A non-commutative geometry approach to the representation theory of reductive $p$-adic groups: Homology of Hecke algebras, a survey and some new results

Abstract

We survey some of the known results on the relation between the homology of the {\em full} Hecke algebra of a reductive $p$-adic group $G$, and the representation theory of $G$. Let us denote by $\CIc(G)$ the full Hecke algebra of $G$ and by $\Hp_*(\CIc(G))$ its periodic cyclic homology groups. Let $\hat G$ denote the admissible dual of $G$. One of the main points of this paper is that the groups $\Hp_*(\CIc(G))$ are, on the one hand, directly related to the topology of $\hat G$ and, on the other hand, the groups $\Hp_*(\CIc(G))$ are explicitly computable in terms of $G$ (essentially, in terms of the conjugacy classes of $G$ and the cohomology of their stabilizers). The relation between $\Hp_*(\CIc(G))$ and the topology of $\hat G$ is established as part of a more general principle relating $\Hp_*(A)$ to the topology of $\Prim(A)$, the primitive ideal spectrum of $A$, for any finite typee algebra $A$. We provide several new examples illustrating in detail this principle. We also prove in this paper a few new results, mostly in order to better explain and tie together the results that are presented here. For example, we compute the Hochschild homology of $\maO(X) \rtimes Γ$, the crossed product of the ring of regular functions on a smooth, complex algebraic variety $X$ by a finite group $Γ$. We also outline a very tentative program to use these results to construct and classify the cuspidal representations of $G$. At the end of the paper, we also recall the definitions of Hochschild and cyclic homology.