On the realization of Riemannian symmetric spaces in Lie groups II

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On the realization of Riemannian symmetric spaces in Lie groups II

Jinpeng An

, Zhengdong Wang

School of Mathematical Science, Peking University, Beijing, 100871, PR China Received 14 April 2005; received in revised form 11 January 2006; accepted 11 January 2006

Abstract

In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 1008–1015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.

©2006 Elsevier B.V. All rights reserved.

MSC:primary 22E15; secondary 57R40, 53C35

Keywords:Symmetric space; Involution; Embedded submanifold

1. Introduction

Suppose G is a connected Lie group with an involution σ. Then the Lie algebra g of G has a canonical de- compositiong=k⊕p, wherekandpare the eigenspaces ofdσ ingwith eigenvalues 1 and−1, respectively. Let G^σ = {g∈G|σ (g)=g}, and supposeKis an open subgroup ofG^σ. Suppose moreover that AdG(K)|pis compact, that is,G/Khas a structure of Riemannian symmetric space. In the particular case thatK=G^σ, it was proved in [1]

thatP =exp(p)is a closed submanifold ofG, and there is a natural isomorphismG/G^σ∼=P. This gave a realization of the symmetric spaceG/G^σ inG. In this paper we generalize this result to the case of arbitrary symmetric space G/K, that is, the case thatKis an arbitrary open subgroup ofG^σ such that AdG(K)|pis compact.

In Section 2 we will make some preparation for this generalized realization. Some properties of the subgroupG^σ will be examined. We will prove the following results.

• There exists a covering groupGofGwith covering homomorphismπsuch that there is an involutionσonG withπ◦σ=σ◦π, and such thatπ⁻¹(K)=(G)^σ.

✩ This work is supported by the 973 Project Foundation of China (#TG1999075102).

* Corresponding author.

E-mail addresses:anjinpeng@gmail.com (J. An), zdwang@pku.edu.cn (Z. Wang).

0166-8641/$ – see front matter ©2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.topol.2006.01.002

• The quotient group G^σ/G^σ₀ is isomorphic to(Z2)^r for some non-negative integerr, where G^σ₀ is the identity component ofG^σ.

• G^σ is connected ifπ₁(G)is finite with odd order.

In Section 3 we will give the precise statement of the realization of arbitrary symmetric spaces in Lie groups. Briefly speaking, a symmetric spaceG/Kis diffeomorphic to a closed submanifoldPof a covering groupGofG, whereG is chosen such thatπ⁻¹(K)=(G)^σ. The idea of the proof is thatG/K∼=G/π⁻¹(K)=G/(G)^σ∼=exp_G(p)=P. 2. The subgroups of fixed points of involutions

For a Lie groupH with an automorphismθ, we always denoteH^θ= {h∈H|θ (h)=h}, and denote the identity component ofH^θ byH₀^θ. In this section we prove the following two theorems. The realization of arbitrary symmetric spaces in Lie groups will be based on Theorem 2.1.

Theorem 2.1.LetGbe a connected Lie group with an involutionσ, K an open subgroup ofG^σ. Then there exists a covering groupGofGwith covering homomorphismπsuch that there is an involutionσonGwithπ◦σ=σ◦π, and such thatπ⁻¹(K)=(G)^σ.

Theorem 2.2.LetGbe a connected Lie group with an involutionσ. Then the quotient groupG^σ/G^σ₀ is isomorphic to(Z2)^r for some non-negative integerr, andr=0ifπ1(G)is finite with odd order.

First we introduce some notations. LetGbe a connected Lie group with an involutionσ. Forg, h∈G, denote the set of all continuous pathsγ:[0,1] →Gwithγ (0)=gandγ (1)=hbyΩ(G, g, h), and denoteπ₁(G, g, h)= {[γ] | γ∈Ω(G, g, h)}, where[γ]is the homotopy class relative to endpoints determined byγ. Note thatπ₁(G, g, g)is the fundamental groupπ₁(G, g)ofGwith basepointg. Forg∈G^σ andγ_g∈Ω(G, e, g), letγ_g^σ ∈Ω(G, e, e)be defined as

γ_g^σ(t )=

γ_g(2t ), t∈ [0,¹₂];

σ (γ_g(2−2t )), t∈ [¹₂,1].

The setπ_g= {[γ_g^σ] |γ_g∈Ω(G, e, g)}is a subset ofπ₁(G, e). Forg₁, g₂∈G^σ, if they belong to the same coset space ofG^σ₀, it is obvious thatπ_g1=π_g2. So we can defineπ_[g]=π_gforg∈G^σ, where[g]denotes the coset spacegG^σ₀. We denote the identity element ofπ₁(G, e)bye.

