Let $G<\rm{GL}_n(\mathbb{k})$ be a linear group, where $\mathbb{k}$ is an algebraically closed field. Assume that the linear action of $G$ on $\mathbb{k}^n$ is strongly-irreducible (i.e. there are no $H$-invariant proper subspaces of $\mathbb{k}^n$, except $0$, for any $H< G$ of finite index). Equip $\mathbb{k}^n$ with the Zariski topology. Could there be a proper non-empty open subset $U\subset\mathbb{k}^n$ which is $G$-invariant, and $\mathbb{k}^n\setminus U\neq \{0\}$?

Thanks for any help.

`$G$`

is an algebraic group, your condition on subgroups of finite index will be taken care of most easily if`$G$`

isconnectedin the Zariski topology. (The answers given don't emphasize that part of the question.) $\endgroup$