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) 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∞ (Tᵤ) W) , where Tᵤ 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.