Classification of momentum proper exact Hamiltonian group actions and the equivariant Eliashberg cotangent bundle conjecture

Let G be a compact and connected Lie group. The Hamiltonian G-model functor maps the category of symplectic representations of closed subgroups of G to the category of exact Hamiltonian G-actions. Based on previous joint work with Y. Karshon, the restriction of this functor to the momentum proper subcategory on either side induces a bijection between the sets of isomorphism classes. This classifies all momentum proper exact Hamiltonian G-actions (of arbitrary complexity). As an extreme case, we obtain a version of the Eliashberg cotangent bundle conjecture for transitive smooth actions. As another extreme case, the momentum proper Hamiltonian G-actions on contractible manifolds are exactly the symplectic G-representations, up to isomorphism.


The main result and applications
Let G be a compact and connected Lie group. We call a Hamiltonian G-action momentum proper iff every momentum map for the action is proper. The purpose of this article is to classify the momentum proper exact Hamiltonian G-actions in terms of the momentum proper symplectic representations of closed subgroups of G. To this end I provide a bijection between the sets of equivalence classes of such representations and of such Hamiltonian actions. (See Corollary 1.11 below.) The bijection is induced by the G-model functor. This is a functor between the category of symplectic representations of closed subgroups of G and the category of exact Hamiltonian G-actions.
• For every object (H, ρ) of SympRep ≤G we define Model G (H, ρ) = Y ρ , ω ρ , ψ ρ ) to be the centred Hamiltonian G-model action induced by (H, ρ). 7 This action is defined as follows. (For details see [KZ18,Section 3].) We define ψ D ρ to be the diagonal H-action on T * G × V induced by the right translation on G and by ρ. We denote by g, h the Lie algebras of G, H and by (1.4) ν ρ : V → h * the unique momentum map for ρ that vanishes at 0. 8 For a ∈ G and ϕ ∈ g * we denote by aϕ ∈ T * a G the image of ϕ under the derivative of the left translation by a. We define 9 (1.5) µ D H,ρ := µ D ρ : T * G × V → h * , µ D ρ a, aϕ, v := −ϕ|h + ν ρ (v). This is a momentum map for ψ D ρ . The pair (Y ρ , ω ρ ) is defined to be the symplectic quotient of ψ D ρ at 0 w.r.t. µ D ρ . This means that (1.6) Y H,ρ := Y ρ = (µ D ρ ) −1 (0)/ψ D ρ . (The subgroup H is compact, since it is closed and G is compact. Therefore, the restriction of ψ D ρ to (µ D ρ ) −1 (0) is proper. Since it is also free, the symplectic quotient is well-defined.) The left translation by G on G induces a G-action on T * G and hence on T * G × V . Since left and right translation commute, this action preserves (µ D ρ ) −1 (0) and descends to a G-action ψ ρ on Y ρ , the symplectic quotient of T * G × V by the diagonal H-action. 10 This defines Model G (H, ρ) = Y ρ , ω ρ , ψ ρ ).
• For every g ∈ G we denote by R g : G → G, R g (a) := ag, the right translation by g, and by R g * : T * G → T * G the induced map. The map Model G assigns to every morphism (g, T ) : (H, ρ) → (H ′ , ρ ′ ) of SympRep ≤G the morphism Model G (g, T ) of Ham ex G given by (1.7) Model G (g, T )(y) := R g −1 * (a, aϕ), T v , where (a, aϕ, v) is an arbitrary representative of y. (Here on the right hand side we denote by a ′ , a ′ ϕ ′ , v ′ the equivalence class of a ′ , a ′ ϕ ′ , v ′ .) The main result is the following. (As always, we assume that G is compact and connected.)

Theorem (Hamiltonian G-model functor). (i) (well-definedness on objects) The map
Model G is well-defined on objects, i.e., ψ ρ is indeed an exact Hamiltonian G-action. 7 In the article [KZ18] we used the notation (Y, ω Y ) instead of (Y ρ , ω ρ ). To make the dependence on ρ explicit, I am using (Y ρ , ω ρ ) here. To help the reader navigate through this article, I have included a list of symbols at the end of this article.
