Lax Formalism for Gelfand-Tsetlin Integrable Systems

Lax formalism for Gelfand-Tsetlin integrable systems

Eder M. Correa*1 and Lino Grama†2

1UFMG, Avenida Antonioˆ Carlos, 6627, 31270-901 Belo Horizonte - MG, Brazil. 2IMECC-Unicamp, Departamento de Matematica.´ Rua Sergio´ Buarque de Holanda, 651, Cidade Universitaria´ Zeferino Vaz. 13083-859, Campinas - SP, Brazil.

Abstract In the present work, we study Hamiltonian systems on (co)adjoint orbits and propose a Lax pair formalism for Gelfand-Tsetlin integrable systems deﬁned on (co)adjoint orbits of the compact Lie groups U(n) and SO(n). In the particular setting of (co)adjoint orbits of U(n), by means of the associated Lax matrix we construct a family of algebraic curves which encodes the Gelfand-Tsetlin integrable systems as branch points. This family of algebraic curves enables us to explore some new insights into the relationship between the topology of singular Gelfand-Tsetlin ﬁbers, singular algebraic curves and vanishing cycles. Further, we provide a new description for Guillemin and Sternberg’s action coordinates in terms of hyperelliptic integrals.

Contents

1 Introduction 2 1.1 Main results ...... 2

2 Collective Hamiltonians and Gelfand-Tsetlin integrable systems 4 2.1 Collective Hamiltonian systems ...... 4 2.2 Thimm’s trick and Gelfand-Tsetlin integrable systems ...... 6

3 Lax pairs and spectral equation 7

4 A Lax pair formalism for Gelfand-Tsetlin integrable systems 9 4.1 Lax equation and collective Hamiltonians ...... 9 4.2 Thimm’s trick and spectral invariants ...... 11 4.2.1 Proof of Theorem 1 ...... 14 arXiv:1810.08848v3 [math.SG] 20 May 2021 5 Gelfand-Tsetlin integrable systems and algebraic curves 16 5.1 Gelfand-Tsetlin integrable systems on U(n)-adjoint orbits ...... 16 5.2 Generalities on Abelian integrals and algebraic curves ...... 18 5.2.1 Proof of Theorem 2 ...... 20 5.2.2 Examples and ﬁnal comments ...... 23

*Eder M. Correa was supported by CNPq grant 150899/2017-3. †Lino Grama is partially supported by FAPESP grant 2018/13481-0, and by CNPq grant 2019/305036-0.

1 1 Introduction

The notion of Lax pair is a new emergent language used in the study of integrability, and one of the most important features of this concept is its relation with the classical r-matrix, see [12]. The classical r-matrix was introduced in late 1970’s by Sklyanin [52], as a part of a vast research program launched by L. D. Faddeev, which culminated in the discovery of the Quantum Inverse Scattering Method and of Quantum Groups, e.g. [17]. A Lax pair for a Hamiltonian system (M,ω,H), consists of two matrix-valued functions (L,P) on the phase space (M,ω) of the system, such that the Hamiltonian evolution equation of motion associated to a Hamiltonian H ∈ C∞(M) can be written as a “zero curvature equation”, also known as Lax equation, see for instance [39] and [59]. For more details about the relationship between the notion of zero curvature equation and the Lax equation, see for instance [26, Chapter 9]. In a broad sense, in which one allows the Lax pairs depending on a complex parameter ζ, called the spectral parameter, i.e., L = L(ζ) and P = P(ζ), through the notion of spectral curves1 we have a very rich interplay between integrable Hamiltonian systems and algebraic curves. Interesting examples in this setting include periodic Toda lattices based on simple Lie algebras, the Neumann problem of the motion of a point on the sphere under a linear force, the free motion of a point on an ellipsoid, the Euler, Lagrange, and Kovalevskaya tops, and other integrable systems, see for instance [25], [1], [43], [8], [7], and references therein. In this paper, we deal with the problem related to the formulation of the Gelfand-Tsetlin integrable systems on coadjoint orbits in terms of Lax pairs. The Gelfand-Tsetlin system was introduced in the 1980’s by Guillemin-Sternberg [29]. They showed that for coadjoint orbits of the compact unitary Lie group U(n) the set of Poisson-commuting functions provided by a suitable application of Thimm’s trick [55] deﬁnes a completely integrable system. The key point for the integrability of the Gelfand-Tsetlin systems is that the coadjoint action of U(n − 1) on a coadjoint orbit of U(n) is coisotropic, see for instance [28]. This last property also holds for coadjoint orbits of SO(n) (see also [31]) and, recently, a modern proof for the integrability of Gelfand-Tsetlin systems on coadjoint orbits of U(n) and SO(n) was given in [47]. A remarkable feature of the Gelfand-Tsetlin systems is their relation with representation theory of Lie groups, Lie algebras, and geometric quantization, see for instance [27]. Also, this class of integrable systems has been playing an important role as a concrete model for the study of Lagrangian Floer theory on non-toric manifolds, e.g. [45], [46], [58]. As mentioned in [35], the algebraic geometric framework based on the notion of a Lax representation has proved to be very powerful not only for constructing new examples and explicit integration, but also for studying topological properties of integrable systems (e.g., [5], [6]). Motivated by these ideas and following the results provided in [58] and [11], we also explore some new insights in the study of the relationship between the topology of singular Gelfand-Tsetlin ﬁbers, singular algebraic curves and vanishing cycles.

1.1 Main results Our first result is concerned with the Lax pair formalism for the Gelfand-Tsetlin systems. The main contribution of this result is to provide a canonical and concrete way to assign Lax pairs to Gelfand-Tsetlin integrable systems for (co)adjoint orbits of U(n) and SO(n). In order to state our first result, let us introduce some basic concepts. Let G be a compact connected Lie group with Lie algebra g. For every adjoint orbit O(Λ) ⊂ g, let (O(Λ),ωO(Λ),G,Φ) be the (canonical) associated Hamiltonian G-space, where ωO(Λ) is the Kirillov-Kostant- Souriau symplectic form, and Φ: (O(Λ),ωO(Λ)) → g is the moment map defined by the natural Hamiltonian action of G on (O(Λ),ωO(Λ)). In this setting, we prove the following result:

1As mentioned in [13], the notion of spectral curves arose historically out of the study of differential equations of Lax type. Following Hitchin’s work [32], they have acquired a central role in understanding the moduli spaces of vector bundles and Higgs bundles on a curve.

2 Theorem 1. Given a Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ), where G is either U(n) or SO(n), then there ∞ exist H ∈ C (O(Λ)), and a pair of matrix-valued functions L,P: (O(Λ),ωO(Λ)) → gl(r,R), satisfying the Lax equation d L(ϕ (Z)) + L(ϕ (Z)),P(ϕ (Z)) = 0, (1.1) dt t t t for all Z ∈ O(Λ), where ϕt (Z) denotes the ﬂow of the Hamiltonian vector ﬁeld XH through Z ∈ O(Λ). Moreover, the solutions of the spectral equation det w1r − L = 0, (1.2) provide a maximal set of conserved quantities in involution for the Hamiltonian system (O(Λ),ωO(Λ),H) which coincides with the Gelfand-Tsetlin integrable system on O(Λ).

The result above establish a description for Gelfand-Tsetlin integrable systems using only techniques in- volving Lax matrices. The ideas developed in our construction are quite natural and generalize in a suitable sense some results introduced in [23] for regular (co)adjoint orbits of U(n), see Remark 4.8. As pointed out in [58], the Gelfand-Tsetlin system resembles a toric moment map in the sense that its image is a convex polytope ∆Λ and the fiber over every interior point of ∆Λ is a Lagrangian torus. The notable difference is that non-torus Lagrangian fibers may appear at some boundary strata of ∆Λ. In the general setting, as observed in [10], the classical Arnold-Liouville theorem completely describes the Liouville foliation defined by an integrable system in a neighborhood of its regular leaves, but says almost nothing about its structure near singular leaves. In some sense, all topological properties of the system are determined by the structure of its singularities. The local properties of singularities of integrable systems were studied by several authors through normal forms, e.g. [56], [34], [15], [42]. In particular, several classical examples, such as Kovalevskaya top [19], Euler top [20], the pendulum [54] and the Clebsch top [21], provide a very rich interplay between certain elliptic fibrations and the Birkhoff normal forms, see also [44]. In this sense, as an application of the ideas introduced in Theorem 1, our second result aims to provide some new tools to investigate the behavior of the Liouville foliation defined by Gelfand-Tsetlin systems through certain families of complex algebraic curves. Based on the classification of Gelfand-Tsetlin fibers provided in [58] by means of the combinatorics of ladder diagrams, we prove the following result:

Theorem 2. Let (O(Λ),ωO(Λ),H ) be the Gelfand-Tsetlin integrable system associated to some adjoint orbit O(Λ) ⊂ u(n) and let ∆Λ be the corresponding Gelfand-Tsetlin polytope. Then, there exists a family of complex algebraic curves n C[w,ζ] o MΛ := CZ = Spec 2 Z ∈ O(Λ) , (1.3) hζ −PΛ(w;Z)i such that PΛ(w;Z) := rΛ(w)det(w1r − L(Z)), for every Z ∈ O(Λ), and rΛ(w) ∈ C(w), satisfying the following:

1) For all Z ∈ O(Λ), we have PΛ(w;Z) = µΛ(w)P1(w;Z)···Pn−1(w;Z), such that µΛ(w) is the minimal polynomial of Λ and Pk(w;Z) ∈ C[w], ∀k = 1,...,n − 1;

−1 2) Given u ∈ ∆Λ, then H (u) is a Lagrangian torus if and only if CZ is a smooth hyperelliptic curve, for some Z ∈ H −1(u);

−1 3) If u ∈ int(F ), for some d-dimensional face F of ∆Λ, then CZ is singular for all Z ∈ H (u); 0 Pk(w;Z) 4) In the setting of item 3), if 0 ≤ Res ;α ≤ 1, for all (α,0) ∈ Sing(CZ), and for all 1 ≤ k ≤ n − 1, Pk(w;Z) then H −1(u) =∼ T d;

−1 5) If u ∈ int(F ), for some d-dimensional face F of ∆Λ, and H (u) is a non-torus ﬁber, then there exists −1 some (α,0) ∈ Sing(CZ), for some Z ∈ H (u), such that multα (Pk(w;Z)) > 1, for some 1 ≤ k ≤ n − 1;

3 H 1 6) For all Z ∈ O(Λ) there exists an Abelian differential Ω(Z) ∈ M (CZ), with poles p jk(Z) ∈ CZ, ( j,k) ∈ I (Λ), such that ! 1 I H (Z) = √ Ω(Z) , (1.4) 2π −1 c jk ( j,k)∈I (Λ)

where c jk is a small loop around the point p jk(Z) ∈ CZ, for every ( j,k) ∈ I (Λ).

The family of algebraic curves MΛ introduced in the above theorem enables us to explore by means of the concrete low dimensional standard examples CP1, Fl(3) and Gr(2,4), some new insights into the relationship between the topology of singular Gelfand-Tsetlin fibers, singular algebraic curves and vanishing cycles. More precisely, we classify Gelfand-Tsetlin fibers for the aforementioned examples using elementary tools coming from the theory of algebraic curves and we provide a characterization for singular Gelfand-Tsetlin fibers in terms of the vanish of certain homology cycles of a generic smooth element in MΛ. Our analysis suggests that questions concerned with the topology and dynamics of the Gelfand-Tsetlin systems, specially those related to the singular locus of the underlying Liouville foliation, can be approached using the theory of algebraic curves and related topics.

2 Collective Hamiltonians and Gelfand-Tsetlin integrable systems

In this section, we provide a brief overview about Hamiltonian systems deﬁned by collective Hamiltonians. The main purpose is to cover the basic material in this topic and explain Thimm’s trick [55].

2.1 Collective Hamiltonian systems Let (M,ω) be a symplectic manifold and τ : G → Diff(M) be a smooth action. The action τ is said to be Hamiltonian if and only if it admits a moment map Φ: (M,ω) → g∗, see for instance [49, §10.1]. We say that the moment map Φ associated to τ is equivariant if it satisﬁes

Φ(τ(g)p) = Ad∗(g)Φ(p), for every p ∈ M, X ∈ g and g ∈ G. In this work we are concerned with the following setting.

Deﬁnition 2.1. A Hamiltonian G-space (M,ω,G,Φ) is composed of:

• A symplectic manifold (M,ω) and a connected Lie group G, with Lie algebra g.

• A Hamiltonian (left) Lie group action τ : G → Diff(M), with associated inﬁnitesimal action δτ : g → X(M).

• A moment map Φ: (M,ω) → g∗.

Deﬁnition 2.2. A Hamiltonian system is deﬁned by a triple (M,ω,H), where (M,ω) is a symplectic manifold and H ∈ C∞(M).

Given a compact connected Lie group G, with Lie algebra g, we are interested in the study of a certain class 2 ∗ of Hamiltonian systems associated to Hamiltonian G-spaces (O(λ),ωO(λ),G,Φ), such that O(λ) ⊂ g is the ∗ ∗ coadjoint orbit through some λ ∈ g , ωO(λ) is the Kirillov-Kostant-Souriau symplectic form and Φ: O(λ) → g is deﬁned by the natural inclusion map O(λ) ,→ g∗.

2For more details see [53, §6.1.1-6.1.2]

4 Remark 2.1. By ﬁxing an Ad-invariant inner product (−,−) on g, we obtain an isomorphism g ' g∗. From this, given (O(λ),ωO(λ),G,Φ), we can always consider (O(Λ),ωO(Λ)), where λ = (Λ,−), for some Λ ∈ g, such that O(Λ) ⊂ g is the adjoint orbit through Λ ∈ g, and

ωO(Λ)(ad(X)Z,ad(Y)Z) = −(Z,[X,Y]), ∀X,Y ∈ g. In the above setting, the adjoint action on O(Λ) defines a Hamiltonian action and, from the isomorphism g ' g∗, we have an equivariant moment map Φ: O(Λ) → g∗ defined by the natural inclusion map O(Λ) ,→ g. Remark 2.2. In the setting of Hamiltonian G-spaces defined by (co)adjoint orbits, throughout this work we shall assume that G is compact. Therefore, the identification described in the above remark will be used without many explanations. Also, we shall assume some basic results about the theory of compact Lie groups, e.g. [50].

Let g be the Lie algebra of a compact and connected Lie group G. We have a Poisson bracket {·,·}g∗ on ∗ 3 ∞ ∗ ∗ the manifold g defined as follows . Given F1,F2 ∈ C (g ) and ξ ∈ g , we set F1,F2 g∗ (ξ) = − ξ, (dF1)ξ ,(dF2)ξ , ∗ ∗ ∼ ∗ where we see (dFi)ξ , i = 1,2, as elements of g via the natural identification Tξ g = g. Also, for a fixed ξ ∈ g , it will be convenient to denote by ∇F(ξ) the element of g which satisfies the pairing

(dF)ξ (η) = η,∇F(ξ) ,

∞ ∗ ∗ for every F ∈ C (g ) and every η ∈ Tξ g . From this, we can rewrite the previous expression of {·,·}g∗ as follows: F1,F2 g∗ (ξ) = − ξ, ∇F1(ξ),∇F2(ξ) . ∗ With the bracket above, the pair (g ,{·,·}g∗ ) deﬁnes a Poisson manifold, e.g. [12, Example 1.1.3]. Given a Hamiltonian system (M,ω,H), denoting by ·,· M the Poisson structure induced by the symplectic structure ω, we are interested in the following concept of integrability.

Deﬁnition 2.3 (Liouville integrability). Let (M,ω,H) be a Hamiltonian system. We say that such a system is ∞ 1 integrable if there exist H1,...,Hn : (M,ω) → R, such that Hi ∈ C (M), for each i = 1,...,n = 2 dimR(M), with H = H1, satisfying • Hi,Hj M = 0, for all i, j = 1,...,n,

• dH1 ∧ ··· ∧ dHn 6= 0, in an open dense subset of M.

n Remark 2.3. In the setting of the above deﬁnition, by considering the induced map H : (M,ω) → R , such that H = (H1,...,Hn), we denote an integrable system by (M,ω,H ). From the ﬁrst item of the integrability condition described above, in order to study integrable systems it will be useful to consider the following concept.

∞ Deﬁnition 2.4. Let (M,{·,·}M) be a Poisson manifold. A smooth function C ∈ C (M), is called a Casimir function if it satisﬁes C,F M = 0, for every F ∈ C∞(M).

3In general, given a vector space V, there exists a correspondence between Lie algebra structures on V and linear Poisson structures on V ∗, see for instance [49, p. 367].

5 ∗ Example 2.1. Consider the Poisson manifold (g ,{·,·}g∗ ) described previously. In this particular case, we have ∗ ∗ that the Casimir functions of (g ,{·,·}g∗ ) are exactly the Ad -invariant functions. For a more general discussion about Casimir functions with respect to the Lie-Poisson bracket, see for instance [41, p. 463]. ∗ ∞ ∗ Ad∗ ∗ Remark 2.4. In what follows, given (g ,{·,·}g∗ ), we shall denote by C (g ) the subalgebra of Ad -invariant functions deﬁned on g∗. In this work we are interested in studying Hamiltonian systems deﬁned by the following special class of functions.

Deﬁnition 2.5. Let (M,ω,G,Φ) be a Hamiltonian G-space. A collective Hamiltonian on M is a function of the form H = Φ∗(F) ∈ C∞(M), where F ∈ C∞(g∗).

