A note on restricted representations of the Witt superalgebras

Let F be an algebraically closed field of prime characteristic p > 3, and W(n) the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W(n) are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W(n) is proved to have one block.