Kac–Wakimoto-type conjecture for m-traces and atypicality

Let ${\mathfrak{g}}$ be a basic classical Lie superalgebra, and let $L(\lambda)$ be a simple ${\mathfrak{g}}$-supermodule of highest weight $\lambda$. Write $\operatorname{atyp}(\lambda)$ for the atypicality of $\lambda$, and let $\mathcal{I}_{L(\lambda)}$ be the ideal of supermodules that occur as direct summands of $L(\lambda)\otimes W$ for some finite-dimensional ${\mathfrak{g}}$-supermodule $W$. An m-trace and atypicality conjecture. The module $L(\lambda)$ admits an m-trace ${\operatorname{\mathsf{t}}}$ on $\mathcal{I}_{L(\lambda)}$. If $L(\mu)$ is another simple supermodule with

$$\operatorname{atyp}(\mu)\leq\operatorname{atyp}(\lambda),$$

then $L(\mu)$ is an object of $\mathcal{I}_{L(\lambda)}$, and

$$\operatorname{atyp}(\mu)\leq\operatorname{atyp}(\lambda)$$

if and only if

$${\operatorname{\mathsf{t}}}_{L(\mu)}(\operatorname{Id}_{L(\mu)})\neq 0.$$

This is presented as a generalization of the Kac–Wakimoto conjecture for basic classical Lie superalgebras. It links the existence of an m-trace on the ideal generated by a simple module with the ordering of simple modules by atypicality and the nonvanishing of the trace on their identity morphisms; the supplied source does not establish whether it has been resolved.