Abstract:

This talk is a report on a joint work with Wilberd van der Kallen based on https://arxiv.org/abs/2407.13653.
Given a split simply connected reductive group scheme G over a field k and a parabolic subscheme P of G, we
construct G-linear semiorthogonal decompositions of the bounded derived category of rational P-modules, which
are finitely generated over k. Each piece of such a semiorthogonal decomposition is equivalent, as a triangulated
category, to the bounded derived category of rational G-modules, which are finitely generated over k. The ensuing
semiorthogonal decompositions are compatible with the Bruhat order on cosets of the Weyl group of P in the Weyl
group of G. Our 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 generalized flag
varieties G/P.

As a corollary, one obtains full exceptional collections in the bounded derived categories of coherent sheaves
on generalized flag schemes G/P of Chevalley group schemes over the integers.