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

Abstract

Abstract. We give a detailed calculation of the Hochschild and cyclic homology of the algebra Ꮿ ∞ 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 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.

Restriction to compact subgroups in the cyclic homology of reductive p -adic groups

“…In Proposition 3.7 we compute the Jacquet restriction map r : H(H(G)) → H(H(M )), and show that it is equal to one defined by Nistor in [35]. Nistor suggested that his map be considered an analogue of parabolic induction, and our computation makes this analogy precise.…”

“…These results, combined with Proposition 3.7, give: Corollary 3.13. (cf [35,. Proposition 7.4]) Let G = SL 2 (F ), and let M be the subgroup of diagonal matrices.…”

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

“…Having discussed the relation between HP * (C ∞ c (G)) and the admissible spectrumĜ = Prim(C ∞ c (G)), let us recall the explicit calculation of HP * (C ∞ c (G)) from [47]. The calculation of HP * (C ∞ c (G)) in [47] follows right away from the calculation of the Hochschild homology groups of C ∞ c (G). The calculations of presented in this section complement the results on the cyclic homology of p-adic groups in [45,48].…”

“…To state the main result of [47] on the Hochschild homology of the algebra C ∞ c (G), we need to introduce first the concepts of a "standard subgroup" and of a "relatively regular element" of a standard subgroup. For any group G and any subset A ⊂ G, we shall denote…”

“…The isomorphism of this theorem was obtained by identifying the E ∞ -term of a spectral sequence convergent to HH q (C ∞ c (G)), and hence it is not natural. This isomorphism can be made natural by using a generalization of the Shalika germs [47] .…”