Analysis on simple Lie groups and lattices

TL;DR

This paper introduces a straightforward analytical tool for studying simple Lie groups and lattices, leveraging subgroup restrictions to explore properties like the Baum-Connes conjecture and strong property (T).

Contribution

It presents a simplified method for analysis of groups such as SL(n,R) and SL(n,Z), building on Lafforgue's approach to non-unitary representations.

Findings

Applicable to operator algebras and Fourier analysis

Useful in geometry of Banach spaces and dynamics

Facilitates understanding of subgroup embeddings within groups

Abstract

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