The Index of a Lie Algebra, the Centraliser of a Nilpotent Element, and the Normaliser of the Centraliser

This definition goes back to J.Dixmier, see [Di, 11.1.6]. He considered index because of its importance in Representation Theory. The problem of computing the index may also be treated as part of Invariant Theory. For, if q is an algebraic Lie algebra and Q is a corresponding algebraic group, then ind q equals the transcendence degree of the field of Q-invariant rational functions on q. If q is reductive, then q and q are isomorphic as qmodules and hence ind q = rk q. It is therefore interesting to study index for non-reductive Lie algebras. On the other hand, studying the index for all Lie algebras is too pretentious. Therefore I think that the most promising approach is to look at the index for various natural classes of non-reductive subalgebras of semisimple Lie algebras. There are at least two such classes: a) parabolic subalgebras and their ‘relatives‘ (nilpotent radicals, seaweeds, etc.); b) centralisers of elements and their ‘relatives’. Some recent results on a) are found in [Pa4], whereas the present paper deals mainly with b).