Simple Modules of Exceptional Groups with Normal Closures of Maximal Torus Orbits

TL;DR

This paper classifies simple modules of exceptional algebraic groups where the closure of every maximal torus orbit is normal, extending previous work on classical groups using a combinatorial normality criterion.

Contribution

It identifies all simple rational modules of exceptional groups with normal orbit closures, providing a complete classification for these cases.

Findings

All such modules for exceptional groups are classified.

Examples of modules with non-normal orbit closures are provided.

A combinatorial criterion for normality based on weights is used.

Abstract

Let G be an exceptional simple algebraic group, and let T be a maximal torus in G. In this paper, for every such G, we find all simple rational G-modules V with the following property: for every vector v in V, the closure of its T-orbit is a normal affine variety. For all G-modules without this property we present a T-orbit with the non-normal closure. To solve this problem, we use a combinatorial criterion of normality formulated in the terms of weights of a simple G-module. This paper continues two papers of the second author, where the same problem was solved for classical linear groups.