Equivariant deformation quantization for the cotangent bundle of a flag manifold

Brylinski, Ranee. "Equivariant deformation quantization for the cotangent bundle of a flag manifold." Annales de l’institut Fourier 52.3 (2002): 881-897. <http://eudml.org/doc/115998>.

@article{Brylinski2002,
abstract = {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 $\{\mathcal \{R\}\}=R(T^\star X)$ which corresponds to a $G$-equivariant quantization of symbols into twisted differential operators acting on half-forms on $X$. We construct, when $\{\mathcal \{R\}\}$ is generated by the momentum functions $\mu ^x$ for $G$, a preferred choice of $\star $ where $\mu ^x\star \phi $ has the form $\mu ^x\phi +\{1\over 2\}\lbrace \mu ^x,\phi \rbrace t+\Lambda ^x(\phi )t^2$. Here $\Lambda ^x$ are operators on $\{\mathcal \{R\}\}$. In the known examples, $\Lambda ^x$ ($x\ne 0$) is not a differential operator, and so the star product $\mu ^x\star \phi $ is not local in $\phi $. $\{\mathcal \{R\}\}$ acquires an invariant positive definite inner product compatible with its grading. The completion of $\{\mathcal \{R\}\}$ is a new Fock space type model of the unitary representation of $G$ on $L^2$ half-densities on $X$.},
affiliation = {22 Overlook Drive, PO Box 1089, Truro MA 02666-1089 (USA)},
author = {Brylinski, Ranee},
journal = {Annales de l’institut Fourier},
keywords = {deformation quantization; flag manifold; unitary representation; Poisson bracket},
language = {eng},
number = {3},
pages = {881-897},
publisher = {Association des Annales de l'Institut Fourier},
title = {Equivariant deformation quantization for the cotangent bundle of a flag manifold},
url = {http://eudml.org/doc/115998},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Brylinski, Ranee
TI - Equivariant deformation quantization for the cotangent bundle of a flag manifold
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 3
SP - 881
EP - 897
AB - 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 ${\mathcal {R}}=R(T^\star X)$ which corresponds to a $G$-equivariant quantization of symbols into twisted differential operators acting on half-forms on $X$. We construct, when ${\mathcal {R}}$ is generated by the momentum functions $\mu ^x$ for $G$, a preferred choice of $\star $ where $\mu ^x\star \phi $ has the form $\mu ^x\phi +{1\over 2}\lbrace \mu ^x,\phi \rbrace t+\Lambda ^x(\phi )t^2$. Here $\Lambda ^x$ are operators on ${\mathcal {R}}$. In the known examples, $\Lambda ^x$ ($x\ne 0$) is not a differential operator, and so the star product $\mu ^x\star \phi $ is not local in $\phi $. ${\mathcal {R}}$ acquires an invariant positive definite inner product compatible with its grading. The completion of ${\mathcal {R}}$ is a new Fock space type model of the unitary representation of $G$ on $L^2$ half-densities on $X$.
LA - eng
KW - deformation quantization; flag manifold; unitary representation; Poisson bracket
UR - http://eudml.org/doc/115998
ER -