On the holomorphic differential forms of the Siegel modular variety (Q1264174)

The Siegel modular variety, \({\mathbb{H}}_ g/\Gamma_ g\), where \({\mathbb{H}}_ g\) is the Siegel upper-half space of degree g and \(\Gamma_ g=Sp(2g, {\mathbb{Z}})\), which operates transitively on \({\mathbb{H}}_ g\), is an open dense subset of a projective algebraic variety \(X_ g.\) If now \(X^ 0_ g\) is the regular part of \(X_ g\) and \(\tilde X_ g\) a smooth compactification of \(X^ 0_ g\) then \[ \Omega^{\nu}(\tilde X_ g)=\Omega^{\nu}(X^ 0_ g)=\Omega^{\nu}({\mathbb{H}}_ g/\Gamma_ g)=\Omega^{\nu}({\mathbb{H}}_ g)^{\Gamma_ g}\quad for\quad g\geq 3\quad and\quad \nu \leq g(g+1) \] where \(\Omega^{\nu}(\tilde X_ g)\) are the sections of the sheaf of alternating holomorphic differential forms on \(\tilde X_ g\) of degree \(\nu\) and \(\Omega^{\nu}({\mathbb{H}}_ g)^{\Gamma_ g}\) is the space of \(\Gamma_ g\)-invariant differential forms on \({\mathbb{H}}_ g\). The author proves that \[ \Omega^{[4\mu]}({\mathbb{H}}_ g)^{\Gamma_ g}\neq 0\quad for\quad g=4k+3,\quad 1\leq \mu \leq k\quad and\quad [\mu]:=\mu g-\mu (\mu -1). \] Before this, it was known that \[ \Omega^{[g-1]}({\mathbb{H}}_ g)^{\Gamma_ g}\neq 0,\quad for\quad g=4k+1\quad and\quad g\neq 13 \] and \[ \Omega^{[4]}({\mathbb{H}}_ g)^{\Gamma_ g}\neq 0,\quad for\quad g\geq 7\quad and\quad g\neq 9. \]