Upper half-plane in the Grassmanian $Gr(n;2n)$

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.