Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization
aa r X i v : . [ m a t h . S G ] M a r Action-angle coordinates on coadjoint orbits and multiplicity freespaces from partial tropicalization
Anton Alekseev Benjamin Hoffman Jeremy Lane Yanpeng Li
Abstract
Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of sym-plectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spacesis a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction ofGelfand-Zeitlin integrable systems for the groups K = U( n ) , SO( n ) . Extending these results to groupsof other types is one of the goals of this paper.Partial tropicalizations are Poisson spaces with constant Poisson bracket built using techniquesof Poisson-Lie theory and the geometric crystals of Berenstein-Kazhdan. They provide a bridge be-tween dual spaces of Lie algebras Lie( K ) ∗ with linear Poisson brackets and polyhedral cones whichparametrize the canonical bases of irreducible modules of G = K C .We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, andapply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K ,we construct an exhaustion by symplectic embeddings of toric domains. As a by product we arriveat a conjectured formula for Gromov width of regular coadjoint orbits. We prove similar results formultiplicity free K -spaces. Contents K ∗ and cluster algebras . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Partial tropicalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 The cone C and the Berenstein-Kazhdan potential . . . . . . . . . . . . . . . . . . 91.1.6 Proof sketch for coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Keywords:
Poisson-Lie groups, coadjoint orbits, symplectic geometry Background 11 K and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Lattices and Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 SL triples and the Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.4 Anti-holomorphic involutions on G . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.5 Generalized minors and double Bruhat cells . . . . . . . . . . . . . . . . . . . . . 132.1.6 Langlands dual groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.7 Comparison map and bilinear form on g . . . . . . . . . . . . . . . . . . . . . . . 152.2 Hamiltonian K -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 The Thimm torus action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Toric manifolds and multiplicity free spaces . . . . . . . . . . . . . . . . . . . . . 172.3 Hamiltonian Poisson-Lie group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 Compact Poisson-Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Explicit formula for the Poisson bracket on K ∗ . . . . . . . . . . . . . . . . . . . 202.3.3 Hamiltonian K -manifolds with K ∗ -valued moment maps . . . . . . . . . . . . . . 212.3.4 Delinearization of Hamiltonian K -manifolds . . . . . . . . . . . . . . . . . . . . 212.3.5 Legendre transforms on K ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Cluster algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.1 Seeds and cluster mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Cluster algebra structures on double Bruhat cells . . . . . . . . . . . . . . . . . . 262.4.3 Homogeneous cluster algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.4 Homogeneity of cluster variables on double Bruhat cells . . . . . . . . . . . . . . 28 G w ,e . . . . . . . . . . . . . . . . . . . . . 323.3 Domination of functions on G w ,e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Geometric crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Polyhedral parametrizations of canonical bases . . . . . . . . . . . . . . . . . . . . . . . 363.6 Comparison map on double Bruhat cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Partial tropicalization 39 K ∗ from cluster charts on G w ,e . . . . . . . . . . . . . . . . . . . . . . . 394.2 The definition of partial tropicalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Partial tropicalization as a limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Structure maps and symplectic leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Properties of the partial tropicalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5.1 Brackets in special charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5.2 Brackets in general charts and relationship with canonical bases . . . . . . . . . . 45 K × ˚ t ∗ + . . . . . . . . . . . . . . . 475.2 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Big action-angle coordinate charts on multiplicity free spaces . . . . . . . . . . . . . . . . 58 There is a dichotomy in symplectic geometry between local and global coordinates. Whereas Darboux’stheorem tells us that symplectic manifolds have no local invariants, the problem of finding large coordinatecharts often relates to subtle properties of symplectic manifolds. Most famously, Gromov’s non-squeezingtheorem demonstrates that the volume of certain coordinate charts on a symplectic manifold may have anupper bound strictly less than the total volume of the symplectic manifold [22].Action-angle coordinates are a type of coordinate chart on symplectic manifolds that originate from thestudy of commutative completely integrable systems in classical mechanics. The domains of action-anglecoordinates are products of the form U × T n , where U is an open subset of R n and T n is a direct productof n circles, S × · · · × S . Such domains carry a canonical symplectic form, ω std = n X i =1 dϕ i ∧ dλ i , (1)where λ i are coordinates on R n and ϕ i are coordinates on T n . The Liouville-Arnold theorem guaranteesexistence of local action-angle coordinates in a neighbourhood of compact regular fibers of commutativecompletely integrable systems [8]. A compact symplectic toric manifold of dimension n with Delzantpolytope △ has a dense subset symplectomorphic to (˚ △ × T n , ω std ) , where ˚ △ denotes the interior of △ .However, there are also many interesting examples of action-angle coordinates on dense subsets that do notarise from a toric structure, such as Gelfand-Zeitlin systems [26], Goldman systems on moduli spaces offlat connections [21, 57], bending flow systems on moduli spaces of polygons [37], and integrable systemsconstructed by toric degeneration on smooth projective varieties [38, 41, 42].Multiplicity free spaces are the natural non-abelian generalization of toric manifolds. A multiplicityfree space ( M, ω, µ ) is a symplectic manifold ( M, ω ) equipped with a Hamiltonian action of a compactLie group K generated by an equivariant moment map µ : M → k ∗ = Lie( K ) ∗ with the property that thenon-empty symplectic reduced spaces are all zero dimensional [27]. Compact multiplicity free spaces are3lassified by the quotient M/K , which is identified with a convex polytope, together with the principalisotropy subgroup L , which is a subgroup of the centralizer of a torus subgroup of K [43]. In this paper,we consider the case where L is a subgroup of the maximal torus, T . In this case, the range of possibledimensions of a multiplicity free space is dim K − dim T dim M dim K + dim T. (2)Unlike toric manifolds, multiplicity free spaces are not known to have action-angle coordinates on densesubsets, except for several examples. The first and oldest example is that of SO(3) acting by rotationon S , equipped with a rotation invariant area form. The action of a maximal torus T SO (3) gives S the structure of a symplectic toric manifold and the action-angle coordinates are simply cylindricalcoordinates. The second major example is Gelfand-Zeitlin systems, which define action-angle coordinateson dense subsets of coadjoint orbits and multiplicity free spaces of compact Lie groups of type A, B, and D[26]. The final example is that of spherical varieties. If a multiplicity free space embeds equivariantly as aprojective variety and the symplectic structure is induced by the embedding, then it is a spherical variety anddense action-angle coordinates can be constructed by toric degeneration [38]. However, most multiplicityfree spaces are not spherical varieties (in fact, a compact multiplicity free space need not admit an invariantcompatible complex structure [58]).The main result of this paper is a construction of action-angle coordinates on large subsets of compactmultiplicity free spaces whose principal isotropy group is contained in a maximal torus. For any subset C of a Euclidean space, let C ( δ ) denote the set of points in C that have distance more than δ from theboundary of C . Theorem 1.1.
Let K a compact connected Lie group and ( M, ω, µ ) a compact multiplicity free space for K of dimension n such that the principal isotropy subgroup is contained in the maximal torus of K . Then,there is a convex polytope △ M ⊂ R n of dimension n such that for all δ > , there exists a symplecticembedding ( △ M ( δ ) × T n , ω std ) ֒ → ( M, ω ) . (3) Moreover, for all ε > , there exists δ > such that the symplectic volumes satisfy Vol( △ M ( δ ) × T n , ω std ) > Vol(
M, ω ) − ε. Additionally, the embeddings (3) intertwine naturally defined Hamiltonian T × T actions; see Theo-rem 5.15 for a precise statement.In fact, we construct many polytopes △ M which satisfy the conclusions of Theorem 1.1. These poly-topes are parameterized by several choices, including a choice of seed for the cluster algebra structure onthe big double Bruhat cell in a Borel subgroup of the complexification of K .One particularly important instance of Theorem 1.1 is the extremal case where dim M = dim K − dim T . In this case ( M, ω, µ ) is a coadjoint orbit, diffeomorphic to K/T , and ω is the canonical Kostant-Kirillov-Souriau symplectic form [44, 45, 56]. Coadjoint orbits diffeomorphic to K/T , also known asregular coadjoint orbits, are parameterized by elements λ in the interior of the positive Weyl chamber. Thecoadjoint orbit parameterized by λ along with its Kostant-Kirillov-Souriau form is denoted ( O λ , ω λ ) . Theorem 1.2.
Let O λ be a regular coadjoint orbit of a compact connected Lie group K . Then, there is aconvex polytope △ λ of dimension n = (dim K − dim T ) such that for all δ > there exists a symplecticembedding ( △ λ ( δ ) × T n , ω std ) ֒ → ( O λ , ω λ ) . Moreover, for all ε > , there exists δ > such that Vol( △ λ ( δ ) × T n , ω std ) > Vol( O λ , ω λ ) − ε. B n ( r ) ⊂ R n denote the open ball of radius r > equipped with the standard symplectic structure on R n , ω std = dx ∧ dy + · · · + dx n ∧ dy n . (4)The Gromov width of a connected symplectic manifold ( M, ω ) of dimension n , denoted GWidth( O λ , ω λ ) ,is supremum of all cross-sectional areas πr such that ( B n ( r ) , ω std ) embeds symplectically into ( M, ω ) .Combining Theorem 1.2 with some results from [18] regarding the geometry of the polytopes △ λ yieldsthe following. Theorem 1.3.
Let ( O λ , ω λ ) be a regular coadjoint orbit of a compact connected simple Lie group K . Then, GWidth( O λ , ω λ ) > min { π h√− λ, α ∨ i | α ∈ R + } , (5) where R + is the set of positive roots of K and α ∨ is the coroot of α ∈ R + . It follows from the upper bounds of [14] that (5) is an equality. This was already known to be anequality in several cases. The case where K is type A, B, or D and λ is arbitrary was proved using Gelfand-Zeitlin systems in [53]. The case where K is arbitrary type and λ is a positive scalar multiple of a dominantintegral weight was proved using toric degenerations by [18] (see [18, Section 2] for a detailed survey ofearlier results). In fact, [14] provides upper bounds for Gromov width of non-regular orbits as well. Themethods developed in this paper do not apply to those orbits. Remark 1.4.
The authors BH and JL will present an alternate approach to the problem of constructingaction-angle coordinates on multiplicity free spaces in a forthcoming paper [35]. Their approach uses agradient-Hamiltonian flows inside the total space of toric degenerations of the base affine space G (cid:12) N , G = K C , which were constructed in [13]. These produce integrable systems with dense action-anglecoordinates on the symplectic implosion of the cotangent bundle of K (cf. [23]). These integrable systemsmay then be combined with symplectic contraction (cf. [36]) to construct completely integrable systemson arbitrary compact multiplicity Hamiltonian K -manifolds that have dense action-angle coordinates (noassumption about the principal isotropy subgroup is needed). The polytopes obtained by this constructionare similar to the polytopes obtained by [7] in the case of spherical varieties. We expect these results willbe sufficient to prove tight lower bounds for the Gromov width of the remaining non-regular coadjointorbits (modulo combinatorial results about string polytopes). It would be interesting to better understandthe relationship between the toric degeneration and Ginzburg-Weinstein approaches to this problem. Our results are obtained by a new method of independent value that combines new findings with previousresults from [2, 3, 4]. The main idea is that to each coadjoint orbit O λ one can associate a family ofsymplectic spaces D exp( sλ ) called dressing orbits. The dressing orbits are symplectomorphic to O λ for allvalues of the parameter s ∈ R × . For s small, D exp( sλ ) resembles of O λ , and there is a natural way toinclude O λ in the family at s = 0 . For s ≪ large, there are coordinates on D exp( sλ ) coming from clusteralgebra theory which make its symplectic structure (exponentially) close to the constant one. Using this,one may construct action-angle coordinates on D exp( sλ ) , and hence on O λ , which exhaust the symplectic5olume as s → −∞ . We call these charts big , in the sense that by picking s ≪ , their volume may bemade arbitrarily close to the volume of O λ .In what follows we briefly outline the main points used in our construction. While we focus hereon coadjoint orbits, in the main body we extend this treatment to allow for arbitrary compact regularmultiplicity-free spaces. We make use of some standard Lie-theoretic notation, which is introduced indetail in Section 2.1. The space k ∗ = Lie( K ) ∗ carries a canonical linear Poisson structure π k ∗ defined by formula { f ξ , f η } = f [ ξ,η ] , where ξ, η ∈ k and f ξ ( x ) = h x, ξ i for x ∈ k ∗ . Coadjoint orbits are symplectic leaves of π k ∗ .Consider the complex Lie group G = K C . It admits the Iwasawa decomposition G = AN − K , where N − is the maximal nilpotent subgroup and A = exp( √− t ) . Denote K ∗ = AN − , and observe that Lie( K ∗ ) may be identified with k ∗ = Lie( K ) ∗ under the pairing h x, ξ i := 2Im ( x, ξ ) g , x ∈ Lie( K ∗ ) , ξ ∈ k . Here ( · , · ) g is a fixed nondegenerate, Ad -invariant bilinear form on g = k C .The group K ∗ is naturally isomorphic to the symmetric space G/K . Hence, it carries a K -action knownas the dressing action. Furthermore, it has a unique Poisson structure π K ∗ given by the Lu formula π ♯K ∗ ( h θ R , ξ i ) = − ξ, ∀ ξ ∈ k , where θ R is the right-invariant Maurer-Cartan form on K ∗ and ξ is the fundamental vector field of ξ . Thegroup ( K ∗ , π K ∗ ) is the dual Poisson-Lie group to K . Symplectic leaves of the Poisson structure π K ∗ arethe orbits of the K -action on K ∗ , which are called dressing orbits .See Section 2.3 for further exposition of these notions. There is an intimate link between the Poisson spaces k ∗ and K ∗ given by the Ginzburg-Weinstein Theorem: Theorem 1.5. [33] There is a Poisson isomorphism γ : k ∗ → K ∗ which restricts to the exponential mapon t ∗ ∼ = √− t . The choice of the Ginzburg-Weinstein isomorphism γ is non unique. Indeed, it can be pre-composedwith any Poisson isomorphism of k ∗ preserving t ∗ or composed with any Poisson isomorphism of K ∗ preserving A .The Poisson structure π k ∗ on k ∗ is linear, and so under scaling transformations A s : x → sx it alsoscales linearly: ( A s ) ∗ π k ∗ = sπ k ∗ . Therefore, the map γ s = γ ◦ A s : x γ ( sx ) is a Poisson isomorphism between ( k ∗ , π k ∗ ) and ( K ∗ , sπ K ∗ ) . In particular, for each s = 0 the map γ s restricts to a symplectomorphism between the coadjoint orbit O λ ⊂ k ∗ through an element λ of the positiveWeyl chamber t ∗ + , and the dressing orbit D exp( sλ ) ⊂ K ∗ through γ s ( λ ) ∈ γ s ( t ∗ + ) . Therefore, we may study D exp( sλ ) for arbitrary s ∈ R × instead of O λ .A construction of γ s is described in Section 2.3.4 and Remark 2.17.6 .1.3 Coordinates on K ∗ and cluster algebras The dual Poisson-Lie group K ∗ is a real subgroup of the Borel subgroup of G : K ∗ = AN − ⊂ B − . Recall that the double Bruhat cell G w ,e = Bw B ∩ B − ⊂ B − is dense in B − . Double Bruhat cells carry cluster structures which provide an infinite set of distinguishedcoordinate systems called toric charts , which are parametrized by cluster seeds. These coordinate systemsrestrict to K ∗ ∩ G w ,e and define dense charts on K ∗ . When restricted to K ∗ , these coordinate systemscombine complex and real coordinates. Toric charts on G w ,e and K ∗ are described in Sections 3.2 and 4.1,respectively.For the sake of exposition, we focus here on a finite number of toric charts known as factorizationcoordinates. They are parametrized by reduced expressions in the Weyl group of G representing the longestelement w . The results below apply to all (twisted) cluster charts, see Sections 3 and 4 for a more detaileddiscussion. Assume K is semisimple of rank r . For a simple root α i of g , denote by α ∨ i ∈ h the correspondingcoroot and by e i ∈ n and f i ∈ n − the corresponding root vectors. Let x i ( t ) = exp( tf i ) be a 1-parameter subgroup of N − . For a fixed reduced word w = s i · · · s i m , introduce the following map L s : R m + r × ( S ) m → K ∗ ( λ − r , . . . , λ − , λ , . . . , λ m , e √− ϕ , . . . , e √− ϕ m ) exp s r X i =1 λ − i α ∨ i ! x i ( e sλ + √− ϕ ) · · · x i m ( e sλm + √− ϕ m ) . The map L s is injective for s ∈ R × , and its image is open and dense in K ∗ . We regard the λ i , ϕ j ascoordinates on K ∗ which depend on the parameter s . The map L s is given in general in Definition 4.2.In the main body of the paper, we consider a wider class of toric charts, so-called cluster charts and twisted cluster charts ; see Definition 3.9. Coordinates in twisted cluster charts are similar to factorizationparameters described above while coordinates in cluster charts are generalizations of minors of a matrix.Furthermore, along with the double Bruhat cell G w ,e we need to consider its Langlands dual ( G ∨ ) w ,e andits (twisted) cluster charts.We denote π s = ( L s ) − ∗ ( sπ K ∗ ) = ( L − s ◦ γ s ) ∗ π k ∗ wherever L − s is defined. While L − s is exponentially contracting, the map γ s is exponentially expanding.For s ≪ , their composition possesses extraordinary properties. Strictly speaking, the following example is related to a certain twisted cluster chart on G w ,e by a monomial change ofcoordinates given by [20, Theorem 1.9]. .1.4 Partial tropicalization Our strategy is to consider the limit s → −∞ . A priori, such a limit does not make sense since the Poissonstructure sπ K ∗ linearly diverges. However, after the coordinate change L s , the Poisson structure π s doesadmit a limit which is described by the following theorem. It makes use of a map hw P T : R m + r × ( S ) m → t ∗ + to the positive Weyl chamber called the highest weight map . This map, and its origin, are described inSections 4.4 and 3.4, respectively. The following summarizes the main results of Section 4. Theorem 1.6.
There is a unique open polyhedral cone C ⊂ R N such that:1. The limit lim s →−∞ (cid:0) π s | C × ( S ) m (cid:1) exists and is a constant Poisson structure π −∞ on C × ( S ) m .2. The symplectic leaves of ( C × ( S ) m , π −∞ ) are the fibers of hw P T : C × ( S ) m → t ∗ + .3. For each regular weight λ ∈ ˚ t ∗ + , the symplectic volume of (hw P T ) − ( λ ) is equal to the symplecticvolume of O λ .4. For a given λ ∈ t ∗ + , the fiber (hw P T ) − ( λ ) ∩ C × ( S ) m is of the form ˚∆ λ × ( S ) m , where ∆ λ is a convex polytope.5. After a linear change of variables, π −∞ acquires the form π −∞ = m X i =1 ∂∂λ i ∧ ∂∂ϕ i . The Poisson manifold ( C × ( S ) m , π −∞ ) is called the partial tropicalization of the dual Poisson-Lie group K ∗ . A priori, its definition depends on the coordinate map L s corresponding to a fixed toricchart. However, one can show that, for all (twisted) cluster charts, the construction gives rise to isomorphicPoisson spaces .Cluster coordinates on K ∗ give a good control of the Poisson bracket π s , and they allow to establishparts 1, 2, 4 and 5 of Theorem 1.6. Twisted cluster coordinates on G w ,e and on its Langlands dual ( G ∨ ) w ,e give a tool for volume estimates and are needed in proving part 3 of Theorem 1.6. Remark 1.7.
Theorem 1.6 hints that if the map L − s ◦ γ s had a limit for s → −∞ , the limit map γ ∞ : O λ → (hw P T ) − ( λ ) would give rise to densely defined action-angle coordinates on the coadjoint orbit. Thisscenario is realized in the case of K = U( n ) [5]. However, for arbitrary compact K it is not known how tochoose the Ginzburg-Weinstein isomorphism γ giving rise to a convergent map lim s →−∞ ( L − s ◦ γ s ) . Remark 1.8.
The real Poisson-Lie group K ∗ has a complex form, denoted G ∗ . The group K ∗ is a con-nected component of the fixed point set of an anti-holomorphic involution of G ∗ . There have been severalrecent approaches to constructing systems of coordinates z , . . . , z n on G ∗ whose Poisson brackets are logcanonical [30, 54, 55], meaning that { z i , z j } G ∗ = c i,j z i z j for some c i,j ∈ C . Strictly speaking, this isomorphism is defined only off a union of hyperplanes in R m ; see Definition 4.1 and Theorem 4.6. z i = e λ i + √− ϕ i , the restriction of the functions λ i , ϕ i to K ∗ would give a Darboux chart on K ∗ . However, this is not the case. Assume we are given a C -valued log canonical coordinate system on a real manifold. This doesn’t necessarily give rise to an R -valuedlog canonical coordinate system or to a Darboux chart. Indeed, let z, z ′ : M → C such that { z, z ′ } = czz ′ . If z = e λ + iϕ , w = e λ ′ + iϕ ′ , then { λ, λ ′ } − { ϕ, ϕ ′ } = ℜ ( c ) { λ, ϕ ′ } + { ϕ, λ ′ } = ℑ ( c ) but we cannot conclude whether or not the Poisson bracket is constant in coordinates λ, λ ′ , ϕ, ϕ ′ .In our particular example, if K = SU(2) , then G ∗ = (cid:26)(cid:18)(cid:18) a b a − (cid:19) , (cid:18) a − c a (cid:19)(cid:19) : a ∈ C × , b, c ∈ C (cid:27) ⊂ SL(2) × SL(2) . Then K ∗ is the set of points where a ∈ R > and b = − c . A system of log canonical coordinates constructedin [30] is z = a, z = ab, z = − bc + a + a − . To extract a Darboux coordinate system, following thestrategy of [2, Section 6], for instance, one would need the Poisson brackets { z , z } K ∗ , { z , z } K ∗ , and { z , z } K ∗ to be log canonical on K ∗ . However, z = z z − z − z on K ∗ , and the Poisson bracket { z , z } is not log canonical. We will explore the connection between thepartial tropicalization of K ∗ and log canonical coordinate systems on G ∗ in future work. C and the Berenstein-Kazhdan potential In Theorem 1.6, the cone C appears as an unexpected and mysterious element of the construction. In fact,it is a version of a well known object in the representation theory of reductive complex Lie groups. Fora dominant integral weight λ ∈ P + , let V λ be the irreducible G -module with high weight λ . Then, thecanonical basis of C [ G ] N ∼ = ⊕ λ ∈ P + V λ can be parametrized as the integral points of the closure C .The defining inequalities of C are determined by the tropicalization of a distinguished function Φ BK : G w ,e → C , called the Berenstein-Kazhdan potential . On K ∗ , the potential Φ BK can be written as a Laurent polynomialin the factorization coordinates e sλi + √− ϕ i introduced in the previous section. The coefficients of thisLaurent polynomial are positive integers. The function Φ BK is a main technical tool in the theory thatfollows; see for instance Corollary 3.18.It is a surprising fact that the s → −∞ behavior of all Poisson brackets on K ∗ is determined (dom-inated) by the behavior of a single function Φ BK . At the moment, this phenomenon has no conceptualexplanation. The function Φ BK was introduced in the context of geometric crystals, which are brieflydescribed in Section 3.4. We sketch a proof of Theorem 1.2. For s < , we have a Ginzburg-Weinstein isomorphism γ s : O λ → D exp( sλ ) . L − s ( D exp( sλ ) ) ⊂ R m + r × ( S ) m has the following property: the symplectic volume of ( L − s ( D exp( sλ ) )) ∩ ( C × ( S ) m ) can be made arbitrarily close to the volume of L − s ( D exp( sλ ) ) by fixing s ≪ . This manifold is very nearto the symplectic leaf (hw P T ) − ( λ ) of π −∞ , and their symplectic forms are also very close to each other(in the appropriate sense). This allows one to use the Moser argument and to construct an embedding (∆ λ ( δ ) × ( S ) m , π −∞ ) ֒ → ( L − s ( D exp( sλ ) ) , π s ) ∼ = ( D exp( sλ ) , sπ K ∗ ) ∼ = ( O λ , π k ∗ ) which is an action-angle coordinate chart on O λ . Everything in this argument is equivariant with respect tothe maximal torus of K and thus the resulting embeddings are also T -equivariant.The constant δ > depends on s , and δ → as s → −∞ . And, the difference in symplectic volumes Vol( O λ ) − Vol(∆ λ ( δ ) × ( S ) m ) > approaches as δ → . So, by picking s ≪ , one arrives at the desired embedding.The proof of Theorem 1.1 requires some additional work. Naïvely, one would like to take actioncoordinates constructed as above, on subsets of k ∗ , and pull them back via the moment map to a multiplicityfree space. However, pullbacks of functions generating periodic flows on k ∗ do not necessarily generateperiodic flows: there may be so-called “nutation effects” that cause the resulting flow to be aperiodic [25].Thus, one of the main steps in the proof of Theorem 1.1 is to show that our coordinates generate periodicflows on multiplicity free spaces. This is achieved in Section 5 by first constructing coordinates that generateapproximately periodic flows for large s , then deforming (via a Moser trick) to coordinates that generateperiodic flows. The construction presented in the paper requires background material from different fields. Sections 2, 3and 4 contain an extensive review of this material. We also recall our previous results, and in some casesupgrade them to the level needed for the present paper. Sections 5 and 6 contain the main results of thepaper and their proofs.We can imagine several ways to read the present text. One possibility is to first check the introductorysections and then read the main body of the paper which are Sections 5 and 6. Another interesting option isto start directly with those two sections and then consult the introductory sections for missing details.In what follows we describe the content of each section with an accent on elements which are eithernew or require non-standard presentation.In Section 2, we collect background material on Lie theory, the theory of Hamiltonian and Poissongroup actions and on cluster structures. Section 2.1 is devoted to Lie theory including the theory of doubleBruhat cells and generalized minors. We also touch upon comparison maps between the group G and itsLanglands dual G ∨ . Section 2.2 summarizes the theory of Hamiltonian K -manifolds with focus on Thimmtorus actions and multiplicity free Hamiltonian spaces. Section 2.3 is a reminder of the theory of Poisson K -actions and of the dual Poisson-Lie group K ∗ . An interesting new element is the notion of Legendretransform on K ∗ . Section 2.4 provides background on cluster algebras and on cluster structures on doubleBruhat cells. A somewhat non-standard part of the discussion concerns homogeneous cluster algebras .Section 3 describes positivity and tropicalization of double Bruhat cells. Section 3.1 is devoted togeneralities on positive varieties with potential and their tropicalizations. We also discuss comparison mapsbetween a cluster variety and its Langlands dual. Section 3.2 contains applications of the material of the10revious section to double Bruhat cells G w ,e . In particular, we recall the definition of the Berenstein-Kazhdan (BK) potential Φ BK . In Section 3.3, we discuss functions dominated by potential, and we givecriteria for domination by Φ BK . Section 3.4 contains elements of the geometric crystal theory. In fact, inthis paper we do not need crystal operations. So, we only review the highest weight and weight maps ongeometric crystals. In Section 3.5, we discuss tropicalization of double Bruhat cells and reduced doubleBruhat cells. The corresponding polyhedral cones will serve as targets for action variables. In Section 3.6,we recall the comparison map between double Bruhat cells G w ,e and G ∨ ; w ,e in the group G and itsLanglands dual G ∨ which is the main tool in volume calculations.Section 4 defines partial tropicalizations. In Section 4.1 , we introduce coordinate systems on thedual Poisson-Lie group K ∗ induced by the cluster structure on the double Bruhat cell G w ,e . Then, inSection 4.2 we define the partial tropicalization P T ( K ∗ ) which carries a constant Poisson structure π −∞ .In Section 4.3, we explain how to obtain the Poisson structure of the partial tropicalization as an s → −∞ limit of a family of Poisson structures on K ∗ . Section 4.4 is the description of symplectic leaves and oftorus actions on the partial tropicalization. Section 4.5 is devoted to explicit computations of π −∞ in clustercharts of Section 4.1. We prove a version of Theorem 1.6 which establishes a Darboux normal form for π −∞ .Section 5 is devoted to the proof of Theorem 1.1 and Theorem 1.2. Section 5.1 states the main technicalresult of the paper which is an extension of partial tropicalization to the universal multiplicity free space K × ˚ t ∗ + . Section 5.2 is devoted to the proof of this result. This section is the core of the paper. In Section 5.3,we construct big action-angle coordinate charts on multiplicity free spaces which is one of our main results.Section 6 contains the proof of Theorem 1.3 on the Gromov width of coadjoint orbits. We are indebted to A. Berenstein for introducing us to the notion of potential and for his invaluable advice.We would like to thank A. Pelayo for his interesting suggestions.Research of AA and YL was supported in part by the National Center for Competence in Research(NCCR) SwissMAP and by the grants number 178794 and 178828 of the Swiss National Science Founda-tion. JL thanks the Fields Institute as well as the organizers of the Toric Topology and Polyhedral Productsthematic program for the support of a Fields Postdoctoral Fellowship during the writing of this paper. BHwas supported in part by the National Science Graduate Research Fellowship, grant number DGE-1650441.
