Fusion Product Structure of Demazure Modules

TL;DR

This paper proves that the fusion product of certain Demazure modules and irreducible modules for affine Lie algebras results in Demazure modules, advancing understanding of module structures and supporting conjectures in representation theory.

Contribution

It demonstrates that fusion products of specific Demazure modules and irreducible modules produce Demazure modules, providing new insights into module structures for affine Lie algebras.

Findings

Fusion product of an irreducible module and Demazure modules is a Demazure module.

Constructs the module structure of irreducible modules as semi-infinite fusion products.

Provides evidence for a conjecture on Schur positivity generalization.

Abstract

Let g be a finite-dimensional complex simple Lie algebra. Fix a non-negative integer l, we consider the set of dominant weights {\lambda} of g such that l{\Lambda}_0+{\lambda} is a dominant weight for the corresponding untwisted affine Kac-Moody algebra. For these special family of dominant weights, we show that the fusion product of an irreducible g-module V({\lambda}) and a finite number of special family of g-stable Demazure modules of level l (considered in [15] and [16]), for the current algebra g[t] associated to g, again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the g[t]-module structure of the irreducible module V(l{\Lambda}_0 + {\lambda}) as a semi-infinite fusion product of finite dimensional g[t]-modules as conjectured in [16]. As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see [7]).