4 DANIEL K. NAKANO
In order to use the reciprocity theorem information about the decomposition of the "generalized
Verma modules" is needed. In Chapter 2 we decompose these Verma modules for type W. This
was done independently of Shen's [Sh-3] recent work where he decomposes the Verma modules for
types W, S and H. Since he takes three papers to obtain these results we will prove, for the sake
of continuity and self-containment, results about the decomposition of generalized Verma modules
and Verma modules for Lie algebras of type W.
In Chapter 3 we use the results in the first two chapters to study the projective modules over Lie
algebras of types W and K. First we prove a theorem which allows us to write the Cartan matrix
as a product of matrices involving the linkage of the classical Go component and the decomposition
of generalized Verma modules. From this result it follows that the u-algebras for Lie algebras of
types W and K have precisely one block. Moreover, this decomposition of the Cartan matrix allows
us to compute Cartan invariants for W(l, 1) and W(2,1). We also look at the connection between
Cartan invariants for W(m,n) and W(m, 1).
I would like to thank my advisor George B. Seligman for his patience, encouragement and
guidance during this work. I am also grateful to Randy Holmes for his willingness to share and
contribute his ideas in Chapters 1 and 2. Conversations with, and suggestions from Monty Mc-
Govern, Walter Feit, Brian Parshall, and James E. Humphreys were helpful. Lastly I would like
to thank the referee for several useful comments. Acknowledgement of financial support should be
given to Yale University and NSF grant DMS-88 06371.
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