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Geometric quantization for proper moment maps (2009)

by X Ma, W Zhang
Venue:I
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Geometric quantization on Kähler and symplectic manifolds

by Xiaonan Ma - International Congress of Mathematicians , 2010
"... We explain various results on the asymptotic expansion of the Bergman ker-nel 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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We explain various results on the asymptotic expansion of the Bergman ker-nel 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.

Formal geometric quantization II

by Paul-emile Paradan , 2009
"... In this paper we pursue the study of formal geometric quantization of non-compact Hamiltonian manifolds. Our main result is the proof that two quantization process coincide. This fact was obtained by Ma and Zhang in the preprint Arxiv:0812.3989 by completely different means. ..."
Abstract - Cited by 3 (0 self) - Add to MetaCart
In this paper we pursue the study of formal geometric quantization of non-compact Hamiltonian manifolds. Our main result is the proof that two quantization process coincide. This fact was obtained by Ma and Zhang in the preprint Arxiv:0812.3989 by completely different means.

Quantization commutes with REDUCTION IN THE NON-COMPACT SETTING: THE CASE OF HOLOMORPHIC DISCRETE SERIES

by Paul-emile Paradan , 2012
"... In this paper we show that the multiplicities of holomorphic discrete series representations relatively to reductive subgroups satisfy the credo “Quantization commutes with reduction”. ..."
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In this paper we show that the multiplicities of holomorphic discrete series representations relatively to reductive subgroups satisfy the credo “Quantization commutes with reduction”.

GEOMETRIC QUANTIZATION AND FAMILIES OF INNER PRODUCTS

by Peter Hochs, Varghese Mathai
"... ar ..."
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...ts index is not well-defined; • if the compact group K is replaced by a noncompact group G, the finite-dimensional representations of G, which make up R(G), are not the interesting ones. Ma and Zhang =-=[17, 18]-=- solved an extended version of Vergne’s conjecture [30] on quantization commutes with reduction for compact groups G = K acting on noncompact manifolds, and later on Paradan [26, 27] gave a new proof ...

FORMAL GEOMETRIC QUANTISATION FOR PROPER ACTIONS

by Peter Hochs, Varghese Mathai
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...geometric quantisation is functorial with respect to Cartesian products and restriction to subgroups. These two properties imply that it is compatible with the ring structure on R−∞(K). Ma and Zhang (=-=[12]-=- and [13]), and also Paradan [17] proved that quantisation commutes with reduction, in the sense that (0.1) QK(N, ν) = Q −∞ K (N, ν), for a certain definition of the quantisation QK(N, ν) ∈ R −∞(K). O...

Quantisation of presymplectic . . .

by Peter Hochs , 2014
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KOMABA, TOKYO, JAPANTORUS FIBRATIONS AND LOCALIZATION OF INDEX III- EQUIVARIANT VERSION AND ITS APPLICATIONS

by Hajime Fujita, Mikio Furuta, Takahiro Yoshida, Hajime Fujita, Mikio Furuta, Takahiko Yoshida , 2010
"... fibrations and localization of index III – equivariant version and its applications by ..."
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fibrations and localization of index III – equivariant version and its applications by
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...s not necessarily compact, Vergne [15] formulated a version of quantization conjecture for general compact Lie group G in terms of transversally elliptic operator [1], which is proved by Ma and Zhang =-=[7]-=- and Tian and Zhang [12]. We expect that there is a relation between our G-invariant index RRG K (M, V, L) and the index of the transversally elliptic operator for the case of torus action. We, howeve...

FORMAL GEOMETRIC QUANTIZATION II

by Paul-Emile Paradan , 2009
"... In this paper we pursue the study of formal geometric quantization of non-compact Hamiltonian manifolds. Our main result is the proof that two quantization process coincide. This fact was obtained by Ma and Zhang in the preprint arXiv:0812.3989 by completely different means. ..."
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In this paper we pursue the study of formal geometric quantization of non-compact Hamiltonian manifolds. Our main result is the proof that two quantization process coincide. This fact was obtained by Ma and Zhang in the preprint arXiv:0812.3989 by completely different means.
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...ment map. There is another way, denoted Q Φ , of quantizing proper Hamiltonian manifolds by localizing the index of the Dolbeault Dirac operator on the critical points of the square of the moment map =-=[15, 19, 20]-=-. The main purpose of this paper is to provide a geometric proof that the quantization process Q −∞ and Q Φ coincide. This fact was proved by Ma and Zhang in the recent preprint [15] by completely dif...

Transversal Index and

by Xiaonan Ma, Weiping Zhang
"... Abstract. For manifolds with boundary, we present a self-contained proof of Braverman’s result which gives an alternative interpretation of the transversal index through certain kind of ..."
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Abstract. For manifolds with boundary, we present a self-contained proof of Braverman’s result which gives an alternative interpretation of the transversal index through certain kind of
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...n the sense of Atiyah [1] and Paradan [9], and these indices coincide with the indices of Spin...

TORUS FIBRATIONS AND LOCALIZATION OF INDEX III- EQUIVARIANT VERSION AND ITS APPLICATIONS

by Hajime Fujita, Mikio Furuta, Takahiko Yoshida
"... ar ..."
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...s not necessarily compact, Vergne [15] formulated a version of quantization conjecture for general compact Lie group G in terms of transversally elliptic operator [1], which is proved by Ma and Zhang =-=[7]-=- and Tian and Zhang [12]. We expect that there is a relation between our G-invariant index RRGK(M,V, L) and the index of the transversally elliptic operator for the case of torus action. We, however, ...

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