Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Abstract.

Let $\tau: {\cal D} \rightarrow{\cal D}^\prime$ be an equivariant holomorphic map of symmetric domains associated to a homomorphism ${\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime$ of semisimple algebraic groups defined over ${\Bbb Q}$. If $\Gamma\subset {\Bbb G} ({\Bbb Q})$ and $\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q})$ are torsion-free arithmetic subgroups with ${\bf\rho} (\Gamma) \subset \Gamma^\prime$, the map τ induces a morphism φ: $\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime$ of arithmetic varieties and the rationality of τ is defined by using symmetries on ${\cal D}$ and ${\cal D}^\prime$ as well as the commensurability groups of Γ and Γ′. An element $\sigma \in {\rm Aut} ({\Bbb C})$ determines a conjugate equivariant holomorphic map $\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma}$ of τ which induces the conjugate morphism $\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma$ of φ. We prove that τ^σ is rational if τ is rational.