aa r X i v : . [ m a t h . S G ] S e p A DUAL PAIR FOR THE CONTACT GROUP
STEFAN HALLER AND CORNELIA VIZMAN
Abstract.
Generalizing the canonical symplectization of contact manifolds, we con-struct an infinite dimensional non-linear Stiefel manifold of weighted embeddings intoa contact manifold. This space carries a symplectic structure such that the contactgroup and the group of reparametrizations act in a Hamiltonian fashion with equivari-ant moment maps, respectively, giving rise to a dual pair, called the EPContact dualpair. Via symplectic reduction, this dual pair provides a conceptual identification ofnon-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of thecontact group. Moreover, the EPContact dual pair gives rise to singular solutions forthe geodesic equation on the group of contact diffeomorphisms. For the projectivizedcotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pairdue to Holm and Marsden, and leads to a geometric description of some coadjointorbits of the full diffeomorphism group. Introduction
Every contact manifold gives rise to a symplectic manifold in a canonical way. Ifthe contact structure is described by a 1-form α on P , then this symplectic manifoldcan be described as P × ( R \
0) with the symplectic form d ( tα ), where t denotes theprojection onto the second factor. Regarding the contact structure as a subbundle ofhyperplanes, ξ ⊆ T P , and denoting the corresponding line bundle over P by L := T P/ξ ,this symplectization can be described more naturally as M = L ∗ \ P , with the symplecticform induced from the canonical symplectic form on T ∗ P via the natural vector bundleinclusion L ∗ ⊆ T ∗ P .The group of contact diffeomorphisms, Diff( P, ξ ), acts on M in a natural way, pre-serving the symplectic structure. This action is in fact Hamiltonian and admits anequivariant moment map. This moment map identifies (unions of) connected compo-nents of the symplectization M with certain coadjoint orbits of the contact group.1.1. The EPContact dual pair.
In this paper we will introduce a natural infinitedimensional generalization M of the symplectization M = L ∗ \ P with similar features.To this end we fix a closed manifold S , we denote by | Λ | S its line bundle of densities, andwe consider the space M of line bundle homomorphisms from | Λ | ∗ S → S to L ∗ → P whichrestrict to a linear isomorphism on each fiber. Every volume density on S provides anidentification M ∼ = C ∞ ( S, M ) and permits to regard elements Φ ∈ M as pairs consistingof a map ϕ : S → P together with a contact form for ξ along this map. This space M can be equipped with the structure of a Fr´echet manifold in a natural way, and admitsa canonical (weakly non-degenerate) symplectic form. The symplectization M can berecovered by choosing S to be a single point. Mathematics Subject Classification.
Key words and phrases.
Contact manifold; Contact diffeomorphism group; Coadjoint orbit; Dualpair; Homogeneous space; Symplectic manifold; Symplectization; Manifold of mappings; Infinite di-mensional manifold; Non-linear Grassmannian; Non-linear Stiefel manifold.
The contact group acts on M in a natural way, preserving the symplectic structure.This action is Hamiltonian and admits an equivariant moment map, see Proposition 2.4.Furthermore, the group of reparametrizations, Diff( S ), acts on M in a Hamiltonianfashion, also admitting an equivariant moment map. On the non-linear Stiefel manifoldof weighted embeddings, E ⊆ M , the latter action is free. We show that the restrictionsof these moment maps to E , X ( P, ξ ) ∗ J E L ←−−−−− E J E R −−−−−→ Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ , (1)constitute a symplectic dual pair in the sense of Weinstein [33], see Theorem 2.6. Here X ( P, ξ ) denotes the Lie algebra of contact vector fields on P , X ( S ) denotes the Lie algebraof all vector fields on S , and Ω ( S, | Λ | S ) denotes the space of smooth 1-form densitieson S . The moment maps are given by h J E L (Φ) , X i = R S Φ( X ◦ ϕ ) for all X ∈ X ( P, ξ ),and h J E R (Φ) , Z i = R S Φ( T ϕ ◦ Z ) for all Z ∈ X ( S ).Actually, we will show a stronger statement: The group Diff( S ) acts freely and tran-sitively on the fibers of J E L , and the group Diff( P, ξ ) acts locally transitive on the levelsets of J E R , see Proposition 4.2 and Theorem 3.5. Moreover, we will see that the levelsets of both moment maps are smooth submanifolds of E . The dual pair in (1) will bereferred to as the EPContact dual pair , because the left leg provides singular solutionsof the EPContact equation, i.e., the Euler–Poincar´e equation associated with the groupof contact diffeomorphisms.Recall that the projectivized cotangent bundle of a manifold Q admits a canonicalcontact structure. The EPContact dual pair corresponding to the projectivized cotan-gent bundle of Q is closely related to the EPDiff dual pair, due to Holm–Marsden [17],associated to the action of Diff( Q ) and Diff( S ) on T ∗ Emb(
S, Q ), the cotangent bun-dle of embeddings of S into Q , see Section 5. This comparison leads to a geometricinterpretation of some coadjoint orbits of Diff( Q ), see Corollary 5.5.1.2. Coadjoint orbits of the contact group.
The EPContact dual pair will beused to identify coadjoint orbits of the contact group via symplectic reduction for thereparametrization action, following the general principle: Symplectic reduction on oneleg of a dual pair of moment maps leads to coadjoint orbits of the other group. The sameprinciple was used in [13], where symplectic reduction on the right leg of the ideal fluiddual pair due to Marsden and Weinstein [25] led to coadjoint orbits of the Hamiltoniangroup consisting of weighted isotropic submanifolds of the symplectic manifold [34, 20].To make this more precise, consider the non-linear Grassmannian of weighted sub-manifolds, G = E / Diff( S ), consisting of pairs ( N, γ ) where N is a submanifold of type S in P and γ : | Λ | ∗ N → L | ∗ N is an isomorphism of line bundles which may be regarded asbeing akin to a trivialization of the contact structure along N . This space G is a Fr´echetmanifold in a natural way and the projection E → G is a smooth principal bundle withstructure group Diff( S ). The moment map J E L descends to a Diff( P, ξ )-equivariant in-jective immersion
G → X ( P, ξ ) ∗ , which permits to identify orbits of the contact groupin G with coadjoint orbits. Each 1-form density ρ ∈ Ω ( S, | Λ | S ) gives rise to a reducedspace G ρ ⊆ G given by G ρ = ( J E R ) − ( O ρ ) / Diff( S ) = ( J E R ) − ( ρ ) / Diff(
S, ρ ) , where O ρ denotes the Diff( S )-orbit through ρ , and Diff( S, ρ ) is the isotropy group of ρ . DUAL PAIR FOR THE CONTACT GROUP 3
Reduction works best for the zero level. The corresponding reduced space G coincideswith the subset of weighted isotropic submanifolds, G iso ⊆ G . We will see that G iso is asmooth submanifold of G and that the action of the contact group on G iso admits localsmooth sections. In particular, this action is locally transitive. Hence, the restrictionof the moment map, G iso → X ( P, ξ ) ∗ , identifies (unions of) connected components of G iso with coadjoint orbits of the contact group. Moreover, this identification intertwinesthe Kostant–Kirillov–Souriau symplectic form with the reduced symplectic form on G iso .These facts are summarized in Theorem 4.12.The situation is more delicate with regard to reduction at more general levels. In thiscase the reduced spaces are more singular subsets of G and it is unclear, if the contactgroup acts locally transitive on them. If ρ is a contact 1-form density on S , i.e., ifker ρ is a contact structure on S , then the reduced space G ρ consists of certain weightedcontact submanifolds of P which are of type ( S, ker ρ ). This is an open condition onthe submanifold in view of Gray’s stability theorem. The condition on the weight,however, is rather singular: The space of all admissible (for G ρ ) weights on a fixedcontact submanifold may be identified with the Diff( S, ker ρ )-orbit of ρ . The situationis tamer if we specialize to 1-dimensional S , see Example 4.19. In particular, (unionsof) connected components in the spaces of weighted transverse knots of fixed length ina contact 3-manifold, may be identified with coadjoint orbits of the contact group.1.3. Singular solutions of the Euler–Poincar´e equation.
Another motivation forstudying the EPContact dual pair is the construction of singular solutions of the geo-desic equation on the group of contact diffeomorphisms equipped with a right invariantRiemannian metric. This works analogous to the EPDiff equation, where the EPDiffdual pair has been used by Holm and Marsden [17] to construct singular solutions forthe geodesic equation on the full diffeomorphism group. Similarly, point vortices in twodimensional ideal fluids, a geodesic equation on the group of volume preserving diffeo-morphisms, have been described using a dual pair by Marsden–Weinstein [25], see theappendix. The same kind of argument has been applied for the Vlasov equation inkinetic theory by Holm–Tronci [18] using the ideal fluid dual pair, and for the Euler–Poincar´e equations on the group of automorphisms of a principal bundle in [10] usingthe EPAut dual pair [12].In all these cases the singular solutions of the system are obtained, via a momentmap, from a collective Hamiltonian dynamics on a symplectic manifold, referred to asClebsch variables. This moment map turns out to be the left leg of a dual pair associatedto commuting actions on the manifold of embeddings, while the right leg moment mapgives conserved quantities by Noether’s theorem. We show that for the group of contactdiffeomorphisms the situation is similar.To describe this in more detail, let us start by briefly reviewing the geodesic equationon a Lie group with respect to a right invariant Riemannian metric. We write the innerproduct on the Lie algebra g in the form ( u, v ) = h Qu, v i , where the inertia operator Q : g → g ∗ is symmetric and strictly positive. Formally, the right trivialized geodesicequation on the Lie algebra g is the Euler–Arnold equation, ddt u = − ad ⊤ u u, (2) STEFAN HALLER AND CORNELIA VIZMAN where the adjoint of the adjoint action can be characterized by (ad ⊤ u v, w ) := ( v, ad u w )for all u, v, w ∈ g . In other words, ad ⊤ u = Q − ad ∗ u Q , where ad ∗ u : g ∗ → g ∗ denotes thecoadjoint action characterized by h ad ∗ u m, v i = h m, ad u v i for u, v ∈ g and m ∈ g ∗ .Via Legendre transformation, using the momentum m := Qu , the Euler–Arnold equa-tion (2) becomes the Lie–Poisson equation, ddt m = − ad ∗ u m, (3)which is the Hamiltonian equation on the Poisson manifold g ∗ for the Hamiltonian h : g ∗ → R , h ( m ) := h m, Q − m i . Its solutions are confined to coadjoint orbits, the symplectic leaves of g ∗ .Let us now turn to the group of contact diffeomorphisms on a contact manifold ( P, ξ ).Recall that its Lie algebra can be canonically identified with the space of contact vectorfields, X ( P, ξ ) = Γ ∞ ( L ), where L = T P/ξ . For simplicity, we will assume P to be closed.We consider X ( P, ξ ) ∗ = Γ −∞ ( L ∗ ⊗ | Λ | P ), the space of distributional sections of L ∗ ⊗ | Λ | P ,where | Λ | P denotes the bundle of densities on P . We assume that the inertia operator, Q : Γ ∞ ( L ) → Γ ∞ ( L ∗ ⊗ | Λ | P ), is a pseudo-differential operator of real order s which issymmetric, strictly positive, invertible, and its inverse, Q − : Γ ∞ ( L ∗ ⊗ | Λ | P ) → Γ ∞ ( L ),is a pseudo-differential operator of order − s . Hence, the corresponding inner product,( u, v ) = h Qu, v i , generates the Sobolev H s/ topology on Γ( L ). Using elliptic theory, suchinertia operators can be easily constructed. For instance, we may use Q = φ (1 + ∆) s/ ,where ∆ is a Laplacian acting on Γ( L ) which is non-negative and formally self-adjointwith respect to a volume density on P and a fiberwise Euclidean metric on L , and φ : L → L ∗ ⊗ | Λ | P denotes the isomorphism of line bundles provided by these geometricchoices.The Hamiltonian function h ( m ) = h m, Q − m i is well defined on Γ − s/ ( L ∗ ⊗ | Λ | P ),the space of sections which are of Sobolev class − s/
2. Note that the Sobolev spaceΓ − s/ ( L ∗ ⊗ | Λ | P ) is invariant under the coadjoint action of Diff( P, ξ ). If k ∈ Γ −∞ ( L ⊠ L )denotes the Schwartz kernel of Q − , then h ( m ) = h k, m ⊠ m i = 12 Z ( x,y ) ∈ P × P m ( x ) k ( x, y ) m ( y )extends continuously (regularization) to m ∈ Γ − s/ ( L ∗ ⊗ | Λ | P ). Assuming s > dim P − dim S, (4)the moment map J E L : E → X ( P, ξ ) ∗ takes values in Γ − s/ ( L ∗ ⊗ | Λ | P ) = Γ s/ ( L ) ∗ . Indeed,for Φ ∈ E the distribution J E L (Φ) is the push forward of a smooth section on S alonga smooth embedding S → P , cf. Remark 2.9. According to a standard property ofthe trace map on Sobolev spaces, see for instance [30, Proposition 1.6 in Chapter 4], itthus provides a continuous functional on Γ s/ ( L ). The map J E L is actually smooth intoΓ − s/ ( L ∗ ⊗ | Λ | P ). Hence, the pull back of the Hamiltonian h to E , H : E → R , H := h ◦ J E L , is smooth. Although the symplectic form on E is only weakly non-degenerate, thefunction H gives rise to a Hamiltonian vector field X H on (and tangential to) E , cf. thediscussion in [5, Section 4.2.2]. Indeed, since J E L is a moment map, we formally have DUAL PAIR FOR THE CONTACT GROUP 5 X H (Φ) = ζ E Q − J E L (Φ) (Φ) and thus X H (Φ) = ζ L ∗ Q − J E L (Φ) ◦ Φ , (5)where ζ E and ζ L ∗ denote the infinitesimal Diff( P, ξ )-actions on E and L ∗ , respectively,cf. (25) and (20) below. By microlocal regularity, Q − J E L (Φ) is smooth along the sub-manifold N in P determined by Φ, see for instance [31, Corollary 9.4 in Chapter 7] or[15, Proposition 3.11 in Chapter IV § ζ L ∗ : Γ ∞ ( L ) → Γ ∞ ( T L ∗ ) isessentially given by a first order differential operator, it extends to distributional sec-tions, and ζ L ∗ Q − J E L (Φ) is smooth along L ∗ | N . In particular, the latter is smooth along Φand thus X H (Φ) is a tangent vector to E at Φ, cf. (5).Every solution Φ t ∈ E of the Hamilton equation ddt Φ t = X H (Φ t ) (6)provides a singular (peakon) solution u t := Q − J E L (Φ t ) ∈ Γ s/ ( L ) of the Euler–Arnoldequation (2) with momentum m t := J E L (Φ t ) ∈ Γ − s/ ( L ∗ ⊗ | Λ | P ). The support of thedistributional momentum m t coincides with the smooth submanifold determined by Φ t ,and this also coincides with the singular support of u t . Due to the dual pair property,each solution Φ t of (6) remains in a level of the other moment map, J E R : E → X ( S ) ∗ ,and is thus confined to a Diff( P, ξ ) orbit in E . Hence, its momentum m t = J E L (Φ t ) isconstrained to a coadjoint orbit.If S is a single point, then the assumption in (4) implies that the distributional kernel k of Q − is continuous. In this case we have E = L ∗ \ P and H is given by the (smooth)restriction of k to the diagonal.The initial value problem for the EPContact equation has been studied by Ebin andPreston in [5]. They consider inertia operators of the form Q = 1 + ∆, where theLaplacian is with respect to a Riemannian metric which is associated with the contactstructure.It appears to be interesting [4] to replace the class of inertia operators considered abovewith operators in the Heisenberg calculus [3, 29, 27], a calculus of pseudo-differentialoperators which is closely linked to the contact geometry on P . Using the Rocklandtheorem, one can construct pseudo-differential operators Q : Γ ∞ ( L ) → Γ ∞ ( L ∗ ⊗ | Λ | P )of Heisenberg order s which are symmetric, strictly positive, invertible, and such thatthe inverse, Q − : Γ ∞ ( L ∗ ⊗ | Λ | P ) → Γ ∞ ( L ), is of Heisenberg order − s . For instance,we may use Q = φ (1 + ∆) s/ , where ∆ is a subLaplacian. Everything mentioned aboveremains valid, provided the Sobolev spaces are being replaced with the correspondingspaces in the Heisenberg Sobolev scale and the assumption (4) is replaced by the strongercondition s/ > dim P − dim S .1.4. Structure of the paper.
The remaining part of the paper is organized as follows.In Section 2 we construct the EPContact dual pair. In Section 3 we show that thelevel sets of the right moment map are submanifolds on which the contact group actslocally transitive. In Section 4 we study the reduced spaces obtained by factoring outthe group of reparametrizations. In Section 5 we compare the EPContact dual pair forthe projectivized cotangent bundle with the EPDiff dual pair of Holm and Marsden. Inthe appendix we provide a comparison with a dual pair due to Marsden and Weinsteinfor the Euler equation of an ideal fluid.
STEFAN HALLER AND CORNELIA VIZMAN
Acknowledgments.
The first author would like to thank the West University ofTimi¸soara for the warm hospitality and Shantanu Dave for a helpful reference. Hegratefully acknowledges the support of the Austrian Science Fund (FWF): project num-bers P31663-N35 and Y963-N35. The second author was partially supported by CNCSUEFISCDI, project number PN-III-P4-ID-PCE-2016-0778.2.
Weighted non-linear Stiefel manifolds
The aim of this section is to construct the EPContact dual pair, see Theorem 2.6.2.1.
Canonical symplectization of contact manifolds.
In this section we set up ournotation and recall some well known facts about the symplectization of contact mani-folds. We emphasize the structure that will be generalized in the subsequent sections.For more details we refer to [1, Appendix 4.E] and [24, Section 12.3].Consider a contact manifold (
P, ξ ) where ξ ⊆ T P denotes the contact subbundle. Wewrite L := T P/ξ for the corresponding line bundle. The vector bundle projection of thedual line bundle will be denoted by π L ∗ : L ∗ → P . The canonical projection T P → L permits to regard the dual bundle as a subbundle of the cotangent bundle, L ∗ ⊆ T ∗ P .We denote by θ L ∗ ∈ Ω ( L ∗ ) the pull back of the canonical 1-form on T ∗ P . Hence, thedefining equation for θ L ∗ is θ L ∗ β ( V ) = β ( T β π L ∗ · V ) , (7)where β ∈ L ∗ x , x ∈ P , and V ∈ T β L ∗ . The pairing in (7) can be viewed either as apairing between L ∗ x and L x by considering the class of T β π L ∗ · V in L x = T x P/ξ x , or asa pairing between T ∗ P and T P by considering β an element of L ∗ x ⊆ T ∗ x P . It is wellknown that the closed 2-form ω L ∗ := dθ L ∗ ∈ Ω ( L ∗ )restricts to a symplectic form on M := L ∗ \ P , which will be denoted by ω M = dθ M .The symplectic manifold ( M, ω M ) is called the symplectization of the contact manifold( P, ξ ). Note that both forms are homogeneous of degree one with respect to the fiberwisescalar multiplication δ t : L ∗ → L ∗ , that is δ ∗ t θ L ∗ = tθ L ∗ and δ ∗ t ω L ∗ = tω L ∗ for all t ∈ R . The action by the contact group.