Some of the key statements of this Section are Example 2.3 which introduces the universal multiplicity freespace K × ˚ t ∗ + , the Delinearization Theorem 2.14 which explains how to replace k ∗ valued moment mapswith K ∗ valued ones, Example 2.18 which is an application of this technique to K × ˚ t ∗ + , Proposition 2.26which gives a criterion for cluster algebras to be homogeneous and Corollary 2.28 which states that doubleBruhat cells are homogeneous cluster varieties. In this section we fix notation which will be used throughout the article. Much of the material was developedin [10, 20]. 11 .1.1 The groups K and G In all that follows, K will be a compact connected Lie group, and G will be its complexification. Then G isa connected reductive complex algebraic group. Pick a pair of opposite Borel subgroups B, B − ⊂ G , andlet H = B ∩ B − be the Cartan subgroup. Let T = H ∩ K be the maximal torus of K , and let N ⊂ B and N − ⊂ B − be the unipotent radicals. We denote the Lie algebras of these groups with fraktur letters, so forinstance Lie( K ) = k and Lie( B − ) = b − .Let t ∗ be the R -linear dual of t . The projection k ∗ → t ∗ induces a linear isomorphism ( k ∗ ) T ∼ = t ∗ , where ( k ∗ ) T is the subspace of T -invariant elements of k ∗ . We will identify these two spaces so that t ∗ ⊂ k ∗ . Let h ∗ be the C -linear dual of h . Then t ∗ ⊗ C ∼ = h ∗ .The operation of taking the center of a Lie group (resp. algebra) is denoted by Z ( · ) (resp. z ( · ) ). TheLie algebras g and k split as the direct sums g = z ( g ) ⊕ [ g , g ] , k = z ( k ) ⊕ [ k , k ] , where [ g , g ] and [ k , k ] are semisimple. Let r denote the rank of g , defined as the rank of the semisimple partof g . Let ˜ r = dim C ( h ) . Recall the functors X ∗ = Hom( C × , − ) and X ∗ = Hom( − , C × ) . In particular, for a complex torus S ofdimension ˜ r , X ∗ ( S ) = Hom( S, C × ) ∼ = Z ˜ r , X ∗ ( S ) = Hom( C × , S ) ∼ = Z ˜ r are the lattices of characters and cocharacters of S , respectively. They come with the natural evaluationpairing h· , ·i . When H is as in the previous section, we make the standard identifications X ∗ ( H ) ⊗ Z R = √− t ∗ , X ∗ ( H ) ⊗ Z C = h ∗ , X ∗ ( H ) ⊗ Z C = h . For a character γ ∈ X ∗ ( H ) and h ∈ H , write h h γ for the evaluation of γ at h .Let R ⊂ X ∗ ( H ) be the set of roots of G , and R ∨ ⊂ X ∗ ( H ) the set of coroots. The choice ofpositive Borel subgroup B determines the positive roots R + ⊂ R . Fix an enumeration of the simple roots α , . . . , α r ∈ R + and simple coroots α ∨ , . . . , α ∨ r ∈ R ∨ . We write [1 , r ] = { , . . . , r } for their index set.The Lie algebra g can be decomposed into root spaces g = h ⊕ L α ∈ R g α , where g α is the α -weight spacefor the adjoint representation of g . Let ω , . . . , ω r be the fundamental weights . By definition, h ω i , α ∨ j i = δ i,j , h ω i , z ( t ) i = 0 , ∀ i, j ∈ [1 , r ] . The positive Weyl chamber t ∗ + ⊂ k ∗ is the intersection of half spaces defined by simple coroots α ∨ , . . . , α ∨ r , t ∗ + = { ξ ∈ t ∗ | h√− ξ, α ∨ i i > , ∀ i = 1 , . . . , r } . For each J ⊂ [1 , r ] define σ J = { ξ ∈ t ∗ + | h√− ξ, α ∨ i i > if and only if i ∈ J } . The subsets σ J , J ⊂ [1 , r ] , define a stratification of t ∗ + . The maximal stratum of t ∗ + (with respect to theinclusion partial order) is the relative interior of t ∗ + , denoted ˚ t ∗ + .The fundamental weights and simple roots of g generate the weight lattice P and root lattice Q , respec-tively. Let P + = √− t ∗ + ∩ P and X ∗ + ( H ) = √− t ∗ + ∩ X ∗ ( H ) denote the sets of dominant weights anddominant characters, respectively. 12 .1.3 SL triples and the Weyl group The Weyl group W = Norm G ( H ) /H acts on the character lattice X ∗ ( H ) by h wγ = ( ˜ w − h ˜ w ) γ , (6)where h ∈ H, w ∈ W, γ ∈ X ∗ ( H ) , and ˜ w ∈ Norm G ( H ) is any lift of w . The action (6) does not dependon the choice of lift ˜ w . The Weyl group is generated by the simple reflections s i , i ∈ [1 , r ] , whose actionon γ ∈ X ∗ ( H ) is given by s i ( γ ) = γ − γ ( α ∨ i ) α i . Let w be the longest element in W , whose length ℓ ( w ) in terms of the simple reflections s i is m := ℓ ( w ) .Consider the Chevalley generators e i , f i , α ∨ i of [ g , g ] . Define x i ( a ) := exp( ae i ) ∈ N, y i ( a ) := exp( af i ) ∈ N − . We lift the Weyl group to
Norm G ( H ) by setting s i = x i ( − y i (1) x i ( − . If w ∈ W and w = s i · · · s i l is any reduced expression for w , define w = s i · · · s i l ∈ G . The s i ’s satisfythe Coxeter relations of W , so the definition of w does not depend on the choice of reduced expression. G Consider the C -antilinear involution ( · ) † : g → g which fixes a and has e † i = f i for all Chevalley generators e i . This lifts to an anti-holomorphic Lie group anti-involution ( · ) † : G G : g g † . The fixed points of g ( g − ) † form the compact subgroup K of G .Additionally, consider the C -antilinear involution ( · ) : g → g which fixes a and has e i = e i and f i = f i for all Chevalley generators e i , f i . This lifts to an anti-holomorphic Lie group involution ( · ) : G → G which restricts to an involution of K . An element x ∈ G is Gaussian decomposable if x ∈ G = N − HN . The set G is an open subvariety of G . For x ∈ G , write x = [ x ] − [ x ] [ x ] + , where [ x ] − ∈ N − , [ x ] ∈ H, [ x ] + ∈ N. Similarly, write [ x ] > = [ x ] [ x ] + and [ x ] = [ x ] − [ x ] .Let γ ∈ X ∗ + ( H ) be a dominant character. The principal minor ∆ γ ∈ C [ G ] is the regular functionuniquely determined by ∆ γ ( x ) = [ x ] γ , for ∀ x ∈ G . Now, let u, v ∈ W . The generalized minor ∆ uγ,vγ ∈ C [ G ] is defined to be ∆ uγ,vγ ( x ) := ∆ γ ( u − xv ) . h, h ′ ∈ H , the generalized minors satisfy ∆ uγ,vγ ( hxh ′ ) = h uγ ∆ uγ,vγ ( x ) h ′ vγ (7)for all x ∈ G . When G = SL n ( C ) , the generalized minors are minors.For each pair of Weyl group elements ( u, v ) , a double Bruhat cell and a reduced double Bruhat cell aredefined respectively by G u,v := BuB ∩ B − vB − ; L u,v := N uN ∩ B − vB − . A point x ∈ G u,v is contained in L u,v if and only if [ u − x ] = 1 . (8)Multiplication in G induces a biregular isomorphism H × L u,v ∼ = G u,v .If p : b G → G is a covering group of G , then (8) implies that the covering morphism induces biregularisomorphisms p : b L u,v → L u,v (9)from the reduced double Bruhat cells of b G to those of G . For any weight γ ∈ P of G , if γ is not containedin X ∗ ( H ) , we can still make sense of the generalized minor ∆ uγ,vγ ∈ C [ L u,v ] by identifying L u,v with the reduced double Bruhat cell of any covering group b G for which γ is a character.In what follows we will focus on the double Bruhat cell G w ,e , which is an open subvariety of B − . Werecall a special case of the twist map of [20]. Our twist map is the biregular involution ζ : G w ,e → G w ,e , x ([ w − x ] > ) θ , (10)where g g θ is a Lie group involution of G determined by x i ( t ) θ = y i ( t ) , for i ∈ [1 , r ]; h θ = h − , for h ∈ H. For a generalized minor ∆ γ,δ , let ∆ ζγ,δ := ∆ γ,δ ◦ ζ be the twisted minor . For γ ∈ X ∗ + ( H ) , ∆ ζw γ,γ = ∆ − γ,γ , ∆ ζγ,γ = ∆ − w γ,γ . (11) Let G ∨ be the Langlands dual group of G . For a fixed Cartan subgroup H ∨ ⊂ G ∨ , one has X ∗ ( H ) = X ∗ ( H ∨ ); X ∗ ( H ∨ ) = X ∗ ( H ) . The group G ∨ is important in describing the representation theory of G ; see Section 3.4 below. As with H ∨ , we superscript with ∨ those varieties and maps associated with G ∨ (as opposed to G ). So for instancethe reduced double Bruhat cells of G ∨ are L ∨ ; u,v rather than L u,v .14 .1.7 Comparison map and bilinear form on g The Cartan matrix A of G is the r × r matrix with entries A i,j = h α j , α ∨ i i . Let D = diag( d , . . . , d r ) be a r × r diagonal matrix, with entries d i ∈ Z , so that AD = ( AD ) T . The matrix D is then a symmetrizer of A . Fix a non-degenerate, g -invariant bilinear form ( · , · ) g : g ⊗ C g → C , as follows. Pick γ ∨ , . . . γ ∨ ˜ r − r ∈ X ∗ ( H ) ∩ z ( g ) so that α ∨ , . . . , α ∨ r , γ ∨ , . . . , γ ∨ ˜ r − r forms a C -basis of h . Then ( · , · ) g is uniquely determinedby: ( α ∨ i , α ∨ j ) g = A i,j d j , ( γ ∨ i , γ ∨ j ) g = δ i,j , z ( g ) = [ g , g ] ⊥ . Denote the induced bilinear form on g ∗ by ( · , · ) g ∗ . If V is a (real or complex) linear subspace of g or g ∗ , denote the restriction of the bilinear form to V by ( · , · ) V . When it is clear from context we omit thesubscripts.The comparison map ψ g : g → g ∗ (12)is the C -linear map determined by ( · , · ) g : h ψ g ( X ) , Y i = ( X, Y ) g , ∀ X, Y ∈ g . For k ⊂ g and h ⊂ g , we have similarly defined maps ψ k : k → k ∗ , ψ h : h → h ∗ (13)Then ψ h ( α ∨ i ) = d i α i . We will assume we have chosen D so that ψ h restricts to a Z -module homomor-phism X ∗ ( H ) → X ∗ ( H ) = X ∗ ( H ∨ ) , which is possible by [3, Proposition 2.2]. This induces a grouphomomorphism Ψ H : H → H ∨ . K -manifolds This section recalls basic facts and conventions regarding Hamiltonian group actions. More details may befound in [28].The Hamiltonian vector field of a smooth function f ∈ C ∞ ( M ) on a symplectic manifold ( M, ω ) is thevector field X f defined by the equation ι X f ω = − df . Given a Lie group K with Lie algebra k and a smoothleft action of K on a manifold M , the fundamental vector field of X ∈ k is the vector field X ∈ X ( M ) defined point-wise as X m = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( − tX ) · m, m ∈ M. Definition 2.1 (Hamiltonian K action) . Suppose that ( M, ω ) is a symplectic manifold and K is a connectedLie group. A smooth left action of K on M is Hamiltonian if there is a K -equivariant map µ : M → k ∗ such that ι X ω = − d h µ, X i for all X ∈ k . The map µ is called a moment map for the action of K . The dataof ( M, ω ) together with a Hamiltonian action of K and a moment map µ is a Hamiltonian K -manifold ,denoted ( M, ω, K, µ ) or, when it is unambiguous, simply ( M, ω, µ ) .A map of Hamiltonian K -manifolds ( M, ω, µ ) and ( M ′ , ω ′ , µ ′ ) is a K -equivariant symplectic map F : M → M ′ such that µ ′ ◦ F = µ . If such an F is also a symplectomorphism, then we say that F is an isomorphism of Hamiltonian K -manifolds. 15et K be a compact connected Lie group with positive Weyl chamber t ∗ + as in Section 2.1. For ev-ery connected Hamiltonian K -manifold ( M, ω, µ ) there is a unique stratum σ ⊂ t ∗ + , called the principalstratum of ( M, ω, µ ) , with the property that µ ( M ) ∩ σ is dense in µ ( M ) ∩ t ∗ + [28]. The principal stratumof ( M, ω, µ ) is the maximal stratum of t ∗ + with the property that µ ( M ) ∩ σ is non-empty. If the principalstratum of ( M, ω, µ ) is ˚ t ∗ + , then the symplectic cross-section theorem can be stated as follows. Theorem 2.2 (Symplectic cross-section theorem) . [28, Theorem 26.7, Proposition 41.2] Let K be a com-pact connected Lie group and let ( M, ω, µ ) be a connected Hamiltonian K -manifold with principal stratum ˚ t ∗ + . Then,(i) The preimage, µ − (˚ t ∗ + ) , is a non-empty, symplectic, T -invariant submanifold of M .(ii) The action of T on µ − (˚ t ∗ + ) is Hamiltonian with moment map µ . Example 2.3 (Symplectic cross-section of T ∗ K ) . Let T ∗ K denote the cotangent bundle of K . Fix thestandard trivializations, T ∗ K ∼ = K × k ∗ and T ( T ∗ K ) ∼ = K × k ∗ × k × k ∗ by left invariance. The canonicalsymplectic form on T ∗ K is ω can = − dθ taut , where θ taut is the tautological 1-form.There is a (left) action of K × K on T ∗ K by the cotangent lift of left and right multiplication. In termsof the trivialization, the action is ( k , k ) · ( k, ξ ) = ( k kk − , Ad ∗ k ξ ) . This action is Hamiltonian withmoment map ( µ L , µ R ) : K × k ∗ → k ∗ × k ∗ , ( k, ξ ) (Ad ∗ k ξ, − ξ ) . (14)The symplectic cross-section of T ∗ K with respect to the action of the right copy of K and the oppositeWeyl chamber − t ∗ + is µ − R ( − ˚ t ∗ + ) = K × ˚ t ∗ + . It is a Hamiltonian K × T manifold. This section recalls the definition of an additional Hamiltonian T -action, called the Thimm torus action ,that is defined on certain dense subsets of Hamiltonian K -manifolds and commutes with the action of K .See [26].Let ( M, ω, µ ) be a connected Hamiltonian K -manifold with principal stratum ˚ t ∗ + . Consider the sym-plectic cross-section of ( M, ω, µ ) with respect to t ∗ + and the symplectic cross-section of ( T ∗ K, ω can , µ R ) with respect to the opposite Weyl chamber − t ∗ + (see Example 2.3). Their symplectic product, ( K × ˚ t ∗ + × µ − (˚ t ∗ + ) , ω can ⊕ ω ) , is a Hamiltonian K × T × T -manifold, with action ( k ′ , t , t ) · ( k, ξ, x ) = ( k ′ kt − , ξ, t · x ) and momentmap ( µ L , µ R , µ ) : K × ˚ t ∗ + × µ − (˚ t ∗ + ) → k ∗ × t ∗ × t ∗ , ( k, ξ, x ) (Ad ∗ k ξ, − ξ, µ ( x )) . The diagonal action of T by t · ( k, ξ, x ) = ( kt − , ξ, t · x ) is Hamiltonian with moment map ( k, ξ, x ) µ ( x ) − ξ . The symplectic reduction by the diagonal T -action is a smooth symplectic manifold, K × T µ − (˚ t ∗ + ) := ( K × µ − (˚ t ∗ + )) /T ∼ = ( K × ˚ t ∗ + × µ − (˚ t ∗ + ) , ω can ⊕ ω ) (cid:12) T. (15)The isomorphism arises from the T -equivariant map ( k, x ) ( k, µ ( x ) , x ) .Let [ k, x ] ∈ K × T µ − (˚ t ∗ + ) denote the T -equivalence class of ( k, x ) . The reduced space (15) is aHamiltonian K × T -manifold, with action ( k ′ , t ) · [ k, x ] = [ k ′ kt, x ] = [ k ′ k, t · x ] and moment map ( µ L , µ R ) : K × T µ − (˚ t ∗ + ) → k ∗ × t ∗ , µ L ([ k, x ]) = Ad ∗ k µ ( x ) , µ R ([ k, x ]) = − µ R ( x ) = µ ( x ) . We include this assumption only for the sake of simplified exposition; it is not necessary in order to define a Thimm torusaction on a dense subset of M . ϕ : K × T µ − (˚ t ∗ + ) → M, [ k, x ] k · x (16)is a K -equivariant symplectomorphism onto a dense, K -invariant subset of M and satisfies µ ◦ ϕ = µ L .Denote the image of ϕ by U . As a consequence of this construction, one has a Hamiltonian action of K × T on U .The construction of an extra Hamiltonian T action on U can be rephrased in the following way. Definea map S : k ∗ → t ∗ + by letting S ( ξ ) be the unique element of the intersection O ξ ∩ t ∗ + . By K -equivariance, U = ( S ◦ µ ) − (˚ t ∗ + ) . Define an action of T on U by t ∗ x = ( k − tk ) · x, (17)where k ∈ K such that Ad ∗ k µ ( x ) ∈ ˚ t ∗ + . This action is well-defined since the stabilizer subgroup of elementsin µ − (˚ t ∗ + ) is contained in T . The restriction of S ◦ µ to U is a moment map for this action (in particular,its restriction to U is smooth) [26]. The preceding discussion can be summarized as follows. Proposition 2.4.
Let K be a compact connected Lie group, let ( M, ω, µ ) be a connected Hamiltonian K -manifold with principal stratum ˚ t ∗ + , and let U = ( S ◦ µ ) − (˚ t ∗ + ) . Then, the map K × T µ − (˚ t ∗ + ) → U, [ k, x ] k · x is an isomorphism of Hamiltonian K × T -manifolds, i.e. it is a K × T -equivariant symplectomorphism andthe following diagrams commute. K × T µ − (˚ t ∗ + ) U k ∗ k ∗ µ L µ = K × T µ − (˚ t ∗ + ) U t ∗ + k ∗ µ R µ S (18)The extra Hamiltonian T action on U is often referred to as the Thimm torus action . It does not ingeneral extend to a well-defined action of T on M . A Hamiltonian T -manifold ( M, ω, µ ) is completely integrable if T acts locally transitively on fibers of µ .If the action of T is also effective, then ( M, ω, µ ) is a toric manifold (or toric T -manifold if we want toemphasize the rôle of the torus). Example 2.5.
As in Example 2.3, the cotangent bundle T ∗ T of a compact torus T has a canonical sym-plectic structure, ω std = − dθ taut . With respect to the trivialization T ∗ T = t ∗ × T , the action of T on T ∗ T by cotangent lift of left multiplication is t · ( λ, t ′ ) = ( λ, tt ′ ) and the moment map is pr( λ, t ) = λ (cf. theformula for µ L in Equation (14)). For any open subset U ⊂ t ∗ , ( U × T, ω std , pr) is a toric manifold. Forthe torus T n = S × · · · × S we fix the identification Lie( S ) ∗ = R such that ω std is given by Equation(1). By equivariance of the moment map, a Hamiltonian K -manifold ( M, ω, µ ) is a multiplicity free space ifand only if the action of the maximal torus T ⊂ K on the symplectic cross-section is completely integrable. Although we like to call it the Thimm torus action, we note that to our knowledge this torus action originated in the work ofGuillemin and Sternberg [26]. ( M, ω, µ ) be a compact, connected multiplicity free space with principal stratum ˚ t ∗ + , Kirwan poly-tope △ = µ ( M ) ∩ t ∗ + , and principal isotropy subgroup L T . Let ˚ △ denote the relative interior of △ .Then W = µ − (˚ △ ) is an open, dense, T -invariant subset of the symplectic cross-section µ − (˚ t ∗ + ) . Thekernel of the action of T on W is the subgroup L and the action of T /L on W is free. The convex set ˚ △ spans an affine subspace of t ∗ that is a translation of the subspace of t ∗ given by the image of the map Lie(
T /L ) ∗ → t ∗ dual to the quotient map t → Lie(
T /L ) . In particular, the dimension of ˚ △ and T /L areequal and ˚ △ is open as a subset of this affine subspace [43]. Fix a linear identification of Lie(
T /L ) ∗ withthe affine subspace spanned by ˚ △ .With respect to the identification of Lie(
T /L ) ∗ with the affine subspace spanned by ˚ △ , the map µ : W → ˚ △ is a moment map for the free action of T /L on W . The tuple ( W, ω, µ ) is thus a toric T /L -manifold.Define the toric
T /L -manifold (˚ △ × ( T /L ) , ω std , pr) as in Example 2.5, taking care to note that Lie(
T /L ) ∗ is identified with the affine subspace of t ∗ spanned by ˚ △ . It follows from compactness of M that µ is proper as a map to ˚ △ . By the classification of non-compact toric manifolds with proper mo-ment maps, ( W, ω, µ ) is isomorphic as a toric T /L -manifold to (˚ △ × ( T /L ) , ω std , pr) [39]. The precedingdiscussion is summarized as follows. Note that the commutative diagram (19) is stated in terms of maps to t ∗ . This is due to our identification of Lie(
T /L ) ∗ with an affine subspace of t ∗ . Lemma 2.6.
