Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices

Abstract

We show that Haar measures of connected semisimple groups, embedded via a representation into a matrix space, have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled. This is still true if the Haar measure of the semisimple group is replaced by the Haar measure of an irreducible lattice of the group, and the asymptotic measure is the same. In the case of an almost simple group of rank greater than $2$, a remainder term is also obtained. This extends and makes precise anterior results of Duke, Rudnick, and Sarnak [DRS] and Eskin and McMullen [EM] in the case of a group variety