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Equivalence of domains arising from duality of orbits on flag manifolds


Author: Toshihiko Matsuki
Journal: Trans. Amer. Math. Soc. 358 (2006), 2217-2245
MSC (2000): Primary 14M15, 22E15, 22E46, 32M05
DOI: https://doi.org/10.1090/S0002-9947-05-03824-9
Published electronically: October 21, 2005
MathSciNet review: 2197441
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Abstract: S. Gindikin and the author 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 non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open $K_{\mathbb C}$-orbit $S$ on an ``arbitrary'' flag manifold generalizing the result of Barchini. This conjecture for closed $S$ was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.


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Additional Information

Toshihiko Matsuki
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Email: matsuki@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-05-03824-9
Keywords: Flag manifolds, symmetric spaces, Stein extensions
Received by editor(s): October 6, 2003
Received by editor(s) in revised form: July 12, 2004
Published electronically: October 21, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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