on the right hand side of (2.25) is adjusted by the positive part of the central character of η i . When we consider the map on Hochschild homology induced by (2.25), the range does not depend on i, but the O(X nr (L)) W (L,s) -module structure still does. Like in [Sol8,(1.33) and (2.14)], we can combine the maps on Hochschild homology induced by the homomorphisms (2.25) a C-linear map The maps HH n (φ * u ), for various u ∈ X unr (L), are our main tools to describe HH n (H(G, K) s ).
Recall that the formal completion of a commutative algebra A with respect to a finite set of characters X is denoted A X . With that notation, for u ′ ∈ U u there are algebra isomorphisms Proposition 2.6. For u ′ ∈ U u the following modules over the formal completion (2.27) are isomorphic: The isomorphism between (a) and (d) is induced by HH n (F s ).
The character ♮ g s : Z W (L,s)u (g) → C × figuring in parts (d) and (e) is defined as (2.28) Let I u ′ ⊂ O(X nr (L)) be the maximal ideal of functions vanishing at u ′ . As O(X nr (L))/I m u ′ ∼ = C an (W (L, s)U u )/I m u ′ C an (W (L, s)U u ) for any m ∈ N, the algebras O(X nr (L)) W (L,s) and C an (W (L, s)U u ) W (L,s) have the same formal completion at u ′ . It follows that in the process described between (2.6) and (2.8) the analytic localization steps do not change the formal completions of the involved algebras (at u ′ and log u (u ′ ) respectively). Then Proposition 2.3 yields the isomorphism between (a),(b) and (c).
The isomorphism between (c) and (d) is a consequence of [Sol8, Theorem 2.8]. As By Theorem 2.1 this restricts to an isomorphism between (d) and (e).
The isomorphism between (c) and (d) (a) and (e) can be constructed from that between HH n (End G (Π s ) op ) and (e) by composing with (2.28), which is induced by a Morita equivalence. Thus the isomorphism between (a) and (e) is given by evaluating H(G, K) s at the families F(M i , η i ). In other words, it is given by HH n (F s ), while ignoring the U u -irrelevant families.
We will lift Proposition 2.6 to a statement about HH n (H(G, K) s ) on the whole of X nr (L).
Lemma 2.7. The map HH n (F s ) is an injection from HH n (H(G, K) s ) to the set of ω ∈ ns i=1 Ω n (X nr (M i )) such that ∀u ∈ X unr (L). Since H(G, K) s has finite rank as a module over the Noetherian algebra Z(H(G, K) s ), so does HH n (H(G, K) s ). Consider a nonzero x ∈ HH n (H(G, K) s ). In view of (2.29), the Z(H(G, K) s )-submodule generated by x has at least one nonzero formal completion, say at W (L, s)u ′ . Then x is nonzero in that completion, and by Proposition 2.6 the image of HH n (F s )x (in a formal completion) is nonzero. Hence HH n (F s ) is injective. Proposition 2.6 shows that the specialization of HH n (F s )x at any central character W (L, s)u ′ ⊂ W (L, s)U u has the property involving HH n (φ * u ). Hence HH n (F s )x satisfies the stated condition, at least on U u . For each g, the required property extends from U u ∩ Ω n (uX nr (L) g,• ) to Ω n (uX nr (L) g,• ) because U u is Zariski-dense and the g-component of HH n (F s )x is an algebraic differential form. Thus the image of HH n (F s ) is contained in the set specified in the statement.

The injection is
To attain surjectivity in Lemma 2.7, we have to take the relations between specialization at u and at wu into account. This is where the algebras from Proposition 2.3 show their usefulness. Let HH n (φ * u ) be the map HH n (φ * ) from [Sol8,(2.17)], for H G W (L,s)u . According to [Sol8,Proposition 2.16] there is a C-linear bijection .
Here HH n (F 1 ) is a version of HH n (F s ) for H G W (L,s)u , see [Sol8,around (2.23)]. .
Like in Lemma 2.7, it follows that the image of HH n (F s ) is contained in (2.31) for all u ∈ X unr (L). The advantage is that now the behaviour at the entire W (L, s)-orbit of u ′ is captured by (2.31). Consider the intersection of the spaces (2.31), over all u ∈ X unr (L). Divide that by the image of HH n (F s ). Proposition 2.6 and (2.30) tell us that the quotient is a O(X nr (L)) W (L,s) -module all whose formal completions are zero. As each O(X nr (M i )) is a finitely generated O(X nr (L)) W (L,s) -module, so are ns i=1 Ω n (X nr (M i )) and its submodules. Hence we may apply (2.29), which says that the quotient under consideration is the zero module. In other words, the image of HH n (F s ) is precisely the intersection of the spaces (2.31). (b) Here restriction means that we only consider the Ω n (X nr (M i )) with X nr (M i ) ∩ W (L, s)U u = ∅.
Suppose that x ∈ HH n (F s )HH n (H(G, K) s ) is nonzero on W (L, s)U u . Pick a u ′ ∈ U u at which x is nonzero. Then Proposition 2.6 shows that HH n (φ * u )x cannot be zero. This proves the injectivity.
The map HH n (φ * u ) is O(X nr (L)) W (L,s) -linear if we let that algebra act on ns i=1 Ω n (X nr (M i )) via the maps (2.32) Fix a character λ ∈ W (L, s)U u of O(X nr (L)) W (L,s) . There are only finitely many H(G, K) s -representations π(Q i , η i , λ i ) with χ η i λ i ∈ W (L, s)λ, so together these support only finitely many central characters. By (2.22) all those central characters lie in W (L, s)U u . Then Proposition 2.6 and (2.30) imply that HH n (F s )HH n (H(G, K) s ) and (2.31) have isomorphic formal completions at W (L, s)λ, with the respect to the O(X nr (L)) W (L,s) -module structure coming from (2.32).
