On Representations of Quantum Conjugacy Classes of GL(n)

Let O be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by T the maximal torus of diagonal matrices in GL(n). With every a is an element of O boolean AND T we associate a highest weight module M-a over the quantum group U-q(gl(n) and an equivariant quantization C-(h) over bar,(a)[O] of the polynomial ring C [O] realized by operators on M-a. All quantizations C-(h) over bar,(a)[O] are isomorphic and can be regarded as different exact representations of the same algebra, C-(h) over bar[O]. Similar results are obtained for semisimple adjoint orbits in gl(n) equipped with the canonical GL(n)-invariant Poisson structure.