Reduced C*-algebras and K-theory for reductive $p$-adic groups

TL;DR

This paper computes the K-theory of the reduced C*-algebra of reductive p-adic groups, using Morita equivalences and spectral sequences, with detailed analysis for Sp_4.

Contribution

It introduces a method to calculate K-theory of these algebras via Morita equivalences to twisted crossed products and spectral sequences, extending previous results.

Findings

K-theory of C*_r(G) is a direct sum of twisted equivariant K-theories.

Each summand is Morita equivalent to a twisted crossed product.

The method applies to groups like Sp_4 and recovers known results.

Abstract

We calculate the $K$-theory of the reduced $C^*$-algebra $C^*_r(G)$ of a reductive $p$-adic group $G$. To do so, we show that each direct summand in Plymen's Plancherel decomposition of $C^*_r(G)$ is Morita equivalent to a twisted crossed product for an action of a finite group on the blow-up of a compact torus along the zero-locus of a certain Plancherel density. It follows that the $K$-theory of $C^*_r(G)$ is the direct sum of the twisted equivariant $K$-theory groups of these blow-ups, which can be computed using an Atiyah-Hirzebruch spectral sequence. As an illustration, the case of $\operatorname{Sp}_4$ is treated in some detail. Our main result is obtained from a more general study of $C^*$-algebras of compact operators on twisted equivariant Hilbert modules, from which we also recover results due to Wassermann for real groups, and to Afgoustidis and Aubert in the $p$-adic case.