Vector-valued Siegel modular forms of degree 2 (awaiting review)

The irreducible representations of $\GL(2,\C)$ are given by $\rho_{k,j}:=\det(St)^{\otimes k}\otimes \text{Sym}^{j}(St)$ where $St$ stands for the standard representation of $\GL(2,\C)$ and $(k,j)\in \Z\times\Z_{\geqslant 0}$. The space of Siegel modualr forms of weight $\rho_{k,j}$ on any arithmetic subsgroup $\Gamma$ of $\GSp(4,\Q)$ will be denoted by $M_{k,j}(\Gamma)$.