Equivariant homology for SL(2) of a p-adic field

Abstract

Let F be a p-adic field and let G = SL(2) be the group of unimodular 2 × 2 matrices over F. The aim of this paper is to calculate certain equivariant homology groups attached to the action of G on its tree. They arise in connection with a theorem of M. Pimsner on the K-theory of the C∗-algebra of G [12], and our purpose is to explore the representation theoretic content of Pimsner’s result. The outcomes of our calculations are given in Theorems 5.4 and 6.1. In Sections 8 and 9 of the paper we re-examine Pimsner’s theorem in the light of these new results. The first author and A. Connes have formulated a very general conjecture [1] describing the K-theory of the reduced C∗-algebra of any locally compact group. For a semisimple group over a p-adic field it asserts, roughly speaking, that the cohomology of the space of tempered representations of G is isomorphic to the equivariant homology of the affine Bruhat-Tits building of G. For SL(2) and other split rank one groups the conjecture amounts to Pimsner’s theorem, but for groups of higher rank the conjecture is not yet proved. In a sequel to this article we shall study the representation theoretic aspects of the conjecture for p-adic groups (we note that the arguments in Sections 5 and 8 readily extend to this general case). Our homology groups are very closely related to the cyclic homology groups of the convolution algebra of smooth compactly supported functions on G, and the results of our calculations are similar to some of P. Blanc and J-L. Brylinski in [3]. But the methods we employ are different, and we hope they complement rather than duplicate those of Blanc and Brylinski. The connection between the two will be explored elsewhere

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