Equidistribution of translates of a homogeneous measure on the Borel--Serre boundary
arXiv:2012.07468
Abstract
Let G be a semisimple linear algebraic group defined over rational numbers, K be a maximal compact subgroup of its real points and Γ be an arithmetic lattice. One can associate a probability measure μ(H) on Γ\G for each subgroup H of G defined over Q with no non-trivial rational characters. As G acts on Γ\G from the right, we can push-forward this measure by elements from G. By pushing down these measures to Γ\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 Γ\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.
Correct typos; 19 pages