By Denis S. Arnon, Bruno Buchberger

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An involutive automorphism θ of G is a Cartan involution, if the Lie subgroup g )} Gθ (R) := {g ∈ G(C)|g = θ(¯ of G(C) is compact. 10. A connected R-algebraic group is reductive, if and only if it has a Cartan involution. Any two Cartan involutions are conjugate by an inner automorphism. Proof. By [54], I. 3, each connected R-algebraic reductive group has a Cartan involution and the Cartan involutions are conjugate. Let θ be a Cartan involution on the connected R algebraic group G. Thus Gθ (R) is compact.

In the next section we use Shimura data to construct complex manifolds, which will be used for the construction of quasi-projective varieties, which are the Shimura varieties. 4 Hermitian symmetric domains 35 a Shimura datum (G, h) of Hodge type. 26. Let (V, h, Q) be a polarized Q-Hodge structure of type (1, 0), (0, 1). h), h) is a Shimura datum. Proof. 20, the Mumford-Tate group MT(V, h) is reductive. The inner automorphism given by g → h(i)gh−1 (i) descends to a Cartan involution θ on MTad (V, h)R = Hgad (V, h)R .

The following lemma concerns in particular Gad (R). 8. If G is a semisimple connected Lie group with trivial center, then it is isomorphic to a direct product of simple groups with trivial centers. 28 1 An introduction to Hodge structures and Shimura varieties Proof. By [27], II. 2, the group G coincides with Gad ∼ = G/Z(G). Since the Lie algebra g of G is the direct sum of simple Lie algebras, g is the Lie algebra of a certain direct product of simple groups, too. Without loss of generality one can assume that these simple Lie groups have trivial centers.