Gromov width of non-regular coadjoint orbits of U(n), SO(2n) and SO(2n+1)
GGROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ) , SO (2 n ) AND SO (2 n + 1) . MILENA PABINIAK
Abstract.
Let G be a compact connected Lie group G and T its maximal torus. The coadjointorbit O λ through λ ∈ t ∗ is canonically a symplectic manifold. Therefore we can ask the questionabout its Gromov width. In many known cases the Gromov width is exactly the minimum overthe set {(cid:104) α ∨ j , λ (cid:105) ; α ∨ j a coroot, (cid:104) α ∨ j , λ (cid:105) > } . We show that the Gromov width of coadjoint orbits ofthe unitary group and of most of the coadjoint orbits of the special orthogonal group is at least theabove minimum. The proof uses the torus action coming from the Gelfand-Tsetlin system. Contents
1. Introduction 11.1. Reformulation of the main result for the unitary group. 31.2. Reformulation of the main result for the special orthogonal group. 51.3. Organization of the paper and acknowledgements. 72. Technical ingredients. 73. Gelfand-Tsetlin torus action. 114. Lower bounds for Gromov width of U ( n ) coadjoint orbits. 125. Lower bounds for Gromov width of SO (2 n ) and SO (2 n + 1) coadjoint orbits. 16References 221. Introduction
Let (
M, ω ) be a symplectic manifold. Non-degeneracy of ω implies that every symplectomorphismis a volume preserving transformation. However, Gromov’s Non-squeezing theorem proves that a The author was supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT, Portugal) grantSFRH/BPD/87791/2012 during the final stage of this research. a r X i v : . [ m a t h . S G ] F e b MILENA PABINIAK group of symplectomorphisms is a proper subset of the group of volume preserving transformations.The theorem says that a ball B N ( r ) of radius r , in a symplectic vector space R N with the usualsymplectic structure, can be symplectically embedded into B ( R ) × R N − only if r ≤ R . Thismotivated the definition of the invariant called Gromov width.
Consider a ball of capacity aB Na = (cid:110) z ∈ C N (cid:12)(cid:12)(cid:12) π N (cid:88) i =1 | z i | < a (cid:111) , with the standard symplectic form ω std = (cid:80) dx j ∧ dy j . The Gromov width of a 2 N -dimensionalsymplectic manifold ( M, ω ) is the supremum of the set of a ’s such that B Na can be symplecticallyembedded in ( M, ω ).In this work we focus on the Gromov width of coadjoint orbits of Lie groups. A Lie group G actson itself by conjugation G (cid:51) g : G → G, g ( h ) = ghg − . Derivative of the above map taken at the identity element gives the action of G on its Lie algebra g ,called the adjoint action. This induces the action of G on g ∗ , the dual of its Lie algebra, called thecoadjoint action. Each orbit O of the coadjoint action is naturally equipped with the Kostant-Kirillovsymplectic form: ω ξ ( X, Y ) = (cid:104) ξ, [ X, Y ] (cid:105) , ξ ∈ g ∗ , X, Y ∈ g . For example, when G = U ( n ) the group of (complex) unitary matrices, a coadjoint orbit can beidentified with the set of Hermitian matrices with a fixed set of eigenvalues. With this identification,the coadjoint action of G on an orbit O is simply the action by conjugation. It is Hamiltonian, andthe momentum map is just inclusion O (cid:44) → g ∗ . We recall the notions of Hamiltonian actions andmomentum maps in Section 2.Choose a maximal torus T ⊂ G and a positive Weyl chamber t ∗ + . Every coadjoint orbit intersectsthe positive Weyl chamber in a single point. Therefore there is a bijection between the coadjointorbits and points in the positive Weyl chamber. Points in the interior of the positive Weyl chamberare called regular points. The main result of this paper describes a lower bound for Gromov widthof the coadjoint orbits of the unitary group. Theorem 1.1.
Let M := O λ be the coadjoint orbit of G = U ( n ) through a point λ ∈ ( t G ) ∗ + (regularor not) or of G = SO (2 n + 1) , SO (2 n ) through a point λ ∈ ( t G ) ∗ + satisfying condition ( ∗ ) stated below. ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 3 The Gromov width of M is at least r G ( λ ) := min { (cid:104) α ∨ , λ (cid:105) ; α ∨ a coroot and (cid:104) α ∨ , λ (cid:105) > } . To state the condition ( ∗ ) we need to review the root system of the special orthogonal groups andfix the notation. Therefore we delay the explanation of ( ∗ ) till Theorem 1.7. Here we only note thatall regular orbits satisfy condition ( ∗ ).This particular lower bound is important because in many known cases it describes the Gromovwidth, not only its lower bound. Karshon and Tolman in [KT05] showed that the Gromov width ofcomplex Grassmannians is given by the above formula. Zoghi in [Zog10] analyzed orbits satisfyingsome additional integrality conditions. He called an orbit O λ indecomposable if there exists a simpleroot α such that for each root α (cid:48) there exists a positive integer k (depending on α (cid:48) ) such that k (cid:104) α ∨ , λ (cid:105) = (cid:104) ( α (cid:48) ) ∨ , λ (cid:105) . Theorem 1.2. [Zog10, Proposition 3.16]
For compact connected simple Lie group G the formula min { |(cid:104) α ∨ , λ (cid:105)| ; α ∨ a coroot } gives an upper bound for Gromov width of regular indecomposable G -coadjoint orbit through λ . Combinining these results we obtain
Theorem 1.3.
The Gromov width of a regular indecomposable U ( n ) - or SO ( n ) -coadjoint orbit through λ is exactly min { |(cid:104) α ∨ , λ (cid:105)| ; α ∨ a coroot } . The result of Zoghi was recently extended by Caviedes in [Cas13] to some non-regular U ( n ) orbits.We quote his result in the U ( n ) subsection below.1.1. Reformulation of the main result for the unitary group.
