PROJECTIVE MODULES 3

even though s/(2) has a maximal p-nilpotent subalgebra of degree 1, the u-algebra is of infinite,

but tame representation type. Recently Pfautsch, Voigt, [Pf-V] and Rickard [Ri] have shown that

the other restricted simple Lie algebras are of wild representation type. The problem with drawing

these comparisons manifests itself in the induction-restriction procedure. With group algebras when

one "induces" then "restricts" one has a direct sum decomposition. In the case of u-algebras this

doesn't always happen and Pollack provides a sufficient condition for this to occur. One interesting

consequence of this, which we will see later in this paper, is that for u-algebras a projective module

can be obtained by inducing from a non-projective module. This phenomenon cannot occur with

group algebras.

In this paper I will attempt to draw upon the analogy between the representation theory of group

algebras and u-algebras in order to compare the representation theory between the u-algebras of

classical and "non-classical" Lie algebras. The representation theory for classical Lie algebras has

been extensively studied over the last 20 years. In 1971 Humphreys [Hu-1] wrote a paper which

provides a detailed account of how linkage between Verma modules holds in the modular setting

and its effect on the block structure of the u-algebra. However, the major contribution of his paper

was the first statement and proof of the BGG reciprocity principle: [P(A) : F(/i)] = [V"(/x) : L(X)]

where P(A) is the projective cover of the irreducible module L(X) and V(fi) is a Verma module.

This reciprocity law is similar to Brauer reciprocity for finite groups.

In Chapter 1 we present a general BGG reciprocity law for graded restricted Lie algebras. Since

every restricted simple Lie algebra is endowed with the structure of a graded restricted Lie algebra

the reciprocity law will hold for both classical Lie algebras and Lie algebras of Cartan type. The

techniques of Jantzen and Holmes will be used to prove the existence of filtrations by V^- modules.

Moreover, a new proof of the reciprocity law will be given which does not use an ordering of the

weights. A more general setting to this theory can be found in [Ho-N].