Stratified Gradient Hamiltonian Vector Fields and Collective Integrable Systems
aa r X i v : . [ m a t h . S G ] O c t CANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS
BENJAMIN HOFFMAN AND JEREMY LANEA
BSTRACT . Let K be a non-abelian compact connected Lie group. We show that every Hamiltonian K -manifold M admits a Hamiltonian torus action of the same complexity on a connected open densesubset. The moment map for this torus action extends continuously to all of M . This generalizes thecelebrated Gelfand-Zeitlin integrable systems introduced by Guillemin and Sternberg.In the process we develop a general framework for integrating gradient Hamiltonian vector fieldson degenerations. This framework can be applied to degenerations of varieties that are possiblysingular or non-compact. It recovers a result of Harada and Kaveh as a special case. Our main resulton Hamiltonian K -manifolds is obtained by applying this framework to the base affine space of thecomplexification of K . C ONTENTS
1. Introduction 12. Background 73. Stratified gradient Hamiltonian flows 134. From valuations to stratified gradient Hamiltonian flows 225. Toric degenerations of G (cid:12) N K -spaces 37Appendix A. Proofs for Section 3 48Appendix B. Proofs for Section 5 56References 581. I NTRODUCTION
Let K be a compact connected Lie group with Lie algebra k . Fix a maximal torus T ⊂ K anda closed positive Weyl chamber t ∗ + ⊂ k ∗ . A Hamiltonian K -manifold is a connected symplecticmanifold M equipped with an action of K which is generated by a K -equivariant moment map µ : M → k ∗ .An important class of Hamiltonian K -manifolds are those which are proper , i.e. for which thereexists a K -invariant set τ ⊂ k ∗ containing µ ( M ) such that τ ∩ t ∗ + is convex and µ : M → τ is a proper map. For any proper Hamiltonian K -manifold M , the set µ ( M ) ∩ t ∗ + is a convex,locally rational polyhedral set and the fibers of µ are connected [57, Theorem 1.1 and Remark5.2]. If M is compact, then µ ( M ) ∩ t ∗ + is a convex rational polytope and this recovers Kirwan’s non-abelian convexity theorem [51]. If additionally K is a torus, then this recovers the famousAtiyah-Guillemin-Sternberg convexity theorem [9, 30].A useful invariant of a Hamiltonian K -manifold M is its complexity. If K = T is a torus and T ker is the kernel of the action of T , then the complexity of M is defined as(1) c T ( M ) = 12 dim M − dim T + dim T ker . Complexity 0 torus manifolds with T ker = 0 are more commonly known as symplectic toric man-ifolds . Delzant famously showed that compact symplectic toric manifolds are classified up to iso-morphism by their moment polytopes [20]. More generally, proper complexity 0 torus manifoldsare classified by their moment map image and T ker [44, Proposition 6.5]. There are many resultsabout torus manifolds of complexity > including classification results, see e.g. [45, 46] and thereferences within.If K is non-abelian, then every Hamiltonian K -manifold M contains a Hamiltonian T -manifold S called the principal symplectic cross-section [34]. The complexity of a K -manifold M is thecomplexity of its principal symplectic cross-section S as a Hamiltonian T -manifold:(2) c K ( M ) = c T ( S ) . Complexity 0 Hamiltonian K -manifolds, also known as multiplicity free K -manifolds , were stud-ied by Guillemin and Sternberg in the 1980’s in relation to geometric quantization and integrablesystems [31, 33, 32]. The classification of proper multiplicity free K -manifolds, also known asthe Delzant conjecture, was eventually proved in [52] following the classification of smooth affinespherical varieties [60]. Hamiltonian K -manifolds of a given complexity are more complicatedwhen K is not a torus. Not much is known about complexity > Hamiltonian K -manifolds for K non-abelian with a few exceptions in low dimension [15].Given K , denote m = (dim K − dim T ) and let T = ( S ) m × T . In 1983, Guillemin and Sternberggave a beautiful construction which proves the following theorem . Theorem 1.1. [31]
Let K be a compact Lie group, and assume the simple factors of k are each oftype A , B , or D . For any Hamiltonian K -manifold M , there exists a connected open dense subset D ⊂ M and a Hamiltonian T -action on D such that c T ( D ) = c K ( M ) . The T -equivariant moment map µ T : D → Lie( T ) ∗ extends to a continuous map µ T : M → Lie( T ) ∗ . Furthermore, if M is a proper Hamiltonian K -manifold, then D is a proper Hamiltonian T -manifold. The construction and the resulting T -actions used to prove Theorem 1.1 are known as Gelfand-Zeitlin systems . As part of the construction of Gelfand-Zeitlin systems, Guillemin and Sternbergshow that µ T ( M ) has an explicit description in terms of µ ( M ) ∩ t ∗ + and certain inequalities knownas “interlacing inequalities.”In the case that M is proper and has complexity 0, then D is classified as a proper complexity 0 T -manifold by its moment map image and T ker . Furthermore, the Gelfand-Zeitlin system equipsa dense subset of D with explicit action-angle coordinates. This coordinatization of D has appli-cations in quantitative symplectic topology [66]. Gelfand-Zeitlin systems have also been used to Although Guillemin and Sternberg only discuss the case of complexity 0 in [31], their construction yields this state-ment for arbitrary complexity. See [55] for the statement about properness.
ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 3 construct new examples of non-displaceable Lagrangian submanifolds [17, 16, 64, 65] and studythe structure of Poisson-Lie groups [6]. They motivated results on toric degenerations, integrablesystems, and geometric quantization [35, 37, 53, 64] as well as Poisson-Lie groups and the devel-oping theory of partial tropicalization [1, 2, 3, 4, 5].The primary drawback of the Gelfand-Zeitlin construction is that it only applies to groups withsimple factors of type A , B , or D . It is a long standing problem to generalize Theorem 1.1 toarbitrary K . Results have been obtained in some cases [36, 37, 4]. Our main result is a full solutionto this problem. Theorem 1.2.
Let K be a compact Lie group. For any Hamiltonian K -manifold M , there exists aconnected open dense subset D ⊂ M and a Hamiltonian T -action on D such that c T ( D ) = c K ( M ) . The T -equivariant moment map µ T : D → Lie( T ) ∗ extends to a continuous map µ T : M → Lie( T ) ∗ . Furthermore, if M is a proper Hamiltonian K -manifold, then D is a proper Hamiltonian T -manifold. If M is proper, then µ T ( M ) is a convex, locally polyhedral set . The significance of this fact isdiscussed in Section 1.2 below.The moment map µ T for the Hamiltonian T -action is constructed by composing the moment mapfor the K -action with a continuous map F :(3) µ T : M µ −→ k ∗ F −→ Lie( T ) ∗ . Our construction of F , which depends on several choices, is outlined below. Although F is notsmooth everywhere, there exists a connected dense open subset D ⊂ M such that F ◦ µ : D → Lie( T ) ∗ is smooth and generates a Hamiltonian T -action. Hamiltonian torus actions constructedby composing µ with another moment map are known as collective integrable systems . Gelfand-Zeitlin systems are also examples of collective integrable systems.We briefly give an example to describe how the scope of our result differs from those in [31]and [37]. Let M be a coadjoint orbit of a compact Lie group K . Then M is a complexity K -manifold, with respect to the coadjoint action of K . The moment map of M is inclusion into k ∗ , and the symplectic form on M is the standard Kostant-Kirillov-Souriau symplectic form. Theapproach of [31] produces a densely defined complexity 0 torus action only when the simple factorsof k are each of type A , B , or D . On the other hand, the approach of [37] produces a denselydefined complexity 0 torus action only when the symplectic form on M comes from an embeddingin projective space, which only occurs when M passes through a (scalar multiple of an) integralweight of K . By contrast, Theorem 1.2 applies to arbitrary coadjoint orbits of arbitrary compactLie groups.1.1. Outline of the construction of F . Let G = K C be the complexification of K . The base affinespace of G is the categorical quotient of G by a maximal unipotent subgroup N , denoted G (cid:12) N .It is a singular affine variety which plays a central role in representation theory and algebraic group One should note that this is not an immediate consequence of the convexity theorems mentioned above since µ T : M → Lie( T ) ∗ is not a smooth moment map. Some authors refer to the homogeneous space
G/N as base affine space. The variety G (cid:12) N is the affine closure of G/N . BENJAMIN HOFFMAN AND JEREMY LANE actions. We construct F as part of a commuting diagram of the following form; see Sections 4 and5 for details.(4) G (cid:12) N X S k ∗ Lie( T ) ∗ . φ/T Ψ F In this diagram, S is the value semigroup of a fixed valuation on the coordinate ring C [ G (cid:12) N ] ,and X S is the affine toric variety whose coordinate ring is the semigroup algebra of S . The chosenvaluation on C [ G (cid:12) N ] must satisfy a number compatibility conditions, and the existence of such avaluation is discussed below.Both G (cid:12) N and X S are equipped with a decomposition as a disjoint union of smooth complexmanifolds. By embedding G (cid:12) N and X S as subvarieties of a complex inner product space E ,each of their smooth pieces acquires a Kähler structure. There is a unitary representation of T on E which preserves the smooth pieces of X S . As a result, each smooth piece of X S is a Hamiltonian T -manifold, with moment map equal to the restriction of a quadratic moment map Ψ : E → Lie( T ) ∗ .The construction of φ has two main ingredients. The first ingredient is a toric degeneration. Fromthe fixed valuation on C [ G (cid:12) N ] , a standard construction [8] produces a flat family of varieties π : X → C . The fiber π − (0) is isomorphic to X S , and the fibers over t = 0 are each isomorphic to G (cid:12) N . The family π : X → C is then an example of a toric degeneration of G (cid:12) N to X S . Awayfrom the toric fiber π − (0) , the family X → C decomposes as a union of smooth subfamilies X σ .These subfamilies are indexed open faces σ of the positive Weyl chamber.The second ingredient in the construction of φ is a gradient Hamiltonian vector field. The family X is embedded as a subvariety of a complex inner product space. Each X σ inherites a Kähler structurefrom the embedding. On each X σ , the gradient Hamiltonian vector field is defined to be V σπ = − ∇ℜ π ||∇ℜ π || where ℜ π denotes the real part of π . The significance of gradient Hamiltonian vector field is thatits flow, when defined, induces a symplectomorphism of fibers of π , ϕ σt : X σ ∩ π − (1) → X σ ∩ π − (1 − t ) . We combine these flows ϕ σt into a piece-wise defined map ϕ t : π − (1) → π − (1 − t ) .The map φ is then constructed as the limit of the maps ϕ t as t → .Gradient Hamiltonian flows on toric degenerations were used previously by [64, 37] to build mapsfrom smooth projective varieties to toric varieties. In our case, we follow the general strategyof [37], but there are some notable complications. First, the map π : X → C is not proper, andso there is no guarantee that the flow ϕ σt exists, even when t < . Second, φ is defined piecewiseusing the flows ϕ σt on the decomposed variety X . There is no guarantee that these assemble to acontinuous map. We build a general framework for integrating gradient Hamiltonian flows in thisnew setting in Section 3. It turns out that both these problems can be resolved in the presence ofa sufficiently nice Hamiltonian torus action on X . Applying this framework to the degeneration of G (cid:12) N we arrive at the following (cf. Theorem 5.7 and Proposition 6.8). Theorem 1.3.
The limit as t → of the gradient Hamiltonian flow ϕ t exists and defines a contin-uous, proper, surjective map φ : G (cid:12) N → X S . Each smooth piece in G (cid:12) N has an open densesubset D such that φ | D is a symplectomorphism onto its image in X S . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 5
The map on the left in (4) is a quotient map for a certain action of the maximal torus T . Once φ hasbeen constructed, the map F in (4) is descends from Ψ ◦ φ which is T -invariant.The key step in the remainder of the proof of Theorem 1.2 is a new construction which we call toric contraction . Toric contraction combines the toric variety X S , symplectic implosion [29], andsymplectic contraction [39] to produce a singular Hamiltonian T -space from an arbitrary Hamilton-ian K -manifold M . Although this construction is entirely new, it bears resemblance to the earlierconstructions of [64, 39]. It is described in Section 6.1.2. Connections with canonical bases and representation theory.
Throughout the constructionoutlined above, there are a number of choices that must be made. Chief among them is a valuation v on C [ G (cid:12) N ] , which satisfies a number of compatibility conditions. The complete set of choicessatisfying these conditions is summarized in our notion of a good valuation (Definition 4.10). Toprove Theorem 1.2, it remains to show that there exists a good valuation.One way to construct a good valuation is using the string valuations constructed in [48]. Fix areduced word i for the longest element of the Weyl group of G and let Λ + ⊂ t ∗ + denote the setof dominant integral weights. Let S i ⊂ Z m × Λ + denote the extended string cone associated to i [10, 59]. The cone S i is an affine semigroup, and it generates a convex rational polyhedral cone, cone( S i ) ⊂ R m × t ∗ [59]. It is possible to realize S i as the value semigroup of a valuation on C [ G (cid:12) N ] known as a string valuation [48, Proposition 6.1]. In Section 5.2 we show the stringvaluation is a good valuation.Applying our construction to the string valuation produces a continuous map F i : k ∗ → Lie( T ) ∗ = R m × t ∗ whose image is cone( S i ) . If M is a Hamiltonian K -manifold, then the composition(5) M µ −→ k ∗ F i −→ cone( S i ) ⊂ Lie( T ) ∗ generates a Hamiltonian torus action on a dense open subset D ⊂ M . Let pr denote the projection R m × t ∗ → t ∗ . If M is a proper K -manifold, then image of (5) is the convex locally rationalpolyhedral set. It is equal to the intersection of cone( S i ) with the preimage under pr of the Kirwanpolytope of M : F i ◦ µ ( M ) = cone( S i ) ∩ pr − ( µ ( M ) ∩ t ∗ + ) . Moreover, the open dense subset D ⊂ M can be described in terms of the geometry of F i ◦ µ ( M ) (see Proposition 6.17). For example, if M is the coadjoint orbit through a weight λ ∈ t ∗ + , then F i ( M ) = cone( S i ) ∩ pr − ( λ ) is a convex rational polytope. In this case the subset D ⊂ M is the pre-image under F i of thesmooth locus of F i ( M ) . If λ is integral, then F i ( M ) is commonly known as the string polytope associated to λ and i .Every irreducible unitary representation V of K has a natural decomposition into weight spaces forthe maximal torus. A canonical basis for V is a decomposition of the weight spaces into 1 dimen-sional subspaces that is in some way “canonical,” i.e. it depends on very few choices. The oldest andmost well-known example is Gelfand-Zeitlin canonical bases for K = U ( n ) . The Gelfand-Zeitlin We learned this heuristic definition from Yael Karshon who informs us that it was also used in Michael Grossberg’sthesis.
BENJAMIN HOFFMAN AND JEREMY LANE systems are related to Gelfand-Zeitlin canonical bases via Bohr-Sommerfeld quantization [31]. Thecollective integrable systems (5) are similarly related to canonical bases for representations of K . There are other examples of valuations in the literature that are similarly related to G (cid:12) N and therepresentation theory of G , such as those described by Fujita-Naito (which recover the Nakashima-Zelevinsky polytopes) [24], those described by Kiritchenko and Feigin-Fourier-Littelmann (whichrecover the Feigin-Fourier-Littelmann-Vinberg polytopes) [50, 23] or the cluster-theoretic valua-tions recently studied by Fujita-Oya in [25]. We expect these valuations will also be amenable toour machinery.1.3. Future directions.
One motivation for this work arises from quantitative symplectic topol-ogy. It is a conjecture due to Yael Karshon that the Gromov width of a coadjoint orbit O λ of acompact simple Lie group K parameterized by λ ∈ t ∗ + is given by a simple formula involving λ and the positive coroots of K [22, Equation 1.1]. Tight upper bounds were proved for all λ and K using the theory of J -holomorphic curves in [14]. The general approach to tight lower boundsis to use Hamiltonian torus actions. In particular, if a subset D ⊂ O λ admits the structure of acomplexity 0 proper torus manifold, then one may construct embeddings of symplectic balls into D using the geometry of its moment map image [66, Proposition 2.5]. This approach has beenapplied to establish tight lower bounds in all the cases where it was previously known that one canequip a big subset of O λ with a proper complexity 0 torus action (see [22] for a survey of knownresults, as well as the more recent results in [4]). Since Theorem 1.2 constructs dense subsets withproper complexity 0 torus actions in all the remaining cases, we expect it will be sufficient to closethe conjecture. We hope to resolve the details of this application in a future paper.A second application is to reduced products of coadjoint orbits. Given several coadjoint orbits O λ , . . . , O λ k of a compact Lie group K , one may form the symplectic reduction of O λ ×· · ·× O λ k by a diagonal action of K . Through geometric quantization, these spaces are linked with the prob-lem of counting multiplicities in tensor products of irreducible G -modules. If K = SU (2) , thenthese reduced spaces are known as polygon spaces. Polygon spaces have complexity 0 Hamilton-ian torus actions known as bending flow systems [43]. For K = SU (3) and generic parameters ( λ , λ , λ ) these reduced spaces are isomorphic to a symplectic 2-sphere. It was recently shownhow this 2-sphere can be equipped with a complexity 0 Hamiltonian circle action [42, 41]. In futurework we will apply the techniques developed in this article to construct complexity 0 Hamiltoniantorus actions on dense subsets of these reduced spaces for arbitrary K and λ , . . . , λ k . We antici-pate that this can be achieved using the valuations constructed on the related affine variety knownas affine configuration space in [62].Toric degenerations play an important role in the rapidly developing theory of cluster varieties [26,11]. In particular, both G (cid:12) N and the configuration space are closely related to well-known clustervarieties. It would be interesting to understand how toric degenerations of cluster varieties interactwith the theory developed here. Heuristically, one might expect these bases to be the Kashiwara-Lusztig dual canonical bases [47, 61], which areparametrized by the lattice points of S i . However, this analogy is not completely apt because the Kashiwara-Lusztigbases are not typically orthogonal, whereas the bases coming from Bohr-Sommerfeld quantization are. Nevertheless,much of the literature on the string cones and valuations is motivated by finding parametrizations for Kashiwara-Lusztig dual canonical bases, and we found this to be a useful example to keep in mind. Thanks to Allen Knutson forexplaining these details to us. ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 7
Finally, in recent work with Anton Alekseev and Yanpeng Li, the authors constructed Hamiltoniantorus actions from Hamiltonian group actions using entirely different techniques involving Poisson-Lie groups, Ginzburg-Weinstein isomorphisms, and the emerging theory of partial tropicalization[5, 4]. The relationship between these two approaches (partial tropicalization and toric degenera-tion) remains an open and interesting question.1.4.
Organization.
The organization of the remainder of the article is as follows. Section 2 fixesnotation and recalls useful background on singular spaces, Hamiltonian group actions, and toricvarieties. Section 3 contains our general framework for integrating gradient Hamiltonian vectorfields on degenerations of singular varieties. Section 4 introduces the notion of a good valuation onan affine variety. From this data, we describe how to construct a toric degeneration and apply theresults of Section 3 to integrate the gradient Hamiltonian flow (Theorem 4.21). Section 5 appliesthis construction to the base affine space and proves Theorem 1.3. Finally, Section 6 combines theseresults to prove Theorem 1.2. Several proofs from Section 3 and 5 are relegated to appendices.1.5.
Acknowledgements.
The authors would like to thank Anton Alekseev, Megumi Harada, YaelKarshon, Kiumars Kaveh, Allen Knutson, Chris Manon, Daniele Sepe, and Reyer Sjamaar forhelpful suggestions and conversations throughout the course of this project. J.L. would like to thankthe Fields Institute and the organizers of the thematic program on Toric Topology and PolyhedralProducts for the support of a Fields Postdoctoral Fellowship during writing of this paper.2. B
ACKGROUND
Decompositions and singular spaces.
This section defines decomposed spaces. We directthe reader to [67] for background.
Definition 2.1. A weakly decomposed space is a paracompact, Hausdorff, second countable topo-logical space X equipped with a locally finite partition by locally closed subspaces X σ ⊂ X , called pieces , such that each piece X σ is a connected smooth manifold in the subspace topology.A map of weakly decomposed spaces is a continuous map f : X → Y such that the image of eachpiece X σ ⊂ X is contained in a piece Y τ ⊂ Y and the restricted maps f σ = f | X σ : X σ → Y τ aresmooth [67, 1.1.6]. Weakly decomposed spaces and maps of weakly decomposed spaces form acategory. Finite products and coproducts are defined in the category of weakly decomposed spacesin the obvious way [67, 1.1.7]. Definition 2.2. [67, Definition 1.1.1] A decomposed space is a weakly decomposed space whosepartition satisfies the frontier condition : If X σ ∩ X τ = ∅ , then X σ ⊂ X τ .Decomposed spaces form a full subcategory of the category of weakly decomposed spaces. Thissubcategory is closed under finite products and coproducts. The set of pieces of a decomposedspace has a naturally defined partial order, which we denote ≺ . If X σ and X τ are pieces of adecomposed space X , then:(6) X σ ≺ X τ if and only if X σ ⊂ X τ . We will often identify the set of pieces of a weakly decomposed space with an indexing set Σ such that an element σ ∈ Σ corresponds to a piece X σ ⊂ X . If the partition satisfies the frontiercondition, then Σ inherits the partial order ≺ . BENJAMIN HOFFMAN AND JEREMY LANE
One of the primary examples of decomposed spaces in this paper will be varieties. By a variety wemean an irreducible quasi-projective variety over the complex numbers as in [38, Chapter 1].
Definition 2.3.
We say that a variety X is a decomposed variety if it is equipped with a partitionby finitely many smooth irreducible subvarieties X σ ⊂ X which endows X with the structure of adecomposed space with respect to its analytic topology.2.2. Convex geometry. A lattice is a free Z -module of finite rank. Given a lattice L , let L ∨ =Hom( L, Z ) denote the dual lattice and let L R = L ⊗ Z R . The cone generated by a set A ⊂ L R is(7) cone( A ) = (X a ∈ A c a a | c a ∈ R ≥ , c a = 0 for all but finitely many a ∈ A ) . A convex polyhedral cone in L R is a subset of the form cone( A ) for some finite A ⊂ L R . Apolyhedral cone is strongly convex if it does not contain any non-trivial subspaces of L R . If A ⊂ L ,then cone( A ) is a rational convex polyhedral cone.A subset P ⊂ L R is a rational convex polyhedron if it can be represented as an intersection offinitely many affine half-spaces whose normal vectors are elements of L ∨ . Every closed face F of P is an intersection of facets F , . . . , F k . A face F is smooth if the primitive normal vectors ofthe facets F , . . . , F k can be extended to a Z -basis for L ∨ . If F ′ ⊂ F and F ′ is smooth, then F issmooth. A point p in P is smooth if the smallest face of P containing p is smooth. The smoothlocus of P is the set of all smooth points and the singular locus is the complement of the smoothlocus. The singular locus of P is a union of faces of codimension ≥ .A subset V ⊂ L R is locally rational polyhedral if for all p ∈ V , there is a neighborhood U p of p and a rational convex polyhedron P such that U p ∩ V = U p ∩ P . If p happens to be a smooth pointof P , then we say p is a smooth point of V . The smooth locus of a locally rational polyhedral set V is the set of smooth points of V and the singular locus is its complement.As an example, open sets in L R are locally rational polyhedral, as are rational convex polyhedra.Note that locally rational polyhedral sets need not be convex.2.3. Lie theory.
