Schubert varieties and cycle spaces

Huckleberry, Alan; Wolf, Joseph A.

doi:10.1215/s0012-7094-03-12021-9

Cited by 14 publications

(21 citation statements)

References 23 publications

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“…The only refinement is that, instead of Σ ∩ κ being finite, we now show that it consists of just one point. This is implicit in [HW1] (see §5 in that paper). Let us repeat the relevant details.…”

Section: Basic Trialitymentioning

confidence: 89%

Cycles in G-orbits in G ? -flag manifolds

Huckleberry

Ntatin

2003

manuscripta mathematica

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There is a natural duality between orbits γ of a real form G of a complex semisimple group G C on a homogeneous rational manifold Z = G C /P and those κ of the complexification K C of any of its maximal compact subgroups K: (γ, κ) is a dual pair if γ ∩ κ is a K-orbit. The cycle space C(γ) is defined to be the connected component containing the identity of the interior of {g : g(κ)∩γ is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C(γ) is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C(γ) is a certain explicitly computable universal domain * Research partially supported by Schwerpunkt "Global methods in complex geometry" and SFB-237 of the Deutsche Forschungsgemeinschaft.† Supported by a stipend of the Deutsche Akademische Austauschdienst.

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Section: Basic Trialitymentioning

confidence: 89%

Cycles in G-orbits in G ? -flag manifolds

Huckleberry

Ntatin

2003

manuscripta mathematica

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“…We want to prove the statement of Theorem 7.1, namely, that G{D} • agrees with Ω. Equivalently, we will prove that MD agrees with Ξ. A. Huckleberry and J. Wolf [16]…”

Section: The Schubert Domainmentioning

confidence: 95%

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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“…Also let ( ) denote the closure of the C -orbit dual to the -orbit . It was shown in [8] that the intersection ( )∩Σ is finite. The following is a refinement of this result.…”

Section: Schubert Varieties and Slicesmentioning

confidence: 99%

“…We extend results and methods originally developed in [2], which is the author's PhD dissertation) for lower-dimensional orbits to the case of the unique closed orbit. For instance, Proposition 7 is implicit in [8] where it was only proven that the intersection of the base cycle with a Schubert slice is finite. Here we prove that this intersection is a single point.…”

Section: Introductionmentioning

confidence: 99%

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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Schubert varieties and cycle spaces

Cited by 14 publications

References 23 publications

Cycles in G-orbits in G ? -flag manifolds

Cycles in G-orbits in G ? -flag manifolds

Real Group Orbits on Flag Manifolds

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

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