Spherical birational sheets in reductive groups

Abstract

We classify the spherical birational sheets in a complex simple simply-connected algebraic group. We use the classification to show that, when G is a connected reductive complex algebraic group with simply-connected derived subgroup, two conjugacy classes O1, O2 of G, with O1 spherical, lie in the same birational sheet, up to a shift by a central element of G, if and only if the coordinate rings of O1 and O2 are isomorphic as G-modules. As a consequence, we prove a conjecture of Losev for the spherical subvariety of the Lie algebra of G.