Joseph and Hodges–Levasseur (in the A case) described the spectra of all quantum function algebras R q [ G ] R_q[G] on simple algebraic groups in terms of the centers of certain localizations of quotients of R q [ G ] R_q[G] by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. We determine the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it we deduce a more explicit description of all prime ideals of R q [ G ] R_q[G] than the previously known ones and an explicit parametrization of S p e c R q [ G ] \mathrm {Spec} R_q[G] . We combine the latter with a result of Kogan and Zelevinsky to obtain in the complex case a torus equivariant Dixmier type map from the symplectic foliation of the group G G to the primitive spectrum of R q [ G ] R_q[G] . Furthermore, under the general assumptions on the ground field and deformation parameter, we prove a theorem for separation of variables for the De Concini–Kac–Procesi algebras U ± w \mathcal {U}^w_\pm , and classify the sets of their homogeneous normal elements and primitive elements. We apply those results to obtain explicit formulas for the prime and especially the primitive ideals of U ± w \mathcal {U}^w_\pm lying in the Goodearl–Letzter stratum over the { 0 } \{0\} -ideal. This is in turn used to prove that all Joseph’s localizations of quotients of R q [ G ] R_q[G] by torus invariant prime ideals are free modules over their subalgebras generated by Joseph’s normal elements. From it we derive a classification of the maximal spectrum of R q [ G ] R_q[G] and use it to resolve a question of Goodearl and Zhang, showing that all maximal ideals of R q [ G ] R_q[G] have finite codimension. We prove that R q [ G ] R_q[G] possesses a stronger property than that of the classical catenarity: all maximal chains in S p e c R q [ G ] \mathrm {Spec} R_q[G] have the same length equal to G K d i m R q [ G ] = dim ⁡ G \mathrm {GK \, dim} R_q[G]= \dim G , i.e., R q [ G ] R_q[G] satisfies the first chain condition for prime ideals in the terminology of Nagata.