Siberian Federal University

We investigate the complex geometry of a multidimensional generalization D(n) of the upper-half-plane, which is homogeneous relative the group G = SL(2n;R). For n > 1 it is the pseudo Hermitian symmet- ric space which is the open orbit of G = SL(2n;R) on the Grassmanian GrC(n; 2n) of n-dimensional subspaces of C2n. The basic element of the construction is a canonical covering of D(n) by maximal Stein submanifolds — horospherical tubes