Volumes for $${mathrm{SL}}_N({mathbb {R}})$$ SL N ( R ) , the Selberg Integral and Random Lattices

There is a natural left and right invariant haar measure associated with the matrix groups gl $${}_n(mathbb {r})$$ and sl $${}_n(mathbb {r})$$ 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 $${}_n(mathbb {r})$$ , by restricting to a fundamental domain. we compute the asymptotic volumes associated with the haar measure for gl $${}_n(mathbb {r})$$ and sl $${}_n(mathbb {r})$$ 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 $${}_n(mathbb {r})$$ imply an asymptotic counting formula for matrices in sl $${}_n(mathbb {z})$$ . we discuss too the sampling of sl $${}_n(mathbb {r})$$ matrices from the truncated sets. by then using lattice reduction to a fundamental domain, we obtain histograms approximating the probability density functions of the lengths and pairwise angles of shortest length bases vectors in the case $$n=2$$ and 3, or equivalently of shortest linearly independent vectors in the corresponding random lattice. in the case $$n=2$$ these distributions are evaluated explicitly.