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Conformally equivariant quantization : existence and uniqueness

Christian Duval, Pierre Lecomte, Valentin Ovsienko (1999)

Annales de l'institut Fourier

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold ( M , g ) . In other words, we establish a canonical isomorphism between the spaces of polynomials on T * M and of differential operators on tensor densities over M , both viewed as modules over the Lie algebra o ( p + 1 , q + 1 ) where p + q = dim ( M ) . This quantization exists for generic values of the weights of the tensor densities and we compute the critical values of the weights yielding...

Equivariant deformation quantization for the cotangent bundle of a flag manifold

Ranee Brylinski (2002)

Annales de l’institut Fourier

Let X be a (generalized) flag manifold of a complex semisimple Lie group G . We investigate the problem of constructing a graded star product on ℛ = R ( T ☆ X ) which corresponds to a G -equivariant quantization of symbols into twisted differential operators acting on half-forms on X . We construct, when ℛ is generated by the momentum functions μ x for G , a preferred choice of ☆ where μ x ☆ φ has the form μ x φ + 1 2 { μ x , φ } t + Λ x ( φ ) t 2 . Here Λ x are operators on ℛ . In the known examples, Λ x ( x ≠ 0 ) is not a differential operator, and so the star product μ x ☆ φ ...

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