8 Viewing the symplectic vector space (V, σ) as a symplectic manifold, the representation ρ is a Hamiltonian G-action on V . Hence it admits a momentum map. In the article [KZ18] we used the notation µ V instead of ν ρ . To make the dependence on ρ explicit, I am using ν ρ here. 9 In the article [KZ18] we used the notation µ D for this map. For simplicity I have dropped the tilde here. 10 This can be seen as part of symplectic reduction in stages.
(ii) (well-definedness on morphisms) The map Model G is well-defined on morphisms, i.e., (1.9) , the right hand side of (1.7) does not depend on the choice of a representative (a, aϕ, v), and Model G (g, T ) is a morphism of Ham ex G . (iii) (functoriality) The map Model G is a covariant functor. (iv) (essential injectivity) The map between the sets of isomorphism classes induced by Model G is injective. (v) (morphisms) Let (H, ρ) and (H ′ , ρ ′ ) be objects of SympRep ≤G , and (g, T ), ( g, T ) be morphisms between these objects. Model G maps these morphisms to the same morphism if and only if  KZ18] with Y. Karshon into a categorical framework. Namely, part (viii) of this theorem (essential surjectivity of Model G ) was proved in [KZ18, 1.5. Theorem], without introducing the categorical setup used in the present article.
The other parts of Theorem 1.8 will be proved in the next section. The proof of (iv) (essential injectivity) is based on Lemma 2.19, which provides criteria under which the symplectic quotient representation of the model action Model G (H, ρ) at a given point is isomorphic to (H, ρ). We also use the fact that if two compact subgroups of a Lie group are conjugate to subgroups of each other then they are conjugate to each other. (This follows from Lemma 2.8 below.) Remark. Naively, in the definition of a morphism of SympRep ≤G , one could try to weaken the condition (1.1) to either the condition c g (H) ⊆ H ′ or c g (H) ⊇ H ′ . With this modification the model functor would no longer be well-defined on morphisms. ("⊇" is needed in order for (1.9) to hold and "⊆" is needed for the right hand side of (1.7) not to depend on the choice of a representative. See the proof of Theorem To see this, let Q be a connected compact manifold of positive dimension, without boundary. We define ω to be the canonical symplectic form on T * Q and ψ to be the trivial G-action on T * Q.
We claim that the isomorphism class of (T * Q, ω, ψ) does not lie in the image of Model G . To see this, assume that H, V, σ, ρ is an object of SympRep ≤G for which ψ ρ is trivial. Then H = G and therefore, Y ρ is canonically diffeomorphic to V . If Y ρ , ω ρ , ψ ρ is isomorphic to T * Q, ω, ψ then it follows that Q is a singleton. This proves the claim. • Many classification results are known for Hamiltonian group actions whose complexity is low. (By definition, the complexity is half the dimension of a generic non-empty reduced space. For references see [KZ18].) What makes Corollary 1.11 special is that it classifies Hamiltonian actions of arbitrary complexity.
By considering the extreme case of the full subgroup H = G, this corollary implies that the momentum proper Hamiltonian G-actions on contractible manifolds are exactly the momentum proper symplectic G-representations, up to isomorphism. See Corollary 1.17 below. On the other hand, by considering the extreme case in which the vector space V is trivial, using Corollary 1.11, we can classify the critical momentum proper exact Hamiltonian G-actions in terms of transitive G-actions on manifolds.
