  After some normalization, the logarithms of the ordered singular values of
Brownian motions on $GL(N,\mathbb F)$ with $\mathbb F=\mathbb R, \mathbb C$
form Weyl-group invariant Heckman-Opdam processes on $\mathbb R^N$ of type
$A_{N-1}$. We use classical elementary formulas for the spherical functions of
$GL(N,\mathbb C)/SU(N)$ and the associated Euclidean spaces $H(N,\mathbb C)$ of
Hermitian matrices, and show that in the $GL(N,\mathbb C)$-case, these
processes can be also interpreted as ordered eigenvalues of Brownian motions on
$H(N,\mathbb C)$ 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 Browniam motion of Biane. We also show how this approach works
for the root systems $B_N, C_N, D_N$.
