Isometries of Hermitian symmetric spaces
Isometries of Hermitian symmetric spaces
Isometries of Hermitian symmetric spaces
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Lemma (I f)∗(Jo) ∈ {±Jo}.<br />
Pro<strong>of</strong>. Since Jo ∈ C(ko) and (I f)∗ is a Lie algebra auto-<br />
morphism, we have (I f)∗(Jo) ∈ C(ko) and ad((I f)∗(Jo)) 3 =<br />
−ad((I f)∗(Jo)).<br />
If k ∈ Ko = {g ∈ G | g(o) = o}, I f(k)(o) = f ◦ k ◦ f −1 (o) =<br />
o. Hence I f| Ko : Ko → Ko is an automorphism <strong>of</strong> Ko.<br />
Thus (I f)∗(C(ko)) = C(ko) = RJo. So (I f)∗(Jo) ∈ {±Jo}.<br />
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