A generalized Steinberg section and branching rules for quantum groups at roots of 1

In this paper we construct a generalization of the classical Steinberg section for the quotient map of a semisimple group with respect to the conjugation action. We then give various applications of our result including the construction of a sort of Gelfand–Zeitlin basis for a generic irreducible representation of $U_q(\mathrm{GL}(n))$ when $q$ is a primitive odd root of unity.