Stein extensions of Riemann symmetric spaces and dualities oforbits on flag manifolds

Simon Gindikin and Toshihiko Matsuki

\def\bc{\mathbb C} \def\br{\mathbb R} It is known that $K_\bc$-orbits $S$ and $G_\br$-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 $K_\bc$ by some conjugate $xK_\bc x^{-1}$ so that the correspondence is preserved. We will show that this replacement is related to the domain introduced by Akhiezer and Gindikin.