Invariant spaces of holomorphic functions on the Siegel upper half-space

n this paper we consider the (ray) representations of the group Aut of biholomorphisms of the Siegel upper half-space U defined by Us(φ)f=(f∘φ−1)(Jφ−1)s/2, s∈R, and characterize the semi-Hilbert spaces H of holomorphic functions on U satisfying the following assumptions: (a) H is strongly decent; (b) Us induces a bounded ray representation of the group Aff of affine automorphisms of U in H. We use this description to improve the known characterization of the semi-Hilbert spaces of holomorphic functions on U satisfying (a) and (b) with Aff replaced by Aut. In addition, we characterize the mean-periodic holomorphic functions on U under the representation U0 of Aff.