Tempered representations and K-theory

kt.k-theory-and-homology oa.operator-algebras rt.representation-theory
Start Date
End Date
Institut Henri Poincare 
Meeting Type
Contact Name
Haluk Sengun 
2024-04-16 18:52:10 
2024-04-16 19:05:31 


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 and 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 GxG, and more generally of the tempered spectrum of symmetric spaces.


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