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The Mackey bijection as a stratified equivalence

Alexandre Afgoustidis, Pierre Clare

Abstract

This paper is about the Mackey analogy between the tempered representation theory of a real reductive group and that of its Cartan motion group. We consider the embedding of reduced C*-algebras constructed recently in connection with the Mackey bijection, and study its behavior on certain natural stratifications of the tempered duals. We formulate our result using a notion of stratified equivalence inspired by the study of the smooth dual of $p$-adic groups via the structure of Hecke algebras, in particular by the work of Aubert, Baum, Plymen and Solleveld. We derive related new topological properties of the Mackey bijection. We also analyze the behavior of the Mackey embedding on a stratification of reduced C*-algebras attached to a partition of the tempered dual into particularly elementary pieces, introduced in recent work of Bradd, Higson and Yuncken.

The Mackey bijection as a stratified equivalence

Abstract

This paper is about the Mackey analogy between the tempered representation theory of a real reductive group and that of its Cartan motion group. We consider the embedding of reduced C*-algebras constructed recently in connection with the Mackey bijection, and study its behavior on certain natural stratifications of the tempered duals. We formulate our result using a notion of stratified equivalence inspired by the study of the smooth dual of -adic groups via the structure of Hecke algebras, in particular by the work of Aubert, Baum, Plymen and Solleveld. We derive related new topological properties of the Mackey bijection. We also analyze the behavior of the Mackey embedding on a stratification of reduced C*-algebras attached to a partition of the tempered dual into particularly elementary pieces, introduced in recent work of Bradd, Higson and Yuncken.
Paper Structure (28 sections, 23 theorems, 67 equations)

This paper contains 28 sections, 23 theorems, 67 equations.

Table of Contents

  1. The tempered dual and the Mackey bijection
  2. Parameters for the tempered dual of G
  3. Imaginary part of the infinitesimal character; tempiric representations
  4. Lowest K-types
  5. Parametrization of the tempered dual
  6. The unitary dual of G0 and the Mackey bijection
  7. Unitary dual of the Cartan motion group
  8. Construction of unitary representations of $G_0$.
  9. The spectral extended quotient $\mathfrak{p}^\ast\!/\!\!/ K$ and its topology.
  10. The Mackey bijection
  11. The Mackey embedding of C*-algebras
  12. The Mackey embedding as a stratified equivalence
  13. Basics on the Jacobson and Fell topologies
  14. Primitive ideals and the Fell topology
  15. Containment of K-types as an open condition
  16. ...and 13 more sections

Key Result

Theorem 1.6

For fixed $I\subset S$ and $\nu\in\mathfrak{a}_{I, +}^\ast$, the map determines a bijection between the set $\widehat{M}_{I, \mathrm{tempiric}}$ of unitary equivalence classes of tempiric representations $\sigma$ of $M_I$, on the one hand, and the set of unitary equivalence classes of irreducible tempered representations $\pi$ of $G$ for which $\operatorname{Im\,inf\

Theorems & Definitions (57)

  • Remark 1.1
  • Definition 1.5
  • Theorem 1.6: See BHY
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.10: Vogan Vogan81; see Vogan07, Theorem 1.2
  • Remark 1.13
  • Theorem 1.14
  • Lemma 1.15
  • proof
  • ...and 47 more