Generalized Steinberg section and application to branching rules

Let $G$ be a reductive group with simply connected semisimple part, ${\mathfrak g}=Lie\,G$. We are going to show how a suitable generalization of the Steinberg section of the adjoint quotient $G\to G//G$ can be used to obtain information on how a generic irreducible representation of the quantized enveloping algebra $U_q({\mathfrak g})$ ($q$ is a primitive root of $1$) decomposes when restricted to the quantized enveloping algebra of a Levi factor. In the special case of $G=GL(n)$ this yields a kind of Gelfand–Zeitlin phenomenon.