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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 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?

Thanks for any help.

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Maybe I missed something here, but isn't $U=\mathbb{k}^n-0$ a candidate?

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    Yeah, well, it's clearly an example, but could there be such a $U$ when it's compliment is not $0$?2011-04-29