Cycles in G-orbits in G ? -flag manifolds

Huckleberry, Alan; Ntatin, B.

doi:10.1007/s00229-003-0405-1

Cited by 3 publications

(10 citation statements)

References 13 publications

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“…Furthermore, if ∈ ( ) is not open (i.e., if is a lower-dimensional orbit), then the following theorem has been proved for not of Hermitian type. Theorem 14 (see [1]). If is not of Hermitian type, then the cycle space associated with ( ) = Ω (18) for all ∈ ( ).…”

Section: Schubert Varieties and Slicesmentioning

confidence: 99%

“…In the sequel, we will follow the notation introduced in [1] or [2], and let ( ) (resp., ( C )) denote the set of all -orbits (resp., C -orbits) in . It is known that these sets are finite [3].…”

Section: Introductionmentioning

confidence: 99%

“…Observe that if is an open -orbit, then ( , ) is a dual pair if and only if ⊂ ; moreover, such a is unique [3]. This duality relation has been extended to include all -and C -orbits [1,[4][5][6][7]. For every ∈ ( ) there exists a unique ∈ ( C ) such that ( , ) is a dual pair and vice versa.…”

Section: Introductionmentioning

confidence: 99%

“…This helps in Section 5 to characterize the nature of the unique closed orbit in the full flag manifold in terms of signatures of a Hermitian form (Proposition 19). In Section 6, we use known results from [1,2,9] together with a result about cycle ordering (Proposition 21), to prove our main result (Theorem 24) that the cycle space of the closed orbit coincides with the universal domain Ω .…”

Section: Introductionmentioning

confidence: 99%

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The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

Ntatin

2019

Journal of Mathematics

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The orbits of a real form G of a complex semisimple Lie group GC and those of the complexification KC of its maximal compact subgroup K acting on Z=GC/Q, a homogeneous, algebraic, GC-manifold, are finite. Consequently, there is an open G-orbit. Lower-dimensional orbits are on the boundary of the open orbit with the lowest dimensional one being closed. Induced action on the parameter space of certain compact geometric objects (cycles) related to the manifold in question has been characterized using duality relations between G- and KC-orbits in the case of an open G-orbit and more recently lower-dimensional G-orbits. We show that the parameter space associated with the unique closed G-orbit in Z agrees with that of the other orbits characterized as a certain explicitly defined universal domain.

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Section: Schubert Varieties and Slicesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

Ntatin

2019

Journal of Mathematics

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“…The proof in [27] uses combinatorial description of the inclusion relations between the closures of K-orbits on the flag manifolds of G. As a corollary, we get that C(O) • is an open set, which is not clear a priori. If this is known, then Theorem 7.2 follows from [15] or from Theorem 12.1.3 in [11]. The latter asserts that the connected component of the interior of C(O), containing the neutral element e ∈ G, coincides with Ω.…”

Section: Cycle Spacesmentioning

confidence: 99%

Real Group Orbits on Flag Manifolds

Akhiezer

2013

Progress in Mathematics

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In this survey, we gather together various results on the action of a real form G0 of a complex semisimple group G on its flag manifolds. We start with the finiteness theorem of J. Wolf implying that at least one of the G0-orbits is open. We give a new proof of the converse statement for real forms of inner type, essentially due to F.M. Malyshev. Namely, if a real form of inner type G0 ⊂ G has an open orbit on a complex algebraic homogeneous space G/H then H is parabolic. In order to prove this, we recall, partly with proofs, some results of A.L. Onishchik on the factorizations of reductive groups. Finally, we discuss the cycle spaces of open G0-orbits and define the crown of a symmetric space of noncompact type. With some exceptions, the cycle space agrees with the crown. We sketch a complex analytic proof of this result, due to G. Fels, A. Huckleberry and J. Wolf. real form G0 acting locally transitively on an affine homogeneous space G/H is either SO1,7 or SO3,5. Moreover, in that case G/H = SO8/Spin7 and the action of G0 is in fact transitive (Corollary 4.7). This very special homogeneous space of a complex group G has on open orbit of a real form G0, the situation being typical for flag manifolds. One can ask what homogeneous spaces share this property. It turns out that if a real form of inner type G0 ⊂ G has an open orbit on a homogeneous space G/H with H algebraic, then H is in fact parabolic and so G/H is a flag manifold. We prove this in Section 5 (see Corollary 5.2) and then retrieve the result of F.M. Malyshev of the same type in which the isotropy subgroup is not necessarily algebraic (Theorem 5.4). It should be noted that, the other way around, the statement for algebraic homogeneous spaces can be deduced from his theorem. Our proof of both results is new.

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Actions on Flag Manifolds: Related Cycle Spaces

Huckleberry

Global Aspects of Complex Geometry

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Cycles in G-orbits in G ? -flag manifolds

Cited by 3 publications

References 13 publications

The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

Real Group Orbits on Flag Manifolds

Actions on Flag Manifolds: Related Cycle Spaces

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