# Equivariant deformation quantization for the cotangent bundle of a flag manifold

Ranee Brylinski^{[1]}

- [1] 22 Overlook Drive, PO Box 1089, Truro MA 02666-1089 (USA)

Annales de l’institut Fourier (2002)

- Volume: 52, Issue: 3, page 881-897
- ISSN: 0373-0956

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topBrylinski, 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 -

## References

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