Group algebras of reductive $p$-adic groups, their representations and their noncommutative geometry

Abstract

This is a survey paper about representation theory and noncommutative geometry of reductive p-adic groups G. The main focus points are: 1. The structure of the Hecke algebra H(G), the Harish-Chandra-Schwartz algebra S(G) and the reduced C*-algebra $C_r^{\ast }(G)$. 2. The classification of irreducible G-representations in terms of supercuspidal representations. 3. The Hochschild homology and topological K-theory of these algebras. In the final part we prove one new result, namely we compute $K_{\ast }(C_r^{\ast }(G))$ including torsion elements, in terms of equivariant K-theory of compact tori.