To explain the latter application, we call an object (M, ω, ψ) of Ham ex,prop For every manifold Q we denote by ω Q the canonical symplectic form on T * Q. We define the G-cotangent functor T * G to be the canonical functor from the category of G-actions on manifolds and G-equivariant diffeomorphisms to the category of Hamiltonian G-actions and G-equivariant symplectomorphisms. It takes an object (Q, θ) to (T * Q, ω Q ) together with lifted G-action θ * , and a morphism f : We define Act trans G to be the category whose objects are the transitive smooth G-actions on connected closed manifolds and whose morphisms are the G-equivariant diffeomorphisms. • By Corollary 1.14 the restriction of the functor T * G to the category Act trans G of transitive G-actions is essentially injective. This proves an equivariant version of the Eliashberg cotangent bundle conjecture. In fact, Corollary 1.14 provides more information, namely it also specifies the image of the class of objects of Act trans G under T * G , up to isomorphism.
• The philosophy behind this application is that symmetry makes problems more accessible. In the present situation it allows for a classification of the structures at hand (transitive G-actions and critical Hamiltonian G-actions). The same philosophy was for example used recently in [FPP18], where the authors used Delzant's classification of symplectic toric manifolds to prove that certain equivariant symplectic capacities are (dis-)continuous. (Without symmetry the question whether a given symplectic capacity is continuous is hard in general.) We will prove Corollary 1.14 in Section 3.
As another application of Corollary 1.11, we now classify the momentum proper Hamiltonian G-actions on contractible manifolds. Here we consider another extreme case, in which the subgroup H equals G. We denote by SympRep G the category whose objects are symplectic G-representations and whose morphisms are G-equivariant linear symplectic maps (possibly not surjective), and by SympRep prop Remarks.
• It follows from part (i) of this corollary and Remark 1.16(ii) that ι G,prop * is injective. Using (ii), this map is bijective.
• Part (ii) means that every momentum proper Hamiltonian G-action on a contractible symplectic manifold is symplectically linearizable 13 . • The statement of Corollary 1.17 means that the momentum proper Hamiltonian G-actions on contractible symplectic manifolds agree with the momentum proper symplectic G-representations, up to isomorphism. This classifies these actions. • Assume that G is non-abelian. In contrast with part (ii) the map ι G * is not surjective. This follows from [KZ18,Corollary 8.4].
For the proof of Corollary 1.17(ii) we need the following.
This follows from a straightforward argument.
Proof of Corollary 1.17. (i): Let R and R ′ be isomorphism classes of SympRep G that are mapped to the same class under ι G * . We choose representatives (V, σ, ρ), Since ρ is linear, the canonical identification between V and T 0 V is an isomorphism from ρ to dρ(0) in SympRep G . Similarly, ρ ′ is isomorphic to dρ ′ (Φ(0)). Combining these three isomorphisms, it follows that ρ and ρ ′ are isomorphic in SympRep G , i.e., R = R ′ . Hence the map ι G * is injective. This proves (i).

Remarks.
• (This remark will be used in the next one.) We define SympRep G to be the category with objects the symplectic G-representations and morphisms between ρ, ρ ′ given by pairs (g, T ), where g ∈ G and T : V → V ′ is a linear symplectic map, such that (1.2) holds. The composition is defined by (1.3). We define the functor i G : SympRep G → SympRep ≤G , i G (ρ) := (G, ρ), i G = identity on morphisms.
We may view SympRep G as a full subcategory of SympRep ≤G via this functor. We define the map F G : SympRep G → SympRep G , F G = identity on objects, F G (g, T ) := ρ ′ g −1 T. A straightforward argument shows that this map is a covariant functor.
• Part (i) of Corollary 1.17 can alternatively be deduced from Theorem 1.8(iv) as follows. Let R, R ′ be isomorphism classes of SympRep G that are mapped to the same class under ι G * . We choose representatives ρ, ρ ′ of R, R ′ . Then ι G (ρ) and ι G (ρ ′ ) are isomorphic. Using Remark 1.18, it follows that Model G •i G (ρ) and Model G •i G (ρ ′ ) are isomorphic. Hence by Theorem 1.8(iv) there exists an isomorphism (g, T ) in This shows that ι G * is injective, i.e., part (i) of Corollary 1.17. • A straightforward argument shows that the map I G : ρ → I ρ G is a natural isomorphism between the functors ι G • F G and Model G •i G , This means that for every morphism (g, commutes, and that I ρ G is an isomorphism for every object ρ of SympRep G . In other words the map Model G (g, T ) is given by

Proof of Theorem 1.8(i-vii) (Hamiltonian G-model functor)
For the proof of Theorem 1.8(i) we need the following. We denote by Ad and Ad * the adjoint and coadjoint representations of G. We define the map This is a momentum map for the lifted left-translation action of G on T * G. We denote by pr 1 : T * G × V → T * G the canonical projection. Since left and right translations commute, µ L is preserved by the lifted right translation action of H on T * G. Hence the map µ L • pr 1 descends to a map 14 µ ρ : Y ρ → g * .
