Dirac Operators on Quadratic Lie Superalgebras

Assume that r is a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, p is a finite dimensional complex super vector space with a non-degenerate super-symmetric bilinear form, and nu : r -> osp(p) is a homomorphism of Lie superalgebras. In this paper, we give a necessary and sufficient condition for r circle plus p to be a quadratic Lie superalgebra. Then, we define the cubic Dirac operator D(g, r) on g and give a formula of (D(g, r))(2.) Finally, we get the Vogan's conjecture for quadratic Lie superalgebras by D(g, r).