We use the idea of free fields to obtain highest weight representations for the extended affine Lie algebra gl 2 (C q ) coordinatized by the quantum torus C q and go on to construct a contravariant hermitian form. We further give a necessary and sufficient condition such that the contravariant hermitian form is positive definite.

One of the most fundamental problems in complex geometry is to determine when two bounded domains in C n are biholomorphically equivalent. Even for complete Reinhardt domains, this fundamental problem remains unsolved completely for many years. Using the Bergmann function theory, we construct an infinite family of numerical invariants from the Bergman functions for complete Reinhardt domains in C n . These infinite family of numerical invariants are actually a complete set of invariants if the domains are pseudoconvex with C 1 boundaries. For bounded complete Reinhardt domains with real analytic boundaries, the complete set of numerical invariants can be reduced dramatically although the set is still infinite. As a consequence, we have constructed the natural moduli spaces for these domains for the first time.