Let ( M, ω, µ ) be a compact, connected, multiplicity free Hamiltonian K -manifold with prin-cipal stratum ˚ t ∗ + , Kirwan polytope △ = µ ( M ) ∩ t ∗ + , and principal isotropy subgroup L T . Let ( W, ω, µ ) and (˚ △ × ( T /L ) , ω std , pr) be the toric T /L -manifolds defined as above. Then ( W, ω, µ ) and (˚ △ × ( T /L ) , ω std , pr) are isomorphic as Hamiltonian T /L -manifolds, i.e. there is a T -equivariant (there-fore also T /L -equivariant) symplectomorphism ( W, ω ) ∼ = (˚ △ × ( T /L ) , ω std ) such that the following dia-gram commutes. W ˚ △ × ( T /L ) t ∗ t ∗ µ ∼ = pr= (19) This section recalls the relevant details of compact connected Poisson-Lie groups and their Hamiltonianactions from [1, 6, 51]. For further background on Poisson-Lie groups the reader may consult [15, 16, 46].In general, a
Poisson-Lie group , ( K, π ) , is a Lie group K equipped with a Poisson bivector field π suchthat the group multiplication map K × K → K is a Poisson map (with respect to the product Poissonstructure on K × K ). The bivector field π necessarily vanishes at the group identity and therefore has alinearization δ π : k → k ∧ k . The linearization is a Lie algebra 1-cocycle and the dual map, δ ∗ π : k ∗ ∧ k ∗ → k ∗ , isa Lie bracket. In other words, the tuple ( k , [ · , · ] k , δ π ) is a Lie bialgebra. Two Lie bialgebras, ( k , [ · , · ] k , δ k ) and ( l , [ · , · ] l , δ l ) , are dual to one another if there is a dual pairing, i.e. a non-degenerate bilinear map h· , ·i : k × l → R , such that h [ X, Y ] k , ξ i = h X ∧ Y, δ l ( ξ ) i and h X, [ η, ξ ] l i = h δ k ( X ) , η ∧ ξ i ∀ X, Y ∈ k , η, ξ ∈ l . (20)Two Poisson-Lie groups are dual to one another if their linearizations are dual to each other as Lie bialge-bras. It is common in the literature to define the dual Poisson-Lie group of a Poisson Lie group ( K, π ) to be the unique simplyconnected dual Poisson-Lie group. Manin triple is a tuple of Lie algebras ( g , k , l ) along with a non-degenerate ad -invariant bilinear formon g such that: i) k and l are Lie subalgebras of g , ii) g = k ⊕ l as vector spaces, and iii) k and l are isotropicwith respect to the bilinear form. Given a Manin triple ( g , k , l ) the bilinear form on g defines a dual pairingbetween k and l . If δ k and δ l denote the maps defined by Equation (20) with respect to this pairing, then ( k , [ · , · ] k , δ k ) and ( l , [ · , · ] l , δ l ) are dual Lie bialgebras. Let K be a compact connected Lie group with complexification G . For all s ∈ R , s = 0 , define thereal-valued bilinear form ⟪ · , · ⟫ s := 2 s Im( · , · ) g . Denote the real Lie subalgebra a ⊕ n − ⊂ g by an − and the corresponding connected subgroup of G by AN − . The tuple ( g , k , an − ) along with ⟪ · , · ⟫ s is a Manin triple. The dual Lie bialgebras defined by thisManin triple are ( k , [ · , · ] k , δ k ,s ) and ( an − , [ · , · ] an − , δ an − ,s ) , where δ k ,s and δ an − ,s are defined by Equation (20).Denote δ k := δ k , and δ an − := δ an − , , and observe that δ k ,s = sδ k and δ an − ,s = sδ an − . The Lie bialgebras ( k , [ · , · ] k , sδ k ) and ( an − , [ · , · ] an − , sδ an − ) integrate to dual Poisson-Lie groups ( K, sπ K ) and ( AN − , sπ K ∗ ) .For this reason, we often denote K ∗ := AN − . Remark 2.7. If b G → G is a covering group of G , then the covering map restricts to a Lie group isomor-phism from d AN − ⊂ b G to AN − ⊂ G . For this reason, we often assume that G is of the form G ∼ = Z × G sc , (21)where G sc is semisimple and simply connected, and Z is an algebraic torus with Lie algebra z ( g ) . Remark 2.8.
For K , which is not assumed to be simply connected, the integration of δ k is achieved bydefining an anti-symmetric r -matrix Λ ∈ k ∧ k such that the Lie algebra 1-cocycle δ k is a coboundary, i.e. ∂ Λ = δ k . The formula for this r -matrix is the same as in the semisimple or simple case (see e.g. [16]).The Poisson-Lie group structure on K is then defined as π K := Λ L − Λ R , where Λ L and Λ R denote theright and left invariant bivector fields on K with Λ L ( e ) = Λ R ( e ) = Λ . Remark 2.9.
Although there is some choice in the definition of the bilinear form ( · , · ) g , the Poisson-Liegroups K and AN − only depend on the restriction of ( · , · ) g to [ g , g ] , since δ k | z ( k ) = 0 and δ an − | i z ( k ) = 0 . Definition 2.10.
For all k ∈ K and b ∈ AN − , the equation kb = b k k b (22)uniquely determines elements b k ∈ AN − and k b ∈ K by the Iwasawa decomposition, G = AN − K . Thisdefines a left action K × AN − → AN − , ( k, b ) b k , and a right action, AN − × K → K , ( k, b ) k b .These are the dressing actions of K and AN − on each other.Note that since T normalizes AN − , the dressing action of T on AN − is simply conjugation. Most references, such as [51], prefer to use an + in their definition of the standard Manin triple on g . Our convention isequivalent: the map g → g , x
7→ − x † , defines an isomorphism of Manin triples ( g , k , an − , ⟪ · , · ⟫ s ) ∼ = ( g , k , an + , ⟪ · , · ⟫ − s ) . .3.2 Explicit formula for the Poisson bracket on K ∗ The real Poisson structure π K ∗ can be described in terms of a coboundary Poisson-Lie group structure on G as follows.Fix a C -basis of g , consisting of E α ∈ g α , α ∈ R , and X j ∈ a , j = 1 , . . . , ˜ r , subject to ( E α , E − α ) g = 1 , ∀ α ∈ R, ( X i , X j ) g = δ i,j , i, j ∈ [1 , ˜ r ] . (23)Taking tensors over R , define r = 12 ˜ r X j =1 ( √− X j ) ⊗ X j + 12 X α ∈ R + (cid:0) E α ⊗ ( √− E − α ) + ( √− E α ) ⊗ E − α − E − α ⊗ √− E − α + ( √− E − α ) ⊗ E − α (cid:1) . (24)Let π G = r L − r R , where r L and r R denote the left and right-invariant tensor fields on G satisfying r L ( e ) = r R ( e ) = r . If G is viewed as a complex manifold and π G a section of the complexification of ∧ T G , then the formula simplifies, taking tensors over C : r = √− ˜ r X j =1 X j ⊗ X j + √− X α ∈ R + E α ⊗ E − α . (25)The dual Poisson Lie group ( AN − , π K ∗ ) is an anti-Poisson submanifold of ( G, π G ) . The following isimmediate from the expression (24) for r . Here and in what follows, we follow the convention that, for X ∈ b − , and f ∈ C ∞ ( B − , C ) , the action of X is ( X · f )( b ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f (exp( − tX ) b ) , ( f · X )( b ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( b exp( tX )) . Lemma 2.11.
Let f and g be complex valued functions defined on an open subset of B − . If f is holomorphicand g is anti-holomorphic, then { f | K ∗ , g | K ∗ } K ∗ = √− ˜ r X j =1 (( X j · f )( X j · g ) − ( f · X j )( g · X j )) | K ∗ + √− X α ∈ R + (( E − α · f )( E − α · g ) − ( f · E − α )( g · E − α )) | K ∗ . Proof.
Computing directly from the definition (24), one finds for b ∈ K ∗ , { f | K ∗ , g | K ∗ } K ∗ ( b ) = −{ f, g } G ( b ) = ι dg | b ι df | b (r b − b r)= ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 √− ˜ r X j =1 ( f (exp( tX j ) b ) g (exp( tX j ) b ) − f ( b exp( tX j )) g ( b exp( tX j )))+ √− X α ∈ R + ( f (exp( tE − α ) b ) g (exp( tE − α ) b ) − f ( b exp( tE − α )) g ( b exp( tE − α ))) . where in the last line we use that df is C -linear and dg is C -antilinear. The result then immediately follows.20 .3.3 Hamiltonian K -manifolds with K ∗ -valued moment maps Let ( K, π ) be a Poisson-Lie group and let ( M, Π) be a Poisson manifold. A smooth left action of K on M is a Poisson action if the action map K × M → M is a Poisson map (with respect to the product Poissonstructure on K × M ). Let θ R ∈ Ω ( AN − , an − ) denote the right invariant Maurer-Cartan form. Definition 2.12 (Hamiltonian K -manifold with K ∗ -valued moment map) . Let s = 0 . A Poisson actionof ( K, sπ K ) on a symplectic manifold ( M, ω ) is Hamiltonian with K ∗ -valued moment map if there is a K -equivariant map Ψ : M → AN − such that ι X ω = − Ψ ∗ ⟪ θ R , X ⟫ s , ∀ X ∈ k . The map Ψ is a K ∗ -valued moment map . The tuple ( M, Ω , Ψ) is a Hamiltonian K -manifold with K ∗ -valued moment map .A K ∗ -valued moment map is a Poisson map to ( AN − , sπ K ∗ ) [49, Theorem 4.8]. Hamiltonian K -manifolds with K ∗ -valued moment maps can be defined more generally for Poisson actions of K under theassumption that dual Poisson-Lie groups K and K ∗ admit a double [49]. K -manifolds Let ( K, sπ K ) and ( AN − , sπ K ∗ ) be as in Section 2.3.1. For all s ∈ R , define a map E s : k ∗ → AN − as follows. Recall from (13) the comparison map ψ k : k → k ∗ given by the choice of form ( · , · ) g . For all λ ∈ t ∗ , define E s ( λ ) := exp( s √− ψ − k ( λ )) . Extend the definition of E s to all of k ∗ by K -equivariancewith respect to the coadjoint action and the dressing action, i.e. E s (Ad ∗ k λ ) := E s ( λ ) k ∀ k ∈ K. For s = 0 , E s is a K -equivariant diffeomorphism.Let θ L ∈ Ω ( AN − , an − ) denote the left invariant Maurer-Cartan form. Following [1], for all s ∈ R ,define β s := 12 √− H ( E ∗ s ( θ L ∧ ( θ L ) † ) g ) ∈ Ω ( k ∗ ) (26)where H : Ω • ( k ∗ ) → Ω •− ( k ∗ ) is the Poincaré homotopy operator defined by the homotopy [0 , × k ∗ → k ∗ , ( t, λ ) tλ , and ( θ L ) † ∈ Ω ( AN − , an + ) denotes the anti-involution ( · ) † applied to the values of θ L . Lemma 2.13.
The 2-form β s is invariant with respect to the coadjoint action of T .Proof. Since the homotopy used to define H is Ad ∗ -equivariant and the map E s is equivariant, it sufficesto show that ( θ L ∧ ( θ L ) † ) g is invariant with respect to the dressing action of T .The dressing action of T on AN − coincides with the action by conjugation, i.e. b t = tbt − for all t ∈ T and b ∈ AN − (see Definition 2.10). Let C t denote conjugation by t , and let Ad t θ L denote the adjointaction of t on the values of θ L . Then C ∗ t θ L = Ad t θ L , and C ∗ t ( θ L ∧ ( θ L ) † ) g = (( C ∗ t θ L ) ∧ ( C ∗ t θ L ) † ) g = ((Ad t θ L ) ∧ (Ad t θ L ) † ) g . A short calculation shows that ((Ad t θ L ) ∧ (Ad t θ L ) † ) g = (( θ L ) ∧ ( θ L ) † ) g .21et M be a smooth manifold equipped with a smooth left action of K . Given a symplectic form ω ∈ Ω ( M ) and a map µ : M → k ∗ , define Ψ s := E s ◦ µ, Ω s := ω + µ ∗ dβ s . (27)For all s = 0 the forms Ω s are symplectic [1]. Theorem 2.14 (Linearization/Delinearization Theorem [1]) . Let M be a smooth manifold equipped with asmooth left action of K . Let ω, Ω s ∈ Ω ( M ) and Ψ s , µ be related by (27) . Then, for all s = 0 , ( M, Ω s , Ψ s ) is a Hamiltonian ( K, sπ K ) -manifold with ( AN − , sπ K ∗ ) -valued moment map if and only if ( M, ω, µ ) is aHamiltonian K -manifold. Let π k ∗ denote the canonical Poisson structure on k ∗ and let π σ s k ∗ denote the gauge transformation of π k ∗ by σ s = dβ s . Define a vector field Y s on k ∗ by the equation Y s = ( π σ s k ∗ ) ♯ (cid:18) − ∂∂s β s (cid:19) (28)and let ψ s denote the flow of Y s . By Moser’s trick for Poisson manifolds, ( ψ s ) ∗ π k ∗ = π σ s k ∗ for all s such that ψ s is defined. On any Hamiltonian K -manifold ( M, ω, µ ) , define a Moser vector field X s by the equation ι X s Ω s = − ∂∂s µ ∗ β s , s ∈ R . (29)and let φ s denote the flow of X s . By Moser’s trick, φ ∗ s Ω s = ω for all s such that φ s is defined.Let wt : AN − → A be given by wt( x ) = [ x ] . If ( M, Ω , Ψ) is a Hamiltonian ( K, sπ K ) -manifold with K ∗ = AN − -valued moment map, then ( M, Ω , E − s ◦ wt ◦ Ψ) is a Hamiltonian T -manifold, where T actsas the maximal torus of K . Proposition 2.15.
The flows ψ s and φ s have the following properties (valid for all s such that ψ s and φ s are defined).(i) π σ s k ∗ = ( E − s ) ∗ sπ K ∗ .(ii) The flow ψ s preserves coadjoint orbits. In particular, ψ s is defined for all s .(iii) ψ s ◦ µ = µ ◦ φ s .(iv) Both ψ s and φ s are equivariant with respect to the action of T as the maximal torus of K .(v) The flow φ s is equivariant with respect to the Thimm torus action defined on the dense subset U ⊂ M (see Section 2.2.1).(vi) The following diagrams commute. M M t ∗ + t ∗ + S ◦ µ φ s S ◦ µ = M M k ∗ K ∗ t ∗ A µ φ s Ψ s pr t ∗ wt E s (30) See e.g. [12] for more details on gauge transformations of Poisson structures. roof. (i) This is [1, Theorem 1] (see also the remark at the end of [1]).(ii) The first claim follows from (i) and the definition of Y s . The second claim follows by the EscapeLemma since coadjoint orbits are compact.(iii) It suffices to prove that µ ∗ X s = Y s . Letting (Ω s ) − denote the Poisson structure on M defined by thesymplectic structure Ω s , observe that (29) can be re-written equivalently as X s = ((Ω s ) − ) ♯ (cid:18) − ∂∂s µ ∗ β s (cid:19) . Since Ψ s = E s ◦ µ is a Poisson map with respect to sπ K ∗ , it follows by (i) that µ = E − s ◦ Ψ s is a Poissonmap with respect to π σ s k ∗ . Thus µ ∗ X s = ( π σ s k ∗ ) ♯ (cid:18) − ∂∂s β s (cid:19) = Y s . (iv) Follows by Lemma 2.13 along with Equations (27), (28), (29), and equivariance of µ .(v) The map µ is equivariant as a map from U to k ∗ with respect to the Thimm torus action of T on U andthe trivial action of T on k ∗ . Since β s is invariant with respect to the trivial action of T on k ∗ , it follows bythe same argument as (iv) that φ s is equivariant with respect to the Thimm torus action of T on U .(vi) The first diagram is a direct consequence of (iii) since S is constant on coadjoint orbits.The action of T on ( M, ω ) as the maximal torus of K is Hamiltonian with moment map pr t ∗ ◦ µ . Theaction of T on ( M, Ω s ) as the maximal torus of K is Hamiltonian with moment map E − s ◦ wt ◦ Ψ s . Since M is connected and φ s : ( M, ω ) → ( M, Ω s ) is a T -equivariant symplectomorphism, it follows by uniquenessof moment maps that there is an element ξ ∈ t ∗ such that E − s ◦ wt ◦ Ψ s ◦ φ s = pr t ∗ ◦ µ + ξ. The set µ − ( t ∗ ) is non-empty by K -equivariance of µ . Let m ∈ µ − ( t ∗ ) . Then it follows by (27) and(iii) that E − s ◦ wt ◦ Ψ s ◦ φ s ( m ) = E − s ◦ wt ◦ E s ◦ µ ( φ s ( m ))= E − s ◦ wt ◦ E s ◦ ψ s ( µ ( m )) . By (ii) and (iv), ψ s fixes elements of t ∗ . Thus E − s ◦ wt ◦ Ψ s ◦ φ s ( m ) = E − s ◦ wt ◦ E s ◦ µ ( m )= µ ( m ) = pr t ∗ ◦ µ ( m ) , and so ξ = 0 . Thus the second diagram commutes.The flow φ s is defined for all s if M is compact. However, one does not need such a strong assumption. Corollary 2.16.
Let ( M, ω, µ ) be a Hamiltonian K -manifold such that the connected components of thefibers of µ are compact. Then, the symplectic forms ω and Ω s are isotopic for all s = 0 . In particular, if ( M, ω, µ ) is multiplicity free, then ω and Ω s are isotopic for all s .Proof. Since µ is K -equivariant and K is compact, connected components of fibers of µ are compact ifand only if the connected components of µ − ( O ) are compact for every coadjoint orbit O . Let m ∈ M . ByProposition 2.15 (ii) and (iii), the flow φ s ( m ) is contained in a connected component of µ − ( K · µ ( m )) .The result then follows by the Escape Lemma for flows of smooth vector fields. The final claim followssince K acts transitively on connected components of µ − ( O ) when ( M, ω, µ ) is multiplicity free.23 emark 2.17. Using a similar argument, one can show that E ◦ ψ : ( k ∗ , π k ∗ ) → ( K ∗ , π K ∗ ) is well defined.It is therefore a Ginzburg-Weinstein isomorphism in the sense of Theorem 1.5. Example 2.18.
Recall the Hamiltonian K × T -manifold ( K × ˚ t ∗ + , ω can , ( µ L , µ R )) from Example 2.3.Since T is a torus, the Poisson-Lie group structure sπ K × T defined in Section 2.3.1 equals sπ K and the dualPoisson-Lie group is ( AN − × A, sπ K ∗ ) . The map E s for K × T is (cid:18) E s , exp (cid:18) s √− ψ − k (cid:19)(cid:19) : k ∗ × t ∗ → AN − × A (31)Since T is a torus, the 2-form β s ∈ Ω ( k ∗ × t ∗ ) coincides with the pullback of the 2-form β s ∈ Ω ( k ∗ ) under projection k ∗ × t ∗ → k ∗ . Thus, Ψ s = (cid:18) E s ◦ µ L , exp (cid:18) s √− ψ − k ◦ µ R (cid:19)(cid:19) , Ω s = ω can + µ ∗ L dβ s . (32)The action of T on K × ˚ t ∗ + coincides with the action of the Thimm torus of K . It follows by Proposition 2.15(ii), (iii), and (v) that the action of T on ( K × ˚ t ∗ + , Ω s ) is also Hamiltonian with moment map µ R .An explicit formula for the Poisson structure defined by Ω s can be given as follows. Fix the same basisof g consisting of elements X j ∈ h and root vectors E α ∈ g α as in Section 2.3.2. Let P j ∈ h ∗ denote thebasis dual to X j . Then, as a section of ∧ T C ( K × ˚ t ∗ + ) , (Ω s ) − = X α ∈ R + s √− E α ∧ E − α ) L exp(2 s √− h ξ, [ E α , E − α ] i ) − X j X Lj ∧ P j + sπ K (33)where for any X ∈ ∧ k g , X L denotes the left-invariant multivector field on K whose value at e equals X .This formula follows by combining the explicit formula for sπ K ∗ given in [50, Proposition 5.12], the factthat (Ψ s ) ∗ (Ω s ) − = sπ K ∗ , and the definition of Ω s . K ∗ The
Legendre transform of f ∈ C ∞ ( K ∗ ) is the map L f : K ∗ → k uniquely defined by the equation ⟪ θ R , L f ⟫ s = df. (34)Note that if β ∈ Ω ( K ∗ ) , then L β is defined as in (34) by replacing L f with L β and df with β . Thisis the analogue for K ∗ -valued moment maps of the Legendre transform on k ∗ used in [25, 26] to studyHamiltonian flows of collective functions (i.e. functions of the form f ◦ µ where µ is a moment map). Themain property of Legendre transforms on k ∗ generalizes directly to the setting of Hamiltonian spaces with K ∗ -valued moment maps as follows. Proposition 2.19. If ( M, Ω , Ψ) is Hamiltonian K -manifold with K ∗ -valued moment map, then ( X f ◦ Ψ ) m = L f (Ψ( m )) m , ∀ m ∈ M. (35) Proof.
By (34), the moment map equation, and the definition of X f ◦ Ψ : for all m ∈ M , − ι L f (Ψ( m )) m Ω = (Ψ ∗ ⟪ θ R , L f ⟫ s ) m = (Ψ ∗ df ) m = − ι ( X f ◦ Ψ ) m Ω . The proposition then follows by non-degeneracy of Ω .24 .4 Cluster algebras In this section, we first recall some basic definitions from cluster theory. Secondly, we recall the construc-tion of the cluster algebra structure on the double Bruhat cell G u,v . At the end, we recall the homogeneityof the cluster algebra structure on G u,v . In the rest of the paper, we mainly focus on the cell G w ,e . Moredetail can be found in [9, 31]. A seed σ = ( I, J, M ) consists of a finite set I , a subset J ⊂ I and an integer matrix M = [ M ij ] i,j ∈ I which is skew-symmetrizable, i.e. there exists an I × I diagonal matrix D with positiveinteger entries, called a skew-symmetrizer, such that M D = − ( M D ) T . The principal part of M is givenby M = [ M ij ] i,j ∈ J .Consider the ambient field F of rational functions over C in | I | independent variables. A labeled seed in F is a pair ( z σ , σ ) where σ is a seed and z σ := { z i ∈ F | i ∈ I } is a set of | I | elements forming a freegenerating set of F . We refer to z σ as a cluster and the z i as cluster variables . Denote z σ := { z i | i ∈ J } ,which is set of unfrozen variables .If ( I, J, M ) is a seed, the mutation µ k ( M ) of M in direction k ∈ J is the I × I matrix with entries: µ k ( M ) ij = − M ij , if k ∈ { i, j } ; M ij + 12 (cid:16) | M ik | M kj + M ik | M kj | (cid:17) , otherwise . If D is a skew-symmetrizer of M , then D is a skew-symmetrizer of µ k ( M ) . A seed mutation in direction k ∈ J transforms a labeled seed ( z σ , σ ) into a new labeled seed µ k ( z σ , σ ) = ( z σ ′ , σ ′ ) , where σ ′ =( I, J, µ k ( M )) and the new cluster z σ ′ ⊂ F contains the cluster variables: µ k ( z i ) = z i , if i = k ; z − k Y M jk > z M jk j + Y M jk < z − M jk j , if i = k. (36)Two seeds will be called mutation equivalent if they are related by a sequence of mutations. The equivalenceclass of a labeled seed ( z σ , σ ) is denoted by | σ | . Definition 2.21.
