Actions of Lie Groups and Homogeneous Manifolds

Let(V, ω)be a symplectic space; in Definition 1.4.10 we have introduced the symplectic groupSp(V, ω). We have thatSp(V, ω)is a closed subgroup of GL(V); its Lie algebra consists of those linear endomorphismsXofV such thatω(X·,·)is a symmetric bilinear form, that is:

(2.1.6) ω(X(v), w) =ω(X(w), v), v, w∈V.

In terms of the linear operatorω:V → V^∗, formula (2.1.6) is equivalent to the identity:

(2.1.7) ω◦X=−X^∗◦ω.

Ifω is the canonical symplectic form ofIR²ⁿ, then we writeSp(IR²ⁿ, ω) = Sp(2n, IR)andsp(IR²ⁿ, ω) = sp(2n, IR). The matrix representations of ele-ments ofSp(V, ω)with respect to a symplectic basis are described in formulas (1.4.7) and (1.4.8). Using (2.1.7) it is easy to see that the matrix representation of elements ofsp(V, ω)in a symplectic basis is of the form:

A B C −A^∗

, B, Csymmetric, whereA^∗denotes the transpose ofA.

2.1.2. Actions of Lie Groups and Homogeneous Manifolds. In this

for allm ∈M. The action is effective if the homomorphismg 7→ γg is injective, i.e., ifT

m∈MG_m ={1}.

IfHis a subgroup ofG, we will denote byG/Hthe set of left cosets ofHin G:

G/H =

gH:g∈G ,

wheregH ={gh:h∈H}is the left coset ofg∈G. We have a natural action of GonG/Hgiven by:

(2.1.10) G×G/H 3(g1, g2H)7−→(g1g2)H ∈G/H;

this action is called action by left translation of G in the left cosets ofH. The action (2.1.10) is always transitive.

IfGacts onM andGmis the isotropy group of the elementm∈M, then the mapβ_mof (2.1.9) passes to the quotient and defines a bijection:

(2.1.11) β¯m :G/Gm−→G(m)

given byβ¯m(gGm) = g·m. We therefore have the following commutative dia-gram:

G

q

βm

##F

FF FF FF FF

G/G_m ^∼=

β¯m

//M

whereq:G→G/G_mdenotes the quotient map.

2.1.5. DEFINITION. Given actions of the groupGon setsM andN, we say that a mapφ:M →N isG-equivariant if the following identity holds:

φ(g·m) =g·φ(m),

for allg∈ Gand allm ∈M. Ifφis an equivariant bijection, we say thatφis an equivariant isomorphism; in this caseφ⁻¹is automatically equivariant.

The bijection (2.1.11) is an equivariant isomorphism when we consider the action ofGonG/Gmby left translation and the action ofGonG(m)obtained by the restriction of the action ofGonM.

2.1.6. REMARK. It is possible to define also a right action of a groupGon a setM as a map:

(2.1.12) M×G3(m, g)7−→m·g∈M

that satisfies(m·g₁)·g₂ = m·(g₁g₂)andm·1 = mfor allg₁, g₂ ∈ Gand all m∈M. A theory totally analogous to the theory of left actions can be developed for right actions; as a matter of facts, every right action (2.1.12) defines a left action by(g, m)7→m·g⁻¹. Observe that in the theory of right actions, in order to define properly the bijectionβ¯_min formula (2.1.11), the symbolG/Hhas to be meant as the set of right cosets ofH.

Let’s assume now thatGis a Lie group and thatMis a manifold; in this context we will always assume that the map (2.1.8) is differentiable, and we will say that Gacts differentiably on M. If H is a closed subgroup of G, then there exists a unique differentiable structure in the setG/Hsuch that the quotient map:

q :G−→G/H

is a differentiable submersion (see Remark 2.1.3). The kernel of the differential dq(1)is precisely the Lie algebrahofH, so that the tangent space toG/H at the point1H may be identified with the quotient spaceg/h. Observe that, sinceq is open and surjective, it follows that G/H has the quotient topology induced byq from the topology ofG.

By continuity, for allm ∈M, the isotropy groupG_m is a closed subgroup of G, hence we get a differentiable structure onG/Gm; it can be shown that the map gG_m 7→g·mis a differentiable immersion, from which we obtain the following:

2.1.7. PROPOSITION. IfGis a Lie group that acts differentiably on the mani-foldM, then for allm∈ M the orbitG(m)has a unique differentiable structure that makes (2.1.11) into a differentiable diffeomorphism; with such structureG(m) is an immersed submanifold ofM, and the tangent spaceTmG(m)coincides with the image of the map:

dβm(1) :g−→TmM,

whereβ_mis the map defined in (2.1.9).

2.1.8. REMARK. If we choose a different pointm⁰ ∈G(m), so thatG(m⁰) = G(m), then it is easy to see that the differentiable structure induced onG(m) by β¯_m⁰ coincides with that induced byβ¯_m.

