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.

Surprising properties of centralisers in classical Lie algebras

“…Then, as shown in Section 8, all irreducible components of N(e) are of dimension dim g e − rk g. In type A the same result is proved in [15,Section 5] for all nilpotent elements.…”

“…The most interesting fibre of this quotient morphism is the one containing zero, the so called null-cone N(e). In type A it is equidimensional by [15,Section 5]. Contrary to the expectations, see [15,Conjecture 5.1], the null-cone is not reduced (as a scheme).…”

“…In type A it is equidimensional by [15,Section 5]. Contrary to the expectations, see [15,Conjecture 5.1], the null-cone is not reduced (as a scheme). A counterexample is provided by e ∈ N(gl 6 ) with partition (4,2).…”

“…* , with a i ∈ F × being pairwise distinct, and β := [15,Section 3]. To prove that the codimension of (g *…”

“…For this centraliser, F a is never maximal among Poisson-commutative subalgebras of S(g e ). The general construction of [15] allows us to write down the invariants. They are H 1 = ξ ge is not maximal.…”