### 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.