There is not only symplectic Dirac operator of Habermann, but also a symplectic Dirac operator on complex symplectic spinors which exist on any symplectic manifolds and not only on those whose first Chern class is even. They are parallel to the Dirac operators on Spin^c-structures. When the manifold is a homogeneous space, the spectral problem reduces to representation theory. From its point of view, in the case of complex symplectic spinors, the spectra of complex symplectic Diracs are connected to the spectra of the classical Diracs -  conformal weight action.