A uniqueness property for spherical homogeneous spaces.

Let $G$ be a connected reductive group. A homogeneous $G$-space $X$ is called spherical if a Borel subgroup $B$ of $G$ has a dense orbit on $X$. To a spherical homogeneous space one assigns certain
combinatorial invariants: the weight lattice, the set of $B$-divisors and the valuation cone. In this talk we discuss the following uniqueness result: there is at most one spherical homogeneous space with given combinatorial invariants. This result was conjectured by Luna.