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.



Existence of proper invariant subset in an irreducible action
Let $Q$ be a non-degenerate quadratic form on $\mathbb{K}^n$, and $G=O(Q)$ its orthogonal group. I think that the set $U$ of vectors $x\in\mathbb{K}^n$ with $Q(x)\neq 0$ does the job.


