Higher level BGG reciprocity for current algebras

Abstract

We establish a higher-level analogue of the Bernstein--Gelfand--Gelfand (BGG) reciprocity for twisted current algebras at every positive integer level, extending the level-one results of Bennett et al. We show that these reciprocity relations form a hierarchy linking adjacent levels, thereby providing a systematic framework to relate representations across arbitrary levels by iteration. Our formulation of the BGG reciprocity naturally includes infinite-dimensional (thick) Demazure modules. Consequently, our main theorem provides the first effective means to analyze the module-theoretic structure of thick Demazure modules at any level. As an application, we realize level-restricted generalized Kostka polynomials as branching polynomials in the language of symmetric polynomials. The resulting level-dependent picture also naturally incorporates theta functions and modular forms. Furthermore, we prove that every finite-dimensional Demazure module admits a filtration by Demazure modules of the underlying simple Lie algebra, resolving a long-standing speculation in non-simply-laced types.