Ornstein-Uhlenbeck operators and holomorphic representations for a class of bounded symmetric domains

Abstract

Let G be the group of holomorphic automorphisms of a bounded symmetric domain D. We discuss the correspondence between unitarizing measures for scalar-valued holomorphic discrete series representations of G and Ornstein-Uhlenbeck operators on D. We complement the results obtained for the domains in [1] and [6] by showing that for Cartan domain II of n×n symmetric complex matrices and domain III of skew-symmetric complex matrices Z such that ZZ⁎<I, the vector fields in the infinitesimal holomorphic representation are the entries of Hua matrices. Similar results stay valid for some Kähler submanifolds M of D, assuming that the entries of the matrices in M depend linearly on fixed independent entries and M is isomorphic to a product of Cartan domains.