Higher Symmetries of the Laplacian via Quantization

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 Šilhan. 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.

22 pages, minor improvements

References in corpus (4)

• Second Order Symmetries of the Conformal Laplacian
• Conformally Equivariant Quantization - a Complete Classification
• Conformally invariant quantization -- towards complete classification
• Symmetries of CR sub-Laplacian

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