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

TL;DR

This survey develops a comprehensive framework for the representation theory and noncommutative geometry of reductive $p$-adic groups, tying together three central algebras: the Hecke algebra $\mathcal{H}(G)$, the Harish-Chandra--Schwartz algebra $\mathcal{S}(G)$, and the reduced $C^*$-algebra $C_r^*(G)$. It explains how Bernstein decomposition partitions the representation theory into blocks, and shows how parabolic induction, Jacquet restriction, and Langlands-type classifications organize irreducibles, including the pivotal ABPS conjecture that links smooth representations to crossed products and graded Hecke algebras. The noncommutative-geometric aspects are developed via Hochschild homology and topological $K$-theory, with explicit computations for cuspidal and non-cuspidal Bernstein blocks, and a key new result expressing $K_*(C_r^*(G))$ in terms of twisted $W$-equivariant $K$-theory of compact tori: $K_*(C_r^*(G)) \cong K_*^{W}(T^{u}_{\mathfrak s})$, up to twists; this provides a concrete bridge between representation theory and noncommutative geometry. The framework yields exact descriptions of $HH_*(\mathcal{H}(G))$ and $HH_*(\mathcal{S}(G))$ in terms of (twisted) extended quotients and (twisted) crossed-product algebras, and clarifies how topological localization informs the study of topological invariants. Collectively, the work highlights how ABPS-type correspondences reduce complex $G$-representation problems to graded-Hecke-algebra computations and equivariant topological invariants, enabling precise structural and computational insights into reductive $p$-adic groups.

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.

Theorems & Definitions (181)

• Example 1.1
• Definition 1.2
• Proposition 1.3
• proof
• Lemma 1.4
• proof
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• Definition 1.8