Homogeneous Spaces Defined by Lie Group Automorphisms. Ii

We will drop the compactness hypothesis on G in the results of §6, doing this in such a way that problems can be reduced to the compact case. This involves the notions of reductive Lie groups and algebras and Cartan involutions. Let © be a Lie algebra. A subalgebra S c © is called a reductive subaU gebra if the representation ad%\® of ίΐ on © is fully reducible. © is called reductive if it is a reductive subalgebra of itself, i.e. if its adjoint representation is fully reducible. It is standard ([11, Theorem 12.1.2, p. 371]) that the following conditions are equivalent: (7.1a) © is reductive, (7.1b) © has a faithful fully reducible linear representation, and (7.1c) © = ©' © 3 , where the derived algebra ©' = [©, ©] is a semisimple ideal (called the "semisimple part") and the center 3 of © is an abelian ideal. Let © = ©' Θ 3 be a reductive Lie algebra. An automorphism σ of © is called a Cartan involution if it has the properties (i) σ = 1 and (ii) the fixed point set ©" of σ\$r is a maximal compactly embedded subalgebra of ©'. The whole point is the fact ([11, Theorem 12.1.4, p. 372]) that (7.2) Let S be a subalgebra of a reductive Lie algebra ©. Then S is reductive in © if and only if there is a Cartan involution σ of © such that σ(ft) = ft. Let G be a Lie group. We say that G is reductive if its Lie algebra © is reductive. Let K be a Lie subgroup of G. We say that K is a reductive subgroup if its Lie algebra ^ is a reductive subalgebra of ©. Let a be an automorphism of G. We say that σ is a Cartan involution of G if a induces a Cartan involution of ©. Let G be a reductive Lie group, and K a closed reductive subgroup such that G acts effectively on X = G/K. Choose a Cartan involution σ of © which preserves S, and consider the decomposition into ( ± l)-einspaces of σ: