Let $G$ be an algebraic closed subgroup of $SL(n,\mathbb{R})$ whose action on $\mathbb{R}^n$ is strongly irreducible, i.e. there is no finite union of proper nonzero linear subspaces of $\mathbb{R^n}$ which is invariant under $G$. Is it true that $G$ is semisimple?

## 1 Answer

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First of all, $G$ has to be redective since the fiexed point set of the unipotent radical would be a non-trivial $G$-invarinant subspace.

Secondly, as noticed by the commentators, one cannot conclude that $G$ is semisimple since the center of $G^0$ my act via a non-trivial character to $SO(2)$. This problem occurs in all even dimensions. Counterexamples are the unitary groups $G=U(n)\subseteq SL(2n,\mathbb R)$.