Cuspidality in higher genus

We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus n to all the rational boundaries of lower degrees. This global operator reduces to Siegel's Φ operator when considering only the maximal standard cusps of degree n - 1. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus n > 1 and to bridge this new notion with the classical one found in the literature.