  A version of quantum orbit method is presented for real forms of equal rank
of quantum complex simple groups. A quantum moment map is constructed, based on
the canonical isomorphism between a quantum Heisenberg algebra and an algebra
of functions on a family of quantum G-spaces. For the series $A$, we construct
some irreducible $*$-representations of $U_q{\frak g}$ which correspond to the
semi-simple dressing orbits of minimal dimension in the dual Poisson Lie group.
It is shown that some complimentary series representations correspond to some
quantum 'tunnel' G-spaces which do not have a quasi-classical analog.