Hence the cokernel of HH n (φ * u ) is a finitely generated O(X nr (L)) W (L,s) -module all whose formal completions at points of W (L, s)U u are zero. Thus Furthermore cokerHH n (φ * u ) is of form O(X nr (L)/W (L, s)) r /N for some submodule N of O(X nr (L)/W (L, s)) r . Then (2.33) entails O(X nr (L)/W (L, s)) r = I m λ O(X nr (L)/W (L, s)) r + N for all λ ∈ W (L, s)U u and all m ∈ Z >0 . With the Zariski-density of W (L, s)U u , it follows that N = O(X nr (L)/W (L, s)) r . Hence cokerHH n (φ * u ) = 0 and HH n (φ * u ) is surjective.
Recall that HH 0 (H(G)) and HH 0 (H(G, K) s ) were already computed in [BDK]. We will now recover those results via families of representations.
By Morita equivalence, we may replace R(H(G, K) s ) with R(G) s . Conversely, for every λ ∈ C ⊗ Z R(H(G, K) s ) * reg the canonical pairing with F M i ,η i produces a regular function on X nr (M i ), so λ comes from an element of ns i=1 O(X nr (M i )). Let ∆ s G be a set of representatives for the inertial equivalence classes of squareintegrable modulo centre representations δ of standard Levi subgroups M of G, such that I G P (δ) ∈ Rep(G) s . From Theorem 1.2 we see that the category of tempered representations in Rep(G) s decomposes as where Rep t (G) d is the full subcategory generated by the subquotients of I G P (δ ⊗ χ) with χ ∈ X unr (M ). With Theorem 2.1 and the same arguments as in the proof of [Sol8, Theorem 2.2], (2.34) induces a decomposition By Proposition 2.3 R t (H G W (L,s)u ) decomposes in the same way. It is known from [Sol7,Proposition 9.5] that the equivalence of categories in Theorem 2.1 sends square-integrable modulo centre representations to tempered essentially discrete series representations. With that and the same process that madeη out of η, described around (2.12), we can associate to d = [M, δ] ∈ ∆ s G a discrete series representationδ of H M . Thus (2.35) is a decomposition of the kind considered in [Sol8, Theorem 2.2 and (2.25)]. We define (b) Select χ δ ∈ X unr (L), t + δ ∈ X + nr (L) such that χ δ t + δ represents the Z(H(G, K) scharacter of δ. The map is O(X nr (L)) W (L,s) -linear if we let O(X nr (L)) W (L,s) act on the target such that: • if g(uX nr (L) w,• ) ⊂ χ δ X nr (M ), then it acts at guχ with χ ∈ X nr (L) w,• via the character W (L, s)uχt + δ , • in the same situation O(X nr (L)) W (L,s) acts at huχ, where h ∈ W (L, s) and χ ∈ X nr (L) w,• , also via the character W (L, s)uχt + δ , annihilates Ω n (g(uX nr (L) w,• )).
Proof. (a) This follows from [Sol8, Lemma 2.12, (2.25), Corollary 2.13] and Theorem 2.8. (b) The condition in the third bullet means that for i ≺ d no map φ w,i : uX nr (L) w,• → χ δ X nr (M ) can exist. In that case λ w,i = 0 and the image of HH n (H(G, K) s ) d in Ω n (g(uX nr (L) w,• ) is 0. From that and (2.25) we see that, for each i separately, there exists such a O(X nr (L)) W (L,s) -module structure as indicated, only with t + η i ∈ X + nr (L) instead of t + δ .
By [Sol8,Corollary 2.13], HH n (φ * u ) • HH n (F d ) is O(X nr (L)) W (L,s) -linear if we let it act according to the central characters of the virtual representations ν d g,w,v from [Sol8,(2.26)]. That means that the natural module structure is adjusted by a representative cc(δ) ∈ t R of the central character of δ (as representation of H G u ). So in that setting log(t + η i ) and log(cc(δ)) represent the same central character, for all i ≺ d. We have translate these to H(G, K) s -representations with Theorem 2.1 and Proposition 2.3. Then cc(δ) becomes t + δ . Hence W (L, s)t + η i = W (L, s)t + δ for all i ≺ d.
Thus the O(X nr (L)) W (L,s) -module structures for the i ≺ d agree, and combine to make HH n (φ * u ) • HH n (F d ) a module homomorphism with the indicated character shift.

Hochschild homology for one entire Bernstein component.
We would like to combine the local conditions involving HH n (φ * u ) to a smaller set of conditions that describe HH n (H(G, K) s ) globally on X nr (L). This is difficult because the algebras H G u and H G W (L,s)u do not vary continuously with u ∈ X unr (L). To compensate for that, we relate the local conditions coming from u, u ′ ∈ X unr (L) that are close. When W (L, s) u ′ ⊂ W (L, s) u , we define (b) Part (a) also holds for u ∈ u ′ X unr (L) W (L,s) u ′ ,• .
Proof. (a) Notice that W (L, s) u ′ ⊂ W (L, s) u by the conditions on U u . We may choose U u ′ so small that it is contained in U u . From the proof of [Sol8, Proposition 2.16.a] we know that .
By construction HH n (φ * u ′ ) −1 u ′ of the left hand side equals HH n (φ * u ′ ) −1 of the right hand side. By Theorem 2.8.b, this describes precisely the restriction of HH n (F s )HH n (H(G, K) s ) to W (L, s)U u ′ . Similarly describes precisely the restriction of HH n (F s )HH n (H(G, K) s ) to W (L, s)U u . Restricting that further W (L, s)U u ′ means that we remove the summands for w ∈ W (L, s) u that do not fix u ′ , because for those U u ′ ∩X nr (L) w = ∅ and w(U u ′ )∩U u ′ = ∅, by the properties of U u ′ . That leaves us with (b) Pick a path p from u ′ to u in u ′ X unr (L) W (L,s) u ′ ,• . We even assume that W (L, s) y = W (L, s) u ′ for all y on p, because that condition holds on an open dense subset of X unr (L). By the compactness of X unr (L), we can choose a finite subset Y of p, such that the U y with y ∈ Y cover p. Choose a finite sequence y 1 , y 2 , . . . , y m in Y , such that u ′ ∈ U y 1 , u ∈ U ym and For 1 ≤ i < m we pick z i ∈ U y i ∩ U y i+1 ∩ p. We follow the new sequence u ′ , y 1 , z 1 , y 2 , z 2 , . . . , z m−1 , y m , u.
