Higher level BGG reciprocity for current algebras

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.

66pp, v4: Substantially revised. Streamlined exposition and added explanations at many places. Added Section 11 on (not affine) highest weight structures, and Appendix A on higher level $q$-Cauchy kernel identities