Polar representations of compact groups and convex hulls of their orbits

Gichev, V. M.

doi:10.1016/j.difgeo.2010.05.005

Cited by 11 publications

(10 citation statements)

References 11 publications

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“…Set E = Conv(F p (G · x)). By Lemma 7 in [26] we get π a (E) =F a (A · π(v)) and so E = KF a (A · v) = CP I . By Theorem 0.3 in [12], any face of E is exposed.…”

Section: Real Reductive Representations On Projective Spacesmentioning

confidence: 90%

“…Since F a = π a • F p and F a (G · x) is convex, it follows π a (E) = π a (F p (G · x)) = F a (G · x). By Lemma 7 in [26] we have E = KC.…”

Section: Convexity Theorems For Real Reductive Representationsmentioning

confidence: 94%

“…The authors applyed the Kirwan'Theorem [27] for the action U × T on the cotangent bundle T * U , where T is a maximal torus of U . Our proof uses a result of Gichev [15]. Theorem 5.5.…”

Section: Thenmentioning

confidence: 99%

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Convexity properties of gradient maps associated to real reductive representations

Biliotti

2020

Journal of Geometry and Physics

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Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson [38]. A similar result is proved for the gradient map associated to the natural G action on P(V ). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic result.2010 Mathematics Subject Classification. 22E45,53D20; 14L24.

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“…Set E = Conv(F p (G · x)). By Lemma 7 in [26] we get π a (E) =F a (A · π(v)) and so E = KF a (A · v) = CP I . By Theorem 0.3 in [12], any face of E is exposed.…”

Section: Real Reductive Representations On Projective Spacesmentioning

confidence: 90%

“…Since F a = π a • F p and F a (G · x) is convex, it follows π a (E) = π a (F p (G · x)) = F a (G · x). By Lemma 7 in [26] we have E = KC.…”

Section: Convexity Theorems For Real Reductive Representationsmentioning

confidence: 94%

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Convexity properties of gradient maps associated to real reductive representations

Biliotti

2020

Journal of Geometry and Physics

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“…The convex hull of an orbit of an isometric representation is called orbitope [60]. These convex bodies have been largely studied in [9,10,11], see also [28,50,60], when in particular the authors proved a reduction Theorem in the sense of polar representations.…”

Section: Introductionmentioning

confidence: 99%

“…Since P = µ a (O) [10], by a theorem of Kostant [53] it is a polytope and so any face of P is exposed [62]. Then, any face of E is exposed as well and E = Kµ a (O) [10], see also [11,28]. The compact group K acts on the set of the faces of E and the Weyl group W = N K (a)/K a acts on the set of faces of P , where N K (a) = {k ∈ K : Ad(k)(a) = a} is the normalizer of a and K a = {k ∈ K : Ad(k)(z) = z, for any z ∈ a} is the centralizer of a in K. Let F (E) and F (P ) denote the sets of faces of E and P , respectively.…”

Section: Introductionmentioning

confidence: 99%

Satake-Furstenberg compactifications and gradient map

Biliotti¹

2020

Preprint

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Let G be a real semisimple Lie group with finite center and let g = k⊕p be a Cartan decomposition of its Lie algebra. Let K be a maximal compact subgroup of G with Lie algebra k and let τ be an irreducible representation of G on a complex vector space V . We denote by µp : P(V ) −→ p the G-gradient map and by O the unique closed orbit of G in P(V ) which is a K-orbit [33,40]. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of G induced by τ are completely determined by the facial structure of the polar orbitope E = conv(µp(O)). Moreover, any parabolic subgroup of G admits a unique closed orbit which is well-adapted to O and µp respectively. These results are new also in the complex reductive case. The connection between E and τ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure γ on O, we construct a map Ψγ from the Satake compactification of G/K associated to τ and E . If γ is a K-invariant measure then Ψγ is an homeomorphism of the Satake compactification and E . Finally, we prove that for a large class of measures the map Ψγ is surjective.

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Homogeneous Spaces with Inner Metric and with Integrable Invariant Distributions

Berestovskiî

Горбацевич

2015

J Math Sci

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Polar representations of compact groups and convex hulls of their orbits

Cited by 11 publications

References 11 publications

Convexity properties of gradient maps associated to real reductive representations

Convexity properties of gradient maps associated to real reductive representations

Satake-Furstenberg compactifications and gradient map

Homogeneous Spaces with Inner Metric and with Integrable Invariant Distributions

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