Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

摘要

Let $tau: {cal D} rightarrow{cal D}^prime$ be an equivariant holomorphic map of symmetric domains associated to a homomorphism ${bfrho}: {Bbb G} rightarrow{Bbb G}^prime$ of semisimple algebraic groups defined over ${Bbb Q}$ . If $Gammasubset {Bbb G} ({Bbb Q})$ and $Gamma^prime subset {Bbb G}^prime ({Bbb Q})$ are torsion-free arithmetic subgroups with ${bfrho} (Gamma) subset Gamma^prime$ , the map τ induces a morphism φ: $Gammabackslash {cal D} rightarrowGamma^prime backslash {cal D}^prime$ 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 $tau^sigma: {cal D}^sigma rightarrow{cal D}^{primesigma}$ of τ which induces the conjugate morphism $phi^sigma: (Gammabackslash {cal D})^sigma rightarrow(Gamma^prime backslash {cal D}^prime)^sigma$ of φ. We prove that τσ is rational if τ is rational.