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2025-T1 Representation Theory and Noncommutative Geometry

Tempered representations and K-theory

Institut Henri Poincaré
Amphithéâtre Hermite
11 rue Pierre et Marie Curie
75005 Paris

Representation Theory and Noncommutative Geometry

Workshop: Tempered representations and K-theory

February 24 to 28, 2025 - IHP, Paris


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

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