Vector valued Siegel modular forms of degree 2 taking values in three-dimensional space (awaiting review)

The module of vector-valued Siegel modular forms of degree 2, taking values in a three-dimensional space, with respect to the full modular group.

Let $\psi_4$, $\psi_6$, $\chi_{10}$, $\chi_{12}$ be generators of $M(\Sp(4,\Z))$, the Siegel modular forms of degree 2 with respect to the full modular group. We will write $\Gamma_2=\Sp(4,\Z)$ for short. For $f\in M_k(\Gamma_2)$ and $g\in M_j(\Gamma_2)$, define the Satoh bracket [MR:0816719] $$[f,g] = \frac 1{2\pi i}\left(\frac1k g\frac{d}{dZ}f-\frac1j f\frac{d}{dZ}g\right).$$ Then $[f,g]\in M_{k,2}(\Gamma_2)$, and for even integers $k$, $$ \begin{aligned} M_{k,2}(\Gamma_2) =& M_{k-10}(\Gamma_2) [\psi_4,\psi_6] \oplus M_{k-14}(\Gamma_2) [\psi_4,\chi_{10}] \\ &\oplus M_{k-16}(\Gamma_2) [\psi_4,\chi_{12}] \oplus V_{k-16}(\Gamma_2) [\psi_6,\chi_{10}]\\ &\oplus V_{k-18}(\Gamma_2) [\psi_6,\chi_{12}]\oplus V_{k-22}(\Gamma_2) [\chi_{10},\chi_{12}], \end{aligned} $$ where $$V_k(\Gamma_2)=M_k(\Gamma_2)\cap \C[\psi_6,\chi_{10},\chi_{12}],$$ $$W_k(\Gamma_2)=M_k(\Gamma_2)\cap \C[\chi_{10},\chi_{12}].$$

We write - $A=\psi_4$, - $B=\psi_6$, - $C=\chi_{10}$, - $D=\chi_{12}$, for short in the modular form sample pages.

Dimension tables are taken from Tsushima [MR:0816719].