Siegel cusp form (awaiting review)

We have the Siegel phi map from degree $g$ down to degree $g-1$: $$\Phi: M_k^{(g)}\to M_k^{(g-1)}$$ defined by $$\Phi(f)(\Omega) = \lim_{t\to\infty} f\left( \begin{pmatrix}\Omega&0\\0&it\end{pmatrix}\right).$$

We say $f$ is a cusp form if for all $\gamma\in \Sp(2g,\Q)$, we have $$\Phi(f|\gamma)=0$$ where $|$ is the slash operator.