A computational approach to 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

Abstract

Let g be a simple complex Lie algebra and let e be a nilpotent element of g. It was conjectured by Premet in [P07i] that the finite W-algebra U(g; e) admits a 1-dimensional representation, and further work [L10, P08] has reduced this conjecture to the case where g is of exceptional type and e lies in a rigid nilpotent orbit in g. Using the PBW-theorem for U(g; e) we give an algorithm for determining a presentation for U(g; e) which allows us to determine the 1-dimensional representations for U(g; e). Implementing this algorithm in GAP4 we verify the conjecture in the case that g is of type G2, F4 or E6. Using a result of Premet in [P08], we can use these results to deduce that reduced enveloping algebras of those types admit representations of minimal dimension, and using the explicit presentations we can determine for which characteristics this will hold. Further, we show that we can determine the 1-dimensional representations of U(g; e) from a smaller set of relations than is required for a presentation. From calculating these sets of relations, we show that in the case that g is of type E7 and e lies in any rigid nilpotent orbit, or in the case that g is of type E8 and e lies in one of 14 (out of 17) rigid nilpotent orbits, that U(g; e) admits a 1-dimensional representation.

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ORCID for Koeck Bernhard: orcid.org/0000-0001-6943-7874

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