Dirac Cohomology for the Cubic Dirac Operator

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{r} \subset \mathfrak{g}$ be any reductive Lie subalgebra such that B∣ $\mathfrak{r}$ is nonsingular where B is the Killing form of $\mathfrak{g}$ . Let Z(r) and Z(g) be, respectively, the centers of the enveloping algebras of $\mathfrak{r}$ and $\mathfrak{r} \subset \mathfrak{g}$ . Using a Harish-Chandra isomorphism one has a homomorphism $\eta :Z(\mathfrak{g}) \to Z(\mathfrak{r})$ which, by a well-known result of H. Cartan, yields the the relative Lie algebra cohomology H($(\mathfrak{g},\mathfrak{r})$.