At each step part (a) guarantees that the relevant preimages under HH n (φ * ? ) u ′ do not change.
For c ∈ π 0 (X nr (L) w ), we denote the corresponding connected component of wfixed points by X nr (L) w c . Then W (L, s) acts naturally on the set of such components, and on the set of pairs (w, c). We denote the stabilizer of (w, c) by W (L, s) w,c , this is a subgroup of Z W (L,s) (w). We register these connected components with the list of pairs (w, c), where w ∈ W (L, s) and c ∈ π 0 (X nr (L) w ). We write We ready to reorganize the conditions that describe H n (F s )HH n (H(G, K) s ) in Theorem 2.8. This is done with decreasing induction on the dimension of the connected components X nr (L) w c , or equivalently on the pairs (w, c). Construction 2.12. (i) We start with w = 1 and X nr (L) w c = X nr (L). Pick Ω n (X nr (M i )) → Ω n (u 1 X nr (L)) = Ω n (X nr (L)), and it sends HH n (F s )HH n (H(G, K) s ) to Ω n (X nr (L)) W (L,s) . By Lemma 2.11, this completely describes the restriction of HH n (F s )HH n (H(G, K) s ) to the subset of X nr (L) not fixed by any nontrivial element of W (L, s). Remove (1, c) from the list of pairs. (ii) Assume that for some connected components X nr (L) w c we have already chosen a map (2.36) HH n (φ * w,c ) : of the form ns i=1,Uu-rel λ w,i HH n (χ −1 η i φ * w,i ) coming from HH n (φ * u ) for some u = u w,c with u w,c ∈ X nr (L) w c but not in any connected component of smaller dimension. Assume that the set of pairs (w, c) for which this has been done is closed under passing to larger pairs. Assume that all those pairs have been removed from the list. Finally and most importantly, we assume that for all those pairs (w, c) the restriction of HH n (F s )HH n (H(G, K) s ) to X nr (L) w c without the connected components of smaller dimension equals (2.37) (iii) From the list of remaining pairs, pick a (g, c) with X nr (L) g c of maximal dimension. Select u = u g,c in X unr (L) g c but not in any connected component of a part of HH n (φ * u ). If there are other h ∈ W (L, s) with X nr (L) h c = X nr (L) g c , then we take u h,c = u g,c and we define HH n (φ * h,c ) in the same way. We need to check that This follows from Lemma 2.11, which says that all the parts HH n (φ * u ) u ′ with u ′ ∈ U u and u ′ / ∈ X nr (L) g c are accounted for by the (w ′ , c ′ ) > (g, c). Lemma 2.11 also tells us that (2.39) and (2.40) describe exactly the restriction of HH n (F s )HH n (H(G, K) s ) to X nr (L) g c without the components of smaller dimension.
(iv) For components (g ′ , c ′ ) in the W (L, s)-orbit of (g, c) or any of the (h, c), we define the maps HH n (φ * g ′ ,c ′ ) by imposing W (L, s)-equivariance (where the group acting involves the characters ♮ g s ). This construction ensures that .
(v) Remove (g, c) and the pairs (h, c) ∼ (g, c) from the list of pairs. Stop if there are no pairs left, otherwise return to step (iii).
With (2.38) we associate to (w, χ) the virtual H(G, K) s -representation In other words, the specialization of HH n (φ * w,c ) at χ ∈ X nr (L) w c corresponds to the map on Hochschild homology induced by ν 1 w,χ . This means that (2.43) HH n (φ * w,c ) • HH n (F s ) : HH n (H(G, K) s ) → Ω n (X nr (L) w c ) is induced by the algebraic family of virtual representations {ν 1 w,χ : χ ∈ X nr (L) w c }. From [Sol8, Lemma 2.5.a] (translated to the current setting with Theorem 2.1) and step (iv) above we see that w,χ as virtual G-representation via the equivalence of categories (2.7), we deduce from (2.38) and (2.23) that (2.45) ζ ∨ (ν 1 w,χ ) = ν w,χ . The above procedure gives rise to a description of HH n (H(G, K) s ) that is more concrete than Theorem 2.8.
Theorem 2.13. For each w ∈ W (L, s) and each c ∈ π 0 (X nr (L) w ), let HH n (φ * w,c ) be as above. We define .
(c) For d ∈ ∆ s G , the restriction of HH n (φ * s ) • HH n (F s ) to the direct summand HH n (H(G, K) s ) d of HH n (H(G, K) s ) becomes O(X nr (L)) W (L,s) -linear if we endow the target with the same module structure as in Lemma 2.10.
Proof. (a) Recall that the specialization of HH n (φ * w,c ) at χ ∈ X nr (L) w c came from a virtual representation of H G χ|χ| −1 , translated to a virtual representation ν 1,w,χ of H(G, K) s via Theorem 2.1 and Proposition 2.3.
Consider one u ∈ X unr (L). From (2.37) and Theorem 2.8 we see that the ν 1,w,χ with central characters in W (L, s)U u span the same set of virtual representations with central characters in W (L, s)U u as all the virtual representations ν g,u ′ defined via H G u . The latter collection spans the entire part of C ⊗ Z R(H(G, K) s ) with central characters in W (L, s)U u , so the ν w,χ span that as well. It follows that the specialization of HH n (φ * s )x at χ ∈ W (L, s)U u is zero if and only if the specialization of HH n (φ * u )x at χ is zero. From [Sol8, Lemmas 1.12 and 2.7] we know that HH n (φ * u ) and HH n (φ * u ) are injective, for twisted graded Hecke algebras and for H(V, G, k, ♮) as in [Sol8,Paragraph 2.3], and hence also for the algebras H G u and H G W (L,s) . By the above considerations with virtual representations, HH n (φ * s ) contains at least as much information as HH n (φ * u ). Hence HH n (φ * s )x is nonzero as soon as x ∈ ns i=1 Ω n (X nr (M i )) does not vanish on U u . That holds for for every u ∈ X unr (L), so HH n (φ * s )x is injective. (b) We already know from Lemma 2.7 and part (a) that HH n (φ * s ) • HH n (F s ) is injective. From Theorem 2.8, (2.37) and (2.41) we know that the assertion holds locally. Hence .