Now we will provide an expression for the Hamiltonian vector field XH ∈ X(M), associated to a collective Hamiltonian H = Φ∗(F) ∈ C∞(M), for more details see [30, p. 241]. ∗ At first, note that by fixing a basis {Xi} for g and denoting by {Xi } its dual, we have

i ∗ ∗ Φ = ∑Φ Xi , and DΦ = ∑dhΦ,XiiXi , i i

i where each component function Φ = hΦ,Xii satisﬁes the equation dhΦ,Xii + ιδτ(Xi)ω = 0. Recall that δτ denotes the inﬁnitesimal action associated to the Hamiltonian action τ : G → Diff(M). Therefore, given H = Φ∗(F) ∈ C∞(M), we have

dH = dF ◦ D = dF( dh ,X iX∗) = − hX∗,( F) ◦ i = − . Φ ∑ Φ i i ∑ i ∇ Φ ιδτ(Xi)ω ιδτ((∇F)◦Φ)ω i i Hence, we obtain XH = δτ((∇F) ◦ Φ). (2.1) Let us illustrate the ideas above by means of an example which will be useful afterwards in this work. Further discussions about the Hamiltonian ﬂow of collective Hamiltonians can be found in [30, p. 241-242].

Example 2.2. Consider now the Hamiltonian G-space (O(λ),ωO(λ),G,Φ). If we take a collective Hamiltonian ∗ ∞ ∗ H = Φ (F) ∈ C (O(λ)), from Eq. 2.1 we have XH = ad ((∇F) ◦ Φ). Thus, since Φ is just the inclusion map ∗ ∗ ∞ O(λ) ,→ g , it follows that the dynamical system deﬁned by XH , such that H = Φ (F) ∈ C (O(λ)), can be ∗ understood through the Hamiltonian system (g ,{·,·}g∗ ,F).

2.2 Thimm’s trick and Gelfand-Tsetlin integrable systems Now we will describe how to obtain conserved quantities in involution when we consider Hamiltonian systems defined by collective Hamiltonians. In what follows, unless otherwise stated, given a Hamiltonian G-space (M,ω,G,Φ), we shall assume the equivariance condition for the moment map Φ: (M,ω) → g∗. Under this ∗ assumption, we have that the map Φ: (M,{·,·}M) → (g ,{·,·}g∗ ) is a Poisson map, see for instance [49, p. 497]), so we obtain F ◦ Φ,I ◦ Φ M(p) = F,I g∗ (Φ(p)), for every p ∈ M and F,I ∈ C∞(g∗). Therefore, from Chevalley’s theorem one can find rank(G) independent ∗ Poisson commuting functions from the Casimir functions of (g ,{·,·}g∗ ). However, it is often the case that 1 rank(G) < 2 dimR(M). In order to find additional independent Poisson commuting functions, one can proceed following the ideas of the well-known Thimm’s trick [55, Proposition 4.1]. More precisely, if we consider a closed and connected Lie subgroup K ⊂ G, we have a natural Hamiltonian action of K on (M,ω) induced by

6 ∗ restriction. Thus, we obtain a Hamiltonian K-space (M,ω,K,ΦK), where the moment map ΦK : (M,ω) → k = Lie(K)∗, is given by ΦK = πK ◦ Φ, ∗ ∗ such that πK : g → k is the projection induced by the inclusion k ,→ g. From this, if we take two collective ∗ ∗ ∞ Hamiltonians Φ (F),ΦK(I) ∈ C (M), we obtain F ◦ Φ,I ◦ ΦK M = F ◦ Φ,I ◦ πK ◦ Φ M = F,I ◦ πK g∗ ◦ Φ. ∗ ∗ Thus, all collective Hamiltonians obtained from the Casimir functions of (g ,{·,·}g∗ ) and (k ,{·,·}k∗ ) are ∗ conserved quantities in involution for the Hamiltonian system (M,ω,ΦK(I)), e.g. [55], [30], [29]. As an application of the above ideas, we obtain the following general construction:

• Consider the Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ) as in Remark 2.1. Given a nested chain of closed and connected subgroups G = K0 ⊃ K1 ⊃ ··· ⊃ Ks, associated to each Hamiltonian Ki-space (l) (l) (O(Λ),ωO(Λ),Ki,Φi), we have Φi = πi ◦ Φl, where πi : kl → ki denotes the projection induced by the inclusion ki ,→ kl, 0 ≤ l < i ≤ s.

∗ ∞ • From the above data, if one considers a Hamiltonian system (O(Λ),ωO(Λ),Φs (I)), for some I ∈ C (ks), applying Thimm’s trick iteratively we obtain the following set of conserved quantities in involution

(i) ∗ (i) (i) ∞ Ad Hj = Φi (Ij ), Ij ∈ C (ki) , (2.2)

where 1 ≤ j ≤ ri, ri = rank(Ki), 1 ≤ i ≤ s.

When the integrability condition holds4 for the set of conserved quantities in involution described above, the integrable system is called Gelfand-Tsetlin integrable system [29]. In concrete cases it is convenient to work with the action coordinate of the Gelfand-Tsetlin integrable system. In order to obtain theses action coordinates one can proceed as follows: Let T ⊂ G be a Cartan subgroup and let G = T0 ⊃ T1 ⊃ ··· ⊃ Ts be a sequence of subgroups of T such that Ti ⊂ Ki is a Cartan subgroup, i = 1,...s. Now by taking a positive Weyl chamber (ti)+ ⊂ ti, for all i = 1,...,s, one can consider the (continuous) sweeping map

si : ki → (ti)+, (2.3) which is defined by letting si(X) be the unique element of the set (Ad(Ki)X)∩(ti)+. For each i = 1,...,s, chose a basis ξi1,...,ξiri of the integer lattice in ti and define + Λi j := l ◦ si ◦ Φi, (2.4) ξi j + where l is the restriction to (ti)+ of the linear functional l = (−,ξi j), here we consider some fixed Ad- ξi j ξi j invariant inner product (−,−) on g. The set of functions {Λi j} defines Guillemin and Sternberg’s action coordinates on O(Λ), see for instance [29, p. 119] and [38]. We also shall refer to the set of functions {Λi j} as Gelfand-Tsetlin integrable system. Further, a more detailed exposition about the above construction for the particular case of adjoint orbits of U(n) will be carried out afterward in Section 5.1.

3 Lax pairs and spectral equation

In this section, we will introduce some basic ideas about the concept of Lax pairs and describe their relation with the study of integrability in the context of Hamiltonian systems. More details about this topic can be found in [7], and [49, p. 578].

4For more details we suggest [47].

7 Definition 3.1. A Lax pair for a Hamiltonian system (M,ω,H) is defined by a pair of matrix-valued smooth functions L,P: (M,ω) → gl(r,R), such that the equation of motion associated to H ∈ C∞(M) is equivalent to the equation d L + L,P = 0. (3.1) dt Remark 3.1. We observe that the derivative in the definition above is taken when we consider the composition of L with the Hamiltonian flow of XH ∈ X(M). The advantage of dealing with Eq. 3.1 instead of the equation of motion induced by H ∈ C∞(M) is that it can be easily solved. Actually, if we consider the initial value problem d L = P,L, with L(0) = L , dt 0 −1 its solution is given by L(t) = g(t)L0g(t) , where g: (−ε,ε) → GL(r,R) is determined by the initial value problem dg = P(t)g(t), with g(0) = 1. dt Example 3.1 (Harmonic Oscillator). A basic example to illustrate the previous discussion is provided by the harmonic oscillator. Consider the Hamiltonian system (R2,dp ∧ dq,H), where the Hamiltonian function is given by 1 H(q, p) = p2 +C2q2, (3.2) 2 such that C ∈ R\{0}. A straightforward computation shows that 2 dH + ιXH (dp ∧ dq) = 0 ⇐⇒ XH = p∂q −C q∂p. From this, we obtain the following equations of motion dq dp = p, and = −C2q. dt dt

We have a Lax pair L,P: (R2,dp ∧ dq) → gl(2,R) for the Hamiltonian system (R2,dp ∧ dq,H) deﬁned by p Cq 1 0 −C L = , and P = . Cq −p 2 C 0 In fact, from a straightforward computation one can check that ( d dq = p, L + L,P = 0 ⇐⇒ dt dt dp 2 dt = −C q. 1 1 Notice that H(q, p) = − det(L) = Tr(L2). 2 4 The key point which makes the existence of Lax pairs an important tool in the study of a Hamiltonian systems is the following. Suppose we have a Lax pair (L,P) for a Hamiltonian system (M,ω,H). If we consider a smooth function F : gl(r,R) → R which is invariant by the adjoint action, i.e. −1 F(gXg ) = F(X), for all g ∈ GL(r,R), and X ∈ gl(r,R), and consider the composition I = F ◦ L ∈ C∞(M). Then, we obtain a function which is constant over the Hamiltonian ﬂow of XH ∈ X(M). In fact, we have

−1 I(t) = F(L(t)) = F(g(t)L0g(t) ) = F(L0) = constant.