We also have the following:

2.1.9. COROLLARY. IfGacts transitively onM, then for allm∈M the map (2.1.11) is a differentiable diffeomorphism of G/G_m onto M; in particular, the

mapβmof (2.1.9) is a surjective submersion.

In the case of transitive actions, when we identifyG/G_m withM by the dif-feomorphism (2.1.11), we will say thatmis the base point for such identification;

we then say thatM (orG/G_m) is a homogeneous manifold.

2.1.10. COROLLARY. LetM, N be manifolds and let Gbe a Lie group that acts differentiably on bothM andN. If the action ofGon M is transitive, then every equivariant mapφ:M →N is differentiable.

PROOF. Choosem∈M; the equivariance property ofφgives us the following commutative diagram:

G

βm

βφ(m)

B

BB BB BB B M φ //N

and the conclusion follows from Corollary 2.1.9 and Remark 2.1.3.

In some situations we will need to know if a given orbit of the action of a Lie group is an embedded submanifold. Let us give the following definition:

2.1.11. DEFINITION. LetXbe a topological space; a subsetS⊂Xis said to be locally closed ifSis given by the intersection of an open and a closed subset of X. Equivalently,Sis locally closed when it is open in the relative topology of its closureS.

Exercise 2.4 is dedicated to the notion of locally closed subsets.

We have the following:

2.1.12. THEOREM. LetGbe a Lie group acting differentiably on the manifold M. Givenm∈M, the orbitG(m)is an embedded submanifold ofM if and only ifG(m)is locally closed inM.

PROOF. See [47, Theorem 2.9.7].

We conclude the subsection with a result that relates the notions of fibration and homogeneous manifold.

2.1.13. DEFINITION. Given manifolds F, E andB and a differentiable map p:E →B, we say thatpis a differentiable fibration with typical fiberF if for all b∈Bthere exists a diffeomorphism:

α:p⁻¹(U)−→U ×F

such thatπ1◦α=p|_p⁻¹_(U), whereU ⊂Bis an open neighborhood ofbinBand π₁ :U ×F → U is the projection onto the first factor. In this case, we say thatα is a local trivialization ofparoundb.

2.1.14. THEOREM. Let Gbe a Lie group and H, K closed subgroups ofG withK ⊂H; then the map:

p:G/K −→G/H

defined byp(gK) =gH is a differentiable fibration with typical fiberH/K. PROOF. It follows from Remark 2.1.3 thatp is differentiable. GivengH ∈ G/H, lets:U → Gbe a local section of the submersionq :G→G/H defined in an open neighborhoodU ⊂G/HofgH; it follows thatq◦sis the inclusion of U inG/H. We define a local trivialization ofp:

α:p⁻¹(U)−→U ×H/K

by settingα(xK) = (xH, s(xH)⁻¹xK). The conclusion follows.

2.1.15. COROLLARY. Under the assumptions of Corollary 2.1.9, the mapβm

given in (2.1.9) is a differentiable fibration with typical fiberG_m. 2.1.16. COROLLARY. Letf:G→ G⁰ be a Lie group homomorphism and let H⊂G,H⁰⊂G⁰be closed subgroups such thatf(H)⊂H⁰; consider the map:

f¯:G/H−→G⁰/H⁰

induced fromf by passage to the quotient, i.e.,f¯(gH) =f(g)H⁰for allg∈G. If f¯is surjective, thenf¯is a differentiable fibration with typical fiberf⁻¹(H⁰)/H.

PROOF. Consider the action ofGonG⁰/H⁰ given by

G×G⁰/H⁰3(g, g⁰H⁰)7−→(f(g)g⁰)H⁰ ∈G⁰/H⁰.

The orbit of the element1H⁰ ∈ G⁰/H⁰ is the image off¯, and its isotropy group is f⁻¹(H⁰); since f¯is surjective, it follows from Corollary 2.1.9 that the map fˆ:G/f⁻¹(H⁰) → G⁰/H⁰ induced fromf by passage to the quotient is a diffeo-morphism. We have the following commutative diagram:

G/H

p

yyrrrrrrrrrr _f¯

##H

HH HH HH HH G/f⁻¹(H⁰)

fˆ

//G⁰/H⁰

wherepis induced from the identity ofGby passage to the quotient; it follows from Theorem 2.1.14 thatp is a differentiable fibration with typical fiberf⁻¹(H⁰)/H.

This concludes the proof.

A differentiable covering is a differentiable fibering whose fiber is a discrete manifold (i.e., zero dimensional). We have the following:

2.1.17. COROLLARY. Under the assumptions of Corollary 2.1.16, if H and f⁻¹(H⁰)have the same dimension, thenf¯is a differentiable covering.

2.1.18. REMARK. Given a differentiable fibration p : E → B with typical fiberF, then every curveγ : [a, b]→ Bof classC^k,0≤ k≤+∞, admits a lift γ : [a, b]→E(i.e.,p◦γ =γ) which is of classC^k:

E

p

[a, b]

˜ γ ==

γ //B

The proof of this fact is left to the reader in Exercise 2.9.

2.1.3. Linearization of the Action of a Lie Group on a Manifold. In this