Multiple products of meromorphic functions

Let $\mathfrak{g}$ be an infinite-dimensional Lie algebra, and $G$ be the algebraic completion of a $\mathfrak{g}$-module. Using the geometric model of Schottky uniformization of Riemann sphere to obtain a higher genus Riemann surface, we construct a family of parametric extensions of coboundary operators for the double complexes of meromorphic functions depending on elements of $G$ that possess prescribed analytic properties.