8 Therefore, H,I M = XH (I) = 0. Hence, one can obtain conserved quantities from the procedure described above. More precisely, if we consider a Hamiltonian system (M,ω,H) which admits a Lax pair L,P: M → gl(r,R), then the coefﬁcients of the characteristic polynomial of L, namely,

r r−1 det(w1r − L) = fr(L)w + fr−1(L)w + ... + f1(L)w + f0(L), (3.3) provide a set of conserved quantities for the Hamiltonian system (M,ω,H). Moreover, if L = UΛU−1, where U : (M,ω) → GL(r,R) and Λ: M → gl(r,R) is a diagonal matrix of the form

Λ = diag(Λ1,...,Λr), it follows that the functions deﬁned by the eigenvalues of L are conserved quantities for the Hamiltonian system (M,ω,H). The equation det(w1r − L) = 0, (3.4) is called the spectral equation associated to L, and we refer to the constants of motion obtained from the characteristic polynomial of L as spectral invariants of L. Remark 3.2. It is worth mentioning that the involution property for the eigenvalue of L is equivalent to the existence of a r-matrix on the phase space, see for instance [7, §2.5].

4 A Lax pair formalism for Gelfand-Tsetlin integrable systems

In this section, we prove our ﬁrst result (Theorem 1), which consists of providing a Lax pair formulation for the Gelfand-Tsetlin integrable system. In order to do this, we reformulate the construction described in Section 2.2 purely in terms of Lax equations, see for instance Theorem 4.3 and Corollary 4.1.

4.1 Lax equation and collective Hamiltonians Let us start by describing the relationship between collective Hamiltonians and the Lax equation5. As we have seen in the Section 2.2, the functions which compose Gelfand-Tsetlin systems are given by collective Hamiltonians, i.e., if we consider a Hamiltonian G-space (M,ω,G,Φ), we can take F ∈ C∞(g∗) and consider the smooth function given by ∗ Φ (F) = F ◦ Φ: M → R. The Hamiltonian vector ﬁeld associated to a function deﬁned as above has the following expression

XΦ∗(F)(p) = δτ(∇F(Φ(p)))p, for every p ∈ M, see for instance Eq. 2.1. If we consider the Hamiltonian G-space (O(λ),ωO(λ),G,Φ) and take a collective Hamiltonian Φ∗(F) ∈ C∞(O(λ)), we have ∗ XΦ∗(F)(ξ) = ad (∇F(Φ(ξ)))ξ, for every ξ ∈ O(λ). By means of an Ad-invariant isomorphism g∗ =∼ g given by some Ad-invariant inner product, and the identiﬁcation between coadjoint and adjoint orbits O(λ) =∼ O(Λ), we obtain from the ordinary differential equation associated to XΦ∗(F) the following expression d ϕ (Z) = X ∗ (ϕ (Z)) = ad(∇F(Φ(ϕ (Z))))ϕ (Z), dt t Φ (F) t t t 5For more details we suggest [48] and references therein.

9 for every initial condition Z ∈ O(Λ). Since the moment map Φ: O(Λ) → g is just the inclusion map, we have the following equation for every Z ∈ O(Λ)

d Φ(ϕ (Z)) = ∇F(Φ(ϕ (Z)),Φ(ϕ (Z)). dt t t t

Notice that, if we denote X = ∇F(Φ(ϕt (Z)) and Y = ϕt (Z), we have d Φ(ϕ (Z)) = (DΦ) (ad(X)Y) = ad(X)Φ(Y) = X,Φ(Y). dt t Y From this, we have the following proposition:

Proposition 4.1. Given the Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ), the dynamic associated to any collective Hamiltonian is completely determined by a zero curvature equation d L + L,P = 0, (4.1) dt where L,P: (O(Λ),ωO(Λ)) → g are Lie algebra-valued smooth functions. Proof. Let Φ∗(F) ∈C∞(O(Λ)) be a collective Hamiltonian associated to some F ∈C∞(O(Λ)). From the Hamil- tonian ﬂow of XΦ∗ (F) ∈ X(O(Λ)), we have the following O.D.E. d ϕ (Z) = X ∗ (ϕ (Z)) = ad(∇F(Φ(ϕ (Z))))ϕ (Z), (4.2) dt t Φ (F) t t t for every Z ∈ O(Λ). We deﬁne the following pair of Lie algebra-valued functions L: Z ∈ O(Λ) 7→ Φ(Z) ∈ g, and P: Z ∈ O(Λ) 7→ ∇F(Φ(Z)) ∈ g. From the previous comments, we obtain d d L(ϕ (Z)) = Φ(ϕ (Z)) = ∇F(Φ(ϕ (Z))),Φ(ϕ (Z)). (4.3) dt t dt t t t

d Hence, from the definition of L and P we have that Eq. 4.2 is exactly the equation dt L + [L,P] = 0. Remark 4.1. Notice that, since the adjoint action of a compact Lie group G on its Lie algebra is a proper action, we have that O(Λ) ⊂ g is a compact embedded submanifold of g. In particular, it follows that C∞(O(Λ)) = Φ∗(C∞(g)), see for instance [57, p. 29]. Thus, given ψ ∈ C∞(O(Λ)), we have some I ∈ C∞(g), such that ∗ ∗ ∞ ψ = Φ (I). Therefore, given a Hamiltonian system (O(Λ),ωO(Λ),Φ (F)), for some F ∈ C (O(Λ)), a straightforward computation shows that d d ψ(ϕ (Z)) = ∇I(L(ϕ (Z))), L(ϕ (Z)), (4.4) dt t t dt t for every Z ∈ O(Λ), where L = Φ and P = ∇F(Φ). Hence, the action of the Hamiltonian vector field XΦ∗(F) ∈ X(O(Λ)) on every ψ ∈ C∞(O(Λ)) is completely determined by the Lax equation d L(ϕ (Z)) + L(ϕ (Z)),P(ϕ (Z)) = 0. dt t t t Remark 4.2 (Euler’s equation). It is worth pointing out that Eq. 4.1 is a generalization of the well-known Euler’s equation of rotation of a rigid body about a fixed point [16]. Rotation of the n-dimensional rigid body is also described by this type of equation as was first shown by Arnold, see for instance [4], [3].

10 4.2 Thimm’s trick and spectral invariants In what follows, we investigate the relationship between Thimm’s trick and the Lax equation 4.1. Given a

Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ), let K ⊂ G be a closed connected Lie subgroup of G. By restriction, one can consider the Hamiltonian K-space (O(Λ),ωO(Λ),K,ΦK), where

ΦK : O(Λ) → k, such that ΦK = πK ◦ Φ.

∞ Here we denote by πK : g → k the orthogonal projection map. By taking F ∈ C (k), we consider the collective ∗ ∞ Hamiltonian ΦK(F) ∈ C (O(Λ)). Note that

∗ ΦK(F) = Φ (F ◦ πK).

∗ ∗ From this, we denote Fe = F ◦ πK and consider ΦK(F) = Φ (Fe) also as a collective Hamiltonian associated to the Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ). As we have seen in the previous section, the dynamic associated to Φ∗(Fe) is completely determined by the Lax equation d L(ϕ (Z)) + L(ϕ (Z)),P(ϕ (Z)) = 0, dt t t t for every Z ∈ O(Λ), where L = Φ and P = ∇Fe(Φ). Now we consider the following result.

∞ Lemma 4.1. In the setting above, we have (∇F) ◦ πK = ∇(F ◦ πK), ∀F ∈ C (k).

Proof. It follows directly from the fact that (DπK)Z = πK, for all Z ∈ g.

From the result above, we see that for the Lax pair L = Φ and P = ∇Fe(Φ) associated to Fe = F ◦ πK, since ΦK = πK ◦ Φ, we have

P = (∇F) ◦ Φ = (∇F) ◦ ΦK =⇒ X ∗ = X ∗ . e Φ (Fe) ΦK (F) Furthermore, we have the following equation d Φ (ϕ (Z)) = ∇F(Φ (ϕ (Z))),Φ (ϕ (Z)), (4.5) dt K t K t K t where ϕ (Z) is the Hamiltonian ﬂow of X . In fact, if we denote W = ϕ (Z) ∈ O(Λ) and Y = ∇F(Φ (ϕ (Z))) ∈ t Φ∗(Fe) t K t k, we have d Φ (ϕ (Z)) = (DΦ ) (ad(Y)W) = ad(Y)Φ (W). (4.6) dt K t K W K Note that the equality on the right-hand side of Eq. 4.6 follows from the fact that ΦK is equivariant and ∞ Y = ∇F(ΦK(ϕt (Z))) ∈ k. Now if we take I ∈ C (k) and consider the collective Hamiltonian

∗ ∞ ψ = ΦK(I) ∈ C (O(Λ)),

∗ from the equation of motion associated to the Hamiltonian system (O(Λ),ωO(Λ),ΦK(F)) we have d ψ(ϕ (Z)) = Φ∗ (F),Φ∗ (I) (ϕ (Z)). (4.7) dt t K K O(Λ) t The left-hand side of the above equation can be written as d d d I(Φ (ϕ (Z))) = (dI) Φ (ϕ (Z)) = ∇I(Φ (ϕ (Z))), Φ (ϕ (Z)). (4.8) dt K t ΦK (ϕt (Z)) dt K t K t dt K t From this, we obtain the following theorem:

11 Theorem 4.3. Given a Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ), and given a closed and connected Lie sub- ∞ ∗ group K ⊂ G, for every F ∈ C (k), the Hamiltonian system (O(Λ),ωO(Λ),ΦK(F)) is completely determined by the Lax equation d L + L ,P = 0, (4.9) dt K K K where LK = ΦK, and PK = ∇F(ΦK). Moreover, for every ﬁnite dimensional unitary representation ρ : K → GL(V), the following facts hold:

1. The spectral equation det w1V − ρ∗(LK) = 0, (4.10) ∗ is preserved by the Hamiltonian ﬂow of ΦK(F) through every Z ∈ O(Λ);

2. The imaginary part of all eigenvalues of ρ∗(LK) are conserved quantities in involution for the Hamilto- ∗ nian system (O(Λ),ωO(Λ),ΦK(F)) on an open dense subset UK ⊂ O(Λ).

