Let k be an algebraically closed field of characteristic p > 3, and W (n) the superversion of the Witt algebra over k, i.e. the Lie superalgebra of superderivations of the Grassmann algebra of rank n over k. The simple modules and projective indecomposable modules in the restricted supermodule category for W (n) are studied. The character formulas for such simple modules are given, and the Cartan invariants for this category are presented.

Let L be a restricted simple Lie algebra of Cartan type. In this paper, we construct the simple L-modules having nonsingular characters and some simple modules with singular characters.  2005 Elsevier Inc. All rights reserved. IntroductionLet L be a simple restricted Lie algebra of Cartan type over an algebraically closed field F with characteristic p 5. Let χ ∈ L * = Hom F (L, F ). A L-module M has character χ provided that D p m − D [p] m = χ(D) p m for D ∈ L and m ∈ M. By [7, Theorem 2.5, p. 207], every simple module has a character. If M is a L-module with χ = 0, then we call M a restricted L-module; if χ = 0, then M is called a nonrestricted L-module.Let L = i L i be the standard grading on L and put L i = j i L j . The height ht(χ) of the character χ is defined by: ht(χ) = min i −1 χ L i = 0 .

Simple modules for the restricted Witt superalgebra W(m, n, 1) are considered. Conditions are provided for the restricted and nonrestricted Kac modules to be simple.

Every simple module having character height at most one for a restricted Cartantype Lie algebra g can be realized as a quotient of a module obtained by starting with a simple module S for the homogeneous component of degree zero in the natural grading of g, extending the action trivially to positive components and inducing up to g. It is shown that if S is not restricted, or if it is restricted and its maximal vector does not have exceptional weight, then the induced module is already simple.

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