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₍ (H (G) ) and HH₍ (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) 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) 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^ (Tₔ) W), where Tₔ 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.