Higher Symmetries of the Laplacian via Quantization

TL;DR

This paper introduces a quantization-based approach to analyze higher symmetries of conformally invariant powers of the Laplacian, connecting geometric, algebraic, and representation-theoretic perspectives.

Contribution

It establishes a novel quantization framework linking Hamiltonian symmetries of geodesic flow with higher Laplacian symmetries, extending prior results and providing algebraic characterizations.

Findings

Correspondence between Hamiltonian symmetries and higher Laplacian symmetries

Quantization of the minimal nilpotent coadjoint orbit of the conformal group

Identification of symmetry algebra with the quotient of the universal enveloping algebra by the Joseph ideal

Abstract

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.