Uniqueness property for spherical homogeneous spaces

Abstract

Let $G$ be a connected reductive group. Recall that a homogeneous $G$-space $X$ is called spherical if a Borel subgroup $B\subset G$ has an open orbit on $X$. To $X$ one assigns certain combinatorial invariants: the weight lattice, the valuation cone, and the set of $B$-stable prime divisors. We prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic. Further, we recover the group of $G$-equivariant automorphisms of $X$ from these invariants