Proof. Consider LK = ΦK and PK = ∇F(ΦK). The first assertion follows directly from Lemma 4.1, Eq. 4.5, and Eq. 4.8. For the item 1, fix a maximal torus T ⊂ K, with Lie algebra t, take a closed positive Weyl chamber t+ ⊂ t, and consider the sweeping map sK : k → t+, which is defined by letting sK(X) be the unique element of the set (Ad(K)X) ∩ t+, ∀X ∈ k. Notice that sK is not smooth everywhere on k. From this, given Z ∈ O(Λ), it follows that

LK(Z) = Ad(k)sK(LK(Z)), for some k ∈ K. Hence, given a finite dimensional unitary representation ρ : K → GL(V), by considering the corresponding induced Lie algebra representation ρ∗ : k → gl(V), since K = exp(k), we obtain m m m Tr [ρ∗(LK(Z))] = Tr Ad(ρ(k))[ρ∗(sK(LK(Z)))] = Tr [ρ∗(sK(LK(Z)))] , ∗ for all Z ∈ O(Λ) and for every m ∈ N. By considering the Hamiltonian flow ϕt (Z) of ΦK(F) through some Z ∈ O(Λ), it follows that ϕt (Z) = Ad(g(t))Z, (∀t ∈ R) (4.11) where g: R → K is a solution of the initial value problem dg = v , with g(0) = e, dt g(t) for the vector field v ∈ X(K), such that vg = (Rg)∗(PK(Ad(g)Z)), where Rg : K → K denotes the right translation, see for instance [30, p. 241-242]. From this, since LK is equivariant and sK is Ad-invariant, given Z ∈ O(Λ), it follows that d Tr[ρ (s (L (ϕ (Z)))]m = Tr[ρ (s (L (Z)))]m =⇒ Tr[ρ (L (ϕ (Z)))]m = 0. ∗ K K t ∗ K K dt ∗ K t Thus, once the coefficients of the characteristic polynomial det w1V − ρ∗(LK(Z)) are given in terms of func- m tions of the form Tr [ρ∗(LK(Z))] , m ∈ N, we obtain the item 1. For item 2, we observe that associated to the Hamiltonian K-space (O(Λ),ωO(Λ),K,ΦK) we have the principal stratum σK ⊂ t+, which satisfies the property that

ΦK(O(Λ)) ∩ t+ ⊂ σK and ΦK(O(Λ)) ∩ σK 6= /0,

12 −1 see for instance [40, Theorem 3.1]. Moreover, we have that ΦK (ΣσK ) ⊂ O(Λ), where ΣσK = Ad(K)σK, deﬁnes an Ad-invariant connected open dense subset in O(Λ), see for instance [38, Proposition 1]. From these, we −1 can set UK = ΦK (ΣσK ). Since sK : ΣσK → σK is smooth, by choosing a basis {ξ1,...,ξrK } for t, where rK = rank(K), and considering the induced dual basis {φ1,...,φrK }, it follows that

sK ◦ LK = (φ1 ◦ sK ◦ LK)ξ1 + ··· + (φrK ◦ sK ◦ LK)ξrK , (4.12)

∞ Ad Since LK is equivariant and sK is Ad-invariant, it follows from the definition of UK that φi ◦sK ◦LK ∈C (UK) , ∗ for all i = 1,...,rK. Hence, from the definition of the Hamiltonian flow of ΦK(F) (see Eq. 4.11) we have that

φ1 ◦ sK ◦ LK,...,φrK ◦ sK ◦ LK are conserved quantities in involution for the Hamiltonian system deﬁned by ∗ (UK,ωO(Λ)|UK ,ΦK(F)|UK ). Therefore, since ρ∗(ξ1),...,ρ∗(ξrK ) are diagonal matrices and det w1V − ρ∗(LK) = det w1V − ρ∗(sK ◦ LK) , on UK, we conclude that the imaginary part of the eigenvalues of ρ∗(LK) are conserved quantities in involution ∗ for the Hamiltonian system (UK,ωO(Λ)|UK ,ΦK(F)|UK ). Remark 4.4. Notice that, in the setting of the previous theorem, ﬁxing some K-invariant Hermitian product on V, ∀Z ∈ O(Λ) we have that ρ∗(LK(Z)) is a skew-Hermitian linear operator, so its eigenvalues are all purely imaginary (and possibly zero). In what follows, given a compact Lie group G, we will denote by Gb the set of equivalence classes of irreducible (unitary) representations of G. Thus, given an irreducible representation ρ : G → GL(V), we shall denote by [ρ] ∈ Gb its corresponding class.

Corollary 4.1. Given a Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ), and a chain of closed connected Lie subgroups G = K0 ⊃ K1 ⊃ ··· ⊃ Ks.

If we denote by Φi the moment map associated to the Hamiltonian action (by restriction) of each Lie subgroup ∞ Ki on O(Λ), i = 0,1,...s, then for every F ∈ C (ks), the following hold:

∗ 1. We can associate to the Hamiltonian system (O(Λ),ωO(Λ),Φs (F)) a system of Lax equations

d L + L ,P = 0, (i = 1,...,s) dt i i i

i i such that Li = Φi, and Pi = ∇(F ◦ πs)(Φk), where πs : ki → ks is the projection map;

2. For every s-tuple ρ = (ρ1,...,ρs), where [ρi] ∈ Kbi, ∀i = 1,...,s, there exist a pair of matrix-valued functions Lρ ,Pρ : O(Λ) → gl(r,R), satisfying the Lax equation d L + L ,P = 0. (4.13) dt ρ ρ ρ

Moreover, the spectrum σ(Lρ ) of Lρ deﬁnes a set of conserved quantities in involution for the Hamilto- ∗ nian system (O(Λ),ωO(Λ),Φs (F)) on an open dense subset UΛ ⊂ O(Λ).

i Proof. The ﬁrst item is a direct consequence of the last theorem, we just need to consider Ki ⊂ G and F ◦ πs ∈ ∞ ∞ C (ki), i = 1,...,s, for every F ∈ C (ks). In this setting, applying the last theorem we get d L + L ,P = 0, dt i i i

13 i where Li = Φi and Pi = ∇(F ◦πs)(Φi), ∀i = 1,...,s. For the second item, given ρ = (ρ1,...,ρs), where [ρi] ∈ Kbi, ∀i = 1,...,s, we set s s M M Lρ := (ρi)∗(Li), and Pρ := (ρi)∗(Pi). (4.14) i=1 i=1

From this, since (ρi)∗ is a Lie algebra homomorphism, ∀i = 1,...,s, we have s s h i Mh i M d L (ϕ (Z)),P (ϕ (Z)) = (ρ ) (L (ϕ (Z))),(ρ ) (P (ϕ (Z))) = (ρ ) (− L (ϕ (Z))). ρ t ρ t i ∗ i t i ∗ i t i ∗ dt i t i=1 i=1 ∗ for all Z ∈ O(Λ), where ϕt (Z) denotes the Hamiltonian ﬂow of Φs (F). Therefore, by linearity of (ρi)∗, i = 1,...,s, we have d h i L (ϕ (Z)) + L (ϕ (Z)),P (ϕ (Z)) = 0. dt ρ t ρ t ρ t

By taking a suitable r ∈ N (large enough), we have Lρ ,Pρ : O(Λ) → gl(r,R). For the last statement of item Ts 2, we consider UΛ = i=1 UKi , where UKi ⊂ O(Λ) is obtained from Theorem 4.3, ∀i = 1,...,s. Since the spectrum σ(Lρ ) of Lρ is determined by the spectrum of each (ρi)∗(Li), i = 1,...,s, we have from Remark 4.4, and Theorem 4.3, that σ(Lρ ) defines a set of conserved quantities in involution for the Hamiltonian system ∗ given by (UΛ,ωO(Λ)|UΛ ,Φs (F)|UΛ ). Definition 4.1. Under the hypothesis of Corollary 4.1, we define the Gelfand-Tsetlin commutative Poisson ∞ subalgebra ΓO(Λ) ⊂ C (O(Λ)) as being D E ∗ Ad ΓO(Λ) := Φk S (kk) k = 1,...,s , (4.15)

Ad where S (kk) denotes the subalgebra of Ad-invariant polynomial functions.

