Demazure Character Formulas for Generalized Kac--Moody Algebras

TL;DR

This paper extends Demazure character formulas to generalized Kac--Moody algebras by constructing specific submodules of highest weight modules and providing a new character formula that generalizes existing results.

Contribution

It introduces a family of submodules for generalized Kac--Moody algebras and derives a generalized Demazure character formula for these modules.

Findings

Modules are spanned by their global basis.

Derived a character formula generalizing Demazure formulas.

Applicable to a broad class of generalized Kac--Moody algebras.

Abstract

For a dominant integral weight $lambda $, we introduce a family of $U_q ^+ (mathfrak{g})$-submodules $V_w (lambda)$ of the irreducible highest weight $U_q (mathfrak{g})$-module $V(lambda)$ of highest weight $lambda $ for a generalized Kac--Moody algebra $mathfrak{g}$. We prove that the module $V_w (lambda)$ is spanned by its global basis, and then give a character formula for $V_w (lambda)$, which generalizes the Demazure character formula for ordinary Kac--Moody algebras.