Throughout, K denotes a compact Lie group and G denotes the complex formof K . Fix a maximal complex algebraic torus H ⊂ G and let T be the maximal compact torus H ∩ K ⊂ K . Lie algebras are often denoted with fraktur letters, e.g. Lie( G ) = g .Let Λ ⊂ t ∗ denote the lattice of real weights of T . We use the convention that each λ ∈ Λ corresponds to the character T → S , t = exp( ξ ) t λ = e √− h λ,ξ i , for all ξ ∈ t .Fix a set R + of positive roots for G and let Λ + ⊂ Λ denote the semigroup of dominant real weights.Let m denote the number of positive roots of G , let n denote the number of simple roots, and let r be the dimension of T . Let N and N − be the opposite unipotent radical subgroups of G with Liealgebras n = M α ∈ R + g α , and n − = M α ∈ R + g − α . Here g α is the α -weight subspace of g under the adjoint representation.The positive Weyl chamber t ∗ + ⊂ t ∗ is the convex rational polyhedral cone generated by Λ + . Wemake the canonical identification of t ∗ with the subspace ( k ∗ ) T ⊂ k ∗ of T -fixed points for thecoadjoint action. Let k ∗ /K denote the quotient topological space for the coadjoint action of K . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 9
The restriction to t ∗ + of the quotient map k ∗ → k ∗ /K defines a homeomorphism t ∗ + ∼ = k ∗ /K . The sweeping map is the continuous map S : k ∗ → k ∗ /K ∼ = t ∗ + .2.4. Hamiltonian group actions.
Recall that k and k ∗ are equipped with the adjoint and coad-joint representations of K . The Lie-Poisson
Poisson structure on k ∗ is the Poisson structure whosebracket on linear functions is given by the formula { ξ, η } k ∗ = [ ξ, η ] for all ξ, η ∈ k , where ξ, η ,and [ ξ, η ] are all viewed as linear functions on k ∗ . The symplectic leaves of the canonical Poissonstructure are precisely the orbits of the coadjoint action of K .Given a smooth action of K on a smooth manifold M , we define the generating vector field map, k → X ( M ) , ξ ξ M , with the sign convention so that it is a Lie algebra anti-homomorphism. Anaction of K on a Poisson manifold M is Hamiltonian if there is a K -equivariant moment map , i.e. a K -equivariant map µ : M → k ∗ , such that for all ξ ∈ k ,(8) {h µ, ξ i , ·} M = ξ M . Since our moment maps are K -equivariant, they are Poisson maps. If M is symplectic with sym-plectic form ω , then (8) is equivalent to the moment map equation ι ξ M ω = d h µ, ξ i . A Hamiltonian K -manifold is a tuple ( M, ω, µ ) where ( M, ω ) is a connected symplectic manifoldequipped with a Hamiltonian action of K and µ : M → k ∗ is an equivariant moment map. Amap of Hamiltonian K -manifolds ( M , ω , µ ) → ( M , ω , µ ) is a K -equivariant symplectic map ϕ : M → M such that µ ◦ ϕ = µ . Example 2.4 (Representations) . Suppose that E is finite dimensional real vector space equippedwith a linear symplectic form ω . Let ρ : K → Sp ( E, ω ) be a representation of K on E by linearsymplectic transformations. The action of K on ( E, ω ) is Hamiltonian with moment map µ : E → k ∗ defined by the condition(9) ω (( ξ E ) v , v ) = h µ ( v ) , ξ i , ∀ v ∈ E, ξ ∈ k . If E is a finite dimensional complex vector space, equipped with a complex inner product h , then ω = −ℑ h is a linear symplectic structure on E . Any unitary representation of K on ( E, h ) issymplectic and thus is also Hamiltonian with moment map (9). Remark . In general, the moment map of a Hamiltonian group action is only unique up totranslation by an element of ( k ∗ ) K . For representations we will always take the moment mapdefined by (9). It is uniquely determined by the property that µ (0) = 0 . Example 2.6 (Torus representations) . Suppose K = T is a compact torus, ( E, h ) is a finite di-mensional complex inner product space, and ρ : T → U ( E, h ) is a unitary representation. Suppose v ∈ E is a weight vector of real weight λ ∈ Λ , i.e. e ξ · v = e √− h λ,ξ i v for all ξ ∈ t . Then themoment map µ : E → t ∗ is given by(10) µ ( v ) = − || v || λ. All group actions we consider are left actions.
The vector space E decomposes orthogonally as a direct sum of weight subspaces E = ⊕ λ ∈ Λ E λ .If v = P λ v λ is the orthogonal decomposition of v , v λ ∈ E λ , then(11) µ ( v ) = − X λ ∈ Λ || v λ || λ. Suppose z , . . . , z n is a basis of E ∗ dual to an orthonormal basis of ( E, h ) . If each z i is a weightvector for the dual representation, of real weight λ i , then(12) µ = 12 n X i =1 | z i | λ i . In particular, µ ( E ) = cone( { λ , . . . , λ n } ) .We end this section with several useful properties of the moment map µ from Example 2.6. Lemma 2.7.
Let a compact torus T act by unitary transformations on a finite dimensional complexinner product space ( E, h ) and let µ : E → Lie( T ) ∗ be the moment map as in Example 2.6. Then:(i) For any connected subtorus T ′ ⊂ T , let ann(Lie( T ′ )) ⊂ Lie( T ) ∗ denote the annihilator of Lie( T ′ ) . Then, (13) Fix(
E, T ′ ) = µ − (ann(Lie( T ′ ))) , where Fix(
E, T ′ ) is the fixed point set for the action of T ′ on E .(ii) Suppose that the set of real weights of the representation is { λ , . . . , λ n } ⊂ Λ . Then, µ : E → Lie( T ) ∗ is a proper map if and only if cannot be written as a non-trivial linearcombination of λ , . . . , λ n with non-negative coefficients. Hamiltonian actions on singular symplectic spaces.
There are several definitions of singu-lar symplectic spaces in the literature. We use the following definition.
Definition 2.8. A singular symplectic space is a locally compact weakly decomposed space X such that each piece X σ is equipped with a symplectic structure, denoted ω σ .Symplectic manifolds are singular symplectic spaces with respect to the trivial decomposition. Theproduct of two singular symplectic spaces X and Y is the product decomposed space X × Y whereeach piece X σ × Y τ is equipped with the product symplectic structure, ω σ ⊕ ω τ . Definition 2.9. A Hamiltonian K -space is a pair ( X, µ ) where X is singular symplectic spaceequipped with a continuous action of K and µ : X → k ∗ is a continuous map such that, for all σ ∈ Σ , the action of K preserves X σ and ( X σ , ω σ , µ | X σ ) is a Hamiltonian K -manifold. A map ofHamiltonian K -spaces f : ( X, µ ) → ( Y, ψ ) is a map of decomposed spaces f : X → Y such thatthe restricted maps f σ : ( X σ , ω σ , µ | X σ ) → ( Y τ , ω τ , ψ | Y τ ) are maps of Hamiltonian K -manifolds.Embedding a decomposed variety X into a smooth Kähler variety M produces a singular sym-plectic space since each piece X σ inherits a symplectic structure from the embedding. As we willfrequently use this construction, we formalize it in the following definition. A map of topological spaces f : X → Y is proper if f − ( C ) is compact for every C ⊂ Y compact. ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 11
Definition 2.10. A decomposed Kähler variety is a tuple ( X, M, ω ) where:(i) X is a decomposed variety,(ii) M is a smooth variety equipped with a Kähler form ω (compatible with the complex struc-ture on M ), and(iii) X is equipped with an embedding into M as a (not necessarily closed) subvariety. Theembedding is implicit in the tuple ( X, M, ω ) .If M is a vector space and ω is linear, then we say that ( X, M, ω ) is a decomposed affine Kählervariety .If ( X, M, ω ) is a decomposed Kähler variety, ( M, ω ) is equipped with a Hamiltonian K action withmoment map µ : M → k ∗ , and the action of K preserves the pieces of X , then X is a Hamiltonian K -space with moment map µ | X : X → k ∗ .Let J, K be compact Lie groups. If ( X, µ ) is a Hamiltonian K -space and ( Y, ψ ) is a Hamiltonian J -space, then the product singular symplectic space X × Y is a Hamiltonian K × J -space with momentmap µ × ψ . If ( X, µ ) is a Hamiltonian K -space and ϕ : J → K is a Lie group homomorphism,then ( X, ( dϕ ) ∗ ◦ µ ) is a Hamiltonian J -space.The symplectic reduction of a Hamiltonian K -space ( X, µ ) at a point λ ∈ k ∗ is the quotient topo-logical space X (cid:12) λ K := µ − ( λ ) /K λ , where K λ is the stabilizer subgroup of λ for the coadjointaction. We end this section by noting two standard facts about symplectic reduction of singularspaces which are straightforward to check. Lemma 2.11.
Let K be a connected Lie group and let ( X, µ ) be a Hamiltonian K -space. If theaction of K on X is proper , then for all λ ∈ k ∗ , the symplectic reduced space X (cid:12) λ K is a singularsymplectic space. If X is a symplectic manifold, then the pieces of the decomposition of X (cid:12) λ K were describedby Sjamaar and Lerman in [70]. In general, X (cid:12) λ K partitions into subsets X σ (cid:12) λ K for eachsymplectic piece X σ . The decomposition of X (cid:12) λ K is then defined by further partitioning each X σ (cid:12) λ K à la Sjamaar and Lerman.If ( X, ( µ, ψ )) is a Hamiltonian K × J -space, then the action of J and the moment map ψ descendto the symplectic reduced spaces X (cid:12) λ K . In more detail, if [ x ] ∈ X (cid:12) λ K denotes an equivalenceclass of an element x ∈ µ − ( λ ) , then j · [ x ] = [ j · x ] well-defines a continuous action of J on X (cid:12) λ K and [ x ] ψ ( x ) well-defines a continuous map X (cid:12) λ K → j ∗ . Lemma 2.12.
Suppose that ( X, ( µ, ψ )) is a Hamiltonian K × J -space and the action of K on X is proper. Then, X (cid:12) λ K is a Hamiltonian J -space with respect to the action of J and the momentmap induced from the action of J on X and the map ψ as above. Affine G -varieties. If X is a closed subvariety of C n , we write I ( X ) ⊂ C [ C n ] for the ideal offunctions vanishing on X . Conversely, if I ⊂ C [ C n ] , then we write V ( I ) ⊂ C n for the vanishinglocus of I . All the algebras we consider will be integral domains. Given an algebra A we make nodistinction between the affine scheme Spec A and its set of closed points.Let G be a reductive algebraic group with (real) weight lattice Λ and let X be an affine G -variety.Let Λ( X ) ⊂ Λ denote the semigroup of highest weights that appear in the G -module C [ X ] . Let A continuous action of a topological group K on topological space X is proper if the map K × X → X × X , ( k, x ) ( x, kx ) is proper. Γ( X ) be the rational convex polyhedral cone cone Λ( X ) ⊂ t ∗ . The cone Γ( X ) has a canonicaldecomposition by the relative interiors of its closed faces. Denote the set of pieces of this de-composition by Σ( X ) (i.e. σ ∈ Σ( X ) is the relative interior of the closed face σ ⊂ Γ( X ) ). Weabbreviate Γ = Γ( X ) , and Σ = Σ( X ) , when the meaning is clear from context.For the remainder of this subsection, let G = H be an algebraic torus with (real) weight lattice Λ , maximal compact torus T , and let X be an affine H -variety. The coordinate ring C [ X ] is a Λ -graded algebra. We write | f | Λ ∈ Λ to denote the Λ -homogeneous degree of a homogeneouselement of C [ X ] . We abbreviate | f | = | f | Λ when the meaning is clear from context. Then Λ( X ) isthe set of degrees of homogeneous elements in C [ X ] .For each σ ∈ Σ( X ) , let T σ ⊂ T denote the connected subtorus such that span R ( σ ) = ann(Lie( T σ )) .Let H σ denote the algebraic subtorus of H with maximal compact torus equal to T σ . Denote(14) X σ = Fix( X, H σ ) , X σ = X σ \ [ τ ≺ σ X τ . where Fix(
X, H σ ) is the fixed point set for the action of H σ . Suppose X is embedded H -equivariantlyas a closed subvariety of a complex vector space E equipped with a representation of H . Assumethat Λ( E ) = Λ( X ) . Denote(15) E σ = Fix( E, H σ ) , E σ = E σ \ [ τ ≺ σ E τ . By definition, X σ = X ∩ E σ as varieties. In fact, slightly more is true. Lemma 2.13.
The scheme-theoretic intersection X ∩ E σ is reduced, i.e. X σ = X ∩ E σ as schemes.Proof. We want to show that I ( X ) + I ( E σ ) is a radical ideal. Let f ∈ C [ E ] such that f k ∈ I ( X ) + I ( E σ ) for some k . We must show that f ∈ I ( X ) + I ( E σ ) . It is a straightforward exerciseto show that it suffices to prove the Lemma for f which is homogeneous.Let us fix a basis K = { z , . . . , z J } of E ∗ consisting of Λ -homogeneous elements. Note that(16) I ( E σ ) = ( z j ∈ K | | z j | / ∈ σ ) . Suppose f k is homogeneous. Then f = g + h , for some homogeneous g ∈ I ( X ) and h ∈ I ( E σ ) .If | f k | ∈ σ , then | h | ∈ σ . The only way this can happen is if h = 0 , so f k ∈ I ( X ) . Since I ( X ) is radical, we have that f ∈ I ( X ) ⊂ I ( X ) + I ( E σ ) . On the other hand, if | f k | / ∈ σ then eachmonomial term of f k contains some z j with | z j | / ∈ σ . Then f k vanishes on E σ so f k ∈ I ( E σ ) . Thisideal is also radical, so f ∈ I ( E σ ) ⊂ I ( X ) + I ( E σ ) . (cid:3) Suppose that E is equipped with a complex inner product h and the representation of T on ( E, h ) isunitary. Let µ : E → Lie( T ) ∗ denote the moment map as in Example 2.6. Then part (i) of Lemma2.7 can be restated in the notation of this section as(17) µ − ( σ ) = E σ . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 13
Affine toric varieties.
We recall some notions from the theory of toric varieties; see [19] forfurther details.Let L be a lattice and let H be the algebraic torus with maximal compact subtorus T whose latticeof real weights is L ⊂ Lie( T ) ∗ . A subset S ⊂ L is an affine semigroup if it is closed under addition,contains , and is finitely generated. The semigroup algebra of a semigroup S is the commutativealgebra of formal linear combinations C [ S ] = (X s ∈ S a s χ s | a s ∈ C , a s = 0 for all but finitely many s ∈ S ) . Here χ s , s ∈ S , are formal variables. Their multiplication is induced by semiring addition, i.e. χ s · χ s ′ = χ s + s ′ . The semigroup algebra C [ S ] is the coordinate ring of an affine toric variety denoted X S . The algebraic action of H on X S is dual to the representation of H on C [ S ] by t · χ a = t a χ a for all a ∈ L . If S ⊂ L is saturated, then X S is normal.Let C be a rank( L ) -dimensional rational convex polyhedral cone in L R . The intersection S = C ∩ L is a saturated affine semigroup, and C = cone( S ) . The torus of the normal affine toric variety X S is H . The fan of X S (considered as an abstract toric variety) is the dual cone of C ; since we onlyconsider affine toric varieties we will not focus on the fan of X S .Given a closed face F of C , let T F ⊂ T denote the connected subtorus such that span R ( F ) =ann(Lie( T F )) . Let H F denote the complex subtorus of H with maximal compact torus equalto T F . The fixed point set X F S = Fix( X S , H F ) is an H orbit-closure. As a toric variety, it isisomorphic to X S F where S F = S ∩ F . Lemma 2.14.
Let C and S = C ∩ L be as above. Suppose X S is embedded H -equivariantly as asubvariety of a complex inner product space ( E, h ) such that the action of T on ( E, h ) is unitary.Let Ψ : E → Lie( T ) ∗ denote the moment map as in Example 2.6. For every closed face F of C :(i) Ψ( X F S ) = F , and X F S = Ψ − ( F ) ∩ X S .(ii) The image under Ψ of the smooth locus of X F S is the smooth locus of F .Sketch of Proof. The set of weights of the H -module C [ X F S ] is S ∩ F . Since X F S is normal, Ψ( X F S ) = cone( S ∩ F ) = F [69, Theorem 4.9]. In particular, Ψ( X S ) = C . Combining thiswith (17), X F S = E F ∩ X S = Ψ − ( F ) ∩ X S . This proves the first claim. The second claim follows by dualizing [19, Proposition 11.1.2]. (cid:3)
3. S
TRATIFIED GRADIENT H AMILTONIAN FLOWS
This section describes how the gradient Hamiltonian flows of degenerations can be integrated undercertain nice conditions. We record these conditions as a list of assumptions (GH1)–(GH6) givenbelow. Section 3.1 recalls basic notions about gradient Hamiltonian vector fields. Section 3.2states our key lemma for integrating gradient Hamiltonian flows on families with non-compactfibers. The main result of this section, Theorem 3.6, is described in Section 3.3. Its proof is givenin Appendix A.2. Section 3.4 establishes several results about gradient Hamiltonian vector fieldson affine varieties which will be used in the sequel.
Gradient Hamiltonian vector fields.
We begin by recalling the definition and elementaryproperties of gradient Hamiltonian vector fields. We refer the reader to [37, Section 2.2] and [64]for more details.Let M be a Kähler manifold and let π : M → C be a holomorphic submersion. Let ℜ π, ℑ π : M → R denote the real and imaginary parts of π , i.e. π = ℜ π + √− ℑ π . Let ∇ ( ℜ π ) denote the gradientvector field of ℜ π with respect to the Kähler metric and let X ℑ π denote the Hamiltonian vectorfield of ℑ π with respect to the Kähler form. Since π is holomorphic and M is Kähler, it followsthat ∇ ( ℜ π ) = − X ℑ π . The gradient Hamiltonian vector field of π is(18) V π := X ℑ π || X ℑ π || = − ∇ℜ π ||∇ℜ π || where || · || denotes norm with respect to Kähler metric. The vector field V π is defined everywhereon M since π is a submersion and therefore ∇ ( ℜ π ) is non-vanishing.Let M z = π − ( z ) denote the fiber of π over z ∈ C . Let ϕ t ( x ) denote the flow of V π through x ∈ M at time t . It follows from (18) that if x ∈ M z and ϕ t ( x ) is defined, then ϕ t ( x ) ∈ M z − t . Thus, if ϕ t ( x ) is defined for all x ∈ M z , then it gives a map ϕ t : M z → M z − t . Since π is a holomorphic submersion, each M z is a smooth Kähler submanifold of M and dim C M z =dim C M − . The following fact is well-known. For completeness, we recall its proof in AppendixA.1. Lemma 3.1.
If it is defined, the map ϕ t : M z → M z − t is symplectic with respect to the restrictedKähler forms on M z and M z − t . If π : M → C is proper, then the flow ϕ t ( x ) through any x ∈ M is defined for all time t . Inwhat follows we will deal with the situation where π is not proper. One of our main tools will bethe following lemma. Its proof – an application of Noether’s theorem – is given in Appendix A.1.Similar versions of this lemma previously appeared in, e.g. [37, Section 2.6] and [39, Lemma 5.1]. Lemma 3.2.
Let M be a Kähler manifold with Kähler form ω and Kähler metric g , and let π : M → C be a holomorphic submersion. Assume there is a Hamiltonian action of a connected Lie group K on ( M, ω ) with moment map ψ : M → k ∗ such that the action of K preserves the fibers of π andthe Kähler metric g . Then, the flow of V π is K -equivariant and preserves fibers of ψ . We now extend the definition of gradient Hamiltonian vector fields to a stratified setting. Let M be a Kähler manifold as before. Let Y be a decomposed space and suppose Y is embedded in M so that each smooth piece Y σ is a submanifold. The stratified tangent bundle of Y is the disjointunion of tangent bundles(19) T Y = [ σ ∈ Σ T Y σ . It inherits a subspace topology from
T M . See e.g. [67, Section 2.1] for more details.Suppose π : M → C is a holomorphic map, each Y σ is a Kähler submanifold of M , and eachrestricted map π : Y σ → C is a submersion. For each σ ∈ Σ , let V σπ : Y σ → T Y σ denote thegradient Hamiltonian vector field of the holomorphic submersion π : Y σ → C . The stratifiedgradient Hamiltonian vector field on Y is the section(20) V π : Y → T Y, V π ( x ) = V σπ ( x ) for x ∈ Y σ . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 15
Note that V π may fail to be continuous. It is a stratified vector field in the sense of [67, 2.1.5].For x ∈ Y , consider the initial value problem(21) ddt ϕ t ( x ) = V π ( ϕ t ( x )) , ϕ ( x ) = x. A solution of this initial value problem is given by combining the flows of the vector fields V σπ in apiecewise manner. Let ϕ σt denote the flow of V σπ . Define ϕ t ( x ) so that(22) ϕ t ( x ) = ϕ σt ( x ) if x ∈ X σ \ X σ and ϕ σt ( x ) is defined . We call ϕ t ( x ) the stratified gradient Hamiltonian flow . Provided that each ϕ σt can be integrated totime t , this defines a map (of sets) ϕ t : Y z → Y z − t .3.2. Integrating gradient Hamiltonian vector fields.
We now turn to a more algebraic setting.Let X be a variety and let π : X → C be a morphism. Denote the fiber of π over z ∈ C by X z .Let Z ⊂ X denote the union of the singular set of X and the critical set of π . Then X \ Z is acomplex manifold and π : X \ Z → C is a holomorphic map. For all z ∈ C , denote U z = X z \ Z .Throughout this section, assume that Z ⊂ X and U is non-empty. Then U z = X z for all z = 0 .Since π : X \ Z → C is a submersion, each U z is a complex submanifold of X \ Z of complexcodimension 1.Assume further that X \ Z is equipped with a Kähler structure. Then we may define the gradientHamiltonian vector field V π on X \ Z as in (18). We are interested in the following questions.(i) For what value of t = 0 can the gradient Hamiltonian flow be integrated to a map ϕ − t : U → X t ?(ii) Assuming that ϕ − t is defined on all of U as in the previous item, under what assumptionsis the image ϕ − t ( U ) dense in X t ?One possible answer to these questions was provided by Harada and Kaveh (see [37, Lemma 2.7and Corollary 2.11]). We now describe a slightly different answer. The key difference between ouranswer and that of Harada and Kaveh is that we do not assume π is proper.Suppose that a Lie group K acts in a Hamiltonian fashion on X \ Z with moment map ψ : X \ Z → k ∗ .For all z ∈ C , let ω z denote the Kähler form on the Kähler submanifold U z ⊂ X \ Z . If the actionof K preserves the submanifolds U z , then the tuples ( U z , ω z , ψ | U z ) are Hamiltonian K -manifolds.Recall that the Liouville volume form on a connected symplectic manifold ( M, ω ) of real dimension n is the top degree form ω n n ! . The Duistermaat-Heckman measure of a Hamiltonian K -manifold ( M, ω, µ ) is the pushforward to k ∗ by µ of the measure on M defined by the Liouville volume form.The proof of the following lemma is given in Appendix A.1. It is an application of Lemma 3.2. Lemma 3.3.
