Abstract
In this paper I investigate the question of linkage and block theory for Lie algebras of Cartan
type. The BGG reciprocity theorem is generalized for graded restricted Lie algebras with two
types of "generalized Verma modules" playing the role of the intermediate modules. Therefore,
the reciprocity theorem will not only hold for classical Lie algebras, but also for Lie algebras of
Cartan type. The "generalized Verma modules" for Lie algebras of Cartan type turn out to be
quite important because most of them are irreducible when induced from irreducible Go modules,
and thus most of them are projective indecomposable when induced from projective covers for GQ,
In the classical case this occurs only with the Steinberg representation.
The second part of the paper deals mainly with block structure and projective modules of
Lie algebras of types W and K. In order to use BGG reciprocity the generalized Verma modules
for Lie algebras of type W are decomposed, and this leads to a classification of all the restricted
irreducible modules. The Lie algebras of types W and K are shown to be of the one block type; that
is, their u-algebras have precisely one block. Furthermore, a procedure is provided for computing
the Cartan invariants for Lie algebras of types W and K, given knowledge about the decomposition
of the generalized Verma modules and about the Jantzen matrix of the classical Go component. In
particular, explicit composition factors for projectives for the restricted hulls of the Lie algebras
W(l,n) and W(2,n) are determined.
Key Words and Phrases: restricted Lie algebras, projective modules
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