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

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 a 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, and Eskin-McMullen in the case of a group variety.

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