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55
LOCALIZATION OF THE RIEMANNROCH CHARACTER
, 2005
"... We present a Ktheoretic approach to the GuilleminSternberg conjecture [17], about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken [28, 29] and TianZhang [35]. Besides providing a new proof of this conjecture for the full nonabelian group act ..."
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Cited by 42 (11 self)
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We present a Ktheoretic approach to the GuilleminSternberg conjecture [17], about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken [28, 29] and TianZhang [35]. Besides providing a new proof of this conjecture for the full nonabelian group action case, our methods lead to a generalization for compact Lie group actions on manifolds that are not symplectic; these manifolds carry an invariant almost complex structure and an abstract moment map.
The quantization conjecture revisited
 Ann. of Math
"... Version: 8/5/98 ABSTRACT: I prove the following strong version of the quantization conjecture of Guillemin and Sternberg: for a reductive group action on a smooth, compact, polarized variety ( X, ℓ), the cohomologies of ℓ over the GIT quotient are equal to the invariant parts of the cohomologies ove ..."
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Cited by 40 (7 self)
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Version: 8/5/98 ABSTRACT: I prove the following strong version of the quantization conjecture of Guillemin and Sternberg: for a reductive group action on a smooth, compact, polarized variety ( X, ℓ), the cohomologies of ℓ over the GIT quotient are equal to the invariant parts of the cohomologies over X. This generalizes the theorem of [GS], which concerned the spaces of global sections, and strengthens its subsequent extensions ([M], [V]) to RiemannRoch numbers. A remarkable byproduct is the rigidity of cohomology of certain vector bundles over the GIT quotient under a small change in the defining polarization. One application is a new proof of the BorelWeilBott theorem of [T] for the moduli stack M of Gbundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Also studied are equivariant holomorphic forms and the equivariant Hodgetode Rham spectral sequences for X and its strata, whose collapse at E1 is shown. Collapse of the Hodgetode Rham sequence for M is hence deduced.
Symplectic Geometry And The Verlinde Formulas
, 1998
"... The purpose of this paper is to give a proof of the Verlinde formulas by applying the RiemannRochKawasaki theorem to the moduli space of flat Gbundles on a Riemann surface # with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli spa ..."
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Cited by 14 (1 self)
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The purpose of this paper is to give a proof of the Verlinde formulas by applying the RiemannRochKawasaki theorem to the moduli space of flat Gbundles on a Riemann surface # with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli space to be an orbifold, and the strata are described as moduli spaces for semisimple centralizers in G. The contribution of the strata are evaluated using the formulas of Witten for the symplectic volume, methods of symplectic geometry, including formulas of WittenJeffreyKirwan, and residue formulas.
GEOMETRIC QUANTIZATION FOR PROPER MOMENT MAPS
, 812
"... Abstract. We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a noncompact symplectic manifold such that the associated moment map is proper. In particular, we resolve the conjecture of Vergne in this noncompact setting. The famous geometric quant ..."
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Cited by 12 (2 self)
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Abstract. We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a noncompact symplectic manifold such that the associated moment map is proper. In particular, we resolve the conjecture of Vergne in this noncompact setting. The famous geometric quantization conjecture of Guillemin and Sternberg [9] states that for a compact prequantizable symplectic manifold admitting a Hamiltonian action of a compact Lie group, the principle of “quantization commutes with reduction ” holds. This conjecture was first proved independently by Meinrenken [14] and Vergne [23] for
Quantization formula for singular reductions
, 1997
"... We extend the recently proved holomorphic quantization formula of Teleman to cases of singular reductions. §0. Introduction and the statement of main results Let (M, ω, J) be a compact Kähler manifold with the Kähler form ω and the complex structure J. Let g TM denote the corresponding Kähler metric ..."
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Cited by 11 (3 self)
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We extend the recently proved holomorphic quantization formula of Teleman to cases of singular reductions. §0. Introduction and the statement of main results Let (M, ω, J) be a compact Kähler manifold with the Kähler form ω and the complex structure J. Let g TM denote the corresponding Kähler metric. We make the assumption that there exists a Hermitian line bundle L over M admitting a Hermitian connection ∇ L such that √ −1
Geometric quantization on Kähler and symplectic manifolds
 International Congress of Mathematicians
, 2010
"... We explain various results on the asymptotic expansion of the Bergman kernel on Kähler manifolds and also on symplectic manifolds. We also review the “quantization commutes with reduction ” phenomenon for a compact Lie group action, and its relation to the Bergman kernel. ..."
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Cited by 10 (5 self)
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We explain various results on the asymptotic expansion of the Bergman kernel on Kähler manifolds and also on symplectic manifolds. We also review the “quantization commutes with reduction ” phenomenon for a compact Lie group action, and its relation to the Bergman kernel.
Unitarity in “quantization commutes with reduction”
, 2007
"... Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M /G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515–538] have shown, under suitable regularity assumptions, that there ..."
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Cited by 10 (2 self)
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Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M /G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515–538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M /G and the Ginvariant subspace of the quantum Hilbert space over M. Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction ” and show that in this setting there is still a natural invertible map between the Hilbert space over M /G and the Ginvariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M /G.
Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids
 Int. J. Geom. Methods Mod. Phys
"... We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a p ..."
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Cited by 9 (1 self)
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We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a MarsdenWeinstein quotient for our setting and prove a “quantization commutes with reduction ” theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Finally, we investigate the functorial behaviour of these constructions under generalized morphisms of groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of families of Lie groups, and foliations, as well as some general constructions from differential geometry.