Dirac Cohomology for the Cubic Dirac Operator

Let g be a complex semisimple Lie algebra and let r ⊂ g be any reductive Lie subalgebra such that B|r is nonsingular where B is the Killing form of g. Let Z(r) and Z(g) be, respectively, the centers of the enveloping algebras of r and g. Using a Harish-Chandra isomorphism one has a homomorphism η : Z(g) → Z(r) which, by a well-known result of H. Cartan, yields the the relative Lie algebra cohomology H(g, r). Let V be any gmodule. For the case where r is a symmetric subalgebra, Vogan has defined the Dirac cohomology Dir(V ) of V . Using the cubic Dirac operator we extend his definition to the case where r is arbitrary subject to the condition stated above. We then generalize results of Huang-Pandžić on a proof of a conjecture of Vogan. In particular Dir(V ) has a structure of a Z(r)-module relative to a “diagonal” homomorphism γ : Z(r) → EndDir(V ). In case V admits an infinitesimal character χ and I is the identity operator on Dir(V) we prove