Semistable points with respect to real forms

Heinzner, Peter; Stötzel, Henrik

doi:10.1007/s00208-006-0063-1

Cited by 27 publications

(30 citation statements)

References 2 publications

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“…The above proposition is related with well-known results on convexity properties of the moment map (see, for instance [HSto,Proposition 3]). The argument used in the proof shows that it is independent of complex case (which is a difference with analogous results).…”

Section: Generalization Of Nikolayevsky's Nice Basis Criteriummentioning

confidence: 60%

On distinguished orbits of reductive representations

Fernández-Culma

2013

Journal of Algebra

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Abstract. Let G be a real reductive Lie group and let τ ∶ G → GL(V ) be a real reductive representation of G with (restricted) moment map mg ∶ V ∖ {0} → g. In this work, we introduce the notion of nice space of a real reductive representation to study the problem of how to determine if a G-orbit is distinguished (i.e. it contains a critical point of the norm squared of mg). We give an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in our particular case and we use it to give an easy-to-check sufficient condition for a G-orbit of a element in a nice space to be distinguished. In the case where G is algebraic and τ is a rational representation, the above condition is also necessary (making heavy use of recent results of M. Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterium. We also provide useful characterizations of nice spaces in terms of the weights of τ . Finally, some applications to ternary forms and minimal metrics on nilmanifolds are presented.

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Section: Generalization Of Nikolayevsky's Nice Basis Criteriummentioning

confidence: 60%

On distinguished orbits of reductive representations

Fernández-Culma

2013

Journal of Algebra

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“…Both g 0 (8) and g −1 (8) are Einstein nilradicals, the coefficient vector c can be taken as 1 28 (4, 2, 3, 1, 2, 2, 3, 2, 2, 2, 5) t for α=0 and as 1 28 (2, 2, 2, 4, 3, 1, 3, 3, 2, 3, 3) t for α= − 1. The exceptional values for g α (9) are α = −2, −1, 0. The algebra g −2 (9) is not an Einstein nilradical, as for any linear combination of the vectors from F representing p, c 7 16 < 0.…”

Section: Algebras Of the Class Amentioning

confidence: 98%

“…. , n − 1 Algebra n Extension of n b(6) m 2 (5) b (8) g −5/2 (7) b 1 (10) g −1 (9) b 2 (10) g −3 (9) b ± (12) g α (11), α = −4± √ 10 2 for any X, Y, Z ∈ n , where σ is the sum of the cyclic permutations. The fact that n is filiform implies that ω = 0.…”

Section: Algebras Of the Class Amentioning

confidence: 99%

“…, 11, the set F for the algebras g α (n) from Table 2 is the same as for the corresponding algebra V(n), so each of them is an Einstein nilradical. We treat the exceptional values below by either giving the coefficient vector c = (c k i j ) of a convex linear combination p = c k i j α k i j (the vectors α k i j are always ordered lexicographically), or otherwise, by showing that some coefficient of any (9) relations for g α (8) and such linear combination is nonpositive. According to Theorem 1, the corresponding algebra is an Einstein nilradical in the former case, and is not in the latter one.…”

Section: Algebras Of the Class Amentioning

confidence: 99%

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Einstein solvmanifolds with a simple Einstein derivation

Nikolayevsky

2008

Geom Dedicata

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The structure of a solvable Lie group admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-tocheck necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).

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“…A class of examples of gradient manifolds is given by coadjoint orbits (see [10]). Let α ∈ u * and let Z = U · α be the coadjoint orbit of α. Identifying u * with iu as before, α corresponds to an element ξ ∈ iu and Z corresponds to the orbit of ξ of the adjoint action of U on iu.…”

Section: Coadjoint Orbitsmentioning

confidence: 99%

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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Semistable points with respect to real forms

Abstract: Abstract. We consider actions of real Lie subgroups G of complex reductive Lie groups on Kählerian spaces. Our main result is the openness of the set of semistable points with respect to a momentum map and the action of G.

Cited by 27 publications

References 2 publications

On distinguished orbits of reductive representations

On distinguished orbits of reductive representations

Einstein solvmanifolds with a simple Einstein derivation

Spherical gradient manifolds

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