Conjugates of rational equivariant holomorphic maps of symmetric domains (Q1881644)

Let \(\rho: G\to G'\) be a homomorphism of semisimple algebraic groups defined over \(\mathbb Q\), and let \(\tau: D\to D'\) be an equivariant holomorphic map of symmetric domains associated to \(\rho\). If \(\Gamma\subset G(\mathbb Q)\) and \(\Gamma'\subset G'(\mathbb Q)\) are torsion-free arithmetic subgroups with \(\rho(\Gamma)\subset \Gamma'\), then the map \(\tau\) induces a morphism \(\phi:\Gamma\setminus G\to \Gamma'\setminus D'\) of arithmetic varieties, and the rationality of \(\tau\) is defined by using symmetries on \(D\) and \(D'\) as well as the commensurability groups of \(\Gamma\) and \(\Gamma'\). An element \(\sigma\in\text{Aut}(\mathbb C)\) determines a conjugate equivariant holomorphic map \(\tau^{\sigma}:D^\sigma\to {D'}^{\sigma}\) of \(\tau\) which induces the conjugate morphism \(\phi^{\sigma}:(\Gamma\setminus D)^{\sigma}\to(\Gamma'\setminus D')^{\sigma}\) of \(\phi\). The author proves that \(\tau^\sigma\) is rational if \(\tau\) is rational.