Let X be a variety and let π : X → C be a morphism of algebraic varieties such that Z is contained in X , U is non-empty, and X \ Z is Kähler. Let K be a connected Lie group andlet ψ : X → k ∗ be a continuous map. Assume:(i) There is a Hamiltonian action of K on X \ Z with moment map ψ | X \ Z such that the actionof K preserves the fibers of π and the Kähler metric.(ii) The map ( π, ψ ) : X → C × k ∗ is proper as a map to its image.Then, the flow ϕ t ( x ) is defined for all x ∈ U and t ∈ R . For t = 0 fixed, ϕ − t : ( U , ω , ψ ) → ( X t , ω t , ψ ) is a map of Hamiltonian K -manifolds. If additionally: (iii) The Duistermaat-Heckman measures of ( U , ω , ψ ) and ( X t , ω t , ψ ) are equal.Then, ϕ − t : U → X t is a symplectomorphism onto a dense subset of X t . The remainder of this section recalls useful notions from algebraic geometry and [37]. Recall thata morphism π : X → C is a flat family of varieties if it is flat and the fibers of π are all reduced asschemes. Proposition 3.4. [37, Proposition 2.8, Corollary 2.10]
Let π : X → C be a flat family of varieties.Then:(a) If p ∈ X z is a smooth point of X z , then it is a smooth point of X .(b) If p ∈ X z is a smooth point of X and a critical point of π : X → C , then it is a singularpoint of X z . By Proposition 3.4, if π is a flat family of varieties, then U z is precisely the smooth locus of X z . Inparticular, U z is dense in X z . Definition 3.5. A degeneration of a variety X is a flat family of varieties π : X → C such that:(i) There is an algebraic isomorphism ρ : X × C × → X \ X such that π ◦ ρ = pr . Such anisomorphism is called a trivialization away from 0 .(ii) The fiber X is non-empty.A degeneration π : X → C is a toric degeneration if X is a toric variety.If π : X → C is a degeneration, then π : X \ Z → C is a submersion onto C . If the variety X issmooth, then it follows by trivialization away from that Z is contained in X .3.3. Stratified gradient Hamiltonian vector fields.
Let ( X, M, ω ) be a decomposed Kähler va-riety (Definition 2.10) and let π : X → C be a degeneration of X . The trivialization ρ and thedecomposition of X allows us to define subfamilies(23) X σ := ρ ( X σ × C × ) , X σ := X σ \ [ τ ≺ σ X τ . The trivialization also yields isomorphisms X σ \ X ∼ = X σ × C × and X σ \ X ∼ = X σ × C × . Denote(24) X σz := X z ∩ X σ , and X σz := X z ∩ X σ for all z ∈ C . Note that X σ , X σ , X σz , and X σz are subvarieties of X for all σ and z ∈ C .We now give a list of assumptions (GH1)–(GH6) that we will place on our degenerations. Theseare partially inspired by Harada and Kaveh’s assumptions (a)–(d) [37, p. 932].(GH1) The restricted maps π : X σ → C are flat families of varieties.(GH2) The family X is embedded into M × C as a subvariety such that the map π : X → C coincides with restriction to X of the projection M × C → C .(GH3) The embedding X = X × { } ∼ = ρ X ⊂ M × { } = M given by assumption (GH2)coincides with the embedding X ֒ → M of the decomposed Kähler variety ( X, M, ω M ) . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 17
For each σ , define Z σ ⊂ X σ to be the union of the critical set of π : X σ → C and the singular setof X σ . Denote U σ = X σ \ Z σ . It follows by assumption (GH1) and Proposition 3.4 that U σ isprecisely the smooth locus of X σ , i.e.(25) U σ = ( X σ ) sm . (see Lemma A.1). With only assumptions (GH1)–(GH3) it is possible that X σ (and therefore also U σ ) is empty. We will add an assumption below which implies that U σ is non-empty. In particular,the subfamilies π : X σ → C will all be degenerations.Equip C with its standard Kähler structure and M × C with the product Kähler structure. Inparticular, the Kähler form is the product symplectic structure ω = ω M ⊕ ω std .(GH4) There is a Hamiltonian action of a compact torus T on M × C with moment map ψ : M × C → t ∗ such that:I) The action of T on M × C preserves the each of the subvarieties X σ \ Z σ , the fibers of π , and the Kähler metric on M × C .II) The map ( π, ψ ) : X → C × t ∗ is proper.III) The subfamilies X σ are saturated by the restricted maps ψ : X → t ∗ .Note that by assumptions (GH4)II) and (GH4)III), the map ( π, ψ ) : X σ → C × t ∗ is proper as a mapto its image for every subfamily X σ .Denote the restriction of ω to the symplectic submanifolds X σ \ Z σ , X σz for z = 0 , and U σ by ω σ , ω σz , and ω σ , respectively. The action of T preserves each of these submanifolds. The restrictedaction is therefore also Hamiltonian with moment map given by the restriction of ψ . The followingassumption is sufficient to conclude that the time-1 gradient Hamiltonian flow on X σ \ Z σ definesa symplectomorphism from a dense subset of ( X σ , ω σ ) onto ( U σ , ω σ ) .(GH5) The Duistermaat-Heckman measures of ( U σ , ω σ , ψ ) and ( X σ , ω σ , ψ ) are equal for all σ ∈ Σ . In particular, U σ is non-empty.The partition of X \ X by the manifolds X σ \ X σ = ρ ( X σ × C × ) is a decomposition. Each X σ \ X σ is a Kähler submanifold of M × C and the restricted maps π : X σ \ X σ → C are holomorphicsubmersions. Thus, we may define a stratified gradient Hamiltonian vector field as in Section 3.1,(26) V π : X \ X → T ( X \ X ) , V π ( x ) = V σπ ( x ) for x ∈ X σ \ X σ . The following is sufficient to prove that its gradient Hamiltonian flow is continuous.(GH6) The stratified vector field V π : X \ X → T ( X \ X ) is continuous.Combining the assumptions above, we arrive at the following. Its proof is given in Appendix A.2. Theorem 3.6.
Let π : X → C be a degeneration of a stratified Kähler variety ( X, M, ω M ) thatsatisfies assumptions (GH1)–(GH6) above. Let T and ψ be the compact torus and moment map ofassumption (GH4).Then, for all x ∈ X , the limit φ ( x ) = lim t → − ϕ t ( x ) exists and defines a continuous, T -equivariant, proper, surjective map φ : X → X . Moreover: A subset A ⊂ X is saturated by a map f : X → Y if it is a union of fibers of f . (a) For all σ ∈ Σ , there is a dense open subset D σ ⊂ X σ such that the time-1 flow of V σπ existsand defines a symplectomorphism ϕ σ : ( D σ , ω σ ) → ( U σ , ω σ ) , where U σ ⊂ X σ is defined asabove.(b) For all σ ∈ Σ , φ | D σ = ϕ σ .(c) ψ ◦ φ = ψ .Remark . Digging into the proof of Theorem 3.6 slightly, note that for each σ ∈ Σ , the opendense subset D σ = ϕ σ − ( U σ ) . In particular, if X σ is smooth, then D σ = X σ , i.e. the time-1 gra-dient Hamiltonian flow ϕ σ exists on all of X σ and defines a symplectomorphism ϕ σ : ( X σ , ω σ ) → ( X σ , ω σ ) . This is useful, for instance, in Section 2.6. Remark . The statement of Theorem 3.6 is specialized to the case t = 1 for simplicity. The sameargument shows that one can define a map φ : X t → X with all the properties above for any t = 0 . Remark . The construction of the map φ has two main components: proving that ϕ t : X → X − t is continuous when t < , and proving that the limit lim t → − ϕ t ( x ) exists and is continuous.The proof that ϕ t is continuous for t < relies on assumptions (GH4) and (GH6) and Lemma 3.2.It is worth mentioning that there are various other frameworks for studying flows of vector fieldson stratified spaces (such as Mather’s control theory [63] or the notion of rugose vector fields [71,Definition (1.4)]). However, we did not use those frameworks.The proof that the limit lim t → − ϕ t ( x ) exists and is continuous follows the same outline as the proofof [37, Theorem 2.12], with some minor modifications to adapt their techniques to our stratifiedsetting.Our primary application of Theorem 3.6 is to toric degenerations. If the zero fiber of the degener-ation X carries a Hamiltonian action of a torus T generated by a moment map Ψ : X → Lie( T ) ∗ ,then the composition of Ψ with φ : X = X → X generates a torus action on a dense subset ofeach piece of X as follows. Corollary 3.10.
Let π : X → C be a toric degeneration of a stratified Kähler variety ( X, M, ω M ) that satisfies assumptions (GH1)–(GH6) above and let T denote the compact torus of the toricvariety X . Assume that:(i) The action of T on X extends to an action on M × { } = M that is Hamiltonian withmoment map Ψ : M → Lie( T ) ∗ .(ii) The subvarieties U σ are T -invariant.Then for each σ ∈ Σ , the restriction of the map Ψ ◦ φ to D σ is a moment map for a Hamiltonian T -action on ( D σ , ω σ ) .Remark . In [37, Definition 2.1], Harada and Kaveh give a slightly non-standard definitionof integrable systems on varieties whose smooth locus is equipped with a symplectic structure.We now give a similar (but not identical) non-standard definition. A collection of real valuedcontinuous functions f , . . . , f n on a smooth connected symplectic manifold M is an integrablesystem if:(i) There exists an open dense subset D ⊂ M such that the restricted functions f i | D are allsmooth and the rank of the Jacobian of F = ( f , . . . , f n ) equals dim( M ) on a densesubset of D . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 19 (ii) The restricted functions f i | D pairwise Poisson commute, i.e. { f i | D , f j | D } = 0 for all ≤ i, j ≤ n .A collection of real valued continuous functions f , . . . , f n on a singular symplectic space X is an integrable system if their restriction to each symplectic piece of X is an integrable system in thesense defined above. Corollary 3.10 produces an integrable system in this sense. Remark . Suppose X is a smooth variety equipped the trivial decomposition, M is a projectivespace equipped with the Fubini-Study Kähler form, the torus T is trivial, and X is a toric degener-ation. Then assumptions (GH4)–(GH6) are satisfied automatically. In particular, (GH5) reduces tothe statement that the symplectic volumes of U and X are equal, which follows by flatness of X (see the proof of [37, Corollary 2.11]). Thus, Corollary 3.10 reduces to [37, Theorem A].3.4. Gradient Hamiltonian flows on decomposed affine Kähler varieties.
We now turn ourattention to gradient Hamiltonian flows on affine varieties. Proposition 3.13 is a useful tool forverifying that (GH6) holds. An application of this result is Theorem 3.15, which says that often thesymplectic structure on a decomposed affine Kähler variety is independent of the embedding intoan ambient affine space. Throughout this section, H is an algebraic torus with maximal compacttorus T and real weight lattice Λ .3.4.1. Continuity of gradient Hamiltonian vector fields.
Let E be a finite dimensional H -module.Extend the action of H to E × C by letting H act trivially on C . Let X be a H -invariant closedsubvariety of X ⊂ E × C . Import all the notation from Section 2.6. Let π denote the projection E × C → C as well as its restriction to X . Denote X = π − (0) ∩ X . Assume:(D1’) The partition of X \ X into X σ \ X , σ ∈ Σ( X ) , gives X \ X the structure of a decomposedvariety.In order to state our second assumption, we briefly recall the Whitney condition (A). Given a k -dimensional submanifold N of a vector space V and a sequence of points { x i } i ∈ N ⊂ N , let lim i →∞ T x i N denote the limit of tangent spaces in the Grassmannian of k -dimensional subspacesof V . A pair ( N ′ , N ) of submanifolds of V satisfies the Whitney condition (A) if:(A) For any sequence { x i } i ∈ N ⊂ N that converges to some x ∈ N ′ , if lim i →∞ T x i N exists, then T x N ′ ⊂ lim i →∞ T x i N .A decomposed space embedded in a vector space satisfies the Whitney condition (A) if each pair ofits pieces satisfies the Whitney condition (A) [67, 1.4.3]. We also assume:(D2’) The decomposition of X \ X in (D1’) satisfies the Whitney condition (A) with respect tothe embedding into E × C .Suppose E is equipped with a complex inner product h E and the action of T is unitary. Equip E × C with the product complex inner product h E ⊕ h C (where h C is the standard complex innerproduct). This equips the submanifolds X σ \ X with a Kähler structure. Provided that the restrictedmaps π : X σ \ X → C are submersions, we may define a stratified gradient Hamiltonian vectorfield V π : X \ X → T ( X \ X ) as in (20). The goal of this section is to prove the following. Proposition 3.13.
Under the assumptions (D1’) and (D2’) above, V π : X \ X → T ( X \ X ) is con-tinuous. We first establish a preliminary result. Throughout, identify E × C = T x ( E × C ) for all x ∈ E × C . Lemma 3.14.
Let σ, τ ∈ Σ( X ) with σ ≺ τ . Let { x j } ⊂ X τ \ X be a sequence of points convergingto x ∈ X σ \ X . If lim j →∞ T x j X τ exists, then lim j →∞ T x j X τ ⊂ T x X σ ⊕ ( E σ × C ) ⊥ . Proof.
Let ˆ g , . . . , ˆ g J be a set of Λ -homogeneous generators of I ( X ) . The tangent space of X at x is T x X = { v ∈ E × C | ( d ˆ g j ) x ( v ) = 0 for all j ∈ [1 , J ] } . Since lim j →∞ T x j X τ ⊂ T x X , it suffices to show T x X ⊂ T x X σ ⊕ ( E σ ⊕ C ) ⊥ .By Lemma 2.13 (and since X σ \ X is an open subset of X σ \ X ), T x X σ = T x X σ = T x X ∩ ( E σ × C ) . It follows by (16) that if | ˆ g j | / ∈ σ , then ˆ g j vanishes on E σ × C . Thus, T x X ∩ ( E σ × C ) = { v ∈ E × C | ( d ˆ g j ) x ( v ) = 0 , for all j ∈ [1 , J ] such that | ˆ g j | ∈ σ } ∩ ( E σ × C ) . Combining these equalities, we arrive at T x X σ = { v ∈ E × C | ( d ˆ g j ) x ( v ) = 0 , for all j ∈ [1 , J ] such that | ˆ g j | ∈ σ } ∩ ( E σ × C ) . Finally, let v ∈ T x X and write v = v ′ + v ′′ , where v ′ ∈ E σ × C and v ′′ ∈ ( E σ × C ) ⊥ . If | ˆ g j | ∈ σ ,then ˆ g j vanishes on ( E σ × C ) ⊥ and so ( d ˆ g j ) x ( v ′′ ) = 0 . It follows from the description of T x X and T x X σ above that v ′ = v − v ′′ ∈ T x X σ . Thus, v ∈ T x X σ ⊕ ( E σ × C ) ⊥ . (cid:3) Proof of Proposition 3.13.
The gradient of ℜ π | X σ \ X is computed with respect to the metric on X σ \ X which is the restriction of the fixed Kähler metric on E × C . Let x ∈ X σ \ X and takea sequence { x j } j ∈ N ∈ X \ X converging to x . By passing to a subsequence, we may assume that { x j } j ∈ N ⊂ X τ \ X for some τ ∈ Σ( X ) with σ ≺ τ , and that lim j →∞ T x j X τ exists. By definitions(18) and (20) and Lemma A.4, it suffices to show that lim j →∞ ∇ x j ( ℜ π | X τ ) = ∇ x ( ℜ π | X σ ) . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 21
In what follows, if W is a linear subspace of E × C , then pr W denotes orthogonal projection to W .We have lim j →∞ ∇ x j ( ℜ π | X τ ) = lim j →∞ pr T xj X τ (cid:18) − ∂∂t (cid:19) = pr lim j →∞ T xj X τ (cid:18) − ∂∂t (cid:19) = pr lim j →∞ T xj X τ ◦ pr T x X σ ⊕ ( E σ × C ) ⊥ (cid:18) − ∂∂t (cid:19) ( by Lemma 3.14 )= pr lim j →∞ T xj X τ (cid:18) pr T x X σ (cid:18) − ∂∂t (cid:19) + pr ( E σ × C ) ⊥ (cid:18) − ∂∂t (cid:19)(cid:19) = pr lim j →∞ T xj X τ (cid:18) pr T x X σ (cid:18) − ∂∂t (cid:19)(cid:19) = pr T x X σ (cid:18) − ∂∂t (cid:19) ( by assumption (D2’) )= ∇ x ( ℜ π | X σ ) . This proves the claim. (cid:3)
Isomorphisms between decomposed affine Kähler varieties.
Let X be an affine H -varietyand import the notation from Section 2.6. Suppose that X embeds as a H -invariant closed sub-variety of a H -module E . We introduce the following conditions, which are analogues of (D1’)and (D2’) for X :(D1) The partition of X into X σ , σ ∈ Σ( X ) , gives X the structure of a decomposed variety.(D2) The decomposition of X in (D1) satisfies the Whitney condition (A) with respect to theembedding into E .We now give the first application of Proposition 3.13. Let ( E, h E ) and ( E ′ , h E ′ ) be two unitary T -modules, with symplectic forms ω E , ω E ′ and T -equivariant moment maps µ, µ ′ , respectively.Let i : X ֒ → E and i ′ : X ֒ → E ′ be two closed H -equivariant embeddings. As usual, we assumewithout loss of generality that Λ( X ) = Λ( E ) = Λ( E ′ ) . The embeddings i and i ′ each endow X with the structure of a Hamiltonian T -space. Denote these Hamiltonian T -spaces ( X E , µ ) and ( X E ′ , µ ′ ) respectively. Theorem 3.15.
Assume X satisfies (D1) and the embeddings i, i ′ both satisfy (D2). If the mo-ment maps µ : E → t ∗ and µ ′ : E ′ → t ∗ are proper, then ( X E , µ ) is isomorphic to ( X E ′ , µ ′ ) as aHamiltonian T -space.Proof. Consider the unitary T -module ( E × E ′ × C , h = h E ⊕ h E ′ ⊕ h C ) with symplectic structure ω = − Im h . The action of T is Hamiltonian with moment map ψ = µ ◦ pr E + µ ′ ◦ pr E ′ . Considerthe trivial degeneration X = X × C of X . Embed X into E × E ′ × C according to the map X × C → E × E ′ × C ( x, t ) ( ti ( x ) , (1 − t ) i ′ ( x ) , t ) . Its image is a closed H -invariant subvariety of E × E ′ × C . It has smooth pieces X σ = X σ × C =( X × C ) ∩ ( E σ × E ′ σ × C ) .We apply Theorem 3.6 to the degeneration X × C → C . To do so, we need to check the conditions(GH1)–(GH6). The conditions (GH1), (GH2), and (GH3), and (GH4)I) are satisfied automatically.To show (GH4)II), it suffices to show that ( ψ, π ) : E × E ′ × C → t ∗ × C is proper. By Lemma 2.7,because µ and µ ′ are assumed to be proper, the cone Γ( E ) = Γ( E ′ ) is strongly convex. It followsthat the map µ ◦ pr E + µ ′ ◦ pr E ′ : E × E ′ → t ∗ is proper. Thus ( ψ, π ) : E × E ′ × C → t ∗ × C isproper. The condition (GH4)III) holds because X σ = X ∩ ( E σ × E ′ σ × C ) = X ∩ ψ − ( σ ) . Here the last equality is a consequence of (17).In contrast with the general setup of Theorem 3.6, there are no singular points of X σ × C . As aresult, the stratified gradient Hamiltonian flow ϕ t is defined for all t ∈ R , for all points of X × C .The condition (GH5) is then satisfied automatically.Finally, it remains to verify condition (GH6). To do this, we apply Proposition 3.13. To applyProposition 3.13, we need to check that (D1’) and (D2’) hold for X ⊂ E × E ′ × C . This isa straightforward consequence of the fact that, by assumption, the partition of X into X σ satis-fies (D1), and that each embedding X ֒ → E and X ֒ → E ′ satisfies (D2).After applying Theorem 3.6, we have a T -equivariant continuous map φ : X E → X E ′ which, foreach σ ∈ Σ , restricts to a symplectomorphism ϕ σ : D σ → U σ from an open dense subset D σ of X σE to an open dense subset U σ of X σE ′ . Since X σE ′ is smooth,it follows that D σ = X σE and U σ = X σE ′ (cf. Remark 3.7). In other words, ϕ σ defines a symplec-tomorphism of X σE and X σE ′ . What is more, ψ ◦ φ = ψ . Thus, we have a map of Hamiltonian T -spaces φ : ( X E , µ ) → ( X E ′ , µ ′ ) . The map ϕ − is an inverse to φ , so φ is an isomorphism ofHamiltonian T -spaces. (cid:3)
4. F
ROM VALUATIONS TO STRATIFIED GRADIENT H AMILTONIAN FLOWS
In this section, after recalling some standard theory in commutative algebra, we introduce thenotion of a good valuation. Out of a good valuation, one may construct a toric degeneration of anaffine variety which satisfies most of the assumptions of Theorem 3.6. The exact statement is inTheorem 4.21, which is the main result of this section.4.1.
Toric degenerations from valuations.
This section recalls the Rees algebra construction oftoric degenerations. In Section 4.1.1 we recall the definition and basic properties of valuations.In Section 4.1.2 we describe the Rees algebra construction for affine varieties equipped with avaluation and a compatible action of a torus. We refer the readers to [48, Sections 1, 2, and 6], [37],and [8] for more details.
ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 23
Valuations.
Definition 4.1.
Let L be a lattice equipped with a total order > which respects addition. Let A bean algebra over C . A valuation on A (with values in L ) is a function v : A \ { } → L , such that:(1) v ( f + g ) ≤ max { v ( f ) , v ( g ) } for all nonzero f, g ∈ A , (2) v ( cf ) = v ( f ) for all nonzero f ∈ A and c ∈ C \ { } ,(3) v ( f g ) = v ( f ) + v ( g ) for all nonzero f, g ∈ A .The image v ( A \ { } ) ⊂ L is a semigroup called the value semigroup . It is denoted S v .Throughout Section 4, we assume that the value semigroup S v of a valuation v generates L as a Z -module. In particular, we only consider valuations such that S v ⊂ L is saturated. We write S = S v when the meaning is clear from context.For s ∈ S , let A ≤ s = { f ∈ A | v ( f ) ≤ s or f = 0 } . Define A ) -graded filtration of A . The associated gradedalgebra is the S -graded algebra gr A = M s ∈ S A ≤ s /A ) and one-dimensionalleaves. Suppose A and v have a finite Khovanskii basis K . We would like to use the subductionalgorithm as in [37, Proposition 3.12] to show that K generates A as an algebra. However, theconstruction described in loc. cit. is specialized to the case where A is the homogeneous coordinatering of a projective variety and v is a valuation that has been extended by homogeneous degree asin [37, Equation (3.4)]. We now introduce a similar version of their construction sufficient for ourneeds.To this end, suppose there exists a lattice Λ and surjective a linear map w : L → Λ such that:(v1) The total order > on L descends under w to a total order on Λ .(v2) The image w ( S v ) contains a minimal element with respect to this order.(v3) The fibers w − ( λ ) ∩ S v are all finite.With these assumptions in place, we have the following proposition. The proof of [37, Proposition3.12] carries through in this adapted setting. Proposition 4.3. K generates A as an algebra. Some authors require instead that v ( f + g ) ≥ min { v ( f ) , v ( g ) } . Thus, a Khovanskii basis K for A and v produces an embedding(27) X ֒ → E, where E is the dual vector space to the vector space span C K ⊂ A .Assumptions (v1)-(v3) have the following consequence which will be useful later on. Lemma 4.4.