Remark 4.5. It is worth observing that the motivation for the deﬁnition above of Gelfand-Tsetlin Poisson subalgebra is the concept of the Gelfand-Tsetlin subalgebras of the universal enveloping algebras, which are examples of Harish-Chandra subalgebras [14, p. 87], [22], see also [47].

4.2.1 Proof of Theorem 1

From the ideas and results provided in the previous section, we have the following theorem:

Theorem 4.6. Given a Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ), where G is either U(n) or SO(n), then there ∞ exist H ∈ C (O(Λ)), and a pair of matrix-valued functions L,P: (O(Λ),ωO(Λ)) → gl(r,R), satisfying the Lax equation d L(ϕ (Z)) + L(ϕ (Z)),P(ϕ (Z)) = 0, (4.16) dt t t t for all Z ∈ O(Λ), where ϕt (Z) denotes the ﬂow of the Hamiltonian vector ﬁeld XH through Z ∈ O(Λ). Moreover, the solutions of the spectral equation det w1r − L = 0, (4.17) provide a maximal set of conserved quantities in involution for the Hamiltonian system (O(Λ),ωO(Λ),H) which coincides with the Gelfand-Tsetlin integrable system on O(Λ).

Proof. Consider the Hamiltonian G-space (O(Λ),ωO(Λ),G,Φ) where G is U(n) or SO(n). In this setting, we have a chain of subgroups U(n) ⊃ U(n − 1) ⊃ ··· ⊃ U(1) or SO(n) ⊃ SO(n − 1) ⊃ ··· ⊃ SO(2),

14 where U(k) is regarded as a Lie subgroup of U(n) through the identiﬁcation

U(k) 0 ! U(k) =∼ ⊂ U(n), (4.18) 0 1n−k

∀k = 1,...,n − 1, and SO(k) is similarly embedded in the top lefthand corner of SO(n). From this, by taking ρ0 = (ρ1,...,ρs), such that ρi is either the canonical representation

ρi :U(n − i) → GL(n − i,C) or ρi : SO(n − i) → GL(n − i,C), where i = 1,...,s, with s = n − 1 or s = n − 2, we can apply Corollary 4.1, and obtain L,P: O(Λ) → u(r), for some suitable r ∈ N, such that L = Lρ0 , and P = Pρ0 , satisfying the Lax equation: d L(ϕ (Z)) + L(ϕ (Z)),P(ϕ (Z)) = 0, dt t t t for all Z ∈ O( ), where (Z) denotes the Hamiltonian ﬂow of H = ∗(F) ∈ C∞(O( )), for some F ∈ √ Λ ϕt √ Φs Λ C∞( −1R), here we consider u(1) =∼ −1R =∼ so(2). Now we observe that, given Z ∈ O(Λ), since s M L(Z) = Li(Z) = diag Li(Z) i = 1,...,s , i=1 such that Li = Φn−i, i = 1,...,s, with s = n − 1 or s = n − 2, it follows that

s 1 1 det w r − L = ∏det w n−i − Li . (4.19) i=1 Therefore, the solutions of the spectral equation associated to L: O(Λ) → u(r) is determined by the solutions of the spectral equation associated to each Li, i = 1,...,s, with s = n − 1 or s = n − 2. From this last fact, we have that the spectrum σ(L,Λ) of L generates the Gelfand-Tsetlin commutative Poisson subalgebra ΓO(Λ), see Deﬁnition 4.1. Hence, since the action of U(k) on adjoint orbits of U(k + 1) is coisotropic, and the same holds for the action of SO(k) on adjoint orbits of SO(k + 1), see for instance [47], [28], we have that σ(L,Λ) deﬁnes an integrable system on O(Λ), which turns out to be the Gelfand-Tsetlin integrable system.

Remark 4.7. In the construction provided in the last theorem, the maximal subset of algebraically independent Poisson-commuting functions of σ(L,Λ) can be obtained by looking at the Gelfand-Tsetlin patterns defined by Λ. Also, it is worth observing that the integrable system provided by σ(L,Λ) defines a map dim (O(Λ)) R H H : (O(Λ),ωO(Λ)) → R 2 , which is a smooth submersion on an open dense subset O(Λ) := UΛ ⊂ O(Λ), see item 2 of Corollary 4.1, and continuous on the whole orbit O(Λ). In fact, the components of H are defined by Guillemin and Sternberg’s action coordinates (Eq. 2.4). Remark 4.8. The result of Theorem 4.6 generalizes the constriction introduced in [23, Proposition 1.4] in the following sense: In the particular setting given by (O(Λ),ωO(Λ),U(n),Φ), such that O(Λ) ⊂ u(n) is some regular orbit, from the last theorem one can consider the Hamiltonian system √ k k O(Λ),ωO(Λ),H = ±( −1) Tr(L ) .

It follows trivially that {σ(L,Λ),H}O(Λ) = 0, i.e., the Lax pair formulation of the last theorem gives rise to the n(n−1) 2 integrals of motion for the Hamiltonian system above. For the particular case that k = 2, the Hamiltonian system deﬁned by H = −Tr(L2) coincides with the Hamiltonian system studied in [23, Proposition 1.4]. In order to recover the result [23, Theorem 1.8] from Theorem 4.6, we need to take a parametric spectral deformation on L given by L(ζ;Z) := ζ1r + L(Z), (Z ∈ O(Λ)) (4.20)

15 for some complex parameter ζ ∈ C. By observing that the coefﬁcients of the characteristic polynomial of L(ζ;Z) are determined by the coefﬁcients of the characteristic polynomial of L(Z), for all Z ∈ O(Λ), see for instance [23, Lemma 1.5], we recover the desired result.

5 Gelfand-Tsetlin integrable systems and algebraic curves

The aim of this section is to prove Theorem 2. In order to do this, we start by reviewing some results of [58] concerning with the topology of the ﬁbers of Gelfand-Tsetlin integrable systems on U(n)-adjoint orbits. After that, following [36], [51], [24], [37], [9], [18], we shall review some basic generalities on Abelian integrals, algebraic curves and Riemann surfaces.

5.1 Gelfand-Tsetlin integrable systems on U(n)-adjoint orbits

n From now on, for all n ∈ N, we shall consider the standard maximal torus T ⊂ U(n), with Lie algebra tn ⊂ u(n) defined by the diagonal matrices. Moreover, in order to avoid problems with signs related to imaginary √ √ eigenvalues, we will identify X ∈ u(n) with − −1X ∈ −1u(n). Since under this last identification we have ∼ n tn = R , we will consider the following realization for the closed positive Weyl chamber: n o (tn)+ = diag λ1,...,λn} ∈ tn λ1 ≥ λ2 ≥ ··· ≥ λn . Also, we shall fix the nested chain of subgroups given by U(n) ⊃ U(n − 1) ⊃ ··· ⊃ U(1), where U(k) is regarded as subgroup of U(n) as in Eq. 4.18, for every k = 1,...,n − 1. We also shall consider the following data: In the setting of Theorem 4.6, given an adjoint orbit O(Λ) ⊂ u(n), we have L: O(Λ) → u(r), n(n−1) where r = 2 , such that L(Z) = diag Li(Z) i = 1,...,n − 1 , for all Z ∈ O(Λ). Since Li : O(Λ) → u(i), i = 1,...,n − 1, we have 1 (i) (i) det w i − Li(Z) = w − λ1 (Z) ··· w − λi (Z) , (5.1) for all Z ∈ O(Λ), and for all i = 1,...,n − 1. From these, we have the spectrum σ(L,Λ) of L given by the functions (k) σ(L,Λ) = λ j 1 ≤ j ≤ k, 1 ≤ k ≤ n − 1 . (5.2) Notice that from Cauchy’s interlace theorem [33], we have that (k) (k−1) (k) (k) (k−1) (k) λ1 ≥ λ1 ≥ λ2 ≥ ··· ≥ λk−1 ≥ λk−1 ≥ λk , (5.3) for all 1 ≤ k ≤ n − 1. The inequalities above define the Gelfand-Tsetlin patterns (Eq. 5.4).

