Appears in collection : Symmetry in Geometry and Analysis

We consider the regular representation on the Hilbert space of square integrable functions on a homogeneous space. We address the question about when this unitary representation is tempered.

Surprisingly, this question triggers a new and rich connection of representation theory with some topics from algebra, topology, analysis, and geometry.

We shall explore such features through the new methods introduced in the proof of the temperedness criterion and a classification theory.

The talk will be based on a series of joint papers with Y. Benoist.