Both sides are finitely generated O(X nr (L)) W (L,s) -modules (for natural module structure, not the module structure determined by the characters of the underlying virtual representations). We consider the quotient module M . Since the two sides of (2.46) are isomorphic locally, all formal completions of M with respect to characters of O(X nr (L)) W (L,s) are zero. By (2.29) M = 0, so the inclusion (2.46) is an equality.
(c) This follows from Lemma 2.10, since every component of HH n (φ * s ) occurs as component of a HH n (φ * u ). Recall that H(G, K) s is Morita equivalent with End G (Π s ) op and that in (2.16) we fixed an isomorphism of O(X nr (L)) W (L,s) -algebras Recall the bijection ζ ∨ from (2.18) and Theorem 2.5.
Theorem 2.14. There exists a unique C-linear bijection . It is not canonical, but from [Sol8,(1.15) and (1.17)] we know that the non-canonicity is limited to one scalar factor for each direct summand indexed by a conjugacy class in W (L, s). We can fix these scalar factors by requiring that (2.47) on the summand indexed by w is induced by the algebraic family of virtual representations (2.48) {ν w,χ : χ ∈ X nr (L) w }.
Indeed, the bijection (2.47) is recovered in that way in [Sol8,Theorem 1.13.a]. The only issue is that [Sol8, §1.2] applies not to tori like X nr (L), but to complex vector spaces. Fortunately [Sol8,Theorem 1.13] can easily be extended to our setting by localization of O(X nr (L)) W (L,s) to sets of the form W (L, s)U u /W (L, s). Thus we make (2.47) canonical. By Lemma 2.7 and Theorem 2.13.b, is a C-linear bijection. We define HH n (ζ ∨ ) as the composition of (2.47) with HH n (φ * s ) • HH n (F s ) −1 . From (2.43), (2.45) and (2.48) we see that whenever F is one of the families {ν 1 w,χ : χ ∈ X nr (L) w c } with c ∈ π 0 (X nr (L) w ). Since every such F is a linear combination of algebraic families of H(G, K) s )representations, (2.49) is implied the condition in the theorem. Hence HH n (ζ ∨ ) is unique.
It remains to check that for an arbitrary algebraic family in Rep(G) s . By [Sol8,Lemma 1.9] the virtual representations ν w,χ with w ∈ [W (L, s)] and v ∈ X nr (L) w /Z W (L,s) (w) such that ♮ w s (W (L, s) v ∩ Z W (L,s) (w)) = 1 form a basis of (2.51) Hence there exist coefficients c(w, v, χ) ∈ C such that Then (2.45) and the bijectivity of Theorem 2.5 imply Recall from (2.42) that ν 1 w,v is a linear combination of the F(M i , η i , v i ). With Theorem 2.5 that can be transferred to (2.51). Hence there exist The same argument for all χ ∈ X nr (M ) simultaneously yields (2.50).
Theorem 2.14 is a homological counterpart to [Sol7,Theorem 9.9], which matches the irreducible representations of H(G, K) s with those of O(X nr (L))⋊C[W (L, s), ♮ s ].

The Schwartz algebra of G
The Harish-Chandra-Schwartz algebra of a reductive p-adic group is an inductive limit of Fréchet spaces, but itself not a Fréchet algebra. To do homological algebra with such topological algebras, we have to agree on a suitable topological tensor product. The best choice is to work in the category of complete bornological vector spaces, with the complete bornological tensor product [Mey, Chapter I]. We denote it by⊗, which is reasonable since for Fréchet algebras it agrees with the projective tensor product [Mey,Theorem I.87].
The Hochschild homology of a complete bornological algebra A is defined as working in the category of complete bornological A-modules. When A is unital, HH * (A) can be computed with the completed bar-complex C n (A) = A⊗ n+1 and the usual differential b n (a 0 ⊗ · · · ⊗ a n ) = n−1 i=0 a 0 ⊗ · · · ⊗ a i a i+1 ⊗ · · · ⊗ a n + (−1) n a n a 0 ⊗ a 1 ⊗ · · · ⊗ a n−1 .
Under additional conditions, these functors HH n are continuous: Lemma 3.1. Suppose that A = lim − →i A i is a strict inductive limit of nuclear Fréchet algebras (where strict means that the transition maps A i → A j are injective and have closed range). Then there is a natural isomorphism Proof. In [BrPl1,Theorem 2] this was shown with respect to the inductive tensor product. Under the assumptions of the lemma, inductive tensor products agree with completed bornological tensor products, for the A i and for A [Mey,Theorem I.93].
Recall that S(G) is the inductive limit of the algebras S(G, K), where K runs over the compact open subgroups of G. As S(G, K) is a closed subspace of S(G, K ′ ) when K ′ ⊂ K, S(G) is even a strict inductive limit. The Plancherel isomorphism from Theorem 1.2 shows that each S(G, K) is nuclear Fréchet algebra. Thus Lemma 3.1 applies and says that (3.2) HH n (S(G)) ∼ = lim − →K HH n (S(G, K)).
The decomposition (2.1) induces a decomposition of the Schwartz algebra of G as a direct sum of two-sided ideals: and each HH n (S(G) s ) is isomorphic with HH n (S(G, K) s ) when s ∈ B(G, K).

Topological algebraic aspects.
