MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS
Let G be a simple algebraic group defined over ℂ. Let e be a nilpotent element in $$ \mathfrak{g} $$ = Lie(G) and denote by U ($$ \mathfrak{g} $$, e) the finite W-algebra associated with the pair ($$ \mathfrak{g} $$, e). It is known that the component group Γ of the centraliser of e in G acts on the set ℰ of all one-dimensional representations of U ($$ \mathfrak{g} $$, e). In this paper we prove that the fixed point set ℰΓ is non-empty. As a corollary, all finite W-algebras associated with $$ \mathfrak{g} $$ admit one-dimensional representations. In the case of rigid nilpotent elements in exceptional Lie algebras we find irreducible highest weight $$ \mathfrak{g} $$-modules whose annihilators in U ($$ \mathfrak{g} $$) come from one-dimensional representations of U ($$ \mathfrak{g} $$, e) via Skryabin’s equivalence. As a consequence, we show that for any nilpotent orbit $$ \mathcal{O} $$ in $$ \mathfrak{g} $$ there exists a multiplicity-free (and hence completely prime) primitive ideal of U ($$ \mathfrak{g} $$) whose associated variety equals the Zariski closure of $$ \mathcal{O} $$ in $$ \mathfrak{g} $$.