Let us write Diff(
P, ξ ) for the group of contact dif-feomorphisms. Since contact diffeomorphisms preserve the contact subbundle ξ , theDiff( P, ξ )-action on P lifts to an action on the total space of L ∗ . For g ∈ Diff(
P, ξ ),we let Ψ L ∗ g ∈ Diff( L ∗ ) denote the corresponding (fiberwise linear) diffeomorphism on L ∗ . Clearly, π L ∗ ◦ Ψ L ∗ g = g ◦ π L ∗ , δ t ◦ Ψ L ∗ g = Ψ L ∗ g ◦ δ t , and Ψ L ∗ g g = Ψ L ∗ g Ψ L ∗ g for all g, g , g ∈ Diff(
P, ξ ) and t ∈ R . Moreover, the contact group action preserves θ L ∗ and ω L ∗ , that is (Ψ L ∗ g ) ∗ θ L ∗ = θ L ∗ and (Ψ L ∗ g ) ∗ ω L ∗ = ω L ∗ for all g ∈ Diff(
P, ξ ). Noticing thatthe symplectic piece M ⊆ L ∗ is invariant under the contact group action, we write Ψ Mg for the restricted action.Let X ( P, ξ ) denote the Lie algebra of contact vector fields. Via the projection
T P → L ,every (contact) vector field gives rise to a section of L which may in turn be regarded as afiberwise linear function on the total space of L ∗ . This provides canonical identifications, X ( P, ξ ) = Γ ∞ ( L ) = C ∞ lin ( L ∗ ) , X ↔ X mod ξ ↔ h X , (8) If ξ = ker α and L ∗ ∼ = P × R denotes the trivialization provided by α , then θ L ∗ = t ( π L ∗ ) ∗ α , where t denotes the projection onto the factor R . DUAL PAIR FOR THE CONTACT GROUP 7 where h X ∈ C ∞ lin ( L ∗ ) is the fiberwise linear function given by h X ( β ) = β ( X x ) for β ∈ L ∗ x and x ∈ P . Clearly, this identification is equivariant, i.e.,(Ψ L ∗ g ) ∗ h X = h g ∗ X (9)for all g ∈ Diff(
P, ξ ) and X ∈ X ( P, ξ ).For X ∈ X ( P, ξ ), we denote the corresponding fundamental vector field (infinitesimalaction) on the total space of L ∗ by ζ L ∗ X ∈ X ( L ∗ ). Clearly, T π L ∗ ◦ ζ L ∗ X = X ◦ π L ∗ , (10) T δ t ◦ ζ L ∗ X = ζ L ∗ X ◦ δ t , (Ψ L ∗ g ) ∗ ζ L ∗ X = ζ L ∗ g ∗ X , and [ ζ L ∗ X , ζ L ∗ X ] = ζ L ∗ [ X ,X ] for all X, X , X ∈ X ( P, ξ ), g ∈ Diff(
P, ξ ) and t ∈ R . From the definition of θ L ∗ in (7) one immediately gets i ζ L ∗ X θ L ∗ = h X (11)for X ∈ X ( P, ξ ). Invariance of θ L ∗ and ω L ∗ yields infinitesimal invariance L ζ L ∗ X θ L ∗ = 0and L ζ L ∗ X ω L ∗ = 0, respectively, for all X ∈ X ( P, ξ ). Using Cartan’s formula and (11), weobtain i ζ L ∗ X ω L ∗ = − dh X (12)as well as the following formula for the bracket of contact vector fields, h [ X,Y ] = ζ L ∗ X · h Y = − ζ L ∗ Y · h X = ω L ∗ ( ζ L ∗ X , ζ L ∗ Y ) , (13)for all X, Y ∈ X ( P, ξ ).Over the symplectic piece M = L ∗ \ P the Hamiltonian vector field corresponding to h MX := h X | M coincides with ζ MX := ζ L ∗ X | M , see (12). Moreover, (13) implies h M [ X,Y ] = { h MX , h MY } , (14)where the right hand side denotes the Poisson bracket on C ∞ ( M ). The formulas (12)and (9) above imply that the action of Diff( P, ξ ) on M is Hamiltonian with equivariantmoment map J M : M → X ( P, ξ ) ∗ , h J M ( β ) , X i := h MX ( β ) = β ( X ) , (15)where β ∈ M and X ∈ X ( P, ξ ). Remark . A slightly more explicit, yet less natural description is possible if the contactstructure is described by a contact form α ∈ Ω ( P ), that is, if ξ = ker α . Such acontact form provides a trivialization P × R ∼ = L ∗ ⊆ T ∗ P , ( x, t ) ↔ tα x . Via thisidentification we have θ L ∗ = t ( π L ∗ ) ∗ α , and the fiberwise linear function h X from (8)becomes h X ( x, t ) = t ( i X α )( x ) where x ∈ P and t ∈ R . A diffeomorphism g of P is acontact diffeomorphism iff it preserves the contact form up to a conformal factor, i.e.,iff there exists a (nowhere vanishing) function a g on P such that g ∗ α = a g α . Similarly,a vector field X on P is a contact vector field iff it satisfies L X α = λ X α , for a conformalfactor λ X ∈ C ∞ ( P ). Both, the group action of Diff( P, ξ ) and the Lie algebra action of X ( P, ξ ) on L ∗ , written in the trivialization L ∗ ∼ = P × R , involve the conformal factors.More explicitly, we have Ψ L ∗ g ( x, t ) = ( g ( x ) , ta g ( x )) and ζ L ∗ X ( x, t ) = ( X ( x ) , tλ X ( x ) ∂ t ). STEFAN HALLER AND CORNELIA VIZMAN
Coadjoint orbits.
It is well known that each connected component of a symplectic man-ifold is equivariantly symplectomorphic to a coadjoint orbit of its Hamiltonian group,see for instance [13]. We will now formulate a similar statement for the group Diff c ( P, ξ )of compactly supported contact diffeomorphisms which can be considered as a specialcase of Theorem 4.12 below.For β ∈ M , the isotropy subgroup Diff c ( P, ξ ; β ) is a closed Lie subgroup of Diff c ( P, ξ ).Moreover, the map provided by the action, Diff c ( P, ξ ) → M , g Ψ Mg ( β ), admits a localsmooth right inverse defined in a neighborhood of β . In particular, the group Diff c ( P, ξ )acts locally and infinitesimally transitive on M , and the Diff c ( P, ξ )-orbit through β isopen and closed in M . Denoting this orbit by M β , the map Diff c ( P, ξ ) → M β is a smoothprincipal bundle with structure group Diff c ( P, ξ ; β ). Hence, M β = Diff c ( P, ξ ) / Diff c ( P, ξ ; β )may be regarded as a homogeneous space. The moment map (15) induces an equivariantdiffeomorphism between M β and the coadjoint orbit of Diff c ( P, ξ ) through J M ( β ) ∈ X ( P, ξ ) ∗ . By infinitesimal equivariance of J M and (13), this diffeomorphism intertwinesthe Kostant–Kirillov–Souriau symplectic form ω KKS with ω M . Indeed, for β ∈ M and X, Y ∈ X ( P, ξ ), we get(( J M ) ∗ ω KKS )( ζ MX ( β ) , ζ MY ( β ))= ω KKS (cid:16) ζ X ( P,ξ ) ∗ X ( J M ( β )) , ζ X ( P,ξ ) ∗ Y ( J M ( β )) (cid:17) = h J M ( β ) , [ X, Y ] i (15) = h M [ X,Y ] ( β ) (13) = ω M ( ζ MX ( β ) , ζ MY ( β )) , whence ( J M ) ∗ ω KKS = ω M .In particular, each connected component of M is equivariantly symplectomorphic to acoadjoint orbit of the identity component in Diff c ( P, ξ ). If P connected and the contactstructure is not coorientable, then M is connected, hence a coadjoint orbit of Diff c ( P, ξ ).2.2.
Moment maps on a manifold of weighted maps.
In this section we introducean infinite dimensional generalization L of L ∗ that also carries a canonical 1-form θ L which is invariant under a natural Diff( P, ξ )-action.To this end, we fix a closed manifold S . We let | Λ | S denote the line bundle of densities[21, Chapter 16] on S , and we write π | Λ | S : | Λ | S → S for the corresponding vector bundleprojection. Recall that sections of | Λ | S can be integrated over S in a natural way. Everyorientation of S provides an isomorphism of line bundles | Λ | S ∼ = Λ dim( S ) T ∗ S . A nowherevanishing density, i.e., a section in Γ ∞ ( | Λ | S \ S ), will be referred to as a volume density. We denote the space of line bundle homomorphisms from | Λ | ∗ S → S to L ∗ → P by L := C ∞ lin ( | Λ | ∗ S , L ∗ ) := n Φ ∈ C ∞ ( | Λ | ∗ S , L ∗ ) (cid:12)(cid:12)(cid:12) ∀ t ∈ R : Φ ◦ δ | Λ | ∗ S t = δ L ∗ t ◦ Φ o . There is a canonical map π L : L → C ∞ ( S, P ), characterized by π L ∗ ◦ Φ = π L (Φ) ◦ π | Λ | ∗ S , (16)for all Φ ∈ L . For the fiber over ϕ ∈ C ∞ ( S, P ) we have a canonical identification, L ϕ := ( π L ) − ( ϕ ) = Γ ∞ ( | Λ | S ⊗ ϕ ∗ L ∗ ) . (17) DUAL PAIR FOR THE CONTACT GROUP 9
The contact group Diff(
P, ξ ) acts from the left on L , and the reparametrization groupDiff( S ) acts on L from the right in an obvious way. More explicitly, these actions aregiven by Ψ L g (Φ) := Ψ L ∗ g ◦ Φ and ψ L f (Φ) := Φ ◦ ψ | Λ | ∗ S f , (18)where Φ ∈ L , g ∈ Diff(
P, ξ ), f ∈ Diff( S ), and ψ | Λ | ∗ S f ∈ Diff( | Λ | ∗ S ) denotes the induced(fiberwise linear) action of Diff( S ) on the total space of | Λ | ∗ S . The two actions on L commute, and the map π L intertwines them with the corresponding actions on C ∞ ( S, P )given by Ψ C ∞ ( S,P ) g ( ϕ ) = g ◦ ϕ and ψ C ∞ ( S,P ) f ( ϕ ) = ϕ ◦ f, where g ∈ Diff(
P, ξ ), f ∈ Diff( S ), and ϕ ∈ C ∞ ( S, P ). More explicitly, we have Ψ L g g =Ψ L g ◦ Ψ L g , π L ◦ Ψ L g = Ψ C ∞ ( S,P ) g ◦ π L , ψ L f f = ψ L f ◦ ψ L f , π L ◦ ψ L f = ψ C ∞ ( S,P ) f ◦ π L , andΨ L g ◦ ψ L f = ψ L f ◦ Ψ L g , for g, g , g ∈ Diff(
P, ξ ) and f, f , f ∈ Diff( S ). Remark . Let µ ∈ Γ ∞ ( | Λ | S \ S ) be a volume density on S , i.e., a nowhere vanishingsmooth section of | Λ | S . Such a volume density provides an identification L ∼ = C ∞ ( S, L ∗ ) , Φ ↔ φ = Φ ◦ ˆ µ, where ˆ µ ∈ Γ ∞ ( | Λ | ∗ S ) denotes the section dual to µ , that is ˆ µ ( µ ) = 1. In this picture theactions on L take the formΨ L g ( φ ) = Ψ L ∗ g ◦ φ and ψ L f ( φ ) = f ∗ µµ · ( φ ◦ f ) , where φ ∈ C ∞ ( S, L ∗ ), g ∈ Diff(
P, ξ ) and f ∈ Diff( S ).The space L can be equipped with the structure of a smooth Fr´echet manifold suchthat the identification L ∼ = C ∞ ( S, L ∗ ) in Remark 2.2 becomes a diffeomorphism, for eachchoice of volume density µ . The map π L : L → C ∞ ( S, P ) is a smooth vector bundle.The tangent space at Φ ∈ L can be canonically identified as T Φ L = (cid:26) η ∈ C ∞ ( | Λ | ∗ S , T L ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) π T L ∗ ◦ η = Φ and ∀ t ∈ R : η ◦ δ | Λ | ∗ S t = T δ L ∗ t ◦ η (cid:27) . (19)The actions of Diff( P, ξ ) and Diff( S ) on L are smooth. For X ∈ X ( P, ξ ) and Z ∈ X ( S ),the corresponding fundamental vector fields are ζ L X (Φ) = ζ L ∗ X ◦ Φ and ζ L Z (Φ) = T Φ ◦ ζ | Λ | ∗ S Z (20)where Φ ∈ L and ζ | Λ | ∗ S Z ∈ X ( | Λ | ∗ S ) denotes the fundamental vector field of the Diff( S )-action on the total space of | Λ | ∗ S . Note that T π | Λ | ∗ S ◦ ζ | Λ | ∗ S Z = Z ◦ π | Λ | ∗ S and (cid:0) δ | Λ | ∗ S t (cid:1) ∗ ζ | Λ | ∗ S Z = ζ | Λ | ∗ S Z . (21)Clearly, (Ψ L g ) ∗ ζ L X = ζ L g ∗ X , ζ L [ X ,X ] = [ ζ L X , ζ L X ], T π L ◦ ζ L X = ζ C ∞ ( S,P ) X ◦ π L , ( ψ L f ) ∗ ζ L Z = ζ L f ∗ Z , ζ L [ Z ,Z ] = − [ ζ L Z , ζ L Z ], T π L ◦ ζ L Z = ζ C ∞ ( S,P ) Z ◦ π L , and [ ζ L X , ζ L Z ] = 0, where g ∈ Diff(
P, ξ ), X, X , X ∈ X ( P, ξ ), f ∈ Diff( S ), Z, Z , Z ∈ X ( S ). The canonical -form. Consider the 1-form θ L on L defined by θ L ( η ) := Z S θ L ∗ ( η ) , (22)where η ∈ T Φ L and Φ ∈ L . Note here that, because of (19), inserting η into θ L ∗ leadsto a fiberwise linear map θ L ∗ ( η ) : | Λ | ∗ S → R which, when regarded as a section of | Λ | S ,may be integrated over S . By invariance of θ L ∗ , the 1-form θ L is invariant under bothactions, i.e., we have (Ψ L g ) ∗ θ L = θ L and ( ψ L f ) ∗ θ L = θ L for all g ∈ Diff(
P, ξ ) and f ∈ Diff( S ). The corresponding infinitesimal invariance reads L ζ L X θ L = 0 and L ζ L Z θ L = 0 , where X ∈ X ( P, ξ ) and Z ∈ X ( S ).Moreover, we introduce the 2-form ω L := dθ L on L . By invariance of θ L , this 2-form is invariant under both actions too. More explicitly, we have (Ψ L g ) ∗ ω L = ω L and( ψ L f ) ∗ ω L = ω L for g ∈ Diff(
P, ξ ) and f ∈ Diff( S ), as well as infinitesimal invariance L ζ L X ω L = 0 and L ζ L Z ω L = 0 for X ∈ X ( P, ξ ) and Z ∈ X ( S ). Clearly, see [32, 9], ω L ( η , η ) = Z S ω L ∗ ( η , η ) (23)where η , η ∈ T Φ L and Φ ∈ L . As before, the fiberwise linear function ω L ∗ ( η , η ) on | Λ | ∗ S may be regarded as a section of | Λ | S which can be integrated over S .The exact 2-form ω L = dθ L is not (weakly) non-degenerate, because ω L ∗ is not sym-plectic on all of L ∗ . In the subsequent section, we will restrict to an invariant opensubset of L on which ω L is (weakly) symplectic. On this symplectic part, both actionsare Hamiltonian with equivariant moment map. This is a well known formal consequenceof the fact that these actions preserve the 1-form θ L , see for instance [24, Section 12.3].The corresponding Hamiltonian functions and moment maps are given by contractionof the fundamental vector fields with the canonical 1-form. However, these geometricobjects make sense on all of L . Hence, we will now formulate their fundamental relationson L . The left moment map.
For X ∈ X ( P, ξ ), consider the function h L X : L → R defined by i ζ L X θ L =: h L X . (24)Using the infinitesimal invariance, L ζ L X θ L = 0, we obtain i ζ L X ω L = − dh L X (25)analogous to (12), as well as h L [ X,Y ] = ζ L X · h L Y = − ζ L Y · h L X = ω L ( ζ L X , ζ L Y ) , (26)for all X, Y ∈ X ( P, ξ ), cf. (13). From the invariance of θ L we obtain, cf. (9)(Ψ L g ) ∗ h L X = h L g ∗ X and ( ψ L f ) ∗ h L X = h L X (27)for all f ∈ Diff( S ), g ∈ Diff(
P, ξ ), and X ∈ X ( P, ξ ). We introduce a smooth map J L L : L → X ( P, ξ ) ∗ (28) DUAL PAIR FOR THE CONTACT GROUP 11 by putting h J L L , X i := h L X , that is, h J L L (Φ) , X i := h L X (Φ) = θ L ( ζ L X (Φ)) , (29)where Φ ∈ L and X ∈ X ( P, ξ ). The equations in (27) may be written in the form h J L L ◦ Ψ L g , X i = h Ψ L g , g ∗ X i and J L L ◦ ψ L f = J L L , (30)where g ∈ Diff(
P, ξ ), X ∈ X ( P, ξ ), and f ∈ Diff( S ). Combining (24), (20), (22), (7),(10), and (16), we obtain h L X (Φ) = Z S Φ( X ◦ ϕ ) , (31)where ϕ ∈ C ∞ ( S, P ), Φ ∈ L ϕ = Γ ∞ ( | Λ | S ⊗ ϕ ∗ L ∗ ), and X ∈ X ( P, ξ ) = Γ ∞ ( L ), cf. (17)and (8). Here we use the canonical contraction between L ∗ ⊆ T ∗ P and T P to obtainthe density Φ( X ◦ ϕ ) ∈ Γ ∞ ( | Λ | S ). More explicitly, the verification of (31) reads: h L X (Φ) (24) = θ L ( ζ L X (Φ)) (20) = θ L ( ζ L ∗ X ◦ Φ) (22) = Z S θ L ∗ ( ζ L ∗ X ◦ Φ) (7) = Z S Φ( T π L ∗ ◦ ζ L ∗ X ◦ Φ) (10) = Z S Φ( X ◦ π L ∗ ◦ Φ) (16) = Z S Φ( X ◦ ϕ ◦ π | Λ | ∗ S ) = Z S Φ( X ◦ ϕ ) . The right moment map.
