Stein extensions of real symmetric spaces and the geometry of the flag manifold

Barchini, L.

doi:10.1007/s00208-003-0419-8

Cited by 21 publications

(16 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The inclusion Ω AG ⊂ Ω I was proved by complex analytic methods (see [22]), but now there is an algebraic proof (see [33]) which may be more appropriate, because the situation would apriori seem to be algebraic in nature. The inclusion Ω I ⊂ Ω AG was shown in [2]. Thus,…”

Section: Introductionmentioning

confidence: 95%

Characterization of cycle domains via Kobayashi hyperbolicity

Fels¹,

Huckleberry²

2005

Bul. Soc. Math. France

View full text Add to dashboard Cite

Abstract. -A real form G of a complex semi-simple Lie group G C has only finitely many orbits in any given G C -flag manifold Z = G C /Q. The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits D generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of D which, with very few well-understood exceptions, are parameterized by the Wolf cycle domains Ω W (D) in G C /K C , where K is a maximal compact subgroup of G. Thus, for the various domains D in the various ambient spaces Z, it is possible to compare the cycle spaces Ω W (D). The main result here is that, with the few exceptions mentioned above, for a fixed real form G all of the cycle spaces Ω W (D) are the same. They are equal to a universal domain Ω AG which is natural from the the point of view of group actions and which, in essence, can be explicitly computed. The essential technical result is that if b Ω is a G-invariant Stein domain which contains Ω AG and which is Kobayashi hyperbolic, then b Ω = Ω AG . The equality of the cycle domains follows from the fact that every Ω W (D) is itself Stein, is hyperbolic, and contains Ω AG . FELS (G.) & HUCKLEBERRY (A.)Résumé (Caractérisation de domaines de cycles par l'hyperbolicité au sens de Kobayashi) Une forme réelle G d'un groupe de Lie semi-simple G C n'admet qu'un nombre fini d'orbites dans toute G C -variété de drapeaux Z = G C /Q. La géométrie complexe de ces orbites est intéressante, par exemple pour la théorie de la représentation associée. Les fonctions holomorphes sur les orbites ouvertes D de G sont constantes en général ; les objets géométriques importants liésà ces orbites sont des sous-variétés complexes de D de dimension positives qui,à quelques rares exceptions bien comprises, sont paramétrées par les domaines de cycles de Wolf Ω W (D) ∈ G C /K C , où K est un sous-groupe maximal compact de G. Alors, pour les domaines D dans les variétés ambiantes Z, il est possible de comparer les domaines de cycles Ω W (D

show abstract

Section: Introductionmentioning

confidence: 95%

Characterization of cycle domains via Kobayashi hyperbolicity

Fels¹,

Huckleberry²

2005

Bul. Soc. Math. France

View full text Add to dashboard Cite

show abstract

“…Then the following assertions hold: (1) The spherical function φ λ has a unique holomorphic extension to . (2) The spherical function φ λ has a unique continuation to a K C -bi-invariant func-…”

Section: Applications To Spherical Functionsmentioning

confidence: 99%

“…[3,9,13], [2,11], [4,10,12] and [16]). Using the complex convexity theorem from [6] one can significantly improve on 1.2, namely…”

Section: Introductionmentioning

confidence: 99%

A convexity property for the SO (2,C)-double coset decomposition of SL (2,C) and applications to spherical functions

Krötz

Otto

2004

Mathematische Zeitschrift

View full text Add to dashboard Cite

show abstract

“…The K 0 -invariant gradient field ∇ f + of the norm function f + := µ K 0 2 of the moment map for the K 0 -action on Z , determined by the G u -invariant metric, is tangent to both the G 0 -and K -orbits. (2) A pair (γ , κ) satisfies duality if and only if their intersection is nonempty and contains a point of {∇ f + = 0}. If the pair does not satisfy duality, p ∈ γ ∩ κ, g = g(t) is the 1-parameter group associated to ∇ f + , and > 0 is sufficiently small, then…”

Section: Dualitymentioning

confidence: 99%

Schubert varieties and cycle spaces

Huckleberry¹,

Wolf²

2003

Duke Math. J.

View full text Add to dashboard Cite

Let G 0 be a real semisimple Lie group. It acts naturally on every complex flag manifold Z = G/Q of its complexification. Given an Iwasawa decomposition G 0 = K 0 A 0 N 0 , a G 0 -orbit γ ⊂ Z , and the dual K -orbit κ ⊂ Z , Schubert varieties are studied and a theory of Schubert slices for arbitrary G 0 -orbits is developed. For this, certain geometric properties of dual pairs (γ , κ) are underlined. Canonical complex analytic slices contained in a given G 0 -orbit γ which are transversal to the dual K 0orbit γ ∩ κ are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space W (D) is a Stein domain that contains the universally defined Iwasawa domain I . This is one of the main ingredients in the proof that W (D) = AG for all but a few Hermitian exceptions. In the Hermitian case, W (D) is concretely described in terms of the associated bounded symmetric domain. IntroductionLet G be a connected complex semisimple Lie group, and let Q be a parabolic subgroup. We refer to Z = G/Q as a complex flag manifold. Write g and q for the respective Lie algebras of G and Q. Then Q is the G-normalizer of q. Thus we can view Z as the set of G-conjugates of q. The correspondence is z ↔ q z , where q z is the Lie algebra of the isotropy subgroup Q z of G at z.Let G 0 be a real form of G, and let g 0 denote its Lie algebra. Thus there is a homomorphism ϕ : G 0 → G such that ϕ(G 0 ) is closed in G and dϕ : g 0 → g is an isomorphism onto a real form of g. This gives the action of G 0 on Z .It is well known (see [21]) that there are only finitely many G 0 -orbits on Z . Therefore at least one of them must be open.Consider a Cartan involution θ of G 0 , and extend it as usual to G, g 0 , and g. Thus of a complex analytic nature, provide detailed information on the q-convexity of D and the Cauchy-Riemann geometry of the lower-dimensional G 0 -orbits.

show abstract

Stein extensions of real symmetric spaces and the geometry of the flag manifold

Cited by 21 publications

References 4 publications

Characterization of cycle domains via Kobayashi hyperbolicity

Characterization of cycle domains via Kobayashi hyperbolicity

A convexity property for the SO (2,C)-double coset decomposition of SL (2,C) and applications to spherical functions

Schubert varieties and cycle spaces

Contact Info

Product

Resources

About