The Kac–Wakimoto character formula for the general linear Lie superalgebra

Abstract

We prove the Kac–Wakimoto character formula for the general linear Lie superalgebra $\mathfrak{g}\mathfrak{l}\left(m|n\right)$, which was conjectured by Kac and Wakimoto in 1994. This formula specializes to the well-known Kac–Weyl character formula when the modules are typical and to the Weyl denominator identity when the module is trivial. We also prove a determinantal character formula for KW-modules.

In our proof, we demonstrate how to use odd reflections to move character formulas between the different sets of simple roots of a Lie superalgebra. As a consequence, we show that KW-modules are precisely Kostant modules, which were studied by Brundan and Stroppel, thus yielding a simple combinatorial defining condition for KW-modules and a classification of these modules.