On Mpc-structures and Symplectic Dirac Operators

Abstract

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabilizer of a Lagrangian subspace) in the group M p c and classify G-invariant M p c -structures on symplectic spaces with a Gaction. We prove a variant of Parthasarathy's formula for the commutator of two symplectic Dirac-type operators on a symmetric symplectic space.2 Subgroups of Sp(V, Ω) and lifts to M p c (V, Ω, j) The symplectic Clifford algebraLet (V, Ω) be a finite-dimensional real symplectic vector space of dimension 2n. The symplectic Clifford Algebra Cl(V, Ω) is the associative unital complex algebra generated by V with the relationwhere h is a positive real number and = h 2π . A symplectic spinor space is a vector space carrying a representation of the symplectic Clifford algebra. This representation, called Clifford multiplication and denoted by cl, is derived from an irreducible unitary representation of the Heisenberg group. There are many