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Isometries of Hermitian Symmetric Spaces

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Isometries of Hermitian Symmetric Spaces Isometries of Hermitian symmetric spaces Makiko Sumi Tanaka The 6th OCAMI-KNUGRG Joint Differential Geometry Workshop on Submanifold Theory in Symmetric Spaces and Lie Theory in Finite and Infinite Dimensions Osaka City University February 1{3, 2013 1 Joint with Jost-Hinrich Eschenburg and Peter Quast Contents 1 Introduction 2 Extrinsically symmetric spaces 3 Isometries of Hermitian symmetric spaces 4 Another result 2 1. Introduction A Riemannian symmetric space M is called a symmetric R-space if M is realized as an orbit of a linear isotropy representation of a Riemannian symmetric space of compact type. Every Hermitian symmetric space M of compact type is a symmetric R-space since M is realized as an adjoint orbit of a compact semisimple Lie group. 3 Every symmetric R-space is realized as a real form of a Hermitian symmetric space of compact type and vice- versa (Takeuchi 1984). Here a real form of a Hermitian symmetric space M is the fixed point set of an involutive anti-holomorphic isometry of M. Every real form is con- nected and a totally geodesic Lagrangian submanifold of M. 4 M : a Riemannian manifold τ : an involutive isometry of M A connected component of F (τ; M) := fx 2 M j τ(x) = xg is called a reflective submanifold. Reflective submanifolds in simply connected irreducible Riemannian symmetric spaces of compact type are clas- sified by Leung (1974, '75 and '79). 5 Every symmetric R-space is considered as a reflective submanifold of a Hermitian symmetric space of compact type. Q. Is every reflective submanifold of a symmetric R-space a symmetric R-space ? If a symmetric R-space is simply connected and irre- ducible, we know it is true by Leung's classification. 6 2. Extrinsically symmetric spaces M ⊂ Rn : a submanifold of Euclidean space M is called extrinsically symmetric if it is preserved by ? the reflection ρp at the affine normal space p + (TpM) n n for every p 2 M. Here ρp : R ! R is the affine isometry with ρ (p) = p, ρ j = −Id and ρ j ? = Id. p p TpM p (TpM) Every extrinsically symmetric submanifold is a Rieman- nian symmetric space w.r.t. the induced metric. In fact, the geodesic symmetry sp at p 2 M is given by sp = ρpjM . 7 Theorem (Ferus 1974, Eschenburg-Heintze 1995) Every symmetric R-space is extrinsically symmetric and every full compact extrinsically symmetric submanifold of Euclidean space is a symmetric R-space. 8 P = G=K : a Riemannian symmetric space of compact type G = I0(M), K = fg 2 G j g(o) = og, o 2 P g = k ⊕ p : the canonical decomposition ∼ g = Lie(G), k = Lie(K), p = ToP Take ξ(=6 0) 2 p with ad(ξ)3 = −ad(ξ), then M = AdG(K)ξ ⊂ p : a symmetric R-space 9 τ : an involutive isometry of M L : a reflective submanifold determined by τ If we assume 9T : a linear isometry of p with T (M) ⊂ M s.t. T jM = τ, L is an extrinsically symmetric submanifold in a linear subspace F (T; p) of p. 10 I(M) : the group of isometries of M T(M) : the transvection group of M, i.e., a subgroup of I(M) generated by fsp ◦ sq j p; q 2 Mg, which is coincides with the identity component I0(M) of I(M) If f 2 T(M), f can be extended to a linear isometry of p since every geodesic symmetry sp has the extension ρp. But what about arbitrary isometry? 11 Remark We have a list of I(M)=I0(M) for irreducible M in p.156 of Loos's book \Symmetric spaces II". 12 3. Isometries of Hermitian symmetric spaces P : a semisimple Hermitian symmetric space (i.e., of compact type or noncompact type) I(P ) : the isometry group of P A(P ) : the holomorphic isometry group of P T(P ) : the transvection group of P G := I0(P ) = A0(P ) = T(P ) g = Lie(G) : semisimple J : the complex structure of P p 2 P 13 ! 