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

Abstract

We calculate the $K$-theory of the reduced $C^{\ast }$-algebra $C_r^{\ast }(G)$ of a reductive $p$-adic group $G$. To do so, we show that each direct summand in Plymen's Plancherel decomposition of $C_r^{\ast }(G)$ 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 $C_r^{\ast }(G)$ 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 ${\mathrm{Sp}}_4$ is treated in some detail. Our main result is obtained from a more general study of $C^{\ast }$-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, as well as a new description of the tempered dual of $G$ as a topological space.

67 pages, minor modifications. Added Section 4.5 on the relationship between our results and the work of Aubert-Baum-Plymen-Solleveld