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 G-invariant
Mpc-structures on symplectic manifolds with a G-action. We prove a variant of
Parthasarathy’s formula for the commutator of two symplectic Dirac-type operators on
general symmetric symplectic spaces.