Moment maps and Riemannian symmetric pairs

O'Shea, Luis; Sjamaar, Reyer

doi:10.1007/pl00004408

Cited by 43 publications

(63 citation statements)

References 20 publications

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“…Biss, Guillemin and the second author [6] proved that these moment map components can also be used to understand the equivariant topology of the real locus with respect to the restricted T R action. In addition, Sjamaar and O'Shea have generalized the results of Duistermaat for Hamiltonian actions of nonabelian Lie groups [16]. The principle behind these results is that real loci should behave in a fashion similar to the symplectic manifolds of which they are submanifolds.…”

Section: Introductionmentioning

confidence: 94%

Real loci of symplectic reductions

Goldin

Holm

2004

Trans. Amer. Math. Soc.

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Abstract. Let M be a compact, connected symplectic manifold with a Hamiltonian action of a compact n-dimensional torus T . Suppose that M is equipped with an anti-symplectic involution σ compatible with the T -action. The real locus of M is the fixed point set M σ of σ. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction M//T has the same ordinary cohomology as its real locus (M//T ) σ red , with degrees halved. This extends Duistermaat's original result on real loci to a case in which there is not a natural Hamiltonian torus action.

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Section: Introductionmentioning

confidence: 94%

Real loci of symplectic reductions

Goldin

Holm

2004

Trans. Amer. Math. Soc.

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“…Moreover, if it is non-empty, M τ is a Lagrangian submanifold, called the real locus of M . For general work on such involutions together with a Hamiltonian group action, see [8] and [22].…”

Section: Preliminariesmentioning

confidence: 99%

Conjugation spaces

Hausmann¹,

Holm

Puppe

2005

Algebr. Geom. Topol.

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There are classical examples of spaces X with an involution τ whose mod2-comhomology ring resembles that of their fixed point set X τ : there is a ring isomorphism κ :. Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism κ is part of an interesting structure in equivariant cohomology called an H * -frame. An H * -frame, if it exists, is natural and unique. A space with involution admitting an H * -frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C k with the complex conjugation. A compact symplectic manifold, with an anti-symplectic involution compatible with a Hamiltonian action of a torus T , is a conjugation space, provided X T is itself a conjugation space. This includes the co-adjoint orbits of any semi-simple compact Lie group, equipped with the Chevalley involution. We also study conjugateequivariant complex vector bundles ("real bundles" in the sense of Atiyah) over a conjugation space and show that the isomorphism κ maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.

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“…its image under the T moment map, and its equivariant cohomology) have been extensively studied; see [Dui83], [OS00], [BGH01], [Sch01]. Most of the known results use the assumption, in addition to certain technical conditions about the T action, that M (and therefore Q) is compact.…”

Section: The Real Locusmentioning

confidence: 99%

The equivariant cohomology of hypertoric varieties and their real loci

Harada¹,

Holm²

2005

Communications in Analysis and Geometry

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Abstract. Let M be a Hamiltonian T space with a proper moment map, bounded below in some component. In this setting, we give a combinatorial description of the T -equivariant cohomology of M, extending results of Goresky, Kottwitz and MacPherson and techniques of Tolman and Weitsman. Moreover, when M is equipped with an antisymplectic involution σ anticommuting with the action of T , we also extend to this noncompact setting the "mod 2" versions of these results to the real locus Q := M σ of M. We give applications of these results to the theory of hypertoric varieties.

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Moment maps and Riemannian symmetric pairs

Cited by 43 publications

References 20 publications

Real loci of symplectic reductions

Real loci of symplectic reductions

Conjugation spaces

The equivariant cohomology of hypertoric varieties and their real loci

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