On Mpc-structures and symplectic Dirac operators

We prove that the kernels of the restrictions of the symplectic Dirac operator and one of the two symplectic Dirac–Dolbeault operators on natural sub-bundles of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute these kernels explicitly for complex projective spaces and show that the remaining Dirac–Dolbeault operator has infinite dimensional kernels on these finite rank sub-bundles. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabiliser of a Lagrangian subspace) in the Mpc group and classify GG-invariant Mpc-structures on symplectic manifolds with a GG-action. We prove a variant of Parthasarathy’s formula for the commutator of two symplectic Dirac-type operators on general symmetric symplectic spaces.