Rank one summands of Frobenius pushforwards of line bundles on G/P

Abstract

Let $X=G/P$ be a partial flag variety, where $G$ is a semi-simple, simply connected algebraic group defined over an algebraically closed field $K$ of positive characteristic. Let $\mathsf{F}:X\to X$ be the absolute Frobenius morphism. Given a line bundle $\mathcal{L}$ on $X$ and an integer $r\ge 1$, we describe all line bundles that are direct summands of the pushforward ${\mathsf{F}}_{\ast }^r\mathcal{L}$. For $\mathcal{L}$ corresponding to a dominant weight, we also compute, for $r$ sufficiently large, the multiplicity of ${\mathcal{O}}_X$ as a summand of ${\mathsf{F}}_{\ast }^r\mathcal{L}$. As an application we answer a question of Gros-Kaneda about the multiplcity of $\mathcal{L}(-\rho )$ as a direct summand of ${\mathsf{F}}_{\ast }{\mathcal{O}}_{G/B}$.

version 2: Minor corrections, fixed a mistake in Example 6.3. version 1: 9 pages, comments welcome