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-$p$.



What is a closed orbit that is not separable? (ANSWERED)
BCnrd writes: "This happens for 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→X which is not separable precisely when H is not smooth. The simplest example is G=GL(1) acting on X=G via $t\cdot x=t^px$, whose orbit map through $x=1$ is $t\mapsto t^p$." 


