Hochschild homology of reductive $p$-adic groups
Maarten Solleveld
TL;DR
The paper computes the Hochschild and cyclic homology of the Hecke algebra H(G) and the Harish-Chandra–Schwartz algebra S(G) for reductive p-adic groups, by constructing algebraic families of G-representations and exploiting Morita equivalences to twisted crossed products. It expresses HH_n(H(G)^s) and HH_n(S(G)^s) as modules of differential forms on twisted extended quotients built from Bernstein components, with precise descriptions via twisted actions and centers. The work extends to periodic cyclic homology and topological K-theory, linking noncommutative invariants to geometric objects on X_nr(L) and X_unr(L), and provides explicit computations in key examples such as SL_2 and GL_n. The results support and clarify the ABPS conjectures at the level of Hochschild and cyclic homology, bridging representation theory, noncommutative geometry, and topological invariants.
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.