Unitarising measures for the representations of affine group and associated invariant operators

Let G denote the group of affine transformations on Cn, or one of its subgroups such as the translation group or the Euclidean motion group. In this paper, we study unitarisability of holomorphic representations T of G into the space of square integrable C-valued functions with respect to a positive real measure μ with support in a domain D in Cn. Moreover, we investigate the infinitesimal representation approach of a unitary representation (Tg,μ) on the Lie algebra G of G in order to determine a class of second-order differential operators (called of Ornstein–Uhlenbeck type) invariant with respect to the unitarising measures μ.