On L^p boundary representations for lattices in SL(2,R).

An interesting notion in Harmonic Analysis on groups is the notion of irreducibility of representations. A way to produce such a representation is to consider the action of a group on "its boundary" endowed with a nice class of quasi-invariant measures  and to construct the corresponding unitary representation on the L2 space of the boundary: this is the so-called quasi-regular representation also called Koopman representation. Somehow, irreducibility in this framework generalizes the notion of ergodicity of a group action. 

In this talk, I will discuss the notion for "L^p boundary representations" in the setting of lattices in SL(2,R). These representations are not anymore unitary representations but can be thought of as a deformation of the unitary one. The irreducibility of such representations rely on the study of a Riesz operator together with decay of matrix coefficients associated with boundary representations and some equisitribution results à la Roblin-Margulis.