The cones cone( S v ) and cone( w ( S v )) are strongly convex.Proof. The total order on Λ defined by (v1) must respect addition since w is linear and the totalorder on L respects addition. The minimal element of w ( S v ) , which exists by (v2), must thereforeequal 0. It follows that cone( w ( S v )) cannot contain any rational lines, so it is strongly convex.Since cone( w ( S v )) is strongly convex, any line ℓ ⊂ L R that is contained in cone( S v ) must becontained in the kernel of the linear map w : L R → Λ R . It follows by (v3) that w − ( { } ) ∩ S v = { } .Thus, cone( S v ) ∩ ker( w ) = { } . Thus cone( S v ) is strongly convex. (cid:3) The Rees algebra construction.
Before stating the Rees algebra construction, we will needthe following technical notion.
Definition 4.5.
Let S ⊂ L be an affine semigroup, let < be a total order on L , and let w : L → Λ be a linear map to a lattice Λ . The tuple ( S , >, w ) is refinable if for any two finite sets { a , . . . , a N } , { b , . . . , b N } ⊂ S such that w ( a i ) = w ( b i ) and a i > b i for all i ∈ [1 , N ] , there exists a linear map e : L → Z such that(28) e ( S ) ⊂ N , and e ( a i ) > e ( b i ) for all i ∈ [1 , N ] . Remark . See [7, 13] for sufficient conditions that ( S , >, w ) be refinable.Let H be a algebraic torus with (real) weight lattice Λ and let A = C [ X ] be the coordinate ringof an affine H -variety X . Let v be a valuation on A with values in ( L, > ) and one-dimensionalleaves such that the value semigroup S = S v is finitely generated. Suppose there exists a linearmap w : L → Λ that satisfies (v1)-(v3) and the following.(v4) If f ∈ A is homogeneous of degree λ ∈ Λ , then λ = w ( v ( f )) .(v5) The tuple ( S , >, w ) is refinable.We now give the setup for the Rees algebra construction. Let X ֒ → E be an H -equivariant embed-ding of X into a finite dimensional H -module E . Let { z i } ni =1 be a system of Λ -homogeneous linearcoordinates on E and let f i ∈ A denote the image of z i under the dual map C [ E ] → A . Assumethat K = { f i } ni =1 is a Khovanskii basis for A and v (cf. the discussion following Proposition 4.3).In particular, f i = 0 for all i .Let H denote the algebraic torus with (real) weight lattice L . Define an H -module structure on E ∗ by letting h · z i = h v ( f i ) z i for all h ∈ H and i = 1 , . . . , n . Equip E with the dual H -modulestructure. The linear map w determines a homomorphism of algebraic tori H → H . Along with the H -module structure, this endows E with an H -module structure. By (v4), this H -module structurecoincides with the original H -module structure on E .The discussion above produces surjective algebra homomorphisms: C [ E ] → A = C [ X ]; C [ E ] → gr A (29) z i f i ; z i f i mod A < v ( f i ) . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 25
The first is dual to the embedding
X ֒ → E . It is a map of Λ -graded algebras. The second is a mapof L -graded algebras dual to an H -equivariant embedding Spec(gr A ) ֒ → E . Lemma 4.7.
Let g , . . . , g J ∈ C [ E ] be L -homogeneous generators of the ideal ker( C [ E ] → gr A ) .Then, there exist Λ -homogeneous generators g , . . . , g J ∈ C [ E ] of the ideal ker( C [ E ] → A ) whichhave the form (30) g j = g j + p j , v ( g j ( f , . . . , f n )) > v ( p j ( f , . . . , f n )) . The Λ -homogeneous degree of g j is | g j | Λ = w ( | g j | L ) = w ( v ( g j ( f , . . . , f n ))) . The proof of Lemma 4.7 is a direct analogue of the first half of the proof of [37, Theorem 3.13]. Itsoutline is as follows.
Sketch of proof.
Let s j = | g j | L denote the L -homogeneous degree of g j as an element of the L -graded algebra C [ E ] . Since g j is in the kernel of C [ E ] → gr A , it follows that g j ( f , . . . , f n ) ∈ A v ( p j ( f , . . . , f n )) and such that g j = g j + p j is in the kernel of C [ E ] → A . One achieves this by the subduction algorithm as in the proof of [37, Proposition 3.12].The Λ -homogeneous degree of g j is w ( s j ) . It follows by (v4) that w ( s j ) = w ( v ( g j ( f , . . . , f n ))) .The subduction algorithm produces a polynomial p j that is also Λ -homogeneous of degree w ( s j ) .Finally, as in [7, Proposition 2.2], the polynomials g , . . . , g J generate the kernel of the map C [ E ] → A . (cid:3) Let g j , g j , and p j , j = 1 , . . . , J be as in Lemma 4.7. Let s j = | g j | L denote the L -homogeneousdegree of g j as an element of C [ E ] . Write p j = P L j l =1 M j,l where each M j,l ∈ C [ E ] is a Λ -homogeneous monomial in z , . . . , z n , of degree | M j,l | Λ = w ( s j ) . Fix a Z -linear map e : L → Z such that:(31) e ( S ) ⊂ N , and e ( v ( g j ( f , . . . , f n ))) > e ( v ( M j,l ( f , . . . , f n ))) ∀ j, l. This exists by (v5).We are now in a position to define the Rees algebra. For all k ≥ , define A ≤ k = { f ∈ A | e ( v ( f )) ≤ k or f = 0 } . The subspaces A ≤ k define a N -graded filtration of A . The Rees algebra of this filtration is(32) R = M k ≥ A ≤ k ⊗ t k ⊂ A ⊗ C [ t ] . The algebra R inherits a Λ -grading from A (where t is defined to be homogeneous of degree 0).The following collects standard facts about R ; see for instance [21, Corollary 6.11]. Proposition 4.8.
Let R be as in (32) . Then,(1) R is finitely generated.(2) The C -algebra homomorphism C [ t ] → R , t t makes R into a flat C [ t ] -algebra.(3) R /t R ∼ = gr A ∼ = C [ S ] .(4) R [ t − ] ∼ = A ⊗ C [ t, t − ] . The geometric interpretation of the Rees algebra R is as follows. Let X = Spec R . Then X is anaffine H -variety. Dualizing the map C [ t ] → R from Proposition 4.8 gives a flat morphism X → C ,which is a degeneration of X to the toric variety Spec gr A .The variety X can be embedded H -equivariantly into E × C . Define an algebra homomorphism(33) C [ E ] ⊗ C [ t ] → A ⊗ C [ t ] , z i t e ( v ( f i )) f i , t t. It is a map of Λ -graded algebras ( t is homogeneous of degree 0 in both algebras). Its image precisely R . We then have an H -equivariant embedding(34) X ֒ → E × C . The image of X in E × C is the subvariety cut out by the Λ -homogeneous polynomials(35) ˆ g j = g j ( z , . . . , z n ) + L j X l =1 t m j,l M j,l ( z , . . . , z n ) j = 1 , . . . , J. where m j,l = e ( v ( g j ( f , . . . , f n )) − v ( M j,l ( f , . . . , f n ))) .We conclude by making the following observations about subfamilies of X . We adopt the notationof Section 2.6. Proposition 4.9.
Let σ be a closed face of Γ( X ) .(1) I ( X ∩ ( E σ × C )) is generated by { ˆ g j | w ( s j ) ∈ σ } ∪ { z i | w ( v ( f i )) / ∈ σ } . Consequently, C [ X ∩ ( E σ × C )] is isomorphic to the subalgebra of C [ X ] = R generated by { t } ∪ { t e ( v ( f i )) f i | w ( v ( f i )) ∈ σ } . Additionally, C [ X ∩ ( E σ × C )] is flat as a C [ t ] -module.(2) Identify E × { } = E . Then I ( X ∩ ( E σ × { } )) ⊂ C [ E ] is generated by { g j | w ( s j ) ∈ σ } ∪ { z i | w ( v ( f i )) / ∈ σ } . Consequently, C [ X ∩ ( E σ × { } )] is isomorphic to the subalgebra of C [ X ] generated by { f i | w ( v ( f i )) ∈ σ } .(3) Identify E × { } = E . Then I ( X ∩ ( E σ × { } )) ⊂ C [ E ] is generated by { g j | w ( s j ) ∈ σ } ∪ { z i | w ( v ( f i )) / ∈ σ } . Consequently, C [ X ∩ ( E σ × { } )] is isomorphic to the subalgebra of C [ S ] generated by { χ v ( f i ) | w ( v ( f i )) ∈ σ } .Proof. By Lemma 2.13, I ( X ∩ ( E σ × C )) = I ( X ) + I ( E σ × C )) . It follows by (16) that if | ˆ g j | / ∈ σ ,then ˆ g j vanishes on E σ × C . The description of I ( X ∩ ( E σ × C )) immediately follows. The C [ t ] -module C [ X ∩ ( E σ × C )] is flat because C [ X ∩ ( E σ × C )] ⊂ R is a torsion-free C [ t ] module, and C [ t ] is a principal ideal domain [21, Corollary 6.3].The second and third items follow from the first, by putting t = 0 and t = 1 . (cid:3) ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 27
Good valuations.
We now introduce the notion of a good valuation, which is the initial dataneeded to perform the construction in the next section.
Definition 4.10.
Let H a and H c be algebraic tori with compact forms T a and T c , respectively. Let X be an affine H a × H c -variety. A good valuation on X is a tuple ( X, E, h E , v , a , c ) consisting of:(i) A finite dimensional complex inner product space ( E, h E ) equipped with a unitary repre-sentation of T a × T c and a H a × H c -equivariant embedding X ֒ → E of X as a closedsubvariety.(ii) A valuation v : A \{ } → L on A = C [ X ] , with values in a lattice with total order ( L, > ) .We require that v has one dimensional leaves, and that S = S v is an affine semigroup.(iii) Surjective Z -linear maps a : L → Λ a , c : L → Λ c . where Λ a (resp. Λ c ) is the character lattice of H a (resp. H c ).Let Λ( X ) ⊂ Λ c be the semigroup of weights of the H c -module C [ X ] , let Γ = Γ( X ) = cone Λ( X ) ,and let Σ = Σ( X ) be the face poset of Γ . The data must satisfy two compatibility conditions:(GV1) (Compatibility of the valuation) The valuation v and the map c satisfy conditions (v1)-(v5)(with w = c ). Additionally, the valuation v and the map a satisfy condition (v4) (with w = a ).(GV2) (Compatibility of the decomposition) The partition of X by the subvarieties X σ , σ ∈ Σ ,defined by the H c action equips X with the structure of a decomposed variety (D1). The de-composition satisfies the Whitney condition (A) with respect to the embedding into E (D2).The actions of T a and T c on E are Hamiltonian with moment maps ψ a and ψ c , respectively (Exam-ple 2.6). Remark . Given a good valuation, the tuple ( X, E, ω E ) is a decomposed affine Kähler variety(Definition 2.10) with respect to the symplectic form ω E = −ℑ h E and the decomposition of X described in (GV2). The action of T a × T c endows X with the structure of a Hamiltonian T a × T c -space. Remark . The letters a and c stand for auxiliary and control . The action of the control torus H c is necessary for the construction in Section 4.3. We include the data of the auxiliary torus so thatthe construction in Section 4.3 can be performed in the presence of an additional group action. Thisauxiliary group action is not necessary for the construction. If there is no additional group actionto keep track of, one may put H a = { e } . Remark . In all that follows we assume without loss of generality that X is not contained inany proper affine subspace of E . That is, we assume that no f ∈ E ∗ \{ } restricts to a constantfunction on X . In particular, the semigroups of H c -weights Λ( X ) and Λ( E ) are equal.Indeed, let hull( X ) denote the affine hull of X in E . Then both hull( X ) and the subspace hull( X ) ⊥ are T a × T c -invariant. Composing the embedding X ֒ → E with the T a × T c -equivariantlinear projection E → E/ hull( X ) ⊥ produces a good valuation ( X, E/ hull( X ) ⊥ , h E/ hull( X ) ⊥ , v , a , c ) The affine hull of X is the smallest affine subspace of E which contains X . where h E/ hull( X ) ⊥ is the complex inner product induced by h E . The image of X under this embed-ding is not contained in any proper affine subspace of E/ hull( X ) ⊥ . Lemma 4.14.
Let X be an affine H a × H c -variety, and let ( X, E, h E , v , a , c ) be a good valuationon X . Then the map ψ c : E → Lie( T c ) ∗ is proper.Proof. Let Π denote the set of weights of the dual representation of T c on E ∗ . By Lemma 2.7, ψ c isproper if and only if 0 cannot be written as a non-trivial linear combination of elements of Π withnon-negative coefficients.Let { z i } ni =1 ⊂ E ∗ be a basis of T c -weight vectors. By our assumption that hull( X ) = E , therestricted functions f i = z i | X are all non-constant Λ c -homogeneous elements of A . The embedding X ֒ → E is T c -equivariant, and so by (v4) the weight of z i ∈ E ∗ equals c ( v ( f i )) . Thus, Π is a subsetof c ( S v ) . Since cone( c ( S v )) is strongly convex (Lemma 4.4), 0 can be written as a non-trivial linearcombination of elements of Π with non-negative coefficients if and only if ∈ Π .Finally, recall from the proof of Lemma 4.4 that c − (0) ∩ S v = { } . The f i are not constant and v has one-dimensional leaves. This implies that Π . (cid:3) Proposition 4.15.
Let X be an affine H a × H c -variety, and let ( X, E, h E , v , a , c ) be a good valu-ation on X . Then, there exists an inner product space ( E ′ , h E ′ ) with unitary T a × T c action, and a H a × H c -equivariant embedding X ֒ → E ′ , so that(1) There exists a basis of E ′∗ which restricts to a Khovanskii basis for A = C [ X ] and v .(2) ( X, E ′ , h E ′ , v , a , c ) defines a good valuation on X .(3) Let X E and X E ′ denote the two Hamiltonian T a × T c -space structures on X coming fromthe embeddings i and i ′ , respectively. Then X E is isomorphic to X E ′ as a Hamiltonian T a × T c -space.Proof. The image of the linear map E ∗ → C [ X ] generates C [ X ] as an algebra. Pick a finite Kho-vanskii basis of C [ X ] . By picking large enough N , one can ensure that the image of the naturalmap L Nk =1 Sym k ( E ∗ ) → C [ X ] contains this finite Khovanskii basis. Define E ′ = L Nk =1 Sym k ( E ) .Then there is a natural H a × H c action on E ′ , and ( E ′ ) ∗ is canonically isomorphic to L Nk =1 Sym k ( E ∗ ) .The linear map L Nk =1 Sym k ( E ∗ ) → C [ X ] determines a surjection of algebras C [ E ′ ] → C [ X ] and,in turn, an embedding X ֒ → E ′ . This map is H a × H c -equivariant.Put a T a × T c -invariant inner product h E ′ on E ′ . We check that the tuple ( X, E ′ , h E ′ , v , a , c ) satisfies(GV2). Condition (D1) holds as it does not depend on the embedding of X . To check (D2), we takethe natural H a × H c -equivariant surjection C [ E ′ ] → C [ E ] , which realizes E as a smooth subvarietyof E ′ . The map C [ E ′ ] → C [ X ] factors as C [ E ′ ] → C [ E ] → C [ X ] . Since X ⊂ E is Whitney A,and E is a smooth subvariety of E ′ , it follows that X ⊂ E ′ is Whitney A.Finally, in order to eliminate elements of E ′∗ which are constant on X , we may replace E ′ with asubspace of E ′ , as in Remark 4.13. The resulting embedding X ֒ → E ′ then satisfies items 1 and 2.We will apply Theorem 3.15 in order to show that X E is isomorphic to X E ′ . (The proof of equiv-ariance with respect to T a follows exactly as the proof for T c , since the Hamiltonian action of T a on E × E ′ × C preserves X × C ). We only need to verify that the moment maps ψ c : E → Lie( T c ) ∗ and ψ ′ c : E ′ → Lie( T c ) ∗ for the action of T c are proper. But this is Lemma 4.14. This proves item3. (cid:3) ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 29
Proposition 4.16.
Let v be a good valuation on A = C [ X ] . Assume that there exists a basis K ′ = { z ′ , . . . , z ′ n } of E ∗ which restricts to a Khovanskii basis for A and v . Then, there exists a H a × H c -weight basis K = { z , . . . , z n } of E ∗ such that: 1) K restricts to a Khovanskii basis for A and v , and 2) the dual basis of K is an orthonormal basis of E .Proof. Let { y , . . . , y n } be a H a × H c -weight basis of E ∗ . By (GV2), each y i is a weight vectorwith weight ( a , c ) ◦ v ( y i ) . For each λ ∈ Λ a × Λ c , let I λ = { i ∈ [1 , n ] | ( a , c ) ◦ v ( y i ) = λ } . Write z ′ j = n X i =1 a i y i = X λ ∈ Λ a × Λ c X i ∈ I λ a i y i , a i ∈ C . Then each term P i ∈ I λ a i y i is a weight vector of weight λ . What is more, v ( z ′ j ) ≤ max λ ( v X i ∈ I λ a i y i !) . By (v4), each term v (cid:0)P i ∈ I λ a i y i (cid:1) is contained in ( a , c ) − ( λ ) , and so each of these terms is distinct.By elementary properties of valuations, it follows that v ( z ′ j ) = max λ ( v X i ∈ I λ a i y i !) . By applying ( a , c ) to both sides of this equation, we find that the right hand side must be containedin ( a , c ) − ( v ( z ′ j )) . This is impossible unless v ( z ′ j ) = v X i ∈ I ( a , c )( v ( z ′ j )) a i y i . Let z ′′ j = X i ∈ I ( a , c )( v ( z ′ j )) a i y i . Then by the preceding argument, the linear functions z ′′ j satisfy v ( z ′′ j ) = v ( z ′ j ) and therefore restrictto a Khovanskii basis for A and v .After possibly discarding functions from K ′ and reindexing we may assume that the functions z ′ , . . . , z ′ n ′ , where n ′ ≤ n , have values v ( z ′ j ) which are all distinct. (The functions z ′ , . . . , z ′ n ′ may fail to be a basis for E ∗ , but they will still restrict to a Khovanskii basis for A and v ). Then,the values v ( z ′′ ) , . . . , v ( z ′′ n ′ ) are all distinct, and therefore the functions z ′′ , . . . , z ′′ n ′ are linearlyindependent. Adding in weight vectors y ′′ n ′ +1 , . . . , y ′′ n from E ∗ as necessary, we arrive at a weightbasis K ′′ = { z ′′ , . . . , z ′′ n ′ , y ′′ n ′ +1 , . . . , y ′′ n } of E ∗ which restricts to a Khovanskii basis for A and v .Finally, following the Gram-Schmidt argument of [37, Lemma 3.23], one can replace K ′′ with abasis K = { z , . . . , z n } for E ∗ which satisfies all the desired properties. (cid:3) In summary of Propositions 4.15 and 4.16, given a good valuation v on A = C [ X ] , we can alwaysassume that the ambient affine space E has a system of linear coordinates K ⊂ E ∗ which restrictsto a Khovanskii basis for A and v , and such that the dual basis of K is an orthonormal weight basisof E . Good valuations and gradient Hamiltonian flows.
We are now in a position to constructtoric degenerations from good valuations and apply Theorem 3.6 to construct integrable systems inthe sense defined in Remark 3.11.The focal point of this construction is an affine H a × H c -variety X equipped with a decompo-sition and the structure of a Hamiltonian T a × T c -space. Throughout, we fix a good valuation ( X, E, h E , v , a , c ) that endows X with its Hamiltonian T a × T c -space structure (Remark 4.11) andimport all the notation from Definition 4.10. Throughout this section, we fix a basis K ⊂ E ∗ that satisfies the conclusions of Proposition 4.16. As demonstrated by Propositions 4.15 and 4.16,we may assume there exists K with these properties without changing the underlying Hamiltonian T a × T c -space structure on X .As in Section 4.1.2, from v and K one can construct a toric degeneration of X which embeds into E × C . We describe this degeneration in Section 4.3.1. We describe how this degeneration interactswith the symplectic structure in Section 4.3.2. This is combined with Theorem 3.6 in Section 4.3.3to construct an integrable system on X .4.3.1. Application of the Rees algebra construction.
Recall that we fixed ( X, E, h E , v , a , c ) and K above. Let S = S v denote the value semigroup of v and let X S denote the associated affinetoric variety. Applying the Rees algebra construction of Section 4.1.2 to v and K produces a toricdegeneration π : X → C of X to X S with the following properties. Proposition 4.17.
A toric degeneration π : X → C of X to X S , constructed from v and K as inSection 4.1.2, has the following properties:(1) X is embedded as a closed subvariety of E × C such that π : X → C coincides with therestriction of the projection E × C → C .(2) The fiber X ⊂ E × { } ∼ = E coincides with the image of the embedding X ֒ → E of thegood valuation. In other words, it is cut out by the kernel of the natural map C [ E ] → C [ A ] described in (29) .(3) For each σ ∈ Σ , the subfamily X σ defined as in (23) using the decomposition of X satisfies X σ = X ∩ ( E σ × C ) . (4) For each σ ∈ Σ , let S σ = c − ( σ ) ∩ S . Then each subfamily X σ → C is a toric degenerationof X σ to X S σ .(5) For each σ ∈ Σ , the action of H a × H c on E × C (where H a × H c acts trivially on C )preserves X σ .Proof. We follow the notation of Section 4.1.2. Fix an enumeration K = { z , . . . , z n } ⊂ E ∗ , let H = H a × H c , and let w = ( a , c ) . Construct the Rees algebra R as in Section 4.1.2, choosing alinear map e : L → Z as in (31). Let X = Spec R and fix the H -equivariant embedding X ֒ → E × C as in (34). Then items 1 and 2 are satisfied by construction.Let C × act on E × C by t · ( z , . . . , z n , t ′ ) = ( t e ( v ( z )) z , . . . , t e ( v ( z n )) z n , tt ′ ) The trivialization away from zero of the toric degeneration is written using this action as ρ : X × C × → X \ X , ρ ( z, t ) = t · ( z, . (36)This action of C × preserves E σ , so ρ ( X σ × C × ) = X ∩ ( E σ × C × ) . Taking closures establishesitem 3. By Proposition 4.9, there is an isomorphism C [ X ∩ E σ ] ∼ = C [ S σ ] and X σ → C is a flat ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 31 morphism. This establishes item 4. Finally, the map (33) preserves the grading by Λ a × Λ c , and sothe action of H a × H c on E × C preserves X . This action also preserves E σ ; putting these two factstogether gives item 5. (cid:3) We record that X \ X satisfies the assumption (D2’). Lemma 4.18.