Let ( z σ , σ ) be a labeled seed. Let P be the C -algebra generated by { z i , z − i | i ∈ I \ J } .The cluster algebra A | σ | over P associated with the labeled seed ( z σ , σ ) is the P -subalgebra of F generatedby the set { z ∈ F | z ∈ z σ ′ , σ ′ ∈ | σ |} of all unfrozen cluster variables. In this case ( z σ , σ ) is the initial seed of A | σ | . Definition 2.22 ([19]) . The (Langlands) dual seed of a seed σ = ( I, J, M ) is σ ∨ := ( I, J, − M T ) . If ( z σ , σ ) is a labeled seed, let F ∨ be the ambient field of rational functions over C , in the variables z ∨ i , i ∈ I .The dual labeled seed is ( z ∨ σ , σ ∨ ) , where z ∨ σ = { z ∨ i ∈ F ∨ | i ∈ I } . For a labeled seed ( z σ , σ ) , denoteby A ∨| σ | := A | σ ∨ | the cluster algebra associated with ( z ∨ σ , σ ∨ ) , which we call the (Langlands) dual clusteralgebra of A σ . 25f µ is a sequence of mutations at indices j , . . . , j n ∈ J which takes a labeled seed ( z , σ ) to ( z ′ , σ ′ ) ,let µ ∨ denote the sequence of mutations at indices j , . . . , j n applied to the dual seed ( z ∨ , σ ∨ ) . It isstraightforward to verify that µ ∨ takes the dual seed ( z ∨ , σ ∨ ) to ( z ′∨ , σ ′∨ ) .Let ( z σ , σ = ( I, J, M )) be a labeled seed, and assume a skew-symmetrizer D of M has been fixed,with diagonal entries D ii = d i . For each labeled seed ( z σ ′ , σ ′ ) ∈ | σ | , define a C -algebra homomorphism Ψ ∗ σ ′ : A ∨| σ | → A | σ | , z ∨ i z d i i , z ∨ i ∈ z σ ′ . (37)Note that Ψ σ ′ = Ψ σ in general. In this section, we assume that G is a connected reductive algebraic group of rank r as before. We willrecall how to construct a cluster algebra structure on the coordinate algebra of a reduced double Bruhat cell L u,v and double Bruhat cell G u,v , for any pair ( u, v ) ∈ W × W .A double reduced word i = ( i , . . . , i n ) for ( u, v ) is a shuffle of a reduced word for u , written inthe alphabet {− r, . . . , − } , and a reduced word for v , written in the alphabet { , . . . , r } , where n = ℓ ( u ) + ℓ ( v ) . For k ∈ − [1 , r ] ∪ [1 , n ] , we denote by k + = min { j | j > k, | i j | = | i k |} . (38)If | i j | 6 = | i k | for all j > k , we set k + = n + 1 . An index k is i -exchangeable if both k, k + ∈ [1 , n ] . Let e ( i ) denote the set of all i -exchangeable indices.Fix a double reduced word i of ( u, v ) . Let I = [ − r, − ∪ [1 , n ] , J = e ( i ) , and L := [ − r, − ∪ e ( i ) ⊂ I .Construct a I × I matrix M ( i ) as in [9, Remark 2.4]: For k, l ∈ I , set p = max { k, l } and q = min { k + , l + } ,and let ǫ ( k ) be the sign of k . Then following [9, Remark 2.4], let M ( i ) kl = − ǫ ( k − l ) · ǫ ( i p ) , if p = q ; − ǫ ( k − l ) · ǫ ( i p ) · A | i k | , | i l | , if p < q and ǫ ( i p ) ǫ ( i q )( k − l )( k + − l + ) > , otherwise . (39)Here and throughout we use the convention that i − k = k , for k ∈ [1 , r ] . Recall that A is the Cartan matrixof g . Denote by M ( i ) := [ M ( i ) kl ] k,l ∈ L the L × L submatrix of M ( i ) .Recall the matrix D , which is the symmetrizer of the Cartan matrix A fixed in Section 2.1.7. Considerthe diagonal I × I matrix D with entries D j,j = D | i j | , | i j | = d | i j | ∈ Z . (40)Let D be the submatrix of D with rows and columns indexed by L . Then M ( i ) is skew-symmetrizable,with skew-symmetrizer D , and M ( i ) is skew-symmetrizable, with skew-symmetrizer D .Define the following seeds: σ ( i ) := ( I, J, M ( i )) , σ ( i ) := ( L, J, M ( i )) . If σ is obtained by mutating σ ( i ) at indices j , . . . , j n , let σ denote the seed obtained by mutating σ ( i ) atthe same indices. For a double reduced word i of ( u, v ) and k ∈ [1 , n ] , denote u k := Y l =1 ,...,ki l < s | i l | , v k := Y l = n,...,k +1 i l > s i l , where the index is increasing (resp. decreasing) in the product on the left (resp. right). Denote ∆ k := ∆ u k ω | ik | ,v k ω | ik | for k ∈ [1 , n ]; ∆ k := ∆ ω | k | ,v − ω | k | for k ∈ [ − r, − . (41)26 heorem 2.23. [9, Theorem 2.10] For every double reduced word i for ( u, v ) , the map ϕ u,v : A | σ ( i ) | → C [ L u,v ] , z k ∆ k (cid:12)(cid:12) L u,v , for k ∈ L is an isomorphism of algebras. If G is simply connected, the map φ u,v : A | σ ( i ) | → C [ G u,v ] , z k ∆ k , for k ∈ I is an isomorphism of algebras. Here z k ∈ z σ ( i ) (resp. z k ∈ z σ ( i ) ) is in the initial seed of A | σ ( i ) | (resp. A | σ ( i ) | ). Remark 2.24.
Theorem 2.10 in [9] is stated for non-reduced double Bruhat cells; the statement for reduceddouble Bruhat cells follows by passing to a cover as in (9) and specializing the frozen variables ∆ w ω i ,ω i =1 . Strictly speaking, in [9] they show there is an isomorphism from C [ G w ,e ] to the upper cluster algebra of σ ( i ) . We ignore this detail here for two reasons: First, it is known that in the case of double Bruhat cells, thecluster algebra and upper cluster algebra coincide (see for instance the introduction of [32]). Second, weonly make use of the Laurent phenomenon of A | σ ( i ) | in what follows, and so all our results hold if “cluster”is replaced by “upper cluster”.Let us apply Theorem 2.23 to the Langlands dual group G ∨ of G . Let i be a double reduced word for ( u, v ) as before. Let M ∨ ( i ) be the matrix defined in (39), with respect to the Cartan matrix A ∨ = A T of G ∨ . Denote by M ∨ ( i ) the submatrix of M ∨ ( i ) formed by taking the [ − r, − ∪ e ( i ) rows and columns.Direct computation shows that M ∨ ( i ) = − M T ( i ) . Thus C [ L ∨ ; u,v ] is the Langlands dual cluster algebra to C [ L u,v ] . In this section, we recall the notion of homogeneous cluster algebras. For a more detailed discussion, see[31, 34].
Definition 2.25.
A cluster algebra A | σ | with initial seed ( z σ , σ ) is graded by an abelian group G if thealgebra C [ z ± σ ] is graded by G and the initial cluster variables z i are homogeneous for i ∈ I . We say agraded cluster algebra is homogeneous if all cluster variables are homogeneous with respect to the grading.Denote by | z | the degree of a homogeneous element z ∈ C [ z ± σ ] . Proposition 2.26. A G -graded cluster algebra A | σ | with initial seed ( z σ , σ = ( I, J, M )) is homogeneousif and only if X i ∈ I | z i | M ij = 0 , ∀ j ∈ J, (42) where z i ∈ z σ and | z i | is the degree of z i .Proof. If A | σ | is homogeneous, the equation (42) follows from the fact that the cluster variables z ′ k ∈ z σ ′ of the seed σ ′ = µ k ( σ ) are homogeneous. To be more precise, the variable z ′ k is homogeneous if and onlyif the monomials in (36) have the same degree. Then we have: X j : M jk > | z j | M jk = − X j : M jk < | z j | M jk , which is equivalent to (42). 27or the other direction, by induction, all we need to show is X i ∈ I | µ k ( z i ) | µ k ( M ) ij = 0 , ∀ k, j ∈ J. (43)First all, note that µ k ( z k ) has degree: | µ k ( z k ) | = −| z k | + 12 X i ∈ I | z i || M ik | . Then for j = k , noting that M kk = 0 , we have: X i ∈ I | µ k ( z i ) | µ k ( M ) ij = 2 X i = k | z i | µ k ( M ) ij + 2 | µ k ( z k ) | µ k ( M ) kj = X i = k | z i | (2 M ij + | M ik | M kj + M ik | M kj | ) − X i ∈ I | z i || M ik | − | z k | ! M kj = X i = k | z i | M ik | M kj | = | M kj | X i ∈ I | z i | M ik = 0 . The relation (42) is used when moving from the second line to the third, and again moving from the thirdline to the fourth. For j = k , we have X i ∈ I | µ k ( z i ) | µ k ( M ) ik = − X i = k | z i | M ik = 0 . Thus we get (43).
Let G be a simply connected semisimple Lie group. Consider the action of H × H on C [ G ] , where ( h, h ′ ) · f ( x ) = f ( hxh ′ ) for f ∈ C [ G ] and ( h, h ′ ) ∈ H × H . Then C [ G u,v ] has a natural P × P -grading,where the P × P -homogenous elements are H × H -eigenvectors in C [ G u,v ] .Thus for a fixed double reduced word i of ( u, v ) ∈ W × W , the generalized minor ∆ γ,δ naturally hasa degree ( γ, δ ) ∈ P × P : | ∆ k | := ( u k ω | i k | , v k ω | i k | ) for k ∈ [1 , n ]; | ∆ k | := ( ω | k | , v − ω | k | ) for k ∈ [ − r, − . Proposition 2.27. [31, Lemma 4.22] Let ( z σ ( i ) , σ ( i ) = ( I, J = e ( i ) , M ( i ))) be the initial seed as inSection 2.4.2. For any k ∈ J , we have: X j u j ω | i j | M ( i ) jk = 0 , X j v j ω | i j | M ( i ) jk = 0 . By Theorem 2.23, we define a grading | · | for the cluster algebra A | σ ( i ) | by: | z k | := | ∆ k | for k ∈ [ − r, − ∪ [1 , n ] . Thus it follows immediately: 28 orollary 2.28.
Let i be a double reduced word of ( u, v ) and ( z σ , σ ( i ) = ( I, J, M ( i ))) be the initial seedas in Section 2.4.2. The P × P -graded cluster algebra ( A | σ ( i ) | , | · | ) is homogeneous. Now assume ( u, v ) = ( w , e ) . Recall that following (10) we defined the twisted minors ∆ ζuω i ,ω := ∆ uω i ,ω i ◦ ζ, Note that the twisted minors ∆ ζuω i ,ω i ’s are homogeneous: Proposition 2.29.
Let f ∈ C [ G w ,e ] be homogenous of degree ( γ, δ ) ∈ P × P . Then f ◦ ζ is homogenousof degree ( − w γ, − δ ) . In particular, the function ∆ ζuω i ,ω i is of degree ( − w uω i , − ω i ) .Proof. By the uniqueness of the Gauss decomposition, we have for all g ∈ G and h ∈ H : [ hg ] > = h [ g ] > , since hg = [ hg ] − [ hg ] > = h [ g ] − h − · h [ g ] > ;[ gh ] > = [ g ] > h, since gh = [ gh ] − [ gh ] > = [ g ] − · [ g ] > h. Thus one computes for x ∈ G w ,e and h ∈ H , f ◦ ζ ( hx ) = f ([ w − hx ] θ > ) = f (( w − hw ) − [ w − x ] θ > ) = h − w γ ( f ◦ ζ )( x ); f ◦ ζ ( xh ) = f ([ w − xh ] θ > ) = f ([ w − x ] θ > · h − ) = f ◦ ζ ( x ) h − δ , which gives us the degree of f ◦ ζ .Thus by Corollary 2.28 and Proposition 2.29, if f ∈ C [ G w ,e ] is a cluster variable or the twist of acluster variable, then f is a H × H -eigenvector. In this section we briefly recall the notions of a positive structure and the tropicalization functor, and de-scribe a fragment of the theory of geometric crystals. More detailed discussions of these subjects can befound in Sections 3.1, 4.2, and 6.3 of [10]. In the next section we will connect these notions to our moreanalytic perspective, of scaled families of functions on the real manifold AN − .Highlights of this Section are Definition 3.3 of a positive variety with potential, Definition 3.7 whichintroduces the notion of tropicalization, Definition 3.14 of the Berenstein-Kazhdan (BK) potential on adouble Bruhat cell G w ,e , Corollary 3.18 explaining that cluster variables on G w ,e are dominated by theBK potential, Remark 3.24 stating that tropicalization of G w ,e in twisted cluster variables gives rise to thesting cone of G ∨ and Theorem 3.26 which establishes an isomorphism (over R ) of sting cones for G and G ∨ .We will make the abbreviation C × n := ( C × ) n . We fix a reductive algebraic group G with compactform K , as in previous sections. Throughout, we frequently view characters γ ∈ X ∗ ( H ) of a complex algebraic torus H as regular functionson H . In particular, the standard coordinates z , . . . , z m on C × m are identified with the standard basis of X ∗ ( C × m ) . 29 efinition 3.1. A toric chart on irreducible complex algebraic variety A is an open embedding θ : H → A of a complex algebraic torus H . Given a toric chart θ : C × m → A , the function z i ◦ θ − ∈ C ( A ) is a θ -coordinate on A . We often write z i = z i ◦ θ − . Definition 3.2.
Let H and S be complex algebraic tori. A rational function f ∈ C ( H ) is positive if it canbe written in a subtraction free form f = f ′ f ′′ , f ′ = X γ ∈ X ∗ ( H ) A γ γ, f ′′ = X δ ∈ X ∗ ( H ) B δ δ (44)for A γ , B δ > with all but finitely many of the coefficients A γ , B δ equal to zero. A rational map F : H → S is positive if F γ is positive for all γ ∈ X ∗ ( S ) .Characters are positive functions and homomorphisms are positive maps, but not conversely. Definition 3.3. A positive variety with potential is a triple ( A, Φ A , Θ A ) , where A is an irreducible complexalgebraic variety, Φ A is a rational function on A called a potential , and Θ A is a non-empty set of toric chartson A , such that:(1) there exists a toric chart θ ∈ Θ A , such that Φ A ◦ θ is positive; and(2) for any pair θ, θ ′ ∈ Θ A , the compositions θ − ◦ θ ′ and ( θ ′ ) − ◦ θ are positive.A positive rational function on a positive variety with potential ( A, Φ A , Θ A ) is a rational function f ∈ C ( A ) such that f ◦ θ is positive for some (equivalently, any) θ ∈ Θ A .Toric charts θ, θ ′ on A that satisfy (2) are positive equivalent toric charts . Positive equivalence is anequivalence relation on the set of all toric charts on A . Thus, condition (2) requires that Θ A is a subset ofan equivalence class.The following special cases deserve their own names. If Φ A = 0 , then ( A, Θ A ) = ( A, , Θ A ) is a positive variety . If Θ A = { θ } is a singleton, then ( A, Φ A , θ ) = ( A, Φ A , { θ } ) is a framed positive varietywith potential . If Φ A = 0 and Θ A = { θ } , then ( A, θ ) = ( A, , { θ } ) is a framed positive variety . Everycomplex algebraic torus has a natural framed positive variety structure. Example 3.4.
Let A be an irreducible complex algebraic variety and assume that C [ A ] is isomorphic to acluster algebra A | σ | . Let ( z σ ′ , σ ′ ) ∈ | σ | be a labeled seed for A | σ | . By the Laurent Phenomenon [9], thereis an inclusion of rings C [ A ] ֒ → C [ z ± σ ] = C [ z ± , . . . , z ± n ] that induces a toric chart θ σ ′ : C × n → A . Since the mutation equations (36) are subtraction free, toric charts θ σ , θ σ ′ associated to mutation equivalent seeds ( z σ , σ ) and ( z σ ′ , σ ′ ) are positive equivalent. Definition 3.5.
Let ( A, Φ A , Θ A ) be a positive variety with potential. A rational function f ∈ C ( A ) is dominated by Φ A , denoted f ≺ Φ A , if there exists positive rational functions f + , f − on ( A, Φ A , Θ A ) anda polynomial p (Φ A ) in Φ A with positive real coefficients, such that f = f + − f − and both p (Φ A ) − f + , and p (Φ A ) − f − are positive with respect to Θ A . Definition 3.6. A map of positive varieties with potential f : ( A, Φ A , Θ A ) → ( B, Φ B , Θ B ) is a rationalmap f : A → B such that: 301) for some (equivalently, any) θ A ∈ Θ A and θ B ∈ Θ B , θ − B ◦ f ◦ θ A is positive, and(2) f ∗ Φ B ≺ Φ A .When Φ A = 0 and Φ B = 0 , then the second condition holds automatically.We now recall the tropicalization construction; for a full discussion see [10]. The tropicalization of aframed positive variety ( A, θ A : H → A ) is ( A, θ A ) t = X ∗ ( H ) . The tropicalization of a positive rational function f on a framed positive variety ( A, θ A ) is f t : ( A, θ A ) t → Z , f t = min γ ∈ X ∗ ( H ) A γ =0 { γ } − min δ ∈ X ∗ ( H ) B δ =0 { δ } where f ◦ θ A is assumed to have the form (44) and H is the domain of θ A . The tropicalization of a map offramed positive varieties f : ( A, θ A ) → ( B, θ B ) is the map f t : ( A, θ A ) t → ( B, θ B ) t uniquely defined bythe property that h γ, f t i = ( f γ ) t for all γ ∈ X ∗ ( S ) , where S is the domain of θ B . If f is a homomorphism,then f t = X ∗ ( f ) .If k is a nonnegative integer, then f t ( kp ) = kf t ( p ) for all p ∈ X ∗ ( H ) . Define the real extension of f t , f t : X ∗ ( H ) ⊗ Z R → X ∗ ( S ) ⊗ Z R , f t ( p ⊗ x ) = (cid:26) f t ( p ) ⊗ x if x > ,f t ( − p ) ⊗ ( − x ) else.Here we have overloaded the notation f t , but the meaning will be clear from context. This map f t ispiecewise R -linear and the linearity chambers are cones. Definition 3.7.
Let ( A, Φ , θ A : H → A ) be a framed positive variety with potential. The tropicalization of A (and its real points) are the cones ( A, Φ , θ A ) t = { x ∈ X ∗ ( H ) | Φ t ( x ) > } ; (45) ( A, Φ , θ A ) t R = { x ∈ X ∗ ( H ) ⊗ Z R | Φ t ( x ) > } . (46)For any δ > , the δ -interior of the tropicalization is ( A, Φ , θ A ) t R ( δ ) = { x ∈ X ∗ ( H ) ⊗ Z R | Φ t ( x ) > δ } . (47)The space ( A, Φ , θ A ) t R has a natural topology defined as follows. An isomorphism H ∼ = C × m inducesan isomorphism of sets ( A, θ A ) t R ∼ = R m . The induced topology on ( A, θ A ) t R is independent of the choiceof isomorphism H ∼ = C × m . The cone ( A, Φ , θ A ) t R ⊂ ( A, θ A ) t R has the subspace topology. With respectto this topology, the set ( A, Φ , θ A ) t R (0) is the topological interior of ( A, Φ , θ A ) t R . If Φ ◦ θ A is a regularfunction on H , then ( A, Φ , θ A ) t is a polyhedral cone.If f : ( A, Φ A , θ A ) → ( B, Φ B , θ B ) is a map of framed positive varieties with potential, then the imageof ( A, Φ A , θ A ) t under f t is contained in ( B, Φ B , θ B ) t . The tropicalization of f is the resulting piecewise Z -linear map f t : ( A, Φ A , θ A ) t → ( B, Φ B , θ B ) t . The real extension of f t defines a piecewise R -linear map f t : ( A, Φ A , θ A ) t R → ( B, Φ B , θ B ) t R .Tropicalization is functorial, in that it respects composition of maps of framed positive varieties withpotential [10, Theorem 4.12]. 31 xample 3.8. Extending Example 3.4, let A ∨ be a variety with C [ A ∨ ] ∼ = A ∨| σ | , and assume a skew sym-metrizer D of the mutation matrix of σ has been fixed. Let Θ and Θ ∨ denote the collection of toric chartson A and A ∨ arising from cluster seeds of A | σ | and A ∨| σ | . Then, as in (37), for each ( z σ ′ , σ ′ ) ∈ | σ | , wehave a map of positive varieties ( A, Θ) Ψ σ ′ −−→ ( A ∨ , Θ ∨ ) . Due to [3, Proposition 4.7], the maps Ψ σ ′ and Ψ σ coincide, after tropicalization. In other words, for any θ ∈ Θ and θ ∨ ∈ Θ ∨ , the diagram commutes: ( A, θ ) t ( A ∨ , θ ∨ ) t ( A, θ ) t ( A ∨ , θ ∨ ) t . Ψ tσ Ψ tσ ′ (48) G w ,e In this section we introduce several important toric charts on G w ,e and define the Berenstein-Kazhdan(BK) potential. This gives us a positive variety with potential ( G w ,e , Φ BK , Θ( G w ,e )) .Assume temporarily that G is of the form (21). Let i be a double reduced word for ( w , e ) (i.e. areduced word for w written in the alphabet {− , . . . , − r } ). Let ( z σ , σ ) ∈ | σ ( i ) | . Then z σ = { z − r , . . . , z − , z , . . . , z m } , where the frozen cluster variables with negative indices are the principal minors z − i = ∆ ω i . Extendthe cluster variables z i ∈ z σ to functions on Z × G sc by setting z i ( z, g ) = z i ( g ) . Choose characters γ , . . . , γ ˜ r − r ∈ X ∗ + ( H ) ∩ z ( g ) ∗ so that the collection ω , . . . , ω r , γ , . . . , γ ˜ r − r (49)forms a Z -basis for X ∗ ( H ) . We use the notation z − ( r + j ) := ∆ γ j ∈ C [ G w ,e ] . By Example 3.4, thebirational evaluation map G w ,e → C ˜ r + m : g ( z − ˜ r ( g ) , . . . , z − r − ( g ) , z − r ( g ) , . . . , z − ( g ) , z ( g ) , . . . , z m ( g )) induces a toric chart which we denote θ σ : C × (˜ r + m ) ֒ → G w ,e . (50)Recall the twist map ζ : G w ,e → G w ,e introduced in (10). For any θ σ , we get a toric chart θ ζσ = ζ ◦ θ σ (51)which is positively equivalent to θ σ . Definition 3.9.
For G of the form (21), any toric chart θ : C × (˜ r + m ) → G w ,e of the form (50) is a clusterchart on G w ,e . Any toric chart θ of the form (51) is a twisted cluster chart on G w ,e . If θ is either a clusterchart or a twisted cluster chart, we will say that θ is a (twisted) cluster chart. Remark 3.10.
Let i be any double reduced word for ( w , e ) . It follows from the definition of A | σ ( i ) | thatthe functions ∆ w ω i ,ω i and ∆ ω i , i ∈ [ i, ˜ r ] , are θ σ -coordinates for any σ ∈ | σ ( i ) | . By (11), the functions ∆ − w ω i ,ω i and ∆ − ω i , i ∈ [ i, ˜ r ] , are θ ζσ -coordinates for any σ ∈ | σ ( i ) | .32ow, let G be any connected reductive algebraic group. Identify H × L w ,e ∼ = G w ,e using multipli-cation in G . For a double reduced word i of ( w , e ) , let ( z σ , σ ) be a labeled seed of the cluster algebra A | σ ( i ) | ⊆ C [ L u,v ] as in Theorem 2.23. Define the toric chart θ σ : C × ˜ r × C × m → H × L w ,e (52)which is the product of an isomorphism C × ˜ r ∼ = H (53)and the toric chart θ σ : C × m → L w ,e . In what follows we will assume a choice of isomorphism (53) hasbeen fixed. Definition 3.11.
Any toric chart θ : C × ˜ r × C × m → G w ,e of the form (52) is a reduced cluster chart . If θ = ζ ◦ θ σ for some reduced cluster chart θ σ , then θ is a twisted reduced cluster chart . In this case, theLanglands dual of θ is defined to be ζ ∨ ◦ θ σ ∨ , the twist of the Langlands dual of θ σ . Remark 3.12.
