C H A P T E R 1

B R A U E R - B G G R E C I P R O C I T Y F O R G R A D E D LIE A L G E B R A S

§1. Introduction and Notation .

The BGG-Brauer reciprocity theorem for Lie algebras was first motivated by Pollack's [Po]

study of A\ . In his thesis Pollack constructed the projective indecomposables for 5/(2) and showed

how they factor into Verma modules. Humphreys [Hu-1], inspired by the ideas of Brauer and by

Pollack's example, discovered a reciprocity law for g where $ is a classical Lie algebra over a field

of characteristic p. Later, in 1976, Bernstein, Gelfand, and Gelfand [BGG] proved, through the use

of category 0 , an analogous reciprocity law for complex-semisimple Lie algebras. More recently,

the BGG-Brauer reciprocity law has been proved in many other settings. Some examples can be

found in [Jan], [Cl-P-S], [Ro-W] and [M-V]. The results in this chapter represent joint work with

Randall Holmes.

In this chapter a BGG-Brauer reciprocity will be proved for restricted graded Lie algebras. We

will use some of the techniques of Jantzen [Jan] in order to prove that the projective indecom-

posables have a filtration by Verma modules. However, we will present a proof of the reciprocity

theorem which does not rely on an ordering of the weights, but rather a "Frobenius reciprocity"

for u-algebras. This proof is different from the proofs given in most of the papers mentioned above.

These results not only generalize Humphreys' and Jantzen's work, but apply to Lie algebras of

Cartan type. (In the situation with a Lie algebra of Cartan type the decomposition may not be

symmetric, as with classical Lie algebras.) Moreover, the reciprocity law proved in this setting will

work for classical Lie algebras, $ with a decomposition

fl

= N~ 0 Go © iV+

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