Spherical birational sheets in reductive groups

TL;DR

This paper classifies spherical birational sheets in complex simple algebraic groups and establishes a criterion linking conjugacy classes and their coordinate rings, confirming a conjecture of Losev for spherical subvarieties.

Contribution

It provides a classification of spherical birational sheets and proves a new criterion relating conjugacy classes to their coordinate rings in reductive groups.

Findings

Classification of spherical birational sheets in simple algebraic groups

Conjugacy classes are characterized by isomorphic coordinate rings as G-modules

Proof of Losev's conjecture for spherical subvarieties of Lie algebras

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 $\mathcal{O}_1$, $\mathcal{O}_2$ of $G$ lie in the same birational sheet, up to a shift by a central element of $G$, if and only if the coordinate rings of $\mathcal{O}_1$ and $\mathcal{O}_2$ are isomorphic as $G$-modules. As a consequence, we prove a conjecture of Losev for the spherical subvariety of the Lie algebra of $G$.