Let $k$ be an algebraically closed field and define algebraic groups to be affine $k$-group schemes of finite type.
Does anyone know of an example of an algebraic group $G$, a normal algebraic subgroup $N \subseteq G$ and an (algebraic) $G$-representation $V$ such that the sum $W \subseteq V$ of all eigenspaces $V^{\chi}$ of characters $\chi \colon N \to \mathbf{G}_m$ is not a $G$-subrepresentation?
One can show that $W$ is always fixed by every $g \in G(k)$ so that in an example as described above $G$ necessarily has to be non-smooth (in particular $k$ has to be of positive characteristic). The reason this question came up is because in the construction of quotients of algebraic groups by normal subgroups, Milne (see https://www.jmilne.org/math/CourseNotes/iAG200.pdf, the proof of Lemma 5.25) apparently needs to take some special care when dealing with non-smooth groups, precisely because of the phenomenon that I would like to see an example for.