Choose as the maximal torus T of U ( n ) a the subgroup of diagonal matrices. We use the following indentifications:- the exponential map exp : Lie ( S ) → S is given by t → e πit ,- u ( n ) is identified with the set of n × n Hermitian matrices,- the pairing in u ( n ), ( A, B ) = trace( AB ) gives us the identification of u ∗ ( n ) with u ( n ),- t ∗ and t are identified with diagonal Hermitian matrices, and then with R n (by mapping a diagonal MILENA PABINIAK matrix to its diagonal entries);Kernel of the exponential maps forms a lattice in t and thus induces a lattice in t ∗ . Choose thefollowing chamber ( t ∗ ) + := { ( λ , λ , . . . , λ nn ); λ ≥ λ ≥ . . . ≥ λ nn } to be the positive Weyl chamber. Fix any λ ∈ ( t ∗ ) + , regular or not and denote by O λ the U ( n )-coadjoint orbit through λ . Recall that the root system of U ( n ) consists of vectors ± ( e j − e k ), j (cid:54) = k ,of lattice length 2. The pairing of λ with a coroot ( e j − e k ) ∨ gives (cid:104) ( e j − e k ) ∨ , λ (cid:105) = 2 (cid:104) e j − e k , λ (cid:105)(cid:104) e j − e k , e j − e k (cid:105) = ( λ j − λ k ) . Note that the real dimension of a U ( n ) coadjoint orbit O λ through a point λ with λ = . . . = λ n >λ n +1 = . . . = λ n > . . . > λ n m +1 = . . . = λ n isdim R ( O λ ) = n ( n − − ( k ( k −
1) + . . . + k m +1 ( k m +1 −
1) )where k j = n j +1 − n j , n m +1 = n .Now we restate the main theorem for the unitary group in more explicit form. Theorem 1.4.
Let λ = ( λ , . . . , λ n ) ∈ ( t ∗ ) + and let m, n , . . . , n m be integers such that λ = . . . = λ n > λ n +1 = . . . = λ n > . . . > λ n m +1 = . . . = λ n . The Gromov width of O λ , U ( n ) coadjoint orbit through λ , is at least r U ( n ) ( λ ) := min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n m − λ n m +1 } . Caviedes in [Cas13] proved the following result.
Theorem 1.5. [Cas13, Theorem 5.4]
Let λ = ( λ , . . . , λ n ) ∈ ( t ∗ ) + and suppose that there are indicies i, j ∈ { , . . . , n } such that for any i (cid:48) , j (cid:48) ∈ { , . . . , n } the difference λ i (cid:48) − λ j (cid:48) is an integer multiple of λ i − λ j . Then the Gromov width of O λ is at most | λ i − λ j | . Note that in that case we have r U ( n ) ( λ ) = | λ i − λ j | . Combining these results together we cancalculate the actual Gromov width.
ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 5 Theorem 1.6.
Let λ = ( λ , . . . , λ n ) ∈ ( t ∗ ) + and suppose that there are indicies i, j ∈ { , . . . , n } suchthat for any i (cid:48) , j (cid:48) ∈ { , . . . , n } the difference λ i (cid:48) − λ j (cid:48) is an integer multiple of λ i − λ j . Then theGromov width of O λ is exactly | λ i − λ j | . Reformulation of the main result for the special orthogonal group.
We identify the Liealgebra so ( m ), and its dual so ( m ) ∗ with the vector space of skew symmetric matrices of appropriatesize. Throughout the paper we use the notation R ( α ) = cos( α ) − sin( α )sin( α ) cos( α ) , L ( a ) = − aa and make the following choices of maximal tori T SO (2 n +1) = R ( α ) R ( α ) ... R ( α n ) 1 , T SO (2 n ) = R ( α ) R ( α ) ... R ( α n ) where α j ∈ S . The corresponding Lie algebra duals are t ∗ SO (2 n +1) = L ( a ) L ( a ) ... L ( a n ) 0 , t ∗ SO (2 n ) = L ( a ) L ( a ) ... L ( a n ) . We identify these duals with R n and denote their elements simply by ( a , a , . . . , a n ) whenever it isclear from the context whether we are in SO (2 n +1) or SO (2 n ) case. We are using the convention thatthe exponential map exp : t SO (2) → T SO (2) is given by L ( a ) → R (2 πa ) , that is S ∼ = R / Z . Moreoverwe choose the positive Weyl chambers to consist of matrices with a ≥ a ≥ a ≥ . . . ≥ a n ≥ G = SO (2 n + 1), and a ≥ a ≥ a ≥ . . . ≥ a n − ≥ | a n | in the case G = SO (2 n ).Note that the real dimension of the coadjoint orbit through a point λ = ( λ , . . . , λ n ) with λ = . . . = λ n > λ n +1 = . . . = λ n > . . . > λ n m +1 = . . . = λ n , MILENA PABINIAK and λ n (cid:54) = 0 is equal todim O λ = n − ( k ( k −
1) + . . . + k m +1 ( k m +1 −
1) ) if G = SO (2 n + 1) , n ( n − − ( k ( k −
1) + . . . + k m +1 ( k m +1 −
1) ) if G = SO (2 n ) , where k j = n j +1 − n j , n m +1 = n . If λ n = 0 one needs to subtract k m +1 ( k m +1 + 1) in the SO (2 n + 1)case, or subtract k m +1 ( k m +1 −
1) in the SO (2 n ) case.The root system of the group SO (2 n + 1) consists of vectors ± e j , j = 1 , . . . n , of squared length1, and of vectors ± ( e j ± e k ), j (cid:54) = k , of squared length 2 in the Lie algebra dual t ∗ SO (2 n +1) ∼ = R n .Therefore this root system for SO (2 n + 1) is non-simply laced. Note that (cid:104) ( e j ± e k ) ∨ , λ (cid:105) = 2 (cid:104) e j ± e k , λ (cid:105)(cid:104) e j ± e k , e j ± e k (cid:105) = λ j ± λ k and (cid:104) ( e j ) ∨ , λ (cid:105) = 2 (cid:104) e j , λ (cid:105)(cid:104) e j , e j (cid:105) = 2 λ j . The root system for SO (2 n ) is simply laced and consists of vectors ± ( e j ± e k ), j (cid:54) = k , of squaredlength 2. Note that (cid:104) ( e j ± e k ) ∨ , λ (cid:105) = 2 (cid:104) e j ± e k , λ (cid:105)(cid:104) e j ± e k , e j ± e k (cid:105) = λ j ± λ k . Recall that r = r G ( λ ) = min {(cid:104) α ∨ j , λ (cid:105) ; α ∨ j a coroot, (cid:104) α ∨ j , λ (cid:105) > } . Using the above analysis of theroot systems we can calculate that for λ in the positive Weyl chamber r SO (2 n +1) ( λ ) = min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n m − λ n , λ n } if λ n (cid:54) = 0 , min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n m } if λ n = 0 r SO (2 n ) ( λ ) = min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n m − λ n m +1 , λ n m + λ n m +1 } . Now we are ready to state the main theorem about the Gromov width of coadjoint orbits of thespecial orthogonal group.