For Z ∈ X ( S ), consider the function k L Z : L → R defined by i ζ L Z θ L =: k L Z . (32)Using the infinitesimal invariance, L ζ L Z θ L = 0, we obtain i ζ L Z ω L = − dk L Z (33)as well as k L [ Z ,Z ] = ζ L Z · k L Z = − ζ L Z · k L Z = ω L ( ζ L Z , ζ L Z ) , (34)for all Z, Z , Z ∈ X ( S ). From the invariance of θ L we obtain(Ψ L g ) ∗ k L Z = k L Z and ( ψ L f ) ∗ k L Z = k L f ∗ Z (35)for all g ∈ Diff(
P, ξ ), f ∈ Diff( S ), and Z ∈ X ( S ). We introduce a smooth map J L R : L → Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ (36)by putting h J L R , Z i := k L Z , that is, h J L R (Φ) , Z i := k L Z (Φ) = θ L ( ζ L Z (Φ)) , (37)where Φ ∈ L and Z ∈ X ( S ). The equations in (35) may be written in the form h J L R ◦ ψ L f , Z i = h ψ L f , f ∗ Z i and J L R ◦ Ψ L g = J L R , (38)where f ∈ Diff( S ), Z ∈ X ( S ), and g ∈ Diff(
P, ξ ). In view of (32), (20), (22), (7), (16),and (21), we have k L Z (Φ) = Z S Φ( T ϕ ◦ Z ) , (39) where ϕ ∈ C ∞ ( S, P ), Φ ∈ L ϕ = Γ ∞ ( | Λ | S ⊗ ϕ ∗ L ∗ ), and Z ∈ X ( S ), cf. (17). As before, weuse the canonical contraction between L ∗ ⊆ T ∗ P and T P to obtain a density Φ(
T ϕ ◦ Z ) ∈ Γ ∞ ( | Λ | S ). More explicitly, the verification of (39) reads: k L Z (Φ) (32) = θ L ( ζ L Z (Φ)) (20) = θ L ( T Φ ◦ ζ | Λ | ∗ S Z ) (22) = Z S θ L ∗ ( T Φ ◦ ζ | Λ | ∗ S Z ) (7) = Z S Φ( T π L ∗ ◦ T Φ ◦ ζ | Λ | ∗ S Z ) (16) = Z S Φ( T ϕ ◦ T π | Λ | ∗ S ◦ ζ | Λ | ∗ S Z ) (21) = Z S Φ( T ϕ ◦ Z ◦ π | Λ | ∗ S ) = Z S Φ( T ϕ ◦ Z ) . It follows from (37) and (39) that J L R (Φ) is indeed a smooth 1-form density as indicatedin (36), i.e., J L R (Φ) ∈ Ω ( S, | Λ | S ). More precisely, we have λ (cid:0) J L R (Φ)( Z ) (cid:1) = (Φ ◦ λ )( T ϕ ◦ Z ) (40)for Z ∈ X ( S ) and λ ∈ Γ ∞ ( | Λ | ∗ S ). Note that J L R (Φ) can also be characterized as thesmooth 1-form density on S corresponding to the 1-homogeneous vertical 1-form Φ ∗ θ L ∗ on the total space of | Λ | ∗ S . More explicitly, we have J L R (Φ) = Φ ∗ θ L ∗ (41)via the canonical identificationΩ ( S, | Λ | S ) = n β ∈ Ω ( | Λ | ∗ S ) (cid:12)(cid:12)(cid:12) β is vertical and (cid:0) δ | Λ | ∗ S t (cid:1) ∗ β = tβ for all t ∈ R o . (42)Here ρ ∈ Ω ( S, | Λ | S ) corresponds to β ∈ Ω ( | Λ | ∗ S ) given by β ( W ) = w ( ρ ( T w π | Λ | ∗ S · W ))where w ∈ | Λ | ∗ S and W ∈ T w | Λ | ∗ S . Remark . Using a volume density µ on S to identify L ∼ = C ∞ ( S, L ∗ ) as in Remark 2.2,the differential forms θ L and ω L become, see (22) and (23), θ L ( η ) = Z S θ L ∗ ( η ) µ and ω L ( η , η ) = Z S ω L ∗ ( η , η ) µ, (43)where φ ∈ C ∞ ( S, L ∗ ) and η, η , η ∈ T φ C ∞ ( S, L ∗ ) = { η ∈ C ∞ ( S, T L ∗ ) : π T L ∗ ◦ η = φ } .For X ∈ X ( P, ξ ) and Z ∈ X ( S ), the fundamental vector fields ζ L X and ζ L Z identify to ζ L X ( φ ) = ζ L ∗ X ◦ φ and ζ L Z ( φ ) = T φ ◦ Z + div µ ( Z ) · ( R ◦ φ ) , (44)where div µ ( Z ) := L Z µµ denotes the µ -divergence, and R := ∂∂t | t =1 δ L ∗ t ∈ X ( L ∗ ) denotesthe Euler vector field of L ∗ . The functions h L X and k L Z become, see (31) and (39), h L X ( φ ) = Z S ( φ ∗ h X ) µ and k L Z ( φ ) = Z S ( φ ∗ θ L ∗ )( Z ) µ. (45)Hence, the maps J L L and J L R identify to J L L : C ∞ ( S, L ∗ ) → C ∞ ( L ∗ ) ∗ → X ( P, ξ ) ∗ , J L L ( φ ) = φ ∗ µ (46) J L R : C ∞ ( S, L ∗ ) → Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ , J L R ( φ ) = φ ∗ θ L ∗ ⊗ µ (47)where φ ∈ C ∞ ( S, L ∗ ) and we use the inclusion X ( P, ξ ) = C ∞ lin ( L ∗ ) ⊆ C ∞ ( L ∗ ), see (8). DUAL PAIR FOR THE CONTACT GROUP 13
The symplectic part.
Let
M ⊆ L = C ∞ lin ( | Λ | ∗ S , L ∗ ) denote the open subset of linebundle homomorphisms | Λ | ∗ S → L ∗ which restrict to a linear isomorphism on each fiber, M := C ∞ lin, inj ( | Λ | ∗ S , L ∗ ) . (48) We will denote the restriction to M of any action, function, form, or vector field on L considered above, by replacing the superscript L with M . Because L ∗ \ P is symplectic,the 2-form ω M = dθ M is (weakly) non-degenerate, whence symplectic, cf. (23).The map π M : M → C ∞ ( S, P ) is a principal fiber bundle with structure group C ∞ ( S, R × ), provided we restrict to the connected components of C ∞ ( S, P ) in the imageof π M . If ϕ is in one of these components, then the fiber M ϕ := ( π M ) − ( ϕ ) may becanonically identified with the space of nowhere vanishing sections of the line bundle | Λ | S ⊗ ϕ ∗ L ∗ , cf. (17). Thus, disregarding the density part, M ϕ may be considered asthe space of contact forms for ξ along the map ϕ : S → P .Clearly, M is invariant under the action of the groups Diff( P, ξ ) and Diff( S ). Sinceboth actions preserve the 1-form θ M , they are Hamiltonian with equivariant momentmaps obtained by contraction of the 1-form with the infinitesimal generators, see forinstance [24, Section 12.3]. We summarize these facts in the following proposition. Proposition 2.4. (a) The action of the group
Diff(
P, ξ ) on M is Hamiltonian with anequivariant moment map J M L : M → X ( P, ξ ) ∗ , given by h J M L (Φ) , X i = ( i ζ M X θ M )(Φ) = h M X (Φ) = Z S Φ( X ◦ ϕ ) , (49) where Φ ∈ M ϕ and X ∈ X ( P, ξ ) . Moreover, the moment map J M L is Diff( S ) -invariant.More explicitly, we have (Ψ M g ) ∗ ω M = ω M , i ζ M X ω M = − d h J M L , X i , h J M L ◦ Ψ M g , X i = h Ψ M g , g ∗ X i , and J M L ◦ ψ M f = J M L where g ∈ Diff(
P, ξ ) , X ∈ X ( P, ξ ) , and f ∈ Diff( S ) .(b) The action of the group Diff( S ) on M is Hamiltonian with an equivariant momentmap J M R : M → Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ , given by h J M R (Φ) , Z i = ( i ζ M Z θ M )(Φ) = k M Z (Φ) = Z S Φ( T ϕ ◦ Z ) , (50) where Φ ∈ M ϕ and Z ∈ X ( S ) . Moreover, the moment map J M R is Diff(
P, ξ ) -invariant.More explicitly, we have ( ψ M f ) ∗ ω M = ω M , i ζ M Z ω M = − d h J M R , Z i , h J M R ◦ ψ M f , Z i = h ψ M f , f ∗ Z i , and J M R ◦ Ψ M g = J M R , where f ∈ Diff( S ) , Z ∈ X ( S ) , and g ∈ Diff(
P, ξ ) .Proof. The statements in (a) follow immediately from (25), (29), (30), and (31). Thestatements in (b) follow immediately from (33), (37), (38), and (39). (cid:3)
Remark . If S is a single point, then we recover the symplectization discussed inSection 2.1. More precisely, in this case the canonical volume density on S provides acanonical isomorphism between the line bundles π L : L → C ∞ ( S, P ) and π L ∗ : L ∗ → P .Up to this identification, we have Ψ L g = Ψ L ∗ g , for all g ∈ Diff(
P, ξ ), θ L = θ L ∗ and ω L = ω L ∗ . Moreover, M = M and J M L = J M . Clearly, the Diff( S )-action is trivial inthis case and J M R = 0. Using a volume density on S to identify L ∼ = C ∞ ( S, L ∗ ) as in Remark 2.2, the space M correspondsto C ∞ ( S, L ∗ \ P ). When ξ = ker α for a contact form α , then the corresponding trivialization L ∗ ∼ = P × R yields a further identification M ∼ = C ∞ ( S, P ) × C ∞ ( S, R × ). A dual pair on the non-linear Stiefel manifold of weighted embeddings.
We will now restrict to an open subset of M on which the Diff( S )-action is free. Let E := Emb lin ( | Λ | ∗ S , L ∗ ) (51)denote the open subset of all (fiberwise linear) embeddings in L = C ∞ lin ( | Λ | ∗ S , L ∗ ). Ele-ments of E are automatically isomorphisms on fibers, so E ⊆ M . We consider E as anon-linear Stiefel manifold of weighted embeddings. We will denote the restriction to E of any action, function, form, or vector field on L considered above, by replacing the superscript L with E . The map π E : E →
Emb(
S, P )is a principal fiber bundle with structure group C ∞ ( S, R × ), provided we restrict to theconnected components of Emb( S, P ) in the image of π E . Since E is open in M , thesymplectic form ω M restricts to a symplectic form ω E on E . Hence, ( E , ω E ) is a (weakly)symplectic Fr´echet manifold.Note that E is invariant under the actions of Diff( P, ξ ) and Diff( S ). In view of Propo-sition 2.4, the restrictions of J M L and J M R to E provide equivariant moment maps X ( P, ξ ) ∗ J E L ←−−−−− E J E R −−−−−→ Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ (52)for the actions of Diff( P, ξ ) and Diff( S ) on E , respectively.A pair of equivariant moment maps for commuting Hamiltonian actions of (infinitedimensional) Lie groups G and H on an (infinite dimensional) symplectic manifold Q , g ∗ J L ←−−−−− Q J R −−−−−→ h ∗ , is called a symplectic dual pair [33] if the distributions ker T J L and ker T J R are symplec-tic orthogonal complements of one another: (ker T J L ) ⊥ = ker T J R and (ker T J R ) ⊥ =ker T J L . Both identities are needed here, due to the weakness of the symplectic form.Let g Q ( x ) := { ζ QX ( x ) | X ∈ g } denote the tangent space to the G -orbit at x ∈ Q . When g Q = h ⊥ Q and h Q = g ⊥ Q , (53)i.e., if the G -orbits and H -orbits are symplectic orthogonal complements of one another,then the actions are said to be mutually completely orthogonal [22]. Since ker T J R = h ⊥ Q ,the first identity in (53) can be rephrased as the transitivity of the g -action on level setsof the moment map J R , and similarly for the second identity.Mutually completely orthogonality of the actions implies that J L and J R form a dualpair. The reverse implication is not always true, due to the weakness of the symplecticform [11]. Theorem 2.6.
The moment mappings J E L and J E R in (52) form a symplectic dual pair,called the EPContact dual pair. Moreover, the commuting actions of Diff(
P, ξ ) and Diff( S ) on E are mutually completely orthogonal, i.e., for each Φ ∈ E we have (cid:8) ζ E X (Φ) (cid:12)(cid:12) X ∈ X ( P, ξ ) (cid:9) = (cid:8) ζ E Z (Φ) (cid:12)(cid:12) Z ∈ X ( S ) (cid:9) ⊥ := (cid:8) A ∈ T Φ E (cid:12)(cid:12) ∀ Z ∈ X ( S ) : ω E Φ ( A, ζ E Z (Φ)) = 0 (cid:9) (54) Using a volume density µ on S to identify L ∼ = C ∞ ( S, L ∗ ) as in Remark 2.2, the subset E correspondsto C ∞ ( S, L ∗ \ P ) ∩ ( π L ) − (Emb( S, P )). If, moreover, ξ = ker α , we get a further identification E ∼ =Emb(
S, P ) × C ∞ ( S, R × ). DUAL PAIR FOR THE CONTACT GROUP 15 as well as (cid:8) ζ E Z (Φ) (cid:12)(cid:12) Z ∈ X ( S ) (cid:9) = (cid:8) ζ E X (Φ) (cid:12)(cid:12) X ∈ X ( P, ξ ) (cid:9) ⊥ := (cid:8) B ∈ T Φ E (cid:12)(cid:12) ∀ X ∈ X ( P, ξ ) : ω E Φ ( ζ E X (Φ) , B ) = 0 (cid:9) . (55) Proof.
Suppose Φ ∈ E . The inclusion (cid:8) ζ E X (Φ) (cid:12)(cid:12) X ∈ X ( P, ξ ) (cid:9) ⊆ (cid:8) ζ E Z (Φ) (cid:12)(cid:12) Z ∈ X ( S ) (cid:9) ⊥ follows immediately from (27) and (25). To show the converse inclusion, suppose A ∈ (cid:8) ζ E Z (Φ) (cid:12)(cid:12) Z ∈ X ( S ) (cid:9) ⊥ . The 1-form β := Φ ∗ i A ω L ∗ ∈ Ω ( | Λ | ∗ S ), given by β ( V ) = ω L ∗ Φ( y ) ( A ( y ) , T y Φ( V )) for all V ∈ T y | Λ | ∗ S , satisfies (cid:0) δ | Λ | ∗ S t (cid:1) ∗ β = tβ , by homogeneity of Φ, A , and ω L ∗ . Thus, for all Z ∈ X ( S ),0 = ω E ( A, ζ E Z (Φ)) (23) = Z S ω L ∗ (cid:0) A, T Φ ◦ ζ | Λ | ∗ S Z (cid:1) = Z S β (cid:0) ζ | Λ | ∗ S Z (cid:1) , where the integrands are fiberwise linear functions on the total space of | Λ | ∗ S , which maybe regarded as sections of | Λ | S and integrated over S . By Lemma 2.7 below, there existsa fiberwise linear function u ∈ C ∞ lin ( | Λ | ∗ S ) such that β = du .Because Φ is a fiberwise linear embedding, one can construct h ∈ C ∞ lin ( L ∗ ), i.e. h ◦ δ L ∗ t = th for all t ∈ R , such that h ◦ Φ = u and dh ◦ Φ = i A ω L ∗ . Indeed, let ˜ u ∈ C ∞ lin ( L ∗ )be any fiberwise linear function with ˜ u ◦ Φ = u and write h = ˜ u + h ′ . Hence, itsuffices to construct h ′ ∈ C ∞ lin ( L ∗ ) which vanishes along Φ and has prescribed derivative dh ′ ◦ Φ = i A ω L ∗ − ( d ˜ u ) ◦ Φ along Φ. This is possible, since Φ ∗ ( i A ω L ∗ ) − Φ ∗ ( d ˜ u ) = β − du = 0.According to the identification (8), there exists a contact vector field X ∈ X ( P, ξ )such that h = − h X , hence i A ω L ∗ = − dh X ◦ Φ (12) = i ζ L ∗ X ◦ Φ ω L ∗ . Since ω L ∗ is non-degenerate over L ∗ \ P , we conclude A = ζ L ∗ X ◦ Φ, and using (20) weget A = ζ E X (Φ), whence (54).It remains to check the other equality (55). The inclusion (cid:8) ζ E Z (Φ) (cid:12)(cid:12) Z ∈ X ( S ) (cid:9) ⊆ (cid:8) ζ E X (Φ) (cid:12)(cid:12) X ∈ X ( P, ξ ) (cid:9) ⊥ follows immediately from (35) and (33), or (54). To show the converse inclusion, supposethat B ∈ (cid:8) ζ E X (Φ) (cid:12)(cid:12) X ∈ X ( P, ξ ) (cid:9) ⊥ . Hence, for all X ∈ X ( P, ξ ),0 = ω E ( ζ E X (Φ) , B ) (20) = ω E ( ζ L ∗ X ◦ Φ , B ) (23) = Z S ω L ∗ ( ζ L ∗ X ◦ Φ , B ) (12) = − Z S ( dh X ◦ Φ)( B ) , and thus R S ( dh ◦ Φ)( B ) = 0, for all h ∈ C ∞ lin ( L ∗ ), cf. (8). This implies that B istangential to ˜ N := Φ( | Λ | ∗ S ). To see this, consider γ : | Λ | ∗ S → ann( T ˜ N ) ⊆ T ∗ L ∗ satisfying π T ∗ L ∗ ◦ γ = Φ and ( T δ L ∗ t ) ∗ ◦ γ ◦ δ | Λ | ∗ S t = γ for all t . Since Φ is a fiberwise linear embedding,there exists h ∈ C ∞ lin ( L ∗ ) with h ◦ Φ = 0 and γ = dh ◦ Φ, hence R S γ ( B ) = 0 for all such γ . We conclude that B is tangential to ˜ N . Consequently, there exists a vector field W on the total space of | Λ | ∗ S such that B = T Φ ◦ W . Clearly, δ ∗ t W = W , for all t ∈ R .Using Lemma 2.8 below, we conclude that there exists Z ∈ X ( S ) such that W = ζ | Λ | ∗ S Z .In view of (20), we obtain B = ζ E Z (Φ). This completes the proof of (55). (cid:3) Lemma 2.7.
Suppose β ∈ Ω ( | Λ | ∗ S ) is a 1-form on the total space of | Λ | ∗ S , such that δ ∗ t β = tβ for all t ∈ R and Z S β (cid:16) ζ | Λ | ∗ S Z (cid:17) = 0 (56) for all Z ∈ X ( S ) . Then β = di R β where R = ∂∂t | t =1 δ t ∈ X ( | Λ | ∗ S ) denotes the radialvector field, i.e., the fundamental vector field of the action δ t .Proof. We fix a volume density µ on S and identify | Λ | ∗ S ∼ = S × R correspondingly.The two canonical projections shall be denoted by p : S × R → S and t : S × R → R ,respectively. The radial vector field becomes R = t∂ t ∈ X ( S × R ). By homogeneity, β ∈ Ω ( S × R ) can be written in the form β = tp ∗ B + ( p ∗ b ) dt where B ∈ Ω ( S ) and b ∈ C ∞ ( S, R ). Moreover, for Z ∈ X ( S ), we have ζ | Λ | ∗ S Z = p ∗ Z + ( p ∗ div( Z )) t∂ t , (57)where L Z µ =: div( Z ) µ and p ∗ Z ∈ X ( S × R ) denotes the vector field which projects to Z on S and 0 on R . Consequently, β (cid:16) ζ | Λ | ∗ S Z (cid:17) = tp ∗ (cid:0) i Z B + b div( Z ) (cid:1) . Using Stokes’ theorem, we obtain Z S β (cid:16) ζ | Λ | ∗ S Z (cid:17) = Z S (cid:0) i Z B + b div( Z ) (cid:1) µ = Z S ( B − db ) ∧ i Z µ. In view of the assumption (56), we conclude that B = db , whence di R β = d ( tp ∗ b ) = tp ∗ db + ( p ∗ b ) dt = tp ∗ B + ( p ∗ b ) dt = β , the desired relation. (cid:3) Lemma 2.8.
