An explicit formula for free multiplicative Brownian motions via spherical functions

Abstract

After some normalization, the logarithms of the ordered singular values of Brownian motions on $GL(N,F)$ with $F=R,ℂ$ form Weyl-group invariant Heckman–Opdam processes on $R^N$ of type $A_{N-1}$. We use classical elementary formulas for the spherical functions of $GL(N,ℂ)/SU\left(N\right)$ and the associated Euclidean spaces $H(N,ℂ)$ of Hermitian matrices, and show that in the $GL(N,ℂ)$-case, these processes can be also interpreted as ordered eigenvalues of Brownian motions on $H(N,ℂ)$ with particular drifts. This leads to an explicit description for the free limits for the associated empirical processes for $N\to \infty$ where these limits are independent from the parameter $k$ of the Heckman–Opdam processes. In particular we get new formulas for the distributions of the free multiplicative Brownian motion of Biane. We also show how this approach works for the root systems $B_N,C_N,D_N$.