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MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS

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Abstract

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 \( \mathfrak{g} \).

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  1. School of Mathematics, The University of Manchester, Manchester, Oxford Road,, M13 9PL, UK

    ALEXANDER PREMET

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Dedicated to Evgenii Borisovich Dynkin on the occasion of his 90th birthday

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PREMET, A. MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS. Transformation Groups 19, 569–641 (2014). https://doi.org/10.1007/s00031-014-9266-9

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