Representation Theory and Noncommutative Geometry, Paris

Tempiric representations, Connes-Kasparov, and pseudodifferential operators on symmetric spaces

5 févr. 2025, 13:45

55m

Amphitheater Darboux (IHP)

Orateur

Nigel Higson (Penn State University)

Description

Let $G$ be a real reductive group, connected for simplicity, with maximal compact subgroup $K$. The Connes-Kasparov isomorphism attaches a single parameter to most, but not all, of the components of the tempered dual of a real reductive group. That parameter is a shifted version of a highest weight for $K$, and every such parameter is attached to a unique component in the tempered dual. On the other hand, Vogan’s theory of minimal $K$-types associates to every component of the tempered dual of $G$ a finite collection of highest weights for $K$. These collections are disjoint, and every highest weight appears in one of them. Most of Vogan’s collections are singletons, but not all of them. The Connes-Kasparov and Vogan correspondences seem to be more similar than they are different, and in the online precursor to the current thematic program at IHP, Vogan asked whether they can be reconciled by adjusting the definition of the reduced C*-algebra of $G$? I shall discuss one answer, involving pseudodifferential operators, and some of the new questions that arise from that answer. This is joint work with Peter Debello.

https://sites.google.com/view/julgfest