The centralisers of nilpotent elements in the classical Lie algebras

The definition of index goes back to Dixmier [3, 11.1.6]. This notion is important in Representation Theory and also in Invariant Theory. By Rosenlicht’s theorem [12], generic orbits of an arbitrary action of a linear algebraic group on an irreducible algebraic variety are separated by rational invariants; in particular, ind g = tr.degK(g). The index of a reductive algebra equals its rank. Computing the index of an arbitrary Lie algebra seems to be a wild problem. However, there is a number of interesting results for several classes of non-reductive subalgebras of reductive Lie algerbas. For instance, parabolic subalgebras and their “relatives” (nilpotent radicals, seaweeds) are considered in [4], [8], [13]. The centralisers of elements form another interesting class of subalgebras. The last topic is closely related to the theory of integrable Hamiltonian systems. Let G be a semisimple Lie group (complex or real), g = LieG, and Gx an orbit of a covector x ∈ g. Let gx denote the stabiliser of x. It is well-known that the orbit Gx possesses a G-invariant symplectic structure. There is a family of commuting with respect to a Poisson bracket polynomial functions on g constructed by the argument shift method such that its restriction to Gx contains 1 2 dim(Gx) algebraically independent functions if and only if ind gx = ind g, [1]. Conjecture (Élashvili). Let g be a reductive Lie algebra. Then ind gx = ind g for each covector x ∈ g. Recall that if g is reductive, then the g-modules g and g are isomorphic. In particular, it is enough to prove the “index conjecture” for stabilisers of vectors x ∈ g. Given x ∈ g, let x = xs + xn be the Jordan decomposition. Then gx = (gxs)xn . The subalgebra gxs is reductive and contains a Cartan subalgebra of g. Hence, ind gxs = ind g = rk g. Thus, a verification of the ”index conjecture” is reduced to the computation of ind gxn for nilpotent elements xn ∈ g. Clearly, we can restrict ourselves to the case of simple g. Note that if x is a regular element, then the stabiliser gx is commutative and of dimension rk g. The “index conjecture” was proved for subregular nilpotents and nilpotents of height 2 [9], and also for nilpotents of height 3 [10]. (The height of a nilpotent element e is the maximal number m such that (ad e) 6= 0.) Recently, Élashvili’s conjecture was proved by Charbonnel [2] for K = C. In the present article, we prove in an elementary way, that for any nilpotent element e ∈ g of a simple classical Lie algebra the index of ge equals the rank of g. We assume that the