Kac–Wakimoto affine character formula for partially integrable tame modules

Let $\widehat{\mathfrak{g}}$ be an affine Lie superalgebra, let $L(\Lambda)$ be a partially integrable tame $\widehat{\mathfrak{g}}$-module, and let $L^\#$ be the relevant lattice of translations $t_\alpha$. Let $R^+$ and $\widehat{R}^+$ denote the finite and affine Weyl denominators, respectively, and let $\widehat{\rho}$, $\rho$, $\bar\Lambda$, and $\operatorname{ch}^+$ have their usual meanings. Kac–Wakimoto's affine character formula. One has

$$\widehat{R}^+\operatorname{ch}^+_{L(\Lambda)}=\sum_{\alpha\in L^\#}t_\alpha\left(e^{\widehat{\rho}+\Lambda-\rho-\bar\Lambda}R^+\operatorname{ch}^+_{L(\bar\Lambda)}\right).$$

The same formula holds for supercharacters after replacing $\widehat{R}^+$ and $R^+$ by $\widehat{R}^-$ and $R^-$ and inserting $\varepsilon_-(t_\alpha)$. This extends finite-dimensional character formulas to the affine setting for partially integrable tame modules.