SU(d)--biinvariant random walks on SL(d,C) and their Euclidean counterparts

Abstract

We establish a deformation isomorphism between the algebras of $SU(d)$-biinvariant compactly supported measures on $SL(d,\comp)$ and $SU(d)$-conjugation invariant measures on the Euclidean space $H_d^0$ of all Hermitian $d\times d$-matrices with trace 0. This isomorphism concisely explains a close connection between the spectral problem for sums of Hermititan matrices on one hand and the singular spectral problem for products of matrices from $SL(d,\comp)$ on the other, which has recently been observed by Klyachko \cite{Kl2}. From this deformation we further obtain an explicit, probability preserving and isometric isomorphism between the Banach algebra of bounded $SU(d)$-biinvariant measures on $SL(d,\comp)$ and a certain (non-invariant) subalgebra of the bounded signed measures on $H_d^0$. We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of $SU(d)$-biinvariant random walks on $SL(d,\comp)$ in a simple way. Our construction relies on deformations of hypergroup convolutions and will be carried out in the general setting of complex semisimple Lie groups.