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5. Cartan decompositions and maximal compact subgroups 37 Suppose that a Cartan decomposition (7) is given. 14), the involutive transformation θ of g0 defined by θ(x + y) = x − y , x ∈ k, y ∈ p. (8) is an automorphism of g0 . Extend θ to an (involutive) automorphism of g = g0 (C) denoted by the same symbol. Clearly, θσ = σθ, where σ is the complex conjugation in g relative g0 . Therefore τ = θσ is an antiinvolution commuting with σ. We claim that it is compact. In fact, for any x ∈ k, y ∈ p we have hτ (x + y, x + y) = −k(x + y, τ (x + y)) = −k(x + y, θ(x + y)) = −k(x + y, x − y) = −k(x, x) + k(y, y) .

If t ∈ Z, then β(t) = αt . The same notation will be used for arbitrary t ∈ R. This is suggested by the following fact: if λi > 0 are the eigenvalues of α, then the operator β(t) can be given by the same eigenspace decomposition as α with the eigenvalues λti . If α,β ∈ P(E) satisfy β = γαγ −1 , where γ ∈ GL(E), then logβ = γ(logα)γ −1 and t β = γαt γ −1 for any t ∈ R. In the complex case the same is true for any invertible antilinear operator γ in E. This can be proved directly, using the eigenspace decompositions.

Hence s = log p = ad z for a certain z ∈ g0 . , z ∈ p. Since the mapping µ(id × exp ad) : K × p → Aut g0 is a homeomorphism, it should map the connected components of K × p onto those of Aut g0 . It follows that K ◦ × p is mapped onto Int g0 . Hence µ(K ◦ × P ) = Int g0 . The decomposition Int g0 = K ◦ P implies the last assertion of the theorem. 40 §5. Cartan decompositions and maximal compact subgroups Corollary. The subgroups K and K ◦ are maximal compact subgroups of the Lie groups Aut g0 and Int g0 , respectively.

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