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

Pierre Clare, Tyrone Crisp

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.

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

Abstract

We calculate the -theory of the reduced -algebra of a reductive -adic group . To do so, we show that each direct summand in Plymen's Plancherel decomposition of 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 -theory of is the direct sum of the twisted equivariant -theory groups of these blow-ups, which can be computed using an Atiyah-Hirzebruch spectral sequence. As an illustration, the case of is treated in some detail. Our main result is obtained from a more general study of -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 -adic case.

Paper Structure

This paper contains 35 sections, 32 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Morita equivalences from twisted equivariant Hilbert modules
  3. Basic definitions
  4. C0(X)⋊gamma W and its dual
  5. The ideal C(X,E,W)
  6. Scalar intertwiners and the completeness condition
  7. Localisation
  8. A special case
  9. Reduced C*-algebras of p-adic reductive groups
  10. Background on parabolic induction and intertwining operators
  11. A twisted action from intertwining operators
  12. The Plancherel theorem for C*r(G)
  13. Scalar intertwiners
  14. Blowing up the parameter space
  15. K-theory of C*r(G)
  16. ...and 20 more sections

Key Result

Lemma 2.5

The definitions eq:xi-bw-definition and eq:BW-valued-ip make $E$ into a right Hilbert $B\rtimes_\gamma W$-module, and we have $\operatorname{K}_{B\rtimes_\gamma W}(E) = \operatorname{K}_B(E)^W.$

Theorems & Definitions (85)

  • Definition 2.1: See § 2.4 in Zeller-Meier
  • Definition 2.2
  • Lemma 2.5
  • proof
  • Remark 2.7
  • Example 2.9
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Lemma 2.13
  • ...and 75 more