Suppose W is a vector field on the total space of | Λ | ∗ S , such that δ ∗ t W = W for all t ∈ R and such that Z S dh ( W ) = 0 , (58) for all smooth, fiberwise linear functions h on the total space of | Λ | ∗ S . Then W is afundamental vector field of the natural Diff( S ) action on | Λ | ∗ S , i.e., there exists Z ∈ X ( S ) such that W = ζ | Λ | ∗ S Z .Proof. As in the proof of the preceding lemma we fix a volume density µ on S , weidentify | Λ | ∗ S ∼ = S × R correspondingly, and we denote the two canonical projections by p : S × R → S and t : S × R → R . Hence, the vector field W can be written in the form W = p ∗ Z + ( p ∗ w ) t∂ t where Z ∈ X ( S ) and w ∈ C ∞ ( S ). Every function ¯ h ∈ C ∞ ( S ) givesrise to a fiberwise linear function h := tp ∗ ¯ h on the total space of | Λ | ∗ S . Then dh ( W ) = tp ∗ (¯ hw + d ¯ h ( Z ))and Stokes’ theorem yields Z S dh ( W ) = Z S (cid:0) ¯ hw + d ¯ h ( Z ) (cid:1) µ = Z S ¯ h (cid:0) w − div( Z ) (cid:1) µ. Note that the integrand is a fiberwise linear function on the total space of | Λ | ∗ S , which may beregarded as a section of | Λ | S and integrated over S . Note that the integrand is a fiberwise linear function on the total space of | Λ | ∗ S , which can beregarded as a section of | Λ | S and integrated over S . DUAL PAIR FOR THE CONTACT GROUP 17
Using the assumption (58), we conclude that w = div( Z ). Consequently, see (57), weobtain W = p ∗ Z + ( p ∗ w ) t∂ t = p ∗ Z + ( p ∗ div( Z )) t∂ t = ζ | Λ | ∗ S Z . (cid:3) Remark . Let us give a more explicit description of the EPContact dual pair if thecontact structure is described by a contact form, ξ = ker α , and a volume density µ on S has been fixed. We have already pointed out before, see footnote 3, that thesechoices provide an identification of the non-linear Stiefel manifold E with Emb( S, P ) × C ∞ ( S, R × ). Via this identification, the actions of Diff( P, ξ ) from the left and Diff( S )from the right areΨ E g ( ϕ, h ) = ( g ◦ ϕ, ( g ∗ αα ◦ ϕ ) h ) and ψ E f ( ϕ, h ) = ( ϕ ◦ f, ( h ◦ f ) f ∗ µµ ) , where g ∈ Diff(
P, ξ ), f ∈ Diff( S ), and ( ϕ, h ) ∈ Emb(
S, P ) × C ∞ ( S, R × ). Using theidentification X ( P, ξ ) = C ∞ ( P ) provided by the contact form α , the EPContact dualpair (52) becomes C ∞ ( P ) ∗ J E L ←−−−−− Emb(
S, P ) × C ∞ ( S, R × ) J E R −−−−−→ Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ (59)with moment maps J E L ( ϕ, h ) = ϕ ∗ ( hµ ) and J E R ( ϕ, h ) = ϕ ∗ α ⊗ hµ. (60)This follows readily from the formulas provided in Remarks 2.2 and 2.3.In view of Theorem 2.6 one might expect [2, 13] that the contact group acts locallytransitive on the level sets of J E R . This is indeed the case, see Theorem 3.5 in the subse-quent section. Moreover, one might expect that a coadjoint orbit O ⊆ X ( S ) ∗ gives riseto a reduced symplectic structure on the quotient ( J E R ) − ( O ) / Diff( S ) which is equivari-antly symplectomorphic to a coadjoint orbit of Diff c ( P, ξ ) via the symplectomorphisminduced by the moment map J E L . Below we will see that this can be made rigorous forcoadjoint orbits corresponding to isotropic embeddings, see Theorem 4.12.3. Level sets of the right moment map
In this section we will show that each level set of the right moment map J E R : E → Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ is a smooth splitting Fr´echet submanifold in E . Furthermore, we will see that the contactgroup acts locally transitive on each level set. More precisely, we will show that thisaction admits local smooth sections. Hence, (unions of) connected components of theselevel sets may be regarded as homogeneous spaces of the contact group. These resultsare summarized in Theorem 3.5 below.A similar transitivity statement has been established in [9, Proposition 5.5] usingmethods quite different from the approach presented here.Let π J L : J L → P denote the 1-jet bundle of sections of L . Recall that each section h ∈ Γ ∞ ( L ) gives rise to a section j h ∈ Γ ∞ ( J L ). We equip the total space of J L with the contact structure uniquely characterized by the following property: A section s ∈ Γ ∞ ( J L ) has isotropic image iff there exists h ∈ Γ ∞ ( L ) such that s = j h . In thiscase h = π J LL ◦ s , where π J LL : J L → L denotes the natural projection. If L ∼ = P × R is a trivialization of L , then J L ∼ = T ∗ P × R , and the contact structure can be describedby the contact form p ∗ θ − dt , where θ denotes the canonical 1-form on T ∗ P , while p : T ∗ P × R → T ∗ P and t : T ∗ P × R → R denote the canonical projections. Consider the line bundle p : hom( p ∗ L, p ∗ L ) → P × P where p , p : P × P → P denotethe two canonical projections. We let P := isom( p ∗ L, p ∗ L ) denote the open subset offiberwise invertible maps. We equip the total space of P with the contact structure ξ P a := (cid:8) A ∈ T a P (cid:12)(cid:12) a (cid:0) ( T a ( p ◦ p ) A ) mod ξ ( p ◦ p )( a ) (cid:1) = ( T a ( p ◦ p ) A ) mod ξ ( p ◦ p )( a ) (cid:9) (61)where a ∈ P . Note that a diffeomorphism g ∈ Diff( P ) is contact if and only if thereexists a smooth map a : P → P with isotropic image satisfying p ◦ p ◦ a = id and p ◦ p ◦ a = g . Moreover, in this case Ψ Lg,x = a ( x ) in hom( L x , L g ( x ) ), for all x ∈ P . HereΨ Lg,x denotes the restriction of Ψ Lg to the fiber L x .It is well known [23, Theorem 1] that there exists a contact diffeomorphism J L ⊇ V Ξ −→ U ⊆ P (62)from an open neighborhood V of the zero section P ⊆ J L onto an open neighborhood U of the diagonal P ⊆ P intertwining the contact structure obtained by restriction from J L with the contact structure obtained by restriction from P . Moreover, for all x ∈ P ,we have Ξ(0 x ) = id L x . (63)It is also well known, see [19, Theorem 43.19] for the coorientable case, that the mapΓ ∞ c ( L ) ⊇ W F −→ Diff c ( P, ξ ) , F ( h ) := p ◦ p ◦ Ξ ◦ j h ◦ (cid:0) p ◦ p ◦ Ξ ◦ j h (cid:1) − , (64)provides a chart for the Lie group Diff c ( P, ξ ) at the identity. Here W is a C ∞ -openneighborhood of zero such that, for each h ∈ W , the image of j h is contained in V and p ◦ p ◦ Ξ ◦ j h as well as p ◦ p ◦ Ξ ◦ j h are diffeomorphisms of P . Clearly, F (0) = id P ,see (63). Moreover, for h ∈ W and x ∈ P , we haveΨ LF ( h ) ,x = (cid:16) Ξ ◦ j h ◦ (cid:0) p ◦ p ◦ Ξ ◦ j h (cid:1) − (cid:17) ( x ) (65)in hom( L x , L F ( h )( x ) ). In particular, j x h = 0 ⇔ F ( h )( x ) = x and Ψ LF ( h ) ,x = id L x . (66) Lemma 3.1.
For Φ ∈ E , the isotropy subgroup Diff c ( P, ξ ; Φ) = { g ∈ Diff c ( P, ξ ) : Ψ E g (Φ) = Φ } is a splitting Lie subgroup of Diff c ( P, ξ ) .Proof. Put ϕ = π E (Φ) ∈ Emb(
S, P ) and N := ϕ ( S ). For the chart F in (64) we obtain F − (cid:0) Diff c ( P, ξ ; Φ) (cid:1) = (cid:8) h ∈ Γ ∞ c ( L ) (cid:12)(cid:12) ∀ x ∈ N : j x h = 0 (cid:9) ∩ W , see (66) and (18). Since N is a closed submanifold in P , the linear space on the righthand side admits a linear complement in Γ ∞ c ( L ). To construct such a complement, let π W : W → N denote the normal bundle of N , where W = T P | N /T N ; fix a tubularneighborhood W ⊆ P of N such that N corresponds to the zero section in W ; andchoose an isomorphism of line bundles L | W ∼ = ( π W ) ∗ L | N . This provides a linear mapΓ ∞ ( L | N ) ⊕ Γ ∞ ( L | N ⊗ W ∗ ) → Γ ∞ ( L | W ) , (67)by regarding sections of L | N as π W -fiberwise constant sections of L | W , and by regardingsections of L | N ⊗ W ∗ as π W -fiberwise linear sections of L | W . Let χ ∈ C ∞ c ( W, R ) be If ξ = ker α , and P ∼ = P × P × ( R × ) denotes the corresponding trivialization, then the contactstructure can be described by the contact form tp ∗ α − p ∗ α on P × P × ( R × ). DUAL PAIR FOR THE CONTACT GROUP 19 a compactly supported bump function such that χ ≡ χ and extension by zero provides a linear map Γ ∞ ( L | W ) → Γ ∞ c ( L ). Composing this with (67), we obtain a linear map we will denoted by χ : Γ ∞ ( L | N ) ⊕ Γ ∞ (cid:0) L | N ⊗ W ∗ (cid:1) → Γ ∞ c ( L ) . (68)The image of χ provides a linear complement of (cid:8) h ∈ Γ ∞ c ( L ) (cid:12)(cid:12) ∀ x ∈ N : j x h = 0 (cid:9) inΓ ∞ c ( L ). Hence, Diff c ( P, ξ ; Φ) is a splitting Lie subgroup of Diff c ( P, ξ ). (cid:3) Suppose Φ , Φ ∈ M , and write ϕ i = π M (Φ i ) ∈ C ∞ ( S, P ). For x ∈ S consider therestrictions to the fibers, Φ ,x : | Λ | ∗ S,x → L ∗ ϕ ( x ) and Φ ,x : | Λ | ∗ S,x → L ∗ ϕ ( x ) , and define asmooth map G (Φ , Φ ) : S → P by G (Φ , Φ )( x ) = (Φ ,x ◦ Φ − ,x ) ∗ ∈ hom( L ϕ ( x ) , L ϕ ( x ) ) (69)for x ∈ S . Clearly, p ◦ p ◦ G (Φ , Φ ) = ϕ and p ◦ p ◦ G (Φ , Φ ) = ϕ . (70) Lemma 3.2.
The map G (Φ , Φ ) : S → P has isotropic image iff J M R (Φ ) = J M R (Φ ) .Proof. Suppose x ∈ S , Z x ∈ T x S , 0 = λ x ∈ | Λ | ∗ S,x , and write a := G (Φ , Φ )( x ). Then: T x G (Φ , Φ ) · Z x ∈ ξ P a (61) ⇔ a (cid:0) T a ( p ◦ p ) T x G (Φ , Φ ) · Z x mod ξ ( p ◦ p )( a ) (cid:1) = T a ( p ◦ p ) T x G (Φ , Φ ) · Z x mod ξ ( p ◦ p )( a )(70) ⇔ a (cid:0) T x ϕ · Z x mod ξ ( p ◦ p )( a ) (cid:1) = T x ϕ · Z x mod ξ ( p ◦ p )( a )(69) ⇔ Φ ∗ ,x (cid:0) T x ϕ · Z x mod ξ ( p ◦ p )( a ) (cid:1) = Φ ∗ ,x (cid:0) T x ϕ · Z x mod ξ ( p ◦ p )( a ) (cid:1) ⇔ λ x (cid:0) Φ ∗ ,x (cid:0) T x ϕ · Z x mod ξ ( p ◦ p )( a ) (cid:1)(cid:1) = λ x (cid:0) Φ ∗ ,x (cid:0) T x ϕ · Z x mod ξ ( p ◦ p )( a ) (cid:1)(cid:1) ⇔ Φ ,x ( λ x )( T x ϕ · Z x ) = Φ ,x ( λ x )( T x ϕ · Z x ) (40) ⇔ λ x (cid:0) J M R (Φ )( Z x ) (cid:1) = λ x (cid:0) J M R (Φ )( Z x ) (cid:1) ⇔ J M R (Φ )( Z x ) = J M R (Φ )( Z x )The lemma follows at once. (cid:3) For ρ ∈ Ω ( S, | Λ | S ) we let E ρ := ( J E R ) − ( ρ ) = (cid:8) Φ ∈ E : J E R (Φ) = ρ (cid:9) denote the corresponding level set of the moment map J E R : E → Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ . Lemma 3.3.
The level set E ρ is a smooth splitting Fr´echet submanifold in E , for each ρ ∈ Ω ( S, | Λ | S ) .Proof. Fix Φ ∈ E ρ , put ϕ := π E (Φ ) ∈ Emb(
S, P ), and consider the submanifold N := ϕ ( S ) of P . Let π W : W → N denote its normal bundle, W := T P | N /T N .Choose a tubular neighborhood W ⊆ P of N in P such that the zero section in W corresponds to N . As in the proof of Lemma 3.1, we fix an isomorphism of line bundles, L | W ∼ = ( π W ) ∗ L | N (71) and a compactly supported bump function χ ∈ C ∞ c ( W, R ) such that χ ≡ X of the zero section in W . The corresponding map (68) extends uniquelyto a linear map ˜ χ such that the following diagram commutes:Γ ∞ c ( L ) j / / Γ ∞ c ( J L )Γ ∞ ( L | N ) ⊕ Γ ∞ (cid:0) L | N ⊗ W ∗ (cid:1) j ⊕ id / / χ O O Γ ∞ (cid:0) J ( L | N ) (cid:1) ⊕ Γ ∞ (cid:0) L | N ⊗ W ∗ (cid:1) ˜ χ O O (72)The line bundle isomorphism in (71) also provides an isomorphismΓ ∞ (cid:0) ( J L ) | N (cid:1) ∼ = Γ ∞ (cid:0) J ( L | N ) (cid:1) ⊕ Γ ∞ (cid:0) L | N ⊗ W ∗ (cid:1) . (73)Using this isomorphism to replace the lower right corner in the diagram (72), we obtainlinear maps γ and Γ ∞ (cid:0) ( J L ) | N (cid:1) → Γ ∞ c ( J L ), s ˜ s , such that the following diagramcommutes: Γ ∞ c ( L ) j / / Γ ∞ c ( J L ) ˜ s Γ ∞ ( L | N ) ⊕ Γ ∞ (cid:0) L | N ⊗ W ∗ (cid:1) γ / / χ O O Γ ∞ (cid:0) ( J L ) | N (cid:1) O O s ❴ O O (74)For every ν ∈ Γ ∞ ( W ) with ν ( N ) ⊆ X we obtain a linear isomorphism˜ ν : Γ ∞ (cid:0) ( J L ) | N (cid:1) → Γ ∞ (cid:0) ν ∗ ( J L ) (cid:1) , ˜ ν ( s ) := ˜ s ◦ ν. (75)Moreover, ˜ ν and its inverse ˜ ν − are given by first order differential operators dependingsmoothly on ν . Furthermore, if ν ( N ) ⊆ X and s ∈ Γ ∞ (( J L ) | N ), then˜ s ◦ ν has isotropic image in J L ⇔ s ∈ img( γ ) . (76)Also note that img( γ ) admits a closed complementary subspace in Γ ∞ (( J L ) | N ). Indeed,the space of smooth sections in the kernel of the canonical projection J ( L | N ) → L | N provides a closed complement for the image of j : Γ ∞ ( L | N ) → Γ ∞ ( J ( L | N )). Takingthe sum with Γ ∞ ( L | N ⊗ W ∗ ) and using (73), we obtain a complementary subspace ofimg( γ ) in Γ ∞ (( J L ) | N ).Let V denote the C ∞ -open neighborhood of zero in Γ ∞ (( J L ) | N ) consisting of all s ∈ Γ ∞ (( J L ) | N ) with the following five properties:(a) the image of ˜ s is contained in V , cf. (62),(b) p ◦ p ◦ Ξ ◦ ˜ s : P → P is a diffeomorphism,(c) p ◦ p ◦ Ξ ◦ ˜ s : P → P is a diffeomorphism,(d) the image of ( p ◦ p ◦ Ξ ◦ ˜ s ) − ◦ ϕ : S → P is contained in X ⊆ W , and(e) ψ s := π W ◦ ( p ◦ p ◦ Ξ ◦ ˜ s ) − ◦ ϕ : S → N is a diffeomorphism.For s ∈ V we define ν s := ( p ◦ p ◦ Ξ ◦ ˜ s ) − ◦ ϕ ◦ ψ − s ∈ Γ ∞ ( W ). Hence, ν s ◦ ψ s = ( p ◦ p ◦ Ξ ◦ ˜ s ) − ◦ ϕ . (77)We will next show that the following map is a diffeomorphismΓ ∞ (cid:0) ( J L ) | N (cid:1) ⊇ V → U ⊆ (cid:8) G ∈ C ∞ ( S, P ) : p ◦ p ◦ G = ϕ (cid:9) s G s := Ξ ◦ ˜ s ◦ (cid:0) p ◦ p ◦ Ξ ◦ ˜ s ) − ◦ ϕ (78) DUAL PAIR FOR THE CONTACT GROUP 21 from V onto the C ∞ -open subset U in (cid:8) G ∈ C ∞ ( S, P ) : p ◦ p ◦ G = ϕ (cid:9) consisting ofall G ∈ C ∞ ( S, P ) with the following five properties:(f) p ◦ p ◦ G = ϕ ,(g) the image of G is contained in U , cf. (62),(h) the image of π J L ◦ Ξ − ◦ G : S → P is contained in X ⊆ W ,(i) ψ G := π W ◦ π J L ◦ Ξ − ◦ G : S → N is a diffeomorphism, and(j) s G := ˜ ν − G (cid:0) Ξ − ◦ G ◦ ψ − G (cid:1) ∈ V , where ν G := π J L ◦ Ξ − ◦ G ◦ ψ − G ∈ Γ ∞ ( W ).To see that (78) is a diffeomorphism, let s ∈ V and observe that (77) and (78) yield G s = Ξ ◦ ˜ s ◦ ν s ◦ ψ s (79)as well as ψ G s = ψ s and ν G s = ν s . Hence, Ξ − ◦ G s ◦ ψ − G s = ˜ s ◦ ν G s and (75) gives s = ˜ ν − G s (cid:0) Ξ − ◦ G s ◦ ψ − G s (cid:1) . (80)We conclude that G s ∈ U and s G s = s , for all s ∈ V , see (j). This shows that the map U → V , G s G , is left inverse to the map (78). To show that it is right inverse too,consider G ∈ U and note that (75) and (j) yield ˜ s G ◦ ν G = Ξ − ◦ G ◦ ψ − G . Hence,Ξ ◦ ˜ s G ◦ ν G ◦ ψ G = G. Composing with p ◦ p and using (b), (f) we obtain, ν G ◦ ψ G = (cid:0) p ◦ p ◦ Ξ ◦ ˜ s G (cid:1) − ◦ ϕ . Combining the latter two equations, we getΞ ◦ ˜ s G ◦ (cid:0) p ◦ p ◦ Ξ ◦ ˜ s G (cid:1) − ◦ ϕ = G. In other words, G s G = G , for all G ∈ U , cf. (78). This shows that (78) is indeed adiffeomorphism. Using (76), (79), and the fact that Ξ is a contact diffeomorphism wefind G s has isotropic image in P ⇔ s ∈ img( γ ) . (81)The construction in (69), cf. also (70), provides a diffeomorphism M ∼ = (cid:8) G ∈ C ∞ ( S, P ) : p ◦ p ◦ G = ϕ (cid:9) , Φ G (Φ , Φ ) . Combining this with the diffeomorphism in (78), we see that the mapΓ ∞ (cid:0) ( J L ) | N (cid:1) ⊇ V → E , s Φ s , (82)characterized by G (Φ , Φ s ) = G s , is a diffeomorphism from V onto a C ∞ -open neighbor-hood of Φ in E . Combining Lemma 3.2 with (81) and J E R (Φ ) = ρ , we obtain J E R (Φ s ) = ρ ⇔ s ∈ img( γ ) . (83)This shows that (82) is a submanifold chart for E ρ in E , centered a Φ . (cid:3) Lemma 3.4.
