Subspaces of degree 2 Siegel modular forms (awaiting review)

The space vector-valued Siegel modular forms $M_{j,k}(\Gamma_2)$ can be decomposed as \[ M_{j,k}(\Gamma_2)=S_{j,k}(\Gamma_2)\oplus N_{j,k}(\Gamma_2) \] where $S_{j,k}(\Gamma_2)$ denotes the subspace of cusp forms on $\Gamma_2$ and $N_{j,k}(\Gamma_2)$ denotes its orthogonal complement with respect to the Petersson inner product. The previous decomposition is stable under the Hecke operators. The subspace $N_{j,k}(\Gamma_2)$ can be in turn decomposed into a direct sum, again stable under the Hecke operators. : \[ N_{j,k}(\Gamma_2)=E_{j,k}(\Gamma_2)\oplus KE_{j,k}(\Gamma_2) \] where $E_{j,k}(\Gamma_2)$ denotes the subspace of Eisenstein series. while $KE_{j,k}(\Gamma_2)$ denotes the subspace of Klingen-Eisenstein series. For $k>4$ and $j>0$, the subspace $KE_{j,k}(\Gamma_2)$ is isomorphic to the space of cusp forms of weight $k+j$ on $\SL(2,\Z)$.