**Representation Theory and Noncommutative Geometry**

## Workshop: Tempered representations and K-theory

### February 24 to 28, 2025 - IHP, Paris

#### Summary

The classification of tempered irreducible representations for real reductive groups was completed in the 1970s by Knapp and Zuckerman, following Harish-Chandra's work on the Plancherel formula. But some aspects of the subject are now undergoing a re-examination, following the discovery of new perspectives. C*-algebras and K-theory are valuable tools in Representation Theory, as shown, for instance, by the Mackey bijection. Indeed, it was the Connes-Kasparov isomorphism in K-theory that motivated the search for a natural bijection between the tempered dual of a real reductive group and the unitary dual of its Cartan motion group, as initially suggested by Mackey in the 1970s.

The meeting will focus on recent developments in which K-theoretic ideas have offered new perspectives on the tempered dual for reductive groups or symmetric spaces, and conversely on new approaches to operator-algebraic problems using contemporary tools in representation theory.

Topics will include:

- New approaches to the Mackey bijection through
*pseudodifferential operator theory*, which has itself undergone an extensive conceptual redesign in the past decade, thanks again to $C^*$-algebra $K$-theory connections. - New perspectives on the the Connes--Kasparov isomorphism using
*Dirac cohomology and cohomological induction;* *Higher orbital intergrals*, which make it possible to go beyond the ``noncommutative topology of the tempered dual'', hinting at something like the ``differential geometry'' of this noncommutative space.- Study of the
*Casselman--Schwartz algebras*and their K-theory via Paley--Wiener theorems, and connections with the Connes--Kasparov isomorphism. - $C^*$-algebraic analysis of the tempered dual from the point of view of $G$ as a symmetric space for $G{\times}G$, and more generally of the tempered spectrum of symmetric spaces.

#### Preliminary list of speakers:

- Anne-Marie Aubert
- Bachir Bekka
- Yves Benoist
- Jacob Bradd
- Peter Hochs
- Vincent Lafforgue
- Roger Plymen
- Angel Román
- Maarten Solleveld
- Xiang Tang
- Robert Yuncken

#### Organising Committee:

- Alexandre Afgoustidis
- Pierre Clare

#### Scientific Committee:

- Tyrone Crisp
- Claire Debord
- Toshiyuki Kobayashi
- Hang Wang