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The properties (Q) and (C). Commuting varieties. (Propriétés (Q) et (C). Variété commutante.) (French) Zbl 1092.14055
It is known that for a reductive Lie algebra $${\mathbf g}$$ the index of $${\mathbf g}$$ equals the rank of $${\mathbf g}$$, and conjectured by Elashvili that the index of the centralizer of an element of a reductive Lie algebra $${\mathbf g}$$ is equal to the rank of $${\mathbf g}$$. Several authors have achieved partial results in favor of this conjecture. The main result of this paper is a complete proof of Elashvili’s conjecture. The author’s approach begins in the general setting of algebraic varieties. Given an algebraic variety $$X$$, complex finite dimensional vector spaces $$E,F$$, and a morphism $$\mu:X\rightarrow \text{Lin} (E,F)$$, for each $$x\in X$$ there is an associated morphism $$\bar \mu _{x}:X\rightarrow \text{Lin}(\text{Ker} \mu (x), F/\text{Im} (\mu (x)))$$ defined by $$\bar \mu _x (y) (v)=\mu (y) (v) + \text{Im} (\mu (x))$$. The morphism $$\mu$$ is said to have property ($${\mathbf R}$$) at $$x$$ if the smallest dimension of $$\text{ker} (\bar \mu _{x} (y))$$ does not exceed the smallest dimension of $$\text{ker} (\mu (z))$$. Under conditions on $$\mu$$, the author proves that distinguished subsets of $$X$$ are of codimension at least $$2$$ if and only if closure of the set of points of $$X$$ for which $$\mu$$ does not have property ($${\mathbf R}$$) has codimension at least $$2$$. This situation is then specialized to the case of a Lie algebra $${\mathbf g}$$. The Lie algebra $${\mathbf g}$$ is said to have property ($${\mathbf Q}$$) at $$v\in {\mathbf g}^*$$ if the coadjoint map from $${\mathbf g}^*$$ to $$\text{Lin} ({\mathbf g}, {\mathbf g}^*)$$ has property ($${\mathbf R}$$) at $$v$$. Using the earlier result together with a result of Dixmier, the author proves that any reductive Lie algebra has property ($${\mathbf Q}$$) at any point of $${\mathbf g}^*$$, and this leads to a proof of Elashvili’s conjecture.

##### MSC:
 14L17 Affine algebraic groups, hyperalgebra constructions 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22E20 General properties and structure of other Lie groups 22E46 Semisimple Lie groups and their representations 17B05 Structure theory for Lie algebras and superalgebras
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