Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras

Let $\frak g$ be a simple finite-dimensional Lie algebra over an algebraically closed field $\mathbb F$ of characteristic 0. We denote by $\operatorname{U}(\frak g)$ the universal enveloping algebra of $\frak g$. To any nilpotent element $e\in \frak g$ one can attach an associative (and noncommutative as a general rule) algebra $\operatorname{U}({\frak g},~e)$ which is in a proper sense a "tensor factor" of $\operatorname{U}(\frak g)$. In this article we consider the case in which $\frak g$ is simple and $e$ belongs of the minimal nonzero nilpotent orbit of $\frak g$. Under these assumptions $\operatorname{U}({\frak g}, e)$ was described explicitly in terms of generators and relations. One can expect that the representation theory of $\operatorname{U}({\frak g}, e)$ would be very similar to the representation theory of $\operatorname{U}(\frak g)$. For example one can guess that the category of finite-dimensional $\operatorname{U}({\frak g}, e)$-modules is semisimple. The goal of this article is to show that this is the case if $\frak g$ is not simply-laced. We also show that, if $\frak g$ is simply-laced and is not of type $A_n$, then the regular block of finite-dimensional $\operatorname{U}({\frak g}, e)$-modules is equivalent to the category of finite-dimensional modules of a zigzag algebra.