Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras

We give a detailed calculation of the Hochschild and cyclic homology of the algebra ∞ (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 of the higher orbital integrals introduced by Blanc and Brylinski (1992) for regular semi-simple elements. Then we extend to higher orbital integrals some results of Shalika (1972). We also investigate the effect of the "induction morphism" on Hochschild homology. 2000 Mathematics Subject Classification. 19D55, 46L80, 16E40. 1. Introduction. Orbital integrals play a central role in the harmonic analysis of reductive p-adic groups; they are, for instance, one of the main ingredients in the Arthur-Selberg trace formula. An orbital integral on a unimodular group G is an im- portant particular case of an invariant distribution on G. Invariant distributions have been used in (3) to prove the irreducibility of certain induced representations of GLn over a p-adic field. Let G be a locally compact, totally disconnected topological group. We denote by ∞ (G) the space of compactly supported, locally constant, and complex valued func- tions on G. The choice of a Haar measure on G makes ∞ (G) an algebra with respect to the convolution product. We refer to ∞ (G) with the convolution product as the (full) Hecke algebra of G.I fG is unimodular, then any invariant distribution on G defines a trace on ∞ (G), and conversely, any trace on ∞ (G) is obtained in this way (this well-known fact follows also from Lemma 3.1). Since the space of traces on an algebra A identifies naturally with the first (i.e., 0th) Hochschild cohomology group of that algebra A, it is natural to ask what are all the Hochschild cohomology groups of ∞ (G). The Hochschild cohomology and homology groups of an algebra A are de- noted in this paper by HH q (A) and, respectively, by HHq(A). Since HH q ( ∞ (G)) is the algebraic dual of HHq( ∞ (G)), it is enough to determine the Hochschild homology groups of ∞ (G). In this paper, G is typically a p-adic group, which, we recall, means that G is the set of F-rational points of a linear algebraic group G defined over a finite extension F of the field Qp of p-adic numbers, p being a fixed prime number. The group G does not have to be reductive, although this is certainly the most interesting case. When we assume G (or G, by abuse of language) to be reductive, we state this explicitly. For us, the most important topology to consider on G is the locally compact, totally