Lemma 2.3.Forg∈G^σ,π_[⁻_g¹_]=π_[g], that is,x∈π_[g]⇐⇒x⁻¹∈π_[g].

Proof. It is obvious from the equation[γ_g^σ]⁻¹= [(σ◦γ_g)^σ]. 2 Lemma 2.4.Forg∈G^σ,π_[g]=π_[g−1].

Proof. For[γ_g^σ] ∈π_[g],[(γ_g⁻¹)^σ] ∈π_[g−1], whereγ_g⁻¹(t )=γ_g(t )⁻¹. But by Lemma 16.7 in [4],[γ_g^σ] · [(γ_g⁻¹)^σ] = [γ_g^σ]·[(γ_g^σ)⁻¹] = [γ_g^σ(γ_g^σ)⁻¹] =e, where(γ_g^σ(γ_g^σ)⁻¹)(t )=γ_g^σ(t )(γ_g^σ)⁻¹(t ). So by Lemma 2.3,[γ_g^σ] = [(γ_g⁻¹)^σ]⁻¹∈ π_[⁻¹

g⁻¹]=π_[g−1]. This provesπ_[g]⊂π_[g−1]. By symmetry, we have the equality. 2 Lemma 2.5.Forg₁, g₂∈G^σ,[γ_g^σ

1] ·π_[g2]=π_[g1g2]for each[γ_g^σ

1] ∈π_[g1]. Henceπ_[g1]·π_[g2]=π_[g1g2]. Proof. Let[γ_g^σ

2] ∈π_[g₂]. By Lemma 16.7 in [4],[γ_g^σ

1] · [γ_g^σ

2] = [γ_g^σ

1γ_g^σ

2] = [(γg₁γg₂)^σ] ∈π_[g₁g₂]. So we have[γ_g^σ

1] · π_[g₂]⊂π_[g₁g₂]. To prove the equality, denotex= [γ_g^σ

1], and consider the maplx:π_[g₂]→π_[g₁g₂]defined bylx(y)= xy. Sincex⁻¹∈π_[⁻_g¹

1]=π_[g₁]=π_[g−1

1 ],x⁻¹·π_[g₁g₂]⊂π_[g₂]. So we can also define the mapl_x−1:π_[g₁g₂]→π_[g₂]by l_x−1(y)=x⁻¹y. Sincelx◦l_x−1 =id,lxis surjective. This meansx·π_[g₂]=π_[g₁g₂], henceπ_[g₁]·π_[g₂]=π_[g₁g₂]. 2 Lemma 2.6.LetKbe an open subgroup ofG^σ. Thenπ_K:=

k∈Kπ_[k]is a subgroup ofπ₁(G, e).

Proof. By Lemma 2.3, eachπ_[k]is closed under the inverse operation, hence so isπ_K. By Lemma 2.5,π_K is also closed under multiplication. 2

By Lemma 2.6, π_[e]=π_(G^σ

0) andπ_(G^σ₎ are subgroups ofπ1(G, e). By Lemma 2.5, for each g∈G^σ and each x ∈π_[g],π_[g]=x·π_[e]. Soπ_[g] is a coset space of π_[e] inπ_(G^σ₎, that is, an element in the quotient groupΠ^σ = π_(G^σ₎/π_[e](note that the fundamental group of a Lie group is always Abelian). Define the mapf:G^σ/G^σ₀→Π^σ by f ([g])=π_[g]. By Lemma 2.5,f is a homomorphism.

Lemma 2.7.LetGbe a connected Lie group with an involutionσ. ThenG^σ has finite many connected components.

If moreoverGis simply connected, thenG^σ is connected.

Proof. The subgroupΣ= {id, σ}of the automorphism group Aut(G)ofGhas a natural action onG, which we also denote byσ. We form the semidirect productG=G×σ Σ. Denotex=(e, σ )∈G. Since the identity component G0ofGis naturally isomorphic toGbyi:(g,id)→g, andσ (i(y))=i(xyx⁻¹),∀y∈G0, to prove the lemma, it is sufficient to show thatG^x₀= {y∈G0|xyx⁻¹=y}has finite many connected components, and is connected whenG0

is simply connected.

