Representations of quantum conjugacy classes of orthosymplectic groups

Let $G$ be the complex symplectic or special orthogonal group and $\mathfrak g$ its Lie algebra. With every point $x$ of the maximal torus $T\subset G$ we associate a highest weight module $M_x$ over the Drinfeld–Jimbo quantum group $U_q(\mathfrak g)$ and a quantization of the conjugacy class of $x$ by operators in $\mathrm{End}(M_x)$. These quantizations are isomorphic for $x$ lying on the same orbit of the Weyl group, and $M_x$ support different representations of the same quantum conjugacy class.