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

2025/10/20 by Maarten Solleveld, Solleveld, Maarten
Mathematics · #16E40 #19K99 #20C08 #20G25 #22E50 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2510.17260

openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

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.

Related

• A non-commutative geometry approach to the representation theory of reductive p-adic groups: Homology of Hecke algebras, a survey and some new results
• Algebraic K-theory of reductive p-adic groups
• Recipes to compute the algebraic K-theory of Hecke algebras of reductive p-adic groups
• The Hecke algebra of a reductive p-adic group: a geometric conjecture
• Restriction to compact subgroups in the cyclic homology of reductive\n p-adic groups