Invariant Spaces of Holomorphic Functions on Symmetric Siegel domains

Abstract: In this paper we consider a symmetric Siegel domain $D$ and some natural representations of the M\"obius group $G$ of its biholomorphisms and of the group $\mathrm{Aff}$ 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 M\"obius-invariant Semi-Hilbert spaces on general domains.