Volumes for SL N (R) , the Selberg Integral and Random Lattices

Abstract

There is a natural left and right invariant Haar measure associated with the matrix groups GL N (R) and SL N (R) due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing a bound on the norm, or in the case of SL N (R) , by restricting to a fundamental domain. We compute the asymptotic volumes associated with the Haar measure for GL N (R) and SL N (R) matrices in the case that the singular values lie between R 1 and 1 / R 2 in the former, and that the 2-norm, or alternatively the Frobenius norm, is bounded by R in the latter. By a result of Duke, Rudnick and Sarnak, such asymptotic formulas in the case of SL N (R) imply an asymptotic counting fo..