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 full Hecke algebra of a reductive p-adic group G, and the representation theory of G. Let us denote by C ∞ c (G) the full Hecke algebra of G and by HP * (C ∞ c (G)) its periodic cyclic homology groups. LetĜ denote the admissible dual of G. One of the main points of this paper is that the groups HP * (C ∞ c (G)) are, on the one hand, directly related to the topology ofĜ and, on the other hand, the groups HP * (C ∞ c (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 * (C ∞ c (G)) and the topology ofĜ 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 O(X) ⋊ 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.

The periodic cyclic homology of crossed products of finite type algebras

“…is acyclic-and thus provides a projective resolution of A with A e := A ⊗ A op modules-we obtain [13,33,51,62,77] Proposition 1.2. For every q ≥ 0, we have a natural isomorphism…”

“…The homology groups of this subcomplex of the Hochschild complex (B q (A ⋊ Γ), b) associated to γ will be denoted by HH * (A ⋊ Γ) γ . We similarly define HC * (A ⋊ Γ) γ and HP * (A ⋊ Γ) γ , see [33,46,62] for instance. This yields the decomposition of Equation (2).…”

“…Let, for any g ∈ Γ, C g denote the centralizer of g in Γ, that is, C g = {γ ∈ Γ, gγ = γg}. Then we have the following result [8,13,15,32,33,62,53,61,72]. Proposition 1.7.…”

“…An approach to this result (as well as to the spectral sequence for the case when Γ is not necessarily finite), using cyclic objects and their generalizations, is contained in [62]. This proposition explains why we are interested in the twisted Hochschild homology groups.…”

“…For γ ∈ Γ, let us denote by γ the conjugacy class of γ ∈ Γ and by Γ the set of conjugacy classes of of Γ. Our first step is to use the decomposition (2) HH q (A ⋊ Γ) ≃ γ ∈ Γ HH q (A ⋊ Γ) γ of the Hochschild homology groups of A ⋊ Γ [5,6,13,33,46,62]. Let Prim(k) be the maximal ideal spectrum of k and denote by supp(A) ⊂ Prim(k) the support of A.…”