Generators for a module of vector-valued Siegel modular forms of degree 2

Abstract

In this paper we will describe all vector-valued Siegel modular forms of degree 2 and weight ${\mathrm{Sym}}^6(\mathrm{St})\otimes {\det }^k(\mathrm{St})$ with $k$ odd. These vector-valued forms constitute a module over the ring of classical Siegel modular forms of degree 2 and even weight and this module turns out to be free. In order to find generators, we generalize certain Rankin-Cohen differential operators on triples of classical Siegel modular forms that were first considered by Ibukiyama and we find a Rankin-Cohen bracket on vector-valued Siegel modular forms.