Theorem 1.7.
Consider the coadjoint orbit O λ of the special orthogonal group passing through apoint λ = ( λ , . . . , λ n ) ∈ t ∗ + in the positive Weyl chamber (chosen above) λ = . . . = λ n > λ n +1 = . . . = λ n > . . . > λ n m +1 = . . . = λ n , satisfying a condition ( ∗ ) ( λ n (cid:54) = λ n − ) ∨ ( λ n = 0) ∨ ( λ n ≥ r G ( λ )) . ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 7 The Gromov width of O λ is at least r G ( λ ) , that is, r SO (2 n +1) ( λ ) = min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n − − λ n , λ n } if λ n (cid:54) = 0 , min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n − } if λ n = 0 if G = SO (2 n + 1) , and r SO (2 n ) ( λ ) = min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n m − λ n m +1 , λ n m + λ n m +1 } . if G = SO (2 n ) . For orbits O λ with λ that do not satisfy the condition ( ∗ ) their Gromov width is atleast λ n = min { r G ( λ ) , λ n } . Organization of the paper and acknowledgements.
Section 2 provides background aboutHamiltonian actions and technical ingredients while Section 3 briefly reviews the Gelfand-Tsetlinaction in the general setting. The main result is proved separately for the unitary group (Section 4)and for the special orthogonal groups (Section 5).The author is very grateful to Yael Karshon for suggesting this problem and helpful conversationsduring my work on this project. The author also would like to thank Tara Holm and AlexanderCaviedes Castro for useful discussions.2.
Technical ingredients.
We obtain technical ingredients needed here by generalizing a result of Lisa Traynor [Tra95, Propo-sition 5.2] to Proposition 2.1 and applying the “classification” result of Karshon and Lerman ([KL10]).While editing this manuscript we found out that equivalent to Proposition 2.1 was already provedin [Sch99, Lemma 3.11]. It was used by Guangcun Lu in [Lu06] to prove one of the claims in hisProposition 1.3. This claim is almost equivalent to our Proposition 2.5 (note that Lu uses a differentnormalization convention). We are still presenting our proof here for completeness, and to show therelation with the result of Lisa Traynor. However we point out that the independent work of Guang-cun Lu, [Lu06], was published before our work, and we encourage the reader to consult this reference.The reader familiar with Proposition 1.3 in [Lu06] may go directly to Section 3.Let 2 N be the dimension of O λ . We start with generalizing a result by Lisa Traynor [Tra95,Proposition 5.2]. Define the following subsets of R N (cid:3) N ( π ) := { < x , . . . , x N < π } , MILENA PABINIAK (cid:52) N ( r ) := { < y , . . . , y N ; y + . . . + y N < r } . Equip their product (cid:3) N ( π ) × (cid:52) N ( r ) with the symplectic form induced from the standard symplecticstructure on R N , namely (cid:80) dx j ∧ dy j . When the dimension N is understood, we simply write (cid:3) ( π )and (cid:52) ( r ). Throughout the paper we use B Kπ r = B K ( r ) = { x ∈ R K ; | x | < r } to denote an open K -dimensional ball of radius √ r , i.e. of capacity π r . (Note that in [Tra95] B denotes closed balls). Proposition 2.1.
For any ρ < r there is a symplectic embedding of N -dimensional ball B N ( ρ ) ofradius √ ρ (i.e. of capacity πρ ) into (cid:3) N ( π ) × (cid:52) N ( r ) . (Both sets are considered as subsets of R N with the standard symplectic form (cid:80) dx j ∧ dy j .)Proof. There is a symplectic embedding Ψ : (cid:3) N ( π ) × (cid:52) N ( r ) → B N ( r ) into the open ball of radius √ r Ψ( x , y , . . . , x N , y N ) = ( √ y cos(2 x ) , −√ y sin(2 x ) , . . . , √ y N cos(2 x N ) , −√ y N sin(2 x N )) . Let SD ( r ) ⊂ B ( r ) be the slit disc radius √ r : SD ( r ) := B ( r ) \ { x ≥ , y = 0 } ⊂ R . Denote by SD N ( r ) the corresponding slit polidisc, SD N ( r ) := SD ( r ) × . . . × SD ( r ) ∈ R N . It is easyto see that Ψ( (cid:3) ( π ) × (cid:52) ( r )) = B N ( r ) ∩ SD N ( r ) . Fix any ρ < r and choose any area preserving diffeomorphism (so also preserving symplectic form) σ ρ : B ( ρ ) → Im σ ρ ⊂ SD ( ρ + 1 N ( r − ρ )) ⊂ SD ( r )such that if x + y ≤ a then | σ ρ ( x, y ) | ≤ a + N ( r − ρ ). Let Ψ ρ be the “product” of N σ ρ ’s:Ψ ρ : B ( ρ ) × . . . × B ( ρ ) → SD N ( ρ + 1 N ( r − ρ )) , Ψ ρ ( x , y , . . . , x N , y N ) = ( σ ρ ( x , y ) , . . . , σ ρ ( x N , y N )) . The map Ψ ρ is symplectic as a product of symplectic maps. Furthermore,Ψ ρ ( B N ( ρ )) ⊂ B N ( r ) ∩ SD N ( r ) = Ψ( (cid:3) ( π ) × (cid:52) ( r )) , ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 9 because if (cid:80) ( x i + y i ) < ρ then (cid:80) | σ ρ ( x i , y i ) | < ρ + N N ( r − ρ ) = r . Therefore Ψ − ◦ Ψ ρ givessymplectic embedding of B N ( ρ ) into (cid:3) ( π ) × (cid:52) ( r ). (cid:3) Corollary 2.2.
