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
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Joint with Jost-Hinrich Eschenburg and Peter Quast
Contents
1 Introduction
2 Extrinsically symmetric spaces
3 Isometries of Hermitian symmetric spaces
4 Another result
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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.
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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.
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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).
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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.
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2. Extrinsically symmetric spaces
M ⊂ R n : 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) ⊥
for every p ∈ M. Here ρp : R n → R n is the affine isometry
with ρp(p) = p, ρp| TpM = −Id and ρp| (TpM) ⊥ = Id.
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.
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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.
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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 = Ad G(K)ξ ⊂ p : a symmetric R-space
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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.
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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?
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Remark
We have a list of I(M)/I0(M) for irreducible M in p.156
of Loos’s book “Symmetric spaces II”.
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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
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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).
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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 .
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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.
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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.
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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.
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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.
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I f : G → G, g ↦→ f ◦ g ◦ f −1 is an automorphism of G
(I f)∗ : g → g is an automorphism of g
⟨ , ⟩ : an inner product on g which is proportional to the
Killing form
⟨(I f)∗(X), (I f)∗(Y )⟩ = ⟨X, Y ⟩ (X, Y ∈ g).
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Lemma (I f)∗(Jo) ∈ {±Jo}.
Proof. Since Jo ∈ C(ko) and (I f)∗ is a Lie algebra auto-
morphism, we have (I f)∗(Jo) ∈ C(ko) and ad((I f)∗(Jo)) 3 =
−ad((I f)∗(Jo)).
If k ∈ Ko = {g ∈ G | g(o) = o}, I f(k)(o) = f ◦ k ◦ f −1 (o) =
o. Hence I f| Ko : Ko → Ko is an automorphism of Ko.
Thus (I f)∗(C(ko)) = C(ko) = RJo. So (I f)∗(Jo) ∈ {±Jo}.
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Let F := ±(I f)∗ if (I f)∗(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.
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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.
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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.
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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.
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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.
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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 ⊂ R n be a full extrinsically symmetric flat torus.
Then P splits extrinsically as a product of round circles,
P = S 1 r1 × · · · × S1 rn ⊂ R2n .
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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).
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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 preserv-
ing 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.
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