Relating Siegel cusp forms to Siegel-Maa{\ss} forms

TL;DR

This paper extends a classical isomorphism linking elliptic cusp forms and Maa{ ext}ss forms to higher-dimensional Siegel modular forms and Siegel-Maa{ ext}ss forms, broadening the understanding of their relationship.

Contribution

It introduces a new isomorphism between Siegel cusp forms and Siegel-Maa{ ext}ss forms for certain subgroups, generalizing known results from the classical setting.

Findings

Established an isomorphism for Siegel cusp and Maa{ ext}ss} forms

Generalized classical results to higher dimensions

Provides a framework for future research in automorphic 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 $\Gamma$.