Tempered representations and K-theory

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
• Yves Benoist
• Jacob Bradd
• Peter Hochs
• Roger Plymen
• Angel Romàn
• Maarten Solleveld
• Robert Yuncken

Organising Committee:

• Alexandre Afgoustidis 
• Pierre Clare 

Scientific Committee:

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

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