On Kac-Weisfeiler modules for general and special linear Lie superalgebras

Let $⋙:={\text{gl}}_{m\mathrm{\mid }n}$ be a general linear Lie superalgebra over an algebraically closed field $\text{mathds}k={\bar{\mathbb{F}}}_p$ of characteristic $p>2$. A module of $⋙$ is said to be of Kac-Weisfeiler if its dimension coincides with the dimensional lower bound in the super Kac-Weisfeiler property presented by Wang-Zhao in \cite{WZ}. In this paper, we verify the existence of the Kac-Weisfeiler modules for ${\text{gl}}_{m\mathrm{\mid }n}$. We also establish the corresponding consequence for the special linear Lie superalgebras ${\mathfrak{s}\mathfrak{l}}_{m\mathrm{\mid }n}$ with restrictions $p>2$ and $p\nmid (m-n)$.