We want to determine the Hochschild homology of the nuclear Fréchet algebra S(G, K) s , which is also the closure of H(G, K) s in S(G). We have to take the topology of S(G, K) s into account, which creates challenges that were absent in the purely algebraic setting of Section 2. Before we start the actual computation, we first settle most issues of topological-algebraic nature.
Recall from Theorem 1.2 that there is an isomorphism of Fréchet algebras By [Hei,Théorème 0.1], (3.5) restricts to an algebra isomorphism Let e d ∈ S(G, K) be the central idempotent corresponding to the direct summand of (3.5) indexed by d. We define S(G, K) d = e d S(G, K), so that by (3.5) Then S(G, K) s = d∈∆ s G S(G, K) d and To analyse modules over this Fréchet algebra, we will make ample use of the following result. It is the specialization of [KaSo,Lemma 3.4] to the affine variety X nr (M ) with the submanifold X unr (M ) and the action of W (M, d).
Proposition 3.2. LetỸ be an affine variety with an embedding ı in X nr (M ). Suppose that: • ı(Ỹ ) is closed in X nr (M ) and isomorphic toỸ , ) is a real analytic Zariski-dense submanifold ofỸ and diffeomorphic to ı(Y ). Let p be an idempotent in the ring of continuous C ∞ (M ) W (M,d) -linear endomorphisms of Ω n sm (Y ), such that p stabilizes Ω n (Ỹ ). Then the natural map The dense subspace e d H(G, K) s of S(G, K) d is a subalgebra because e d is central, but it is not contained in H(G, K). Its irreducible representations are the constituents of the I G P (δ ⊗ χ) with χ ∈ X nr (M ). Its centre is the restriction of Z(H(G, K) s ) to F(M, δ), so Proof. (a) From (3.7) and (3.8) we see that that Z(S(G, K) d )-module S(G, K) is a direct summand of , it is the image of the idempotent the averages over W (M, d). It follows from (3.6) that ,d) .
From this and (3.7),(3.8) and (3.9) we see that we are in the right position to apply Proposition 3.2, which yields exactly the statement. (b) This follows from part (a) and (2.5).
We note that there are natural homomorphisms of Z(H(G, K) s )-modules and the outer sides should be closely related. However, the composed map is in general not bijective, HH n (e d H(G, K) s ) can be more intricate. As discussed around (3.1), we compute the Hochschild homology of Fréchet algebras with respect to the complete projective tensor product. We establish some topological properties of the Hochschild homology groups of S(G, K) s , making use of [KaSo].
Proposition 3.4. HH n (S(G, K) s ) is a quotient of two closed submodules of a finitely generated Fréchet Z(S(G, K) s )-module. In particular HH n (S(G, K) s ) is a Fréchet Z(S(G, K) s )-module.
Proof. To compute HH n (S(G, K) s ) according to the definition (3.1), we can use any (bornological or Fréchet) projective bimodule resolution of S(G, K) s . One such resolution was constructed in [OpSo1,Theorem 4.2], for S(G, K) but that is enough because S(G, K) s is a direct summand of S(G, K). The set of n-chains of that resolution is a finitely generated projective S(G, K) s⊗ S(G, K) s,op -module. By construction this projective resolution contains a projective bimodule resolution of H(G, K) s , namely the set of elements that live in powers of H(G, K) s⊗ H(G, K) s,op .
By tensoring with S(G, K) s over S(G, K) s⊗ S(G, K) s,op , we obtain a differential complex (C * , d * ) that computes HH n (S(G, K) s ). Each term C n is a direct summand of (S(G, for some r ∈ N. By Theorem 1.2 and [KaSo, Theorem 3.1.b], (S(G, K) s ) r and its direct summand C n are finitely generated Fréchet Z(S(G, K) s )-modules. The set of n-cycles Z n is closed in C n (by the continuity of the boundary map) and hence closed in (S(G, K) s ) r . The intersection C ′ n of C n with (H(G, K) s ) r is a finitely generated Z(H(G, K) s )-module, by (2.5). That yields a differential complex (C ′ n , d n ) which computes HH n (H(G, K) s ).
Choose a finite set Y n ⊂ (H(G, K) s ) r that generates C ′ n as Z(H(G, K) s -module. With Lemma 3.3 we see that Y n also generates C n as Z(S(G, K) s )-module. As the boundary map d n is Z(S(G, K) s )-linear, the set of n-boundaries B n = d n (C n−1 ) is generated as Z(S(G, K) s )-module by d n (Y n−1 ). There are inclusions We want to show that B n = Z(S(G, K) s )d(Y n−1 ) is a closed subspace of the right hand side, just like Z n and C n . The right hand side of (3.11) embeds as Via this embedding the elements of d(Y n−1 ) become analytic (in fact algebraic) functions on X unr (M ) × {1, . . . , r ′ }. By [Tou,Corollaire V.1.6], generalized to an W (M, d)-invariant setting in [KaSo,Theorem 1.2], the finite set d(Y n−1 ) generates a closed C ∞ (X unr (M )) W (M,d) -submodule C ∞ (X unr (M )) r ′ . Hence B n is closed in any of the modules from (3.11). Now (3.11) and HH n (S(G, K) s ) = Z n /B n provide the required properties.
We use the same algebraic families F(M i , η i ) in Rep(G) s as in Paragraph 2.2. Recall from Definition 1.1 that F(M i , η i ) naturally contains a tempered algebraic family By Theorem 1.2 it gives rise to a homomorphism of Fréchet Z(S(G, K) s )-algebras Here Z(S(G, K) s ) acts on the right hand side via evaluations at the central characters of the underlying G-representations π(M i , η i , χ i ). In terms of (3.8), the direct summands Z(S(G, K) d ) = C ∞ (X unr (M )) W (M,d) of Z(S(G, K) s ) annihilate the range of (3.12) when i ≺ d. When i ≺ d, pick χ η i ,δ so that η i is a subquotient of I G P (δ ⊗ χ η i ,δ ). Then C ∞ (X unr (M )) W (M,d) acts on the range of (3.12) via the map We note that These Fréchet algebra homomorphisms induce homomorphisms of Fréchet Z(S(G, K) s )-modules We remark that (ii) and (iii) need not be isomorphic for more general central characters (e.g. the central character of π(M, δ, χ) with χ ∈ X nr (M ) not unitary).

