Simple Modules of Exceptional Groups with Normal Closures of Maximal Torus Orbits

Let $G$ be an exceptional simple algebraic group, and let $T$ be a maximal torus in $G$. In this paper, for every such $G$, we find all simple rational $G$-modules $V$ with the following property: for every vector $v\in V$, the closure of its $T$-orbit is a normal affine variety. To solve this problem, we use a combinatorial criterion of normality formulated in terms of weights of simple $G$-modules. This paper continues the works of the second author in which the same problem was solved for classical linear groups.