PROJECTIVE MODULES 7

If L is a finite dimensional restricted Lie algebra then U(L) is a finite dimensional associative

algebra. So U(L) as right U(L) module will break up into a finite direct sum of indecomposable

modules by the Krull-Schmidt theorem. These modules will be called the projective indecomposable

modules [Cu-R-1] or the projective covers [Cu-R-2] of U(L). The irreducible modules for U(L) will

be in one-to-one correspondence with the projective covers.(Each projective cover has a unique

head which corresponds to the irreducible module.) Suppose T is a maximal torus for Go and g.

For each A G T let V{\) (resp. P(A)) denote the projective cover for the irreducible # (resp. Go)

module £(A) (resp. L(X)). We should note that since

N±

are p-nilpotent ideals in B^ any Go

module can be extended to

B±

by letting

N±

act trivially. Furthermore, the irreducible modules

for B^ are just the irreducible modules for Go with N± acting trivially.

There are two types of Verma modules which will play an intermediate role between the projec-

tives and the irreducibles. These are

v±r(\) = L(\)®u(B±)u(s)

The Verma modules here can be called "generalized Verma modules" because they are the finite

dimensional analogs to the generalized Verma modules employed in the BGG reciprocity law proved

in [Ro-W]. Unlike their setting where all finite dimensional modules for U(Go) are semisimple (by

Weyl's theorem), we must consider inducing from both projective covers and irreducible modules

because in our situation U(Go) is semisimple if and only if Go is a torus [Ho].

§2. Equivalence betwee n simple an d projective objects in C an d Cgr.

Let T be the forgetful functor from Cgr to C. Since V^J: • and V($r are graded U(g) modules it

follows that

J7V± • = V± • and TV^ = V *

•* v proj y proj u / f c U ' * y irr v irr'