Multiplicity-free complex manifolds

Huckleberry, Alan; Wurzbacher, Tilmann

doi:10.1007/bf01453576

Cited by 51 publications

(77 citation statements)

References 11 publications

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“…In the special case G = K C it is proven in [5] that for generic x the fiber K µp(x) /K x is a torus. As a generalization we prove the following proposition, which also allows us to extend the notion of "K-spherical" defined in [13] to actions of real-reductive groups. Proposition 1.2.…”

Section: Annales De L'institut Fouriermentioning

confidence: 93%

“…Notice that our proof of Brion's theorem is different from the ones in [4] and [13]. In particular, for every generic element x ∈ X we construct a minimal parabolic subgroup…”

Section: Annales De L'institut Fouriermentioning

confidence: 98%

“…(1) If µ locally almost separates the U -orbits, then µ separates all the U -orbits in Z (see [13]). (2) Since X is Lagrangian, we see that µ k | X ≡ 0, where µ k denotes the moment map for the K-action on Z.…”

Section: Examplesmentioning

confidence: 99%

“…Assume furthermore that the U -action on Z is Hamiltonian, i. e. that there exists a U -equivariant moment map µ : Z → u * where u denotes the Lie algebra of U . In the special case that Z is compact it is shown in [4] (see also [13]) that µ separates the U -orbits if and only if Z is a spherical U C -manifold, which means that a Borel subgroup of U C has an open orbit in Z. Note that µ separates the U -orbits if and only if it induces an injective map Z/U → u/U .…”

Section: Introductionmentioning

confidence: 99%

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Spherical gradient manifolds

Miebach¹,

Stötzel²

2010

Ann. inst. Fourier

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18 pagesNous étudions l'action d'un groupe réel-réductif $G=K\exp(\lie{p})$ sur une sous-variété réel-analytique $X$ d'une variété kählérienne $Z$. Nous supposons que l'action de $G$ peut être prolongée à une action holomorphe du groupe complexifié $G^\mbb{C}$ telle que l'action d'un sous-groupe maximal compact de $G^\mbb{C}$ soit hamiltonienne. L'application moment $\mu$ induit une application gradient $\mu_\lie{p}\colon X\to\lie{p}$. Nous montrons que $\mu_\lie{p}$ separe les orbites de $K$ si et seulement si un sous-groupe minimal parabolique de $G$ possède une orbite ouverte dans $X$. Ce résultat généralise la charactérisation de Brion des variétés kählériennes sphériques qui admettent une application moment

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Section: Annales De L'institut Fouriermentioning

confidence: 93%

“…Notice that our proof of Brion's theorem is different from the ones in [4] and [13]. In particular, for every generic element x ∈ X we construct a minimal parabolic subgroup…”

Section: Annales De L'institut Fouriermentioning

confidence: 98%

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spherical gradient manifolds

Miebach¹,

Stötzel²

2010

Ann. inst. Fourier

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“…We refer the reader to [5] for the notion of a Hamiltonian K-action on M and the notion of a Poisson K-action on M .…”

Section: Definition 42mentioning

confidence: 99%

Spherical nilpotent orbits and the Kostant-Sekiguchi correspondence

King

2002

Trans. Amer. Math. Soc.

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Abstract. Let G be a connected, linear semisimple Lie group with Lie algebra g, and let K C → Aut(p C ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. The Kostant-Sekiguchi correspondence is a bijection between the nilpotent K C -orbits in p C and the nilpotent G-orbits in g. We show that this correspondence associates each spherical nilpotent K C -orbit to a nilpotent G-orbit that is multiplicity free as a Hamiltonian K-space. The converse also holds.

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