The Frobenius morphism on flag varieties, I

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 ${\mathcal RHom}_{{\bf G}/{\bf B}}(-,ω_{{\bf G}/{\bf B}}^{\frac{1}{2}})$ 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 ${\mathbb Z}_{\rm S}$.