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

are p-nilpotent ideals in B^ any Go
module can be extended to

by letting

act trivially. Furthermore, the irreducible modules
for B^ are just the irreducible modules for Go with 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± = and TV^ = V *
•* v proj y proj u / f c U ' * y irr v irr'
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