Sincex=(e, σ )is an element of order two inG, it lies in some maximal compact subgroupLofG. By Theorem 3.1 of Chapter XV in [3], there exist some linear subspacesm1, . . . ,mkof the Lie algebragofG(which is isomorphic to the Lie algebra ofG) such that

(1) g=l⊕m1⊕ · · · ⊕mk, wherelis the Lie algebra ofL;

(2) Ad(l)(mi)=mi,∀l∈L, i∈ {1, . . . , k}(in particular, Ad(x)(mi)=mi);

(3) the mapψ:L0×m1× · · · ×m_k→G0defined byψ (l, X1, . . . , X_k)=le^X1· · ·e^Xk is a diffeomorphism, where L0is the identity component ofL.

Denote L^x₀ = {l ∈ L0 | xlx⁻¹ =l}, m^x_i = {X ∈ m_i | Ad(x)(X) =X}, i = 1, . . . , k. We claim that G^x₀ = ψ (L^x₀×m^x₁× · · · ×m^x_k). In fact, for y∈G^x₀, writey =le^X1· · ·e^Xk, where l∈L0, Xi ∈mi. Then y=xyx⁻¹= (xlx⁻¹)e^Ad(x)(X1)· · ·e^Ad(x)(Xk). By (3), we havexlx⁻¹=l, Ad(x)(Xi)=X_i, i=1, . . . , k. That is,l∈L^x₀,X_i ∈m^x_i. Hence we haveG^x₀⊂ψ (L^x₀×m^x₁× · · · ×m^x_k). The other inclusion is obvious.

Since L0is a connected compact Lie group,L^x₀ has finitely many connected components, hence so doesG^x₀= ψ (L^x₀×m^x₁×· · ·×m^x_k). IfG0is simply connected, by (3),L0is also simply connected. By Theorem 8.2 of Chapter VII in [2],L^x₀is connected, hence so isG^x₀. This proves the lemma. 2

Lemma 2.8.Forg∈G^σ, ife∈π_[g], then[g] =G^σ₀. Proof. We endow G=

g∈Gπ1(G, e, g) with the canonical smooth manifold structure and the canonical group structure such thatGis the universal covering group ofGwith covering homomorphismπ([γ])=γ (1). The induced involution ofσ onGisσ (˜ [γ])= [σ◦γ]. Let g∈G^σ. Ife∈π_[g], by definition ofπ_[g], there is aγg∈Ω(G, e, g) such that[γ_g] = [σ◦γ_g]inπ₁(G, e, g). That is,[γ_g] ∈(G)^σ˜. By Lemma 2.7,(G)^σ˜ is connected. Sog=π([γ_g])∈ π((G)^σ˜)=G^σ₀, that is,[g] =G^σ₀. 2

Proposition 2.9.Each non-trivial element ofΠ^σ has order2, and the homomorphismf:G^σ/G^σ₀→Π^σ is an iso- morphism.

Proof. The first assertion follows from Lemma 2.3, the surjectivity off follows from the definition ofΠ^σ, and the injectivity off follows from Lemma 2.8. 2

In particular, we have

Lemma 2.10.Forg₁, g₂∈G^σ, if[g₁] = [g₂], thenπ_[g1]∩π_[g2]= ∅.

Now we are prepared to prove Theorems 2.1 and 2.2.

Proof of Theorem 2.2. By Lemma 2.7,G^σ/G^σ₀ is a finite group. By Proposition 2.9,G^σ/G^σ₀ is Abelian, and each non-trivial element ofG^σ/G^σ₀ has order 2. By the structure theorem of finitely generated Abelian groups,G^σ/G^σ₀ is isomorphic to(Z2)^rfor some non-negative integerr. Ifπ1(G)is finite with odd order, so areπ_(G^σ₎,Π^σ, andG^σ/G^σ₀. This forcesr=0. 2

Proof of Theorem 2.1. We define an equivalence relation on the set

g∈GΩ(G, e, g) as follows. For γ₁, γ₂∈

g∈GΩ(G, e, g),γ₁∼γ₂if and only ifγ₁(1)=γ₂(1)and[γ₁₂] ∈π_K, whereγ₁₂∈Ω(G, e, e)is defined by γ12(t )=

γ1(2t ), t∈ [0,¹₂];

γ₂(2−2t ), t∈ [¹₂,1]. Denote G=(

g∈GΩ(G, e, g))/∼, and denote the equivalence class of aγ ∈

g∈GΩ(G, e, g)by γ. Endow G with a smooth manifold structure in the canonical way, and define the group operation on G as follows. For γ1,γ2 ∈G, letα(t )=γ1(t )γ2(t )andβ(t )=γ1(t )⁻¹, and defineγ1γ2 = α,γ1⁻¹= β. It is easy to check that these are well defined, makingGa Lie group. Letπ:G→Gbe the mapπ(γ)=γ (1), thenπ is a covering homomorphism. Sinceσ_∗(π_K)=π_K, the mapσ:G→G, σ(γ)= σ◦γis a well defined involution ofG. It is obvious thatπ◦σ=σ◦π, that is,σis just the induced involution ofσ onG. Now we have

