On symmetric invariants of centralisers in reductive Lie algebras

Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let ge be the centraliser of e in g. In this paper we study the algebra S (ge)ge of symmetric invariants of ge. We prove that if g is of type A or C, then S (ge)ge is always a graded polynomial algebra in l variables, and we show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type A we prove that the invariant algebra S (ge)ge is freely generated by a regular sequence in S (ge) and describe the tangent cone at e to the nilpotent variety of g. © 2007 Elsevier Inc. All rights reserved.