Maximal rank subgroups and strong functoriality of the additive eigencone

TL;DR

This paper investigates the structure of the additive eigencone associated with complex Lie groups, focusing on subgroups with strong functoriality properties, especially those from centralizers of torus elements, extending previous work on automorphism-invariant subgroups.

Contribution

It introduces a new class of subgroups from centralizers of torus elements that exhibit strong eigencone functoriality, expanding understanding beyond automorphism-invariant subgroups.

Findings

Identified subgroups with strong eigencone functoriality from centralizers of torus elements.

Extended the class of subgroups known to have this property beyond automorphism-invariant cases.

Provided new insights into the structure of the additive eigencone for complex Lie groups.

Abstract

Let $G$ be a simple connected complex Lie group. The additive eigencone of $G$ is a polyhedral cone containing the set of solutions to the additive eigenvalue problem, a generalization of the Hermitian eigenvalue problem. The additive eigencone is functorial, and for certain subgroups satisfies a stronger functoriality property: the eigencone of the subgroup is determined by the inequalities of the larger eigencone. Belkale and Kumar first studied this property for subgroups invariant under a diagram automorphism of $G$. We study a new class of subgroups arising from centralizers of torus elements that have the strong eigencone functoriality property.