Equivalence of domains arising from duality of orbits on flag manifolds

Matsuki, Toshihiko

doi:10.1090/S0002-9947-05-03824-9

Equivalence of domains arising from duality of orbits on flag manifolds
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Trans. Amer. Math. Soc. 358 (2006), 2217-2245 Request permission

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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