A sufficient condition for irreducibility of a $G$-variety

Question

Let $G$ be an algebraic group over a field $k$ and let $V$ be a variety on which $G$ acts. Suppose $U\subset V$ is a closed, irreducible, $G$-stable subset which intersects every $G$-orbit non-trivially. Does it follow that $V$ is irreducible?

Answer 1

You probably want to assume $G$ itself to be irreducible. Also, do you really want $U$ to be $G$-stable? Then it seems that your assumptions already imply that $U=V$ (see below).

At least over algebraically closed field you could argue as follows: $G\times U$ is irreducible (this would need more care in case of an arbitrary base field) and maps, by your assumption, surjectively onto $V$ through $G\times U\hookrightarrow G\times V\to V$, so $V$ is irreducible as well.