Riesz operators and some spherical representations for hyperbolic groups

TL;DR

This paper introduces the Riesz operator for Gromov hyperbolic groups to study boundary representations, establishing asymptotic Schur relations and defining an analogue of complementary series with proven irreducibility.

Contribution

It presents the Riesz operator in hyperbolic groups, derives asymptotic Schur relations, and constructs an analogue of complementary series with irreducibility results.

Findings

Asymptotic Schur's relations established

Riesz operator plays role similar to Knapp-Stein intertwiner

Irreducibility of the defined complementary series

Abstract

We introduce the Riesz operator in the context of Gromov hyperbolic groups in order to investigate a one parameter family of non unitary boundary Hilbertian representations of hyperbolic groups. We prove asymptotic Schur's relations, the latter being the main result of this paper. Up to normalization, the Riesz operator plays the role in the context of hyperbolic groups of the Knapp-Stein intertwiner for complementary series for Lie groups. Assuming the positivity of the Riesz operator, we define an analogue of complementary series for hyperbolic groups and prove their irreducibility.