Meromorphic Functions Compatible with Homomorphisms of Actions on $\bf C$

Abstract

We consider homomorphisms $H:G_1\longrightarrow G_2$ of holomorphic (group or pseudo-group) actions $G_1$ and $G_2$ on domains $\Omega_1$ and $\Omega_2$ respectively in $\bf C$, together with meromorphic functions $f$ that are compatible with these homomorphisms in the sense that \begin{equation} f(g(z))=H(g)(f(z))\nonumber \end{equation} for every $g\in G_1$ and $z\in\Omega_1$. Such situations are rooted in the cases of elliptic and modular functions, modular and automorphic forms, etc... We investigate various aspects of such cases, such as constructions and correspondences between families of functions compatible with different homomorphisms, that transform one family of functions compatible with one homomorphism to another one compatible with a different homomorphism.