Derived subalgebras of centralisers and finite -algebras

Let g=Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0 . Let e be a nilpotent element of g and let ge=Lie(Ge) where Ge stands for the stabiliser of e in G . For g classical, we give an explicit combinatorial formula for the codimension of [ge,ge] in ge and use it to determine those e∈g for which the largest commutative quotient U(g,e)ab of the finite W -algebra U(g,e) is isomorphic to a polynomial algebra. It turns out that this happens if and only if e lies in a unique sheet of g . The nilpotent elements with this property are called non-singular in the paper. Confirming a recent conjecture of Izosimov, we prove that a nilpotent element e∈g is non-singular if and only if the maximal dimension of the geometric quotients S/G , where S is a sheet of g containing e , coincides with the codimension of [ge,ge] in ge and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element e in a classical Lie algebra g the closed subset of Specm U(g,e)ab consisting of all points fixed by the natural action of the component group of Ge is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.