In these notes we consider the usual Fedosov star product on a symplectic
manifold $(M,\omega)$ emanating from the fibrewise Weyl product $\circ$, a
symplectic torsion free connection $\nabla$ on M, a formal series $\Omega \in
\nu Z^2_{\rm\tiny dR}(M)[[\nu]]$ of closed two-forms on M and a certain formal
series s of symmetric contravariant tensor fields on M. For a given symplectic
vector field X on M we derive necessary and sufficient conditions for the
triple $(\nabla,\Omega,s)$ determining the star product * on which the Lie
derivative $\Lie_X$ with respect to X is a derivation of *. Moreover, we also
give additional conditions on which $\Lie_X$ is even a quasi-inner derivation.
Using these results we find necessary and sufficient criteria for a Fedosov
star product to be $\mathfrak g$-invariant and to admit a quantum Hamiltonian.
Finally, supposing the existence of a quantum Hamiltonian, we present a
cohomological condition on $\Omega$ that is equivalent to the existence of a
quantum momentum mapping. In particular, our results show that the existence of
a classical momentum mapping in general does not imply the existence of a
quantum momentum mapping.