MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS

Abstract

Let G be a simple algebraic group defined over C. Let e be a nilpotent elementin g = Lie(G) and denote by U(g, e) the finite W-algebra associated with the pair (g, e).It is known that the component group G of the centraliser of e in G acts on the set Eof all one-dimensional representations of U(g, e). In this paper we prove that the fixedpoint set EG is non-empty. As a corollary, all finite W-algebras associated with g admitone-dimensional representations. In the case of rigid nilpotent elements in exceptionalLie algebras we find irreducible highest weight g-modules whose annihilators in U(g)come from one-dimensional representations of U(g, e) via Skryabin's equivalence. As aconsequence, we show that for any nilpotent orbit O in g there exists a multiplicity-free(and hence completely prime) primitive ideal of U(g) whose associated variety equals theZariski closure of O in g. © 2014 Springer Science+Business Media New York.