Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let M = Γ \ X be a quotient of X by a torsion-free discrete subgroup Γ of G such that M is of finite volume in the canonical metric. Then, due to the G -equivariant Borel embedding of X into its compact dual X c , the locally symmetric structure of M can be considered as a special kind of a ( G C , Xc )structure on M , a maximal atlas of Xc -valued

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... with locally constant transition maps in the complexified group G C . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the ( G C , Xc ) -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung. Mathematics Subject Classification (2000) . 53C35, 32M15, 22E40.