Analysis on homogeneous spaces and operator algebras

C∗-algebras for real reductive symmetric spaces and K-theory

24 mars 2025, 11:00

1h

Amphithéâtre Hermite (Institut Henri Poincaré)

Orateur

Shintaro Nishikawa

Description

To a real reductive symmetric space $G/H$, we may associate one and often two $C^{\ast }$-algebras. The first corresponds to the support of the Plancherel measure for the regular representation on $L^2(G/H)$, while the second corresponds to the subset of the support consisting of irreducible representations that admit $H$-fixed distributions. The latter $C^{\ast }$-algebra exists for favorable classes of symmetric spaces.

We investigate the structure and properties of these $C^{\ast }$-algebras, leveraging the established Plancherel theory for $G/H$: the Plancherel decomposition developed by Erik van den Ban, Patrick Delorme, and Henrik Schlichtkrull, as well as the theory of discrete series representations, as studied by Flensted-Jensen, Oshima--Matsuki, Schlichtkrull, and others. We also discuss subtle aspects that seem not immediate from these results.

This is joint work with A. Afgoustidis, N. Higson, P. Hochs, and Y. Song.