On the sheaf cohomology of some $p$-adic period domains with coefficients in certain line bundles

Abstract

Let $p$ be a prime. This papers aims at investigating sheaf cohomology of a broader class of $p$-adic period domains, other then the Drinfeld's upper half space (cf. \cite{O2}). Concretely, we let $\mathbf{G}$ be a split connected reductive group over $\mathbb{Q}_p$ and restrict our attention to the $p$-adic period domain $\mathscr{F}^{\mathrm{wa}}$ which parametrizes the weakly admissible filtrations on the trivial $\mathbf{G}$-isocrystal inside a complete flag variety $\mathscr{F}$. Then, we consider sheaf cohomology of $\mathscr{F}^{\mathrm{wa}}$ with coefficients in vector bundles which are induced by restriction of a homogeneous line bundle on $\mathscr{F}$ associated to a dominant weight of $\mathbf{G}$.