Characterize primitive ideals contained in holomorphic-evaluation closures

Characterize precisely which primitive ideals of the quantised coordinate algebra O(G_q) are contained in the Zariski-closed subset Prim(\theta_\pi) associated with a quantum holomorphic evaluation \theta_\pi, for general complex semisimple quantum groups G_q and topologically irreducible *-representations \pi of the polynomial-function algebra on the Drinfeld double.

Background

For a quantum holomorphic evaluation \theta_\pi, the paper identifies Prim(\theta_\pi) with the set of primitive ideals containing the intersection of the kernels of the representations \pi_C\otimes K_{-\nu} over all weights \nu of the finite-dimensional representation associated with \pi. This determines the Zariski closure attached to an evaluation but does not generally identify all primitive ideals lying in it.

The authors explicitly state that the precise primitive ideals contained in Prim(\theta_\pi) are unknown because the gluing of the Joseph strata in the Zariski topology has not been determined. They obtain complete information in the special case G_q=SL_q(2,\mathbb C), but leave the general characterization unresolved.