Generalization of the Bruhat decomposition

The problem of describing adjacency on the set of orbits of a Borel subgroup $B$ of a reductive group $G$ acting on a spherical variety (that is, a $G$-variety with a finite number of $B$-orbits) is considered. The adjacency relation on the set of $B$-orbits generalizes the classical Bruhat order on the Weyl group. For a special class of homogeneous spherical varieties $G/H$, where $H$ is a product of a maximal torus and the commutator subgroup of a maximal unipotent subgroup of the group $G$, a satisfactory description of the set of $B$-orbits with adjacency relation is obtained.