The index of centralizers of elements of reductive Lie algebras

TL;DR

This paper proves that the index of the centralizer of an element in a reductive Lie algebra equals the algebra's rank, confirming a long-standing conjecture in Lie theory.

Contribution

It provides an almost general proof of Elashvili's conjecture regarding the index of centralizers in reductive Lie algebras.

Findings

Confirmed that the index of centralizers equals the rank in reductive Lie algebras

Extended the validity of Elashvili's conjecture to a broader class of cases

Contributed to the understanding of the structure of reductive Lie algebras

Abstract

For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.