Isometries of Hermitian symmetric spaces

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Theorem 1 (Eschenburg-Quast-T., J. of Lie Theory 2013) Let P ⊂ g be a canonically embedded semisimple Hermi- tian symmetric space and let f be an isometry of P , then there exists a linear isometry F of g (w.r.t. a suitable invariant inner product) whose restriction to P coincides with f. 16
Proof: P = P1 × · · · × Pr : a decomposition into irreducible fac- tors G = G1 × · · · × Gr : a decomposition of the transvection group G, where Gj is the transvectin group of Pj g = g1 ⊕ · · · ⊕ gr : the direct sum decomposition of g = Lie(G), where gj = Lie(Gj) is simple ι = ι1 × · · · × ιr where ιj : Pj → gj is the canonical em- bedding of Pj. 17
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: Consider the map ι : P → g, p
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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