Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. exploit this to compute analytically marginals probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for case space Hermitian matrices, where have already appeared physics literature. For when is $m \times m$ positive definite we how efficiently by evaluating Pfaffians at specific values $m$. Equivalently, obtain same result constructing skew polynomials with regards log-normal weight function (skew polynomials). Other spaces studied and type obtained quaternionic case. Moreover, functions a particular diffusion reproducing kernels Karlin-McGregor type, describing non-intersecting Brownian motions, which also processes Weyl chamber Lie groups.