The parabolic Euler-character conjecture for ortho-symplectic Lie superalgebras

Let $\mathfrak{g}=\mathfrak{spo}(2n|2m+1)$, and let $\mathfrak{p}$ be the maximal parabolic subalgebra obtained by removing the simple root $\epsilon_m$, with Levi subalgebra $\mathfrak{l}=\mathfrak{gl}(n|m)$. For a partition $\lambda$ satisfying $\lambda_{n+1}\leq m$, write $\lambda^\sharp$ for the corresponding highest weight, let $L^0(\lambda^\sharp)$ be the associated simple $\mathfrak{l}$-module, and let $E^\mathfrak{p}$ denote the parabolic Euler-character construction. Parabolic Euler-character conjecture. The modules $E^\mathfrak{p}(L^0(\lambda^\sharp))$, as $\lambda$ ranges over all such partitions, form a basis for the complexified Grothendieck group of finite-dimensional $\mathfrak{g}$-modules. Moreover, when $\lambda$ is “not close” to the zero weight, the composition factors of $E^\mathfrak{p}(L^0(\lambda^\sharp))$ are the same as those of the Kac module $K(\lambda^\sharp)$ for $\mathfrak{gl}(n|m)$. The first assertion is a basis statement for the finite-dimensional representation theory of $\mathfrak{spo}(2n|2m+1)$, while the second predicts agreement with the corresponding Kac-module composition factors away from weights near zero; the source does not make “not close” precise or provide a resolution.