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

TL;DR

This paper develops semiorthogonal decompositions for derived categories of representations of parabolic subgroups in split reductive group schemes over Z, leading to full exceptional collections on generalized flag varieties.

Contribution

It constructs G-linear semiorthogonal decompositions compatible with Bruhat order, extending foundational work and producing full exceptional collections over Z.

Findings

Semiorthogonal decompositions compatible with Bruhat order

Full exceptional collections on generalized flag schemes over Z

Extensions of foundational B-module results

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.