Let σ, τ ∈ Σ with σ ≺ τ . Let x ∈ X σ \ X σ , and let ( x j ) j ∈ N ∈ X τ be a sequence ofpoints with lim j →∞ x j = x . Assume that lim j →∞ T x j X τ exists (in the total Grassmannian Gr( E × C ) ). Then T x X σ ⊆ lim j →∞ T x j X τ .Proof. Using the trivialization away from zero (36), it suffices to prove the analogous claim for X ∼ = X ⊂ E . But this is precisely the Whitney condition (A) for the stratification of X , whichholds by assumption (GV2) of Definition 4.10. (cid:3) Symplectic geometry of X . Let π : X → C be a toric degeneration of X constructed from v and K as in the previous subsection. Recall that X is embedded as a subvariety of E × C . Let h C be the standard complex inner product on C , let h E ⊕ h C be the product complex inner product on E × C and let ω := ω E ⊕ ω C be the associated symplectic structure (cf. Example 2.4).As in Section 4.1.2, let H denote the algebraic torus with (real) weight lattice L . Let T denote themaximal compact torus of H . Then v and the basis K ⊂ E ∗ determine a representation of H =( C × ) m on E as described in Section 4.1.2. Since K satisfies the conclusions of Proposition 4.16,the action of T on ( E, h E ) is unitary.Extend the action of H from E to E × C by letting H act trivially on the second factor. As inExample 2.6, the action of T on ( E × C , ω ) is Hamiltonian with moment map(37) Ψ : E × C → Lie( T ) ∗ , Ψ = 12 n X i =1 | z i | v ( z i ) . Recall from Section 4.2, that we similarly extended the actions of H a and H c to E × C . The actionsof T a and T c are Hamiltonian, with moment maps ψ a and ψ c respectively. Extend a and c to R -linear maps Lie( T ) ∗ → Lie( T a ) ∗ and Lie( T ) ∗ → Lie( T c ) ∗ . These maps aredual to homomorphisms T a → T and T c → T . As in Section 4.1.2, since v is compatible with a and c by (GV1), the actions of T a and T c on E × C coincide with the actions defined by thehomomorphisms T a → T and T c → T , and the action of T on E × C . It follows that:(38) ψ a = a ◦ Ψ , ψ c = c ◦ Ψ . The action of H on E × C preserves the fiber X ⊂ E ×{ } . Recall from Proposition 4.17, item (4),that the subvariety X σ = X ∩ X σ is isomorphic to the toric variety X S σ associated to the semigroup S σ = c − ( σ ) ∩ S for each σ ∈ Σ . Proposition 4.19.
Let v be a good valuation as above, let Ψ : E × C → Lie( T ) ∗ be the momentmap (37) , and let σ ∈ Σ . Then:(1) The action of H on E × C preserves X σ .(2) Ψ( X σ ) = cone( S σ ) = Ψ( E σ × C ) for all σ ∈ Σ . In particular, Ψ( X ) = cone( S ) =Ψ( E × C ) . The subspaces E σ × C are saturated by Ψ for all σ ∈ Σ .(3) The fibers of the domain restricted map Ψ : X → Lie( T ) ∗ are connected. These symbols were also used to denote moment maps on E . Here we overload notation. (4) The map (Ψ , π ) : E × C → Lie( T ) ∗ × C is proper. In particular, the domain restricted map Ψ : X → Lie( T ) ∗ is proper.(5) X σ = X ∩ ψ − c ( σ ) , for each σ ∈ Σ .Proof. (1) The map C [ E ] → gr A in (29) is a map of L -graded algebras, so H preserves X .Also, H preserves E σ × C . By Proposition 4.17, item 3, X σ = X ∩ ( E σ × C ) . Thus, H preserves X σ .(2) As an H -module, the semigroup algebra C [ S σ ] splits as a direct sum C [ S σ ] = ⊕ λ ∈ S σ V ( λ ) ,where V ( λ ) is the one dimensional H -module with weight λ ∈ L . Then cone( S σ ) = Ψ( X σ ) by [69, Theorem 4.9]. And cone( S σ ) = Ψ( E σ × C ) by the same reasoning. The subspaces E σ × C are saturated by Ψ is a consequence of Lemma 2.7.(3) Since S is saturated by assumption, the variety X ∼ = X S is normal. The claim then followsimmediately by [69, Corollary 4.13].(4) It suffices to prove that the restricted map Ψ : E = E × { } → Lie( T ) ∗ is proper. Thisfollows immediately from Lemma 4.14, since ψ c = c ◦ Ψ and since the restricted map c : cone( S ) → Lie( T c ) ∗ is proper by (v3). Since X is closed in E , the domain restrictedmap Ψ : X → Lie( T ) ∗ is also proper.(5) The description of X σ follows by Proposition 4.17, item 3: X σ = X ∩ ( E σ × C ) = X ∩ ψ − c ( σ ) . (cid:3) As in Section 3.3, denote X σ = X σ \ ∪ τ ≺ σ X τ and X σ = X ∩ X σ for all σ ∈ Σ . It follows byProposition 4.19 part 2 that Ψ( X σ ) = Ψ( X σ ) \ [ τ ≺ σ Ψ( X τ ) = cone( S σ ) \ [ τ ≺ σ cone( S τ ) . For all τ ≺ σ , each cone( S τ ) is a face of cone( S σ ) . As a consequence the set Ψ( X σ ) is a convex,locally rational polyhedral set. In particular, the smooth locus of Ψ( X σ ) is well defined. We makeuse of this in the following Corollary. Corollary 4.20.
As in Section 3.3, let U σ ⊂ X σ denote the smooth locus of X σ . Then:(1) The image Ψ( U σ ) is the smooth locus of the locally rational polyhedral set Ψ( X σ ) . Inparticular, Ψ( U σ ) is a convex, locally rational polyhedral set.(2) The restricted map Ψ : U σ → Ψ( U σ ) is proper.Proof. The singular locus ( X σ ) sing ⊂ X σ is saturated by Ψ : X σ → Lie( T ) ∗ . The image Ψ(( X σ ) sing ) is the complement of the smooth locus of cone( S σ ) . Since X σ is an open subset of X σ , one has U σ = X σ \ ( X σ ∩ ( X σ ) sing ) . Since ( X σ ) sing is saturated by Ψ , this implies Ψ( U σ ) = Ψ( X σ ) \ Ψ(( X σ ) sing ) . It follows that Ψ( U σ ) is the smooth locus of Ψ( X σ ) . Convexity of Ψ( U σ ) then follows from con-vexity of Ψ( X σ ) . This proves the first claim.By Proposition 4.19, for each τ ≺ σ the restricted map Ψ : X σ → cone( S σ ) is proper and theclosed subset X τ is saturated by Ψ . It follows that U σ = X σ \ (cid:0) ∪ τ ≺ σ X τ ∪ ( X σ ) sing (cid:1) is also saturated by Ψ : X σ → cone( S σ ) . Thus, the restricted map Ψ : U σ → Ψ( U σ ) is proper. Thisproves the second claim. (cid:3) ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 33
Main result.
We are now in a position to give the main result of Section 4. It is a directspecialization of Theorem 3.6.Let v be a good valuation on A = C [ X ] , and let X ⊂ E × C be the toric degeneration constructedas in Proposition 4.17. Let T = T c and ψ = ψ c . Recall condition (GH5) from Section 3.3, whichstates(GH5) The Duistermaat-Heckman measures of (( X σ ) sm , ω σ , ψ ) and ( X σ , ω σ , ψ ) are equal for all σ ∈ Σ .Recall that the symplectic forms ω σ and ω σ come from the embedding X ֒ → E × C . Theorem 4.21.
Let X be as above, and assume condition (GH5) holds. Then, there exists a contin-uous, surjective, T a × T c -equivariant, proper map φ : X = X → X = X S such that(a) For each σ ∈ Σ , there is a dense open subset D σ ⊂ X σ such that φ | D σ defines a symplec-tomorphism of ( D σ , ω σ ) onto the dense, open, H -invariant submanifold ( X σ ) sm .(b) ( ψ a , ψ c ) ◦ φ = ( ψ a , ψ c ) .(c) For each σ ∈ Σ , the map Ψ ◦ φ restricts to the moment map of a Hamiltonian T action on ( D σ , ω σ ) .Proof. We need only check conditions (GH1)-(GH6) and apply Theorem 3.6. (The proof of equiv-ariance with respect to T a follows exactly as the proof for T c , since the Hamiltonian action of T a on E × C preserves X , π , and the Kähler metric, cf. Proposition 3.3). Condition (GH1) holds byitems (3) and (4) of Proposition 4.17. Condition (GH2) holds by item (1) of Proposition 4.17 andthe definition of the action of H . Condition (GH3) holds by item (2) of Proposition 4.17, condi-tion (GV2) of Definition 4.10, and (17). Condition (GH4) follows from Lemma 4.14, (17), anditem (5) of Proposition 4.17. Condition (GH5) holds by assumption. Condition (GH6) holds byLemma 4.18 and Proposition 3.13. (cid:3)
5. T
ORIC DEGENERATIONS OF G (cid:12) N We now apply the results of the previous section to the base affine space of a complex semisimplealgebraic group G . The main result is Theorem 5.7. It is applied in Section 6 to prove Theorem1.2.5.1. The base affine space G (cid:12) N . This section recalls the relevant properties of the base affinespace. In particular, it shows how the base affine space may be equipped with the structure of adecomposed affine Kähler variety and a Hamiltonian K × T -space (cf. Definition 2.10). For moredetails, see e.g. [27, 29].Let G be as in Section 2.3. The coordinate ring C [ G ] has a G × G -module structure, which is dualto the action of G × G on G given by ( g, h ) · k = gkh − , for g, h, k ∈ G . The subalgebra C [ G ] N of N ∼ = 1 × N -invariants inherits a rational G × H -module structure from C [ G ] since H normalizes N . The base affine space, denoted G (cid:12) N , is the affine G × H -variety whose coordinate ring is C [ G (cid:12) N ] = C [ G ] N . Consider G (cid:12) N as a H ∼ = 1 × H -variety. Then the set Λ( G (cid:12) N ) of H -weights appearing in C [ G ] N is equal to the set of dominant weights Λ + . Therefore Γ( G (cid:12) N ) = t ∗ + , and Σ = Σ( G (cid:12) N ) consists of the open faces of t ∗ + . For each σ ∈ Σ , we will consider the subvarieties ( G (cid:12) N ) σ and ( G (cid:12) N ) σ as in Section 2.6.The variety G (cid:12) N also can be partitioned as a union of the G ∼ = G × orbits. This partitioncoincides with the partition into ( G (cid:12) N ) σ , σ ∈ Σ , which was described above. The G -orbits of G (cid:12) N are smooth manifolds, and this partition of G (cid:12) N satisfies (D1).In more detail, for each σ ∈ Σ the piece ( G (cid:12) N ) σ is isomorphic as an algebraic G -homogeneousspace to the quotient G/ [ P σ , P σ ] , where P σ is the parabolic subgroup of G with Lie algebra(39) p σ = h ⊕ n ⊕ M α ∈ R + ,σ g − α , R + ,σ = { α ∈ R + | λ ( α ∨ ) = 0 , ∀ λ ∈ σ } , and [ P σ , P σ ] is its commutator subgroup. The open dense piece of G (cid:12) N is isomorphic to the G -homogeneous space G/N .Fix a finite set Π ⊂ Λ + that generates Λ + as a semigroup and let(40) E = M ̟ ∈ Π V ( ̟ ) . By definition, E is a G × H -module, where × H acts on each summand V ( ̟ ) with weight − ̟ .There is an embedding of G × H -modules,(41) E ∗ = M ̟ ∈ Π V ( ̟ ) ∗ ⊂ M λ ∈ Λ + V ( λ ) ∗ ∼ = C [ G ] N . The isomorphism of modules on the right only depends on the choice of highest weight vectors v ( λ ) ∈ V ( λ ) , and is described in [27, Theorem 12.1]. Since Π generates Λ + as a semigroup, E ∗ generates C [ G ] N as an algebra [27, Theorem 12.6]. Therefore, we have a G × H -equivariantembedding(42) G (cid:12) N ֒ → E which identifies G (cid:12) N with a subvariety of E . Explicitly, if [ P σ , P σ ] denotes the coset of theidentity in G/ [ P σ , P σ ] , then the embedding (42) sends(43) [ P σ , P σ ] X ̟ ∈ Π ∩ σ v ( ̟ ) ∈ E. For each σ ∈ Σ , consider the subvarieties(44) E σ = M ̟ ∈ σ ∩ Π V ( ̟ ) , and E σ = E σ \ [ τ ≺ σ E τ . They coincide with the definition in Section 2.6. The subspaces E σ and E σ are G × H -invariant.The embedding (42) endows G (cid:12) N with the structure of a stratified affine Kähler variety ( G (cid:12) N, E, ω E ) as follows. Let h E denote the unique K × T -invariant complex inner product on E suchthat || v ( ̟ ) || = 1 for all ̟ ∈ Π and let ω E = −ℑ h E . Since the direct summands V ( ̟ ) are × T -weight spaces with distinct weights, the direct sum (40) is orthogonal with respect to h E . The orbitstratification of G (cid:12) N ⊂ E is a Whitney stratification. Thus ( G (cid:12) N, E, ω E ) is a decomposedaffine Kähler variety, and the decomposition satisfies (D2).The action of K × T ⊂ G × H on ( E, h E ) is unitary, so by Example 2.4 it is Hamiltonian withrespect to ω E . In particular, the action of T × T ⊂ K × T is Hamiltonian. This action preservesthe smooth pieces of G (cid:12) N ⊂ E . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 35
We summarize the preceding discussion as follows.
Lemma 5.1.
Let Π ⊂ Λ + be a finite set which generates Λ + , let E be as in (40) , and let G (cid:12) N beembedded in E as in (42) . Then, as a subvariety of E , the decomposition of G (cid:12) N by × H orbittypes satisfies (GV2). With h E as above, the tuple ( G (cid:12) N, E, ω E ) is a decomposed Kähler variety,and G (cid:12) N is a Hamiltonian K × T -space. Gradient Hamiltonian flows for G (cid:12) N . This section is concerned with the construction ofa good valuation on the affine variety G (cid:12) N . Throughout, in the terminology of Definition 4.10,we let H a = H c = H , and consider G (cid:12) N as a H a × H c = H × H -variety, where H × H acts asa subgroup of G × H .The vector space ( E, h E ) , the embedding G (cid:12) N ֒ → E , the inner product h E , and the decompositionof G (cid:12) N for a good valuation were all constructed in the previous section. It remains to constructthe valuation v itself and check (GV1).Section 5.2.1 gives a general construction of suitable valuations on C [ G (cid:12) N ] . Section 5.2.2 appliesthe resulting good valuations to construct integrable systems on G (cid:12) N in the sense defined inRemark 3.11.5.2.1. Valuations on C [ G (cid:12) N ] . This section reviews a general construction of suitable valuationson C [ G (cid:12) N ] from valuations on C ( G/B ) . See [48] for more details.As before, let Λ denote the weight lattice of H and m = dim C ( G/H ) . Fix a total order on Z m × Λ as in [48, p. 2492], as follows. First, fix a refinement of the standard partial order on Λ to a totalorder. Second, equip Z m with the standard lexicographic order. For ( ζ , λ ) and ( ϕ, µ ) in Z m × Λ ,define(45) ( ζ , λ ) > ( ϕ, µ ) if λ > µ, or if λ = µ and ζ > ϕ. Let c : Z m × Λ → Λ denote projection to Λ . By definition, the total order on Z m × Λ descendsunder c to the total order on Λ .Let ν : C ( G/B ) \ { } → Z m be a valuation with one dimensional leaves. Assume that the imageof ν generates Z m as a group. We construct a valuation on C [ G (cid:12) N ] from ν as follows. First,let i : N − → G/B denote the embedding n − n − B . This embedding fixes an isomorphism C [ N − B/B ] ∼ = C [ N − ] such that F ∈ C [ N − B/B ] is identified with ι ∗ F . For F ∈ C [ N − B/B ] ,define(46) ν | N − : C [ N − ] \ { } → Z m , ν | N − ( ι ∗ F ) = ν ( F ) . This defines a valuation with one dimensional leaves on C [ N − ] . Second, consider the composition(47) j : N − × H ֒ → G/N ֒ → G (cid:12) N The first map is ( n, h ) nhN . It identifies N − × H with the open subset B − N ⊂ G/N . Thesecond map is inclusion of the dense G -orbit in G (cid:12) N . With respect to the isomorphism fixed in(41), the dual map(48) j ∗ : M λ ∈ Λ + V ( λ ) ∗ ∼ = C [ G (cid:12) N ] → C [ N − × H ] ∼ = C [ N − ] ⊗ C [ H ] has the property that for z λ ∈ V ( λ ) ∗ , j ∗ z λ = f λ ⊗ χ λ . Here χ λ ∈ C [ H ] denotes the character on H corresponding to λ and f λ ∈ C [ N − ] is given by f λ ( n ) = z λ ( n · v ( λ )) , where v ( λ ) ∈ V ( λ ) is the chosen highest weight vector and n ∈ N − .An arbitrary element z ∈ C [ G (cid:12) N ] can be written as z = P λ ∈ Λ + z λ , z λ ∈ V ( λ ) ∗ , with all butfinitely many z λ being zero. Given z , let λ = max { γ | z γ = 0 } and define(49) v : C [ G (cid:12) N ] \ { } → Z m × Λ , v ( z ) = ( ν | N − ( f λ ) , λ ) . The following is automatic from the definition (see e.g. [48, Proposition 6.1]).
Lemma 5.2.
The function v : C [ G (cid:12) N ] \ { } → Z m × Λ defined in (49) is a valuation with onedimensional leaves. The set S v generates Z m × Λ as a group. Furthermore, c ( S v ) = Λ + . The action of H on G/B defines a Λ -grading of C ( G/B ) . Assume that:(*) There exists a linear map a ′ : Z m → Λ such that if f ∈ C ( G/B ) is Λ -homogeneous ofdegree δ , then a ′ ( ν ( f )) = δ .With this assumption in place, we have the following. The proof is deferred to Appendix B.1. Lemma 5.3.
The valuation v and the map c satisfy (v1)-(v5). The valuation v and the map a = a ′ − c satisfy (v4). Good valuations and toric degenerations of G (cid:12) N . We now describe the general form ofgood valuations (Definition 4.10) from which we will construct toric degenerations G (cid:12) N . Data 5.4.
Fix a tuple ( G (cid:12) N, E, h E , v , a , c ) where: • ( E, h E ) is a finite dimensional complex inner product space defined as in (40), equippedwith an H × H -equivariant embedding G (cid:12) N ֒ → E defined as in (42). • v : C [ G (cid:12) N ] \ { } → Z m × Λ is a valuation of the form (49). As in Section 5.2.1,we assume that: v is constructed from a valuation ν : C ( G/B ) \{ } → Z m with one-dimensional leaves according to (49), the image of ν generates Z m as a group, and ν satisfies(*). Additionally, we assume that S = S v is finitely generated. • a , c : Z m × Λ → Λ , are surjective Z -linear maps where c : Z m × Λ → Λ is the projection tothe second factor and a = a ′ − c as in Lemma 5.3. Lemma 5.5.
Data 5.4 defines a good valuation on G (cid:12) N .Proof. This follows immediately from Lemma 5.1 and Lemma 5.3. (cid:3)
For the remainder of this section, assume we have fixed a choice of Data 5.4. Let H = ( C × ) m × H ,the algebraic torus with (real) weight lattice Z m × Λ . Let X S denote the toric H -variety associated tothe value semigroup S . The good valuation can be used to construct a toric degeneration π : X → C of G (cid:12) N to X S that embeds as a subvariety of E × C (Proposition 4.17). This also fixes anembedding of X S into E such that the action of the compact subtorus T = ( S ) m × T on X S isHamiltonian, generated by the restriction of the moment map Ψ : E × C → Lie( T ) ∗ defined as in(37).The linear action of the compact subtorus T × T ⊂ H × H on E × C is Hamiltonian with momentmap ( ψ a , ψ c ) : E × C → t ∗ × t ∗ . This map satisfies (38). The action of T × T on E × C preserves X ∼ = G (cid:12) N and X ∼ = X S . The restriction of this action to X ∼ = G (cid:12) N coincides with the actionof T × T on G (cid:12) N as the maximal torus of K × T ⊂ G × H .For each open face σ ∈ Σ of t ∗ + , recall X σ = X ∩ E σ . Denote U σ = ( X σ ) sm . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 37
Proposition 5.6.
For all σ ∈ Σ , the Duistermaat-Heckman measures of the Hamiltonian T -manifolds ( U σ , ω σ , ψ c ) and ( X σ , ω σ , ψ c ) are the same. The proof of Proposition 5.6 is given in Appendix B.2.
Theorem 5.7.
Let G be a connected semisimple Lie group and fix a choice of Data 5.4. Let X S denote the toric variety constructed from this data as above. Then, there exists a continuous,surjective, T × T -equivariant, proper map φ : G (cid:12) N → X S such that(a) ( ψ a , ψ c ) ◦ φ = ( ψ a , ψ c ) .(b) For each σ ∈ Σ , D σ = φ − ( U σ ) is an open dense subset of ( G (cid:12) N ) σ and φ : ( D σ , ( ψ a , ψ c )) → ( U σ , ( ψ a , ψ c )) is an isomorphism of Hamiltonian T × T -manifolds.(c) For each σ ∈ Σ , the restricted map Ψ ◦ φ : D σ → Lie( T ) ∗ generates a complexity 0Hamiltonian T -action on D σ .Proof. Combine Lemma 5.5, Proposition 5.6, and Theorem 4.21. (cid:3)
Theorem 5.7 can be applied in the following example.
Example 5.8 (String valuations) . Let i = ( i , . . . , i m ) be a reduced word for the longest elementof the Weyl group of G , expressed in terms of simple reflections associated with simple real roots α i , . . . , α i m . Let ν i : C ( G/B ) \{ } → Z m be the valuation constructed in [48, Section 2.2]. Specif-ically, ν i is defined from the highest-term valuation associated to the standard coordinate chart onthe Bott-Samelson variety associated to i . Define a ′ : Z m → Λ , a ′ ( v , . . . , v m ) = m X j =1 v j α i j . The map a ′ and the valuation ν i satisfy (*) by [48, Proposition 3.8]. Define v i : C [ G (cid:12) N ] \{ } → Z m × Λ using ν = ν i as in (49). By [48, Proposition 6.1], the value semigroup S i = S v i coincideswith the set of integral points of a rational convex polyhedral cone known as the extended stringcone associated to i . The extended string cones were introduced in [10] and [59]. It was provedthat they are rational in [59]. Therefore, the tuple ( G (cid:12) N, E, h E , v i , a , c ) is an example of Data 5.4,and Theorem 5.7 holds with X S = X S i .6. C OLLECTIVE INTEGRABLE SYSTEMS ON H AMILTONIAN K - SPACES
In all that follows, K is a compact connected Lie group, T ⊂ K is a maximal torus, and wefix a positive Weyl chamber t ∗ + ⊂ k ∗ . Recall that Hamiltonian K -manifolds are assumed to beconnected. Symplectic implosion.
This section recalls symplectic implosion, introduced in [29].
