Chevalley restriction theorem for vector-valued functions on quantum groups

Abstract:

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.