The reduced and unreduced charts in Definitions 3.9 and 3.11 are used for different purposes.Assume G = Z × G sc of the form (21), and let H sc be the Cartan subgroup of G sc . Then, the unreducedcharts are more convenient to describe the Poisson brackets that arise in Theorem 4.14. On the other hand,the reduced charts are defined on G w ,e for all reductive G , and are needed in Section 3.4.Let ( z σ , σ ) be a seed for C [ G w ,esc ] , and let ( z σ , σ ) be the corresponding reduced seed for C [ L w ,e ] . If θ = θ σ and θ red = θ σ is the corresponding reduced cluster chart on G w ,e ∼ = Z × H sc × L w ,e , then the θ -coordinates are related to the θ red -coordinates by a Laurent monomial change of coordinates as follows.Let i ∗ : C [ G w ,esc ] → C [ L w ,e ] be the projection dual to the inclusion L w ,e ֒ → G w ,esc . Then, define C [ Z ] ⊗ C [ G w ,esc ] → C [ Z ] ⊗ C [ H sc ] ⊗ C [ L w ,e ] f ⊗ z k f ⊗ | z k | ⊗ i ∗ z k , (54)where we write | z k | = ( | z k | , | z k | ) ∈ P × P , using the grading of Corollary 2.28. It is straightforwardto show that (54) is an algebra isomorphism, and does not depend on the choice of seed σ . Similarly, if θ = ζ ◦ θ σ is a twisted cluster chart and θ red = ζ ◦ θ σ , then (54) takes θ -coordinates to Laurent monomialsin θ red coordinates. In particular, in either case the θ red -coordinates z i are P × P -homogenous. Definition 3.13 (Notation) . For a connected reductive algebraic group G , denote by Θ( G w ,e ) the set of allreduced cluster charts arising from all double reduced word i of ( w , e ) . Moreover, if G is of the form (21),we extend Θ( G w ,e ) to include all (twisted) cluster charts and (twisted) reduced cluster charts arising fromall double reduced words i of ( w , e ) , all the choices of characters γ i , and all isomorphisms H ∼ = C × ˜ r . Inparticular, if ( z σ , σ ) is mutation equivalent to ( z σ ( i ) , σ ( i )) , then θ σ ∈ Θ( G w ,e ) . Definition 3.14.
Let G be a connected reductive algebraic group, and assume the character lattice X ∗ ( H ) contains the fundamental weights ω i . The function Φ BK = X i ∈ I ∆ w ω i ,s i ω i + ∆ w s i ω i ,ω i ∆ w ω i ,ω i (55)is a regular function on G w ,e . Following its description in [10, Corollary 1.25], Φ BK is called the Berenstein-Kazhdan (BK) potential .More generally, let G be any connected reductive algebraic group, and let b G → G be a covering groupon which Φ BK is defined. Then the function Φ BK is invariant under translations by the center Z ( b G ) , andso descends to a well defined function on G . This function will also be denoted Φ BK .The following is a restatement of [2, Proposition 4.9] and [10, Lemma 3.36]. Proposition 3.15.
The triple ( G w ,e , Φ BK , Θ( G w ,e )) is a positive variety with potential. By tropicalization, for any θ ∈ Θ( G w ,e ) , we get a polyhedral cone ( G w ,e , Φ BK , θ ) t called a Berenstein-Kazhdan (BK) cone . 33 .3 Domination of functions on G w ,e This section describes a family of functions on G w ,e that are dominated by Φ BK . This technical propertyis exploited to describe the limiting behavior of a family of Poisson brackets in Theorem 4.9.Let G be a reductive algebraic group and g its Lie algebra as before. Recall that N − is the unipotentradical of the Borel B − . A N − × N − -variety is a variety equipped with a left action of N − and a rightaction of N − , such that the two actions commute.Since N − is unipotent, the exponential map exp : n − → N − is algebraic. Thus for any ( F, F ′ ) ∈ n − × n − , the map C × A → A, ( t, a ) exp( − tF ) · a · exp( tF ′ ) (56)is algebraic. The action of the fundamental vector field of ( F, F ′ ) ∈ n − × n − on f ∈ C ∞ ( A, C ) is givenby ( F · f · F ′ )( a ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f (cid:0) exp( − tF ) · a · exp( tF ′ ) (cid:1) . Since the map (56) is algebraic, the action of ( F, F ′ ) restricts to a derivation of C [ A ] . Lemma 3.16.
Let A be a N − × N − -variety and ( A, Φ , Θ) be a positive variety with potential. Let { a i } ni =1 be a set of positive rational functions on ( A, Φ , Θ) and let ( F, F ′ ) ∈ n − × n − . If F · a i · F ′ a i ≺ Φ , ∀ i ∈ [1 , n ] , then ( F · f · F ′ ) /f ≺ Φ for any subtraction free Laurent polynomial f := f ( a , . . . , a n ) in the functions a i .Proof. First of all, since the Lie algebra n − × n − acts by derivations, for a Laurent monomial a m · · · a m n n and any positive real number c > , we have: F · ( ca m · · · a m n n ) · F ′ ca m · · · a m n n = X i m i F · a i · F ′ a i ≺ Φ . Next, denote by f = f + · · · + f m a linear combination of Laurent monomials in the functions a i withpositive coefficients. By the first step, we know ( F · f i · F ′ ) /f i ≺ Φ . In other words, one can write F · f i · F ′ f i = f + i − f − i where p i (Φ) − f + i and p i (Φ) − f − i are positive with respect to Θ , for some polynomial p i in Φ with positivecoefficients. Then we have: F · f · F ′ f = X i f i f · F · f i · F ′ f i = X i f i f · f + i − X i f i f · f − i . Then, putting p (Φ) = P i p i (Φ) , one has p (Φ) − X f i f · f + i = X f i f (cid:0) p (Φ) − f + i (cid:1) , p Φ − X f i f · f − i = X f i f (cid:0) p (Φ) − f − i (cid:1) which is positive with respect to Θ . Thus ( F · f · F ′ ) /f ≺ Φ .Our prototypical example of a N − × N − -variety is the double Bruhat cell G w ,e with action ( u, g, u ′ ) ugu ′ , where g ∈ G w ,e and u, u ′ ∈ N − . 34 roposition 3.17. Let G be an algebraic group of the form (21) . For the positive variety with potential ( G w ,e , Φ BK , Θ( G w ,e )) , we have F · ∆ · F ′ ∆ ≺ Φ BK , F, F ′ ∈ n − for all generalized minors ∆ . In particular, if i is a double reduced word for ( w , e ) , then ( F · z · F ′ ) /z ≺ Φ BK for all θ σ ( i ) -coordinates z .Proof. This was shown in a slightly weaker form in [2, Theorem 4.13]; the proof extends to our contextwith minor modifications.
Corollary 3.18.
Let G be a connected reductive algebraic group, of the form (21) . Let θ ∈ Θ( G w ,e ) , andlet z be a Laurent monomial in θ -coordinates on G w ,e . Then F · z · F ′ z ≺ Φ BK , for all ( F, F ′ ) ∈ n − × n − .Proof. First, assume that θ = θ ζσ ( i ) , for some cluster chart θ σ ( i ) given by a double reduced word i for ( w , e ) . Let { z − ˜ r , . . . , z − , z , . . . , z m } be the set of θ σ ( i ) -coordinates; these are simply generalized minorson G w ,e . By [11, Theorem 5.8], any θ σ ( i ) -coordinate z i can be written as a subtraction free polynomial in“factorization parameters” t i . In turn, by [20, Theorem 1.9] these t i can be written as Laurent monomialsin the θ ζσ ( i ) -coordinates. So we have z i = f ( z − ˜ r ◦ ζ, . . . , z m ◦ ζ ) for some subtraction free Laurent polynomial f . Precomposing both sides by the involution ζ gives z i ◦ ζ = f ( z − ˜ r , . . . , z m ) . Therefore the θ -coordinate z i ◦ ζ is a subtraction free Laurent polynomial in the θ σ ( i ) coordinates, and the claim follows from Lemma 3.16 and Proposition 3.17.Now, assume that θ is any cluster chart. Then by Theorem 2.23 and the Laurent phenomenon for clusteralgebras, the θ -coordinates can be written as subtraction free Laurent polynomials in θ σ ( i ) -coordinates, forsome double reduced word i for ( w , e ) . Similarly, if θ is a twisted cluster chart, then the θ -coordinatesare subtraction free Laurent polynomials in θ ζσ ( i ) -coordinates, for some i . In the first case, the claim thenfollows from Lemma 3.16 and Proposition 3.17. In the second case, the claim follows from Lemma 3.16,Proposition 3.17, and the first paragraph of this proof. The authors of [10] defined geometric crystals as a geometric analogue of Kashiwara’s crystals. A (positivedecorated) geometric crystal is a positive variety with potential ( X, Φ , Θ) with several additional rationalstructure maps. Though our main examples originate from this theory, we will only use a fragment of theadditional structure.For a connected reductive group G , the positive variety with potential ( G w ,e , Φ BK , Θ( G w ,e )) can beenriched with the structure of a geometric crystal, as in [10, Theorem 6.15]. Of the additional geometriccrystal structure maps, there are two we are interested in: hw : G w ,e ∼ = H × L w ,e → H, ( h, z ) h, (57) wt : G w ,e → H, z [ z ] . (58)35hey are called the highest weight and weight maps, respectively. Using (8) they satisfy the followingequalities. hw( g ) w γ = ∆ w γ,γ ( g ) , wt( g ) γ = ∆ γ,γ , (59)for any γ ∈ X ∗ ( H ) , whenever the right hand side is defined. Combining this with Remark 3.10 yields thefollowing lemma. Lemma 3.19.
Let θ ∈ Θ( G w ,e ) . Then, wt ◦ θ and hw ◦ θ are group homomorphisms. Moreover, theinduced maps on character lattices, X ∗ (wt ◦ θ ) and X ∗ (hw ◦ θ ) , are injective with torsion-free cokernels.In particular, hw , wt : ( G w ,e , Φ BK , Θ( G w ,e )) → H are positive maps and their tropicalizations withrespect to any θ ∈ Θ( G w ,e ) are Z -linear. Let G ∨ be the Langlands dual group of G . Then ( G ∨ ; w ,e , Φ ∨ BK , Θ( G ∨ ; w ,e )) together with the structuremaps hw ∨ : G ∨ ; w ,e → H ∨ , wt ∨ : G ∨ ; w ,e → H ∨ captures aspects of the representation theory of G , according to the following theorem. It is a shadow of[10, Theorem 6.15], obtained by forgetting parts of the Kashiwara crystal structure. Theorem 3.20.
Let G be a connected reductive algebraic group, let θ ∨ ∈ Θ( G ∨ ; w ,e ) and consider theframed positive variety with potential ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) . Let λ, ν ∈ X ∗ ( H ) = X ∗ ( H ∨ ) be characters of H . If λ ∈ X ∗ + ( H ) is dominant, then the cardinality of ((hw ∨ ) t ) − ( λ ) ∩ ((wt ∨ ) t ) − ( ν ) ⊂ ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t (60) is equal to the dimension of the ν -weight space in V λ , the irreducible G module with high weight λ . If λ X ∗ + ( H ) , then ((hw ∨ ) t ) − ( λ ) = ∅ . What is more, the sets ((hw ∨ ) t ) − ( λ ) and (60) are the latticepoints of bounded convex polytopes. Part of the last statement of Theorem 3.20 can be seen from Lemma 3.19. In any θ ∨ ∈ Θ( G w ,e ) , maps hw t and wt t extend to linear maps out of ( G ∨ ; w ,e , θ ∨ ) t . The fibers of these maps in ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t are the intersection of an affine hyperplane with the polyhedral cone ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t . A description ofthese polytopes, as well as their connection to canonical bases, will be given in the next section. Let G be a reductive algebraic group. According to [11], for each reduced word i of w there is a polyhedralcone called the string cone . Its integral points parametrize the canonical basis of the quantized universalenveloping algebra U q ( n ) . The name “string cone” comes from the interpretation of points of the cone asstrings of operators on U q ( n ) . In this section, we show how to recover each string cone as the tropicalizationof L w ,e , with respect to a specific toric chart. Definition 3.21.
Let G be a connected reductive group, define the function Φ L = X i ∈ [1 ,r ] ∆ w ω i ,s i ω i . (61)It is a well defined regular function on L w ,e under the identification (9).Next, we will introduce toric charts for L w ,e . Recall (8) that an element x ∈ G w ,e belongs to L w ,e if and only if ∆ w ω i ,ω i ( x ) = 1 for all i ∈ [1 , r ] . Consider any chart θ : C × ( m +˜ r ) → G w ,e in Θ( G w ,e ) asDefinition 3.13. Because the functions ∆ w ω i ,ω i are all Laurent monomials in θ -coordinates, the preimage θ − ( L w ,e ) is a subtorus of C m +˜ r of dimension m . Then θ := θ | θ − ( L w ,e ) : θ − ( L w ,e ) → L w ,e is a toric chart for L w ,e . Denote by Θ( L w ,e ) all toric charts for L w ,e arising this way.36 emark 3.22. If the chart θ ∈ Θ( G w ,e ) is a reduced cluster chart, then by (52) the chart θ for L w ,e isnothing but the cluster chart as in Example 3.4. What is more, by [11, Theorem 4.7, Eq(6.1)], the followingmap η : ( L e,w ) T ֒ → G w ,e ζ −→ G w ,e ∼ = H × L w ,e pr −→ L w ,e is a biregular isomorphism, where pr is the natural projection, ζ is the twist, and ( · ) T : G → G is thetranspose anti-automorphism of [20, Equation 2.1]. The map η is called “twist” in [11]. Therefore, if θ ∈ Θ( G w ,e ) is a twisted reduced chart, θ can be viewed as the composition of a toric chart for ( L e,w ) T and the map η . Proposition 3.23.
The triple ( L w ,e , Φ L , Θ( L w ,e )) is a positive variety with potential. The natural pro-jection pr : G w ,e ∼ = H × L w ,e → L w ,e gives rise to a morphism of positive varieties with potential from ( G w ,e , Φ BK , Θ( G w ,e )) to ( L w ,e , Φ L , Θ( L w ,e )) . For any θ ∈ Θ( G w ,e ) , it induces a surjective map pr t : ( G w ,e , Φ BK , θ ) t ։ ( L w ,e , Φ L , θ ) t . Proof.
It is enough to consider the case when θ is the twisted reduced cluster chart ζ ◦ θ σ ( i ) , as in (52). Write z , . . . , z m for the θ -coordinates which factor through the projection G w ,e → L w ,e . By [11, Theorem 4.8,Theorem 5.8], any generalized minor ∆ γ,δ can be written (as a function on L w ,e ), as ∆ γ,δ ( θ ( z , . . . , z m )) = X π ∈ S γ,δ N π z c ( π )1 · · · z c m ( π ) m , (62)where S γ,δ is a finite index set, N π ∈ R > , and c k ( π ) ∈ Z .The first statement follows immediately from the positive expression (62) for the generalized minors ∆ w ω i ,s i ω i . To show that pr is a morphism of positive varieties with potential, it is enough to show that Φ BK − Φ L ◦ pr is positive with respect to θ . By [3, Proposition 5.16], for any ( h, x ) ∈ H × L w ,e one has Φ BK ( hx ) = X i ∈ [1 ,r ] (cid:0) ∆ w ω i ,s i ω i ( x ) + h − w α i ∆ w s i ω i ,ω i ( x ) (cid:1) . Then Φ BK ( hx ) − Φ L ( x ) = X i ∈ [1 ,r ] h − w α i ∆ w s i ω i ,ω i ( x ) , (63)and so Φ BK − Φ L ◦ pr is positive with respect to θ .It remains to show that pr t : ( G w ,e , Φ BK , θ ) t → ( L w ,e , Φ L , θ ) t is surjective. If ( t , . . . , t m ) ∈ ( L w ,e , Φ L , θ ) t , we must find λ ∨ ∈ X ∗ ( H ) such that ( λ ∨ , t , . . . , t m ) ∈ ( H × L w ,e , Φ BK , θ ) t . By (63),one has ( λ ∨ , t , . . . , t m ) ∈ ( H × L w ,e , Φ BK , θ ) t if and only if m X k =1 c k ( π ) t k − h w α i , λ ∨ i + Φ tL ( t , . . . , t m ) > , (64)for all π ∈ S w s i ω i ,ω i and i ∈ [1 , r ] . Let λ ∨ = P n i ω ∨ i for n i > . Then we have −h w α i , λ ∨ i = P n j h− w α i , ω ∨ j i > since − w α i is a simple root. By picking the n i sufficiently large one ensures theinequalities (64) all hold. 37 emark 3.24. If one chooses θ to be the twisted reduced chart ζ ◦ θ σ ( i ) associated to a double reducedword i of ( w , e ) , then the cone ( L w ,e , Φ L , θ ) t coincides with the string cone associated with i defined in[11, 17, 18], as follows. By Remark 3.22 and [11, Theorem 4.8], the chart θ is positively equivalent to thefactorization chart x i defined in [11, Eq (4.10)]. Moreover, the θ -coordinates are Laurent monomials in thecoordinates of x i , and vice versa. Thus the cone ( L w ,e , Φ L , θ ) t is unimodularly isomorphic to the one in[11, Theorem 3.10]. What is more, for λ ∨ ∈ X ∗ ( H ) , the following maps are injective: (hw t ) − ( λ ∨ ) ֒ → ( G w ,e , Φ BK , θ ) t ։ ( L w ,e , Φ L , θ ) t . (65)So one may view the fibers (hw t ) − ( λ ∨ ) as subsets of the string cone; these are sometimes known as stringpolytopes . In this section, we will discuss comparison maps between G w ,e and G ∨ ; w ,e , in the spirit of Langlandsdual cluster algebras.Fix a double reduced word i for ( w , e ) , and recall that C [ L ∨ ; w ,e ] ∼ = A ∨| σ ( i ) | is the Langlands dualcluster algebra of C [ L w ,e ] ∼ = A | σ ( i ) | . Recall from (40) that the matrix D is a skew-symmetrizer of M ( i ) .Then for each labeled seed ( z σ , σ ) ∈ | σ ( i ) | , one has the map Ψ ∗ σ : C [ L ∨ ; w ,e ] → C [ L w ,e ] : z ∨ j z d | ij | j as in (37). We extend Ψ σ to a map Ψ σ : G w ,e → G ∨ ; w ,e , by setting Ψ ∗ σ := Ψ ∗ H ⊗ Ψ ∗ σ : C [ G ∨ ; w ,e ] ∼ = C [ H ∨ ] ⊗ C [ L ∨ ; w ,e ] → C [ G w ,e ] ∼ = C [ H ] ⊗ C [ L w ,e ] . If θ = θ σ is a reduced cluster chart for G w ,e , and θ ∨ = θ σ ∨ its dual chart, one has a map of positivevarieties Ψ θ := Ψ σ : ( G w ,e , θ ) → ( G ∨ ; w ,e , θ ∨ ) . (66)Note that Ψ tθ : ( G w ,e , θ ) t → ( G w ,e , θ ∨ ) t is linear and lifts the comparison map ψ h of (13), in the sensethat the following diagrams commute. ( G w ,e , θ ) t ( G ∨ ; w ,e , θ ∨ ) t X ∗ ( H ) X ∗ ( H ∨ ) Ψ tθ hw t (hw ∨ ) t ψ h ( G w ,e , θ ) t ( G ∨ ; w ,e , θ ∨ ) t X ∗ ( H ) X ∗ ( H ∨ ) Ψ tθ wt t (wt ∨ ) t ψ h (67)According to the following, the tropicalized comparison maps are compatible with the twist map. Theorem 3.25. [3, Theorem 5.8] Let θ, θ ′ be reduced cluster charts on G w ,e . Let θ ∨ and θ ′∨ be thecorresponding Langlands dual charts. Then, the following diagram commutes. ( G w ,e , θ ) t ( G ∨ ; w ,e , θ ∨ ) t ( G w ,e , θ ) t ( G ∨ ; w ,e , θ ∨ ) tΨ tθ ζ t ( ζ ∨ ) t Ψ tθ Additionally, the tropicalized comparison maps preserves the Berenstein-Kazhdan cone.38 heorem 3.26. [3, Theorem 5.21] Assume θ = θ σ is a reduced cluster chart for G w ,e , and θ ∨ = θ σ ∨ isits dual chart. Then the comparison map Ψ tθ restricts to an injective map Ψ tθ : ( G w ,e , Φ BK , θ ) t ֒ → ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t , which extends to a linear isomorphism of real BK cones: Ψ tθ : ( G w ,e , Φ BK , θ ) t R ∼ −→ ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t R . This section recalls the definition and properties of the partial tropicalization of K ∗ from [2, 3, 4]. In theprocess, it extends the results of [2, 3, 4] to arbitrary (twisted) cluster charts on G w ,e (the results in [2, 3, 4]were only stated for cluster charts associated to reduced words).Some of the key results of this Section are as follows: Lemma 4.8 stating that functions dominated bythe BK potential exponentially decay in the tropical limit, Lemma 4.11 which describes the torus action onthe partial tropicalization, Lemma 4.12 which describes symplectic leaves of the partial tropicalization andTheorem 4.17 which introduces Darboux coordinates for the Poisson bracket π −∞ . K ∗ from cluster charts on G w ,e Fix a choice of θ ∈ Θ( G w ,e ) and consider the framed positive variety with potential ( G w ,e , Φ BK , θ ) .Recall (58) the weight map, wt : G w ,e → H , and (57) the highest weight map, hw : G w ,e → H . ByLemma 3.19, their tropicalizations are Z -linear maps wt t = (wt ◦ θ ) t : ( G w ,e , θ ) t → X ∗ ( H ) , hw t = (hw ◦ θ ) t : ( G w ,e , θ ) t → X ∗ ( H ) . Moreover, wt t = X ∗ (wt ◦ θ ) and hw t = X ∗ (hw ◦ θ ) . Recall that A H is the real Lie subgroup A =exp( a ) . As sets, K ∗ ∩ G w ,e = wt − ( A ) . By Lemma 3.19, θ − ( K ∗ ∩ G w ,e ) = (wt ◦ θ ) − ( A ) is a connected real Lie subgroup of C × (˜ r + m ) . Let T θ denote the maximal compact subtorus of (wt ◦ θ ) − ( A ) .When it is clear from context we write T = T θ . Since the maximal compact subgroup of A is { e } , we have T ⊂ ker(wt ◦ θ ) and dim R T = m .In what follows we will be concerned with the manifold ( G w ,e , θ ) t R × T θ . If z is a positive function on ( G w ,e , θ ) , then we abuse notation and write z t for the map ( G w ,e , θ ) t R × T θ pr −−→ ( G w ,e , θ ) t R z t −→ R . Definition 4.1.
Let θ, θ ′ ∈ Θ( G w ,e ) , and consider the manifolds ( G w ,e , θ ) t R × T θ and ( G w ,e , θ ′ ) t R × T θ ′ .For each open linearity chamber C of the piecewise linear bijection X ∗ ( C × (˜ r + m ) ) ⊗ R = ( G w ,e , θ ) t R Id t −−→ ( G w ,e , θ ′ ) t R = X ∗ ( C × (˜ r + m ) ) ⊗ R consider the unique linear map X ∗ ( C × (˜ r + m ) ) ⊗ R → X ∗ ( C × (˜ r + m ) ) ⊗ R which agrees with (Id t ) | C on C . Let e Id C : C × (˜ r + m ) → C × (˜ r + m ) (wt ◦ Id) t = wt t , we have e Id C ( T θ ) = T θ ′ . Define Id P T = a C ((Id t ) | C × e Id C ) : a C C × T θ → ( G w ,e , θ ′ ) t R × T θ ′ , (68)where the disjoint union ranges over all the open linearity chambers of Id t . The map Id P T is the partiallytropicalized change of coordinates .For all s = 0 , there is an isomorphism of real Lie groups P s : ( G w ,e , θ ) t R × T → (wt ◦ θ ) − ( A ) (69)defined by the property that e s h γ,x i t γ = P s ( x, t ) γ , ∀ γ ∈ X ∗ C × (˜ r + m ) , ( x, t ) ∈ ( G w ,e , θ ) t R × T . (70)Note that since T is contained in ker(wt ◦ θ ) , t h γ, wt t i = 1 , ∀ t ∈ T , γ ∈ X ∗ ( H ) . (71)Recall that the standard coordinates z i on C × (˜ r + m ) are identified with the standard basis of X ∗ ( C × (˜ r + m ) ) .With this identification, define the system of polar coordinates λ i , ϕ i on ( G w ,e , θ ) t R × T : (cid:26) λ i ( x, t ) = h z i , x i ,e √− ϕ i ( x,t ) = t z i . (72)The coordinates ϕ i are defined (modulo π ) for the indices i such that z i takes on values outside of R > onthe subset G w ,e ∩ K ∗ . In other words, ϕ i are defined for indices i such that z i (viewed as a character of C × (˜ r + m ) ) is not in the image of X ∗ ( H ) under X ∗ (wt ◦ θ ) . Definition 4.2.
The detropicalization map , L θs , is the composition L θs := θ ◦ P s : ( G w ,e , θ ) t R × T → G w ,e ∩ K ∗ . Let θ ∈ Θ( G w ,e ) be a (twisted) cluster chart on G w ,e as in the previous section. By Corollary 2.28 andProposition 2.29, there is Z -module homomorphism, | · | : X ∗ ( C × (˜ r + m ) ) → X ∗ ( H ) × X ∗ ( H ) , (73)defined by the property that | z i | = ( α i , β i ) for all i , where ( α i , β i ) is the homogeneous degree of the θ -coordinate z i . If | γ | = ( α, β ) , denote | γ | = α and | γ | = β . Proposition 4.3.
