Anthony W. Knapp

Commentary on "Duality Theorems in Relative Lie Algebra Cohomology"

This set of notes is a preliminary version of the first few chapters in the book of Knapp and Vogan entitled Cohomological Induction and Unitary Representations.  The notes are preliminary in two ways.  One is that the notes work with K finite functions on K, while the book works with K finite distributions on K.  Although the two approaches come to the same thing, the book explains why the use of distributions is more natural.

The other sense in which the notes are preliminary is that the proof of Proposition 1.1 of these notes has a gap.  The proposition is still correct, but the gap is complicated to fill and the missing steps account for some of the complexity of Chapter I of the book.  The key to supplying the missing steps occurs in Proposition 1.68 of the book, which makes critical use of a theorem of Schwartz that is stated and proved in Appendix B.