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

  • Victor Nistor
Part of the Aspects of Mathematics book series (ASMA)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

Authors and Affiliations

  • Victor Nistor
    • 1
  1. 1.Pennsylvania State UniversityUniversity Park

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