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

2005/12/12 by François Maucourant, F. Maucourant, Maucourant, F. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Representation Theory (math.RT) #math.DS #math.RT

paper · pdf · doi:10.48550/arxiv.math/0512240

openalex publication_date 2005/12/12 · arxiv created 2005/12/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 in obtained. This extends and precises anterior results of Duke, Rudnick and Sarnak, and Eskin-McMullen in the case of a group variety.

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