The action of
Diff c ( P, ξ ) on the level set E ρ admits local smooth sections,for each ρ ∈ Ω ( S, | Λ | S ) .Proof. We continue to use the notation set up in the proof of Lemma 3.3. Using thecommutativity of the diagram (74) we obtain a linear map img( γ ) → Γ ∞ c ( L ), s h s ,such that j h s = ˜ s , for all s ∈ img( γ ). Using (65), (a), (b), and (c), see also (64), wefind h s ∈ W and Ψ LF ( h s ) ,ϕ ( x ) = (cid:16) Ξ ◦ ˜ s ◦ (cid:0) p ◦ p ◦ Ξ ◦ ˜ s (cid:1) − (cid:17) ( ϕ ( x )) in hom( L ϕ ( x ) , L F ( h s )( ϕ ( x )) ), for all x ∈ S and s ∈ img( γ ) ∩ V . Hence, see (78) and (82),Ψ LF ( h s ) ,ϕ ( x ) = G (Φ , Φ s )( x ) . Using (69) we obtain Φ ∗ s,x ◦ Ψ LF ( h s ) ,ϕ ( x ) = Φ ∗ ,x , and dualizing yields Ψ L ∗ F ( h s ) ,ϕ ( x ) ◦ Φ ,x = Φ s,x for all x ∈ S and s ∈ img( γ ) ∩ V . Hence, in view of (18), we getΨ E F ( h s ) (Φ ) = Φ s , for all s ∈ img( γ ) ∩ V . As (82) restricts to a chart, img( γ ) ∩ V → E ρ , for the manifold E ρ , the lemma follows. (cid:3) Combining Lemmas 3.1, 3.3, and 3.4 we obtain the following result:
Theorem 3.5.
Suppose ρ ∈ Ω ( S, | Λ | S ) . Then the level set E ρ is a smooth splittingFr´echet submanifold of E . For Φ ∈ E ρ , the isotropy subgroup Diff c ( P, ξ ; Φ) is a closedLie subgroup of
Diff c ( P, ξ ) . Moreover, the map provided by the action, Diff c ( P, ξ ) → E ρ , g Ψ E g (Φ) , admits a local smooth right inverse defined in a neighborhood of Φ in E ρ .In particular, the group Diff c ( P, ξ ) acts locally and infinitesimally transitive on E ρ , andthe Diff c ( P, ξ ) -orbit of Φ is open and closed in E ρ . Denoting this orbit by E ρ Φ , the map Diff c ( P, ξ ) → E ρ Φ is a smooth principal bundle with structure group Diff c ( P, ξ ; Φ) . Hence, E ρ Φ = Diff c ( P, ξ ) / Diff c ( P, ξ ; Φ) may be regarded as a homogeneous space. Weighted non-linear Grassmannians
We continue to consider a manifold P endowed with a contact structure ξ , and a closedmanifold S . Recall that the Diff( S ) action is free on the non-linear Stiefel manifold E ofweighted embeddings. We will now factor out this action and consider the correspondingspace G = E / Diff( S ) of unparametrized weighted submanifolds of P .4.1. Principal bundles over non-linear Grassmannians.
Let Gr S ( P ) denote thenon-linear Grassmannian of all smooth submanifolds of P which are diffeomorphic to S .It is well know that Gr S ( P ) can be equipped with the structure of a Fr´echet manifoldsuch that the canonical map Emb( S, P ) → Gr S ( P ) becomes a principal bundle withstructure group Diff( S ).Consider the space of weighted submanifolds G := (cid:26) ( N, γ ) (cid:12)(cid:12)(cid:12)(cid:12) N ∈ Gr S ( P ) and γ ∈ Γ ∞ ( | Λ | N ⊗ L | ∗ N ) a nowhere vanishing section (cid:27) . (84)The Diff( P, ξ )-actions on P and on L ∗ induce a left action on G . For g ∈ Diff(
P, ξ ) welet Ψ G g denote the corresponding action on G , that is, Ψ G g ( N, γ ) = ( g ( N ) , g ∗ γ ). Remark . If ξ = ker α , then the contact form α provides a trivialization L ∗ ∼ = P × R which permits to identify G with a weighted non-linear Grassmannian, G ∼ = Gr wt S ( P ) := { ( N, ν ) | N ∈ Gr S ( P ) and ν ∈ Γ ∞ ( | Λ | N \ N ) } , (85) DUAL PAIR FOR THE CONTACT GROUP 23 by identifying (
N, ν ) with (
N, ν ⊗ α | N ) ∈ G . The weighted Grassmannian can beequipped with a smooth structure such that the canonical forgetful map Gr wt S ( P ) → Gr S ( P ) is a smooth fiber bundle. Indeed, it can be canonically identified with the bun-dle associated to the principal fiber bundle Emb( S, P ) → Gr S ( P ) via the Diff( S )-actionon the space Γ ∞ ( | Λ | S \ S ) of volume densities on S . Note that the induced smoothstructure on G does not depend on the contact form α for ξ . Via the identification (85),the Diff( P, ξ )-action becomesΨ G g ( N, ν ) = (cid:18) g ( N ) , g ∗ αα (cid:12)(cid:12)(cid:12) g ( N ) g ∗ ν (cid:19) , (86)where g ∈ Diff(
P, ξ ) and (
N, ν ) ∈ Gr wt S ( P ). Indeed, g ∗ ( ν ⊗ α | N ) = g ∗ αα (cid:12)(cid:12) g ( N ) g ∗ ν ⊗ α | g ( N ) .The space G in (84) can be equipped with the structure of a smooth manifold suchthat the canonical forgetful map π G : G → Gr S ( P )becomes a smooth fiber bundle with typical fiber Γ ∞ ( | Λ | S \ S ). Indeed, if ( N, γ ) ∈ G ,then locally around N , the contact structure on P is coorientable and can be describedby a contact form. We can therefore use Remark 4.1 to equip G with a smooth structure.In view of (86) the Diff c ( P, ξ )-action on G is smooth.To an element Φ ∈ E = Emb lin ( | Λ | ∗ S , L ∗ ) over the embedding ϕ = π E (Φ) ∈ Emb(
S, P )we associate a pair (
N, γ ) ∈ G in the following way: N = ϕ ( S ) and γ is the compositionof Φ (corestricted to L ∗ | N ) with the isomorphism | Λ | ∗ ϕ : | Λ | ∗ N → | Λ | ∗ S induced by thediffeomorphism ϕ : S → N . It is easy to see that the map q : E → G , given by q (Φ) =( N, γ ), is a smooth principal bundle with structure group Diff( S ). We summarize thisin the following Diff( P, ξ )-equivariant commutative diagram: E q (cid:15) (cid:15) π E / / Emb(
S, P ) (cid:15) (cid:15) G π G / / Gr S ( P ) (87)By Diff( S ) invariance, see Proposition 2.4(a), the moment map J E L descends to asmooth map J G L : G → X ( P, ξ ) ∗ , J G L ◦ q = J E L . (88)In view of (49) we have the explicit formula h J G L ( N, γ ) , X i = Z N γ ( X | N ) , (89)where ( N, γ ) ∈ G and X ∈ X ( P, ξ ). On the right hand side X is regarded as a sectionof L , see (8), restricted to N and contracted with γ to produce a density on N whichcan be integrated. Proposition 4.2.
The following assertions hold true: Using a contact form α to identify G ∼ = Gr wt S ( P ) as in Remark 4.1, the map (89) is simply h J G L ( N, ν ) , X i = Z N α ( X ) | N ν. (a) The map J G L : G → X ( P, ξ ) ∗ is a Diff(
P, ξ ) -equivariant injective immersion.(b) We have Diff(
P, ξ ; (
N, γ )) = Diff(
P, ξ ; J G L ( N, γ )) , where the left hand side denotesthe isotropy group of ( N, γ ) ∈ G and the right hand side denotes the isotropy groupof J G L ( N, γ ) ∈ X ( P, ξ ) ∗ for the coadjoint action.(c) The group Diff( S ) acts freely and transitively on level sets of J E L : E → X ( P, ξ ) ∗ .Proof. In view of Proposition 2.4(a), the smooth map J G L is Diff( P, ξ )-equivariant. Itfollows from the dual pair symplectic orthogonality condition (55) that J G L is immersive.To check injectivity, suppose ( N , γ ) and ( N , γ ) are two elements in G such that J G L ( N , γ ) = J G L ( N , γ ). Since γ i is nowhere vanishing, we have supp( J G L ( N i , γ i )) = N i ,see (89), whence N = N . Assume, for the sake of contradiction, γ = γ . Thenthere exists ¯ X ∈ Γ ∞ ( L | N ) such that h γ , ¯ X i 6 = h γ , ¯ X i with respect to the canonicalpairing between Γ ∞ ( | Λ | N ⊗ L | ∗ N ) and Γ ∞ ( L | N ). Extending ¯ X to a global section X ∈ Γ ∞ ( L ), we obtain h J G L ( N , γ ) , X i 6 = h J G L ( N , γ ) , X i using (89). Since this contradictsour assumption J G L ( N , γ ) = J G L ( N , γ ), we must have γ = γ . This shows that J G L isinjective.The assertion about the isotropy groups in (b) follows readily from the injectivity andequivariance of J G L . The assertion in (c) also follows from the injectivity statement in(a), since the Diff( S )-action on the fibers of q : E → G is free and transitive. (cid:3)
Right leg symplectic reduction.
In this section we study the spaces obtainedby symplectic reduction for the right moment map J E R : E → Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ . For a1-form density ρ ∈ Ω ( S, | Λ | S ) we put G ρ := q ( E ρ ) , where E ρ = ( J E R ) − ( ρ ). By Diff( S )-equivariance of J E R , and since Diff( S ) acts transitivelyon the fibers of q : E → G , the definition of G ρ may be rephrased equivalently as q − ( G ρ ) = E ρ · Diff( S ) = ( J E R ) − ( ρ · Diff( S )) . (90)Here ρ · Diff( S ) ⊆ Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ denotes the coadjoint orbit through ρ . Note that q induces a bijection G ρ = ( J E R ) − (cid:0) ρ · Diff( S ) (cid:1) / Diff( S ) = E ρ / Diff(
S, ρ ) , (91)where Diff( S, ρ ) = { f ∈ Diff( S ) : f ∗ ρ = ρ } denotes the isotropy group of ρ . Thus, G ρ isthe underlying set of the symplectically reduced space at ρ .We have the following more explicit description of G ρ : Lemma 4.3.
For each ρ ∈ Ω ( S, | Λ | S ) we have G ρ = { ( N, γ ) ∈ G| ( N, ι ∗ N γ ) ∼ = ( S, ρ ) } . Here ι N : N → P denotes the inclusion and the pull back ι ∗ N γ ∈ Ω ( N, | Λ | N ) = Γ ∞ ( | Λ | N ⊗ T ∗ N ) is defined as the composition | Λ | ∗ N γ −→ L | ∗ N ⊆ T ∗ P | N T ∗ ι N −−−→ T ∗ N . Proof.
Consider Φ ∈ E over ϕ := π E (Φ) ∈ Emb(
S, P ) and put (
N, γ ) := q (Φ). Bydefinition of q , we have ϕ ( S ) = N and the “triangle” on the top of the following diagram Because γ is nowhere vanishing, the kernel of ι ∗ N γ : T N → | Λ | N coincides with ξ | N ∩ T N . DUAL PAIR FOR THE CONTACT GROUP 25 commutes: | Λ | ∗ SJ E R (Φ) (cid:15) (cid:15) Φ / / L | ∗ N (cid:15) (cid:15) | Λ | ∗ Nι ∗ N γ (cid:15) (cid:15) γ o o | Λ | ∗ ϕ t t T ∗ S T ∗ P | NT ∗ ϕ o o T ∗ ι N / / T ∗ N T ∗ ϕ j j The left rectangle in this diagram commutes in view of the formula for J E R in (50);the right rectangle commutes in view of the definition of ι ∗ N γ ; and the “triangle” atthe bottom commutes trivially. We conclude that ( N, ι ∗ N γ ) ∼ = ( S, J E R (Φ)) via ϕ . Hence, q (Φ) ∼ = ( S, ρ ) iff (
S, J E R (Φ)) ∼ = ( S, ρ ). The latter, in turn, holds iff there exists f ∈ Diff( S )with J E R (Φ) = f ∗ ρ , i.e., iff Φ ∈ ( J E R ) − ( ρ · Diff( S )). Using the description (90) of G ρ weobtain the lemma. (cid:3) Remark . We have seen in Remark 4.1 that the choice of a contact form α on P permits to identify G with a weighted Grassmannian. Under this identification, thereduced space becomes G ρ ∼ = (cid:8) ( N, ν ) ∈ Gr wt S ( P ) : ( N, ι ∗ N α ⊗ ν ) ∼ = ( S, ρ ) (cid:9) . (92) Remark . A general fiber of the forgetful map π G : G → Gr S ( P ) will intersect severalof the spaces G ρ , for many different ρ . A notable exception are fibers over isotropicsubmanifolds, cf. (97) in Section 4.3 below.Since we do not expect G ρ to be a submanifold in G for general ρ , we will consider G ρ as a Fr¨olicher space with the smooth structure induced from the ambient Fr´echetmanifold G .Recall that a Fr¨olicher space, see [19, Section 23] and [6, 7, 8], is a set X togetherwith a set C X of curves into X and a set of functions F X on X with the following twoproperties:(a) A function f : X → R is in F X if and only if f ◦ c ∈ C ∞ ( R , R ) for all c ∈ C X .(b) A curve c : R → X is in C X if and only if f ◦ c ∈ C ∞ ( R , R ) for all f ∈ F X .A map g : X → Y between Fr¨olicher spaces is called smooth if g ◦ c ∈ C Y for all c ∈ C X .Equivalently, smoothness of g can be characterized by f ◦ g ∈ F X for all f ∈ F Y . Notethat C X coincides with the set of smooth curves into X , and F X coincides with the setof smooth functions on X , provided R is equipped with the standard Fr¨olicher structure C R = C ∞ ( R , R ) = F R . Fr¨olicher spaces and smooth maps between them form a categorywhich is complete, cocomplete and Cartesian closed, see [19, Theorem 23.2].Any subset A of a Fr¨olicher space X admits a unique Fr¨olicher structure such thatthe inclusion A ⊆ X is initial, i.e., a curve into A is smooth iff it is smooth into X .The c ∞ -topology on a Fr¨olicher space X is the strongest topology such that all smoothcurves into X are continuous. If U is a cover of X by c ∞ -open subsets, then a function f on X is smooth iff the restriction f | U is smooth (with respect to the induced Fr¨olicherstructure) for all U ∈ U .Any Fr´echet manifold, together with the usual smooth curves into and smooth func-tions on it, constitutes a Fr¨olicher space. More generally, manifolds modeled on c ∞ -open subsets of convenient vector spaces [19, Section 27] are Fr¨olicher spaces. For Fr´echet man-ifolds the c ∞ -topology coincides with the Fr´echet topology, see [19, Theorem 4.11(1)].We consider G ρ as a Fr¨olicher space with the smooth structure induced from G . Hence,a curve in G ρ is smooth iff it is smooth into G ; and a function on G ρ is smooth iff itis smooth along smooth curves. Moreover, we equip E ρ / Diff(
S, ρ ) with the inducedFr¨olicher structure. Hence, a function on E ρ / Diff(
S, ρ ) is smooth iff the corresponding(fiberwise constant) function on E ρ is smooth, with respect to the Fr¨olicher structureon E ρ considered before; and a curve in E ρ / Diff(
S, ρ ) is smooth iff it is smooth alongsmooth functions. One readily checks that the maps in the commutative diagram E ρ (cid:15) (cid:15) q ρ ! ! E ρ / Diff(
S, ρ ) / / G ρ are all smooth in the sense of Fr¨olicher spaces, where q ρ denotes the restriction of q .If smooth curves in G ρ can be lifted to smooth curves in E ρ , then the horizontal mapprovides a diffeomorphism of Fr¨olicher spaces, E ρ / Diff(
S, ρ ) = G ρ .Any subgroup of Diff c ( P ) or Diff( S ) inherits a Fr¨olicher structure from the ambientLie group, and the group operations are smooth with respect to this Fr¨olicher structure. Proposition 4.6. If q ρ : E ρ → G ρ admits local (with respect to the c ∞ -topology) smoothsections in the sense of Fr¨olicher spaces, then the following hold true:(a) The map q ρ : E ρ → G ρ is a locally trivial smooth principal bundle with structure group Diff(
S, ρ ) in the sense of Fr¨olicher spaces. Moreover, the canonical identification E ρ / Diff(
S, ρ ) = G ρ is a diffeomorphism of Fr¨olicher spaces.(b) The Diff c ( P, ξ ) -action on G ρ admits local (with respect to the c ∞ -topology) smoothsections in the sense of Fr¨olicher spaces. The Diff c ( P, ξ ) -orbit through ( N, γ ) ∈ G ρ is open and closed (with respect to the c ∞ -topology) in G ρ . Denoting this orbit by G ρ ( N,γ ) , the map Diff c ( P, ξ ) → G ρ ( N,γ ) provided by the action is a locally trivial smoothprincipal bundle with structure group Diff c ( P, ξ ; (
N, γ )) in the sense of Fr¨olicherspaces. Hence, G ρ ( N,γ ) = Diff c ( P, ξ ) / Diff c ( P, ξ ; (
N, γ )) may be regarded as a homogeneous Fr¨olicher space.(c) The map J G L restricts to a Diff c ( P, ξ ) -equivariant smooth bijection J G ρ L : G ρ ( N,γ ) → X ( P, ξ ) ∗ (93) onto the coadjoint orbit of Diff c ( P, ξ ) through J G L ( N, γ ) .Proof. Let σ : U → E ρ be a local smooth section of q ρ : E ρ → G ρ , defined on a c ∞ -opensubset U in G ρ . Putting E ρU := ( q ρ ) − ( U ), we obtain a local trivialization, U × Diff(
S, ρ ) → E ρU , ( z, f ) ψ E f ( σ ( z )) . (94)Clearly, this is a Diff( S, ρ )-equivariant smooth bijection. To see that these are actuallydiffeomorphisms of Fr¨olicher spaces, we use the fact that
E → G is a smooth principalbundle. This implies that the map ˜ δ : E × G E →
Diff( S ), implicitly characterized by ψ E ˜ δ (Φ , Φ ) (Φ ) = Φ for all Φ , Φ ∈ E with q (Φ ) = q (Φ ) ∈ G , is smooth. Restricting˜ δ , we obtain a smooth map δ : E ρ × G ρ E ρ → Diff(
S, ρ ), which can be used to express
DUAL PAIR FOR THE CONTACT GROUP 27 the inverse of (94): E ρU → U × Diff(
S, ρ ), Φ ( q (Φ) , δ ( σ ( q (Φ)) , Φ)). This shows thatthe trivialization (94) is a diffeomorphism, whence E ρ → G ρ is a locally trivial smoothprincipal fiber bundle. The remaining assertions in (a) are now obvious.Using local sections of E ρ → G ρ and the fact that the Diff c ( P, ξ )-action on E ρ admitslocal smooth sections, see Theorem 3.5, we readily see that the Diff c ( P, ξ )-action on G ρ admits local smooth sections. The remaining assertions in (b) are then obvious.Part (c) follows at once, see Proposition 4.2(a). (cid:3) Note that the assumption in Proposition 4.6 is trivially satisfied for ρ = 0. Thisisotropic case will be discussed in Section 4.3; and we will obtain a more precise con-clusion than formulated in Proposition 4.6 above, see Theorem 4.12. In particular, wewill show that in this case G is a smooth submanifold of G which inherits a reducedsymplectic form from E . Moreover, the map in (93) is a symplectomorphism onto thecoadjoint orbit equipped with the Kostant–Kirillov–Souriau form.For more general ρ (e.g. ρ of contact type) the situation is more delicate. If we equip G ρ with the trace topology induced from G , then q restricts to principal fiber bundle( J E R ) − ( ρ · Diff( S )) / Diff( S ) → G ρ with structure group Diff( S ), see (91). However, withrespect to this topology, the action of Diff( P, ξ ) on G ρ will in general not admit localsections, see Proposition 4.20 below.The next lemma provides a criterion for the premise in Proposition 4.6 above. Lemma 4.7.
