Isometries of Hermitian symmetric spaces

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Isp : G → G, g ↦→ sp ◦ g ◦ s −1 p : an involutive automorphism of G (Isp)∗ : g → g : an involutive automorphism of g g = kp ⊕ pp, (Isp)∗ = id on kp, (Isp)∗ = −id on pp kp ∼ = Lie(Kp), Kp = {g ∈ G | g(p) = p}, pp ∼ = TpP Jp ∈ C(kp), the center C(kp) of kp (If P is irreducible, Jp generates C(kp).) The action of Jp on TpP ∼ = pp is given by ad(Jp)|pp. Thus ad(Jp) has eigenvalues ±i and 0, so that ad(Jp) 3 = −ad(Jp). 14
Consider the map ι : P → g, p ↦→ Jp, which is a G- equivariant embedding of P into g with the image Ad(G)Jo, the adjoint orbit for a chosen base point o ∈ P . ι is called the canonical embedding of P . 15
Page 1 and 2: Isometries of Hermitian symmetric s
Page 3 and 4: 1. Introduction A Riemannian symmet
Page 5 and 6: M : a Riemannian manifold τ : an i
Page 7 and 8: 2. Extrinsically symmetric spaces M
Page 9 and 10: P = G/K : a Riemannian symmetric sp
Page 11 and 12: I(M) : the group of isometries of M
Page 13: 3. Isometries of Hermitian symmetri
Page 17 and 18: Proof: P = P1 × · · · × Pr : a
Page 19 and 20: Let f ∈ I(P ). Let m be the midpo
Page 21 and 22: Lemma (I f)∗(Jo) ∈ {±Jo}. Proo
Page 23 and 24: Remark It is a known fact that ever
Page 25 and 26: 4. Another result Theorem 2 (Loos 1
Page 27 and 28: Proposition 3 Let M be a compact Ri
Page 29: Further, a meridian M − of X is e

Isometries of Hermitian symmetric spaces

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