## Multiplicity-free primitive ideals associated with rigid nilpotent orbits.(English)Zbl 1353.17009

Let $$G$$ be a complex simple algebraic group and denote by $$\mathfrak{g}$$ the Lie algebra of $$G$$. For a nilpotent element $$e$$ of $$\mathfrak{g}$$ denote by $$U(\mathfrak{g},e)$$ the finite $$W$$-algebra associated with $$(\mathfrak{g},e)$$ and by $$\mathcal{E}$$ the set of all one-dimensional representations of $$U(\mathfrak{g},e)$$. Moreover, let $$G_e$$ be the centralizer of $$e$$ in $$G$$. In the main result of the paper under review the author proves that the subset $$\mathcal{E}^\Gamma$$ of $$\mathcal{E}$$ consisting of all elements fixed by the action of the component group $$\Gamma$$ of $$G_e$$ on $$\mathcal{E}$$ is non-empty. In particular, all finite $$W$$-algebras associated with $$\mathfrak{g}$$ admit one-dimensional representations. For rigid nilpotent elements in exceptional Lie algebras he find irreducible highest weight $$\mathfrak{g}$$-module whose annihilators in the universal enveloping algebra $$U(\mathfrak{g})$$ of $$\mathfrak{g}$$ come from one-dimensional representations of $$U(\mathfrak{g},e)$$ via Skryabin’s equivalence. It is then proved that the Zariski closure of every nilpotent orbit in $$\mathfrak{g}$$ is the associated variety of a multiplicity-free (and so completely prime) primitive ideal of $$U(\mathfrak{g})$$.

### MSC:

 17B35 Universal enveloping (super)algebras 17B08 Coadjoint orbits; nilpotent varieties 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)

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