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Let $G$ be a connected, unipotent, linear algebraic group over $\mathbb{C}$. Let $\rho:G\to \mathrm{GL}(V)$ be a rational representation. Does $\rho$ have at most countably many subrepresentations (up to isomorphism)?

The question is motivated by the following. If $G$ is reductive, then it has at most countably many representations. If $G=\mathbb{G}_a$, then a representation of $G$ on $V$ is a nilpotent linear operator on $V$. From the Jordan normal form, $G$ also has at most countably many representations. But I don't know what happens for noncommutative unipotent groups.

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