Vector-valued Siegel modular forms (awaiting review)

Let $\rho: \GL_g(\mathbb{C})\to \GL(V),$ where $V$ is a finite-dimensional complex vector space, be an irreducible representation. A (vector valued) Siegel modular form of degree $g$, weight $\rho$, with respect to a subgroup $\Gamma\subset \GSp(2g,\Q)$ is a complex analytic function $F:\Bbb{H}_g\to V$ such that for any $\gamma=\begin{pmatrix} A&B\\C&D\end{pmatrix}$ we have $F((AZ+B)(CZ+D)^{-1})=\left(\rho(C Z+D)\right)F(Z)$. If $g=1$, one must also impose the following condition: there exists $\epsilon>0$ such that $$|F(x+iy)|\ll y^\epsilon\quad\text{for }y\geq 1.$$

Consider the case $g=2$. Let $std$ denote the standard representation of $\GL_2(\C)$, i.e. $std$ is the identity map from $\GL_2(\C)$ to $\GL(\C^2)$. To any pair of integers $(k,j)$ one associates the representation $\rho_{k,j}:=Sym^{j}(std)\otimes\det(std)^{k}$ of $\GL_2(\C)$. This enumerates all the irreducible representations of $\GL_2(\C)$.