Quantum surfaces, special functions, and the tunneling effect

The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irred ucible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommuta tive product on the leaves incorporates a closed 2-form and a measure which (in general) are different from the classical symplectic form and the Liou ville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structu res could even occur to be exponentially small with respect to the deformat ion parameter. This is interpreted as a tunneling effect in the quantum geo metry.