Proof. Let
is exact on a large class of Z(S(G, K) d )-modules. This class contains all modules which as topological vector spaces are quotients of S(Z), and all modules that we need here are of that form. This exactness implies that (3.16) By (3.14) at the exactness of F P , the last expression can be identified with the formal completion of HH n (e d H(G, K) s ) at pr(ξ). Hence (i) and (ii) are naturally isomorphic. Next we apply HH n (F t d ) to the last line of (3.17), with image in As F t d (e d ) = 1, we may just as well set F d (e d ) = 1 and apply HH n (F d ). Then the image becomes To the last expression we apply id ⊗ HH n (F s ) −1 (which exists by Lemma 2.7), and we obtain the desired description of (3.17) and of (3.16).
The injectivity of HH n (F t s ) is more subtle for these topological algebras than it was in the earlier purely algebraic settings (e.g. Lemma 2.7).
Lemma 3.6. (a) The continuous Z(S(G, K) s )-linear map Proof. (a) The kernel of HH n (F t s ) is a closed Z(S(G, K) s )-submodule of HH n (S(G, K) s ), so by Proposition 3.4 it is a quotient of two closed submodules of a finitely generated Fréchet Z(S(G, K) s )-module. Using the central idempotents e d , we can decompose Here each HH n (F t d ) is a quotient of two closed submodules of a finitely generated Fréchet C ∞ (X unr (M )) W (M,d) -module. Suppose that ker HH n (F t d ) is nonzero for one specific δ. By [KaSo,Lemma 1.1] at least one of its formal completions is nonzero, say at ξ = W (M, d)(M, δ, χ). By Proposition 3.5 that formal completion of ker HH n (F t d ) can be considered as a submodule of the formal completion of HH n (H(G, K) s ) d at pr(ξ).
From Theorem 2.8 and Lemma 2.10 we know that HH n (F d ) is injective on HH n (H(G, K) s ) d pr(ξ) . That holds for HH n (F t d ) as well, because F t d = F d on these formal completions. Hence HH n (F d )(ker HH n (F t s )) has a nonzero formal completion at ξ, which is clearly a contradiction.
By Lemma 2.7 the map HH n (F s ), is injective, just as HH n (F t s ). Hence the natural map HH n (H(G, K) s ) → HH n (S(G, K) s ) equals the injection HH n (F t s ) −1 • HH n (F s ).
Lemma 3.7. The continuous map is injective.
Proof. This can be shown in the same way as Theorem 2.13.a. Ultimately the argument relies on [Sol8,Lemma 1.12] which holds just as well in a smooth settting, as explained on [Sol8,p. 21].
We fix d = [M, δ] and we represent the central character of δ by χ δ t + δ with χ δ ∈ X unr (L) and t + δ ∈ X + nr (L). We let in the following way: , then Z(S(G, K) d ) acts as zero on Ω n sm (X unr (L) w c ). Lemma 3.8. The following map is Z(S(G, K) d )-linear: Proof. From Lemma 2.10.b and Theorem 2.13.c we know how Z(H(G, K) s ) acts on That action is pointwise, in the sense that upon specialization at any point of X nr (L) w the Z(H(G, K) s )-action goes via evaluation at a character (or is just zero). Via the natural map such an action naturally gives rise to an action of C ∞ (X unr (M )) W (M,d) on w∈W (L,s) Ω n sm (X unr (L) w ), which is pointwise in the same sense. The map When we compare this with Lemma 2.10.b, we see that Z(S(G, K) s ) acts in the way described just before the lemma.
Next we prove the most technical step towards our description of HH n (S(G, K) s ). .
Proof. From Lemma 3.8 we know that HH n (φ * s ) becomes Z(S(G, K) d )-linear if we restrict its domain to the summands Ω n (X unr (M i )) with i ≺ d. We consider (3.20) If we take the direct sum over d ∈ ∆ s G , then by Theorem 2.13 we obtain precisely the term on the right hand side. By continuous extension we find that the direct sum over d ∈ ∆ s G of the spaces With the injectivity and the C ∞ (X unr (M )) W (M,d) -linearity of HH n (φ * s ) we find that the C ∞ (X unr (M )) W (M,d) -module in the statement is generated by .
By Theorem 2.13.b HH n (F d )HH n (H(G, K) s ) contains (3.24), so HH n (F t d )HH n (S(G, K) s ) contains (3.24) as well. Hence the C ∞ (X unr (M )) W (M,d)module HH n (F t d )HH n (S(G, K) s ) contains the module in the statement.
Everything is in place to establish a smooth version of Theorem 2.13. .
Proof. Evaluation of (3.25) HH 0 (φ * s )HH 0 (F s )HH 0 (H(G, K) s ) at (w, χ) corresponds to the map on HH 0 induced by the virtual representation ν 1 w,χ of H(G, K) s from (2.42). If we evaluate at a family of χ's simultaneously, that interpretation becomes valid and nontrivial in degrees n > 0 as well. The W (L, s)-invariance of (3.25) (and its versions in degrees n > 0) in Theorem 2.13.b is a consequence of: • the relations (2.44) between these virtual representations, • the fact the Hochschild homology does not distinguish equivalent virtual representations [Sol8,Lemma 1.7].
Our maps in the smooth setting are basically the same as the earlier maps in an algebraic setting, only restricted to tempered representations and allowing for smooth functions. Therefore HH n (φ * s )HH n (F t s )HH n (S(G, K) s ) also consists of W (L, s)-invariant elements. More explicitly, it is contained in ,s) .
