  We develop a new approach, based on quantization methods, to study higher
symmetries of invariant differential operators. We focus here on conformally
invariant powers of the Laplacian over a conformally flat manifold and recover
results of Eastwood, Leistner, Gover and \v{S}ilhan. In particular, conformally
equivariant quantization establishes a correspondence between the algebra of
Hamiltonian symmetries of the null geodesic flow and the algebra of higher
symmetries of the conformal Laplacian. Combined with a symplectic reduction,
this leads to a quantization of the minimal nilpotent coadjoint orbit of the
conformal group. The star-deformation of its algebra of regular functions is
isomorphic to the algebra of higher symmetries of the conformal Laplacian. Both
identify with the quotient of the universal envelopping algebra by the Joseph
ideal.
