Equivariant deformation quantization for the cotangent bundle of a flag manifold
Annales de l'Institut Fourier, Volume 52 (2002) no. 3, p. 881-897
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(T X) which corresponds to a G-equivariant quantization of symbols into twisted differential operators acting on half-forms on X. We construct, when 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 . In the known examples, Λ x (x0) is not a differential operator, and so the star product μ x φ is not local in φ. acquires an invariant positive definite inner product compatible with its grading. The completion of is a new Fock space type model of the unitary representation of G on L 2 half-densities on X.
Soit X une variété de drapeaux (généralisés) pour un groupe de Lie semisimple complexe G. Nous étudions la construction d’un star produit gradué sur =R(T X) qui correspond à une quantification G-équivariante des symboles vers les opérateurs différentiels tordus agissant sur les demi-formes sur X. Lorsque est engendrée par les fonctions moment μ x pour G, nous construisons de tels star produits , où μ x φ est de la forme μ x φ+1 2{μ x ,φ}t+Λ x (φ)t 2 . Les Λ x sont ici des opérateurs sur . Dans les exemples connus, Λ x (x0) n’est pas un opérateur différentiel, et le star produit μ x φ n’est donc pas local en φ. Alors est munie d’une forme hermitienne invariante définie positive et compatible avec sa graduation. L’espace complété de est un nouveau modèle de type Fock de la représentation unitaire de G sur les demi-densités L 2 sur X.
DOI : https://doi.org/10.5802/aif.1905
Classification:  53D55,  22E46,  17B35,  53D50
Keywords: deformation quantization, flag manifold, unitary representation
@article{AIF_2002__52_3_881_0,
     author = {Brylinski, Ranee},
     title = {Equivariant deformation quantization for the cotangent bundle of a flag manifold},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {3},
     year = {2002},
     pages = {881-897},
     doi = {10.5802/aif.1905},
     zbl = {1010.53067},
     mrnumber = {1907391},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_3_881_0}
}
Brylinski, Ranee. Equivariant deformation quantization for the cotangent bundle of a flag manifold. Annales de l'Institut Fourier, Volume 52 (2002) no. 3, pp. 881-897. doi : 10.5802/aif.1905. http://www.numdam.org/item/AIF_2002__52_3_881_0/

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