Cuspidal components of Siegel modular forms for large discrete series representations of ${\mathrm{S}\mathrm{p}}_4(\mathbb{R})$

In this paper, we consider automorphic forms on ${\mathrm{S}\mathrm{p}}_4({\mathbb{A}}_{\mathbb{Q}})$ which generate large discrete series representations of ${\mathrm{S}\mathrm{p}}_4(\mathbb{R})$ as $({\mathfrak{s}\mathfrak{p}}_4(\mathbb{R}),K_{\mathrm{\infty }})$-modules. We determine the cuspidal components and the structure of the space of such automorphic forms.