Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equivalence of domains arising from duality of orbits on flag manifolds III

Author: Toshihiko Matsuki
Journal: Trans. Amer. Math. Soc. 359 (2007), 4773-4786
MSC (2000): Primary 14M15, 22E15, 22E46, 32M05
Published electronically: April 24, 2007
MathSciNet review: 2320651
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Abstract: In Gindikin and Matsuki 2003, we defined a $ G_{\mathbb{R}}$- $ K_{\mathbb{C}}$ invariant subset $ C(S)$ of $ G_{\mathbb{C}}$ for each $ K_{\mathbb{C}}$-orbit $ S$ on every flag manifold $ G_{\mathbb{C}}/P$ and conjectured that the connected component $ C(S)_0$ of the identity would be equal to the Akhiezer-Gindikin domain $ D$ if $ S$ is of nonholomorphic type. This conjecture was proved for closed $ S$ in Wolf and Zierau 2000 and 2003, Fels and Huckleberry 2005, and Matsuki 2006 and for open $ S$ in Matsuki 2006. It was proved for the other orbits in Matsuki 2006, when $ G_{\mathbb{R}}$ is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed $ K_{\mathbb{C}}$-orbit when $ G_{\mathbb{R}}$ is of Hermitian type. Thus the conjecture is completely solved affirmatively.

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Toshihiko Matsuki
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Keywords: Flag manifolds, symmetric spaces, Stein extensions
Received by editor(s): October 20, 2004
Received by editor(s) in revised form: April 28, 2005
Published electronically: April 24, 2007
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.