Analysis on simple Lie groups and lattices

Abstract

We present a simple tool to perform analysis with groups such as ${\mathrm{SL}}_n(\mathbf{R})$ and ${\mathrm{SL}}_n(\mathbf{Z})$, that has been introduced by Vincent Lafforgue in his study of nonunitary representations and strong property (T), in connection with the Baum–Connes conjecture. It has been later applied in various contexts: operator algebras, Fourier analysis, geometry of Banach spaces, and dynamics. The idea is to first restrict to compact subgroups and then exploit how they sit inside the whole group.