The symplectic Dirac operator

Question

Let $(M,\omega)$ be a symplectic manifold. The symplectic Clifford algebra is:

$$ef-fe=\omega(e,f)$$

I define the symplectic Dirac operator over the symplectic spinors:

$${\cal D}=\sum_{i,j}  \omega (e_i,e_j) e_i.\nabla_{e_j}$$

with $(e_i)$ an orthonormal basis. We have:

$${\cal D}(f\psi )= (df)^{\omega *} .\psi + f {\cal D}(\psi )$$

Have we:

$${\cal D}^2 =\Delta + \alpha$$

with $\alpha$ a scalar?

Comments

Precisely what stops you from working out the answer yourself?