Higher symmetries of the Laplacian via quantization

Michel, Jean-Philippe

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1581-1609.

Nous développons une nouvelle approche, basée sur des méthodes de quantification, pour étudier les symétries supérieures d’opérateurs différentiels invariants. Nous traitons ici le cas des puissances conformes du laplacien sur une variété conformément plate et retrouvons les résultats de Eastwood, Leistner, Gover et Šilhan. En particulier, la quantifciation conformément équivariante établit une correspondence entre l’algèbre des symétries hamiltoniennes du flot géodésique nul et l’algèbre des symétries supérieures du laplacien conforme. Via une réduction symplectique, ceci conduit à une quantification de l’orbite nilpotente minimale du groupe conforme. La star-déformation de son algèbre de fonctions régulières est isomorphe à l’algèbre des symétries supérieures du laplacien conforme. Les deux s’identifient au quotient de l’algèbre enveloppante de l’algèbre de Lie conforme par l’idéal de Joseph.

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.

Classification : 58J10, 53A30, 70S10, 17B08, 53D20, 53D55
Keywords: Symmetry algebra, Laplacian, Quantization, Conformal geometry, Minimal nilpotent orbit, Symplectic reduction.
Mot clés : algèbre de symétries, laplacien, quantification, géometrie conforme, orbite nilpotente minimale, réduction symplectique.

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