Let ρ ∈ Ω ( S, | Λ | S ) be a -form density and assume that the Diff( S ) -action on the orbit through ρ in Ω ( S, | Λ | S ) admits local (with respect to the c ∞ -topology)smooth sections in the sense of Fr¨olicher spaces. Then the map E ρ → G ρ admits local(with respect to the c ∞ -topology) smooth sections in the sense of Fr¨olicher spaces.Proof. Since
E → G admits local sections, each point in G ρ admits an open neighborhood˜ U in G and a smooth section ˜ σ : ˜ U → E such that q ◦ ˜ σ = id ˜ U . Then U := ˜ U ∩ G ρ isa c ∞ -open neighborhood, and the restriction ¯ σ := ˜ σ | U is a smooth section mapping¯ σ : U → ( J E R ) − ( ρ · Diff( S )), see (90). By assumption, after possibly shrinking U , thereexists a smooth map f : U → Diff( S ) such that J E R (¯ σ ( z )) = f ( z ) ∗ ρ for all z ∈ U . Hence, σ : U → E ρ , σ ( z ) := ψ E f ( z ) − (¯ σ ( z )), is the desired local smooth section of E ρ → G ρ . (cid:3) Remark . For a contact 1-form density, the Gray stability theorem [14, Theorem 2.2.2]permits to reformulate the assumption in Lemma 4.7. More precisely, if ρ ∈ Ω ( S, | Λ | S )is a 1-form density such that ker ρ is a contact distribution on S , then the following twostatements are equivalent:(a) The Diff( S )-action on the Diff( S )-orbit through ρ in Ω ( S, | Λ | S ) admits local (withrespect to the c ∞ -topology) smooth sections in the sense of Fr¨olicher spaces.(b) The Diff( S, ker ρ )-action on the Diff( S, ker ρ )-orbit through ρ in Ω ( S, | Λ | S ) admitslocal (with respect to the c ∞ -topology) smooth sections in the sense of Fr¨olicherspaces.We do not know if these (equivalent) properties hold true for all contact 1-form densities.4.3. Weighted isotropic non-linear Grassmannians.
We will now specialize to theisotropic case, ρ = 0. Let us introduce the notation E iso := ( π E ) − (Emb iso ( S, P )) = ( J E R ) − (0) = E , (95) where Emb iso ( S, P ) denotes the space of isotropic embeddings, cf. (41), (47), or (60).This can equivalently be characterized as the elements in E = Emb lin ( | Λ | ∗ S , L ∗ ) whichrestrict to isotropic embeddings | Λ | ∗ S \ S → L ∗ \ P = M . Let Gr iso S ( P ) denote the space of isotropic submanifolds of type S and consider thespace of all weighted isotropic submanifolds of type S , G iso := ( π G ) − (Gr iso S ( P )) (96)= { ( N, γ ) | N ∈ Gr iso S ( P ) , γ ∈ Γ ∞ ( | Λ | N ⊗ L ∗ | N ) nowhere vanishing } . In view of (87) and (95) we have q − ( G iso ) = E iso = ( J E R ) − (0). Hence, G iso coincideswith the reduced space G ρ for ρ = 0, i.e., G = ( J E R ) − (0) / Diff( S ) = G iso = ( π G ) − (Gr iso S ( P )) . (97) Remark . If α is a contact form for ξ , then isotropic submanifolds N are characterizedby ι ∗ N α = 0 and the identification in Remark 4.4 becomes G = G iso ∼ = (cid:8) ( N, ν ) : N ∈ Gr iso S ( P ) and ν ∈ Γ ∞ ( | Λ | N \ N ) (cid:9) . (98) Lemma 4.10.
The subset Gr iso S ( P ) is a smooth splitting submanifold of Gr S ( P ) .Proof. This follows from the tubular neighborhood theorem for contact structures nearisotropic submanifolds, see [14, Theorem 2.5.8] or [23, Theorem 1]. Since we were notable to locate this statement in the literature, we will sketch a proof in the subsequentparagraph.Suppose S ∼ = N ⊆ P is an isotropic submanifold, and let E := T N ⊥ /T N denoteits conformal symplectic normal bundle, see [14, Definition 2.5.3]. Using the relativePoincar´e lemma, one easily constructs a 1-form ε on the total space of E such that (1) ε vanishes along the zero section; (2) i X dε = 0 for every vector X tangent to the zerosection; and (3) such that ( dε ) | N represents the conformal symplectic structure on eachfiber of E , cf. the proof of [23, Proposition in Section 4]. Hence α := p ∗ ε + p ∗ θ + dt isa contact form in a neighborhood of the zero section of E ⊕ T ∗ N × R , where p , p , t denote the canonical projections onto the three summands, and θ denotes the canonical1-form on T ∗ N . Assuming, for simplicity, that the contact structure on P is coorientablenear N , the tubular neighborhood theorem for isotropic submanifolds asserts that thereexists a contact diffeomorphism ψ between an open neighborhood of the zero section in E ⊕ T ∗ N × R and an open neighborhood of N in P which restricts to the identity along N . Using this diffeomorphism, we obtain a manifold chart for Gr S ( P ) centered at N byassigning to a smooth section σ of E ⊕ T ∗ N × R , which is sufficiently C -close to the zerosection, the submanifold ψ ( σ ( N )) in P . As ψ is contact, the part of Gr iso S ( P ) covered bythis chart corresponds to sections σ ∈ Γ ∞ ( E ⊕ T ∗ N × R ) such that σ ∗ α = 0. IdentifyingΓ ∞ ( E ⊕ T ∗ N × R ) = Γ ∞ ( E ) × Ω ( N ) × C ∞ ( N ) and writing σ = ( s, β, f ) accordingly,the latter condition is equivalent to s ∗ ǫ + β + df = 0. Hence, Gr iso S ( P ) corresponds tothe part of the chart domain contained in the splitting linear subspaceΓ ∞ ( E ) × C ∞ ( N ) ⊆ Γ ∞ ( E ) × Ω ( N ) × C ∞ ( N ) = Γ ∞ ( E ⊕ T ∗ N × R ) , ( s, f ) ( s, − s ∗ ε − df, f ) . Using a volume density µ on S to identify L ∼ = C ∞ ( S, L ∗ ) as in Remark 2.2, the subset E iso corresponds to C ∞ ( S, M ) ∩ ( π L ) − (Emb iso ( S, P )). If moreover ξ = ker α , then the correspondingdiffeomorphism L ∼ = C ∞ ( S, P ) × C ∞ ( S ) provides an identification E iso ∼ = Emb iso ( S, P ) × C ∞ ( S, R × ). DUAL PAIR FOR THE CONTACT GROUP 29
This shows that Gr iso S ( P ) is a splitting smooth submanifold of Gr S ( P ). (cid:3) Remark . Lemma 4.10 implies that Emb iso ( S, P ) is a smooth splitting submanifold ofEmb(
S, P ), because the natural map Emb(
S, P ) → Gr S ( P ) is a (locally trivial) smoothprincipal bundle with typical fiber Diff( S ). Since π E : E →
Emb(
S, P ) is a (locallytrivial) smooth fiber bundle, this also implies that E iso is a smooth submanifold of E ,see (95). Using the isotropic isotopy extension theorem for contact manifolds, see [14,Theorem 2.6.2] for instance, one can show that the group Diff c ( P, ξ ) acts locally andinfinitesimally transitive on E iso . Hence, for ρ = 0, Theorem 3.5 is essentially known.As mentioned before, one expects that connected components of G iso , endowed witha reduced symplectic form, are symplectomorphic to coadjoint orbits of Diff c ( P, ξ ) viathe restriction of J G L : G → X ( P, ξ ) ∗ . The following theorem makes this precise. Theorem 4.12. (a) The subset G iso is a smooth splitting submanifold of G . More-over, the map provided by the action, Diff c ( P, ξ ) → G iso , g Ψ G g ( N, γ ) , admits a localsmooth right inverse defined in a neighborhood of ( N, γ ) in G iso . In particular, thegroup Diff c ( P, ξ ) acts locally and infinitesimally transitive on G iso , and the Diff c ( P, ξ ) -orbit of ( N, γ ) is open and closed in G iso . Denoting this orbit by G iso( N,γ ) , the map Diff c ( P, ξ ) → G iso( N,γ ) is a smooth principal bundle with structure group Diff c ( P, ξ ; (
N, γ )) in the sense of Fr¨olicher spaces. Hence, G iso( N,γ ) = Diff c ( P, ξ ) / Diff c ( P, ξ ; (
N, γ )) may be regarded as a homogeneous space in the sense of Fr¨olicher spaces.(b) The projection q restricts to a smooth principal bundle q iso : E iso → G iso with struc-ture group Diff( S ) . The restriction of the symplectic form ω E to E iso descends to a(reduced) symplectic form ω G iso on G iso . The Diff(
P, ξ ) -equivariant injective immersion J G iso L : G iso → X ( P, ξ ) ∗ , h J G iso L ( N, γ ) , X i = Z N γ ( X | N ) , provided by restriction of J G L from (89) , identifies G iso( N,γ ) with the coadjoint orbit through J G L ( N, γ ) of the contact group Diff c ( P, ξ ) , such that ( J G iso L ) ∗ ω KKS = ω G iso , (99) where ω KKS denotes the Kostant–Kirillov–Souriau symplectic form on the coadjoint orbitthrough J G L ( N, γ ) , cf. Remark 4.13 below.Remark . To avoid discussing differential forms on coadjoint orbits, we considerthe Kostant–Kirillov–Souriau form on the coadjoint orbit through J G L ( N, γ ) as a formalobject only. We actually work with its pull back along J G iso L , that is, the well definedsmooth 2-form on G iso characterized by(( J G iso L ) ∗ ω KKS )( ζ G iso X ( N, γ ) , ζ G iso Y ( N, γ )) := h J G iso L ( N, γ ) , [ X, Y ] i , (100)where X, Y ∈ X ( P, ξ ) and (
N, γ ) ∈ G iso . To motivate this definition, recall that for a Liealgebra g the Kostant–Kirillov–Souriau symplectic form on the coadjoint orbit through λ ∈ g ∗ is (formally) given by ω KKS ( ζ g ∗ X ( λ ) , ζ g ∗ Y ( λ )) = h λ, [ X, Y ] i , where X, Y ∈ g and ζ g ∗ X denotes the infinitesimal coadjoint action. Since J G iso L is equi-variant, we are being lead to (100). Proof of Theorem 4.12.
We have already observed that Gr iso S ( P ) is a smooth submanifoldof Gr S ( P ), see Lemma 4.10. Since the forgetful map π G : G → Gr S ( P ) is a smooth fiberbundle, we conclude that G iso is a smooth submanifold of G , see (96). In particular, themap provided by the action Diff c ( P, ξ ) → G iso , g Ψ G g ( N, γ ), is smooth. The remainingassertions in (a) thus follow from Proposition 4.6(b). Note that in the isotropic case theassumption in the latter proposition is trivially satisfied.In view of E iso = q − ( G iso ), the smooth principal bundle q : E → G restricts to a smoothprincipal bundle q iso : E iso → G iso with structure group Diff( S ). By Proposition 4.2 themap J G iso L is a Diff( P, ξ )-equivariant injective immersion. In view of (the trivial inclusionin) Equation (55), we have ω E ( ζ E X , ζ E Z ) = 0 for all X ∈ X ( P, ξ ) and Z ∈ X ( S ). SinceDiff c ( P, ξ ) acts infinitesimally transitive on E iso , the 1-form ω E ( − , ζ E Z ), thus, vanisheswhen pulled back to E iso . Hence, the restriction of ω E to E iso is vertical. We concludethat there exists a unique 2-form ω G iso on G iso such that ( q iso ) ∗ ω G iso coincides with the pullback of ω E to E iso . Clearly, ω G iso is closed. The 2-form ω G iso is (weakly) non-degeneratein view of (the non-trivial inclusion in) Equation (55). From (100), (29), (26) and theequivariance of q we immediately obtain ( q iso ) ∗ ( J G iso L ) ∗ ω KKS = ( q iso ) ∗ ω G iso , whence (99).The remaining assertions are now obvious. (cid:3) Remark . We expect that the isotropy group Diff c ( P, ξ ; (
N, γ )) in Theorem 4.12(a)is a closed Lie subgroup in Diff c ( P, ξ ). If this is the case then G iso( N,γ ) may be regarded asa homogeneous space in the category of smooth manifolds. Example 4.15. If S is the circle S and P is a 3-dimensional contact manifold, then theweighted non-linear Grassmannian G becomes the manifold of weighted (unparametrized)knots in P , and G iso is the (symplectic) manifold of weighted Legendrian knots in P . ByTheorem 4.12, its connected components can be identified with coadjoint orbits of theidentity component of the contact group.4.4. Weighted contact non-linear Grassmannians.
