2024/07/18 by Alexander Samokhin, Samokhin, Alexander, Wilberd van der Kallen +1 · 2 citations
Mathematics · #14F08 #14L30 #14M15 #19L47 #20G05 #20G10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2407.13653
openalex publication_date 2024/07/18 · openalex created_date 2024/09/09 · openalex updated_date 2026/07/28
Given a split simply connected and connected algebraic group scheme \mathbb G over \mathbb Z and a split parabolic subgroup scheme \mathbb P⊂ \mathbb G, this paper constructs semi-orthogonal decompositions of the bounded derived category Db(\mathrm rep( \mathbb P)) of noetherian representations of \mathbb P with each semi-orthogonal component being equivalent to the bounded derived category Db(\mathrm rep( \mathbb G)) of noetherian representations of \mathbb G. The semi-orthogonal components of those decompositions are stable under the monoidal action of Db(\mathrm rep( \mathbb G)) on Db(\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.