Spinors over a Cone, Dirac Operator, and Representations of Spin(4, 4)

The representations of Spin(4, 2)e seem to be of particular physical interest since its quotient SO(4, 2)e is the conformal group of the spacetime. Kostant [7] has considered the Laplacian on a projective cone in R8 and has shown that the kernel H of the Laplacian is an irreducible unitary representation of SO(4, 4)e. Moreover, it is known [9] that there is a Howe pair (SO(2), SO(4, 2)e) in SO(4, 4)e and that the weight spaces of SO(2) in H are irreducible representations of SO(4, 2)e. He conjectured that the analogous facts should hold for the Dirac operator acting on the spinor fields on S3 × S3 ⊆ R8. We introduce a hitherto unknown embedding of the bundle of spinors on S3 × S3 in the bundle of spinors on R8. This embedding makes it possible to obtain a particularly convenient expression for the Dirac operator on S3 × S3 in terms of spinors and vector fields on the underlying R8 in particular the conformal invariance is manifest. We explicitly describe the kernel F of the restriction of the Dirac operator to even spinor fields and show that it is an irreducible representation of Spin(4, 4)e. We also show that it has an invariant sesquilinear scalar product (with is not, however, positive definite). The existence of this scalar product can also be established by showing that F ⊆ S+ ⊗ H where S+ the vector space of even spinors. We identify within Spin(4, 4)e a Howe pair consisting of SO(2) and Spin(4, 2)e and show that the weight spaces of SO(2) in F are irreducible representations of Spin(4, 2)e.