Simple modules of classical linear groups with normal closures of maximal torus orbits

Let $T$ be a maximal torus in a classical linear group $G$. In this paper we find all simple rational $G$-modules $V$ such that for each vector $\mathbf v\in V$ the closure of the $T$-orbit of $\mathbf v$ is a normal affine variety. For every $G$-module without this property we present a $T$-orbit with nonnormal closure. To solve this problem, we use a combinatorial criterion of normality which is formulated in terms of the set of weights of a simple $G$-module. The same problem for $G=SL(n)$ was solved by the author earlier.