Nonexistence of Arithmetic Fake Compact Hermitian Symmetric Spaces of Types

1.1. Let G be a noncompact connected real semi-simple Lie group with trivial center and with no nontrivial compact connected normal subgroups, and g be its Lie algebra. The group Aut(G) (=Aut(g)) of automorphisms of G is a Lie group with finitely many connected components, and G is its identity component. We will denote the identity component of Aut(G) in the Zariski-topology by Int(G). Let X be the symmetric space of G (X is the space of maximal compact subgroups of G), and Xu be the compact dual of X. There is a natural identification of the group of isometries of X with Aut(G). We assume in this paper that X (and hence Xu) is hermitian. Then every holomorphic automorphism of X is an isometry. The group Hol(X) of holomorphic automorphisms of X is a subgroup of finite index of the group Aut(G) of isometries, and it is known (see [Ta], the remark in §5) that Hol(X) ∩ Int(G) = G.