  We show that the exterior derivative operator on a symplectic manifold has a
natural decomposition into two linear differential operators, analogous to the
Dolbeault operators in complex geometry. These operators map primitive forms
into primitive forms and therefore lead directly to the construction of
primitive cohomologies on symplectic manifolds. Using these operators, we
introduce new primitive cohomologies that are analogous to the Dolbeault
cohomology in the complex theory. Interestingly, the finiteness of these
primitive cohomologies follows directly from an elliptic complex. We calculate
the known primitive cohomologies on a nilmanifold and show that their
dimensions can vary depending on the class of the symplectic form.
