Representations and Characters: Revisiting Some Aspects of the Works of Harish-Chandra and Weil

Overview

This workshop is devoted to the theory of representations of reductive groups over local fields.

There are two main types of mutually intertwined problems in Representation Theory: to classify all irreducible (unitary) representations and to decompose a representation into irreducible ones. Both problems have important applications to other areas (e.g. Number Theory, Harmonic Analysis, Mathematical Physics), but despite the significant progress made, they are still wide open. Many different approaches can be used to address these problems: an important one consists in studying representations through their invariants.

Characters and matrix coefficients are among the most important analytic invariants of representations. The modern theory of characters started in the work of Harish-Chandra. They are traces of integrated versions of the representations and are in bijective correspondence with irreducible representations. Just to give some examples of their role in representation theory, Harish-Chandra’s description of discrete series representations was based on the construction of their characters, and characters relate to Fourier transforms of coadjoint orbits (Rossmann, Murnaghan). On the other hand, matrix coefficients allow to locate the representations inside the admissible dual and provide a link of representation theory to many aspects of classical analysis.

This workshop puts special emphasis on characters and matrix coefficients, in connection with the recent developments of representation theory, especially those related to the theta correspondence for the Weil representation or those related to general branching problems of representations.

The meeting aims at providing a successful research opportunity to its participants, to exchange on recent developments, share points of view and ideas and strengthen or promote collaborations. To make the more specialized talks accessible to all the participants, some experts will give a few survey talks during the first week. Moreover, two afternoons in the second week will be devoted to talks of young participants.