Assume G is of the form (21) , and let θ ∈ Θ( G w ,e ) . Let {· , ·} C K ∗ denote the C -linearextension of the Poisson bracket of π K ∗ . Restrict the θ -coordinates z i ∈ C [ G w ,e ] to functions on G w ,e ∩ K ∗ . Then, there exists c z i ,z j , c z i ,z j , c z i ,z j ∈ C , such that { z i , z j } C K ∗ = z i z j c z i ,z j ; { z i , z j } C K ∗ = z i z j c z i ,z j ; { z i , z j } C K ∗ = z i z j ( c z i ,z j + f z i ,z j ) , (74) where f z i ,z j is a C -linear combination of ( E − α · z i )( E − α · z j ) z i z j , ( z i · E − α )( z j · E − α ) z i z j , α ∈ R + . (75) Moreover, c z i ,z j = √− (cid:16) ( | z i | , | z j | ) h ∗ − ( | z i | , | z j | ) h ∗ (cid:17) , c z i ,z j = − c z i ,z j ∈ √− Q . (76)40 roof. First consider brackets of the form { z i , z j } = { z i , z j } C K ∗ . Consider the holomorphic Poisson-Liestructure π holG = (r hol ) L − (r hol ) R on G given by the r -matrix r hol = √− r , where r is given by (25).The bivector π holG restricts to a holomorphic Poisson structure on the double Bruhat cell G w ,e . Therefore,for holomorphic functions on G w ,e , the brackets { z i , z j } C K ∗ and −√− { z i , z j } G w ,e are equal. By [31,Theorem 4.18] and [29, Theorem 3.1], the bracket of two θ -variables on G w ,e is log canonical, with rationalstructure constant.Next, consider brackets of the form { z i , z j } . Since the bivector π K ∗ ∈ Γ( ∧ T K ∗ ) is real, we have { z i , z j } = { z i , z j } . The brackets { z i , z j } are pure imaginary, and so this amounts to c z i ,z j = − c z i ,z j .Finally, consider the mixed brackets { z i , z j } . The claim follows directly from the expression for thebracket in Lemma 2.11, as well as the P × P -homogeneity of (twisted) cluster variables established inCorollary 2.28 and Proposition 2.29. Definition 4.4.
Assume G is of the form (21), and let θ ∈ Θ( G w ,e ) . The partial tropicalization of K ∗ with respect to θ is the Poisson manifold ( P T ( K ∗ , θ ) , π θ −∞ ) , where:(1) P T ( K ∗ , θ ) := ( G w ,e , Φ BK , θ ) t R (0) × T .(2) π θ −∞ is defined such that: { λ i , ϕ j } = √− c z i ,z j − c z i ,z j ) , { ϕ i , ϕ j } = { λ i , λ j } = 0 , (77)where:(a) λ i , ϕ j are the coordinates (72), and(b) c z i ,z j , c z i ,z j , and c z i ,z j are as in (74). Remark 4.5.
By the explicit formula in [31, Corollary 4.21], given a double reduced word i for ( w , e ) , inthe chart θ σ ( i ) , the coefficient c z i ,z j is given by c z i ,z j = − √− (cid:16) ( | z i | , | z j | ) h ∗ − ( | z i | , | z j | ) h ∗ (cid:17) whenever i < j . For i > j , we have c z i ,z j := − c z j ,z i .The following gives an interpretation of coordinate transformations, after partial tropicalization. Itfollows from [2, Theorem 6.23]. Theorem 4.6.
Assume G is of the form (21) , and let θ, θ ′ ∈ Θ( G w ,e ) . Recall the map Id P T defined in (68) .Then, the restriction of Id P T to a densely defined map Id P T : P T ( K ∗ , θ ) → P T ( K ∗ , θ ′ ) has (Id P T ) ∗ ( π θ −∞ ) = π θ ′ −∞ wherever it is defined. For δ > , the δ -interior of P T ( K ∗ , θ ) is P T ( K ∗ , θ, δ ) = ( G w ,e , Φ BK , θ ) t R ( δ ) × T . The following results show that for all δ > , π θs := ( L θs ) ∗ ( sπ K ∗ ) converges uniformly to π θ −∞ on P T ( K ∗ , θ, δ ) as s → −∞ . First, some notation: 41 otation 4.7 (Big-O notation) . A family of differential forms or multi-vector fields on
P T ( K ∗ , θ, δ ) pa-rameterized by s is O ( e sδ ) if its coefficients are O ( e sδ ) as functions on ( −∞ , × P T ( K ∗ , θ, δ ) when it iswritten in the coordinates λ i , ϕ i (72) . Lemma 4.8. [2, Lemma 6.17] Let θ ∈ Θ( G w ,e ) . If f ∈ C ( G w ,e ) is dominated by Φ BK , then ( f ◦ L θs ) | P T ( K ∗ ,θ,δ ) = O ( e sδ ) for all δ > . Theorem 4.9.
Assume G is of the form (21) , and let θ ∈ Θ( G w ,e ) . Then, for all δ > π θs | P T ( K ∗ ,θ,δ ) = π θ −∞ + O ( e sδ ) . Proof.
By Corollary 3.18 the functions f z i ,z j in (74), considered as functions on G w ,e , are dominatedby the potential Φ BK , up to complex conjugation of some terms. The analytical significance of this isexplained by Lemma 4.8, and the proof is then exactly as in [2, Theorem 6.18]. The structure maps of the partial tropicalization are: hw P T : ( G w ,e , θ ) t R × T ( G w ,e , θ ) t R X ∗ ( H ) ⊗ Z R t ∗ , wt P T : ( G w ,e , θ ) t R × T ( G w ,e , θ ) t R X ∗ ( H ) ⊗ Z R t ∗ . pr hw t −√− ψ h pr wt t −√− ψ h (78)Although they are defined on ( G w ,e , θ ) t R × T , hw P T and wt P T will sometimes denote their restrictions tothe subspace
P T ( K ∗ , θ ) . Lemma 4.10.
Let θ ∈ Θ( G w ,e ) . Then, the following diagram commutes for all s = 0 . ( G w ,e , θ ) t R × T K ∗ ∩ G w ,e t ∗ A wt PT L θs wt E s (79) Proof.
For ( x, t ) ∈ ( G w ,e , θ ) t R × T , E s ◦ wt P T ( x, t ) = exp (cid:18) s √− ψ − h ( −√− ψ h ◦ wt t ( x )) (cid:19) = exp (cid:16) s t ( x ) (cid:17) Thus it suffices to prove exp (cid:16) s t ( x ) (cid:17) = wt( L θs ( x, t )) , ∀ ( x, t ) ∈ ( G w ,e , θ ) t R × T . Let γ ∈ X ∗ ( H ) and ( x, t ) ∈ ( G w ,e , θ ) t R × T arbitrary. Applying the definitions, wt( L θs ( x, t )) γ = (wt ◦ θ ◦ P s ( x, t )) γ = P s ( x, t ) h γ, wt t i = exp (cid:16) s t ( x ) (cid:17) γ t h γ, wt t i = exp (cid:16) s t ( x ) (cid:17) γ . The fourth equality follows by (71). The lemma follows since γ is arbitrary.42he degree map (cf. (73)) of a (twisted) cluster chart θ determines a homomorphism ϕ θ : H × H → C × (˜ r + m ) , ϕ θ ( h , h ) γ = h | γ | h | γ | ∀ γ ∈ X ∗ C × (˜ r + m ) . This in turn determines an action of H × H on C × (˜ r + m ) by multiplication such that θ is H × H -equivariant.The image of the anti-diagonal subgroup { ( h, h − ) ∈ H × H | h ∈ H } ∼ = H is contained in the kernel of wt ◦ θ since |h γ, wt t i| = |h γ, wt t i| for all γ ∈ X ∗ ( H ) . Thus, the image of the composition ι θ : T ֒ → H h ( h,h − ) −−−−−−−→ H × H ϕ θ −→ C × (˜ r + m ) is contained in T . Let T act on ( G w ,e , θ ) t R × T by translation on T with respect to ι θ , and consider thedressing (conjugation) action of T on K ∗ ∩ G w ,e . Then, P s and L θs = θ ◦ P s are each T -equivariant.Recall that the action of T on K ∗ by conjugation coincides with the dressing action. For all s = 0 ,the dressing action of T on ( K ∗ , sπ K ∗ ) is Hamiltonian with moment map E − s ◦ [ · ] (see the discussionpreceding Proposition 2.15). According to the following lemma, this Hamiltonian action survives the limit s → −∞ . Lemma 4.11.
Let G be of the form (21) , and let θ ∈ Θ( G w ,e ) . The kernel of the homomorphism ι θ : T → T is the center of K . Moreover:(1) For all s = 0 , the action of T on ( G w ,e , θ ) t R × T defined by ι θ is Hamiltonian with respect to π θs withmoment map wt P T .(2) The action of T on P T ( K ∗ , θ ) defined by ι θ is Hamiltonian with respect to π θ −∞ with moment map wt P T .Proof.
To see that the kernel of ι θ is the center of K , it suffices to observe that L θs is T -equivariant and thekernel of the dressing action of T on K ∗ equals the center of K .The restriction of [ · ] to G w ,e ∩ K ∗ equals wt . Since L θs is T -equivariant, the action of T on ( G w ,e , θ ) t R × T defined by ι θ is Hamiltonian with respect to π θs with moment map E − s ◦ wt ◦ L θs . By (79), E − s ◦ wt ◦ L θs =wt P T , which completes the proof of item 1.Since ι θ does not depend on s , fundamental vector fields X ∈ X (( G w ,e , θ ) t R × T ) , X ∈ t , for theaction defined by ι θ do not depend on s . By Theorem 4.9, X = lim s →−∞ ( π θs ) ♯ ( d h wt P T , X i ) = ( π θ −∞ ) ♯ ( d h wt P T , X i ) at points in P T ( K ∗ , θ, δ ) . This completes the proof of item 2. Lemma 4.12.
Let θ ∈ Θ( G w ,e ) . Then:(1) The fibers of hw P T : P T ( K ∗ , θ ) → t ∗ are the symplectic leaves of ( P T ( K ∗ , θ ) , π θ −∞ ) .(2) The image of hw P T : P T ( K ∗ , θ ) → t ∗ is ˚ t ∗ + .(3) For all λ ∈ ˚ t ∗ + , let ω θ −∞ denote the symplectic structure on (hw P T ) − ( λ ) defined by π θ −∞ . Then, Vol((hw
P T ) − ( λ ) , ω θ −∞ ) = Vol( O λ , ω λ ) . (80) Proof.
The proof of Item 1. is the same as the proof of [3, Proposition 6.3] which handles the special case θ = θ σ ( i ) . Item 2. follows from Theorem 3.20. Item 3. is Theorem 6.5 and Remark 6.6 of [3].43he next lemma is crucial for estimating Hamiltonian flows in Section 5. It was proved for the specialcase θ = θ σ ( i ) in Theorem 6.11 of [3]. The proof in general is the same. Lemma 4.13.
Let θ ∈ Θ( G w ,e ) . Then, For all X ∈ t , h hw P T , X i = h S ◦ E − s ◦ L θs , X i + O ( e sδ ) , (81) as a function on P T ( K ∗ , θ, δ ) . Moreover, d h hw P T , X i = d h S ◦ E − s ◦ L θs , X i + O ( e sδ ) , (82) as linear operators. This section establishes basic results about π θ −∞ . Section 4.5.1 describes the special case θ = θ σ ( i ) or θ ζσ ( i ) ,where i is a double reduced word for ( w , e ) . Versions of these results were proved in [3]. Section 4.5.2describes the case for general θ ∈ Θ( G w ,e ) . Let i = ( i , . . . , i m ) be a double reduced word for ( w , e ) . If θ = θ σ ( i ) , then for all j ∈ [ − r, − ∪ [1 , m ] and k ∈ [1 , m ] : { λ j , ϕ k } = 0 if j > k, { λ j , ϕ k } = ( ω i j , ω i k ) h ∗ − ( ω i j , s i j +1 · · · s i k ω i k ) h ∗ if j < k. If j < − r , then { λ j , ϕ k } = 0 for all k ∈ [1 , m ] . In particular, if z j = ∆ w ω | ij | ,ω | ij | is a θ -coordinate and λ j = z tj , then { λ j , ϕ k } = 0 for all k ∈ [1 , m ] .Proof. We refer to the definition (77) of π θ −∞ and use invariance of ( · , · ) h ∗ under the Weyl group. Thecoefficient c z j ,z k is given in (76). The coefficient c z j ,z k is given in Remark 4.5. The last statement can beshown directly from the previous ones; it also follows from Lemma 4.12(1). Corollary 4.15.
Let i = ( i , . . . , i m ) be a double reduced word for ( w , e ) , and let θ = θ σ ( i ) . For j ∈ [1 , m ] , let j − = max { k ∈ [ − r, − ∪ [1 , m ] | | i k | = | i j |} . Let B denote the m × m matrix with entries B j,k = { λ j − , ϕ k } . Then B = X − Y where Y is an upper triangular unimodular matrix with nonnegative entries and X isthe diagonal matrix with X j,j = d | i j | .Consequently, there is a Lie group automorphism C ϕ : T θ → T θ so that, if ϕ ′ k = ϕ k ◦ C ϕ , then thematrix B ′ with B ′ j,k = { d | i j | λ j − , ϕ ′ k } is equal to the identity matrix.Proof. Then, by Theorem 4.14 and invariance of ( · , · ) h ∗ with respect to the Weyl group, B j,j = { λ j − , ϕ j } = ( ω i j − , ω i j ) h ∗ − ( ω i j , s i j − · · · s i j ω i j ) h ∗ = ( ω i j , ω i j ) h ∗ − ( ω i j , s i j ω i j ) h ∗ = ( ω i j , α i j ) h ∗ = 1 /d | i j | . j > j − > k , then B j,k = 0 by Theorem 4.14. If j > k > j − , then again by invariance B j,k = { λ j − , ϕ k } = ( ω i j − , ω i k ) h ∗ − ( ω i j , s i j − · · · s i k ω i k ) h ∗ = 0 . Finally, if j < k then B j,k = { λ j − , ϕ k } = ( ω i j − , ω i k − s i j − · · · s i k ω i k ) = c ( ω i j , α i j ) h ∗ for some nonnegative integer c , since ω i k − s i j − · · · s i k ω i k is a positive integer linear combination ofsimple roots α i and ( ω i j , α i ) = δ i j ,i . The desired automorphism C ϕ can be constructed by reducing Y tothe identity matrix by multiplying by a unimodular matrix on the right. By the discussion of Section 3.4, the cone factor of
P T ( K ∗ , θ ) is related to the representation theory of G ∨ . Ultimately, we want to connect the Poisson geometry of k ∗ with the representation theory of K ⊂ G . Proposition 4.16.
Assume G is of the form (21) . Let i be a double reduced word for ( w , e ) , let θ = θ σ ( i ) ∈ Θ( G w ,e ) , and let θ ∨ = θ σ ( i ) ∨ ∈ Θ( G ∨ ; w ,e ) . Let λ j , ϕ k and λ ∨ j , ϕ ∨ k be coordinates on ( G w ,e , θ ) t R × T θ and ( G ∨ ; w ,e , θ ∨ ) t R × T θ ∨ , respectively, as in (72) . Identify T = T θ ∨ = T θ by setting ϕ k = ϕ ∨ k for all k where ϕ k is defined.Given the fixed isomorphism C ˜ r ∼ = H as in (53) , let X , . . . , X ˜ r be the associated basis of X ∗ ( H ) . Let x ∨ j = λ ∨ j − for j ∈ [1 , m ] and x ∨− j = h X j , (hw ∨ ) t i . There are coordinates υ ∨ , . . . , υ ∨ m , modulo π , so that(1) Under the constant Poisson structure ( Ψ tθ × Id T ) ∗ ( π θ −∞ ) on ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t R (0) × T , one has { x ∨ j , υ ∨ k } = 0 for all j ∈ [ − ˜ r, − , k ∈ [1 , m ]; { x ∨ j , υ ∨ k } = δ j,k for all j, k ∈ [1 , m ] . (2) x ∨− k ◦ ( Ψ tθ × Id T ) = h X k , hw P T i for all k ∈ [1 , ˜ r ] .Proof. Let ˆ θ = θ σ ( i ) be the unreduced cluster chart on G w ,e which is related to θ as in Remark 3.12. Foreach ˆ θ -coordinate ˆ z j , view the corresponding coordinates ˆ λ j , ˆ ϕ j as functions on P T ( K ∗ , θ ) by precom-posing with Id P T : P T ( K ∗ , θ ) → P T ( K ∗ , ˆ θ ); notice that in this case Id P T is globally defined because the coordinate transformation (54) is Laurentmonomial. Let ˆ ϕ ′ k be related to ˆ ϕ k as in Corollary 4.15, and set υ ∨ k = ˆ ϕ ′ k , for k = 1 , . . . , m .For ˆ z j = ∆ u j ω | ij | ,ω | ij | , write u j ω | i j | = P ri =1 c i w ω i for c i ∈ Z . If z j = ˆ z j | L w ,e , we have λ j = ˆ λ j − r X i =1 c i ∆ tw ω i ,ω i ! . Due to the last statement of Theorem 4.14, the functions ∆ tw ω i ,ω i are Casimirs of π θ −∞ . By Corollary 4.15,if j > then { x ∨ j ◦ Ψ tθ , υ k } = { λ j − ◦ Ψ tθ , ˆ ϕ ′ k } = { d | i j | λ j − , ˆ ϕ ′ k } = δ j,k , as desired. If j < then { x j , υ k } = 0 by (2) together with Lemma 4.12(1). This establishes condition (1).The condition (2) follows from the definition of x ∨− k and the commutivity of the first diagram in (67).45he following extends Proposition 4.16 to include arbitrary (twisted) reduced cluster charts. Theorem 4.17.
Let i = ( i , . . . , i m ) be a double reduced word for ( w , e ) and let ( z σ , σ ) ∈ | σ ( i ) | . Let θ = θ σ or ζ ◦ θ σ , and let θ ∨ = θ σ ∨ or ζ ∨ ◦ θ σ ∨ , respectively. Let λ j , ϕ k be coordinates on ( G w ,e , θ ) t R × T θ as in (72) .For the isomorphism C ˜ r ∼ = H as in (53) , let X , . . . , X ˜ r be the associated basis of X ∗ ( H ) . There arelinear coordinates x − ˜ r , . . . , x − , x , . . . , x m on ( G w ,e , θ ) t R , and coordinates υ , . . . , υ m modulo π on T θ , so that:(1) Under the constant Poisson structure π θ −∞ on P T ( K ∗ , θ ) , { x j , υ k } = 0 for all j ∈ [ − ˜ r, − , k ∈ [1 , m ]; { x j , υ k } = δ j,k for all j, k ∈ [1 , m ] . (2) x − j = h X j , hw P T i for all j ∈ [1 , ˜ r ] .(3) There is a unimodular map R ˜ r + m ∼ = ( G ∨ ; w ,e , θ ∨ ) t R which takes the image of P T ( K ∗ , θ ) under ( q, t ) ( x − ˜ r ( q ) , . . . , x − ( q ) , x ( q ) , . . . , x m ( q )) ∈ R ˜ r + m , ( q, t ) ∈ P T ( K ∗ , θ ) . (83) to ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t R (0) .Proof. Write ˜ θ = θ σ ( i ) and ˜ θ ∨ = θ σ ( i ) ∨ . Identify T θ ∼ = T θ ∨ and ˜ T ∼ = ˜ T ∨ by putting ϕ k = ϕ ∨ k and ˜ ϕ k = ˜ ϕ ∨ k , using notation for coordinate functions as in (72). Due to the commutivity of (3.25), as well asTheorem 48 (if θ is a twisted reduced cluster chart), the diagram commutes: P T ( K, θ ) ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t R (0) × T θ ∨ P T ( K, ˜ θ ) ( G ∨ ; w ,e , Φ ∨ BK , ˜ θ ∨ ) t R (0) × T ˜ θ ∨ Ψ tθ × Id T θ Id PT Id PT Ψ t ˜ θ × Id T ˜ θ (84)wherever the vertical arrows are defined. Let C ⊂ ( G w ,e , Φ BK , θ ) t R (0) be an open linearity chamber ofthe map Id t ◦ Ψ tθ = Ψ t ˜ θ ◦ Id t . Consider the restriction of the map in (84) to C × T θ : C × T θ → ( G ∨ ; w ,e , ˜ θ ∨ ) t R × T ˜ θ ∨ . Linearly extend this to a map F : P T ( K ∗ , θ ) → ( G ∨ ; w ,e , ˜ θ ∨ ) t R × T ˜ θ ∨ . Notice that, by following the two sides of the diagram (84), one can decompose F as F = F ◦ ( Ψ tθ × Id T θ ) = ( Ψ t ˜ θ × Id T ˜ θ ) ◦ F , so that, when restricted to C × T θ (resp. Φ tθ ( C ) × T θ ∨ ), the map F agrees with Id P T (resp. F agreeswith Id P T ). Note that each F i is the product of a unimodular map f i in the first factor with a Lie groupisomorphism e f i in the second. Let x ∨ j and υ ∨ k be the functions on ( G ∨ ; w ,e , ˜ θ ∨ ) t R × T ˜ θ ∨ constructed inProposition 4.16. Let x j = x ∨ j ◦ F, υ k = υ ∨ k ◦ F. We will show that x j and υ k have the desired properties.46irst, consider the unique constant Poisson structure ˜ π on ( G ∨ ; w ,e , ˜ θ ∨ ) t R × T ˜ θ ∨ which coincides with ( Ψ t ˜ θ × Id T ˜ θ ) ∗ ( π ˜ θ −∞ ) on the open subset ( G ∨ ; w ,e , Φ ∨ BK , ˜ θ ∨ ) t R (0) × T ˜ θ ∨ . By Theorem 4.6, the map F is aPoisson map on the open subset C × T θ of its domain. Since F is the product of a linear map and a fixedLie group isomorphism, and because both π θ −∞ and ˜ π are constant, the map F is a Poisson map on its entiredomain. So, by by Proposition 4.16(1), the Poisson brackets { x j , υ k } with respect to π θ −∞ have the desiredform (1).Second, by definition x − j = h X j , (hw ∨ ) t i ◦ F . Restricting to the open set C , one then has x − j = h X j , (hw ∨ ) t i ◦ ( Ψ t ˜ θ × Id T ˜ θ ) ◦ Id P T . By Proposition 4.16(2), x − j = h X k , hw P T i on C × T θ . If two linear maps agree on an open subset of theirdomain, they are equal, and so x − j has the desired form (2) on all of P T ( K ∗ , θ ) .Finally, the map (83) can be decomposed P T ( K ∗ , θ ) pr −−→ ( G w ,e , θ ) t R Ψ tθ −→ ( G ∨ ; w ,e , θ ∨ ) t R (0) f −→ ( G ∨ ; w ,e , ˜ θ ∨ ) t R ∼ = R ˜ r + m . The isomorphism ( G ∨ ; w ,e , ˜ θ ∨ ) t R ∼ = R ˜ r + m given by the choice of coordinates x ∨ j is unimodular, and themap f is unimodular. Also, Ψ tθ ◦ pr ( P T ( K ∗ , θ )) = ( G ∨ ; w ,e , Φ ∨ BK , θ ∨ ) t R (0) . This gives condition (3). We are now able to prove our main results. We first construct action-angle coordinates on the space K × ˚ t ∗ + in Sections 5.1 and 5.2. These are then used in Section 5.3 to build action-angle coordinates on multiplicityfree spaces through some standard reduction arguments. K × ˚ t ∗ + Fix an arbitrary (twisted) reduced cluster chart θ on G w ,e . As in (53) and Theorem 4.17, we have a fixedisomorphism C × ˜ r ∼ = H and denote X , . . . , X ˜ r the associated basis of X ∗ ( H ) . Let: x − ˜ r , . . . , x − , x , . . . , x m , υ , . . . , υ m (85)be the linear coordinates on ( G w ,e , θ ) t R × T given by Theorem 4.17. Let Υ , . . . , Υ ˜ r (86)denote coordinates on T defined modulo π by the dual basis of X , . . . , X ˜ r .As θ is now fixed, we will frequently suppress it from notation (e.g. L θs = L s ). Moreover, denote C := ( G w ,e , Φ BK , θ ) t R (0) , C ( δ ) := ( G w ,e , Φ BK , θ ) t R ( δ ) . With this notation,
P T ( K ∗ , θ ) = C × T . Definition 5.1.
