The Frobenius morphism on flag varieties, I

TL;DR

This paper constructs special semiorthogonal decompositions for derived categories of flag varieties of rank 2 groups, enabling explicit descriptions of Frobenius pushforward bundles and full exceptional collections in certain cases.

Contribution

It introduces new semiorthogonal decompositions compatible with Bruhat order, over localized integers, and constructs explicit decompositions of Frobenius pushforwards for classical groups.

Findings

Decomposition of Frobenius pushforward bundles into indecomposables.

Construction of self-dual t-structures on derived categories.

Existence of full exceptional collections when p > h.

Abstract

In this paper, given a semisimple algebraic group $\bf G$ of rank 2, we construct a special semiorthogonal decomposition in the derived category of coherent sheaves on the flag variety ${\bf G}/{\bf B}$. These decompositions are defined over the localization ${\mathbb Z}_{\rm S}$, where $\rm S$ is the set of bad primes for $\bf G$, while their block structure is compatible with the Bruhat order on Schubert varieties. The non-standard $t$-structures on ${\rm D}^b({\bf G}/{\bf B})$ defined by these decompositions are self-dual with respect to the duality ${\mathcal RHom}_{{\bf G}/{\bf B}}(-,\omega _{{\bf G}/{\bf B}}^{\frac{1}{2}})$ given by the square root of the canonical sheaf of ${\bf G}/{\bf B}$. For the groups of classical type, this allows to construct an explicit decomposition of the higher Frobenii pushforward bundles ${\sf F}^n_{\ast}{\mathcal O}_{{\bf G}/{\bf B}}$ into a direct sum of indecomposable bundles. When $p>h$, the Coxeer number of the corresponding group, this set of indecomposable bundles forms a full exceptional collection in ${\rm D}^b({\bf G}/{\bf B})$ defined over ${\mathbb Z}_{\rm S}$.