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A NONCOMMUTATIVE GEOMETRY APPROACH TO THE REPRESENTATION THEORY OF REDUCTIVE pADIC GROUPS: HOMOLOGY OF HECKE ALGEBRAS, A SURVEY AND SOME NEW RESULTS
, 2004
"... ABSTRACT. We survey some of the known results on the relation between the homology of the full Hecke algebra of a reductive padic 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 ˆ G de ..."
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ABSTRACT. We survey some of the known results on the relation between the homology of the full Hecke algebra of a reductive padic 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 ˆ G 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 ˆ G 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 ˆG 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. CONTENTS
HIGHER ORBITAL INTEGRALS, SHALIKA GERMS, AND THE HOCHSCHILD HOMOLOGY OF HECKE ALGEBRAS
"... Abstract. We give a detailed calculation of the Hochschild and cyclic homology of the algebra C ∞ c (G) of locally constant, compactly supported functions on a reductive p–adic group G. We use these calculations to extend to arbitrary elements the definition the higher orbital integrals introduced b ..."
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Abstract. We give a detailed calculation of the Hochschild and cyclic homology of the algebra C ∞ c (G) of locally constant, compactly supported functions on a reductive p–adic group G. We use these calculations to extend to arbitrary elements the definition the higher orbital integrals introduced by Blanc and Brylinski for regular semisimple elements. Then we extend to higher orbital integrals some results of Shalika. We also investigate the effect of the “induction morphism ” on Hochschild homology. Contents