Relating Siegel cusp forms to Siegel-Maa{\ss} 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 Maa{\ss} 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-Maa{\ss} forms of weight $k$ for $\Gamma$.