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

In this paper we consider the (ray) representations of the group $\mathrm{Aut}$ of biholomorphisms of the Siegel upper half-space $\mathcal U$ defined by $U_s(\varphi) f=(f\circ \varphi^{-1}) (J \varphi^{-1})^{s/2}$, $s\in\mathbb R$, 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 $\mathrm{Aff}$ 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 $\mathrm{Aff}$ replaced by $\mathrm{Aut}$. In addition, we characterize the mean-periodic holomorphic functions on $\mathcal U$ under the representation $U_0$ of $\mathrm{Aff}$.