Proof of Theorem 1.8(i). The map µ ρ is a momentum map for ψ ρ . Hence ψ ρ is a Hamiltonian action, and therefore Model G is well-defined on objects, as claimed.
For the proof of Theorem 1.8(ii) we need the following.
2.1. Remark (product of proper maps). Let X, Y, X ′ , Y ′ be topological spaces, with Y and Y ′ Hausdorff. Let f : X → Y and f ′ : X ′ → Y ′ be proper continuous maps. Then the Cartesian product map f × f ′ : X × X ′ → Y × Y ′ is proper. This follows from an elementary argument. (Hausdorffness ensures that every compact subset of Y × Y ′ is closed.) Proof of Theorem 1.8(ii). Let H, V, σ, ρ and H ′ , V ′ , σ ′ , ρ ′ be objects of SympRep ≤G and (g, T ) a morphism between them. We denote by h and h ′ the Lie algebras of H and H ′ . By (1.1) we have c g −1 (H ′ ) = H. It follows that Ad g −1 (h ′ ) = h. Hence Ad * (g) = Ad * g −1 induces a map from h * to h ′ * , which we again denote by Ad * (g). We have The map ρ ′ • c g : H → isomorphisms of (V ′ , σ ′ ) is a Hamiltonian action with momentum map where ν ρ ′ is as in (1.4). By (1.2) ρ ′ leaves the image T (V ) invariant and T is a symplectic embedding that is equivariant w.r.t. ρ and ρ ′ • c g . It follows that , aϕ, v). The claimed inclusion (1.9) follows. We define It follows that the right hand side of (1.7) does not depend on the choice of the representative (a, aϕ, v), as claimed. We denote by Φ : the map induced by Φ. We show that Φ is a morphism of Ham ex G . The map Φ is smooth, presymplectic, and equivariant w.r.t. the G-actions induced by the left translations on G. It follows that Φ is smooth, symplectic, and equivariant w.r.t. to the G-actions ψ ρ and ψ ρ ′ .
2.4. Claim. The maps T and Φ are proper.
Proof of Claim 2.4. The map T : V → V ′ is linear symplectic and hence injective. Since V is finite-dimensional, it follows that where · , · ′ are arbitrary norms on V, V ′ . This implies that T is proper, as claimed.
We denote by : T * G → T * G is proper, since it is invertible with continuous inverse. Using Remark 2.1 and properness of T , it follows that the Cartesian product map R g −1 * × T : Since this map restricts to Φ on (µ D ρ ) −1 (0), it follows that Φ is proper. Since π −1 ρ ′ (K ′ ) is compact, it follows that the right hand side of (2.6) is compact, hence also the left hand side. Since π ρ maps this set to Φ −1 (K ′ ), it follows that Φ −1 (K ′ ) is compact. This proves Claim 2.4. Using Claim 2.4, it follows that Φ is a G-equivariant proper symplectic embedding, i.e., a morphism of Ham ex G . This proves that the map Model G is well-defined on morphisms. This completes the proof of Theorem 1.8(ii).
Proof of Theorem 1.8(iii). It follows from a straightforward argument that Model G maps the unit morphisms to unit morphisms and intertwines the compositions. Hence it is a covariant functor. This proves Theorem 1.8(iii).
