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

TL;DR

This paper proves that finite W-algebras associated with exceptional Lie algebras always have 1-dimensional representations, confirming a conjecture and linking to minimal representations of modular Lie algebras.

Contribution

It verifies Premet's conjecture for types G2, F4, E6, E7 using algorithmic presentations, establishing the existence of 1-dimensional representations for these W-algebras.

Findings

Confirmed existence of 1-dimensional representations for G2, F4, E6, E7 W-algebras.

Connected these representations to minimal dimension modules of modular Lie algebras.

Identified prime primitive ideals with specific associated varieties in universal enveloping algebras.

Abstract

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 $e$.