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

TL;DR

This survey explores the structure, classification, and noncommutative geometric properties of reductive p-adic groups, culminating in a new computation of the K-theory of their reduced C*-algebras including torsion elements.

Contribution

It provides a comprehensive overview of the algebraic and geometric aspects of reductive p-adic groups and introduces a novel computation of K-theory for their C*-algebras.

Findings

Detailed description of Hecke and Schwartz algebras

Classification of irreducible representations via supercuspidal representations

Computed K-theory of reduced C*-algebras including torsion elements

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 $C_r^* (G)$. 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 $K_* (C_r^* (G))$ including torsion elements, in terms of equivariant K-theory of compact tori.