λ1 λ2 λ3 ··· λn−1 λn ≥ ≥ ≥ ≥ ≥ ≥ (n−1) (n−1) (n−1) λ1 λ2 λn−1 ≥ ≥ ≥ (n−2) (n−2) λ1 λn−2 (5.4) ≥ ≥

. . .. . ≥ ≥ (1) λ1

16 From above we have the following deﬁnition.

Deﬁnition 5.1. Given Λ = diag{λ1,...,λn}, such that λ1 ≥ λ2 ≥ ··· ≥ λn, the Gelfand-Tsetlin polytope ∆Λ is (k) deﬁned by the collection of points u = (u j ) satisfying the Gelfand-Tsetlin patterns. Consider now the following index set

(k) I (Λ) = ( j,k) j,k ∈ Z≥1, λ j is not a constant function on O(Λ) , (5.5)

1 so that |I (Λ)| = 2 dimR(O(Λ)). Then we have the following results.

Proposition 5.1 ([29]). The Gelfand-Tsetlin polytope ∆Λ coincides with the image of the map

|I (Λ)| (k) H : O(Λ) → R , H = (λ j )( j,k)∈I (Λ), (5.6) that is, ∆Λ = H (O(Λ)).

(k) Proposition 5.2 (Proposition 5.3 in [29]). For each ( j,k) ∈ I (Λ), we have that λ j : O(Λ) → R is continuous in every point of O(Λ), and smooth at Z ∈ O(Λ) if

(k+1) (k) (k+1) λ j (Z) > λ j (Z) > λ j+1 (Z). (5.7)

(k) H −1 In particular, H = (λ j )( j,k)∈I (Λ) is smooth on the open dense subset O(Λ) = H (int(∆Λ)). Further- (k) H more, the functions {λ j }( j,k)∈I (Λ) are functionally independent precisely in O(Λ) .

Let F be a d-dimensional face of the Gelfand-Tsetlin polytope ∆Λ. Denoting by int(F ) its relative interior, we have the following result.

Lemma 5.1 (Lemma 6.7 in [58]). For each d-dimensional face F of ∆Λ, there exists I (F ) ⊂ I (Λ), such (k) −1 that |I (F )| = d, and for every ( j,k) ∈ I (F ), the component λ j is smooth on H (int(F )). Furthermore, d −1 (k) for every u ∈ int(F ), we have a ﬁberwise T -action on H (u) generated by {λ j }( j,k)∈I (F ). Theorem 5.1 ([58]). Let u be a point lying on the relative interior of a d-dimensional face of the Gelfand-Tsetlin d polytope ∆Λ. Then there is a T -equivariant diffeomorphism

Ψ: H −1(u) → T d ×Y(u), (5.8) where

1) a free T d-action is given by Lemma 5.1;

2)T d acts freely on the left factor T d ×Y(u), and Y(u) := H −1(u)/T d. Remark 5.2. It is worth pointing out that in the above theorem we also have the following facts:

−1 (i) H (u) = En−1 → En−2 → ··· → E1 → E0 = point (iterated bundle structure);

(ii) Ek → Ek−1 is a Sk-bundle, where Sk is a product of odd-dimensional spheres. Moreover, we have

n−1 −1 dimR(H (u)) = ∑ dimR(Sk); (5.9) k=1

d d d d 1 (iii) T = T 1 × T 2 × ··· × T n−1 , such that dk = “number of S -factors occurring in Sk”;

17 (iv) Y(u) = Yn−1(u) → Yn−2(u) → ··· → Y1(u) → Y0(u) = point (iterated bundle structure), such that

d1+···+dk Yk(u) = Ek/T . (5.10)

d +···+d Moreover, Yk(u) → Yk−1(u) is a Yk-bundle, such that Yk = Sk/T 1 k . The results listed above provide a very rich description for the topology of the ﬁbers of the Gelfand-Tsetlin integrable system, more details can be found in [58, Theorem 4.12 and Theorem 6.11]. As a consequence of the previous facts, we have the following result.

−1 Corollary 5.1 ([58]). For every u ∈ ∆Λ, we have that H (u) is a Lagrangian torus if and only if u ∈ int(∆Λ).

5.2 Generalities on Abelian integrals and algebraic curves

Given a Riemann surface M and an open set U ⊆ M, we will denote by O(U) and by M (U) the ring of holomorphic functions and the field of meromorphic functions on U, respectively. Also, we will denote by M 1(U) the module of Abelian differentials (meromorphic 1-forms) defined on U ⊆ M. Definition 5.2. An Abelian integral is an integral of the form Z R(w,ζ(w))dw, (5.11) γ where γ : [0,1] → C is a continuous path in C,R(w,ζ) is a rational function of two variables w and ζ, and ζ(w) is a continuous function of w defined on the image of the path γ such that ζ(w) and w satisfy a polynomial relation h(w,ζ(w)) = 0, for some h(w,ζ) ∈ C[w,ζ]. In the setting of Definition 5.2 we are interested in the particular case where h(w,ζ) = ζ 2 − P(w), for some P(w) ∈ C[w], such that deg(P) = N, for some N ∈ Z≥0. In order to deal with this case from the modern perspective of Riemann surfaces and algebraic curves, let us collect some general facts about affine curves6 of the form C : ζ 2 − P(w) = 0. In the general setting, we have the following facts about such a curve C : 1. can be regarded as the affine scheme Spec C[w,ζ] ; C hζ 2−P(w)i

2. C is a smooth curve if and only if P(w) = (w − α1)···(w − αN), such that αi 6= α j, ∀i 6= j. In this case, the algebraic curve C is called hyperelliptic curve and the points p j = (α j,0) ∈ C , j = 1,...,N, are called branch points of C ;

3. If C is singular, then its set of singular points Sing(C ) is given by

2 Sing(C ) = (α,0) ∈ C P(α) = 0, multα (P) > 1 ; (5.12)

4. C is irreducible if and only if P(w) is not a square of an element of C[w];

5. If P(w) = (q(w))2, for some q(w) ∈ C[w], then C decomposes into two irreducible smooth components Ci, i = 1,2, such that

C1 : ζ − q(w) = 0 and C2 : ζ + q(w) = 0;

6. C is irreducible if and only if its set of regular points Creg := C \Sing(C ) is connected. In this last case Creg deﬁnes a non-compact Riemann surface;

6For us an afﬁne (algebraic) curve is not necessary irreducible.

18 7. If C is irreducible and Σ is the compact Riemann surface defined by the compactification of C , then we have that w,ζ ∈ M (Σ) and 2 M (Σ) =∼ C(w)[ζ]/hζ − P(w)i, (5.13) in the sense of fields isomorphisms. Here hζ 2 − P(w)i is the ideal generated by h(w,ζ) in C(w)[ζ]. 8. For every f ∈ M (Σ) there exists a rational function R(w,ζ), such that f = R(w,ζ). Conversely, every rational function R(w,ζ) whose denominator is not divisible by h(w,ζ) = ζ 2 − P(w) gives rise to a meromorphic function on Σ.

Let C be an algebraic curve as above and suppose that C is irreducible. By considering Creg as a non- compact Riemann surface, we have Creg = Σ\S, where S ⊂ Σ is a finite set. For any rational function R(w,ζ) whose denominator is not divisible by h(w,ζ) = ζ 2 − P(w), we have an associated abelian differential Ω := R(w,ζ)dw on Creg. From this, given a piecewise-smooth path γ : [0,1] → C, such that P(γ(t)) 6= 0, ∀t ∈ [0,1], and a continuous function ζ(w) of w defined on the image of the path γ, satisfying ζ(w)2 = P(w), we have Z Z Ω = R(w,ζ(w))dw, (5.14) c γ where c is the piecewise-smooth7 path obtained from the composition of γ with the map w 7→ (w,ζ(w)). From the ideas above, we have the following definition. Definition 5.3. A hyperelliptic integral is an integral of the form Z R(w,ζ)dw, (5.15) c where R(w,ζ) is a rational function on a hyperelliptic curve C : ζ 2 − P(w) = 0, and c: I → C is a piecewise- smooth path not passing through any pole of the meromorphic differential R(w,ζ)dw. Let C : ζ 2 − P(w) = 0 be a hyperelliptic curve. Given a rational function R(w,ζ) on C , by using the algebraic equation which defines C , we have that R(w,ζ) can be written in the following form8 A(w) + ζB(w) R(w,ζ) = , (5.16) C(w) such that A(w),B(w),C(w) ∈ C[w]. Conversely, any function as above gives rise to a rational function on C . This last fact enables us to explicitly construct meromorphic differentials on C with certain prescribed residues. The next example will play an important role in the proof of Theorem 2. 2 Example 5.1. Given a hyperelliptic curve C : ζ − P(w) = 0, such that P(w) = (w − α1)···(w − αN), we can consider the following meromorphic differential on C 1 wdw Ω := . (5.17) 2 w − αk for some 1 ≤ k ≤ N. By definition Ω has one pole on C which coincides with the branch point pk = (αk,0) ∈ C . 2 By taking ε > 0 small enough, such that P(w) 6= 0 for all w ∈ B(α j;ε )\{α j}, for every j = 1,...,N, we can 9 define a smooth path c j : [0,2π] → C , such that v √ √ u N √ 2 2t −1 t −1u 2 2t −1 c j(t) := ε e + α j,εe t ∏ (ε e + α j − αi) . (5.18) i6= j,i=1

7 2 d 1 d Since ζ(w) = P(w) and P(γ(t)) 6= 0, ∀t ∈ [0,1], we have dt ζ(γ(t)) = 2ζ(γ(t)) dt P(γ(t)), ∀t ∈ [0,1] where γ is smooth. 8See for instance [51, p. 259]. √ 9 t −1 The curve c j is just the composition of the loop t 7→ e with the standard local chart at p j = (α j,0) ∈ C , see [51, p. 141].