Definition 6.1. [29, Definition 2.1] The symplectic implosion of a Hamiltonian K -space ( X, µ ) with respect to a positive Weyl chamber t ∗ + is the quotient topological space E X := µ − ( t ∗ + ) / ∼ where ∼ is the following equivalence relation: Two points p, p ′ ∈ µ − ( t ∗ + ) are equivalent if and only if λ = µ ( p ) = µ ( p ′ ) and p ′ = k · p ′ for some element k of the commutator subgroup [ K λ , K λ ] ⊂ K λ . The symplectic implosion is a singular symplectic space. If K = T is a torus, then x ∼ y if andonly if x = y , and so E X = X . In general, the action of the maximal torus T preserves µ − ( t ∗ + ) and ∼ , so it descends to a continuous action on E X . The moment map µ also descends to a continuousmap µ : E X → t ∗ . Explicitly, if [ x ] ∈ E X is an equivalence class of an element x ∈ µ − ( t ∗ + ) , then(50) t · [ x ] = [ t · x ] and µ ([ x ]) = µ ( x ) . Together, these endow E X with the structure of a Hamiltonian T -space, ( E X, µ ) .Suppose ( X, µ ) is a Hamiltonian K -space and ( Y, ψ ) is a Hamiltonian J -space. Let T ⊂ K and S ⊂ J be maximal tori and let t ∗ + and s ∗ + be positive Weyl chambers. Recall that we may form theproduct Hamiltonian K × J -space ( X × Y, µ × ψ ) . Then, the symplectic implosion of the product(with respect to the positive Weyl chamber t ∗ + × s ∗ + ) is canonically isomorphic, as a Hamiltonian T × S -space, to the product E X × E Y .Suppose ( X, µ ) is a Hamiltonian K -manifold and σ ∈ Σ is its principal stratum , i.e. σ is the uniqueelement of Σ such that µ ( X ) ∩ σ is non-empty and µ ( X ) ∩ t ∗ + ⊂ σ . The pre-image µ − ( σ ) is a T -invariant symplectic submanifold of X known as the principal symplectic cross-section . Theequivalence relation ∼ is trivial on µ − ( σ ) (meaning that x ∼ y if and only if x = y ). Thus, thequotient map for ∼ identifies the principal symplectic cross-section with a subspace of E X . Withthis identification, µ − ( σ ) is a dense piece of E X . We summarize these results in the following. Proposition 6.2 (Properties of symplectic implosion) . [29] Let ( X, µ ) be a Hamiltonian K -space.Then,(i) The symplectic implosion ( E X, µ ) is a Hamiltonian T -space.(ii) If X is a Hamiltonian K -manifold with principal stratum σ , then E X is connected and hasa dense piece which is identified with the principal symplectic cross-section, µ − ( σ ) . The universal imploded cross-section.
Let T ∗ K denote the cotangent bundle of K and let ω can denote the canonical symplectic structure. Cotangent lift of the action of K × K on K byleft and right multiplication defines a Hamiltonian K × K -action on T ∗ K . With respect to theleft-invariant trivialization T ∗ K ∼ = K × k ∗ , the action of K × K is ( k , k ) · ( k, ξ ) = ( k kk − , Ad ∗ k ξ ) and the moment map is ( µ L , µ R ) : K × k ∗ → k ∗ × k ∗ ( k, ξ ) (Ad ∗ k ξ, − ξ ) . Definition 6.3.
The universal imploded cross-section E T ∗ K is the symplectic implosion of theHamiltonian K -manifold ( T ∗ K, ω can , µ R ) with respect to the Weyl chamber − t ∗ + . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 39
Let Σ denote the poset of open faces of t ∗ + . For each σ ∈ Σ , recall that P σ ⊂ G denotes theparabolic subgroup of G with Lie algebra (39). Let K σ = K ∩ P σ and let [ K σ , K σ ] denote itscommutator subgroup. The decomposition of E T ∗ K by symplectic pieces is E T ∗ K = ( K × t ∗ + ) / ∼ = a σ ∈ Σ ( K/ [ K σ , K σ ]) × σ. We denote the symplectic form on ( K/ [ K σ , K σ ]) × σ by ω σ .The action of K × T on T ∗ K preserves the subset ( S ◦ µ L ) − ( t ∗ + ) = K × t ∗ + , (where S is thesweeping map defined in Section 2.3), and so descends to a continuous action on E T ∗ K . In all thatfollows, it will be convenient to twist the action of the subtorus × T ⊂ K × T by the inverse. Themoment map is then ( µ L , − µ R = S ◦ µ L ) : E T ∗ K → k ∗ × t ∗ Explicitly, if ( k, ξ ) ∈ K × t ∗ + is a representative of [ k, ξ ] ∈ E T ∗ K , and ( k , k ) ∈ K × T , then(51) ( µ L , S ◦ µ L )([ k, ξ ]) = (Ad ∗ k ξ, ξ ) and ( k , k ) · [ k, ξ ] = [ k kk , ξ ] . This endows E T ∗ K with the structure of a Hamiltonian K × T -space, ( E T ∗ K, ( µ L , S ◦ µ L )) .The moment map µ L : E T ∗ K → k ∗ is a quotient map of topological spaces and the fibers of µ L areprecisely the orbits of the action of the subgroup × T ⊂ K × T . Thus, the following diagramcommutes and the bottom map is a homeomorphism.(52) E T ∗ K ( E T ∗ K ) / × T k ∗ / × T µ L ∼ = For each λ ∈ t ∗ + , the identification (52) restricts to an isomorphism of symplectic manifolds:(53) ( E T ∗ K ) / × T k ∗ ( E T ∗ K ) (cid:12) λ × T O λ . ∼ = ∼ = In fact, these are isomorphisms of Hamiltonian K -manifolds. Here ( E T ∗ K ) (cid:12) λ × T denotes thesymplectic reduction of ( E T ∗ K, S ◦ µ L ) at λ (cf. [29, Theorem 3.4, Example 3.6]) and O λ denotesthe coadjoint orbit through λ ∈ t ∗ + ⊂ k ∗ equipped with its canonical symplectic structure.Recall from Section 5.1 that it is possible to equip G (cid:12) N with the structure of a Hamiltonian K × T -space by embedding it into a complex inner product space ( E, h E ) . Theorem 6.4. [29, 39]
Equip G (cid:12) N with the structure of a Hamiltonian K × T -space by embeddingit G × H -equivariantly into a complex inner product space ( E, h E ) as in (40) and (42) . Then, G (cid:12) N and ( E T ∗ K, ( µ L , S ◦ µ L )) are isomorphic as Hamiltonian K × T -spaces. The construction of the Kähler structure on G (cid:12) N depended on a choice of embedding into acomplex inner product space (Section 5.1). Theorem 6.4 implies that the resulting Hamiltonian K × T -space structure on G (cid:12) N is independent of those choices. We remark that this can also beshown using Theorem 3.15. For each σ ∈ Σ , the isomorphism G (cid:12) N ∼ = E T ∗ K restricts to an isomorphisms of symplecticpieces ( G (cid:12) N ) σ ∼ = K/ [ K σ , K σ ] × σ . In what follows we identify G (cid:12) N with E T ∗ K in thismanner.6.3. Symplectic contraction.
This section recalls symplectic contraction, introduced by Hilgert,Manon, and Martens in [39].Let ( X, µ ) be a Hamiltonian K -manifold. Let ( E X, µ ) denote the symplectic implosion of ( X, µ ) with respect to t ∗ + . Let ( E T ∗ K, ( µ L , S ◦ µ L )) denote the universal imploded cross-section. Theproduct ( E X × E T ∗ K, µ × ( µ L , S ◦ µ L )) is a Hamiltonian T × K × T -space. Definition 6.5. [39] The symplectic contraction of a Hamiltonian K -space ( X, µ ) is(54) X sc = ( E X × E T ∗ K ) (cid:12) T, where symplectic reduction is taken with respect to the diagonally included subtorus T → T × K × T , t ( t, e, t − ) . The action of this subtorus is generated by the moment map µ − S ◦ µ L .By Lemma 2.11, X sc is a singular symplectic space. By Lemma 2.12, it is a Hamiltonian K × T -space, ( X sc , ( µ L , S ◦ µ L )) . Elements of X sc may be represented as equivalence classes [[ x ] , [ k, λ ]] ,where [ x ] ∈ E X , [ k, λ ] ∈ E T ∗ K , and µ ([ x ]) = S ◦ µ L ([ k, λ ]) . The moment map for the K × T -action is(55) ( µ L , S ◦ µ L ) : X sc → k ∗ × t ∗ , [[ x ] , [ k, λ ]] (Ad ∗ k λ, λ ) . Hilgert, Martens, and Manon also introduced the symplectic contraction map. It is a map from X to X sc that shares many of the features of time-1 gradient Hamiltonian flows. We use the algebraicdefinition of the symplectic contraction map due to [39]. An alternate geometric description of thismap was examined in [56]. Definition 6.6. [39] The symplectic contraction map is(56) Φ sc : X → X sc , Φ sc ( x ) = [[ h · x ] , [ h − , µ ( h · x )]] where h ∈ K such that µ ( h · x ) ∈ t ∗ + . In particular, this map is well-defined.Important properties of symplectic contraction are summarized in the following. Lemma 6.7 (Properties of symplectic contraction) . [39] Let ( X, µ ) be a Hamiltonian K -space.Then:(i) The symplectic contraction of ( X, µ ) is a Hamiltonian K × T -space ( X sc , ( µ L , S ◦ µ L )) .(ii) The symplectic contraction map is continuous, proper, surjective, K -equivariant, and itsfibers are connected. Moreover, the following diagram commutes. (57) X X sc k ∗ k ∗ Φ sc µ µ L = ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 41 (iii) If ( X, ω, µ ) is a Hamiltonian K -manifold with principal stratum σ , then (( µ ) − ( σ ) × ( K/ [ K σ , K σ ] × σ )) (cid:12) T is a dense symplectic piece of X sc . Moreover, the restricted map Φ sc : (( S ◦ µ ) − ( σ ) , ( µ, S ◦ µ )) → ((( µ ) − ( σ ) × ( K/ [ K σ , K σ ] × σ )) (cid:12) T, ( µ L , S ◦ µ L )) is an isomorphism of Hamiltonian K × T -manifolds. Completely integrable systems on k ∗ . This section constructs integrable systems on k ∗ , where k is an arbitrary compact Lie algebra and k ∗ is equipped with its canonical Lie-Poisson structure(see Section 2.4). See Remark 6.9 below for a discussion of what we mean by integrable systemson k ∗ .Every compact connected Lie group K has the form K = ( K ss × Z ) /D where K ss is a semisimplecompact connected Lie group, Z is a compact connected torus, and D ⊂ K ss × Z is a finite centralsubgroup such that (1 × Z ) ∩ D is the trivial group. Let T ss be a maximal torus of K ss . Then T = ( T ss × Z ) /D is a maximal torus of K . Fix a positive Weyl chamber t ∗ + = ( t ss ) ∗ + × z ∗ .The cotangent bundle T ∗ K is naturally isomorphic to the quotient ( T ∗ ( K ss × Z )) /D , where D actson T ∗ ( K ss × Z ) by cotangent lift of the action of D on K ss × Z by left multiplication. This inducesan identification of Hamiltonian K × T -spaces, E T ∗ K ∼ = E ( T ∗ K ss × T ∗ Z ) /D ∼ = ( E T ∗ K ss × T ∗ Z ) /D. For the remainder of this subsection, let G temporarily denote the complexification of K ss and let H be the maximal torus of G with H ∩ K ss = T ss . (Elsewhere G is the complexification of K .)Let N be the maximal unipotent subgroup of G compatible with H and ( t ss ) ∗ + .Fix a good valuation ( G (cid:12) N, E, h E , v , a , c ) as in Data 5.4. Let X S denote the affine toric varietyassociated to the value semigroup of v and construct a toric degeneration of G (cid:12) N to X S as inSection 4.3. The torus T ss = ( S ) m × T ss is the maximal compact subgroup of the algebraic torusof X S .We equip X S with the structure of a Hamiltonian T ss -space as follows. Recall that Σ denotesthe poset of open faces of ( t ss ) ∗ + . Each closed face σ ⊂ ( t ss ) ∗ + corresponds to a torus orbit-closure X σ S ⊂ X S . Following Whitney [72], inductively define ( X σ S ) = X σ S and ( X σ S ) i +1 =( X σ S ) i \ ( X σ S ) smi . The sets ( X σ S ) smi are disjoint T ss -invariant smooth manifolds. Decomposingfurther into connected components if necessary, this endows X S with the structure of a weaklydecomposed space (cf. Definition 2.1). The vector space E is equipped with a representation of T ss and X S is identified T ss -equivariantly with a subvariety of E . The symplectic structure on E endows each piece ( X σ S ) smi with a symplectic structure. Thus X S is a singular symplectic space inthe sense of Definition 2.8. The action of T ss on E is Hamiltonian generated by a moment map Ψ ss : E → Lie( T ss ) ∗ as in (37). Since each ( X σ S ) smi is T ss -invariant, the restriction of Ψ ss | ( X σ S ) smi is a moment map for the action of T ss . Thus, ( X S , Ψ ss = Ψ ss | X S ) is a Hamiltonian T ss -space(cf. Definition 2.9).We may form the product Hamiltonian T ss × Z -space ( X S × T ∗ Z, Ψ × µ L,Z ) . Let(58) X = ( X S × T ∗ Z ) /D where the action of D is with respect to the inclusion of D into T ss × Z = ( S ) m × T ss × Z as asubgroup of T ss × Z . Denote T = ( S ) m × T = ( S ) m × ( T ss × Z ) /D. The continuous map Ψ ss × µ Z,L : X S × T ∗ Z → Lie( T ss × Z ) ∗ is D -invariant so it descends to acontinuous map Ψ : X → Lie( T ) ∗ such that the following diagram commutes.(59) X S × T ∗ Z Lie( T ss ) ∗ × Lie( Z ) ∗ X Lie( T ) ∗ Ψ ss × µ Z,L /D Ψ Then ( X, Ψ) is a Hamiltonian T -space.Let φ ss : G (cid:12) N → X S denote a map constructed as in Theorem 5.7 from the toric degeneration.Combining with Theorem 5.7, we have a continuous, proper, surjective and T ss × T ss -equivariantmap:(60) φ ss : E T ∗ K ss ∼ = G (cid:12) N φ ss −−→ X S . Let φ : E T ∗ K → X be the map of quotient spaces induced by φ ss × Id T ∗ Z as in the followingdiagram.(61) E T ∗ K ss × T ∗ Z X S × T ∗ Z E T ∗ K X φ ss × Id T ∗ Z /D /Dφ Since φ is T × T -equivariant and Ψ is T × T -invariant, it follows by (52) that the composition Ψ ◦ φ descends to a continuous T -invariant map F : k ∗ → Lie( T ) ∗ such that the following diagramcommutes.(62) E T ∗ K X k ∗ Lie( T ) ∗ φµ L Ψ F Proposition 6.8.
Let X be defined as in (58) , and let maps Ψ : X → Lie( T ) ∗ , φ : E T ∗ K → X ,and F : k ∗ → Lie( T ) ∗ be constructed as in (59) , (61) , and (62) respectively. Then:(i) The map φ : E T ∗ K → X is continuous, proper, surjective and T × T -equivariant.(ii) The following diagrams commute. (63) k ∗ Lie( T ) ∗ k ∗ Lie( T ) ∗ t ∗ t ∗ t ∗ + t ∗ + F pr t ∗ a F S c (iii) The map Ψ is proper and its fibers are connected.(iv) F ( k ∗ ) = Ψ( X ) = cone( S ) × Lie( Z ) ∗ ⊂ Lie( T ) ∗ . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 43
Proof.
These properties all follow immediately from the definitions (58), (59), (61), and (62) alongwith Theorem 5.7 and Proposition 4.19. That F ( k ∗ ) = Ψ( X ) follows since µ L and φ are surjectiveand (62) commutes. (cid:3) Remark . A collection of real valued continuous functions f , . . . , f n on a smooth connectedPoisson manifold M of constant rank r is an integrable system if:(i) There exists an open dense subset D ⊂ M such that the restricted functions f i | D are allsmooth and the rank of the Jacobian of F = ( f , . . . , f n ) equals dim( M ) − r on a densesubset of D .(ii) The restricted functions f i | D pairwise Poisson commute, i.e. { f i | D , f j | D } = 0 for all ≤ i, j ≤ n .The Poisson manifold k ∗ decomposes as a disjoint union of the orbit-type strata for the coadjointaction. The orbit-type strata are constant rank Poisson submanifolds. The restriction of the coordi-nates of the map F : k ∗ → Lie( T ) ∗ to each orbit-type stratum of k ∗ defines an integrable system inthe sense defined above . We view this as an integrable system on the Poisson manifold k ∗ .We end this section with some useful topological details about the maps Ψ and φ . Recall fromSection 4.3.2 that Ψ ss ( X σ S ) = cone( S σ ) and Ψ ss ( X σ S ) = cone( S σ ) \ [ τ ≺ σ cone( S τ ) is a convex, locally rational polyhedral set. Recall that U σ = ( X σ S ) sm and denote U σ = ( U σ × T ∗ Z ) /D. We record several facts about U σ (cf. Corollary 4.20). Proposition 6.10.
Let X be defined as in (58) , and let maps Ψ , φ , and F be constructed as in (59) , (61) , and (62) respectively. For all σ ∈ Σ :(i) The image Ψ( U σ ) is the smooth locus of the locally rational polyhedral set Ψ( X σ S ) × Lie( Z ) ∗ . In particular, it is convex.(ii) The restricted map Ψ : U σ → Ψ( U σ ) is proper.(iii) The pre-image φ − ( U σ ) is a connected open dense T × T -invariant subset of K/ [ K σ , K σ ] × σ and the restricted map φ : ( φ − ( U σ ) , (pr t ∗ ◦ µ L , S ◦ µ L )) → ( U σ , ( s ◦ Ψ , c ◦ Ψ)) is an isomorphism of Hamiltonian T × T -manifolds. Toric contraction and collective integrable systems.
Let ( Y, µ ) be a Hamiltonian K -space.Let T = ( S ) m × T and ( X, Ψ) be defined as in the previous section. We may form the productHamiltonian T × T -space ( E Y × X, µ × Ψ) . Definition 6.11.
The toric contraction of a Hamiltonian K -space ( Y, µ ) with respect to ( X, Ψ) is(64) X Y := ( E Y × X ) (cid:12) T, In fact, even more is true: the restriction of this integrable system to any symplectic leaf of k ∗ is again an integrablesystem in the sense defined in Remark 3.11. where symplectic reduction is taken with respect to the action of T on E Y × X defined via theinclusion T → T × T = T × ( S ) m × T , t ( t, , t − ) . The action of this subtorus is generatedby the moment map µ − c ◦ Ψ .By Lemma 2.11, X Y is a singular symplectic space. By Lemma 2.12, the toric contraction is aHamiltonian T -space ( X Y , Ψ Y ) where Ψ Y denotes the induced map(65) Ψ Y : X Y = ( E Y × X ) (cid:12) T → Lie( T ) ∗ , [[ y ] , z ] Ψ( y ) . Remark . Note that despite terminology, the action of T on X Y is not necessarily a complexity0 action. Rather, we call X Y the “toric contraction” of Y because it is constructed by reductionwith the toric space X . Proposition 6.19 below characterizes when the action of T on X Y hascomplexity 0.By Proposition 6.8, the product map Id E Y × φ : E Y × E T ∗ K → E Y × X. is T × T × T -equivariant with respect to the Hamiltonian T × T × T -actions generated by the momentmaps µ × (pr t ∗ ◦ µ L , S ◦ µ L ) and µ × ( a ◦ Ψ , c ◦ Ψ) respectively. By Proposition 6.8, Id E Y × φ also intertwines these moment maps. (Note however that Id E Y × φ is not a map of Hamiltonian T × T × T -spaces according to our definition because φ is not a map of decomposed spaces.)The map Id E Y × φ therefore descends to a continuous map φ Y : Y sc → X Y such that the followingdiagram commutes.(66) E Y × E T ∗ K E Y × X ( µ − S ◦ µ L ) − (0) ( µ − c ◦ Ψ) − (0) Y sc X Y . Id E Y × φ/T /Tφ Y Both Y sc and X Y inherit Hamiltonian T × T -space structures from the subtorus × T × T ⊂ T × T × T with moment maps denoted (pr t ∗ ◦ µ L , S ◦ µ L ) and ( a ◦ Ψ , c ◦ Ψ) respectively. Definition 6.13.
The toric contraction map of a Hamiltonian K -space ( Y, µ ) with respect to ( X, Ψ) is the composition Φ : Y Φ sc −−→ Y sc φ Y −→ X Y . We now give a series of results about toric contractions. The first result is true for any (possiblysingular) Hamiltonian K -space. Proposition 6.14 (Properties of toric contraction, part I) . Let ( Y, µ ) be a Hamiltonian K -space.Then:(i) The toric contraction map is continuous, surjective, proper, and T -equivariant (with respectto the action of T on Y as the maximal torus T ⊂ K and the action of T on X Y as thesubtorus T × ⊂ T × T ). ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 45 (ii) The following diagram commutes. (67)
Y X Y k ∗ Lie( T ) ∗ Φ µ Ψ Y F Proof. (i) The map φ constructed in (61) is continuous, surjective, proper and T × T -equivariantby Proposition 6.8. It follows that the map φ Y : Y sc → X Y constructed in (66) is continu-ous, surjective, proper, and T × T -equivariant (where T × T acts on Y sc as the subgroup T × T ⊂ K × T and it acts on X Y as the subgroup T × T ⊂ T ). The result then followssince the map Φ sc is continuous, surjective, proper and K -equivariant (Lemma 6.7) and Φ = φ Y ◦ Φ sc .(ii) The diagram (67) is the outer square of the following diagram.(68) Y Y sc X Y k ∗ k ∗ Lie( T ) ∗ Φ sc µ φ Y µ L Ψ Y F The left square commutes by Lemma 6.7. The right square commutes by combining (62)and (66). (cid:3)
Remark . There is an analogue of the commuting diagram (67) for Gelfand-Zeitlin systems. Itwas first constructed for the case where Y is an integral coadjoint orbit of U ( n ) [64]. In that case,the map Φ is the gradient Hamiltonian flow of a toric degeneration of the coadjoint orbit. Later, itwas constructed for all Hamiltonian U ( n ) -spaces Y [39]. In loc. cit. the map Φ is the branchingcontraction map.If the Hamiltonian K -space ( Y, µ ) is a Hamiltonian K -manifold, then its toric contraction hasseveral additional properties. Proposition 6.16 (Properties of toric contraction, part II) . Let ( Y, µ ) be a Hamiltonian K -manifoldwith principal stratum σ . Then:(i) The toric contraction X Y has a dense piece U ∼ = ( µ − ( σ ) × U σ ) (cid:12) T, where U σ ⊂ X σ is defined as in Section 6.4.(ii) The pre-image D = Φ − ( U ) is a connected open dense T -invariant subset of Y and therestricted toric contraction map is an isomorphism of Hamiltonian T × T -manifolds, Φ : ( D, (pr t ∗ ◦ µ, S ◦ µ )) → ( U , ( a ◦ Ψ , c ◦ Ψ)) . Proof.
