  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 Mpc and classify
G-invariant Mpc-structures on symplectic spaces with a G-action. We prove a
variant of Parthasarathy's formula for the commutator of two symplectic
Dirac-type operators on a symmetric symplectic space.
