Uniqueness property for smooth affine spherical varieties

Let G be a connected reductive algebraic group over an

algebraically closed field of characteristic 0. A normal

irreducible G-variety X is called spherical if a Borel

subgroup of G has an open orbit on X. It was conjectured by F.

Knop that two smooth affine spherical G-varieties are

equivariantly isomorphic provided their algebras of regular

functions are isomorphic as G-modules. Knop proved that this

conjecture implies a uniqueness property for multiplicity free

Hamiltonian actions of compact groups on compact real manifolds

(the Delzant conjecture). In the talk I am going to outline my

recent proof of Knop's conjecture (arXiv:math/AG.0612561).