  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 $\mathrm{Aff}$.
