On symmetric invariants of centralisers in reductive Lie algebras

Let 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 and let be the centraliser of e in . In this paper we study the algebra of symmetric invariants of . We prove that if is of type A or C, then 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 is freely generated by a regular sequence in and describe the tangent cone at e to the nilpotent variety of .