I am studying the paper "The Main Conjecture for $GL(2)$" by Skinner and Urban (available here). In this paper, in Section 5.5.3, the authors define a filtration $M_{\underline k}^{n,q}$ on the space of Siegel modular forms of weight $\underline k$ and genus $n$ (to be precise they work with a unitary group rather than the symplectic group, but this does not matter). For $q=0$ we obtain the space of cuspidal forms.
Let $\bar X$ be a toroidal compactification of the relevant Shimura variety $X$ and let $\pi \colon \bar X \to X^\ast$ be the canonical morphism to the minimal compactification. In the sentence after the definition of $M_{\underline k}^{n,q}$ they give an equivalent definition of $M_{\underline k}^{n,q}$. In the case $q=0$ this equivalent definition is $$ H^0(X^\ast, \pi_\ast \omega_{\underline k} \otimes I^0), $$ $\omega_{\underline k}$ is the usual sheaf and $I^0$ is the sheaf given by the boundary of $X^\ast$.
My question is: why are these two definitions equivalent?
Cuspidal forms are usually defined using the toroidal compactification, so it is not clear a priori that we can use the minimal compactification to define them.
More generally, let us denote by $J$ the sheaf of ideals on $\bar X$ given by the boundary. Is it true that $$ \pi_\ast ( \omega_{\underline k} \otimes J ) = \pi_\ast \omega_{\underline k} \otimes I \; ? $$ This would of course imply the above equality, and note that $$ \pi_\ast J = I. $$ (This is proved in the paper "Cohomological vanishing on Siegel modular varieties and applications to lifting Siegel modular forms" by Ghitza and Mullane).
Can someone help me? Thank you!