Comparing this with Lemma 3.9, we deduce that the inclusion in Lemma 3.9 is in fact an equality. Now (3.21) entails that equals the right hand side of (3.26). Thus HH n (φ * s )HH n (F t s ) is a continuous bijection between the Fréchet spaces HH n (S(G, K) s ) and (3.26). By the open mapping theorem, it is an isomorphism of Fréchet spaces.
Like in Proposition 2.9, there is a clearer description in degree n = 0: is an isomorphism of C ∞ (X unr (M )) W (M,d) -modules from HH 0 (S(G, K) d ) to the set of elements of ns i=1,i≺d C ∞ (X unr (M i )) that descend to linear functions on C ⊗ Z R(S(G, K) d ). (b) Part (a) yields an isomorphism of Z(S(G, K) s )-modules (a) From Proposition 2.9.a and Theorem 2.13.b we see that the stated condition on f ∈ ns i=1,i≺d C ∞ (X unr (M i )), only with R(H(G, K) s ), is equivalent to the condition The condition in the statement is local, so can be checked locally in terms of (3.27).
If one restrict to R(S(G, K) d ), only the parts of the condition of the subvarieties X unr (L) w remain. Then we get exactly the description of HH 0 (S(G, K) s ) already established in part (a). (b) This is analogous to Proposition 2.9.b.
We can use that to establish an analogue of Theorem 2.14 for S(G, K) s ). For an algebraic family of O(X nr (L))⋊C[W (L, s), ♮ s ]-representations F(M, η), parametrized by X nr (M ) and on a vector space V M,η , we define Recall from Theorem 2.5 that ζ ∨ restricts to a bijection Theorem 3.12. There exists a unique isomorphism of Fréchet spaces Proof. This is analogous to the proof of Theorem 2.14. For the construction of HH n (ζ ∨ t ) we use (3.28) and Theorem 3.10 instead of [Sol8, Theorem 1.2], Theorem 2.13.b and Lemma 2.7. In all the involved algebraic families of representations F(M, η), temperedness of F(M, η, χ) is equivalent to χ ∈ X unr (L). The uniqueness and further properties of the thus defined map HH n (ζ ∨ t ) can be shown in exactly the same way as for HH n (ζ ∨ ).
Theorems 2.14 and 3.12 relate the Hochschild homology of H(G) s and S(G) s to that of the twisted crossed products O(X nr (L)) ⋊ C[W (L, s), ♮ s ] and C ∞ (X unr (L)) ⋊ C[W (L, s), ♮ s ].
These theorems can be considered as confirmations of the ABPS conjectures [ABPS2] on the level of Hochschild homology.
Finally, we take a closer look at HH n (S(G, K) d ). From the Plancherel isomorphism (3.5) we get By Lemma 3.6.b we can regard HH n (H(G, K) s ) as a subspace of HH n (S(G, K) s ).
Theorem 3.13. (a) The maps HH n (F t d ) and HH n (φ * s ) provide isomorphisms between the Fréchet Z(S(G, K) d )-modules HH n (S(G, K) d ), and Proof. (a) The Z(S(G, K) d )-linearity comes from Lemma 3.8 and the isomorphisms follow immediately from Theorem 3.10. The range of HH n (F t d ) is a Fréchet space because by the continuity of HH n (φ * s ) it is closed in ns i=1,i≺d Ω n sm (X unr (M i )). The range of HH n (φ * s )HH n (F t d ) is Fréchet because as checked directly after (3.22) it is a direct summand of w∈W (L,s) Ω n sm (X unr (L) w ). (b) Recall from Lemma 2.10 that From part (a) and Theorem 2.13.b we see that the closure of To this we apply HH n (F t s ) −1 , which exists and is continuous by part (a). We find that the closure of (3.30) equals After (3.22) we constructed a continuous C ∞ (X unr (M )) W (M,d) -linear idempotent endomorphism p of w∈W (L,s) Ω n sm (X unr (L) w ), with image (2.39). By part (a) HH n (S(G, K) d ) is isomorphic as C ∞ (X unr (M )) W (M,d) -module to the image of p, via the map HH n (φ * s )HH n (F t d ). Similarly Theorem 2.13 tells us that HH n (H(G, Thus we have translated the statement to: the natural map is an isomorphism of Fréchet C ∞ (X unr (M )) W (M,d) -modules. As the action comes from an embedding X unr (L) w → X unr (M ) for each relevant w, that claim is an instance of Proposition 3.2.
Let us record a consequence of Theorem 3.13: as Fréchet Z(S(G, K) s )-modules. However, usually is not isomorphic to HH n (S(G, K) s ) as Z(S(G, K) s )-module. The reason is that the terms with d ′ = d can be nonzero, but do not occur in (3.31).

Cyclic homology
Recall from [Lod,§2.1.7] that the cyclic homology of a unital algebra A can be computed as the total homology of a bicomplex (B (A), b, B). Here and B(A) p,q is zero otherwise. The vertical differential b is the same as in the bar-resolution, so each column of B(A) computes the Hochschild homology of A. The horizontal differential B induces a map B : HH n (A) → HH n+1 (A). When A = O(V ) for a nonsingular complex affine variety or A = C ∞ (V ) for a smooth manifold V , B is the usual exterior differential d : Ω n (V ) → Ω n+1 (V ) [Lod,§2.3.6].
For A = H(G, K) s , we know from Theorem 2.13 that there is an isomorphism , induced by the algebraic families of virtual representations ν 1 w,χ : χ ∈ X nr (L) w c w ∈ W (L, s), c ∈ π 0 (X nr (L) w ).