For the proof of Theorem 1.8(iv) we need the following. Let G be a group, X a set, ψ an action of G on X, and x ∈ X. We denote by 2.7. Remark. Let G be a Lie group, (ρ, H) an object of SympRep ≤G , and y = [a, aϕ, v] ∈ Y ρ . Then 2.8. Lemma. Let G be a topological (finite-dimensional) manifold with a continuous group structure, N, N ′ compact submanifolds of G, and g ∈ G, such that and N ′ is conjugate to some subset of N. Then we have In the proof of this lemma we will use the following. Proof of Lemma 2.8. We choose g ′ ∈ G, such that and define ψ := c g ′ g : G → G. We have is well-defined. This map is injective. Since N is compact, the number of its connected components is finite. It follows that the map (2.12) is surjective. It follows that N ⊆ ψ(N), and therefore, c −1 g ′ (N) ⊆ c g (N). By (2.11) we have N ′ ⊆ c −1 g ′ (N). It follows that N ′ ⊆ c g (N). Combining this with (2.9), it follows that c g (N) = N ′ . This proves Lemma 2.8.
Let G be a Lie group, (M, ω, ψ) a symplectic G-action, and x ∈ M.
Remark. The isotropy representation of ψ at x is by definition the map v. This is a symplectic representation of the isotropy group Stab ψ x . In order to define the symplectic quotient representation of ψ at x, we need the following remarks.
2.13. Remarks (symplectic quotient representation). (i) Let G be a Lie group, (M, ψ) a G-action on a manifold, and x ∈ M. We denote by g the Lie algebra of G and by (2.14) L x := L ψ x : g → T x M the infinitesimal action at x. The equality holds.
(ii) Let (V, σ) be a symplectic vector space and W ⊆ V a linear space. We denote by the symplectic complement of W . Let (M, ω, ψ) be a symplectic G-action and x ∈ M. The form ω x induces a linear symplectic form ω x on the quotient space . This map is a linear symplectic isomorphism w.r.t. ω x and ω ψg(x) .
(i) Since Φ is G-equivariant and injective, we have This map is a linear symplectic isomorphism that intertwines ρ ψ,x and ρ ψ ′ ,x ′ .

Lemma
Let y ∈ Y ρ be a point for which µ ρ (y) is central and Stab ψρ y = c a (H), for some representative (a, aϕ, v) of y. Then ρ = (H, ρ) is isomorphic to ρ ψρ,y .
Remark. The subgroup c a (H) does not depend on the choice of the representative (a, aϕ, v) of y.
In the proof of this lemma we will use the following.
2.23. Claim. The pair a, A ρ y is a morphism from ρ to ρ ψρ,y (the isotropy representation of ψ ρ at y).
Proof of Claim 2.23. The map ι a,ϕ is a symplectic embedding. It follows that A ρ y is linear symplectic. We denote by ψ L : G × T * G × V → T * G × V the action induced by the left-translation on G. Let h ∈ H. For all w ∈ V , we have ι a,ϕ • ρ h (w) = a, aϕ, ρ h w = ah, aϕh, w Using that ρ h is linear, it follows that The statement of Claim 2.23 follows.
Let y ∈ Y ρ . Recall that L y = L ψρ y : g → T y Y ρ denotes the infinitesimal ψ ρ -action.
Here in the last step we used that the map pr 1 •ι a,ϕ is constantly equal to (a, aϕ). It follows that imA ρ y ⊆ ker dµ ρ (y). Using (2.27), the claimed inclusion (2.26) follows. This completes the proof of Claim 2.24.
By part (2.25) of this claim there is a canonical projection pr ρ y : imL y (ωρ)y → V ψρ y = imL y (ωρ)y /imL y .
By part (2.26) the restriction pr ρ y imA ρ y is well-defined. It follows from Claim 2.23 and the equality Stab ψρ y = c a (H) that imA ρ y is invariant under ρ ψρ,y .
