Generalized Joseph's decompositions

We generalize the decomposition of $U_q(\mathfrak{g})$ introduced by A. Joseph and relate it, for $\mathfrak{g}$ semisimple, to the celebrated computation of central elements due to V. Drinfeld. In that case we construct a natural basis in the center of $U_q(\mathfrak{g})$ whose elements behave as Schur polynomials and thus explicitly identify the center with the ring of symmetric functions.