7! ◦ ◦ −1 Isp : G G; g sp g sp : 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 = fg 2 G j g(p) = pg, pp = TpP Jp 2 C(kp), the center C(kp) of kp (If P is irreducible, Jp generates C(kp).) ∼ j The action of Jp on TpP = pp is given by ad(Jp) pp. 3 Thus ad(Jp) has eigenvalues ±i and 0, so that ad(Jp) = −ad(Jp). 14 Consider the map ι : P ! g; p 7! 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 2 P . ι is called the canonical embedding of P . 15 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 I(P ) is generated by I(P1) × · · · × I(Pr) and by all per- mutations of isometric irreducible factors of P . Since permutations of isometric irreducible factors of P extend to permutations of the corresponding simple fac- tors of g (possibly up to sign on some factors), we may assume that P is irreducible. 18 Let f 2 I(P ). Let m be the midpoint of a geodesic arc between o to f(o), then sm ◦ f(o) = o. Since P is an extrinsically symmetric submanifold of g and sm has the extension ρm, we may assume that f(o) = o. 19 −1 If : G ! G; g 7! f ◦ g ◦ f is an automorphism of G (If )∗ : g ! g is an automorphism of g h ; i : an inner product on g which is proportional to the Killing form h(If )∗(X); (If )∗(Y )i = hX; Y i (X; Y 2 g). 20 Lemma (If )∗(Jo) 2 {±Jog. Proof. Since Jo 2 C(ko) and (If )∗ is a Lie algebra auto- 3 morphism, we have (If )∗(Jo) 2 C(ko) and ad((If )∗(Jo)) = −ad((If )∗(Jo)). −1 If k 2 Ko = fg 2 G j g(o) = og, If (k)(o) = f ◦ k ◦ f (o) = j ! o. Hence If Ko : Ko Ko is an automorphism of Ko. Thus (If )∗(C(ko)) = C(ko) = RJo. So (If )∗(Jo) 2 {±Jog. 21 Let F := ±(If )∗ if (If )∗(Jo) = ±Jo, respectively. To prove f = F jP , we show ι ◦ f = F ◦ ι. Since we have (ι ◦ f)(o) = (F ◦ ι)(o) and ι ◦ f ◦ γ = F ◦ ι ◦ γ for arbitrary geodesic γ starting from o, we obtain the conclusion. 22 Remark It is a known fact that every reflective sub- manifold of a semisimple Hermitian symmetric space is a symmetric R-space. Because we know every reflective submanifold of a semisimple Hermitian symmetric space is either a complex submanifold or a real form from the next proposition. 23 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), 2m Gm(C )(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 ⊂ Rn 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 2 M A connected component of F (so;M) is called a polar. For p 2 F (so;M), the connected component of F (sp ◦ so;M) through p is called the meridian. 25 Proposition 1 (Chen-Nagano 1978) If M is a compact Riemannian symmetric space of rank k, then any meridian M− ⊂ M has the same rank k. Proposition 2 A meridian M− in an extrinsically symmetric space M is extrinsically symmetric. 26 Proposition 3 Let M be a compact Riemannian symmetric space which satisfies that every polar is a single point. Then M is the Riemannian product of round spheres and possibly a torus. Proposition 4 Let P ⊂ Rn be a full extrinsically symmetric flat torus. Then P splits extrinsically as a product of round circles, = 1 × · · · × 1 ⊂ R2n. P Sr1 Srn 27 Proof of Theorem 2 (Eschenburg-Quast-T.) We prove it by induction over the dimension. The beginning of the induction is the observation that an extrinsically symmetric flat torus is a product of Eu- clidean circles (Proposition 4). As induction step, we show an alternative for arbitrary compact symmetric spaces X: Either X is a Riemannian product of Euclidean spheres and possibly flat torus or it contains a certain totally geodesic submanifold , a so called meridian (Proposition 1). 28 Further, a meridian M− of X is extrinsically symmetric, then so is M− (Proposition 2).
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