γ ∈(G)^σ⇐⇒ σ◦γ = γ

⇐⇒σ◦γ∼γ

⇐⇒γ (1)∈G^σ,[γ_{γ (1)}^σ ] ∈π_K

⇐⇒γ (1)∈K (by Lemma 2.10)

⇐⇒ γ ∈π⁻¹(K).

That is,(G)^σ=π⁻¹(K). This completes the proof of Theorem 2.1. 2

Remark 2.1.The covering groupG ofGsatisfying the conclusion of Theorem 2.1 is not necessarily unique. For example, letG be a semisimple Lie group with trivial center, and letσ be a global Cartan involution ofG. Then K=G^σ is a maximal compact subgroup ofG. Let G be any finite cover ofG with an involution σ such that π◦σ=σ ◦π, then(G)^σ is a maximal compact subgroup of G. Since π⁻¹(G^σ)is a compact subgroup ofG containing(G)^σ, it must equal(G)^σ. In fact, using the same method as in the proof of Theorem 2.1, it is easy to see that a covering groupG ofG with covering homomorphismπ satisfying π⁻¹(K)=(G)^σ if and only if π_∗(π₁(G, e))∩π_(G^σ₎=π_K, and the groupGthat we constructed in the proof of Theorem 2.1 is just the one satisfying π_∗(π₁(G, e))=π_K.

Remark 2.2.Professor Jiu-Kang Yu pointed out to the first author proofs of Theorems 2.1 and 2.2 using nonabelian cohomology after he read the first draft of the paper.

3. Covering groups and the realization of symmetric spaces

LetGbe a connected Lie group with an involutionσ, and letK be an open subgroup ofG^σ. Letg=k⊕pbe the canonical decomposition of the Lie algebragofGwith respect to the differential of the involutionσ. Suppose AdG(K)|pis compact, that is,G/Khas a structure of Riemannian symmetric space. Supposeπ:G→Gis a covering homomorphism of Lie groups. Then the twisted conjugate actionτofGonG, which is defined byτ_g(h)=ghσ (g)⁻¹, can be lifted to a well defined actionτofGonG, that is,τ_g(h)=ghσ(g)⁻¹, g∈G, h∈G, whereg∈π⁻¹(g), σis the induced involution ofσ onG.

Theorem 3.1.Under the above assumptions, there is a covering groupGofGwith covering homomorphismπsuch thatP=exp_G(p)is a closed submanifold ofG, and such that the mapϕ:G/K→Pdefined byϕ(gK)=gσ(g)⁻¹ is a diffeomorphism, whereg∈π⁻¹(g),σis the induced involution ofσ onG. With respect to the actions ofGby left multiplication onG/Kand by the actionτonP,ϕis equivariant.

Proof. By Theorem 2.1, there is a covering groupG of Gwith covering homomorphism π such thatπ⁻¹(K)= (G)^σ. Soπinduces aG-equivariant diffeomorphismπ¯:G/(G)^σ→G/Kwith respect to left multiplications (note that the left multiplication of GonG/(G)^σ is well defined). Since AdG((G)^σ)|p=AdG(K)|p is compact, by Corollary 2.6 in [1],P=exp_G(p)is a closed submanifold ofG, and the mapϕ:G/(G)^σ →P, ϕ(g(G)^σ)= gσ(g)⁻¹is aG-equivariant diffeomorphism with respect to left multiplication and twisted conjugate action ofG. Letϕ=ϕ◦(π )¯ ⁻¹. Thenϕ(gK)=ϕ(g(G)^σ)=gσ(g)⁻¹. It is obviously aG-equivariant diffeomorphism with respect to left multiplication onG/Kand the actionτonP. 2

Acknowledgements

The authors would like to thank the referee for some suggestion on improving the English usage of the paper.

They also thank the referee of [1], whose suggestion of revision motivated the authors considering the problems of the paper. The first author thank Professor Jiu-Kang Yu for valuable conversation.

References

[1] J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 1008–1015.

[2] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.

[3] G. Hochschild, The Structure of Lie Groups, Holden-Day, San Francisco, 1965.

[4] N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, NJ, 1951.