Noncommutative Geometry and Number Theory pp 301-321 | Cite as

# 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. Let *Ĝ* 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 of *Ĝ* 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 of *Ĝ* 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*) ⋊ Г, the crossed product of the ring of regular functions on a smooth, complex algebraic variety *X* by a finite group Г. 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.

## Keywords

Conjugacy Class Spectral Sequence Noncommutative Geometry Chern Character Cyclic Homology## Preview

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