Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles

TL;DR

This paper generalizes Lusztig's q-analogues of weight multiplicity using weights of B-submodules in finite-dimensional G-modules, linking algebraic group representations with cohomology of line bundles on homogeneous vector bundles.

Contribution

It introduces a new framework for Kostka-Foulkes polynomials by extending Lusztig's q-analogues to broader settings involving B-submodules.

Findings

Established a generalized theory of Kostka-Foulkes polynomials

Connected representation theory with cohomology of line bundles

Provided new combinatorial formulas for weight multiplicities

Abstract

Let $G$ be a semisimple algebraic group and $B$ a Borel subgroup. We consider generalisations of Lusztig's q-analogues of weight multiplicity, where the set of positive roots is replaced with the multiset of weights of a $B$-submodule of an arbitrary finite-dimensional $G$-module.