Let us now consider a 1-formdensity ρ ∈ Ω ( S, | Λ | S ) of contact type, i.e., ker ρ ⊆ T S is assumed to be a contacthyperplane distribution. Then the reduced space G ρ , see (91), consists of weightedcontact submanifolds. More precisely, according to Lemma 4.3 we have G ρ ⊆ ( π G ) − (Gr contact( S, ker ρ ) ( P, ξ )) , (101)where Gr contact( S, ker ρ ) ( P, ξ ) ⊆ Gr S ( P ) denotes the subset of contact submanifolds which are oftype ( S, ker ρ ). In contrast to the isotropic case, see (97), the inclusion (101) is strict.The maps in (87) restrict to a Diff( P, ξ )-equivariant commutative diagram E ρq ρ (cid:15) (cid:15) ∼ = π E ρ / / Emb contact( S, ker ρ ) ( P, ξ ) (cid:15) (cid:15) G ρ π G ρ / / Gr contact( S, ker ρ ) ( P, ξ ) (102)where Emb contact( S, ker ρ ) ( P, ξ ) ⊆ Emb(
S, P ) denotes the subset of contact embeddings inducingthe contact structure ker ρ on S . Lemma 4.16. If ρ ∈ Ω ( S, | Λ | S ) is a contact -form density, then the following holdtrue: DUAL PAIR FOR THE CONTACT GROUP 31 (a) Gr contact( S, ker ρ ) ( P, ξ ) is an open subset of Gr S ( P ) .(b) Emb contact( S, ker ρ ) ( P, ξ ) is an initial Fr´echet submanifold of Emb(
S, P ) .(c) The natural map Emb contact( S, ker ρ ) ( P, ξ ) → Gr contact( S, ker ρ ) ( P, ξ ) is a smooth principal bundle with structure group Diff( S, ker ρ ) .(d) The natural map L| Emb contact( S, ker ρ ) ( P,ξ ) ( π L ,J L R ) −−−−→ Emb contact( S, ker ρ ) ( P, ξ ) × Γ ∞ (cid:0) ( T S/ ker ρ ) ∗ ⊗ | Λ | S (cid:1) is a diffeomorphism of Fr´echet manifolds, providing a Diff( S, ker ρ ) -equivariant triv-ialization of the bundle π L : L → C ∞ ( S, P ) over Emb contact( S, ker ρ ) ( P, ξ ) .(e) The map π E : E →
Emb(
S, P ) restricts to a diffeomorphism of Fr´echet manifolds, E ρ ∼ = Emb contact( S, ker ρ ) ( P, ξ ) . Proof. (a) follows from the Gray stability theorem, see [14, Theorem 2.2.2]. Locallyaround points in Gr contact(
S,ρ ) ( P, ξ ), the Gray stability theorem permits to construct cross sec-tions of the Diff( S )-bundle Emb( S, P ) → Gr S ( P ) which take values in Emb contact( S, ker ρ ) ( P, ξ ).Such a local cross section, defined on an open subset U in Gr S ( P ), provides a local triv-ialization of Diff( S )-bundles, U ×
Diff( S ) ∼ = Emb( S, P ) | U , which maps U ×
Diff( S, ker ρ )onto Emb contact( S, ker ρ ) ( P, ξ ) | U . Recall that Diff( S, ker ρ ) is a Fr´echet Lie group, and the naturalinclusion into Diff( S ) is initial, see [19, Theorem 43.19]. Whence (b) and (c).Since ρ is nowhere vanishing, the map in (d) is a bijection. This map is smooth becausethe inclusion Emb contact( S, ker ρ ) ( P, ξ ) ⊆ Emb(
S, P ) is initial. To see that its inverse is smoothtoo, we fix a vector bundle homomorphism σ : T S/ ker ρ → T S splitting the canonicalprojection
T S → T S/ ker ρ . Let W denote the set of embeddings ϕ ∈ Emb(
S, P ) forwhich the composition
T S/ ker ρ σ −→ T S
T ϕ −→ ϕ ∗ T P → ϕ ∗ L is an isomorphism of line bundles over S . Clearly, W is an open neighborhood ofEmb contact( S, ker ρ ) ( P, ξ ) in Emb(
S, P ). We obtain a smooth map s : W × Γ ∞ (cid:0) ( T S/ ker ρ ) ∗ ⊗ | Λ | S (cid:1) → L , characterized by π L ( s ( ϕ, β )) = ϕ and J L R ( s ( ϕ, β )) ◦ σ = β , for all ϕ ∈ W and β ∈ Γ ∞ (cid:0) ( T S/ ker ρ ) ∗ ⊗ | Λ | S (cid:1) . Its restriction provides the smooth inverse for the map in (d).Restricting the diffeomorphism in (d) to the level set E ρ , we obtain a diffeomorphism E ρ ∼ = Emb contact( S, ker ρ ) ( P, ξ ) × { ρ } , whence (e). (cid:3) The diffeomorphism in Lemma 4.16(e) induces a natural diffeomorphism of Fr¨olicherspaces: E ρ / Diff(
S, ρ ) ∼ = Emb contact( S, ker ρ ) ( P, ξ ) × Diff( S, ker ρ ) Diff( S, ker ρ )Diff( S, ρ ) . Note that the isotropy group Diff(
S, ρ ) is akin to the group of strict contact diffeomor-phisms.The diffeomorphism in Lemma 4.16(d) induces a diffeomorphism G| Gr contact( S, ker ρ ) ( P,ξ ) ∼ = Emb contact( S, ker ρ ) ( P, ξ ) × Diff( S, ker ρ ) Γ ∞ (cid:0) (( T S/ ker ρ ) ∗ ⊗ | Λ | S ) \ S (cid:1) which restricts to a natural diffeomorphism of Fr¨olicher spaces, G ρ ∼ = Emb contact( S, ker ρ ) ( P, ξ ) × Diff( S, ker ρ ) O ρ . (103)Here G ρ is equipped with the Fr¨olicher structure induced from G , and O ρ denotes theDiff( S, ker ρ )-orbit of ρ equipped with the Fr¨olicher structure induced from Ω ( S, | Λ | S )which coincides with the Fr¨olicher structure induced from Γ ∞ (( T S/ ker ρ ) ∗ ⊗ | Λ | S ). Remark . If α is a contact form for ξ , then contact submanifolds N are characterizedby the fact that ι ∗ N α is a contact form on N , and the identification in Remark 4.4 becomes G ρ = (cid:8) ( N, ν ) : N ∈ Gr contact S ( P ), ν ∈ Γ ∞ ( | Λ | N \ N ), and ( N, ι ∗ N α ⊗ ν ) ∼ = ( S, ρ ) (cid:9) . If (
N, ν ) ∈ G ρ then any other weight on N allowed in G ρ must be of the form f ∗ ι ∗ N αι ∗ N α · f ∗ νν · ν for a contact diffeomorphism f ∈ Diff( N, ker ι ∗ N α ). Thus, unlike the isotropic case (98),in the contact case not all weights on a contact submanifold N ∈ Gr contact S ( P ) are allowedin G ρ , i.e., the inclusion in (101) is strict. Remark . Let ρ ∈ Ω ( S, | Λ | S ) be a contact 1-form density. Since G ρ may not be amanifold, we refrain from considering the Kostant–Kirillov–Souriau form on G ρ . How-ever, formally pulling back the Kostant–Kirillov–Souriau form along J E ρ L : E ρ → X ( P, ξ ) ∗ ,we obtain a well defined smooth 2-form ( J E ρ L ) ∗ ω KKS on E ρ , characterized by(( J E ρ L ) ∗ ω KKS )( ζ E ρ X (Φ) , ζ E ρ Y (Φ)) := h J E ρ L (Φ) , [ X, Y ] i , where Φ ∈ E ρ and X, Y ∈ X ( P, ξ ), cf. Remark 4.13 and Theorem 3.5. Proceeding exactlyas in the proof of Theorem 4.12, we see that this coincides with ω E ρ , the pull back ofthe symplectic form ω E to E ρ , i.e., ( J E ρ L ) ∗ ω KKS = ω E ρ . The discussion in the next example shows that the situation is as nice as one couldwish for 1-dimensional S . Subsequently, we will see that the situation is considerablymore delicate in general, see Proposition 4.20. Example 4.19.
Let us specialize to the circle, S = S . In this case, any contact 1-form density ρ ∈ Ω ( S, | Λ | S ) gives rise to an orientation and a Riemannian metric on S . We write p | ρ | for the induced volume density on S , and denote the total volume byvol( ρ ) := R S p | ρ | . Using parametrization by arc length it is easy to see that two contact1-form densities lie in the same Diff( S )-orbit iff they have the same total volume. Inparticular, the Diff( S )-orbits through contact 1-form densities are closed submanifoldsin Ω ( S, | Λ | S ). Moreover, parametrization by arc length provides local smooth sectionsfor the Diff( S )-action on these orbits. In particular, the assumption in Lemma 4.7 issatisfied in this case.Suppose ( P, ξ ) is a contact manifold and let ρ ∈ Ω ( S, | Λ | S ) be a contact 1-formdensity on S = S . Using (103) we conclude that G ρ is a closed submanifold of G .Parametrization by arc length provides local smooth sections of E ρ → G ρ and the latteris a locally trivial smooth principal bundle. Note that the structure group Diff( S, ρ ) ∼ =SO(1) is a closed Lie subgroup of Diff( S ). By Proposition 4.6, the Diff c ( P, ξ )-action on G ρ admits local smooth sections. Moreover, its orbits are open and closed subsets in G ρ DUAL PAIR FOR THE CONTACT GROUP 33 which may be identified with coadjoint orbits of the contact group via the restrictionof J G L . The symplectic form on E gives rise to a reduced symplectic form on G ρ whichcoincides with the pull back of the Kostant–Kirillov–Souriau symplectic form via J G L as in Theorem 4.12(b). If P is 3-dimensional, then G ρ is a (symplectic) manifold ofweighted transverse knots.A slightly more explicit description can be given if the contact structure is admits acontact form, ξ = ker α . Then, via the identification in Remark 4.4, we have G ρ ∼ = (cid:8) ( N, ν ) ∈ Gr wt S ( P ) (cid:12)(cid:12) ι ∗ N α = 0 , vol( ι ∗ N α ⊗ ν ) = vol( ρ ) (cid:9) , for every contact 1-form density ρ .The following result shows that the trace topology on G ρ induced from G is not theappropriate topology for general contact ρ . Proposition 4.20.
There exist a compact contact manifold ( P, ξ ) , a compact manifold S , and a contact -form density ρ ∈ Ω ( S, | Λ | S ) such that the continuous bijection E ρ / Diff(
S, ρ ) → ( J E R ) − ( ρ · Diff( S )) / Diff( S ) (104) induced by the natural inclusion is not a homeomorphism with respect to the quotienttopologies. In particular, the continuous bijection E ρ / Diff(
S, ρ ) → G ρ induced by q is nota homeomorphism where the right hand side is equipped with the trace topology inducedfrom G . Moreover, for ( N, γ ) ∈ G ρ the map provided by the action, Diff(
P, ξ ) → G ρ , g Ψ G g ( N, γ ) , does not admit a continuous local (with respect to the trace topologyinduced from G ) right inverse defined in a neighborhood of ( N, γ ) . The following lemma will be crucial in the proof of Proposition 4.20.
Lemma 4.21.
There exists a compact contact manifold ( S, α ) and a sequence of diffeo-morphisms f n ∈ Diff( S ) with the following properties:(a) f ∗ n α → α with respect to the C ∞ -topology.(b) There does not exist a sequence of diffeomorphisms g n ∈ Diff( S ) such that g ∗ n α = f ∗ n α for all n and g n → id S with respect to the C -topology.Proof. Let (
M, ω ) be a connected compact symplectic manifold with integral symplecticform. Choose a sequence of non-empty open subsets
U, U , U , U , . . . of M such thattheir closures are mutually disjoint. Choose points x ∈ U and x n ∈ U n . Assume thatthe sequences of closures ¯ U n only accumulates at a single point. Choose Hamiltoniandiffeomorphisms h n ∈ Ham(
M, ω ) such that for each n we have(i) h n ( y ) = y for all y ∈ S i = n ¯ U i , and(ii) h n ( x ) = x n .Shrinking U n , we may moreover assume(iii) h − n ( U n ) ⊆ U .For each n let λ n be a compactly supported smooth function on U n such that λ n isconstant and strictly positive in a neighborhood of x n . Let λ : M → R denote thefunction which coincides with λ n on U n and vanishes outside S n U n . Multiplying λ n witha sufficiently fast decreasing sequence of constants, we may assume that the followinghold true:(iv) λ is smooth on M , and(v) h ∗ n λ → λ with respect to the C ∞ topology on M . By construction, we have:(vi) λ is constant and strictly positive on a neighborhood of x n , for each n , and(vii) λ vanishes on U .Let p : S → M be the circle bundle with Chern class [ ω ] and let ˜ α ∈ Ω ( S ) be aprincipal connection 1-form with curvature ω . Hence, ˜ α is a contact form on S . Itis well known that Hamiltonian diffeomorphisms on M can be lifted to strict contactdiffeomorphisms on S . Hence, there exist diffeomorphisms f n ∈ Diff( S ) such that f ∗ n ˜ α =˜ α and p ◦ f n = h n ◦ p . We consider the contact form α := e − p ∗ λ ˜ α on S . From (v) weimmediately obtain f ∗ n α → α , whence (a).To see (b), let E denote the Reeb vector field of α . From (vii) we see that α coincideswith the principal connection ˜ α on p − ( U ). Over p − ( U ), the Reeb vector field E thuscoincides with the fundamental vector field of the principal circle action. For each y ∈ p − ( U ) we thus have Fl Et ( y ) = y ⇔ t ∈ π Z . Hence, if g ∈ Diff( S ) is sufficientlyclose to the identity with respect to the C -topology, thenFl g ∗ Et ( x ) = x ⇔ t ∈ π Z . (105)Note that g ∗ E is the Reeb vector field of g ∗ α , and f ∗ n E is the Reeb vector field of f ∗ n α . For each n there exists a constant 0 < c n < f ∗ n α coincides with c n α on a neighborhood of p − ( x ), see (ii) and (vi). Hence, f ∗ n E coincides with c − n E on aneighborhood of p − ( x ). In particular,Fl f ∗ n Et ( x ) = x ⇔ t ∈ πc n Z . (106)Comparing (105) and (106) and using c n = 1, we conclude g ∗ E = f ∗ n E and thus g ∗ α = f ∗ n α . This shows (b). (cid:3) Proof of Proposition 4.20.
We consider a closed manifold S of dimension 2 k +1, a contactform α on S , and diffeomorphisms f n ∈ Diff( S ) as in Lemma 4.21. Using Gray’sstability result [14, Theorem 2.2.2], we may w.l.o.g. assume that each f n is a contactdiffeomorphism. Hence, there exist smooth functions λ n on S such that f ∗ n α = λ n α .Since f ∗ n α → α , we have λ n →
1, as n → ∞ . In particular, we may assume λ n > µ := | α ∧ ( dα ) k | denote the volume density associated with the volume form α ∧ ( dα ) k . Note that f ∗ n µ = λ k +1 n µ . Moreover, we put ρ := α ⊗ µ ∈ Ω ( S, | Λ | S ).We consider the manifold P := S equipped with the contact structure ξ := ker( α ).Using the volume density µ on S and the contact form α on P , we may identify E =Emb( S, P ) × C ∞ ( S, R × ), see Remark 2.2. Using this identification we define a sequenceΦ n ∈ E by Φ n := (id S , λ k +2 n ). Clearly, Φ n converges to Φ := (id S , ∈ E . Using (47) wefind J E R (Φ) = ρ and J E R (Φ n ) = λ k +2 n α ⊗ µ = f ∗ n ( α ⊗ µ ) = f ∗ n ρ = λ k +2 n ρ. (107)In particular, we have Φ ∈ E ρ , Φ n ∈ ( J E R ) − ( ρ · Diff( S )) and Φ n → Φ, as n → ∞ .We will now show that the corresponding sequence in E ρ / Diff(
S, ρ ) does not converge,cf. (104). Suppose, by contradiction, there exists a sequence of diffeomorphisms g n ∈ Diff( S ) such that ψ E g n (Φ n ) ∈ E ρ and ψ E g n (Φ n ) converges in E ρ / Diff(
S, ρ ). W.l.o.g. wemay moreover assume that ψ E g n (Φ n ) converges in E ρ . In particular, g n converges to adiffeomorphism g ∈ Diff( S ). Using the Diff( S ) equivariance of J E R , the relation ρ = J E R ( ψ E g n (Φ n )) yields g ∗ n ρ = J E R (Φ n ) . (108) DUAL PAIR FOR THE CONTACT GROUP 35
In particular, letting n → ∞ , we obtain g ∈ Diff(
S, ρ ). Replacing g n with g n ◦ g − wemay w.l.o.g. assume that g n → id S . Combining (107) and (108) we see that g n is acontact diffeomorphism. Hence, there exist smooth functions ˜ λ n with g ∗ n α = ˜ λ n α . Weobtain λ k +2 n ρ = g ∗ n ρ = ˜ λ k +2 n ρ and thus ˜ λ k +2 n = λ k +2 n . Since g n converges to the identity,we may assume ˜ λ n >
0. Hence, ˜ λ n = λ n and thus g ∗ n α = f ∗ n α . This contradicts thechoice of f n , see Lemma 4.21(b). Hence, the sequence in E ρ / Diff(
S, ρ ) corresponding toΦ n does not converge.This shows that the continuous bijection (104) is not a homeomorphism. The remain-ing statements follow immediately from the fact that the projection q : E → G admitslocal (smooth) sections. (cid:3) Comparison with the EPDiff dual pair
A pair of moment maps has been introduced by D. D. Holm and J. E. Marsden [17] inrelation to the EPDiff equations describing geodesics on the group of all diffeomorphisms.The left moment map provides singular solutions of these equations, whereas the rightmoment map provides a constant of motion for the collective dynamics of these singularsolutions. It has been shown in [9] that the pair of moment maps, when restricted to anappropriate open subset, do indeed form a symplectic dual pair. In this section we relatethe EPDiff dual pair of a manifold with the EPContact dual pair of its projectivizedcotangent bundle.5.1.
The dual pair for the EPDiff equation.
The (regular) cotangent bundle to thespace of smooth maps from a closed manifold S into a manifold Q can be equipped withthe canonical symplectic structure. Recall that the tangent space at η ∈ C ∞ ( S, Q ) is T η C ∞ ( S, Q ) = Γ ∞ ( η ∗ T Q ). Using the canonical pairing, we regard the space of 1-formdensities along η , Γ ∞ ( | Λ | S ⊗ η ∗ T ∗ Q ) = T ∗ η C ∞ ( S, Q ) reg , (109)as the regular cotangent space at η . In this way we identify the space of smooth fiberwiselinear maps from | Λ | ∗ S to T ∗ Q with the regular cotangent bundle: C ∞ lin ( | Λ | ∗ S , T ∗ Q ) = T ∗ C ∞ ( S, Q ) reg . (110)Via this identification, the canonical 1-form on T ∗ C ∞ ( S, Q ) reg can be written in the form θ T ∗ C ∞ ( S,Q ) reg ( A ) = Z S θ T ∗ Q ( A ) , (111)where A is a tangent vector at Φ ∈ T ∗ C ∞ ( S, Q ) reg , i.e., A ∈ T Φ C ∞ lin ( | Λ | ∗ S , T ∗ Q ) = (cid:26) A ∈ C ∞ ( | Λ | ∗ S , T T ∗ Q ) (cid:12)(cid:12)(cid:12)(cid:12) π T T ∗ Q ◦ A = Φ and ∀ t ∈ R : A ◦ δ | Λ | ∗ S t = T δ T ∗ Qt ◦ A (cid:27) and θ T ∗ Q denotes the canonical 1-form on T ∗ Q . As before, the integrand θ T ∗ Q ( A ) is afiberwise linear function on the total space of | Λ | ∗ S , which may be regarded as a sectionon | Λ | S and integrated over S . The differential dθ T ∗ C ∞ ( S,Q ) reg is the canonical (weaklynon-degenerate) symplectic form on T ∗ C ∞ ( S, Q ) reg .The cotangent lifted actions of the groups Diff( Q ) and Diff( S ) on the manifold C ∞ ( S, Q ) preserve the canonical 1-form θ T ∗ C ∞ ( S,Q ) reg . In particular, these actions are Hamiltonian with equivariant moment maps J Sing : T ∗ C ∞ ( S, Q ) reg → X ( Q ) ∗ , h J Sing (Φ) , Y i = θ T ∗ C ∞ ( S,Q ) reg (cid:16) ζ T ∗ C ∞ ( S,Q ) reg Y (Φ) (cid:17) , (112)and J S : T ∗ C ∞ ( S, Q ) reg → X ( S ) ∗ , h J S (Φ) , Z i = θ T ∗ C ∞ ( S,Q ) reg (cid:16) ζ T ∗ C ∞ ( S,Q ) reg Z (Φ) (cid:17) , (113)respectively, where Φ ∈ T ∗ C ∞ ( S, Q ) reg . Here ζ T ∗ C ∞ ( S,Q ) reg Y and ζ T ∗ C ∞ ( S,Q ) reg Z denote thefundamental vector fields on T ∗ C ∞ ( S, Q ) reg corresponding to the (infinitesimal) actionof Y ∈ X ( Q ) and Z ∈ X ( S ), respectively. More explicitly, using the identification (109),these cotangent moment maps are h J Sing (Φ) , Y i = Z S Φ( Y ◦ η ) and h J S (Φ) , Z i = Z S Φ( T η ◦ Z ) , where η ∈ C ∞ ( S, Q ) and Φ ∈ T ∗ η C ∞ ( S, Q ) reg = Γ ∞ ( | Λ | S ⊗ η ∗ T ∗ Q ). In particular, thesecond formula shows that J S takes values in Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ . More precisely, J S (Φ)is the 1-form density on S corresponding to the 1-homogeneous vertical 1-form Φ ∗ θ T ∗ Q on the total space of | Λ | ∗ S where we regard Φ : | Λ | ∗ S → T ∗ Q , cf. (110).We denote by T ∗ C ∞ ( S, Q ) × reg the open subset of (110) that corresponds to the space C ∞ lin, inj ( | Λ | ∗ S , T ∗ Q ) of smooth maps that are linear and injective on fibers. Restrictingfurther the actions and moment maps to the open subset T ∗ Emb(
S, Q ) × reg , we obtainthe EPDiff symplectic dual pair [9]: X ( Q ) ∗ J Sing ←−−−−−− T ∗ Emb(
S, Q ) × reg J S −−−−−→ Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ . (114)The left moment map J Sing provides the formula for singular solutions of the EPDiffequations, whereas the right moment map J S provides a Noether conserved quantityfor the (collective) Hamiltonian dynamics of these singular solutions in terms of thecanonical variable Φ ∈ T ∗ Emb(
S, Q ) × reg , see [17]. Remark . Fixing a volume density µ on S , we obtain identifications T ∗ C ∞ ( S, Q ) reg ∼ = C ∞ ( S, T ∗ Q ) and T ∗ C ∞ ( S, Q ) × reg ∼ = C ∞ ( S, T ∗ Q \ Q ), cf. (110), as well as Ω ( S, | Λ | S ) ∼ =Ω ( S ). Using these identifications, the moment maps may be written in the form h J Sing ( φ ) , Y i = Z S φ ( Y ) µ and J S ( φ ) = φ ∗ θ T ∗ Q , where φ ∈ C ∞ ( S, T ∗ Q ) and Y ∈ X ( Q ), cf. [9, Section 5].5.2. The projectivized cotangent bundle.