The first order of business is to show that U is a piece of X Y . This follows since µ − ( σ ) × U σ is a piece of E Y × X and the stabilizer subgroup at every point in µ − ( σ ) × U σ for the T -actionequals the kernel of the T -action, so the reduced space is a symplectic manifold.To see (ii), note that φ − Y ( U ) = ( µ − ( σ ) × φ − ( U σ )) (cid:12) T. By Proposition 6.10, this is a connected open dense T × T invariant subset of Y sc and the restrictedmap φ Y : ( φ − Y ( U ) , (pr t ∗ ◦ µ L , S ◦ µ L )) → ( U , ( a ◦ Ψ , c ◦ Ψ)) is an isomorphism of Hamiltonian T × T -manifolds. Item (ii) then follows by Lemma 6.7.Since Φ − ( U ) is dense in Y (ii) and Φ : Y → X Y is surjective (Proposition 6.14), it follows that U is dense in X Y . This completes the proof of (i). (cid:3) Recall that if a Hamiltonian K -manifold ( Y, µ ) is proper, then the set △ Y = µ ( Y ) ∩ t ∗ + = S ◦ µ ( Y ) is a convex locally rational polyhedral set and the fibers of µ are connected [57, Theorem 1.1 andRemark 5.2]. Note that if ( Y, µ ) is proper, then S ◦ µ : Y → △ Y is a proper map. Proposition 6.17 (Properties of toric contraction, part III) . Let ( Y, µ ) be a proper Hamiltonian K -manifold. Then:(i) Ψ Y ( X Y ) = c − ( △ Y ) ∩ Ψ( X ) is a convex, locally rational polyhedral set. If Y is compact,then Ψ Y ( X Y ) is a convex polytope.(ii) The map Ψ Y : X Y → Ψ Y ( X Y ) is proper.(iii) The dense piece ( U , Ψ Y ) is a proper Hamiltonian T -manifold. Moreover, if σ is the princi-pal stratum of ( Y, µ ) , then Ψ Y ( U ) = c − ( △ Y ∩ σ ) ∩ Ψ( U σ ) is the smooth locus of convex locally rational polyhedral set c − ( △ Y ∩ σ ) ∩ Ψ( X σ ) .(iv) The fibers of Ψ Y and Ψ Y ◦ Φ are connected. Our proof of fiber connectedness for Ψ Y ◦ Φ uses the following lemma. Lemma 6.18. [55]
Let Y be a Hausdorff topological space and let f : Y → V be a continuousproper map to a convex subset of a real vector space V . Suppose that S ⊂ Y is a dense subset suchthat: S is saturated by f , f ( S ) is convex, the fibers of f | S are connected, and f | S : S → f ( S ) isan open map (with respect to the subspace topologies). Then, the fibers of f are connected.Proof of Proposition 6.17. (i) The equality Ψ Y ( X Y ) = c − ( △ Y ) ∩ Ψ( X ) follows immediatelyfrom the definitions. Since △ Y is a convex locally rational polyhedral set, c is a linear map,and Ψ( X ) = cone( S ) is a convex rational polyhedral cone by Proposition 6.8, it follows that Ψ Y ( X Y ) is a convex locally rational polyhedral set. If Y is compact, then △ Y is a convexpolytope. It follows that Ψ Y ( X Y ) is a convex polytope since the fibers c − ( λ ) ∩ cone( S ) are compact.(ii) This follows by (i), since Ψ : X → Ψ( X ) is proper (Proposition 6.8) and S ◦ µ : Y → △ Y is proper.(iii) It follows from the definitions that Ψ Y ( U ) = c − ( △ Y ∩ σ ) ∩ Ψ( U σ ) . In particular, Ψ Y ( U ) is convex since it is an intersection of convex sets. It is the smooth locus of c − ( △ Y ∩ σ ) ∩ Ψ( X σ ) since △ Y ∩ σ is smooth and Ψ( U σ ) is the smooth locus of Ψ( X σ ) . Properness of Ψ Y : U → Ψ Y ( U ) follows since S ◦ µ : ( S ◦ µ ) − ( △ Y ∩ σ ) → △ Y ∩ σ is proper and Ψ : U σ → Ψ( U σ ) is proper (Proposition 6.10). Although we cite [57] for a statement of the convexity theorem, it should be noted that convexity for proper mo-ment maps is due to numerous authors including Condevaux, Dazord, Molino, Knop, Birtea, Ratiu, Ortega, Sjamaar,Karshon, and Bjorndahl.
ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 47 (iv) First, we show that the fibers of Ψ Y are connected. Let ξ ∈ Lie( T ) ∗ . Then Ψ − Y ( ξ ) = { ([ y ] , z ) ∈ E Y × X | Ψ( z ) = ξ, µ ([ y ]) = c ( ξ ) } /T = ( µ − ( c ( ξ )) × Ψ − ( ξ )) /T Since ( Y, µ ) is proper, the fibers of µ : Y → k ∗ and thus also µ : E Y → t ∗ + are connected.The fibers of Ψ are connected by Proposition 6.8. Thus, Ψ − Y ( ξ ) is connected since it is aquotient of a product of connected spaces.Fiber connectedness of Ψ Y ◦ Φ is an application of Lemma 6.18. By (i) and (ii), Ψ Y ◦ Φ : Y → Ψ Y ( X Y ) is a continuous proper map to the convex set Ψ Y ( X Y ) . By Proposition6.16, Φ − ( U ) is a dense subset of Y . The set Φ − ( U ) is saturated by Ψ Y ◦ Φ since U issaturated by Ψ Y . By (iii), Ψ Y ( U ) is convex. The fibers of the restriction of Ψ Y ◦ Φ to Φ − ( U ) are connected since the fibers of Ψ Y are connected, U is saturated by Ψ Y , and Φ : Φ − ( U ) → U is a symplectomorphism. Finally, the restricted map Ψ Y ◦ Φ : Φ − ( U ) → Ψ Y ( U ) is open since ( U , Ψ Y ) is proper [18]. (cid:3) Recall that c K ( Y ) denotes the complexity of a Hamiltonian K -manifold Y , defined in (1) and (2). Proposition 6.19 (Properties of toric contraction, part IV) . Let ( Y, µ ) be a Hamiltonian K -manifoldand let ( U , Ψ) be the dense piece of the toric contraction X Y (defined in Proposition 6.16). Then, c T ( U ) = c K ( Y ) . Proof.
Let σ denote the principal stratum of ( Y, µ ) , let S = µ − ( σ ) denote the principal symplecticcross-section, and let T S denote the kernel of the T -action on S . Then, c K ( Y ) = c T ( S ) = 12 dim S − dim T + dim T S . Recall from Proposition 6.16 that U is constructed as the diagonal symplectic reduction of S × U σ by T . Let T σ ⊂ T denote the connected subtorus with ann(Lie( T σ )) = span R ( σ ) . The kernel ofthe diagonal T -action on S × U σ is T σ . In particular, note that T σ ⊂ T S . In fact, the diagonalaction of T /T σ on S × U σ is free. Thus, dim U = dim S + dim U σ − T + 2 dim T σ . The action of T on U descends from the action of T on U σ . The kernel of the action of T on U σ is the subtorus T F , where F = c − ( σ ) is the face of cone( S ) that is the pre-image of σ under theprojection c . By construction, U σ is a complexity 0 Hamiltonian T -manifold, i.e. U σ − dim T + dim T F . By construction, the kernel T ker of the T action on U is the product of the subgroups T F and × T S .The intersection of these subgroups is × T σ , so dim T ker = dim T F + dim T S − dim T σ . Combining these facts, we have that c T ( U ) = 12 dim U − dim T + dim T ker = 12 dim S − dim T + dim T S = c K ( Y ) which completes the proof. (cid:3) Remark . If ( U , Ψ) is a proper complexity 0 T -manifold, then by [44, Proposition 6.5] it isclassified up to isomorphism by its image, the convex locally rational polyhedral set Ψ( U ) = c − ( △ Y ∩ σ ) ∩ Ψ( U σ ) (Proposition 6.17), and the stabilizer subgroup T ker (which is the product ofthe subgroups T F and × T S described in the preceding proof). Proof of Theorem 1.2.
Let D = Φ − ( U ) ⊂ Y . It is an open dense subset of Y (Proposition 6.16).Since it is isomorphic as a symplectic manifold to U (Proposition 6.16), it is equipped with aHamiltonian T -action generated by the moment map Ψ ◦ Φ = F ◦ µ (Proposition 6.14) and c K ( M ) = c T ( D ) (Proposition 6.19). Finally, if ( Y, µ ) is a proper Hamiltonian K -manifold, then ( D, F ◦ µ ) is a proper Hamiltonian T -manifold (Proposition 6.17). (cid:3) Remark . Theorems which guarantee the convexity of the image of amoment map are pervasive in various generalizations of symplectic geometry. In particular, forsingular symplectic spaces, there are convexity theorems for isolated singularities [12], as well assingularities which may arise in symplectic reduction at a regular value of the moment map [68,58, 40].In this section we described the toric contraction of a Hamiltonian K -space. This is a new familyof examples of singular symplectic spaces which come equipped with a Hamiltonian torus actionwhose moment map has a convex image and connected fibers. In this case, convexity and fiberconnectedness of the moment map are immediate consequences of the construction. It is naturalto ask whether there is a category of singular symplectic spaces equipped with Hamiltonian torusactions which includes some or all the aforementioned examples, for which there exists a universalconvexity theorem. A PPENDIX
A. P
ROOFS FOR S ECTION
Proof of Lemmas 3.1, 3.2, and 3.3.
Proof of Lemma 3.1.
Let
X, Y be vector fields on M z . Extend them to vector fields ˜ X, ˜ Y on M bythe condition that ( ϕ t ) ∗ ˜ X = ˜ X and ( ϕ t ) ∗ ˜ Y = ˜ Y . It is sufficient to prove that ( L V π ω )( ˜ X, ˜ Y ) = L V π ( ω ( ˜ X, ˜ Y )) = d ( ω ( ˜ X, ˜ Y ))( V π ) = 0 . First, recall the general identity, ( dω )( ˜ X, ˜ Y , V π ) = d ( ω ( ˜ Y , V π ))( ˜ X ) − d ( ω ( ˜ X, V π ))( ˜ Y ) + d ( ω ( ˜ X, ˜ Y ))( V π ) − ω ([ ˜ X, ˜ Y ] , V π ) + ω ([ ˜ X, V π ] , ˜ Y ) − ω ([ ˜ Y , V π ] , ˜ X ) . Since ω is symplectic, dω = 0 . Also [ ˜ X, V π ] = [ ˜ Y , V π ] = 0 , so this identity can be re-arranged to d ( ω ( ˜ X, ˜ Y ))( V π ) = − d ( ω ( ˜ Y , V π ))( ˜ X ) + d ( ω ( ˜ X, V π ))( ˜ Y ) + ω ([ ˜ X, ˜ Y ] , V π ) . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 49
It remains to show all three terms on the right vanish. For any tangent vector W ∈ ker( π ∗ ) , ω ( W, V π ) = 1 || X ℑ π || ω ( W, X ℑ π )= − || X ℑ π || d ( ℑ π )( W ) = 0 . In particular, ˜ X, ˜ Y , [ ˜ X, ˜ Y ] ∈ ker( π ∗ ) , so the three remaining terms vanish. (cid:3) Proof of Lemma 3.2.
Let ω denote the Kähler form on M and let g denote the Kähler metric. Sincethe action of K preserves the fibers of π : M → C , for all ξ ∈ k , ω ( ξ M , X ℑ π ) = − d ℑ π ( ξ M ) = 0 . Since the action of K is Hamiltonian with moment map ψ : M → k ∗ , for all ξ ∈ k , h dψ ( V π ) , ξ i = ω ( ξ M , V π ) = 1 || X ℑ π || ω ( ξ M , X ℑ π ) = 0 . This completes the proof that the flow of V π preserves the fibers of ψ .For all ξ ∈ k , [ ξ M , V π ] = (cid:20) ξ M , || X ℑ π || X ℑ π (cid:21) = ξ M (cid:18) || X ℑ π || (cid:19) X ℑ π + 1 || X ℑ π || [ ξ M , X ℑ π ]= − ( L ξ M g )( X ℑ π , X ℑ π ) + 2 g ([ ξ M , X ℑ π ] , X ℑ π ) || X ℑ π || X ℑ π + 1 || X ℑ π || [ ξ M , X ℑ π ] . The Lie bracket [ ξ M , X ℑ π ] equals the Hamiltonian vector field of the function ω ( ξ M , X ℑ π ) . Wehave shown above this vanishes on M , so [ ξ M , X ℑ π ] = 0 . By assumption, L ξ M g = 0 . Thus allterms above vanish, and the flow of V π is K -equivariant. (cid:3) Proof of Lemma 3.3.
Since the restricted map π : X \ Z → C is a holomorphic submersion, thegradient Hamiltonian vector field V π is defined everywhere on X \ Z .Let x ∈ U = X \ Z . By assumption (i) and Lemma 3.2, the flow ϕ t ( x ) is contained in ψ − ( ψ ( x )) (for all t such that it is defined). We now recall a slightly modified version of the argument from[37, Lemma 2.7] to show that the flow ϕ t ( x ) is defined for all t ∈ R . We prove that the flow ϕ − t ( x ) is defined for all t > ; the proof for t < is identical. By the fundamental theorem of ODE, theflow ϕ − t ( x ) is defined for all t ∈ [0 , b ) for some b > . Assume for the sake of contradiction thatthe largest such b is finite. By assumption (i) and Lemma 3.2, ϕ − t ( x ) ∈ π − ([0 , b ]) ∩ ψ − ( ψ ( x )) forall t ∈ [0 , b ) . By assumption (ii), the set π − ([0 , b ]) ∩ ψ − ( ψ ( x )) is compact, so there exists a limitpoint x b ∈ X b ∩ ψ − ( ψ ( x )) such that ϕ − t ( x ) → x b as t → b . Since x b is contained in X b ⊂ X \ Z ,the vector field V π is defined at x b . It follows by the fundamental theorem of ODE that ϕ − b ( x ) isdefined and equals x b . This implies that the flow φ t ( x ) is defined for all t ∈ [0 , b ′ ) , for some b ′ > b ,which is a contradiction. Thus, ϕ t ( x ) is defined for all t ∈ R .For the remainder of the proof, fix t = 0 arbitrary. The map ϕ − t : ( U , ω ) → ( X t , ω t ) is symplecticby Lemma 3.1. By Lemma 3.2, ϕ − t is K -equivariant and ψ ◦ ϕ − t = ψ . Thus, ϕ − t : ( U , ω , ψ ) → ( X t , ω t , ψ ) is a map of Hamiltonian K -manifolds. Now, suppose that assumption (iii) holds. Since ϕ − t : ( U , ω ) → ( X t , ω t ) is a symplectomorphismonto its image, it remains to prove that ϕ − t ( U ) is dense in X t .Let c ⊂ c ⊂ . . . be an exhaustion of the manifold X t by compact subsets. Since ψ : X t → k ∗ iscontinuous, the images c ′ m = ψ ( c m ) are compact subsets of k ∗ . By assumption (ii), the sets C m = ( π, ψ ) − ( { t } × c ′ m ) ⊂ X t define an exhaustion of X t by compact subsets. It follows by assumption (iii) that for all m , Z ψ − ( c ′ m ) ∩ U ω n n ! = Z C m ω nt n ! . Since the sets C m ⊂ X t are compact, these volumes are all finite.Since π : X \ Z → C is a submersion, the dimensions of U and X t are the same. Since the map ϕ − t : ( U , ω ) → ( X t , ω t ) is a symplectomorphism onto its image and ψ ◦ ϕ − t = ψ , the volumesare equal: Z ϕ − t ( ψ − ( c ′ m ) ∩ U ) ω nt n ! = Z ψ − ( c ′ m ) ∩ U ω n n ! . Combining the two equalities above, Z ϕ − t ( ψ − ( c ′ m ) ∩ U ) ω nt n ! = Z C m ω nt n ! . Since these volumes are finite, the set ϕ − t ( ψ − ( c ′ m ) ∩ U ) is dense in C m . Since the C m exhaust X t , ϕ − t ( U ) is dense in X t . (cid:3) A.2.
Proof of Theorem 3.6.
Throughout this section, let ( X, M, ω M ) be a decomposed Kählervariety and let π : X → C be a degeneration X that satisfies assumptions (GH1)–(GH6) as inSection 3.3.We begin by recalling and defining some notation. The variety X is decomposed by smooth sub-varieties X σ indexed by elements σ of a poset Σ . For each σ ∈ Σ , the subfamilies X σ , X σ ⊂ X aredefined by the decomposition of X and the trivialization of X away from 0 as in (23). Denote by X z , X σz , and X σz the fiber of π over z ∈ C in X , X σ , and X σ respectively. By definition, X σ ⊂ X σ ⊂ X and X σz ⊂ X σz ⊂ X z for all z . Let Z ⊂ X (respectively Z σ ⊂ X σ and Z σ ⊂ X σ ) denote the unionof the singular locus of X (respectively X σ and X σ ) and the critical set of π viewed as a map withdomain X (respectively X σ and X σ ). Denote U z = X z \ ( X z ∩ Z ) , U σz = X σz \ ( X σz ∩ Z σ ) , and U σz = X σz \ ( X σz ∩ Z σ ) . Lemma A.1.
For all σ ∈ Σ , Z σ is contained in X σ . Moreover, U σ is the smooth locus of X σ .Proof. The fact that Z σ is contained in X σ is a consequence of (GH2) and smoothness of X σ . By(23), X σ is an open subset of X σ . Thus, Z σ = Z σ ∩ X σ and U σ = U σ ∩ X σ . By assumption (GH1)and Proposition 3.4, U σ is the smooth locus of X σ . Since X σ is an open subset of X σ , it followsthat U σ is the smooth locus of X σ . (cid:3) Lemma A.2.
The following statements are true for all σ ∈ Σ .(a) The flow ϕ σ − is defined for all x ∈ U σ .(b) The map ϕ σ − : ( U σ , ω σ , ψ ) → ( X σ , ω σ , ψ ) is a map of Hamiltonian T -manifolds.(c) The set D σ = ϕ σ − ( U σ ) is dense in X σ .(d) The map ϕ σ − : ( U σ , ω σ ) → ( X σ , ω σ ) is a symplectomorphism onto its image. ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 51
Proof.
Fix σ ∈ Σ arbitrary. The proof is a direct application of Lemma 3.3 to the subfamily π : X σ → C . We have Z σ ⊂ X σ by Lemma A.1. The set U σ is non-empty by (GH5). The smoothsubvariety U σ inherits a Kähler structure from the embedding into M × C given by assumption(GH2). Assumptions (i)–(iii) of Lemma 3.3 are precisely assumptions (GH4) and (GH5). (cid:3) Lemma A.3.
The map ϕ t : X → X − t is continuous, for all < t < . Before proving Lemma A.3, we note the following elementary lemma.
Lemma A.4.
