Rank one summands of Frobenius pushforwards of line bundles on G/P
Feliks Rączka
TL;DR
The paper addresses the decomposition of Frobenius pushforwards of line bundles on partial flag varieties $X=G/P$ in positive characteristic. It provides a precise criterion to determine which line bundles $\mathscr{L}^{P}(\lambda)$ can appear as rank-one direct summands of $\mathsf{F}_{*}^{r}\mathscr{L}^{P}(\mu)$, expressed through an explicit inequality involving $\mu$, $\lambda$, and the root data $\rho_{P}$. It further develops a method to compute the multiplicity of the structure sheaf $\mathscr{O}_{X}$ for dominant weights within a suitable $p^{r}$-range, showing that $H^{0}(X,\mathscr{L}^{P}(\mu))\otimes\mathscr{O}_{X}$ appears as a summand for large $r$. The work relies on the Frobenius-kernel framework and key injectivity results in the style of Haboush, and it connects to classical questions about $F$-splitting, $D$-affinity, and the Gros–Kaneda problem for summands in $\mathsf{F}_{*}\mathscr{O}_{G/B}$, while also providing a concrete description in the full flag case $X=G/B$.
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 $\mathsf{F}\colon X\to X$ be the absolute Frobenius morphism. Given a line bundle $\mathscr{L}$ on $X$ and an integer $r\geq1$, we describe all line bundles that are direct summands of the pushforward $\mathsf{F}_{*}^{r}\mathscr{L}$. For $\mathscr{L}$ corresponding to a dominant weight, we also compute, for $r$ sufficiently large, the multiplicity of $\mathscr{O}_{X}$ as a summand of $\mathsf{F}_{*}^{r}\mathscr{L}$. As an application we answer a question of Gros-Kaneda about the multiplcity of $\mathscr{L}(-ρ)$ as a direct summand of $\mathsf{F}_{*}\mathscr{O}_{G/B}$.