On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

Details On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

2009-05-22

arxiv.org/abs/0905.3714v3

Mathematics

Representation Theory

14 pages, minor changes.

Scientific paper

We consider the finite $W$-algebra $U(\g,e)$ associated to a nilpotent element $e \in \g$ in a simple complex Lie algebra $\g$ of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a conjecture of Premet, that $U(\g,e)$ always has a 1-dimensional representation, when $\g$ is of type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in $U(\g)$ whose associated variety is the coadjoint orbit corresponding to $e$.

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