What is an example of an action of a *linearly reductive* group variety acting on an affine variety with the property that there exists a closed orbit that is not separable?

To be more precisely, let's work over a fixed algebraically closed field $k$. Suppose that we are given an affine variety $X$ and a group variety $G$ acting on $X$. Given a closed point $x \in X$, we define the **orbit** $\operatorname{O}(x)$ to be the image of the map $G \to X$ given by $g \mapsto g x$ . We say that the orbit is **separable** if the natural map
$$
G \to \operatorname{O}(x)
$$
given by $g \mapsto gx$ is separable.

This question is only interesting in characteristic $p>0$. In this case, the condition that $G$ is **linearly reductive** is very strong: it implies $G$ is the product of a multiplicative torus and a finite group of order prime-to-$p$.

every$G$ of positive dimension, since all one needs is a non-smooth subgroup scheme (such as kernel of Frobenius). Indeed, if $H$ is a closed $k$-subgroup scheme of $G$ then let $X = G/H$ equipped with the natural left $G$-action (so $X$ is smooth, since $G$ is smooth, regardless of how "bad" $H$ may be). Then the orbit map through $x = 1$ is the natural surjection $G \rightarrow X$ which is not separable precisely when $H$ is not smooth. The simplest example is $G = {\rm{GL}}_1$ acting on $X = G$ via $t.x = t^p x$, whose orbit map through $x = 1$ is $t \mapsto t^p$. – BCnrd Sep 12 '10 at 5:39