Invariant Spaces of Holomorphic Functions on the Siegel Upper Half-Space

In this paper we consider the (ray) representations of the group $\mathrm{A}\mathrm{u}\mathrm{t}$ of biholomorphisms of the Siegel upper half-space $\mathcal{U}$ defined by $U_s(\phi )f=(f\circ {\phi }^{-1})(J{\phi }^{-1}{)}^{s\mathrm{/}2}$, $s\in \mathbb{R}$, and characterize the semi-Hilbert spaces $H$ of holomorphic functions on $\mathcal{U}$ satisfying the following assumptions: (a) $H$ is strongly decent; (b) $U_s$ induces a bounded ray representation of the group $\mathrm{A}\mathrm{f}\mathrm{f}$ of affine automorphisms of $\mathcal{U}$ in $H$. We use this description to improve the known characterization of the semi-Hilbert spaces of holomorphic functions on $\mathcal{U}$ satisfying (a) and (b) with $\mathrm{A}\mathrm{f}\mathrm{f}$ replaced by $\mathrm{A}\mathrm{u}\mathrm{t}$. In addition, we characterize the mean-periodic holomorphic functions on $\mathcal{U}$ under the representation $U_0$ of $\mathrm{A}\mathrm{f}\mathrm{f}$.