Equivariant holomorphic maps of symmetric domains (Q1105742)

Let G be a \({\mathbb{Q}}\)-algebraic simple group. Then G(\({\mathbb{R}})\) has a decomposition (1): \(G({\mathbb{R}})=G_ 1\times...\times G_ m\) where the \(G_ i\) are simple groups which are noncompact for \(i\leq r\) and compact for \(i>r\). Let \(K_ i\) be a maximal compact group in \(G_ i\) for \(i\leq r\). Then G is of Hermitian type when the \(G_ i/K_ i\) are Hermitian symmetric. The symmetric domain associated to G is \(\prod^{r}_{i=1}G_ i/K_ i\). Let G and G' be two groups of Hermitian type and \(\rho\) a homomorphism of G in G'. A holomorphic map form X in X' (the symmetric domains associated to G and G') is called an Eichler map when it is equivariant. In this paper, the author gives a classification of \({\mathbb{Q}}\) symplectic representations for some G, which admits an Eichler map. (For these groups G, the non compact \(G_ i\) in the decomposition (1) are of type II or III.) This gives a partial answer to a problem of Satake and Kuga. The author uses his results to deduce some properties of families of Abelian varieties.