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 connected 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) := {x ∈ M | τ(x) = x} 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 irreducible, 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 ∈ M. Here ρp : R → R is the affine isometry with ρ (p) = p, ρ | = −Id and ρ | ⊥ = 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 ∈ M is given by sp = ρp|M .

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 = {g ∈ G | g(o) = o}, o ∈ P g = k ⊕ p : the canonical decomposition ∼ g = Lie(G), k = Lie(K), p = ToP

Take ξ(≠ 0) ∈ 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 ∃T : a linear isometry of p with T (M) ⊂ M s.t. T |M = 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 {sp ◦ sq | p, q ∈ M}, which is coincides with the identity component I0(M) of I(M)

If f ∈ 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 ∈ 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 = {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. 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 ∈ 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 factors

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 embedding of Pj.

17 I(P ) is generated by I(P1) × · · · × I(Pr) and by all permutations of isometric irreducible factors of P . Since permutations of isometric irreducible factors of P extend to permutations of the corresponding simple factors of g (possibly up to sign on some factors), we may assume that P is irreducible.

18 Let f ∈ 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 ⟨ , ⟩ : an inner product on g which is proportional to the Killing form

⟨(If )∗(X), (If )∗(Y )⟩ = ⟨X,Y ⟩ (X,Y ∈ g).

20 Lemma (If )∗(Jo) ∈ {±Jo}.

Proof. Since Jo ∈ C(ko) and (If )∗ is a Lie algebra auto- 3 morphism, we have (If )∗(Jo) ∈ C(ko) and ad((If )∗(Jo)) =

−ad((If )∗(Jo)). −1 If k ∈ Ko = {g ∈ G | g(o) = o}, If (k)(o) = f ◦ k ◦ f (o) = | → o. Hence If Ko : Ko Ko is an automorphism of Ko.

Thus (If )∗(C(ko)) = C(ko) = RJo. So (If )∗(Jo) ∈ {±Jo}.

21 Let F := ±(If )∗ if (If )∗(Jo) = ±Jo, respectively. To prove f = F |P , 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 submanifold 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 ∈ 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 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). Thus passing to meridi- ans again and again, we will lower the dimension preserving preserving the maximal torus unless we reach a space which is a Riemannian product of spheres and possibly a torus and whose maximal torus is a Riemannian product of circles.