Isometries of Hermitian symmetric spaces

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Proposition (Murakami 1952, Takeuchi 1964) Let P be an irreducible semisimple Hermitian symmetric space. Then I(P )/I0(P ) and A(P )/A0(P ) are given as follows. (1) If P is isometric to Q2m(C) (m ≥ 2), Gm(C 2m ) (m ≥ 2) or their noncompact duals, I(P )/I0(P ) ∼ = Z2 × Z2, A(P )/A0(P ) ∼ = Z2 (2) Otherwise, I(P )/I0(P ) ∼ = Z2, A(P ) = A0(P ). Remark I(P )/A(P ) ∼ = Z2. 24
4. Another result Theorem 2 (Loos 1985) Let M ⊂ R n be a compact extrinsically symmetric space. Then the maximal torus of M is a Riemannian product of round circles. M : a compact Riemannian symmetric space o ∈ M A connected component of F (so, M) is called a polar. For p ∈ F (so, M), the connected component of F (sp ◦ so, M) through p is called the meridian. 25
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 and 14: 3. Isometries of Hermitian symmetri
Page 15 and 16: Consider the map ι : P → g, p
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: Remark It is a known fact that ever
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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