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

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