Stein extensions of symmetric spaces and the orbit correspondence on flag manifolds

1.利用复化对称空间G_C/K_C中的岩泽投影，得到了一个凸集作为紧对称子空间或紧因果对称子空间的像，并证明了它包含G_C/K_C.2的其他实对称子空间的原点的邻域。S.Gindkin证明了星球柯西变换给出了紧对称空间上的Sato超函数空间与凸域上的全纯函数空间之间的同构关系。我们澄清了他几何思想的一部分。我们计算了最简单的秩二示例SU(3)/SO(3)，并研究了对偶星球柯西变换与作为星球柯西变换像的凸域的圈的交集(积分)的拓扑性质。我们研究了紧李群上扭曲共轭类的中心在基本胞上的作用。然后给出了紧李群上的外对合的一个简单分类。我们邀请了S.Gindkin，G.Olafsson和R.Stanton，并讨论了Akhiezer-Gindkin域和相关的表示理论。

1. It is known that we get a convex set as the image of the compact symmetric subspace or the compactly causal symmetric subspace by the Iwasawa projection in the complexified symmetric space G_C/K_C. We showed that the image contains a neighborhood of the origin for other real symmetric subspaces of G_C/K_C.2. S. Gindikin showed that the horospherical Cauchy transform gave an isomorphism between the space of Sato's hyperfunctions on compact symmetric spaces and the space of holomorphic functions on a convex domain. We clarified a part of his geometric idea.3. We computed the simplest rank-two example SU(3)/SO(3) and investigated topological properties of the intersection of cycles (for integration) for the dual horospherical Cauchy transform with the convex domain obtained as the image of the horospherical Cauchy transform.4. We investigated the action of the center on the fundamental cell for twisted conjugacy classes on compact Lie groups. Then we got a simple classification of exterior involutions on compact Lie groups.5. We invited S.Gindikin, G.Olafsson and R. Stanton and discussed the Akhiezer-Gindikin domain and related representation theories.