If there is a symplectic embedding (cid:3) ( π ) × (cid:52) ( r ) (cid:44) → ( M, ω ) , then the Gromov width of M is at least πr , because for any ρ < r we have a symplectic embedding B πρ (cid:44) → M . Therefore to find lower bounds for Gromov width, instead of looking for embeddings of symplecticballs, we can look for embeddings of (cid:3) ( π ) × (cid:52) ( r ). This is exactly how we will proceed. First wereview some properties of momentum maps.An effective action of a torus T on a symplectic manifold ( M, ω M ) is called a Hamiltonian action if there exists a T -invariant map Φ : M → t ∗ , called the momentum map (or moment map), suchthat(1) ι ( ξ M ) ω = d (cid:104) Φ , ξ (cid:105) ∀ ξ ∈ t , where ξ M is the vector field on M generated by ξ ∈ t . Then M is referred to as a Hamiltonian T manifold . The spaces t ∗ , t and R dim T are isomorphic though not canonically. Once a specificisomorphism is chosen one can view a momentum map as a map to R dim T . Throughout the paper weidentify Lie ( S ) with R using the convention that the exponential map exp : Lie ( S ) → R is given by t → e πit ( so S ∼ = R / Z ).If dim T = dim M the Hamiltonian action is called toric. We call M a proper Hamiltonian T space if there exists an open and convex subset T ⊂ t ∗ , containing Φ( M ) and such that the momentmap Φ : M → T is proper as a map to T . In particular if M is compact, then it is also proper. Example 2.3.
Consider the T action on C by( e it , e it )˙( z , z ) = ( e it z , e it z ) . Then Φ : C : → R given by Φ( z , z ) = ( − π | z | , − π | z | )is a momentum map. The image of the momentum map is the (closed) third orthant and the imageof a ball of capacity a (so of radius (cid:112) aπ ) is presented in the figure below. − a − a Compact connected toric manifolds are classified by their momentum map image (Delzant The-orem). Karshon and Lerman generalized this theorem to the case of non compact manifolds withproper momentum map ([KL10]). Using their work we conclude the following proposition.
Proposition 2.4.
For a connected, proper Hamiltonian T N space ( M N , ω M ) , with a momentummap Φ : M → t ∗ , and for a subset S ⊂ int Φ( M ) , S = W ( (cid:52) N ( r )) for some W ∈ GL ( n, Z ) , we havethat (Φ − ( S ) , ω M ) is symplectomorphic to T N × S with the symplectic form ω given by ω ( p,q ) ( v, ξ ) = ξ ( v ) = − ω ( p,q ) ( ξ, v ) , ω ( p,q ) ( v, v (cid:48) ) = 0 = ω ( p,q ) ( ξ, ξ (cid:48) ) , for any ( p, q ) ∈ T N × S , ξ, ξ (cid:48) ∈ T q S = t ∗ , and any v, v (cid:48) ∈ T p T N = t .Proof. The space T N × S with the above symplectic form and the T N action is on the first factorvia the group multiplication is a connected, proper Hamiltonian T N manifold with momentum mapimage S (contractible). The space (Φ − ( S ) , ω M ) is also a connected, proper Hamiltonian T N manifold.According to Proposition 3.9 in [KL10] such manifolds are classified by their momentum map image.Therefore T N × S and Φ − ( S ) are symplectomorphic. Note that the above theorem is true also formore general S , however for our purposes it is enough to consider only S = W ( (cid:52) N ( r )). (cid:3) Recall that we work with the convention S = R / Z . Therefore there exists a symplectic embedding( (0 , N × S, (cid:88) dx j ∧ dy j ) (cid:44) → ( T N × S, (cid:88) dx j ∧ dy j ) ∼ = (Φ − ( S ) , ω M ) . Proposition 2.5. If W ( (cid:52) ( r )) ⊂ int Φ( M ) , for some W with ± W ∈ SL ( N ; Z ) , then for any ρ < r aball B Nρ = B N ( ρ/π ) of capacity ρ embeds symplectically into ( M, ω M ) , and thus the Gromov widthof M is at least r .Proof. First suppose that W ∈ SL ( N ; Z ). According to Proposition 2.4,Φ − ( W ( (cid:52) N ( r )) ) ∼ = ( T N × W ( (cid:52) N ( r )) , ω ) ⊃ ((0 , N × W ( (cid:52) N ( r )) , (cid:88) dx j ∧ dy j ) . ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 11 Notice that the map(
Id, W ) : ((0 , N × (cid:52) N ( r ) , (cid:88) dx j ∧ dy j ) → ((0 , N × W ( (cid:52) N ( r )) , (cid:88) dx j ∧ dy j )is a symplectomophism because det( Id, W ) = det W = 1 . Also (cid:16) (0 , N × (cid:52) ( r ) , (cid:88) dx j ∧ dy j (cid:17) ∼ = (cid:16) (0 , π ) N × (cid:52) ( r/π ) , (cid:88) dx j ∧ dy j (cid:17) are symplectomorphic via ( x, y ) → ( πx, y/π ). Therefore (cid:0) (0 , π ) N × (cid:52) ( r/π ) , (cid:80) dx j ∧ dy j (cid:1) can besymplectically embedded into (cid:0) Φ − ( W ( (cid:52) ( r ))) , ω M (cid:1) . Together with Proposition 2.1 this gives thatfor any ρ < r , a ball of capacity ρ , ( B N ( ρ/π ) , (cid:80) dx j ∧ dy j ), can be symplectically embedded into M .If − W ∈ SL ( N ; Z ) then we obtain a symplectic embedding of ( B N ( ρ/π ) , − (cid:80) dx j ∧ dy j ) into M , but( B N ( ρ/π ) , − (cid:80) dx j ∧ dy j ) and ( B N ( ρ/π ) , (cid:80) dx j ∧ dy j ) are symplectomorphic. (cid:3) Gelfand-Tsetlin torus action.
