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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.
A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q), w
We provide a micro-local necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let $bf G$ be a complex algebraic reductive group, and $bf Hsubset G$ be a spherical algebraic su
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $mathbbm k$ of characteristic zero. We consider the commuting variety $mathcal C(mathfrak u)$ of the nilradical $mathfrak u$ of the Lie algebra $mathfrak b$ o
The commuting variety of a reductive Lie algebra ${goth g}$ is the underlying variety of a well defined subscheme of $gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.