Simple Modules of Classical Linear Groups with Normal Closures of Maximal Torus Orbits

TL;DR

This paper classifies simple modules of classical linear groups where the closure of every maximal torus orbit is normal, extending previous results from SL(n) to other classical groups using combinatorial criteria.

Contribution

It generalizes the classification of modules with normal orbit closures from SL(n) to all classical linear groups using a new combinatorial approach.

Findings

Identifies all simple modules with normal orbit closures for classical groups.

Provides examples of modules with non-normal orbit closures.

Extends previous work on SL(n) to broader classical groups.

Abstract

Let T be a maximal torus in a classical linear group G. In this paper we find all simple rational G-modules V such that for each vector v in V the closure of its T-orbit is a normal affine variety. For every other G-module we present a T-orbit with the non-normal closure. We use a combinatorial criterion of normality formulated in terms of the set of weights of a simple G-module. This work is a continuation of the previous work, where the same problem was solved in the case G=SL(n).