Higher Symmetries of the Laplacian via Quantization

We develop a new approach, based on quantization methods, to study higher
symmetries of invariant di erential operators. We focus here on conformally invariant powers
of the Laplacian over a conformally at manifold and recover results of Eastwood, Leistner,
Gover and ilhan. In particular, conformally equivariant quantization establishes a correspondence
between the algebra of Hamiltonian symmetries of the null geodesic ow 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.