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 $\mathbb Z$.