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 HHn(H(G)) and HHn(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 [J. Algebra 606 (2022), 371-470] that every Bernstein ideal H(G)s of H(G) is closely related to a crossed product algebra of the form O(T) (sic) 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)(sic)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)(congruent to) of S(G) is isomorphic to HH*(C-infinity(Tu)(sic) 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.