We study the quasi-regular representation of a semisimple Lie group on the Hilbert space of square-integrable functions on a homogeneous space. A program initiated by Benoist–Kobayashi is to determine when this unitary representation is tempered, or equivalently, when it achieves the “maximal spectral gap”.

In a joint work with Yves Benoist, we establish a geometric criterion which applies to every homogeneous space of a semisimple Lie group, by relating the spectral gap to decay of matrix coefficients and growth of subgroups. Our result unifies and extends earlier works of Elstrodt, Patterson, Sullivan, Shalom, Benoist–Kobayashi, Edwards–Oh, and Lutsko–Weich–Wolf in various cases.

The talk will include an introduction to induced representations and explain some key ingredients underlying the proof.