Hochschild homology of reductive $p$-adic groups

Abstract

Consider a reductive p -adic group G , its (complex-valued) Hecke algebra \mathcal H (G) , and the Harish-Chandra–Schwartz algebra \mathcal S (G) . We compute the Hochschild homology groups of \mathcal H (G) and of \mathcal 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} (\mathcal H (G)) and HH_{n} (\mathcal 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 \mathcal H (G)^{\mathfrak s} of \mathcal H (G) is closely related to a crossed product algebra of the form \mathcal O (T)\rtimes W . Here \mathcal 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_{*} (\mathcal H (G)^{\mathfrak s}) and HH_{*} (\mathcal O (T)\rtimes W) , although we have to say that in some cases it is necessary to twist \mathbb{C} [W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal \mathcal S (G)^{\mathfrak s} of \mathcal 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_{*} (\mathcal H (G)) and HH_{*} (\mathcal S (G)) , we also show how the Bernstein centre of \mathcal H (G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of \mathcal H (G) and of \mathcal S (G) and we relate that to topological K-theory.