In this Subsection we describe the Gelfand-Tsetlin (sometimes spelled Gelfand-Cetlin, or Gelfand-Zetlin) system of action coordinates, which originally appeared in [GS83]. It is related to the classicalGelfand-Tsetlin polytope introduced in [GT50]. Here we only briefly recall necessary facts about thisaction and refer the reader to [GS83, NNU10, Pab11b, Pab11a, Pab12, Kog00]Let G be a compact, connected Lie group and O λ its coadjoint orbit. Consider a sequence ofsubgroups G = G k ⊃ G k − ⊃ . . . ⊃ G . Inclusion of G j into G gives an action of G j on O λ . Thisaction is Hamiltonian with momentum map Φ j , where Φ j is the composition of the G -momentummap Φ and the projection p j : g ∗ → g ∗ j . Choose maximal tori, T G j , and positive Weyl chambersfor each group G j in the sequence. Every G j orbit intersects the positive Weyl chamber ( t G j ) ∗ + exactly once. This defines a continuous (but not everywhere smooth) map s j : g ∗ j → ( t G j ) ∗ + . LetΛ ( j ) = ( λ ( j )1 , . . . , λ ( j ) rk G j ) denote the composition s j ◦ Φ j : O λ Φ j (cid:47) (cid:47) Λ ( j ) (cid:34) (cid:34) g ∗ j s j (cid:15) (cid:15) ( t G j ) ∗ + The functions { Λ ( j ) } , j = 1 , . . . , k −
1, form the
Gelfand-Tsetlin system which we denote byΛ : O λ → R n ( n − / . Let U denote the subset of O λ on which the Gelfand-Tsetlin functions do not coincide “unnecessarily” U = { A ∈ O λ ; λ ( j ) k ( A ) = λ ( j ) k +1 ( A ) if and only if λ ( j ) k = λ ( j ) k +1 on the whole O λ } . The Gelfand-Tsetlin functions have many useful properties. The ones we are interested in are sum-marized in the following proposition (for more details see for example [GS83, Pab12]).
Proposition 3.1.
In the case of a coadjoint action of G = U ( n ) , SO (2 n + 1) or SO (2 n ) on anorbit O λ through λ ∈ ( t G ) ∗ + (and appropriately chosen sequences of subgroups) the Gelfand-Tsetlinfunctions are smooth on the open dense subset U ⊂ O λ defined above. Moreover, U is equippedwith a Hamiltonian action of a torus T GT , called the Gelfand-Tsetlin torus , of dimension equal tothe complex dimension of O λ . This action makes U into a proper toric manifold. The momentummap consists of those coordinates of Λ | U which are not constant on the whole orbit. The closure ofthe momentum map image, Λ( U ) , is the Gelfand-Tsetlin polytope P (defined carefully below). Inparticular Λ − ( int P ) ⊂ U . Lower bounds for Gromov width of U ( n ) coadjoint orbits. In the case of G = U ( n ) we apply the above procedure to the sequence of subgroups U ( n ) ⊃ U ( n − ⊃ . . . ⊃ U (2) ⊃ U (1) . Then the Gelfand-Tsetlin functions at A ∈ O λ λ ( j )1 ( A ) ≥ λ ( j )2 ( A ) ≥ . . . ≥ λ ( j ) j ( A ) , j = 1 , . . . n − j × j top-left submatrix of A ordered in a non-increasing way (due to our choiceof positive Weyl chamber).The classical min-max principle (see for example Chapter I.4 in [CH62]) implies that λ ( l +1) j ( A ) ≥ λ ( l ) j ( A ) ≥ λ ( l +1) j +1 ( A ) . These inequalities cut out a polytope in R n ( n − / , which we denoted by P , and Λ( O λ ) is containedin this polytope. In fact, Λ( O λ ) is exactly P ([GS83, Pab12, NNU10]). According to Proposition 3.1the number of “non-trivial ”Gelfand-Tsetlin functions, i.e. ones that are not constant on the wholeorbit O λ is equal to N = dim C O λ = dim T GT . These N functions are the coordinates of momentum ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 13 map for the T GT action. Therefore Λ( O λ ) in fact sits in some affine R N ⊂ R n ( n − / and one canview the polytope P as a polytope in t ∗ GT ∼ = R N ⊂ R n ( n − / .Convenient way to visualize the Gelfand-Tsetlin functions for U ( n ) is via the standard ladderdiagram([BCFKvS00, NNU10]). For chosen λ , λ = . . . = λ n > λ n +1 = . . . = λ n > . . . > λ n m +1 = . . . = λ n , let Q = Q λ be an n × n square with squares Q l of size ( n l − n l − ) × ( n l − n l − ), l = 1 , . . . , m + 1, n = 0, n m +1 = n , on the diagonal. The ladder diagram is the set of boxes below the diagonalsquares. Note that the number of boxes in the ladder diagram for λ is equal to N , the number ofnon-trivial Gelfand-Tsetlin functions. To refer to particular boxes we think of Q as sitting in the firstquadrant of R and use Cartesian coordinates. Min-max inequalities imply that for every box in theladder diagram Q its value needs to be between the values of its right neighbor and top neighbor. Thecoordinates of T GT corresponding to these non-constant functions give an effective Hamiltonian torusaction on U . Q Q Q Q a b c d λ (5)1 λ (6)2 λ (7)3 λ (4)1 λ (3)1 λ (1)1 λ (5)2 λ (6)3 λ (7)5 λ (7)7 λ (6)4 λ (6)5 λ (6)6 λ (2)1 λ (2)2 λ (3)3 λ (4)4 λ (5)5 . . . . . .. . . Figure 1.
