Equivariant deformation quantization for the cotangent bundle of a flag manifold

Brylinski, Ranee

Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 881-897.

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 $ℛ=R\left(T^{☆}X\right)$ qui correspond à une quantification $G$-équivariante des symboles vers les opérateurs différentiels tordus agissant sur les demi-formes sur $X$. Lorsque $ℛ$ est engendrée par les fonctions moment ${\mu }^x$ pour $G$, nous construisons de tels star produits $☆$, où ${\mu }^x☆\phi$ est de la forme ${\mu }^x\phi +\frac{1}{2}\{{\mu }^x,\phi \}t+{\Lambda }^x\left(\phi \right)t^2$. Les ${\Lambda }^x$ sont ici des opérateurs sur $ℛ$. Dans les exemples connus, ${\Lambda }^x$ ($x\ne 0$) n’est pas un opérateur différentiel, et le star produit ${\mu }^x☆\phi$ n’est donc pas local en $\phi$. Alors $ℛ$ est munie d’une forme hermitienne invariante définie positive et compatible avec sa graduation. L’espace complété de $ℛ$ est un nouveau modèle de type Fock de la représentation unitaire de $G$ sur les demi-densités $L^2$ sur $X$.

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 $ℛ=R\left(T^{☆}X\right)$ 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 ${\mu }^x$ for $G$, a preferred choice of $☆$ where ${\mu }^x☆\phi$ has the form ${\mu }^x\phi +\frac{1}{2}\{{\mu }^x,\phi \}t+{\Lambda }^x\left(\phi \right)t^2$. Here ${\Lambda }^x$ are operators on $ℛ$. In the known examples, ${\Lambda }^x$ ($x\ne 0$) is not a differential operator, and so the star product ${\mu }^x☆\phi$ is not local in $\phi$. $ℛ$ acquires an invariant positive definite inner product compatible with its grading. The completion of $ℛ$ is a new Fock space type model of the unitary representation of $G$ on $L^2$ half-densities on $X$.

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