Hochschild homology of reductive $p$-adic groups

TL;DR

This paper computes the Hochschild homology of Hecke and Schwartz algebras of reductive p-adic groups, linking algebraic structures to geometric and topological invariants, and clarifies their relation to Bernstein centers and K-theory.

Contribution

It establishes explicit isomorphisms between Hochschild homology of these algebras and crossed product algebras, enhancing understanding of their structure and representation theory.

Findings

Hochschild homology groups are computed for Hecke and Schwartz algebras.

Isomorphisms are established with crossed product algebras involving unramified characters.

Connections to cyclic homology and topological K-theory are elucidated.

Abstract

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.