On symmetric invariants of centralisers in reductive Lie algebras

Abstract

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