SU(d)--biinvariant random walks on SL(d,C) and their Euclidean counterparts
arXiv:math/0309361
Abstract
We establish a deformation isomorphism between the algebras of -biinvariant compactly supported measures on $SL(d,\comp)$ and -conjugation invariant measures on the Euclidean space of all Hermitian -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 -biinvariant measures on $SL(d,\comp)$ and a certain (non-invariant) subalgebra of the bounded signed measures on . We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of -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.
18 pages