2.28. Claim. The pair e, pr ρ y imA ρ y is an isomorphism between the restriction of ρ ψρ,y to imA ρ y and ρ ψρ,y . Proof of Claim 2.28. The projection pr ρ y is presymplectic. Since imA ρ y is symplectic, the restriction pr ρ y imA ρ y is linear symplectic and therefore injective. We have y is linear symplectic, hence injective) = dim pr ρ y imA ρ y (since pr ρ y imA ρ y is injective).
It follows that V ψρ y = pr ρ y imA ρ y , hence pr ρ y imA ρ y is surjective. Hence this map is a linear symplectic isomorphism. It is Stab ψρ y -equivariant. The statement of Claim 2.28 follows. It follows from Claims 2.23 and 2.28 that ρ and ρ ψρ,y are isomorphic. This proves Lemma 2.19.
Proof of Claim 2.32. By (2.30,2.29,2.31) we have Stab . By Remark 2.18(ii) the map µ ρ ′ • Φ is a momentum map for ψ ρ . Since G is connected, the same holds for Y ρ . It follows that µ ρ ′ • Φ − µ ρ is constantly equal to a central element of g * . At y this map attains the value µ ρ ′ (y ′ ) − µ ρ (y) = µ ρ ′ (y ′ ) − 0, which is thus a central element of g * . Hence the hypotheses of Lemma 2.19 are satisfied. Applying this lemma, the statement of Claim 2.32 follows.
Combining this claim with what we already showed, it follows that ρ is isomorphic to ρ ′ . Hence Model G induces an injective map between the sets of isomorphism classes. This proves Theorem 1.8(iv).
Proof of Theorem 1.8(v,vi,vii). (v) follows from a straightforward argument.
(vi): Let ρ and ρ ′ be objects of SympRep ≤G , such that ρ ′ is momentum proper and there exists a morphism (g, T ) from ρ to ρ ′ . Let Q ⊆ h be compact. Equality (2.3) implies that ) . The set Ad * (g)(Q) is compact. By hypothesis ν ρ ′ is proper, and by Claim 2.4 the same holds for T . It follows that ν ρ ′ • T is proper, and therefore, using (2.33), the set ν −1 ρ (Q) is compact. Hence ν ρ is proper, i.e., ρ is momentum proper, as claimed.
Let now (M, ω, ψ) and (M ′ , ω ′ , ψ ′ ) be objects of Ham ex G , such that ψ ′ is momentum proper and there exists a morphism Φ from ψ to ψ ′ . We choose a momentum map µ ′ for ψ ′ . By definition, Φ is a proper G-equivariant symplectic embedding. It follows that µ ′ • Φ is a proper momentum map for ψ. Hence ψ is momentum proper. This proves (vi).
"⇐": Assume that µ ρ is proper. Let Q ⊆ h * be compact. We choose a compact set K ⊆ g * such that i * (K) = Q. (We may e.g. choose a linear complement W ⊆ g * of ker i * and define Stab ψρ y ⊆ c a (H). Since y is ψ ρ -maximal, Stab ψρ y contains some conjugate of Stab ψρ [e,0,0] = H. Using Lemma 2.8, it follows that Stab ψρ y = c a (H). Using that y is ψ ρ -central, the hypotheses of Lemma 2.19 are therefore satisfied. Applying this lemma, it follows that ρ and ρ ψρ,y are isomorphic, as claimed.
The general situation can be reduced to this case, by using Theorem 1.8(viii) (essential surjectivity), the fact that stabilizers are preserved under equivariant injections, and Remark 2.18(ii). This proves (i).
(ii): Consider first the case in which there exists an isomorphism from ψ to ψ ′ that maps x to x ′ . Then it follows from Remark 2.18(i) that ρ ψ,x and ρ ψ ′ ,x ′ are isomorphic.
(iii): Remark 4.3 implies that (4.2) is a left-inverse for (1.12). Since (1.12) is surjective, it follows that (4.2) is also a right-inverse. This proves (iii) and completes the proof of Proposition 4.1.

Acknowledgments
I would like to thank Yael Karshon for some useful comments and the anonymous referee for valuable suggestions.