Uniqueness property for spherical homogeneous spaces

TL;DR

This paper proves that spherical homogeneous G-spaces are uniquely determined by their combinatorial invariants and demonstrates how to recover their automorphism groups from these invariants.

Contribution

It establishes a uniqueness property for spherical homogeneous spaces based on combinatorial invariants and provides a method to determine their automorphism groups.

Findings

Two spherical homogeneous G-spaces with identical invariants are isomorphic.

The automorphism group can be reconstructed from the invariants.

The invariants uniquely characterize the spherical homogeneous spaces.

Abstract

Let G be a connected reductive group. Recall that a G-variety X is called spherical if X is normal and a Borel subgroup of G has an open orbit on X. To a spherical homogeneous G-space 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 show how to recover the group of G-equivariant automorphisms from these invariants.