Let X and Y be metric spaces, let f : X → Y a be map of the underlying sets, and let x ∈ X . Assume that for every sequence { x i } i ∈ N ⊂ X with lim i →∞ x i = x , there is a subsequence { x i j } j ∈ N with the property that lim j →∞ f ( x i j ) = f ( x ) . Then, f is continuous at x .Proof of Lemma A.3. Fix σ ∈ Σ , x ∈ X σ , and T ∈ (0 , arbitrary. We prove that ϕ T : X → X − T is continuous at x . (Note that ϕ t ( x ) is defined for all x ∈ X and t ∈ (0 , by the same argumentas in the proof of Lemma 3.3.)Let { x i } i ∈ N ⊂ X be an arbitrary sequence converging to x . By Lemma A.4, it suffices to find asubsequence { x i j } j ∈ N so that { ϕ T ( x i j ) } j ∈ N converges to ϕ T ( x ) . By passing to a subsequence ifnecessary, we may assume without loss of generality that { x i } i ∈ N ⊂ X τ for some τ ≥ σ . (If τ = σ then the result follows immediately since the restriction of V π to X σ \ X σ is smooth. The remainderof the proof deals with the case τ > σ .)Consider the sequence of paths { ϕ t ( x i ) : [0 , T ] → X τ } i ∈ N . Since ϕ t preserves ψ and x i convergesto x , there is a compact set c ⊂ ψ ( X ) so that ϕ t ( x i ) ∈ ψ − ( c ) ∩ π − ([0 , T ]) for all t ∈ [0 , T ] and all i ∈ N . By assumption (GH4)II), ψ − ( c ) ∩ π − ([0 , T ]) is compact. By a standard diagonalizationargument, by replacing { x i } i ∈ N with a subsequence, we may assume that for each t ∈ Q ∩ [0 , T ] the sequence of points { ϕ t ( x i ) } i ∈ N converges as i → ∞ .We will show below that the sequence of time derivatives(69) dϕ t ( x i ) dt = V π ( ϕ t ( x i )) , i ∈ N , converges uniformly on Q ∩ [0 , T ] . Assuming this to be true for the moment, because Q ∩ [0 , T ] is dense in [0 , T ] it follows that the sequence { V π ( ϕ t ( x i )) } i ∈ N converges uniformly on [0 , T ] . As aconsequence, the paths ϕ t ( x i ) converge uniformly to a C path µ : [0 , T ] → X . For all t ′ ∈ [0 , T ] ,V π ( µ ( t ′ )) = V π ( lim i →∞ ϕ t ′ ( x i )) ϕ t ( x i ) converges to µ ( t ) . = lim i →∞ V π ( ϕ t ′ ( x i )) Assumption (GH6) = lim i →∞ (cid:18) ddt ϕ t ( x i ) (cid:12)(cid:12)(cid:12) t = t ′ (cid:19) Definition of ϕ t . = ddt (cid:16) lim i →∞ ϕ t ( x i ) (cid:17) (cid:12)(cid:12)(cid:12) t = t ′ Uniform convergence of derivatives. = ddt µ ( t ) (cid:12)(cid:12)(cid:12) t = t ′ ϕ t ( x i ) converges to µ ( t ) .Since ϕ t preserves ψ ,(70) ψ ( µ ( t )) = ψ (cid:16) lim i →∞ ϕ t ( x i ) (cid:17) = lim i →∞ ψ ( ϕ t ( x i )) = lim i →∞ ψ ( x ) = ψ ( x ) . Thus, µ ([0 , T ]) is contained in ψ − ( ψ ( x )) . It follows by assumption (GH4)III) that µ ([0 , T ]) iscontained in X σ .In summary, the path µ ( t ) solves the same initial value problem on X σ \ X σ , defined by the smoothvector field V σπ and the initial value µ (0) = x , as the integral curve ϕ σt ( x ) . It follows by uniquenessof solutions that lim i →∞ ϕ t ( x i ) = µ ( t ) = ϕ t ( x ) for all t ∈ [0 , T ] . In particular, this holds for t = T , which completes the proof (modulo the claimthat (69) converges uniformly).It remains to show that (69) converges uniformly on Q ∩ [0 , T ] . For each t ∈ Q ∩ [0 , T ] let µ ( t ) =lim i →∞ ϕ t ( x i ) ; we have already established this limit exists. Assume for the sake of contradictionthat (69) does not converge uniformly to V π ( µ ) , as a function of t ∈ Q ∩ [0 , T ] . Then there is some γ > so that, for all N > , there is i ≥ N and t i ∈ Q ∩ [0 , T ] with(71) γ < || V π ( ϕ t i ( x i )) − V π ( µ ( t i )) || . By passing to a subsequence, we may assume that the sequence { ( x i , t i ) } i ∈ N satisfies (71) for all i ∈ N . By compactness of [0 , T ] , we may also assume that lim i →∞ t i = t ⋆ for some t ⋆ ∈ [0 , T ] . Similarly, by compactness of ψ − ( c ) ∩ π − ( t ⋆ ) (and the same argumentas (70)), we may additionally assume that lim i →∞ ϕ t ⋆ ( x i ) = y for some y ∈ X σ . We first prove three preliminary claims. Claim 1: µ has a unique continuous extension to [0 , T ] . Proof of Claim 1:
Let { s n } n ∈ N ∈ Q ∩ [0 , T ] be a sequence converging to some s ∈ [0 , T ] . Assumethe sequence { µ ( s n ) } n ∈ N is not Cauchy, then there is ǫ > so that for all N > there exist m, n ∈ N with || µ ( s n ) − µ ( s m ) || > ǫ . Since lim n →∞ s n = s , for any L > we find n, m so that | s n − s m | < /L and || µ ( s n ) − µ ( s m ) || > ǫ . For any ǫ ′ > we can pick i ∈ N sufficiently largethat || ϕ s n ( x i ) − µ ( s n ) || < ǫ ′ / , and || ϕ s m ( x i ) − µ ( s m ) || < ǫ ′ / . Then || ϕ s n ( x i ) − ϕ s m ( x i ) || > ǫ − ǫ ′ . By the mean value inequality applied to the path ϕ s ( x i ) , there is s ′ between s m and s n so that || V π ( ϕ s ′ ( x i )) || ≥ || ϕ s n ( x i ) − ϕ s m ( x i ) ||| s n − s m | > L ( ǫ − ǫ ′ ) . Since L and ǫ ′ were arbitrary, this implies that || V π || is unbounded on the compact set ψ − ( c ) ∩ π − ([0 , T ]) . By the assumption (GH6), this is a contradiction. Therefore lim n →∞ µ ( s n ) exists.This extension is also unique; if s n → s and s ′ n → s are two sequences with lim n →∞ µ ( s n ) =lim n →∞ µ ( s ′ n ) , then the sequence µ ( s ) , µ ( s ′ ) , µ ( s ) , µ ( s ′ ) , . . . has no limit, contradicting whatwe have shown above. This proves Claim 1. Claim 2: lim i →∞ ϕ t ⋆ ( x i ) = µ ( t ⋆ ) . ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 53
Proof of Claim 2:
Assume not, and write lim i →∞ ϕ t ⋆ ( x i ) = y as before; then || y − µ ( t ⋆ ) || = ǫ > .Let L > , and pick t ′ ∈ Q ∩ [0 , T ] with | t ′ − t ⋆ | < /L . For any ǫ ′ > , we may choose i ∈ N with || ϕ t ⋆ ( x i ) − y || < ǫ ′ / , and || ϕ t ′ ( x i ) − µ ( t ′ ) || < ǫ ′ / . Then || ϕ t ⋆ ( x i ) − ϕ t ′ ( x i ) || > ǫ − ǫ ′ By the mean value inequality applied to the path ϕ t ( x i ) , there is t ′′ between t ′ and t ⋆ so that || V π ( ϕ t ′′ ( x i )) || ≥ || ϕ t ⋆ ( x i ) − ϕ t ′ ( x i ) ||| t ⋆ − t ′ | > L ( ǫ − ǫ ′ ) . Since L and ǫ ′ were arbitrary, this implies that || V π || is unbounded on the compact set ψ − ( c ) ∩ π − ([0 , T ]) . By the assumption (GH6), this is a contradiction. Thus lim i →∞ ϕ t ⋆ ( x i ) = y = µ ( t ⋆ ) ,which establishes Claim 2. Claim 3: lim i →∞ || ϕ t i ( x i ) − ϕ t ⋆ ( x i ) || = 0 . Proof of Claim 3:
Assume not, then there is some ǫ > so that for all N there exists i > N with || ϕ t i ( x i ) − ϕ t ⋆ ( x i ) || > ǫ . Let L > , and pick N sufficiently large that | t i − t ⋆ | < /L for all i > N . Fix i > N so that || ϕ t i ( x i ) − ϕ t ⋆ ( x i ) || > ǫ . By the mean value inequality applied to thepath ϕ t ( x i ) , there is some t ′ between t ⋆ and t i so that || V π ( ϕ t ′ ( x i )) || ≥ || ϕ t i ( x i ) − ϕ t ⋆ ( x i ) ||| t i − t ⋆ | > Lǫ. Since ǫ was fixed and L arbitrary, this implies that || V π || is unbounded on the compact set ψ − ( c ) ∩ π − ([0 , T ]) . By the assumption (GH6), this is a contradiction. This proves Claim 3.Now, we may complete the proof that (69) converges uniformly on Q ∩ [0 , T ] . One has for all i ∈ N , || ϕ t i ( x i ) − µ ( t ⋆ ) || ≤ || ϕ t i ( x i ) − ϕ t ⋆ ( x i ) || + || ϕ t ⋆ ( x i ) − µ ( t ⋆ ) || . By Claim 2 and Claim 3, both terms on the right hand side go to zero as i → ∞ . By the assump-tion (GH6), it follows that(72) lim i →∞ V π ( ϕ t i ( x i )) = V π ( µ ( t ⋆ )) . Similarly, by Claim 1 and assumption (GH6), one has(73) lim i →∞ V π ( µ ( t i )) = V π ( µ ( t ⋆ )) . At the same time, by (71) < γ < || V π ( ϕ t i ( x i )) − V π ( µ ( t i )) || ≤ || V π ( ϕ t i ( x i )) − V π ( µ ( t ⋆ )) || + || V π ( µ ( t ⋆ ) − V π ( µ ( t i )) || for all i . But by (72) and (73), the right hand side goes to zero as i → ∞ . This is a contradiction.Therefore, the sequence of derivatives (69) converges uniformly on Q ∩ [0 , T ] , as desired. (cid:3) Lemma A.5.
For all x ∈ X , the limit lim t → − ϕ t ( x ) exists. For any open precompact subset A ⊂ ψ ( M ) and ǫ > , there exists ρ > such that for all < t < ρ and x ∈ ψ − ( A ) ∩ X , || ϕ t ( x ) − lim t → − ϕ t ( x ) || < ǫ. A subset of a topological space is precompact if its closure is compact.
The proof of Lemma A.5 closely follows the outline of the proof of [37, Theorem 2.12]. It relieson the following gradient-inequality theorem, which we quote from [37].
Theorem A.6. [54]
Let X be an algebraic subset of a finite dimensional real inner product space E and let f : E → R be a semi-algebraic function. Then for any x ∈ X there is an open neighborhood Y x ⊂ X (in the analytic topology) and constants c x > and < α x < such that for any y ∈ Y x ∩ X sm , (74) ||∇ f ( y ) || ≥ c x | f ( y ) − f ( x ) | α x , where ∇ f denotes the gradient of f | X with respect to the induced metric on X sm .Proof of Lemma A.5. Let A an open precompact subset of ψ ( M ) and denote its closure by A . Byassumption (GH4), ψ − ( A ) ∩ ( M × { } ) is compact. It follows that ψ − ( A ) ∩ X σ is compact forany σ ∈ Σ .Let σ ∈ Σ and let x ∈ X σ . We want to apply Theorem A.6 to the function ℜ π and the point x . Tothis end, we may assume without loss of generality that M is a quasi-affine variety, embedded in acomplex vector space E . Near x ∈ M , we may extend the Kähler metric on M to a Riemannianmetric on E . Following [37, Remark 2.16], an inequality of the form (74) holds near x for anyRiemannian metric placed on E (with a possibly different value of c x ). We conclude that for each x ∈ X σ , there exists an open neighborhood Y σx ⊂ X σ of x along with constants c σx and α σx thatsatisfy the gradient inequality (74).The set ψ − ( A ) ∩ X σ is compact, and so it is possible to choose finitely many points x , . . . , x k σ ∈ ψ − ( A ) ∩ X σ such that the neighborhoods Y σx , . . . , Y σx kσ ⊂ X σ form an open cover of ψ − ( A ) ∩ X σ .For each σ ∈ Σ , choose > r σ > so thatif: y ∈ ψ − ( A ) ∩ X σ ∩ { y ∈ X | | π ( y ) | < r σ } , then: y ∈ k σ [ i =1 Y σx i . It is possible to choose r σ > because ψ − ( A ) ∩ X σ is compact.Let r = min σ ∈ Σ { r σ } . Define Y = ψ − ( A ) ∩ { y ∈ X | | π ( y ) | < r } ; c = min σ ∈ Σ min ≤ i ≤ k σ { c σx i } ; α = max σ ∈ Σ max ≤ i ≤ k σ { α σx i } . Then Y is an open subset of X that contains ψ − ( A ) ∩ X . For any σ ∈ Σ , if y ∈ Y ∩ ( X σ ) sm , then y ∈ S k σ i =1 Y σx i by construction of Y . As a consequence,(75) || V σπ ( y ) || ≤ c |ℜ π ( y ) | − α . Finally, let x ∈ X σ ∩ ψ − ( A ) . For all t , t such that − r < t < t < , || ϕ σt ( x ) − ϕ σt ( x ) || ≤ Z t t || V σπ ( ϕ σt ( x )) || dt. By (75), || ϕ σt ( x ) − ϕ σt ( x ) || ≤ Z t t c |ℜ π ( ϕ σt ( y )) | − α dt = Z t t c | − t | − α dt. ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 55
Therefore, || ϕ σt ( x ) − ϕ σt ( x ) || ≤ c (1 − α ) ((1 − t ) − α − (1 − t ) − α ) . Since α > , lim t → − (1 − t ) − α = 0 . Both claims then follow easily. (cid:3) Finally, we give the proof of Theorem 3.6. The outline of this proof follows the outline of the proofof [37, Theorem 2.12] very closely. Although the outline is the same, one must be careful to notethat we are working with a stratified flow.
Proof of Theorem 3.6.
For each x ∈ X , define φ ( x ) = lim t → − ϕ t ( x ) . Since these limits exist and are elements of X (Lemma A.5), this defines a map φ : X → X .For each σ ∈ Σ , let D σ = ϕ σ − ( U σ ) (the flow ϕ σ − is defined at points in U σ by Lemma A.2). ByLemma A.2, D σ is dense in X σ and ϕ σ : D σ → U σ is a symplectomorphism. In particular, for all x ∈ D σ , φ ( x ) = lim t → − ϕ t ( x ) = lim t → − ϕ σt ( x ) = ϕ σ ( x ) . Thus, the restriction of φ to D σ coincides with ϕ σ .We claim that φ is continuous. Fix some open precompact subset A ⊂ ψ ( M ) and ǫ > . ByLemma A.5, there exists > t > such that for all x ∈ ψ − ( A ) ∩ X , || ϕ t ( y ) − φ ( y ) || < ǫ/ . By Lemma A.3, there exists δ > such that for any x, y ∈ X , if || x − y || < δ , then || ϕ t ( x ) − ϕ t ( y ) || < ǫ/ . Combining these inequalities, we have that for all x, y ∈ µ − ( A ) ∩ X such that || x − y || < δ , || φ ( x ) − φ ( y ) || ≤ || φ ( x ) − ϕ t ( x ) || + || ϕ t ( x ) − ϕ t ( y ) || + || ϕ t ( y ) − φ ( y ) || < ǫ. Thus φ is continuous.By Lemma A.2, the maps ϕ σ : D σ → U σ are T -equivariant and satisfy ψ ◦ ϕ σ = ψ for all σ ∈ Σ .It follows that φ is T -equivariant and satisfies ψ ◦ φ = φ since φ is continuous, and for each σ ∈ Σ it coincides with ϕ σ on the dense subset D σ ⊂ X σ .Let c be a compact subset of X . By assumption (GH4)II), there exists a compact subset c ′ ⊂ t ∗ such that c is contained in ( π, ψ ) − ( { } × c ′ ) . Since ψ ◦ φ = ψ , the pre-image φ − ( c ) is containedin ( π, ψ ) − ( { } × c ′ ) . Since φ − ( c ) is a closed subset of a compact set, it is compact. Thus φ isproper.Let x ∈ X σ for some arbitrary σ . Since U σ is dense in X σ (Lemma A.1) we can find a sequence x i ⊂ U σ such that x i → x as i → ∞ . Since φ is proper, there is a compact subset c ⊂ X suchthat φ − ( x i ) ∈ c for all i ∈ N . Thus, there exists a subsequence x i k and a point y ∈ c such that φ − ( x i k ) → y as k → ∞ . It follows by continuity of φ that φ ( y ) = x . Thus φ : X → X issurjective. (cid:3) A PPENDIX
B. P
ROOFS FOR S ECTION
Proof of Proposition 5.3.
Before giving the proof of Proposition 5.3, we recall some notionsfrom [48, 37, 8] that will also be useful in the next section.Fix λ ∈ Λ + . Let χ λ denote the associated character on B and let L λ = G × B C denote theassociated line bundle over G/B , where ( g, z ) ∼ ( gb, χ λ ( b ) z ) . Holomorphic sections of L λ canbe written in the form σ f ( g ) = ( g, f ( g )) , where f ∈ C [ G ] such that f ( gb ) = χ λ ( b ) f ( g ) . Therepresentation of G on the space of holomorphic sections H ( G/B ; L λ ) is given explicitly by g · σ f = σ g · f where ( g · f )( g ′ ) = f ( g − g ′ ) . As a G -module, H ( G/B ; L λ ) is isomorphic to thehighest weight module V ( λ ) ∗ . Fix τ = σ ˆ f to be the lowest weight vector with ˆ f ( e ) = 1 . Let R λ = L k ≥ H ( G/B ; L ⊗ kλ ) denote the homogeneous coordinate ring of G/B associated to L λ .The valuation ν : C ( G/B ) \ { } → Z m can be used to define a new valuation(76) e ν : R λ \ { } → Z m × N , f ( ν ( f k /τ k ) , k ) where f k is the highest degree homogeneous part of f , and the degree of f k is k . Let N h λ i ⊂ Λ + denote the semigroup generated by λ . Lemma B.1.
The bijection Z m × N → Z m × N h λ i , ( ϕ, k ) ( ϕ, kλ ) restricts to a bijection from S e ν onto S v ∩ ( Z m × N h λ i ) .Proof. Let f be an element of C [ G ] × N which is a × H -weight vector of weight λ . Then j ∗ f = f | N − ⊗ χ λ , so v ( f ) = ( ν | N − ( f | N − ) , λ ) . On the other hand, every element of H ( G/B ; L λ ) equals σ f for some f of this form. Since τ is non-vanishing on N − B/B , the rational function σ f /τ restricts to an element of C [ N − B/B ] ⊂ C ( G/B ) . Let i : N − → G/B be the inclusion n − n − B/B . We have that i ∗ σ f τ ( n − ) = σ f ( n − B ) τ ( n − B ) = f ( n − )ˆ f ( e ) = f | N − ( n − ) . Recalling the definition of ν | N − from (46), we have ν ( σ f /τ ) = ν | N − (cid:16) i ∗ σ f τ (cid:17) = ν | N − ( f | N − ) so e ν ( σ f ) = ( ν ( σ f /τ ) ,
1) = ( ν | N − ( f | N − ) , . Thus, the map ( ϕ, k ) ( ϕ, kλ ) defines a bijection from S e ν ∩ ( Z m × { } ) onto S v ∩ ( Z m × { λ } ) .Since L ⊗ kλ ∼ = L kλ , a similar argument shows that the map defines a bijection from S e ν ∩ ( Z m × { k } ) onto S v ∩ ( Z m × { kλ } ) for all k > . (cid:3) The
Newton-Okounkov body associated to λ and ν is the bounded convex set(77) △ ( R λ , ν ) := cone( S e ν ) ∩ ( R m × { } ) . This polytope is often identified with its projection to the subspace R m . Proof of Proposition 5.3.
The valuation v and the map c satisfy (v1) by definition of the total orderson Z m × Λ and Λ . They satisfy (v2) since c ( S v ) = Λ + and G is semisimple. They satisfy (v3) byLemma B.1 and since △ ( R λ , ν ) is bounded for all λ . They satisfy (v5) by the same argument as in[7, Proposition 2.2] which was adapted from [13, 3.2]. ANONICAL BASES AND COLLECTIVE INTEGRABLE SYSTEMS 57
Finally, we show that the valuation v and the maps c and a ′ − c satisfy (v4) with respect to theactions of × H and H × respectively. Let H × H act on N − × H according to ( h , h ) · ( n, h ) =( h nh − , h hh − ) . Then the embedding j constructed in (47) is H × H -equivariant.Suppose z ∈ C [ G (cid:12) N ] is Λ × Λ -homogeneous of degree ( γ, λ ) . Then z is contained in V ( λ ) ∗ ⊂ C [ G ] N . It follows that j ∗ z = f ⊗ χ λ for some f ∈ C [ N − ] that is Λ -homogeneous of degree γ + λ (with respect to the conjugation action of H × on N − ): ( h · f )( n ) = z ( h − nh · v ( λ )) = h λ z ( h − n · v ( λ )) = h γ + λ z ( n · v ( λ )) . Thus, c ( v ( z )) = c ( ν | N − ( f ) , λ ) = λ and a ( v ( z )) = a ′ ( ν | N − ( f )) − c ( ν | N − ( f ) , λ ) = γ. (cid:3) B.2.
Proof of Proposition 5.6.
Before giving the proof of Proposition 5.6, we recall some de-tails about the Duistermaat-Heckman measures of the two spaces. Throughout this section we usenotation established in Section 6.Fix σ ∈ Σ . Canonically identify span R ( σ ) ⊂ t ∗ with Lie(
T /T σ ) ∗ . Let dm denote the Lebesguemeasure on span R ( σ ) determined by the lattice Λ ∩ span R ( σ ) . Since T σ is connected, this isidentified with the weight lattice of T /T σ . Note that T σ = T ∩ [ K σ , K σ ] . Duistermaat-Heckman measure of ( X σ , ω σ , ψ ) : Denote the Duistermaat-Heckman measure of ( X σ , ω σ , ψ ) by ν . First, recall that we have fixed an isomorphism X ∼ = G (cid:12) N . By Theorem 6.4there is an isomorphism of Hamiltonian T -manifolds ( X σ , ω σ , ψ ) ∼ = ( K/ [ K σ , K σ ] × σ, ω σ , − µ R = S ◦ µ L ) . The torus
T /T σ acts freely on K/ [ K σ , K σ ] × σ . The moment map − µ R is proper as a map to σ andevery λ ∈ σ is a regular value. It follows by the Duistermaat-Heckman theorem that ν = f ( λ ) dm ,where f ( λ ) is the symplectic volume of the symplectic reduced space ( K/ [ K σ , K σ ] × σ ) (cid:12) λ T . By(53), ( K/ [ K σ , K σ ] × σ ) (cid:12) λ T is symplectomorphic to the coadjoint orbit O λ ⊂ k ∗ parameterizedby λ , equipped with its natural Kostant-Kirillov-Souriau symplectic form ω λ . Thus, for all λ ∈ σ , f ( λ ) = Vol( O λ , ω λ ) . Let k = dim R ( K/K σ ) = dim R ( O λ ) . The function f is continuous and has the property that forall α > , f ( αλ ) = Vol( O αλ , ω αλ ) = Vol( O λ , αω λ ) = α k f ( λ ) . Duistermaat-Heckman measure of ( U σ , ω σ , ψ ) : Denote the Duistermaat-Heckman measure of ( U σ , ω σ , ψ ) by ν .The intersection F = c − ( σ ) ∩ cone( S ) is a closed face of cone( S ) . The action of T on U σ is acomplexity 0 action with T ker = T F . Canonically identify span R ( F ) ⊂ Lie( T ) ∗ with Lie( T / T F ) ∗ .Let dM denote the Lebesgue measure on span R ( F ) determined by the lattice ( Z m × Λ) ∩ span R ( F ) .Since T F is connected, this is identified with the weight lattice of T / T F . By Corollary 4.20, Ψ( U σ ) is the smooth locus of the convex, locally rational polyhedral set Ψ( X σ ) ⊂ F . In particular, Ψ( U σ ) is convex. It also follows by Corollary 4.20 that the restricted map Ψ : U σ → Ψ( U σ ) is proper. Itfollows by the classification of proper complexity 0 torus manifolds [44, Proposition 6.5] that theDuistermaat-Heckman measure of ( U σ , ω σ , Ψ) is χdM , where χ is the support function of Ψ( U σ ) . Recall that T = ( S ) m × T . Also recall that we have identified Lie(( S ) m ) ∗ ∼ = R m so that the weightlattice is identified with Z m . Identify (( S ) m × ∩ T F with a subgroup J ⊂ ( S ) m . The projectionof span R F to R m is canonically identified with Lie(( S ) m /J ) ∗ . The weight lattice of ( S ) m /J isidentified with pr R m (span R F ) ∩ Z m . Let dx denote the Lebesgue measure on Lie(( S ) m /J ) ∗ determined by this lattice.The decomposition Lie( T ) ∗ = R m × t ∗ restricts to an isomorphism of subspaces Lie( T / T F ) ∗ ∼ = Lie(( S ) m /J ) ∗ × Lie(
T /T σ ) ∗ . This isomorphism identifies dM with the product measure dx × dm . By Tonelli’s theorem, theDuistermaat-Heckman measure of ( U σ , ω σ , ψ c = c ◦ Ψ) equals g ( λ ) dm where g ( λ ) := Vol(pr R m ( c − ( λ ) ∩ F ) , dx ) . This function is continuous. For all α > , pr R m ( c − ( αλ ) ∩ F ) = α pr R m ( c − ( λ ) ∩ F ) . Since dim(( S ) m /J ) = k , g ( αλ ) = α k g ( λ ) . Lemma B.2.
For all λ ∈ σ ∩ Λ , g ( λ ) = f ( λ ) .Proof. Fix λ ∈ σ ∩ Λ . Recall that P σ denotes the parabolic subgroup of G with Lie algebra (39).Fix the embedding of G/P σ into P ( V ( λ )) associated to the line bundle L λ . Let ω F S,λ denote therestriction to
G/P σ of the Fubini-Study symplectic form on P ( V ( λ )) . As symplectic manifolds, ( O λ , ω λ ) ∼ = ( G/P λ , ω F S,λ ) . It follows by [37, Theorem B] that f ( λ ) = Vol( O λ , ω λ ) = Vol( G/P λ , ω F S,λ ) = Vol( △ ( R λ , ν ) , dx ) , where △ ( R λ , ν ) is the Newton-Okounkov body associated to λ and ν . It follows from the discus-sion in the previous section that pr R m ( c − ( λ ) ∩ F ) = △ ( R λ , ν ) . Thus,
Vol( △ ( R λ , ν ) , dx ) = g ( λ ) which completes the proof. (cid:3) Proof of Proposition 5.6.
By the discussion above, it suffices to show that f ( λ ) = g ( λ ) for all λ ∈ σ . By the previous lemma, f ( λ ) = g ( λ ) for all λ ∈ σ ∩ Λ . The result follows by continuity of Vol( O λ , ω λ ) as a function of λ , and the scaling property shared by f and g . (cid:3) Remark
B.3 . If v is constructed from a string valuation as in Example 5.8, then the equality Vol(
G/P λ , ω F S,λ ) = Vol( △ ( R λ , ν )) used in the proof of Lemma B.2 can be deduced from proper-ties of canonical bases and string polytopes rather than [37, Theorem B].R EFERENCES
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