Relating Siegel cusp forms to Siegel-Maaß forms

Abstract

In this paper we generalize a well-known isomorphism between the space of cusp forms of weight $k$ for a Fuchsian subgroup of the first kind $\Gamma \subset {\mathrm{SL}}_2(\mathbb{R})$ and the space of certain Maaß forms of weight $k$ for $\Gamma$ to an isomorphism between the space of Siegel cusp forms of weight $k$ for a subgroup $\Gamma \subset {\mathrm{Sp}}_n(\mathbb{R})$, which is commensurable with the Siegel modular group ${\mathrm{Sp}}_n(\mathbb{Z})$, and a suitable space of Siegel-Maaß forms of weight $k$ for $\Gamma$.