A q-Analog of the Hua Equations

A necessary condition is established for a function to be in the image of a quantum Poisson integral operator associated to the Shilov boundary of the quantum matrix ball. A quantum analogue of the Hua equations is introduced.Comment: 22 pages, LaTeX2

Function Theory on a q-Analog of Complex Hyperbolic Space

This work deals with function theory on quantum complex hyperbolic spaces. The principal notions are expounded. We obtain explicit formulas for invariant integrals on `finite' functions on a quantum hyperbolic space and on the associated quantum isotropic cone. Also we establish principal series of $U_q \mathfrak{su}_{n,m}$-modules related to this cone, and obtain the necessary conditions for those modules to be equivalent.Comment: 21 page

A Quantum Analogue of the Bernstein Functor

We consider Knapp-Vogan Hecke algebras in the quantum group setting. This allows us to produce a quantum analogue of the Bernstein functor as a first step towards the cohomological induction for quantum groups.Comment: LaTeX2e, 16 pages; some inessential corrections have been introduce