Given the coordinates x − i , x j , υ j , Υ i and C as defined above (for a particular choice of θ ),let ω −∞ := dυ ∧ dx + · · · + dυ m ∧ dx m + d Υ ∧ dx − + · · · + d Υ ˜ r ∧ dx − ˜ r . The symplectic manifold ( C × T × T, ω −∞ ) has the following immediate properties:47A) The map ( C × T × T, ω −∞ ) → ( C × T , π −∞ ) , ( x, t, t ′ ) ( x, t ) , is Poisson. This follows fromTheorem 4.17.(B) The action of T on C × T × T defined by t · ( x, t ′ , t ′′ ) = ( x, t ′ , tt ′′ ) is Hamiltonian with respect to ω −∞ with moment map hw P T (where hw P T ( x, t, t ′ ) := hw P T ( x, t ) ). This follows from Theorem 4.17.(C) The action of T on C × T × T defined by t · ( x, t ′ , t ′′ ) = ( x, ι ( t ) t ′ , t ′′ ) is Hamiltonian with respect to ω −∞ with moment map wt P T (where wt P T ( x, t, t ′ ) := wt P T ( x, t ) ). This follows from Lemma 4.11.(D) The two T -actions commute, so ( C × T × T, ω −∞ , (hw P T , wt P T )) is a Hamiltonian T × T -manifold.Recall the Hamiltonian K × T -manifold ( K × ˚ t ∗ + , ω can , ( µ L , µ R )) from Example 2.3. Theorem 5.2.
For all δ > and any bounded open subset U ⊂ C ( δ ) there exists T × T -equivariantsymplectic embeddings ( U × T × T, ω −∞ ) ֒ → ( K × ˚ t ∗ + , ω can ) such that the following diagrams commute. U × T × T K × ˚ t ∗ + t ∗ + t ∗ +hw PT − µ R = U × T × T K × ˚ t ∗ + t ∗ k ∗ wt PT µ L pr t ∗ (87) Throughout the proof of Theorem 5.2 we use big-O notation analogous to Notation 4.7, now used in refer-ence to the coordinates x − i , x j , Υ k , υ l on C × T × T . Proof of Theorem 5.2:
Fix δ > and let U be a bounded open subset of C ( δ ) as in the statement of Theorem5.2. Step 1 (delinearize K × ˚ t ∗ + ): Let ( K × ˚ t ∗ + , Ω s , Ψ s ) be the delinearization of ( K × ˚ t ∗ + , ω can , ( µ L , µ R )) as described in Example 2.18. Recall that (29) defines an isotopy φ s : K × ˚ t ∗ + → K × ˚ t ∗ + such that φ ∗ s Ω s = ω can . By Proposition 2.16, φ s is defined for all s . By Proposition 2.15, it suffices to construct T × T -equivariant symplectic embeddings of ( U × T × T, ω −∞ ) into ( K × ˚ t ∗ + , Ω s ) such that the followingdiagrams commute. U × T × T K × ˚ t ∗ + t ∗ + t ∗ +hw PT − µ R = U × T × T K × ˚ t ∗ + K ∗ t ∗ A wt PT Ψ s wt E − s (88) Step 2 (trivialize Ψ s ): Consider the T -invariant orthogonal complement t ⊥ ⊂ k of t and define a map t ⊥ × ˚ t ∗ + → K ∗ by sending ( Z, ξ ) E s (Ad ∗ e Z ξ ) . For a sufficiently small neighbourhood V of ∈ t ⊥ , therestriction of this map to V × ˚ t ∗ + is a tubular neighbourhood of the submanifold E s (˚ t ∗ + ) . Note that E s (˚ t ∗ + ) does not depend on s . This tubular neighbourhood embedding is T -equivariant with respect to the coadjointand dressing action: for all t ∈ T and ( Z, ξ ) ∈ t ⊥ × ˚ t ∗ + , t · ( Z, ξ ) = (Ad t Z, ξ ) E s (Ad ∗ exp(Ad t Z ) ξ ) = E s (Ad ∗ t Ad ∗ e Z ξ ) = tE s (Ad ∗ e Z ξ ) t − . Ψ s trivializes over the tubular neighbourhood V × ˚ t ∗ + as follows: ( Z, ξ, t ) ( e Z t, ξ )( Z, ξ, t ) V × ˚ t ∗ + × T K × ˚ t ∗ + ( Z, ξ ) V × ˚ t ∗ + K ∗ ( Z, ξ ) E s (Ad ∗ e Z ξ ) pr Ψ s (89)This trivialization is T × T -equivariant with respect to the actions T × T × V × ˚ t ∗ + × T → V × ˚ t ∗ + × T ; ( t ′ , t ′′ , Z, ξ, t ) (Ad t ′ Z, ξ, t ′ t ( t ′′ ) − ) ,T × T × K × ˚ t ∗ + → K × ˚ t ∗ + ; ( t ′ , t ′′ , k, ξ ) ( t ′ k ( t ′′ ) − , ξ ) . (90) Step 3 (detropicalize):
Let L s = L θs : ( G w ,e , θ ) t R × T → K ∗ be the detropicalization map (Definition4.2). Define L s := L s × Id T : ( G w ,e , θ ) t R × T × T → K ∗ × T. Lemma 5.3. [4, Lemma 4.1, Lemma 4.2, Proposition 4.3] For all δ > and any neighbourhood V of ∈ t ⊥ , there exists s < such that for all s s , L s ( C ( δ ) × T ) ⊂ V × ˚ t ∗ + (91) where V × ˚ t ∗ + has been identified with its image in K ∗ under the embedding from Step 2. What is more, if p ∈ C ( δ ) × T and L s ( p ) = ( Z, ξ ) ∈ V × ˚ t ∗ + under (91) , then Z is O ( e sδ ) . Thus, for s ≪ , we have embeddings such that the following diagram commutes. C ( δ ) × T × T V × ˚ t ∗ + × T K × ˚ t ∗ + C ( δ ) × T V × ˚ t ∗ + K ∗ pr L s pr Ψ s L s (92)Since L s is T -equivariant, the map L s is T × T -equivariant. Step 4 (Poisson bracket estimates):
Let L α : K ∗ → k denote the Legendre transform of α ∈ Ω ( K ∗ ) (see Section 2.3.5). Given β ∈ Ω C ( K ∗ ) , denote β R , β I ∈ Ω ( K ∗ ) such that β = β R + √− β I . Define L C β : K ∗ → g by complexifying the equation defining L β . Then, L C β = L β R + √− L β I . Lemma 5.4.
Let z be a Laurent monomial in the θ -coordinates z i with γ = | z | . Let β = dzz , β ′ = d ∆ γ,γ ∆ γ,γ . Then, as functions C ( δ ) × T → k , L β R ◦ L s = L β ′ ◦ L s + O ( e sδ ) and L β I ◦ L s = O ( e sδ ) . roof. Given X ∈ an − , let X R denote the right invariant vector field on K ∗ with X R ( e ) = X . Bydefinition of the Legendre transform, ⟪ X, L β R − L β ′ ⟫ s + √− ⟪ X, L β I ⟫ s = ⟪ θ R ( X R ) , L β R − L β ′ ⟫ s + √− ⟪ θ R ( X R ) , L β I ⟫ s = β ( X R ) − β ′ ( X R )= X · ∆ γ,γ ∆ γ,γ − X · zz . By (7), if X ∈ a then this function is identically . If X ∈ n − , then applying Corollary 3.18 and Lemma 4.8to z and ∆ γ,γ , (cid:18) X · ∆ γ,γ ∆ γ,γ − X · zz (cid:19) ( L s ( p )) = O ( e sδ ) . The same argument proves a similar result for the pair of forms β = β R − √− β I and β ′ = β ′ . Combiningthe resulting two equations completes the proof. Lemma 5.5.
Let {− , −} s denote the Poisson bracket of L ∗ s Ω s . Then, for all i, k ∈ [1 , ˜ r ] and j ∈ [1 , m ] :(i) { x − i , Υ k } s = δ i,k + O ( e sδ ) ,(ii) { x j , Υ k } s ∈ Z + O ( e sδ ) ,(iii) { υ j , Υ k } s = O ( e sδ ) ,as real-valued functions on C ( δ ) × T × T .Proof. If f is a real valued function defined on C × T × T such that f ( x, t, t ′ ) = f ( x, t ) , then by (92) andProposition 2.19, { f, Υ k } s ( p ) = { f ◦ L − s , Υ k ◦ L − s } Ω s ( L s ( p ))= { f ◦ L − s ◦ Ψ s , Υ k ◦ L − s } Ω s ( L s ( p )) . = d (Υ k ◦ L − s ) (cid:16) X f ◦ L − s ◦ Ψ s (cid:17) L s ( p ) = d (Υ k ◦ L − s ) (cid:16) L f ◦ L − s (Ψ s ( L s ( p )))) L s ( p ) (cid:17) (93)Equation (i): First, consider the case where f = x − i , i ∈ [1 , ˜ r ] . By Lemma 4.13, for all X ∈ t , x − i ◦ L − s = h hw P T ◦ L − s , X i i = h S ◦ E − s , X i i + O ( e sδ ) . (94)Since S ◦ E − s ◦ Ψ s is the moment map for the Thimm torus action on K × ˚ t ∗ + , it follows again by Lemma 4.13that X x − i ◦ L − s ◦ Ψ s = X i + O ( e sδ ) , (95)where X i is the fundamental vector field of X i for the Thimm torus action. Setting f = x − i in (93), { f, Υ k } s ( p ) = d (Υ k ◦ L − s ) (cid:16) X f ◦ L − s ◦ Ψ s (cid:17) L s ( p ) = d (Υ k ◦ L − s )( X i L s ( p ) ) + O ( e sδ )= δ i,k + O ( e sδ ) . (96)50quation (ii): Let z = z j , j ∈ [1 , m ] , γ = | z | , β = dz/z , and β ′ = d ∆ γ,γ / ∆ γ,γ . Then, β R = s d ( x j ◦ L − s ) , β I = d ( υ j ◦ L − s ) , β ′ R = s d ( g ◦ L − s ) , β ′ I = 0 , (97)where g = h γ, wt t i . Note that g ◦ L − s ◦ Ψ s = h γ, wt t ◦ L − s ◦ Ψ s i = h−√− ψ h ◦ wt t ◦ L − s ◦ Ψ s , −√− ψ h γ i = h E − s ◦ wt ◦ Ψ s , −√− ψ h γ i . (98)Since E − s ◦ wt ◦ Ψ s is the moment map for the action of T on K × ˚ t ∗ + as the maximal torus, s L β ′ R (Ψ s ( m )) m = X m , where X m is the fundamental vector field of X = −√− ψ h ( γ ) ∈ t with respect to the same action.Setting f = x j in (93), and combining with Lemma 5.4, { x j , Υ k } s ( p ) = 2 s d (Υ k ◦ L − s ) (cid:16) L β R (Ψ s ( L s ( p ))) L s ( p ) (cid:17) = 2 s d (Υ k ◦ L − s ) (cid:18) L β ′ R (Ψ s ( L s ( p ))) L s ( p ) (cid:19) + O ( e sδ )= d (Υ k ◦ L − s ) (cid:16) X L s ( p ) (cid:17) + O ( e sδ ) ∈ Z + O ( e sδ ) . In the last step, we have that d (Υ k ◦ L − s ) (cid:16) X L s ( p ) (cid:17) ∈ Z since X L s ( p ) is the fundamental vector field ofthe integral element −√− ψ h ( γ ) ∈ t acting as an element of the maximal torus. The action of the maximaltorus on the Upsilon coordinates is as in (90). (In fact, one can easily say exactly which integer this formulaevaluates to, but we do not need a precise value).Equation (iii): The same argument as in the previous case, now applied to β I , shows that { υ j , Υ k } s = O ( e sδ ) . This completes the proof.The following completes the description of the Poisson brackets of L ∗ s Ω s . Lemma 5.6.
Let {− , −} s denote the Poisson bracket of L ∗ s Ω s . Then, for all i, j ∈ [1 , ˜ r ] , { Υ i , Υ j } s = O ( e sδ ) , as real-valued functions on C ( δ ) × T × T .Proof. The brackets { Υ i , Υ j } s can be approximated using the formula for (Ω s ) − provided in (33). First,note that P k ( d Υ i ) = 0 for all i, k ∈ [1 , ˜ r ] . Second, recall from Remark 2.8 that π K = Λ L − Λ R where Λ = P α ∈ R + E α ∧ E − α . Third, note that the denominator of the first term of (33) does not approach zeroas s → −∞ . For a fixed p ∈ C ( δ ) × T and t ∈ T , by Lemma 5.3 we may take s sufficiently negative thatwe can express L s ( p, t ) = ( e Z t, ξ ) ∈ K × ˚ t ∗ + in terms of the trivialization (89). Here Z ∈ t ⊥ and ξ ∈ ˚ t ∗ + are elements which depend on s . Noting that E α , E − α ∈ t ⊥ + √− t ⊥ , it then suffices to show that, forany Y ∈ t ⊥ , ( Y L · Υ i )( e Z t ) = O ( e sδ ) , ( Y R · Υ i )( e Z t ) = O ( e sδ ) , (99)51here Y L and Y R denote the left- and right-invariant vector fields on K whose value at e is Y .There exists unique Z ′ ∈ t ⊥ and X ∈ t so that for q ∈ R , log( e Z e q Ad t Y ) = log( e Z + qZ ′ e qX ) mod q .Due to Lemma 5.3, the element Z is O ( e sδ ) . Therefore, by the Baker-Campbell-Hausdorff formula, Z + q (Ad t Y + O ( e sδ )) = Z + q ( Z ′ + X + O ( e sδ )) mod q . By taking orthogonal projection k → t , we see that X = O ( e sδ ) .Now, computing from the definition, ( Y L · Υ i )( e Z t ) = ddq (cid:12)(cid:12)(cid:12) q =0 Υ i ( e Z te qY )= ddq (cid:12)(cid:12)(cid:12) q =0 Υ i ( e Z e q Ad t Y t )= ddq (cid:12)(cid:12)(cid:12) q =0 Υ i ( e Z e q Ad t Y )= ddq (cid:12)(cid:12)(cid:12) q =0 Υ i ( e Z + qZ ′ e qX )= d Υ i ( X ) = O ( e sδ ) , as desired. The proof for the action of Y R is essentially the same. Corollary 5.7.
The coordinates (85) and (86) can be chosen such that, on C ( δ ) × T × T , L ∗ s Ω s = ω −∞ + O ( e sδ ) . Proof.
By Theorem 4.17 and Theorem 4.9, the coordinates (85) can be chosen so that, under L ∗ s Ω s , { x j , υ k } s = δ j,k + O ( e sδ ) , { x j , x k } s = { x j , x − k } s = { x − j , x − k } s = { υ j , υ k } s = O ( e sδ ) for j, k > . By Lemma 5.5, one has for this choice of coordinates, { x − i , Υ k } s = δ i,k + O ( e sδ ) , { x j , Υ k } s = N j,k + O ( e sδ ) , { υ j , Υ k } s = O ( e sδ ) , where N j,k ∈ Z . Finally, by Lemma 5.6, { Υ i , Υ j } s = O ( e sδ ) . Consider the unimodular change of coordinates given by replacing x j := x j − ˜ r X k =1 N j,k x − k . It is easy to check that the new coordinates x − i , x j , Υ i , υ j satisfy all the above equalities except now { x j , Υ k } s = O ( e sδ ) . This completes the proof.In what follows we will assume without loss of generality we have chosen the coordinates (85) and (86)so that the conclusion of Corollary 5.7 holds.
Step 5 (correction maps):
Unfortunately, the map L s does not make the diagrams (88) commute. It istherefore necessary to precompose L s with a “correction map”.52 emma 5.8. For all δ > and any bounded open subset U ⊂ C ( δ ) , there exists T × T -equivariantembeddings G s : U × T × T → C ( δ ) × T × T such that the following diagrams commute. U × T × T C ( δ ) × T × T K × ˚ t ∗ + U × T K ∗ ˚ t ∗ + k ∗ pr G s L s Ψ s hw PT E − s S (100) U × T × T C ( δ ) × T × T K × ˚ t ∗ + U × T K ∗ ˚ t ∗ + k ∗ pr G s L s Ψ s wt PT wt E − s (101) Moreover, with respect to the coordinates x − i , x k , υ k , Υ i , the Jacobian ( G s ) ∗ has block-form ( G s ) ∗ = I ˜ r × ˜ r + O ( e sδ ) O ( e sδ ) O ( e sδ ) 00 I m × m I m × m
00 0 0 I ˜ r × ˜ r , (102) where O ( e sδ ) denotes a matrix whose entries are O ( e sδ ) .Proof. Since the diagram (92) commutes, it suffices to find a T -equivariant map g s : U × T → C ( δ ) × T such that that the following diagrams commute, U × T C ( δ ) × T K ∗ ˚ t ∗ + k ∗ hw PT g s L s E − s S (103) U × T C ( δ ) × T K ∗ ˚ t ∗ + k ∗ wt PT g s L s wt E − s (104)and, with respect to the coordinates x − i , x k , υ k , the Jacobian of g s has block-form ( g s ) ∗ = I ˜ r × ˜ r + O ( e sδ ) O ( e sδ ) O ( e sδ )0 I m × m
00 0 I m × m . (105)53he map G s = g s × Id T : U × T × T → C ( δ ) × T × T will then have the desired properties.For every bounded open set U ⊂ C ( δ ) , it is possible to find a convex bounded set C ⊂ ˚ t ∗ + such that U × T ⊂ (hw P T ) − ( C ) ∩ ( C ( δ ) × T ) . Thus it is sufficient to construct g s on subsets of the form (hw P T ) − ( C ) ∩ ( C ( δ ) × T ) for any convexbounded set C ⊂ ˚ t ∗ + .Such a map g s was constructed in [4, Lemma 3.1, Lemma 3.2, Proposition 3.3] for the case where C = { p } for an arbitrary element p ∈ ˚ t ∗ + . The same implicit function theorem argument used in [4,Lemma 3.2] can be easily extended to construct g s with the properties described above for any convexbounded open set C ⊂ ˚ t ∗ + .Although it was not mentioned in [4], it is easy to see that (104) commutes. By an argument similar tothe proof of Lemma 4.10, it suffices to show that wt t ◦ g s = wt t . This is easy to see looking at the definitionof g s .To see that the left diagram in (88) commutes, observe that by (100), (27), and since S ◦ µ L = − µ R , hw P T ◦ pr = S ◦ E − s ◦ Ψ s ◦ L s ◦ G s = S ◦ µ L ◦ L s ◦ G s = − µ R ◦ L s ◦ G s . Step 6 (set up Moser’s trick):
Fix δ > and let U be a bounded open subset of C ( δ ) . Let ω s = G ∗ s L ∗ s Ω s ∈ Ω ( U × T × T ) . The goal is to construct a Moser flow that deforms ω s to ω −∞ . For s < , define a closed2-form ω τs on U × T × T by the equation ω τs = (1 − τ ) ω −∞ + τ ω s , τ ∈ [0 , . Lemma 5.9.
For all δ > , τ ∈ [0 , , and s ≪ , the form ω τs is non-degenerate on U × T × T .Proof. By Corollary 5.7, L ∗ s Ω s = ω −∞ + O ( e sδ ) . By Lemma 5.8, ω s = G ∗ s L ∗ s Ω s = G ∗ s ( ω −∞ + O ( e sδ )) = ω −∞ + O ( e sδ ) . Thus ω τs = (1 − τ ) ω −∞ + τ ( ω −∞ + O ( e sδ )) = ω −∞ + O ( e sδ ) is non-degenerate for s ≪ .Define another closed 2-form α s = ω s − ω −∞ on U × T × T . Then α s is O ( e sδ ) . Lemma 5.10.
Let U be an open subset of C ( δ ) . Then α s is exact on U × T × T .Proof. The form ω −∞ is exact, and so it suffices to show that L ∗ s Ω s is exact. Consider the involution K × ˚ t ∗ + → K × ˚ t ∗ + : ( k, ξ ) ( k, ξ ) . (106)By naturality of the canonical symplectic form on T ∗ K , the involution of ( K × t ∗ , ω can ) which maps ( k, ξ ) ( k, − ξ ) is symplectic. Therefore (106) is anti-symplectic on ( K × ˚ t ∗ + , ω can ) . Consider also theinvolution ( · ) : C ( δ ) × T × T → C ( δ ) × T × T, ( x − i , x j , Υ i , υ j ) = ( x − i , x j , − Υ i , − υ j ) . (107)Then L s intertwines the involutions (107) and (106), and so by [6, Section 3.4], the involution (106) isanti-symplectic on ( K × ˚ t ∗ + , Ω s ) . Consequently, one has ( · ) ∗ L ∗ s Ω s = − L ∗ s Ω s .On the other hand, for any cohomology class [ ω ] ∈ H ( C ( δ ) × T × T ) ∼ = H ( T × T ) , one has ( · ) ∗ [ ω ] = [ ω ] . Therefore the class [ L ∗ s Ω s ] ∈ H ( C ( δ ) × T × T ) is equal to , and hence L ∗ s Ω s is exact.54 emma 5.11. Let n be a positive integer, let T = ( S ) n and fix a subtorus T ′ ⊂ T . Assume σ s ∈ Ω k ( T ) be a family of exact smooth T ′ -invariant forms, parametrized by s < . Assume there is some δ > suchthat σ s = O ( e sδ ) . Then, there exists a family γ s ∈ Ω k − ( T ) of smooth T ′ -invariant forms, which satisfy dγ s = σ s and γ s = O ( e sδ ) .Proof. Let us first introduce some notation. For any positive integer l , we denote multi-indices by capitalletters, eg J = ( j , . . . , j l ) . Write e π √− ϕ = ( e π √− ϕ , . . . , e π √− ϕ n ) ∈ T = ( S ) n for points of T , and consider the forms dϕ J = dϕ j ∧ · · · ∧ dϕ j l . For any form ω ∈ Ω l ( T ) , write ω = P J ω J dϕ J . For m = ( m , . . . , m n ) ∈ Z n , let m · ϕ = m ϕ + · · · + m n ϕ n , and let ˆ ω J ( m ) = Z T ω J e − π √− m · ϕ be the m th Fourier coefficient of ω J . Define ω [ m ] = e π √− m · ϕ X J ˆ ω J ( m ) dϕ J . Then ω = P m ∈ Z n ω [ m ] , where the sum on the right converges uniformly because ω is smooth.We consider the form σ s ∈ Ω k ( T ) , writing σ = σ s for simplicity. For each = m = ( m , . . . , m n ) ∈ Z n , choose j ( m ) ∈ { , . . . , n } so that m j ( m ) = 0 . Define the forms γ [ m ] := 12 π √− m j ( m ) ι ∂∂ϕj ( m ) σ [ m ] , and let ˆ γ J ( m ) be the coefficient of dϕ J in e − π √− m · ϕ γ [ m ] , so e − π √− m · ϕ γ [ m ] = X J ˆ γ J ( m ) dϕ J . Because the Fourier series of σ converges uniformly, we have X m =0 | ˆ γ J ( m ) | X m =0 πm j ( m ) X K | ˆ σ K ( m ) | = X K X m =0 | ˆ σ K ( m ) | πm j ( m ) (108) < X K X m =0 | ˆ σ K ( m ) | < ∞ and hence P m =0 γ [ m ] converges uniformly to a smooth k − form. Let us denote this form by γ = P m =0 γ [ m ] .Since the exterior derivative d preserves the Fourier mode m , we have d ( σ [ m ]) = ( dσ )[ m ] = 0 . (109)Applying Cartan’s magic formula and (109), we have dγ [ m ] = σ [ m ] for all = m ∈ Z n . So σ − dγ = σ [0] .Because σ is exact, it follows that σ [0] = P K ˆ σ K (0) dϕ K is exact. But this can only be true if σ [0] = 0 .So σ = dγ . 55et us control the size of γ . We compute as in (108) and use Parseval’s theorem: | γ J | X K X m | ˆ σ K ( m ) | = X K Z T | σ K | . Since σ s is O ( e sδ ) , we conclude P K R T | σ K | is O ( e sδ ) . Hence γ s = γ is O ( e sδ ) . Finally, since dγ s = σ s and σ s is T ′ -invariant, by averaging γ s with respect to translations by T ′ we can assume that γ s is T ′ -invariant as well. Because T and T ′ are compact this does not affect the conclusion that γ s is O ( e sδ ) . Lemma 5.12.
