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,C) and SU(d)-conjugation invariant measures on the Euclidean space Hd0 of all Hermitian dxd-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,C) on the other, which has recently been observed by Klyachko [13]. 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,C) and a certain (non-invariant) subalgebra of the bounded signed measures on Hd0. We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of SU(d)-biinvariant random walks on SL(d,C) 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.