Invariant Spaces of Holomorphic Functions on Symmetric Siegel domains

In this paper we consider a symmetric Siegel domain $D$ and some natural representations of the Möbius group $G$ of its biholomorphisms and of the group $\mathrm{A}\mathrm{f}\mathrm{f}$ of its affine biholomorphisms. We provide a complete classification of the affinely-invariant semi-Hilbert spaces (satisfying some natural additional assumptions) on tube domains, and improve the classification of Möbius-invariant Semi-Hilbert spaces on general domains.