Stein extensions of Riemann symmetric spaces and dualities oforbits on flag manifolds
Simon Gindikin and Toshihiko Matsuki
It is known that $K_{\mathbb C}$-orbits $S$ and $G_{\mathbb R}$-orbits $S'$ on a complex flag manifold are in one-to-one correspondence by the condition that $S\cap S'$ is non-empty and compact. We may replace $K_{\mathbb C}$ by some conjugate $xK_{\mathbb C} x^{-1}$ so that the correspondence is preserved. We will show that this replacement is related to the domain introduced by Akhiezer and Gindikin.