Translate of horospheres and counting problems

TL;DR

This paper investigates the convergence of horospherical measures in semisimple Lie groups and applies the results to geometric counting problems and rational point enumeration on flag varieties.

Contribution

It provides necessary and sufficient conditions for the convergence of translated horospherical measures and applies these results to geometric and number-theoretic counting problems.

Findings

Characterization of convergence of horospherical measures

Determination of limiting measures when convergence occurs

New methods for counting rational points on flag varieties

Abstract

Let G be a semisimple Lie group without compact factors, \Gamma be an irreducible lattice in G. In the first part of the article we give the necessary and sufficient condition under which a sequence of translates of probability "horospherical measures" is convergent. And the limiting measure is also determined when it is convergent (see Theorems 1 and 2 for the precise statements). In the second part, two applications are presented. The first one is of geometric nature and the second one gives an alternative way to count the number of rational points on a flag variety.