Suppose $G$ is a reductive algebraic group with Lie algebra $\mathfrak{g}$ and Cartan $\mathfrak{h} \subseteq \mathfrak{g}$ which is the Lie algebra for a torus $T \subseteq G$.
Is it true that $Hom_G(V \otimes \mathfrak{g}, \mathbb{C})$ is naturally isomorphic to $Hom_{N(T)}(V \otimes \mathfrak{h}, \mathbb{C})$ via the restriction map, for every $G$-module $V$? On the LHS we consider algebraic representations of $G$. I'm pretty sure this is equivalent to the statement that $\mathfrak{g}^*$ is isomorphic as $G$-representations to sections of the vector bundle $G \times_{N(T)} \mathfrak{h}^* \to G/N(T)$ (the induction of $\mathfrak{h}^*$ as a $N(T)$-representation to $G$, where $N(T)$ is the normalizer of $T$ in $G$).
I was thinking this might be true since it's reminiscent of the Chevalley restriction theorem which provides a graded algebra isomorphism $\mathcal{O}(\mathfrak{g})^G \to \mathcal{O}(\mathfrak{h})^W$ between rings of invariant functions on $\mathfrak{g}$ and $\mathfrak{h}$, and in particular gives us an isomorphism $Hom_G(\mathfrak{g}, \mathbb{C}) \cong Hom_W(\mathfrak{h}, \mathbb{C})$. Here $W=N(T)/T$ is the Weyl group of $T$.
This is false. The homogeneous space $G/N(T)$ is not compact so the space of sections of the above vector bundle will be infinite-dimensional.