Highest weight category structures on $rep(B)$ and full exceptional collections on generalized flag varieties over $\mathbb{Z}$

Abstract

Given a split simply connected and connected algebraic group scheme $\mathbb{G}$ over $\mathbb{Z}$ and a split parabolic subgroup scheme $\mathbb{P}\subset \mathbb{G}$, this paper constructs semi-orthogonal decompositions of the bounded derived category $D^b(\mathrm{rep}(\mathbb{P}))$ of noetherian representations of $\mathbb{P}$ with each semi-orthogonal component being equivalent to the bounded derived category $D^b(\mathrm{rep}(\mathbb{G}))$ of noetherian representations of $\mathbb{G}$. The semi-orthogonal components of those decompositions are stable under the monoidal action of $D^b(\mathrm{rep}(\mathbb{G}))$ on $D^b(\mathrm{rep}(\mathbb{P}))$. The decompositions depend on an arbitrarily chosen total order on the Weyl group that refines the Bruhat order. The semi-orthogonal decompositions are also compatible with the Bruhat order on cosets of the Weyl group of $\mathbb{P}$ in the Weyl group of $\mathbb{G}$. Their construction builds upon the foundational results on $\mathbb{B}$-modules from the works of Mathieu, Polo, and van der Kallen, and upon properties of the Steinberg basis of the $\mathbb{T}$-equivariant $K$-theory of $\mathbb{G}/\mathbb{B}$. As a corollary, we obtain full exceptional collections in the bounded derived category of coherent sheaves on generalized flag schemes $\mathbb{G}/\mathbb{P}$ over $\mathbb{Z}$.

Exposition revised