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

In this paper we consider the (ray) representations of group $\mathrm{Aut}$ biholomorphisms 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 semi-Hilbert spaces $H$ holomorphic functions on satisfying following assumptions: (a) is strongly decent; (b) $U_s$ induces a bounded ray representation $\mathrm{Aff}$ affine automorphisms in $H$. We use description to improve known characterization with replaced $\mathrm{Aut}$. addition, mean-periodic under $U_0$ $\mathrm{Aff}$.