Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras

TL;DR

This paper generalizes Kostka-Foulkes polynomials to symmetrizable Kac-Moody algebras, proving their equivalence with Lusztig's t-analog of weight multiplicities and deriving explicit formulas for affine cases.

Contribution

It extends classical polynomials to a broader algebraic setting and establishes their connection with Lusztig's t-analogs, including explicit formulas for affine Kac-Moody algebras.

Findings

Kostka-Foulkes polynomials are generalized to all symmetrizable Kac-Moody algebras.

Proved the polynomials coincide with Lusztig's t-analog of weight multiplicities.

Derived explicit product formulas for affine Kac-Moody algebra string functions.

Abstract

We introduce a generalization of the classical Hall-Littlewood and Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove that these Kostka-Foulkes polynomials coincide with the natural generalization of Lusztig's $t$-analog of weight multiplicities, thereby extending a theorem of Kato. For $g$ an affine Kac-Moody algebra, we define $t$-analogs of string functions and use Cherednik's constant term identities to derive explicit product expressions for them.