The ladder diagram for λ = ( a, a, a, b, b, c, c, d ) and its filling with theGelfand-Tsetlin functions.For what follows it will be more convenient to index the Gelfand-Tsetlin functions with Cartesiancoordinates of the corresponding ladder diagram. Therefore let λ j,k := λ ( j + k − j . We need more careful analysis of the Gelfand-Tsetlin polytope P := Λ( O λ ). Let { e j,k | ( j, k ) a box in the ladder diagram } denote the set of generators of R N ∼ = t ∗ GT . As usually, for x ∈ R N we denote by x j,k its coordinate in e j,k direction. We fix the following ordering of generators to obtain an ordered basis for R N : e j,k proceeds e j (cid:48) ,k (cid:48) iff k (cid:48) > k or k = k (cid:48) and j > j (cid:48) . Q Q Q Q ......... Figure 2.
The ordered basis of R N .For each l = 1 , . . . , n denote by g ( l ) an integer such that the row l intersects the diagonal box Q n g ( l ) . This implies λ n − l +1 = λ n g ( l ) . For example, in the situation presented on Figure 1 we have g (5) = g (4) = 2, g (3) = g (2) = 3, g (1) = 4.Let V = Λ( diag( λ n , . . . , λ )) ∈ P be the image of diag( λ n , . . . , λ ) ∈ u ( n ) ∗ , that is V = (cid:80) V j,k e j,k where V j,k = λ j,k ( diag( λ n , . . . , λ ) = λ n − k +1 = λ n g ( k ) . Note that V is a vertex of P because all Gelfand-Tsetlin functions when evaluated at diag( λ n , . . . , λ )are equal to their lower bounds. The vertex V does not need to be smooth: there might be more than N edges in P starting from V .Take ( s, l ) such that ( s, l )-th box is in the ladder diagram. Define the following N subset of P , E s,l = { x ∈ R N | x s,l ∈ [ λ n g ( l ) , λ n g ( l ) − ] ,x j,k = x s,l for j ≤ s, l ≤ k ≤ n − n g ( l ) − ,x j,k = λ n − k +1 for other ( j, k ) } . ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 15 In other words, E s,l is the set of points with almost all coordinates equal to the coordinates of V . Weallow the coordinate x s,l to be greater than V s,l , and this forces some other coordinates to change aswell due to min-max principle. See Figure 3. λ n λ n λ n λ n x x x x λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n λ n x x x x Figure 3.
Elements of sets E , , ( x ∈ [ λ n , λ n ]) and E , , ( x ∈ [ λ n , λ n ]). Lemma 4.1.
Each subset E s,l is an edge of P starting from V . The lattice length of E s,l is λ n g ( l ) − − λ n g ( l ) .Proof. Let H s,l be an affine subspace of R N defined by x j,k = x s,l for j ≤ s, l ≤ k ≤ n − n g ( l ) − ,x j,k = λ n g ( k ) for other ( j, k ) . Then E s,l = H s,l ∩ P is a face of P because P is fully contained in the closure of one component of R N \ H s,l . Any x ∈ H s,l ∩ P is determined by the value of x s,l ∈ ( λ n g ( l ) , λ n g ( l ) − ) , therefore this face is1-dimensional and of lattice length λ n g ( l ) − − λ n g ( l ) . Each of the N boxes ( s, l ) in the ladder diagramsgives such an edge. Moreover V belongs to each E s,l . (cid:3) Let w s,l denote the primitive vector in the direction of E s,l (starting from V ), that is w s,l = (cid:88) e j,k , where the sum is over j, k such that j ≤ s, l ≤ k ≤ n − n g ( l ) − . Recall that r = r U ( n ) ( λ ) = min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n m − λ n m +1 } . Convexity of P and Lemma 4.1 imply that the following set is a subset of P R := convex hull { V, V + r w s,l | ( s, l ) a box in the ladder diagram } . Lemma 4.2. R is SL ( N, Z ) -equivalent to the closure of an N -dimensional tetrahedron (cid:52) N ( r ) , (cid:52) N ( r ) = { < y , . . . , y N ; y + . . . + y N < r } . Proof.
Edges of R starting from V are given by { r w s,l } . Notice that the first non-zero coordinate of w s,l is equal to 1 and appears on the e s,l -th coordinate. Therefore the matrix of vectors w s,l , orderedthe same way we ordered the basis elements, is an integral (all entries are 0 or 1), lower triangularmatrix, with 1’s on diagonal. Thus it belongs to SL ( N, Z ). (cid:3) Proof. (of Theorem 1.1.)
