Harish-Chandra--Schwartz's algebras associated with discrete subgroups of Semisimple Lie groups

TL;DR

This paper investigates the Harish-Chandra--Schwartz algebra for discrete subgroups of semisimple Lie groups, establishing its density in the reduced $C^*$-algebra and relating their norms, with implications for groups acting on symmetric spaces.

Contribution

It proves the density of the Harish-Chandra--Schwartz space in the reduced $C^*$-algebra and establishes a norm inequality for groups acting on symmetric spaces.

Findings

Harish-Chandra--Schwartz space is dense in the reduced $C^*$-algebra.

The reduced $C^*$-norm is controlled by the Schwartz space norm.

The inequality holds for any discrete group acting on a Riemannian symmetric space.

Abstract

We prove that the Harish-Chandra--Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced $C^*$-algebra of the discrete subgroup. Then, we prove that for the reduced $C^*$-norm is controlled by the norm of the Harish-Chandra--Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.