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

The index of centralizers of elements of reductive Lie algebras