Recall that Λ − ( Int P ) ⊂ U , and U is a proper Hamiltonian T GT -space,dim T GT = N = dim C U . The non-constant coordinates of Λ | U form a momentum map. Lemma 4.2gives that W ( (cid:52) N ( r )) = R ⊂ Int P for some change of basis matrix W ∈ SL ( N ; Z ). Therefore, byProposition 2.5 the Gromov width of O λ is at least r , as claimed. (cid:3) Lower bounds for Gromov width of SO (2 n ) and SO (2 n + 1) coadjoint orbits. In the case of special orthogonal group SO ( m ), m = 2 n or 2 n + 1 we work with the followingsequence of subgroups G m = SO ( m ) ⊃ G m − = SO ( m − ⊃ G m − = SO ( m − ⊃ . . . ⊃ G = SO (2) . and obtain the following Gelfand-Tsetlin functions. For each j < m , λ ( j )1 ( A ) ≥ λ ( j )2 ( A ) ≥ . . . ≥ λ ( j ) (cid:98) m (cid:99) ( A ) , ( ≥ j odd )are numbers such that j × j submatrix of A is SO ( j ) equivalent to ( λ ( j )1 ( A ) , . . . , λ ( j ) (cid:98) m (cid:99) ( A )) ∈ ( t SO ( j ) ) ∗ + . Note that for any λ the number of the Gelfand-Tsetlin functions that are not constant on the wholeorbit O λ is exactly the complex dimension of the orbit.Similarly to the unitary case the following inequalities between the above functions need to besatisfied: λ (2 k )1 ≥ λ (2 k − ≥ λ (2 k )2 ≥ λ (2 k − ≥ . . . ≥ λ (2 k ) k − ≥ λ (2 k − k − ≥ | λ (2 k ) k | ,λ (2 k +1)1 ≥ λ (2 k )1 ≥ λ (2 k +1)2 ≥ λ (2 k )2 ≥ . . . ≥ λ (2 k +1) k ≥ | λ (2 k ) k | , , ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 17 for all indices for which it makes sense. Moreover the image Λ( O λ ) is equal to the polytope P = P λ ,defined by the above set of inequalities (see [Pab12]).To visualize the Gelfand-Tsetlin functions for G = SO (2 n + 1) and SO (2 n ) orbits O λ through λ ∈ ( t G ) ∗ + we add extra boxes from the IV-th quadrant to the U ( n )-ladder diagram for the corresponding λ . In the SO (2 n + 1) case we add ( n − n ) + n boxes and in the SO (2 n ) case add ( n − n ) boxesthe way presented on the Figure 4 (first two pictures). Q Q Q Q Q Q SO (2 n + 1) , λ n = 0 SO (2 n ) Q Q Q SO (2 n + 1) , λ n = 0 Figure 4.
The so-diagrams for SO (2 n + 1) and SO (2 n ).Each diagonal (that is intersection of the diagram with a line of slope −
1) corresponds to functionswith the same superscript. Boxes in diagonal squares Q l contain functions which are constant on thewhole orbit. If λ n = 0 then all the Gelfand-Tsetlin functions corresponding to the boxes below thelast diagonal square are equal to 0 on the whole orbit. We delete these boxes from the diagram (anexample is presented on the third picture in Figure 4). Now there is a one-to-one correspondencebetween the boxes in the diagram and the Gelfand-Tsetlin functions that are not constant on thewhole orbit. Call such a diagram the so-diagram . The subspace of ( t G ) ∗ spanned by the images of non-constant Gelfand-Tsetlin functions has thedimension equal to the complex dimension of the orbit O λ , which we continue to denote by N . We iden-tify this subspace with R N , where the basis of R N consists of elements { e j,k ; ( j, k ) is in the so-diagram } ordered in the following way: e j,k proceeds e j (cid:48) ,k (cid:48) iff j < j (cid:48) or j = j (cid:48) and k > k (cid:48) . Q Q Q
123 15 19 2210 ... ...
Figure 5.
Ordering of the basis.Let V = Λ( λ ). Notice that V = (cid:80) λ j e j,k , with the sum taken over all boxes in the so-diagram. Itis the vertex of P as all Gelfand-Tsetlin functions attain their maximum at λ . Recall the definition of g ( j ) from the previous Section and notice that the column j in the so-diagram intersects the diagonalsquare Q g ( j ) . ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 19 Take ( s, l ) such that ( s, l )-th box is in the diagram. Define the following affine subspaces of R N ,one per each box in the so-diagram H s,l = { x ∈ R N | x s,l ∈ R ,x j,k = x s,l for k ≤ l, s ≤ j ≤ n g ( s ) ,x j,k = λ j for other ( j, k ) } . Recall that r = r G ( λ ) = min {(cid:104) α ∨ j , λ (cid:105) ; α ∨ j a coroot, (cid:104) α ∨ j , λ (cid:105) > } , that is r SO (2 n +1) ( λ ) = min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n − − λ n , λ n } if λ n (cid:54) = 0 , min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n − } if λ n = 0 r SO (2 n ) ( λ ) = min { λ n − λ n +1 , λ n − λ n +1 , . . . , λ n m − λ n m +1 , λ n m + λ n m +1 } . Lemma 5.1.
For each of these affine hyperplanes the intersection H s,l ∩ P is an edge of P . Edges E s,l with g ( s ) = m + 1 (that is n g ( s ) = n ) and s − n < l ≤ in the SO (2 n + 1) case, s − n + 1 < l ≤ in the SO (2 n ) case, if they exist, are of lattice length λ n . All other edges are of lattice length at least r G ( λ ) .Proof. For each of these affine hyperplanes, the polytope P is contained in a closure of one connectedcomponent of R N \ H s,l . Therefore E s,l := H s,l ∩ P is a face of P . A point in E s,l is uniquelydetermined by the value x s,l . Therefore each E s,l is an edge of P . The length of these edges dependson the position of ( s, l ) box, and on whether we are in SO (2 n + 1) or SO (2 n ) case. Examples arepresented in Figures 6 and 7. One can easily check that the lengths are given by the following formulas.For SO (2 n + 1) | E s,l | = λ n g ( s ) if s < n g ( s ) and s − n < l ≤ n g ( s ) − n, λ n g ( s ) if on the “rim” i.e. l = s − n,λ n g ( s ) − λ n g ( l )+1 otherwise . For SO (2 n ) | E s,l | = λ n g ( s ) if s < n g ( s ) and s − n + 1 < l ≤ n g ( s ) − n + 1 , λ n g ( s ) if on the “rim” i.e. l = s − n + 1 ,λ n g ( s ) − | λ n g ( l )+1 | otherwise (recall: λ n can be negative) . All of the above values are positive. The value λ j can be negative if and only if G = SO (2 n ) and λ λ λ λ λ λ x , − ∈ [0 , λ ] x , − x , − x , − λ λ λ x , − λ λ λ λ λ λ λ λ x , − ∈ [ − λ , λ ] λ λ λ λ λ λ λ λ λ x , ∈ [ λ , λ ] x , x , Figure 6.
