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

Abstract: Given a split reductive Chevalley group scheme G over Z and a parabolic subgroup scheme P in G, this paper constructs G-linear semiorthogonal decompositions of the bounded derived category of noetherian representations of P with each semiorthogonal component being equivalent to the bounded derived category of noetherian representations of G. The G-linear semiorthogonal decompositions in question are compatible with the Bruhat order on cosets of the Weyl group of P in the Weyl group of G. Their construction builds upon the foundational results on B-modules from the works of Mathieu, Polo, and van der Kallen, and upon properties of the Steinberg basis of the T-equivariant K-theory of G/B. As a corollary, we obtain full exceptional collections in the bounded derived category of coherent sheaves on generalized flag schemes G/P over Z.