By (2.42) each of these families is a linear combination of algebraic families F ′ (M i , η i ) obtained from F(M i , η i ) by composition with an algebraic map from X nr (L) w c to X nr (M i ). In particular (4.1) is a linear combination of maps (4.2) HH n (F ′ M i ,η i ) : HH n (H(G, K) s ) → HH n O(X nr (L) w c ) ⊗ End C I G P i (η i ) K . By Morita invariance and the Hochschild-Kostant-Rosenberg theorem, the right hand side of (4.2) can be identified with (4.3) HH n O(X nr (L) w c ) ∼ = Ω n (X nr (L) w c ). Via these maps, the natural differential B on HH * (H(G, K) s ) is transformed into the exterior differential d on Ω * (X nr (L) w c ). All the maps in (4.2) and (4.3) (and between them) can be realized on the level of chain complexes. For HH n (F ′ M i ,η i ) that is clear, the Morita equivalence between where BΩ * (V ) is the bicomplex with Ω p−q (V ) in degree (p, q), provided that p ≥ q ≥ 0. From Theorem 2.13 we know that its image is actually smaller, we can restrict it to a morphism of bicomplexes Analogous considerations for S(G, K) s , now using Theorem 3.10, lead to a morphism of bicomplexes By Theorems 2.13 and 3.10, the maps (4.4) and (4.5) induce isomorphisms on the Hochschild homology of the involved bicomplexes. It follows from Connes' periodicity exact sequence that (4.4) and (4.5) also induce isomorphisms on cyclic homology, see [Lod,§2.5]. .
Proof. As explained above, it remains to identify the cyclic homology of the right hand sides of (4.4) and (4.5). By design see [Lod,§2.3]. In our case V = w∈W (L,s) X nr (L) w and the group W (L, s) acts on Ω * (V ), namely by the natural action on the underlying space tensored with the characters ♮ w s . Taking invariants for an action of a finite group commutes with homology, so we may just take the W (L, s)-invariants in (4.6). That yields HC n (H(G, K) s ), and the argument for HC n (S(G, K) s ) is completely analogous.
From Theorem 4.1 we see that HC n (H(G, K) s ) and HC n (S(G, K) s ) stabilize: for n > dim C (X nr (L)) they depend only on the parity of n. By [Lod,Proposition 5.1.9], the periodic cyclic homology is the limit term: .

(4.8)
We point out that the right hand sides of, respectively, (4.7) and (4.8) are naturally isomorphic with the periodic cyclic homology groups of, respectively, That can be derived with similar arguments. Hence (4.7) and (4.8) are the versions of Theorems 2.14 and 3.12 for periodic cyclic homology. We note also that (4.8) relates to the conjectural description of the topological K-theory of S(G, K) s in [ABPS2,Conjecture 5]. In [ABPS2,§4] things are formulated for the C * -completion of S(G) s , which has the same topological K-theory as S(G, K) s by [Sol1,(3.2)]. Since the Chern character K * (S(G, K) s ) ⊗ Z C → HP * (S(G, K) s ) is an isomorphism [Sol1, Theorem 3.2], (4.8) provides a description of K * (S(G, K) s ) modulo torsion. With the comments around (4.9) we can formulate that as an isomorphism (4.10) K * (S(G, K) s ) ⊗ Z C ∼ = HP * C ∞ (X unr (L)) ⋊ C[W (L, s), ♮ s ] .
With elementary Lie theory one sees that X nr (L) w is a finite union of cosets X nr (L) w c of the complex torus X nr (L) w,• . Since X nr (L) w is a commutative Lie group. its tangent spaces at any two points are canonically isomorphic. that are locally constant (as differential forms). The same holds for S(G, K) s .
Proof. First we consider a simpler setting, namely the graded algebra of differential forms Ω * (T ) on a complex algebraic torus T . Write T as a direct product of onedimensional algebraic subtori T i , then H * dR (T ) = i H * dR (T i ), and in combination with (4.11) we find that this is precisely the space of constant differential forms in Ω * (T ).
The above argument uses the structure of T as algebraic variety, not as group, so it applies to all the varieties X nr (L) w c . Further, the action of Z W (L,s) (w, c) on Ω * (X nr (L) w c ) preserve the subspace of constant differential forms. Hence that are locally constant (keeping the canonical identifications of different tangent spaces in mind). Combining that with Theorem 2.13 and (4.7), we get the lemma for H(G, K) s . The above arguments involving T also work for smooth differential forms on compact real tori. With that, Theorem 3.10 and (4.8), we establish the lemma for S(G, K) s .
Assume now that the 2-cocycle ♮ s is trivial, like in most examples. Then (4.7)-(4.8) and Corollary 4.3 simplify to

Examples
The smallest nontrivial example is G = SL 2 (F ), where F is any non-archimedean local field. The Hochschild homology for this group is known entirely from [Sol4], here we work it out in our notations.
For every supercuspidal G-representation V with V K = 0, the corresponding direct summand of H(G, K) is Morita equivalent with C. This contributes a factor C to HH 0 (H(G, K)), and nothing to HH n (H(G, K)) with n > 0. The same applies to HH * (S(G, K)).
Another well-studied example is the general linear group G = GL n (F ). For this G most aspects are simpler than for other reductive p-adic groups. Consider an arbitrary inertial equivalence class s = [L, σ] for G. By picking suitable representatives, we can achieve that where σ i and σ i ′ with i = i ′ are not equivalent up to unramified twists. There are natural isomorphisms Let M be a Levi subgroup of G containing L and let δ ∈ Irr(M ) be square-integrable modulo centre, such that δ ∈ Rep(M ) s . Then d = [M, δ] can be represented by data where ℓ i j=1 d j e i,j = e i and St(d j , σ i ) is the generalized Steinberg representation associated to σ ⊠d j i . Moreover we may assume that St(d j , σ i ) and St(d j ′ , σ i ) do not differ by an unramified twist if j = j ′ . In this case there are natural isomorphisms It is known from [DKV,Théoréme B.2.d] that I G P preserves irreducibility for tempered representations. Hence the intertwining operators by which W (M, d) acts on C ∞ (X unr (M )) ⊗ End C (I G P (δ)) must be scalar at every point of X unr (M ). In particular W (L, s) acts on C ∞ (X unr (L)) ⊗ End C I G P 0 L (σ) K as a group, not via a projective