6 DANIEL K. NAKANO

where Go a reductive subalgebra of g (see [Se-2]).

Let g be a graded restricted Lie algebra

Q = G-S® ...£_! © Go © Gi © ... © Gt

and set

# + = ©:o#i N~ = ei0Gi

B+ = Go © iV+ 5 - = Go © N~.

We will assume that N+ and iV~ are p-nilpotent subalgebras of g. Since 9 is a restricted Lie

algebra we can look at representations of the u-algebra U(g). If T is a maximal torus of g then

there exists a basis T such that t^ = t V t € T. This implies the irreducible modules for U(g) will

be parametrized by

f = HomZp(T0,Zp)

where To is the Zp span of T. These elements can be thought of as Zp-valued characters [Se-1] of

T.

Throughout this chapter all modules will be right modules. Let C denote the category of all

finite dimensional U(g) modules and Cgr denote the category of all finite dimensional graded U(g)

modules. A U(g) module M is graded if M = ©-^nMi with M{Gj C Mi+j. Let T : Cgr 1— C be

the forgetful functor, ie. the functor which forgets all gradings.

Let M be a finite dimensional right U(g) module. By using the Hopf algebra structure of U(g)

one can construct the dual module M* = Hom[p(M,F). The dual module will be a right U(g)

module with the following action:

(/./)(m) = -f(m.l) for f G M*, / G g and m e M.