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

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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.## References

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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
- Received by editor(s): October 20, 2004
- Received by editor(s) in revised form: April 28, 2005
- Published electronically: April 24, 2007
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 4773-4786 - MSC (2000): Primary 14M15, 22E15, 22E46, 32M05
- DOI: https://doi.org/10.1090/S0002-9947-07-04076-7
- MathSciNet review: 2320651