The Index of Centralizers of Elements of Reductive Lie Algebras

For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to A.G. 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. It reduces to the case of nilpotent elements. In \cite{Pa1} and \cite{Pa2}, D.I. Panyushev proved the conjecture for some classes of nilpotent elements (e.g. regular, subregular and spherical nilpotent elements). Then the conjecture has been proven for the classical Lie algebras in \cite{Y1} and checked with a computer programme for the exceptional ones \cite{De}. In this paper we give an almost general proof of that conjecture.

2010 Mathematics Subject Classification: 22E46, 17B80, 17B20, 14L24

Keywords and Phrases: reductive Lie algebra; index; centralizer; argument shift method; Poisson-commutative family of polynomials; rigid nilpotent orbit; Slodowy slice

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