Dirac Operators and Nilpotent Lie Algebra Cohomology

Abstract

Let g be a complex reductive Lie algebra with an invariant symmetric bilinear form B, equal to the Killing form on the semisimple part of g. In this chapter we consider a parabolic subalgebra q = l ⊕ u of g, with unipotent radical u and a Levi subalgebra l. We will denote by

$$ \bar q = \mathfrak{l} \oplus \bar u $$

the opposite parabolic subalgebra. Here the bar notation does not mean complex conjugation in general, but it will be a conjugation in the cases we will study the most, so the notation is convenient.