Let U be a convex bounded open subset of C ( δ ) . Then there exists a T × T -invariant form β s ∈ Ω ( U × T × T ) such that dβ s = α s and β s is O ( e sδ ) .Proof. Fix a point p ∈ U . The set U is convex so we may define a straight line retract H : [0 , × U × T × T → U × T × T ( q, x, υ, Υ) H q ( x, υ, Υ) = ( p + q ( x − p ) , υ, Υ) from U × T × T to { p } × T × T . Let h : Ω • ( U × T × T ) → Ω •− ( U × T × T ) be the homotopy operatorassociated with H , so hα s = Z (cid:16) ι ∂∂q H ∗ α s (cid:17) dq ∈ Ω ( U × T × T ) . Note that H q : U × T × T → U × T × T is T × T -equivariant for all q ∈ [0 , , and that the form and α s is T × T -invariant. What is more, α s is O ( e sδ ) and U is bounded. Therefore the form hα s is T × T -invariantand is O ( e sδ ) . Since h is a homotopy operator and since α s is closed, one has α s = dhα s + H ∗ α s . By Lemma 5.10, the form H ∗ α s is exact. It is T × T -invariant, and is O ( e sδ ) . So by Lemma 5.11, there isa T × T -invariant γ s ∈ Ω ( { p } × T × T ) satisfying dγ s = H ∗ α s , which also satisfies γ s = O ( e sδ ) . Then β s = hα s + γ s has the desired properties. Step 7 (integrate the Moser vector field):
To complete the proof, notice that there exists < δ ′ < δ and a convex bounded open set U ′ ⊂ C ( δ ′ ) so that U ⊂ U ⊂ U ′ . We replace U with U ′ , and all theprevious discussion holds with δ replaced by δ ′ . We will then consider U × T × T as a submanifold of U ′ × T × T ⊂ C ( δ ′ ) × T × T .By Lemma 5.9, the form ω τs ∈ Ω ( U ′ × T × T ) is non-degenerate for all τ ∈ [0 , , for s ≪ . For anysuch s , the equation ι X τs ω τs = − β s (110)defines a τ -dependent vector field X τs , τ ∈ [0 , , on U ′ × T × T . Denote by φ τs the flow of X τs for allpoints p ∈ U ′ × T × T and all τ ∈ [0 , for which it is defined. Lemma 5.13.
The vector field X τs is O ( e sδ ′ ) for all τ .Proof. By (110), ι X τs ω τs = ι X τs ω −∞ + τ ι X τs α s = − β s = O ( e sδ ′ ) . Since α s is O ( e sδ ′ ) and ω −∞ is nondegenerate, the vector field X τs must be O ( e sδ ′ ) . Proposition 5.14.
There exists s ≪ and x ∈ ( G w ,e , θ ) t R such that: . The flow φ τs | U × T × T : U × T × T → U ′ × T × T is defined for all τ ∈ [0 , ;2. The time 1 flow φ s satisfies ( φ s ) ∗ ( ω s ) = ( φ s ) ∗ ( ω s ) = ω −∞ and is T × T -equivariant;3. There exists x ∈ ( G w ,e , θ ) t R so that the map ˜ φ s : U × T × T → ( G w ,e , θ ) t R × T × T ( x, υ ) ( x + x , υ ) has ˜ φ s ( U × T × T ) ⊂ U ′ × T × T , satisfies ( ˜ φ s ) ∗ ( ω s ) = ω −∞ , and additionally satisfies hw P T ◦ ˜ φ s = hw P T , and wt P T ◦ ˜ φ s = wt P T . Proof.
Let R denote the minimum distance from ∂U to ∂U ′ , where we consider the standard Euclideanmetric on ( G w ,e , θ ) t R ∼ = R ˜ r + m . Let c > denote the norm of the linear map (wt P T × hw P T ) : ( G w ,e , θ ) t R ∼ = ( G w ,e , θ ) t R × { } → t ∗ × t ∗ . Fix a linear section σ of this map, and let b = || σ || > . Fix an invariant metric on T × T , and equip ( G w ,e , θ ) t R × T × T with the product metric. Since X τs is O ( e sδ ) , we may choose s ≪ such that || X τs || < min { R bc , R } at all points of U ′ × T × T . Then for all ( x, υ ) ∈ U × T × T and all τ ∈ [0 , ,we have that the distance from φ τs ( x, υ ) to ( x, υ ) is less than R . Therefore the flow of X τs , restricted to U × T × T , does not escape U ′ × T × T , for τ ∈ [0 , . This establishes the first claim.The second claim is due to the standard Moser argument: By (110), one has dι X τs ω τs + dβ s = L X τs ω τs + ∂ω τs ∂τ = 0 and therefore ( φ τs ) ∗ ω τs = ω s = ω −∞ wherever the flow is defined. Finally, since ω τs and β s are both T × T -invariant, the flow of X τs must be T × T -equivariant.For the third claim, notice that the map (wt P T , hw P T ) : C ( δ ) × T × T → t ∗ × t ∗ is a moment mapfor the T × T action, with respect to both ω −∞ and ω s (cf. properties (B) and (C) and Lemma 5.8). Byuniqueness of moment maps there is some ˆ x ∈ t ∗ × t ∗ such that (wt P T ◦ φ s , hw P T ◦ φ s ) + ˆ x = (wt P T , hw P T ) . From || X τs || < R bc we conclude that || ˆ x || < R b . Let ( x ,
1) = σ (ˆ x ) ∈ ( G w ,e , θ ) t R × T × T . Then || x || b || ˆ x || R/ . Therefore, for ( x, υ ) ∈ U × T × T , the distance from ( x, υ ) to ˜ φ s ( x, υ ) is lessthan R/ R/ R , and hence ˜ φ s ( x, υ ) is contained in U ′ × T × T . The form ω −∞ is constant, andso a shift by x preserves ω −∞ . Therefore ( ˜ φ s ) ∗ ( ω s ) = ω −∞ . The final claim is immediate from theconstruction.We then have the following embedding: F : U × T × T U ′ × T × T C ( δ ′ ) × T × T K × t ∗ + . ˜ φ s G s L s It satisfies F ∗ ω can = ω −∞ , and makes the diagrams (87) commute. This completes the proof of Theo-rem 5.2. 57 .3 Big action-angle coordinate charts on multiplicity free spaces Theorem 5.2 can be used to construct action-angle coordinates on big subsets of a large family of compact,connected, multiplicity free Hamiltonian K -manifolds. Before stating the theorem, we define the domainsof our action-angle coordinates.Let ( M, ω, µ ) be a compact, connected multiplicity free space with principal stratum ˚ t ∗ + , Kirwan poly-tope △ = µ ( M ) ∩ t ∗ + , and principal isotropy subgroup L T . Let ˚ △ denote the relative interior of △ . Re-call from Section 2.2.2 that there is an associated toric T /L -manifold (˚ △× T /L, ω std , pr : ˚ △× T /L → ˚ △ ) ,where ω std was defined via a linear identification of Lie(
T /L ) ∗ with the affine subspace of t ∗ spanned by ˚ △ . In what follows, we will view (˚ △ × T /L, ω std , pr : ˚ △ × T /L → ˚ △ ) as a Hamiltonian T -manifold.Let ( C × T × T, ω −∞ , (hw P T , wt P T )) be one of the Hamiltonian T × T -manifolds defined in Section 5.1(depending on the choices described therein). The open submanifolds C ( δ ) × T × T , δ > , are alsoHamiltonian T × T -manifolds. We may form the product Hamiltonian T × T × T -manifold, ( C ( δ ) × T × T × ˚ △ × T /L, ω −∞ ⊕ ω std , (hw P T , wt P T , pr)) . Our model space is the symplectic reduction of this space with respect to the diagonal T action generatedby the moment map pr − hw P T , M ( δ, ˚ △ , L ) := ( C ( δ ) × T × T × ˚ △ × T /L, ω −∞ ⊕ ω std ) (cid:12) T. Provided that ˚ △ ⊂ ˚ t ∗ + and δ > is sufficiently small, M ( δ, ˚ △ , L ) is non-empty. Since the action of T isfree, M ( δ, ˚ △ , L ) is a smooth symplectic manifold. Denote the reduced symplectic form on M ( δ, ˚ △ , L ) by ω red . The moment map (hw P T , wt P T ) and the associated T × T action descend to M ( δ, ˚ △ , L ) so that wehave a Hamiltonian T × T -manifold, ( M ( δ, ˚ △ , L ) , ω red , (hw P T , wt P T )) . Note that we use the same notation for the maps induced on the quotient as for the maps hw P T and wt P T .By definition of the symplectic reduced space, there is a T × T -equivariant diffeomorphism C ( δ, ˚ △ ) × T × ( T /L ) ∼ = −→ M ( δ, ˚ △ , L ) , ( x, t, t ′ L ) [ x, t, t ′ , hw P T ( x ) , , where C ( δ, ˚ △ ) denotes the projection to C ( δ ) of the T × T -invariant subspace (hw P T ) − (˚ △ ) ⊂ C ( δ ) × T × T .Finally, recall from Proposition 2.4 that the dense subset U = ( S ◦ µ ) − (˚ t ∗ + ) ⊂ M is a Hamiltonian K × T -manifold with moment map ( µ, S ◦ µ ) . The extra Hamiltonian T -action on U is the Thimmtorus action. The embeddings constructed in the following theorem are action-angle coordinates by thedescription of M ( δ, ˚ △ , L ) above. Theorem 5.15.
Let ( M, ω, µ ) be a compact , connected, multiplicity free Hamiltonian K -manifold suchthat ˚ △ ⊂ ˚ t ∗ + and let U = ( S ◦ µ ) − (˚ t ∗ + ) . For all ε > , there exists δ > and symplectic embeddings ( M ( δ, ˚ △ , L ) , ω red ) ֒ → ( U , ω ) such that:(i) The symplectic volumes satisfy Vol( M ( δ, ˚ △ , L ) , ω red ) > Vol(
M, ω ) − ε. Those familiar should note that Theorem 5.15 easily extends to the more general setting of convex multiplicity free Hamilto-nian K -manifolds (cf. [43, Definition 2.2]). ii) The embeddings ( M ( δ, ˚ △ , L ) , ω red ) ֒ → ( U, ω ) are embeddings of Hamiltonian T × T -manifolds: ( M ( δ, ˚ △ , L ) , ω red , (hw P T , wt P T )) ֒ → ( U , ω, ( S ◦ µ, pr t ∗ ◦ µ )) . Proof.
Construction of the embeddings and proof of (ii):
The desired embeddings of Hamiltonian T × T -manifolds are constructed by first constructing embeddings of Hamiltonian T × T × T -manifolds, thenreducing by a certain diagonal copy of T .First, we consider the following composition of an embedding and an isomorphism of Hamiltonian T × T × T -manifolds, ( C ( δ, ˚ △ ) × T × T × ˚ △ × T /L, ω −∞ ⊕ ω std , (hw P T , wt P T , pr)) ֒ → ( K × ˚ t ∗ + × ˚ △ × T /L, ω −∞ ⊕ ω std , ( S ◦ µ L , pr t ∗ ◦ µ L , pr)) ∼ = ( K × ˚ t ∗ + × W, ω −∞ ⊕ ω, ( S ◦ µ L , pr t ∗ ◦ µ L , µ )) . (111)The embedding exists by Theorem 5.2: since ˚ △ is the relative interior of a convex polytope and the fibersof hw P T are bounded, the set C ( δ, ˚ △ ) ⊂ C ( δ ) is bounded. The isomorphism follows by Lemma 2.6.Next, we apply a diagonal symplectic reduction to this composition, corresponding to the inclusion T → T × T × T , t ( t − , , t ) . On the left side, this action is generated by the moment map pr − hw P T and on the right it is generated by µ − S ◦ µ L . Since the embeddings, torus actions, and moment mapsabove are all compatible with this reduction, we obtain an embedding, ( C ( δ, ˚ △ ) × T × T × ˚ △ × T /L, ω −∞ ⊕ ω std , (hw P T , wt P T , pr)) (cid:12) T֒ → ( K × ˚ t ∗ + × W, ω −∞ ⊕ ω, ( S ◦ µ L , pr t ∗ ◦ µ L , µ )) (cid:12) T. (112)The reduced space on the right embeds densely as a Hamiltonian T × T submanifold of ( U , ω, ( S ◦ µ, pr t ∗ ◦ µ )) by Proposition 2.4. The space on the left is our model space. Thus, we have constructedthe desired embedding of Hamiltonian T × T -manifolds, ( M ( δ, ˚ △ , L ) , ω red , (hw P T , wt P T )) ֒ → ( U , ω, ( S ◦ µ, pr t ∗ ◦ µ )) . Proof of (i):
Let β M denote the Liouville measure of ( M, ω ) and let β W denote the Liouville measure of ( W, ω ) as a Hamiltonian T -manifold. Recall, e.g. from [24], that Vol(
M, ω ) = Z k ∗ µ ∗ β M = Z t ∗ + Vol( O ξ , ω ξ ) µ ∗ β W . (113)Let dλ denote the measure on hw − ( ξ ) ∩ C defined by the lattice π Z m . By Lemma 4.12, Z hw − ( ξ ) ∩ C dλ = Vol( O ξ , ω ξ ) . By definition of C ( δ ) ⊂ C , there exists a constant c ( ξ ) > , depending continuously on ξ ∈ ˚ t ∗ + , such that Z hw − ( ξ ) ∩ C ( δ ) dλ > Vol( O ξ , ω ξ ) − c ( ξ ) δ. Since the measure µ ∗ β W is compactly supported and c ( ξ ) depends continuously on ξ ∈ ˚ t ∗ + , there exists C > such that C > Z t ∗ + c ( ξ ) µ ∗ β W . β denote the Liouville measure of ( M ( δ, ˚ △ , L ) , ω red ) . By Lemma 2.6, the pushforward of theLiouville measure of (˚ △ × T /L, ω std , pr) to t ∗ by pr equals µ ∗ β W (noting again that we have linearlyidentified Lie(
T /L ∗ ) with the affine subspace of t ∗ spanned by ˚ △ ). The Duistermaat-Heckman measureof ( M ( δ, ˚ △ , L ) , ω red ) as a Hamiltonian T × T -manifold is therefore dλ × µ ∗ β W , which is supported on C ( δ, ˚ △ ) . Combining these facts and applying Fubini’s theorem, Vol( M ( δ, ˚ △ , T W ) , ω red ) = Z C ( δ, ˚ △ ) dλ × µ ∗ β W = Z t ∗ + Z hw − ( ξ ) ∩ C ( δ ) dλ ! µ ∗ β W > Z t ∗ + (Vol( O ξ , ω ξ ) − c ( ξ ) δ ) µ ∗ β W > Z t ∗ + Vol( O ξ , ω ξ ) µ ∗ β W − Cδ = Vol( M, ω ) − Cδ.
Letting δ < ε/C completes the proof.As discussed in the introduction, Theorem 1.2 has a more direct proof than Theorem 5.15 since therecannot be nutation effects in the case of coadjoint orbits. Nonetheless, the following proof shows howTheorem 1.2 can be derived as a corollary of Theorem 5.15.
Proof of Theorem 1.2.
Let ( M, ω, µ ) = ( O λ , ω λ , ι ) be a regular coadjoint orbit of K , equipped with thecoadjoint action of K with moment map the inclusion ι : O λ → k ∗ . Then ( O λ , ω λ , ι ) is a multiplicityfree space with principal isotropy subgroup L = T and Kirwan polytope △ = { λ } . Let ( C × T × T, ω −∞ , (hw P T , wt P T )) be one of the Hamiltonian T × T -manifolds defined in Section 5.1.The model space is M ( δ, ˚ △ , L ) ∼ = ( C ( δ ) × T × T, ω −∞ ) (cid:12) λ T ∼ = ( △ λ ( δ ) × T , ω red ) . Here △ λ ( δ ) is the intersection of the subspace ( −√− ψ h ◦ hw t )( λ ) with C ( δ ) . It follows from the definitionof ω −∞ (Definition 5.1) that the reduced symplectic form ω red equals ω std := dυ ∧ dx + · · · + dυ m ∧ dx m . Thus, Theorem 5.15 gives us the desired embeddings ( △ λ ( δ ) × T , ω std ) ֒ → ( O λ , ω λ ) . Remark 5.16.
By Theorem 5.15, we have constructed embeddings of Hamiltonian T -manifolds ( M ( δ, ˚ △ , L ) , ω red , wt P T ) ֒ → ( M, ω, pr t ∗ ◦ µ ) where one recalls that pr t ∗ ◦ µ is the moment map for the action of T on M as the maximal torus of K .Reduction by T at an arbitrary element ξ ∈ t ∗ yields symplectic embeddings M ( δ, ˚ △ , L ) (cid:12) ξ T ֒ → M (cid:12) ξ T. In the case of coadjoint orbits, the space on the right is a symplectic analogue of a weight variety andthe space on the left has the form △ λ,ξ ( δ ) × T /T where △ λ,ξ ( δ ) is the intersection of △ λ ( δ ) with theaffine subspace ( −√− ψ h ◦ wt t )( ξ ) . It should be straightforward to show using Theorem 3.20 that suchembeddings are volume exhausting, provided the space on the right is non-empty.60 Main Results: Applications to Gromov width
Given two symplectic manifolds ( N, τ ) and ( M, ω ) , a symplectic embedding of ( N, τ ) into ( M, ω ) is aninjective immersion ϕ : N ֒ → M such that ϕ ∗ ω = τ . The Gromov width of a connected symplecticmanifold ( M, ω ) of dimension n is GWidth(
M, ω ) = sup r> (cid:8) πr | ∃ a symplectic embedding ( B n ( r ) , ω std ) ֒ → ( M, ω ) (cid:9) , where ω std is defined as in (4). Following earlier work by [40, 48, 59, 53], Caviedes Castro proved thefollowing. Theorem 6.1. [14] Let K be a compact connected simple Lie group. For all λ ∈ t ∗ + , GWidth( O λ , ω λ ) min { π h√− λ, α ∨ i | α ∈ R + and h√− λ, α ∨ i > } . (114) Example 6.2.
Let K = SU (2) , let λ ∈ ˚ t ∗ + , and let α denote the positive root. In this case, GWidth( O λ , ω λ ) = Vol( O λ , ω λ ) = 2 π ( √− λ, α )( α, α ) = 2 π h√− λ, α ∨ i . Lower bounds for Gromov width of symplectic manifolds with action-angle coordinates U × T n can beobtained by studying the “integral affine geometry” of the domain U as follows. The open simplex of size △ n := (cid:26) x = ( x , . . . , x n ) ∈ R n | x + · · · + x n < π and x i > ∀ i = 1 , . . . , n (cid:27) . (115)The open simplex of size ℓ > is the scaling ℓ △ n (i.e. the set of points ℓ x , x ∈ △ n ). The group of integralaffine transformations of ( R n , Z n ) , denoted Aff( R n , Z n ) , consists of all transformations Ψ : R n → R n of the form Ψ( x ) = A x + b , where A ∈ SL n ( Z ) and b ∈ R n . Given U, U ′ ⊂ R n , an integral affineembedding of U into U ′ is an integral affine transformation Ψ ∈ Aff( R n , Z n ) such that Ψ( U ) ⊂ U ′ . Definition 6.3.
The integral affine width of a set U ⊂ R n is c △ ( U ) := sup ℓ > { ℓ | ∃ Ψ ∈ Aff( R n , Z n ) such that Ψ( ℓ △ n ) ⊂ U } . Integral affine width has two important properties that are immediate from the definition.(i) (conformality) For all β ∈ R and U ⊂ R n , c △ ( βU ) = | β | · c △ ( U ) , where βU = { βu | u ∈ U } .(ii) (monotonicity) If Ψ ∈ Aff( R n , Z n ) and U, U ′ ⊂ R n such that Ψ( U ) ⊂ U ′ , then c △ ( U ) c △ ( U ′ ) .Gromov width has analogous properties. Conformality is the property that GWidth(
M, βω ) = | β | GWidth(
M, ω ) for all β ∈ R . Monotonicity is the property that if ( N, τ ) embeds symplectically into ( M, ω ) , then GWidth(
N, τ ) GWidth(
M, ω ) .We note two additional properties of integral affine width. First, for all U ⊂ R n , c △ ( U ) = c △ ( U int ) ,where U int denotes the interior of U . The second property is continuity. Recall the Hausdorff distancebetween non-empty sets A, B ⊂ R n is d H ( A, B ) := inf ε > { ε | A ⊂ B ε and B ⊂ A ε } , where A ε := { x ∈ R n | d ( x , A ) ε } and d ( x , A ) := inf { d ( x , y ) | y ∈ A } . The Hausdorff distancedefines a metric on the set K n of non-empty compact subsets of R n . Integral affine width is continuous asa real valued function on ( K n , d H ) . 61 emma 6.4. Let U be an open subset of R n . Then, GWidth( U × T n , ω std ) > c △ ( U ) , (116) where ω std denotes the standard symplectic structure on R n × T n defined in (1) .Proof. Let Ψ ∈ Aff( R n , Z n ) and ℓ > such that Ψ( ℓ △ n ) ⊂ U . The map e Ψ : ( R n × T n , ω std ) → ( R n × T n , ω std ) , e Ψ( x , θ ) = (Ψ( x ) , (Ψ − ) T ( θ )) is symplectic. Restriction of e Ψ defines a symplectic embedding of ( ℓ △ n × T n , ω std ) into ( U × T n , ω std ) .It is known that GWidth( ℓ △ n × T n , ω std ) = ℓ (see e.g. [18, Proposition 2.1] and the references therein).By monotonicity of Gromov width, it follows that GWidth( U × T n , ω std ) > ℓ . Taking the supremum overall ℓ such that there exists Ψ ∈ Aff( R n , Z n ) with Ψ( ℓ △ n ) ⊂ U completes the proof.Assume that K is a simple compact Lie group. Fix a double reduced word i for ( w , e ) , and let θ = ζ ◦ θ σ ( i ) , as in Remark 3.24. For λ ∈ ˚ t ∗ + , let △ λ ⊂ R m denote the projection to R m of the fiber (hw P T ) − ( λ ) ⊂ P T ( K ∗ , θ ) ; see also (65). The parameterization ˚ t ∗ + → K m , λ
7→ △ λ is continuous with respect to d H .Note that for β > , △ βλ = β △ λ . Lemma 6.5.
With K and θ as above, for all λ ∈ ˚ t ∗ + , c △ ( △ λ ) > min { π h√− λ, α ∨ i | α ∈ R + } . (117) Proof.
For all λ ∈ ˚ t ∗ + , let ℓ λ := min { π h√− λ, α ∨ i | α ∈ R + } . By Remark 3.24 together with [18,Theorem 7.2] and [52], there exists an integral affine embedding of ℓ λ △ m into △ λ for all λ ∈ √− P ∩ t ∗ + .For all β > and λ ∈ √− P ∩ t ∗ + , conformality of integral affine width implies c △ ( △ β · λ ) = c △ ( β △ λ ) = βc △ ( △ λ ) > βℓ λ = ℓ β · λ . Thus (117) holds for all λ in the set ˚ t ∗ + , Q := (cid:8) β · λ | β > , λ ∈ ˚ t ∗ + such that λ ∈ √− P ∩ t ∗ + (cid:9) . Now let λ ∈ ˚ t ∗ + arbitrary. The set ˚ t ∗ + , Q is dense in ˚ t ∗ + , so we can find a sequence ( λ n ) ∞ n =1 ⊂ ˚ t ∗ + , Q converging to λ ∈ ˚ t ∗ + . Then △ λ n converge to △ λ with respect to d H . It follows by continuity of c △ and ℓ λ that c △ ( △ λ ) = lim n →∞ c △ ( △ λ n ) > lim n →∞ ℓ λ n = ℓ λ . Proof of Theorem 1.3.
Let K be an arbitrary compact connected simple Lie group and fix λ ∈ ˚ t ∗ + arbitrary.By Theorem 6.1, it remains to prove the lower bound.Fix any choice of double reduced word i for ( w , e ) , and let θ be as above. Let ε > . For k ∈ N ,consider the /k -interior △ λ (1 /k ) of △ λ . Since △ λ (1 /k ) converges to △ λ with respect to d H , continuityof c △ implies that for k sufficiently large, c △ ( △ λ (1 /k )) > c △ ( △ λ ) − ε. (118)By Theorem 1.2, there exists a symplectic embedding ( △ λ (1 /k ) × T , ω std ) ֒ → ( O λ , ω λ ) . Combiningthis with monotonicity of Gromov width, Lemma 6.4, Equation (118), and Lemma 6.5, GWidth( O λ , ω λ ) > GWidth( △ λ (1 /k ) × T , ω std ) > c △ ( △ λ (1 /k )) > c △ ( △ λ ) − ε > min { π h λ, α ∨ i | α ∈ R + } − ε. (119)Since ε > was arbitrary, this completes the proof. 62 emark 6.6. As we see, our lower bounds come from an upper semicontinuity property enjoyed by integralaffine width. One might wonder if there is a more direct approach using an upper semicontinuity propertyof the Gromov width of regular coadjoint orbits. To our knowledge, it is an open problem whether Gromovwidth has such a property. See [18][Remark 4.1] for more details.
Remark 6.7.
Note that by Theorem 6.1 and (119), min { π h√− λ, α ∨ i | α ∈ R + } > c △ ( △ λ ) . Thus the inequality of Lemma 6.5 is in fact an equality.
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