Let $k$ be a field and $G$ be algebraic group over $k$ and $U$ be an unipotent subgroup of G (in fact it's the unipotent radical of some parabolic if this helps).
I wonder if the two following properties (which are absolutely false in general) might be true :
Is it true that taking invariants under $U(k)$ is an exact functor from the category of $G$-reps to the category of $k$ vector spaces ?
Is it true that $(V \otimes W)^{U(k)} = V^{U(k)}\otimes W^{U(k)}$ ?