Basic Questions in Group-Theoretic Analysis on Manifolds. Introductory school at CIRM (Marseille): Methods in representation theory and operator algebras. CIRM, Marseille, France, 6-10 January 2025.

The thematic trimester "Representation Theory and Noncommutative Geometry"
Organizers: Alexandre Afgoustidis, Anne-Marie Aubert, Pierre Clare, Haluk Şengün.

In this survey lecture, I will provide a contemporary overview of recent developments in representation theory, focusing on two fundamental questions about global analysis on manifolds $X$ acted upon by real reductive groups $G$.

Problem A: Does the group $G$ sufficiently control the space of functions on $X$?

Problem B: What can we say about the "spectrum" of $L^2(X)$?

We will discuss recent progress, with particular emphasis on the " multiplicity" for Problem A and the "decay of matrix coefficients" for Problem B. Problem A concerns induction, as well as its dual concept, restriction. The latter asks how an irreducible representation of a group behaves when restricted to a subgroup, often referred to as “ branching problems.”

The branching probles will be explored in the mini-course series at IHP from January 13-17, which will focus on Problem A, while Problem B, which employs dynamical approaches, will be discussed in an independent mini-course series at IHP from February 17-21. Both mini-courses are designed for young students and non-experts.

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© Toshiyuki Kobayashi