Geodesic
Upper half-plane in the Grassmanian $Gr(n;2n)$
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 406-411

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We investigate the complex geometry of a multidimensional generalization $\mathcal{D}(n)$ of the upper-half-plane, which is homogeneous relative the group $G=SL(2n; \mathbb{R})$. For $n>1$ it is the pseudo Hermitian symmetric space which is the open orbit of $G=SL(2n; \mathbb{R})$ on the Grassmanian $Gr_\mathbb{C}(n;2n)$ of $n$-dimensional subspaces of $\mathbb{C}^{2n}$. The basic element of the construction is a canonical covering of $\mathcal{D}(n)$ by maximal Stein submanifolds — horospherical tubes.
Keywords: Grassmanian, pseudo Hermitian symmetric space, cycle, horosphere, horospherical tube. 
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     author = {Simon Gindikin},
     title = {Upper half-plane in the {Grassmanian} $Gr(n;2n)$},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
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Simon Gindikin. Upper half-plane in the Grassmanian $Gr(n;2n)$. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 406-411. http://geodesic.mathdoc.fr/item/JSFU_2019_12_4_a0/