Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let $F$ be an algebraically closed field of prime characteristic, and $W(m,n,\textbf{1})$ be the simple restricted Lie superalgebra of Witt type over $F$, which is the Lie superalgebra of superderivations of the superalgebra $\fraka(m;\textbf{1})\otimes \wedge(n)$, where $\fraka(m;\textbf{1})$ is the truncated polynomial algebra with $m$ indeterminants and $\wedge(n)$ is the Grassmann algebra with $n$ indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of $W(m,n,\textbf{1})$ with atypical weights of type I.