Chevalley restriction theorem for vector-valued functions on quantum groups

We generalize Chevalley's theorem about restriction $\operatorname{Res}: \mathbb{C}[\mathfrak{g}]^{\mathfrak{g}} \to \mathbb{C}[\mathfrak{h}]^W$ to the case when a semisimple Lie algebra $\mathfrak{g}$ is replaced by a quantum group and the target space $\mathbb{C}$ of the polynomial maps is replaced by a finite dimensional representation $V$ of this quantum group. We prove that the restriction map $\operatorname{Res}:(O_{q}(G)\otimes V)^{U_{q}(\mathfrak{g})}\to O(H)\otimes V$ is injective and describe the image.