**EDIT** Let $\mathcal{O}$ be the ring of integers in a non-Archimedean local field. Let $GL\_n(\mathcal{O})$ be the (compact) group of $n\times n$ matrices with entries in $\mathcal{O}$ such that its inverse also has entries in $\mathcal{O}$.



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> Consider a continuous finite dimensional representation of $GL\_n(\mathcal{O})$ in a **complex** vector space $V$. Is it true that the image of $GL\_n(\mathcal{O})$ in $GL(V)$ is a finite subgroup?
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A reference would be most helpful.



Image of a finite dimensional complex representations of $GL_n(\mathcal{O})$
Call the representation $\pi$. Let $U$ be a neighbourhood of the identity in $\operatorname{GL}(V)$ that contains no non-trivial subgroup. Then the pre-image of $U$ under $\pi$ is a neighbourhood of the identity there, hence contains some open subgroup. That is, $\ker(\pi)$ contains an open subgroup, hence is open, hence has finite index. The representation thus factors through the finite quotient $\operatorname{GL}\_n(\mathcal O)/\ker(\pi)$.


