Let $\mathbf{U}_q(\mathfrak{g})$ be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we use the Sweedler notation. The locally finite part of $\mathbf{U}_q(\mathfrak{g})$ is the subspace defined by $F_l(\mathbf{U}_q(\mathfrak{g}))=\{x\in \mathbf{U}_q(\mathfrak{g}): dim (ad(\mathbf{U}_q(\mathfrak{g}))(x))<\infty\}$. The locally finite part has a Peter-Weyl decomposition $$F_l(\mathbf{U}_q(\mathfrak{g}))=\bigoplus _{\lambda\in R} ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda}).$$ I wonder if the cyclic $ad$-modules $ad(\mathbf{U}_q(\mathfrak{g}))(K_{-\lambda})$ are invariant under the action of the Lusztig braid group operators.
