A Weyl Module Stratification of Integrable Representations (with an appendix by Ryosuke Kodera)

TL;DR

This paper introduces a filtration on integrable highest weight modules of affine Lie algebras, revealing a connection between Weyl modules, Kostka polynomials, and geometric realizations via Schubert manifolds.

Contribution

It constructs a new filtration on integrable modules, linking graded multiplicities to level-restricted Kostka polynomials and geometric models for Weyl modules.

Findings

Graded multiplicity of Weyl modules equals level-restricted Kostka polynomial.

Global Weyl modules of type ADE are realized via Schubert manifolds.

Provides a geometric interpretation of conformal coinvariants.

Abstract

We construct a filtration on integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each Weyl module there is given by a corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type $\mathsf{ADE}$ in terms of Schubert manifolds of thick affine Grassmanian, as predicted by Boris Feigin.