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49
An analytic proof of the geometric quantization conjecture
 of GuilleminSternberg, Invent. Math 132
, 1998
"... Abstract. We present a direct analytic approach to the GuilleminSternberg conjecture [GS] that ‘geometric quantization commutes with symplectic reduction’, which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods a ..."
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Cited by 55 (8 self)
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Abstract. We present a direct analytic approach to the GuilleminSternberg conjecture [GS] that ‘geometric quantization commutes with symplectic reduction’, which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts. Let M;x be a closed symplectic manifold such that there is a Hermitian line bundle L over M admitting a Hermitian connection rL with the property that ÿ1p
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.
A vector partition function for the multiplicities of slkC
, 2003
"... We use GelfandTsetlin diagrams to write down the weight multiplicity function for the Lie algebra slkC (type Ak−1) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to t ..."
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Cited by 22 (2 self)
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We use GelfandTsetlin diagrams to write down the weight multiplicity function for the Lie algebra slkC (type Ak−1) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the DuistermaatHeckman measure from symplectic geometry, which gives a largescale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the DuistermaatHeckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl4C (A3).
Quantization Of Presymplectic Manifolds And Circle Actions
, 1999
"... We prove several versions of “quantization commutes with reduction” for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin c structure. Our theorems work whenever the quantization data and the reduction data are compatible; this c ..."
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Cited by 17 (0 self)
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We prove several versions of “quantization commutes with reduction” for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin c structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, symplectic) manifold.
Almost complex structures and geometric quantization
 Math. Research Letters
, 1996
"... Abstract. We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin c quantization. We prove the analog of Kodaira vanishing for the Spin c Dirac operator, which shows that the index space of this operator provides an honest (no ..."
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Cited by 16 (1 self)
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Abstract. We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin c quantization. We prove the analog of Kodaira vanishing for the Spin c Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semiclassically. We also introduce a new quantization scheme, based on a rescaled Laplacian, for which we are able to prove strong semiclassical properties. The two quantizations are shown to be close semiclassically.
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