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 $H{H}_n(\mathcal{H}(G))$ and $H{H}_n(\mathcal{S}(G))$ 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 $\mathcal{O}(T)⋊W$. 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 $H{H}_{\ast }(\mathcal{H}(G{)}^{\mathfrak{s}})$ and $H{H}_{\ast }(\mathcal{O}(T)⋊W)$, 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 $H{H}_{\ast }(C^{\infty }(T_u)⋊W)$, where $T_u$ denotes the Lie group of unitary unramified characters of $L$. In these pictures of $H{H}_{\ast }(\mathcal{H}(G))$ and $H{H}_{\ast }(\mathcal{S}(G))$, 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.