The edges E , − , E , − and E , in the polytope for SO (2 n + 1) coadjoint orbit. λ λ λ x , ∈ [0 , λ ] x , x , x , λ λ λ λ x , − ∈ [ − λ , λ ] x , − λ λ λ λ λ λ λ λ λ x , ∈ [ λ , λ ] x , x , Figure 7.
The edges E , , E , − and E , in the polytope for SO (2 n ) coadjoint orbit. j = n . Then λ n (cid:54) = λ n − . Therefore there are no boxes ( s, l ) with g ( s ) = m + 1.Note that the values 2 λ n g ( s ) , λ n g ( s ) − λ n g ( l )+1 , and λ n g ( s ) + λ n g ( l )+1 are equal to (cid:104) α ∨ , λ (cid:105) for somecoroot α ∨ . Moreover, for g ( s ) ≤ mλ n g ( s ) ≥ λ n g ( s ) − | λ n g ( s )+1 | = (cid:104) α ∨ , λ (cid:105) for some coroot α ∨ . Only the edges E s,l with g ( s ) = m + 1 (that is n g ( s ) = n ) and s − n < l ≤ SO (2 n + 1) case, s − n + 1 < l ≤ SO (2 n ) case, if they exist, have lattice length equal to λ n and λ n can be smaller than r . (cid:3) Lemma 5.2. If λ satisfies condition ( ∗ ) then all edges E s,l defined above have lattice length at least r G ( λ ) . ROMOV WIDTH OF NON-REGULAR COADJOINT ORBITS OF U ( n ), SO (2 n ) AND SO (2 n + 1). 21 Proof.
Recall that the condition ( ∗ ) says:( λ n (cid:54) = λ n − ) ∨ ( λ n = 0) ∨ ( λ n ≥ r G ( λ )) . If λ n ≥ r G ( λ ) then Lemma 5.1 proves the claim. If λ n = 0, or if λ n (cid:54) = λ n − and G = SO (2 n ), thenthere are no boxes ( s, l ) with g ( s ) = m + 1 and thus, by Lemma 5.1, all edges have lattice length atleast r G ( λ ). If λ n (cid:54) = λ n − and G = SO (2 n + 1) then the only box ( s, l ) with g ( s ) = m + 1 is the ( n, E n, corresponding to this box have latticelentgh 2 λ n . (cid:3) Let w s,l denote the primitive vector in the direction of E s,l (starting from V ), that is w s,l = − (cid:88) e j,k , where the sum is over j, k such that k ≤ l, s ≤ j ≤ n g ( s ) . Recall that r = r G ( λ ). Let r (cid:48) := r if ( ∗ ) is satisfied ,λ n if ( ∗ ) is not saitisfied (what implies r > λ n (cid:54) = 0) . Denote by R := convex hull { V, V + r (cid:48) w s,l | ( s, l ) a box in the ladder diagram } . Convexity of P and Lemmas 5.1, 5.2 imply that R ⊂ P . Lemma 5.3. R is ± SL ( N, Z ) -equivalent to the closure of an N -dimensional tetrahedron (cid:52) N ( r (cid:48) ) .Proof. Edges of R starting from V are given by { r w s,l } . Notice that the first non-zero coordinate of w s,l is equal to − e s,l -th coordinate. Therefore the matrix of vectors w s,l , orderedthe same way we ordered the basis elements, is an integral (all entries are 0 or − − ± SL ( N, Z ). (cid:3) Proof. (of Theorem 1.7.)
We need to show that the Gromov width is at least r (cid:48) . The proof isanalologous to the unitary case. Subset U ⊂ O λ is a proper Hamiltionan T GT space with dim T GT =dim C U . Moreover Λ − ( R ) ⊂ Λ − ( int P ) ⊂ U and R = W ( (cid:52) ( r (cid:48) )) for some W ∈ ± SL ( N, Z ).Therefore, by Proposition 2.5 the Gromov width of O λ is at least r (cid:48) , as claimed. (cid:3) Remark 5.4.
Note that if λ does not satisfy condition ( ∗ ), then the Gelfand-Tsetlin polytope P iscontained between two affine hyperplanes ( x s,l = 0) and ( x s,l = λ n ) where ( s, l ) = ( n − ,
0) in the SO (2 n + 1) case and ( s, l ) = ( n − ,
0) in the SO (2 n + 1) case. These hyperplanes are lattice distance λ n < r apart and therefore in that case P cannot contain W ( (cid:52) N ( r )) for any W ∈ ± SL ( N, Z ). Thismeans that B r , a ball of capacity r , cannot be equivariantly (with respect to the Gelfand-Tsetlinaction) embedded into U . There might still exist a symplectic though not equivariant embedding of B r . The Gromov width of this orbit might still be equal to r . References [BCFKvS00] Victor V. Batyrev, Ionut¸ Ciocan-Fontanine, Bumsig Kim, and Duco van Straten,
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Milena Pabiniak, CAMGSD, Departamento de Matem´atica, Instituto Superior T´ecnico, Lisboa, Portugal
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