Equidistribution of translates of a homogeneous measure on the Borel--Serre boundary

Abstract

Let G be a semisimple linear algebraic group defined over rational numbers, K be a maximal compact subgroup of its real points and {\Gamma} be an arithmetic lattice. One can associate a probability measure {\mu}(H) on {\Gamma}\G for each subgroup H of G defined over Q with no non-trivial rational characters. As G acts on {\Gamma}\G from the right, we can push-forward this measure by elements from G. By pushing down these measures to {\Gamma}\G/K, we call them homogeneous. It is a natural question to ask what are the possible weak-* limits of homogeneous measures. In the non-divergent case this has been answered by Eskin--Mozes--Shah. In the divergent case Daw--Gorodnik--Ullmo prove a refined version in some non-trivial compactifications of {\Gamma}\G/K for H generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general H with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on SL_n proved by geometry of numbers and a theorem of Kleinbock--Margulis; (2) relations between partial Borel--Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory.