2025/11/23 by Riche, Simon, Situ, Quan
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics
paper · doi:10.48550/arxiv.2511.18518
Let G be a connected reductive algebraic group over an algebraically closed field of positive characteristic, \mathfrakg be its Lie algebra, and B be a Borel subgroup. We prove a formula for the dimensions of extension groups, in the principal block of the category of strongly B-equivariant \mathfrakg-modules (also called modular category O), from a simple object to a costandard object, under the assumption that Lusztig's conjecture holds (which is known in large characteristic). The answer is given by a coefficient of a periodic Kazhdan--Lusztig polynomial associated with the corresponding affine Weyl group. Among other things, the proof uses a torus-equivariant version of the Koszul duality for \mathfrakg-modules constructed by the first author.