Chevalley restriction theorem for vector-valued functions on quantum groups

Abstract

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