In this 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."
Problem B employs dynamical approaches, an idea of geometric quantization, and combinatorial techniques.
Both topics are designed for young students and non-experts.
© Toshiyuki Kobayashi