Generalized Kac–Wakimoto conjecture for modified dimensions

Let ${\mathfrak{g}}$ be a basic classical Lie superalgebra, let $\mathcal{F}$ be the category of finite-dimensional integrable ${\mathfrak{g}}$-supermodules, and let $I_J$ be the full subcategory of supermodules that occur as direct summands of $J\otimes X$ for some $X$ in $\mathcal{F}$. A simple supermodule is ambidextrous if it admits a nonzero ambidextrous trace, and let $\operatorname{\mathsf{d}}_J$ denote the modified dimension function on $I_J$.

Generalized Kac–Wakimoto conjecture. Every simple ${\mathfrak{g}}$-supermodule $J$ is ambidextrous. If $L$ is another simple supermodule satisfying

$$\operatorname{atyp}(L)\leq\operatorname{atyp}(J),$$

then $L$ is an object of $I_J$, and

$$\operatorname{atyp}(L)=\operatorname{atyp}(J)\quad\text{if and only if}\quad \operatorname{\mathsf{d}}_J(L)\ne 0.$$

This generalizes the Kac–Wakimoto criterion by replacing superdimension with modified dimension. The paper proves this conjecture for ${\mathfrak{osp}}(m|2n)$, while it is attributed to Geer, Patureau-Mirand, and Kujawa in the general setting.