Summary: 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 citePa1 and citePa2, 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 citeY1 and checked with a computer programme for the exceptional ones citeDe. In this paper we give an almost general proof of that conjecture.

22E46, 17B80, 17B20, 14L24

reductive Lie algebra, index, centralizer, argument shift method, Poisson-commutative family of polynomials, rigid nilpotent orbit, slodowy slice