Geodesic
Generalization of the Bruhat decomposition
Izvestiya. Mathematics , Tome 45 (1995) no. 2, pp. 339-352

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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. 
@article{IM2_1995_45_2_a5,
     author = {D. A. Timashev},
     title = {Generalization of the {Bruhat} decomposition},
     journal = {Izvestiya. Mathematics },
     pages = {339--352},
     publisher = {mathdoc},
     volume = {45},
     number = {2},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_45_2_a5/}
}
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D. A. Timashev. Generalization of the Bruhat decomposition. Izvestiya. Mathematics , Tome 45 (1995) no. 2, pp. 339-352. http://geodesic.mathdoc.fr/item/IM2_1995_45_2_a5/