Higher level BGG reciprocity for current algebras

TL;DR

This paper develops a higher-level analogue of BGG reciprocity for twisted current algebras, connecting it with theta functions, modular forms, and providing new insights into Demazure modules and Kostka polynomials.

Contribution

It introduces a higher-level BGG reciprocity for twisted current algebras, extending the classical case and linking it with advanced mathematical structures.

Findings

Established higher-level BGG reciprocity for twisted current algebras.

Connected Demazure modules' branching properties with symmetric polynomials.

Provided a new interpretation of level-restricted generalized Kostka polynomials.

Abstract

We exhibit a higher-level analogue of the Bernstein-Gelfand-Gelfand (BGG) reciprocity for twisted current algebras for each positive integer, which recovers the original one (established by Bennett, Berenstein, Chari, Ion, Khoroshkin, Loktev, and Manning) as its level-one case. This work brings theta functions and modular forms into the theory of symmetric polynomials. Furthermore, we establish branching properties for both versions of Demazure modules and provide a new interpretation of level-restricted generalized Kostka polynomials in terms of symmetric polynomials.