Let $G$ be a connected, 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)?

## 1 Answer

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No. Let $G$ be the 2-dimensional unipotent abelian group, and its 3-dimensional representation given by $$(t,s)\mapsto \begin{pmatrix}1 & t & s\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}.$$

Let $G_u$ be the subgroup of $G$ of those $(t,ut)$, $t\in\mathbf{C}$. Then the fixed point set $V_u$ of $G_u$ consists of triples $(x,-uz,z)$, $x,z\in\mathbf{C}$. Then $V_u$ is a subrepresentation of $V$ whose kernel is exactly $G_u$. In particular, the $V_u$, for $u\in\mathbf{C}$, are pairwise non-isomorphic representations.

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