We will compare the EPDiff dual pairdescribed in the preceding paragraph with the EPContact dual pair associated with theprojectivized cotangent bundle. Recall that the projectivized cotangent bundle, P := P ( T ∗ Q ) = ( T ∗ Q \ Q ) / R × p −→ Q, admits a canonical contact structure [1, Appendix 4] given by ξ ℓ = ( T ℓ p ) − (ker β ) , (115)where ℓ ∈ P and β ∈ T ∗ Q is any non-zero element of ℓ . As the natural action of Diff( Q )on P preserves the contact structure ξ , we obtain an injective group homomorphismDiff( Q ) → Diff(
P, ξ ) . DUAL PAIR FOR THE CONTACT GROUP 37
The line bundle L ∗ , see Section 2.1, associated with the projectivized cotangent bundleis naturally isomorphic to the canonical line bundle over P : γ = { ( ℓ, β ) | ℓ ∈ P, β ∈ ℓ } . Indeed, the vector bundle homomorphism χ : γ → T ∗ P over the identity on P , given by χ ( ℓ, β ) := β ◦ T ℓ p , induces an isomorphism of line bundles, χ : γ → L ∗ . Furthermore, χ ∗ θ L ∗ = pr ∗ θ T ∗ Q , (116)where pr : γ → T ∗ Q denotes the canonical projection, i.e., the blow-up of the zerosection in T ∗ Q . We consider the map κ : L ∗ → T ∗ Q , κ := pr ◦ χ − . One readily checks: Lemma 5.2.
The map κ is a vector bundle homomorphism over the bundle projection p , L ∗ κ / / π L ∗ (cid:15) (cid:15) T ∗ Q π T ∗ Q (cid:15) (cid:15) P p / / Q which has the following properties:(a) κ is equivariant over the homomorphism Diff( Q ) → Diff(
P, ξ ) .(b) κ restricts to a diffeomorphism from L ∗ \ P onto T ∗ Q \ Q .(c) κ ∗ θ T ∗ Q = θ L ∗ . Composition with κ provides a map, cf. (110), L = C ∞ lin ( | Λ | ∗ S , L ∗ ) κ ∗ −−−−−→ C ∞ lin ( | Λ | ∗ S , T ∗ Q ) = T ∗ C ∞ ( S, Q ) reg which fits into the following diagram: X ( P, ξ ) ∗ i ∗ (cid:15) (cid:15) L J L L o o κ ∗ (cid:15) (cid:15) J L R / / X ( S ) ∗ X ( Q ) ∗ T ∗ C ∞ ( S, Q ) reg J Sing o o J S / / X ( S ) ∗ (117)Here i ∗ denotes the dual of the Lie algebra homomorphism i : X ( Q ) → X ( P, ξ ) corre-sponding to the homomorphism of groups Diff( Q ) → Diff(
P, ξ ). Clearly, i ∗ is equivariantover the homomorphism Diff( Q ) → Diff(
P, ξ ). Note that via (8) and κ , the Lie algebra X ( P, ξ ) = C ∞ lin ( L ∗ ) may be regarded as the space of homogeneous functions on T ∗ Q \ Q ,while the image of i consists of those which extend to fiberwise linear functions on T ∗ Q . Proposition 5.3.
The diagram (117) commutes. The map κ ∗ is equivariant over thehomomorphism Diff( Q ) → Diff(
P, ξ ) and also Diff( S ) -equivariant. It restricts to a sym-plectic diffeomorphism from M ⊆ L onto T ∗ C ∞ ( S, Q ) × reg .Proof. The map κ ∗ is equivariant over the homomorphism Diff( Q ) → Diff(
P, ξ ) since κ has the same property, see Lemma 5.2(a). Clearly, κ ∗ is Diff( S )-equivariant too. Hence,the fundamental vector fields are κ ∗ -related, that is, T κ ∗ ◦ ζ L i ( Y ) = ζ T ∗ C ∞ ( S,Q ) reg Y ◦ κ ∗ and T κ ∗ ◦ ζ L Z = ζ T ∗ C ∞ ( S,Q ) reg Z ◦ κ ∗ (118)for Y ∈ X ( Q ) and Z ∈ X ( S ). Using Lemma 5.2(c), (22), and (111), we obtain( κ ∗ ) ∗ θ T ∗ C ∞ ( S,Q ) reg = θ L . (119) Combining the latter with the first equation in (118), we see that the square on theleft hand side in (117) commutes, cf. (112) and (29). Combining (119) with the secondequation in (118), we see that the square on the right hand side in (117) commutes, cf.(113) and (37). As κ restricts to a diffeomorphism from L ∗ \ P onto T ∗ Q \ Q , the map κ ∗ restricts to a diffeomorphism from M onto T ∗ C ∞ ( S, Q ) × reg which is symplectic in viewof (119). (cid:3) Coadjoint orbits of the diffeomorphism group.
The first line in (117) becomesa dual pair when restricted to E = Emb lin ( | Λ | ∗ S , L ∗ ). The second line has to be restrictedto T ∗ Emb(
S, Q ) × reg to become a dual pair. The latter is a proper open subset of theimage κ ∗ ( E ). To make this more precise, note that E Q := (cid:8) Φ ∈ E : p ◦ π E (Φ) ∈ Emb(
S, Q ) (cid:9) is a Diff( S ) invariant open subset of E . Since p : P → Q is Diff( Q ) equivariant, E Q isinvariant under Diff( Q ) too. According to Proposition 5.3, the map κ ∗ restricts to aDiff( Q ) and Diff( S ) equivariant symplectomorphism which makes the following diagramcommute: E QJ E QL z z ∼ = κ ∗ (cid:15) (cid:15) J E QR $ $ X ( Q ) ∗ T ∗ Emb(
S, Q ) × reg J Sing o o J S / / X ( S ) ∗ (120)Here J E Q L and J E Q R denote the restrictions of i ∗ ◦ J L L and J L R , respectively, cf. (117).This can be used to obtain a geometric interpretation of some coadjoint orbits ofDiff( Q ). To this end consider the open subset G Q := { ( N, γ ) ∈ G : p | N ∈ Emb(
N, Q ) } of G . Since E Q = q − ( G Q ), the principal Diff( S )-bundle q : E → G restricts to a principalDiff( S ) bundle q Q : E Q → G Q . Restricting i ∗ ◦ J G L : G → X ( Q ) ∗ , see (88), we obtain asmooth map J G Q L : G Q → X ( Q ) ∗ , J G Q L ◦ q Q = J E Q L . (121)In view of (89) we have the explicit formula h J G Q L ( N, γ ) , Y i = Z N γ ( i ( Y ) | N ) , (122)where ( N, γ ) ∈ G Q and Y ∈ X ( Q ). On the right hand side i ( Y ) is regarded as a sectionof L , see (8), restricted to N and contracted with γ ∈ Γ ∞ ( | Λ | N ⊗ L | ∗ N ) to produce adensity on N . Proposition 5.4.
The map J G Q L in (121) is a Diff( Q ) -equivariant injective immersion.Proof. By Proposition (4.2), the map J G Q L is smooth and Diff( Q )-equivariant. It isimmersive because the moment maps at the bottom line of (120) are mutually completelyorthogonal, see [9, Theorem 5.6]. The injectivity follows from the transitivity of theDiff( S )-action on level sets of the left moment map J Sing , which is the content of [9,Proposition 5.2]. (cid:3)
DUAL PAIR FOR THE CONTACT GROUP 39
Recall from Theorem 4.12 that G iso is a closed submanifold of G . Hence, G iso Q := G Q ∩G iso is a closed splitting submanifold of G Q . Consequently, E iso Q := E Q ∩ E iso = q − ( G iso Q ) isa closed splitting submanifold of E Q . The projection q Q : E Q → G Q restricts to a smoothprincipal bundle q iso Q : E iso Q → G iso Q with structure group Diff( S ). Via κ ∗ the manifold G iso Q = E iso Q / Diff( S ) = { ( N, γ ) ∈ G : N ∈ Gr iso S ( P ) , p | N ∈ Emb(
N, Q ) } identifies with J − S (0) / Diff( S ), the reduced space at zero of the right Diff( S )-action onEmb( S, Q ) × reg .According to [9, Proposition 5.5], the group Diff c ( Q ) acts locally transitive on E iso Q and, thus, on G iso Q too. In particular, the Diff c ( Q )-orbit ( G iso Q ) ( N,γ ) through ( N, γ ) ∈ G iso Q is open and closed in G iso Q . From Theorem 4.12 we thus obtain: Corollary 5.5.
The projection q restricts to a smooth principal bundle q iso Q : E iso Q → G iso Q with structure group Diff( S ) . The restriction of the symplectic form ω E to E iso Q descendsto a (reduced) symplectic form ω G iso Q on G iso Q . The Diff( Q ) -equivariant injective immersion J G iso Q L : G iso Q → X ( Q ) ∗ , h J G iso Q L ( N, γ ) , Y i = Z N γ ( i ( Y ) | N ) , provided by restriction of J G Q L from (121) , identifies ( G iso Q ) ( N,γ ) with the coadjoint orbitof Diff c ( Q ) through J G iso Q L ( N, γ ) in such a way that (cid:0) J G iso Q L (cid:1) ∗ ω KKS = ω G iso Q . (123) Here ( N, γ ) ∈ G iso Q and ω KKS denotes the Kostant–Kirillov–Souriau symplectic form onthe coadjoint orbit through J G iso Q L ( N, γ ) ∈ X ( Q ) ∗ .Remark . In the Legendrian case one has a description of the coadjoint orbit that doesnot use contact geometry. A transverse Legendrian submanifold N ⊆ P T ∗ Q projects toa codimension one submanifold N = p ( N ) ⊆ Q , while N has a unique Legendrian liftto the projectivized cotangent bundle N ∋ x ann( T x N ) ∈ P T ∗ x Q. Moreover, the line bundle L = T ( P T ∗ Q ) /ξ restricted to N is canonically isomorphicto the pull back of the normal line bundle, p | ∗ N T N ⊥ , since the contact hyperplane at y ∈ N is ξ y = ( T y p ) − ( T x N ) for x = p ( y ) ∈ N . Hence, the coadjoint orbit of Diff c ( Q )described above can be seen as (cid:26) ( N , γ ) (cid:12)(cid:12)(cid:12)(cid:12) N ⊆ Q has codimension one and γ ∈ Γ ∞ ( | Λ | N ⊗ ( T N ⊥ ) ∗ ) is nowhere vanishing (cid:27) , embedded into X ( Q ) ∗ via Y R N γ ( Y | N mod T N ). Note that we have a canonicalidentification | Λ | N ⊗ ( T N ⊥ ) ∗ = | Λ | Q ⊗ O ⊥ N , where O ⊥ N denotes the orientation bundleof the normal bundle T N ⊥ . Hence, disregarding the latter orientation bundle, we mayregard points in this coadjoint orbit as codimension one submanifolds N in Q , weightedby a volume density of the ambient space Q along N . Appendix A. Comparison with the dual pair for the Euler equation
A dual pair of moment maps associated to the Euler equations of an ideal fluid hasbeen described by J. E. Marsden and A. Weinstein [25]; it justifies the existence ofClebsch canonical variables for ideal fluid motion and also explains the Hamiltonianstructure of point vortex solutions in a geometric way. It has been shown in [9] thatthe pair of moment maps restricted to the open subset of embeddings does indeed forma symplectic dual pair. In this section we relate this dual pair to the EPContact dualpair, see (52).A.1.
The dual pair for the Euler equation.
The space of smooth maps from aclosed manifold S into a symplectic manifold ( M, ω ) can be equipped with a symplecticstructure once a volume density µ ∈ Γ ∞ ( | Λ | S \ S ) has been fixed. Recall that the spaceof maps C ∞ ( S, M ) is a Fr´echet manifold in a natural way. The symplectic form on C ∞ ( S, M ) can be described by ω C ∞ ( S,M ) φ ( U, V ) = Z S ω ( U, V ) µ, (124)where U, V ∈ Γ ∞ ( φ ∗ T M ) = T φ C ∞ ( S, M ) are vector fields along φ ∈ C ∞ ( S, M ). Thegroup of symplectic diffeomorphisms, Diff(
M, ω ), acts on C ∞ ( S, M ) in a natural wayfrom the left, preserving the symplectic form (124). Moreover, the group of volumepreserving diffeomorphisms, Diff(
S, µ ), acts from the right by reparametrization, alsopreserving the symplectic form (124). Clearly, these two actions commute.Suppose the symplectic form on M is exact, that is ω = dθ for some 1-form θ on M .In this case the Diff( S, µ ) action on C ∞ ( S, M ) is Hamiltonian with equivariant momentmap. This moment map, denoted by J C ∞ ( S,M ) R , is given by the composition C ∞ ( S, M ) → Ω ( S ) µ = Ω ( S, | Λ | S ) ⊆ X ( S ) ∗ → X ( S, µ ) ∗ . Here the first arrow is given by pull back of θ ; the second identification is via the volumedensity µ ; the third is the inclusion of smooth sections into distributional sections of T ∗ S ⊗ | Λ | S ; and the fourth map is the dual of the canonical inclusion X ( S, µ ) ⊆ X ( S ).We will write this as J C ∞ ( S,M ) R ( φ ) = φ ∗ θ ⊗ µ or h J C ∞ ( S,M ) R ( φ ) , X i = Z S ( φ ∗ θ )( X ) µ, (125)where φ ∈ C ∞ ( S, M ) and X ∈ X ( S, µ ).Via the Lie algebra homomorphism C ∞ ( M ) → X ham ( M, ω ), the Poisson algebra C ∞ ( M ) acts on C ∞ ( S, M ) in a Hamiltonian fashion with infinitesimally equivariantmoment map J C ∞ ( S,M ) L : C ∞ ( S, M ) → C ∞ ( M ) ∗ given by J C ∞ ( S,M ) L ( φ ) := φ ∗ µ or h J C ∞ ( S,M ) L ( φ ) , h i = Z S ( φ ∗ h ) µ (126)where φ ∈ C ∞ ( S, M ) and h ∈ C ∞ ( M ). This moment map is in fact equivariant withrespect to the natural action of the full symplectic group, Diff( M, ω ). Of course this moment map is even Diff( M )-equivariant, but Diff( M ) does not act symplecticallyon C ∞ ( S, M ) nor does it act by Poisson maps on C ∞ ( M ). DUAL PAIR FOR THE CONTACT GROUP 41
Restricting the actions and moment maps to the open subset Emb(
S, M ) ⊆ C ∞ ( S, M )of embeddings, we obtain a symplectic dual pair, see [9] and [11, Section 4.2]: C ∞ ( M ) ∗ J Emb(
S,M ) L ←−−−−−−−−− Emb(
S, M ) J Emb(
S,M ) R −−−−−−−−−→ X ( S, µ ) ∗ (127)A.2. Comparison with the EPContact dual pair.
We will now specialize to thesymplectization of a contact manifold (
P, ξ ), that is, we consider M = L ∗ \ P equippedwith the symplectic form ω M obtained by restricting the canonical 2-form ω L ∗ on thetotal space of L ∗ , cf. Section 2.1. We will relate the dual pair for the Euler equation(127) with the EPContact dual pair constructed in Theorem 2.6.Recall the symplectic manifold M = C ∞ lin, inj ( | Λ | ∗ S , L ∗ ) in (48), with Hamiltonian ac-tions of the groups Diff( P, ξ ) and Diff( S ). The volume density µ on S provides anidentification ι µ : M → C ∞ ( S, M ) , ι µ (Φ) := Φ ◦ ˆ µ, (128)where ˆ µ ∈ Γ ∞ ( | Λ | ∗ S ) denotes the section dual to µ . Let j : X ( P, ξ ) → C ∞ ( M ), j ( X ) := h MX , denote the Lie algebra homomorphism provided by (8), see also (14). In view of(9), j is equivariant over the homomorphism Diff( P, ξ ) → Diff(
M, ω M ). Note that thecomposition of j with the action C ∞ ( M ) → X ham ( M, ω M ) yields a Lie algebra homo-morphism X ( P, ξ ) → X ham ( M, ω M ) ⊆ X ( M, ω M ) corresponding to the homomorphismof groups Diff( P, ξ ) → Diff(
M, ω M ), see (12). Finally, let i : X ( S, µ ) → X ( S ) denote thenatural inclusion. Clearly, i is equivariant over the inclusion Diff( S, µ ) ⊆ Diff( S ).These maps give rise to the following diagram: X ( P, ξ ) ∗ M J M L o o ∼ = ι µ (cid:15) (cid:15) J M R / / X ( S ) ∗ i ∗ (cid:15) (cid:15) C ∞ ( M ) ∗ j ∗ O O C ∞ ( S, M ) J C ∞ ( S,M ) L o o J C ∞ ( S,M ) R / / X ( S, µ ) ∗ (129)Here i ∗ and j ∗ denote the (equivariant) maps dual to the homomorphisms i and j ,respectively. Proposition A.1.
The diagram (129) commutes. The map ι µ in (128) is a symplecto-morphism which is equivariant over the inclusion Diff(
S, µ ) ⊆ Diff( S ) and equivariantover the homomorphism Diff(
P, ξ ) → Diff(
M, ω M ) . Moreover, ι µ ( E ) = { φ ∈ C ∞ ( S, M ) : π M ◦ φ ∈ Emb(
S, P ) } , where π M : M → P denotes the restriction of the canonical projection π L ∗ : L ∗ → P .Proof. Clearly, ι µ is an equivariant diffeomorphism, see Remark 2.2. It is symplectic inview of (43) and (124). The right hand side of the diagram commutes in view of (47)and (125). The left hand side of the diagram commutes in view of (46) and (126). (cid:3) The first line in (129) becomes a dual pair only when restricted to E , while the secondline needs to be restricted to Emb( S, M ) to become a dual pair. Note that the image ι µ ( E ) is an open subset (strict, in general) of Emb( S, M ). References [1] V. I. Arnold,
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E-mail address : [email protected] Cornelia Vizman, Department of Mathematics, West University of Timis¸oara, Bd.V.Pˆarvan 4, 300223-Timis¸oara, Romania.
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