19 By deﬁnition we have that c j is a small loop around p j = (α j,0) ∈ C , for every j = 1,...,N, and from a straightforward application of the residue theorem it follows that

1 I 1 I wdw 1 w √ Ω = √ = Ind(γ j;α j)Res ;α j , (5.19) 2π −1 c j 4π −1 γ j w − αk 2 w − αk √ 2 2t −1 where γ j : [0,2π] → C is the closed curve γ(t) = ε e + α j. Since

I ( 1 dw w 1 αk, if j = k Ind(γ j;α j) = √ = 2 and Res ;α j = α jRes ;α j = , w−αk w−αk 2π −1 γ j w − α j 0, if j 6= k we conclude that Res Ω; pk = αk and Res Ω; p j = 0, ∀ j 6= k.

5.2.1 Proof of Theorem 2

In order to prove Theorem 2, let us give the general recipe of how to assign a family of algebraic curves to each adjoint orbit of U(n). In the general setting, let Λ ∈ u(n)\{0} be an arbitrary diagonal matrix with distinct eigenvalues λn1 ,...,λns = λn, s > 0, i.e., Λ = diag{λ1,...,λn}, where

λ1 = ··· = λn1 > λn1+1 = ··· = λn2 > ··· > λns−1+1 = ··· = λn, (5.20)

k1 k2 ks such that k1 + ··· + ks = n. In this case, we have ∼ O(Λ) = U(n)/(U(k1) × ··· × U(ks)), (5.21)

2 2 such that dimR(O(Λ)) = n − ∑i ki . If we consider the associated Lax matrix L: O(Λ) → gl(r,R), given by Theorem 4.6, we have the following result.

Lemma 5.2. In the above setting, for every Z ∈ O(Λ), we have

1 m1 ms det(w r − L(Z)) = (w − λn1 ) ···(w − λns ) FΛ(w;Z), (5.22)

(k j−1)k j where m j = 2 , ∀ j = 1,...,s, and FΛ(w;Z) ∈ C[w].

Proof. Given Λ = diag{λ1,...,λn}, suppose that for some ` ∈ N we have

λ1 ≥ ··· ≥ λi > λi+1 = λi+2 = ··· = λi+`−1 = λi+` > λi+`+1 ≥ ··· ≥ λn. (5.23) | {z } `

(n−1) (n−1) Denoting λi+1 = ··· = λi+` = c0, since λ j ≥ λ j ≥ λ j+1, for all 1 ≤ j ≤ n − 1, it follows that λ j = c0, for all i + 1 ≤ j ≤ i + ` − 1, that is

(n−1) (n−1) (n−1) (n−1) (n−1) ··· ≥ λi ≥ λi+1 = λi+2 = ··· = λi+`−1 ≥ λi+` ≥ ··· . (5.24) | {z } `−1

(n−1) (n−2) (n−1) By observing that λ j ≥ λ j ≥ λ j+1 , for all 1 ≤ j ≤ n − 2, from a similar argument as above, we have (n−2) that λ j = c0, for all i + 1 ≤ j ≤ i + ` − 2. By applying the above argument inductively, we conclude that

(n−k) (n−k) (n−k) λi+1 = λi+2 = ··· = λi+`−k = c0, (5.25) | {z } `−k

20 m `(`−1) for all 1 ≤ k ≤ ` − 1. Hence, we have that (w − c0) , such that m = 2 , is a divisor of the polynomial det(w1r − L(Z)) ∈ C[w], for all Z ∈ O(Λ). From this, if Eq. 5.20 holds, we have that

1 m1 ms det(w r − L(Z)) = (w − λn1 ) ···(w − λns ) FΛ(w;Z), (5.26)

(k j−1)k j for all Z ∈ O(Λ), where m j = 2 , ∀ j = 1,...,s, and FΛ(w;Z) ∈ C[w]. n(n−1) In the setting of the previous lemma, since k1 + ··· + ks = n and r = 2 , it follows that

s 2 2 n(n − 1) (k j − 1)k j n − ∑i ki dimR(O(Λ)) deg(FΛ(w;Z)) = − ∑ = = . (5.27) 2 j=1 2 2 2

Actually, from the Gelfand-Tsetlin pattern (Eq. 5.4) and Deﬁnition 5.1, we have that

(k) FΛ(w;Z) = ∏ w − λ j (Z) , (5.28) ( j,k)∈I (Λ) for all Z ∈ O(Λ). By keeping the previous notation, we have the following result.

Lemma 5.3. There exists a rational function rΛ(w) ∈ C(w), such that, for all Z ∈ O(Λ), the following equation holds h i 1 (k) rΛ(w)det(w r − L(Z)) = µΛ(w) ∏( j,k)∈I (Λ) w − λ j (Z) , (5.29) where µΛ(w) = (w − λn1 )···(w − λns ) ∈ C[w] is the minimal polynomial of Λ ∈ u(n).

Proof. As before, let Λ ∈ u(n)\{0} be an arbitrary diagonal matrix with distinct eigenvalues λn1 ,...,λns , with multiplicities given respectively by k1,...,ks, such that s > 0. By setting 1 rΛ(w) := , (5.30) m1−1 ms−1 (w − λn1 ) ···(w − λns )

(k j−1)k j such that m j = 2 , ∀ j = 1,...,s, it follows from Lemma 5.2 that Eq. 5.29 holds for all Z ∈ O(Λ).

From above, for every Z ∈ O(Λ) one can deﬁne a complex algebraic curve CZ by setting

2 CZ : ζ − PΛ(w;Z) = 0, (5.31) where PΛ(w;Z) = rΛ(w)det(w1r − L(Z)). The motivation behind Eq. 5.31 is the following. In the setting of Remark 5.2, let u ∈ ∆Λ be a point lying on the relative interior of a d-dimensional face. Considering the iterated bundle −1 H (u) = En−1 → En−2 → ··· → E1 → E0 = point, (5.32) −1 we have that the description of H (u) provided in Theorem 5.1 depends on each Sk-bundle Ek → Ek−1. −1 By choosing any Z ∈ H (u), for each k = 1,...,n − 1, we have that Sk is identiﬁed with the set of points k (ξ1,...,ξk) ∈ C , satisfying either " #  n−1 |ξ |2 detw1 − Λ = detw1 − L (Z) (w − c ) − i , i f k = n − 1,  n n−1 n−1 n−1 ∑ (n−1)  i=1 w − λ (Z)  i or (5.33)  " #  k |ξ |2 detw1 − L (Z) = detw1 − L (Z) (w − c ) − i , i f 1 ≤ k ≤ n − 2,  k+1 k+1 k k k ∑ (k) i=1 w − λi (Z)

21 where c1,...,cn−1, are constants determined by u ∈ ∆Λ. Therefore, each Sk-bundle Ek → Ek−1 can be described by means of the equations above and the patterns (inequalities) satisﬁed by the components of u ∈ ∆Λ, see for instance [58, §5]. From Eq. 4.19 and Lemma 5.3, we have that the complex algebraic curve CZ introduced in Eq. 5.31 encodes in its set of branch points some fundamental elements which determine the geometry and topology of the iterated bundle giving in Eq. 5.32. In this setting, we have the following theorem:

Theorem 5.3. Let (O(Λ),ωO(Λ),H ) be the Gelfand-Tsetlin integrable system associated to some adjoint orbit O(Λ) ⊂ u(n) and let ∆Λ be the corresponding Gelfand-Tsetlin polytope. Then, there exists a family of complex algebraic curves n C[w,ζ] o MΛ := CZ = Spec 2 Z ∈ O(Λ) , (5.34) hζ −PΛ(w;Z)i such that PΛ(w;Z) := rΛ(w)det(w1r − L(Z)), for every Z ∈ O(Λ), and rΛ(w) ∈ C(w), satisfying the following:

1) For all Z ∈ O(Λ), we have PΛ(w;Z) = µΛ(w)P1(w;Z)···Pn−1(w;Z), such that µΛ(w) is the minimal polynomial of Λ and Pk(w;Z) ∈ C[w], ∀k = 1,...,n − 1; −1 2) Given u ∈ ∆Λ, then H (u) is a Lagrangian torus if and only if CZ is a smooth hyperelliptic curve, for some Z ∈ H −1(u);

H 1 6) For all Z ∈ O(Λ) there exists an Abelian differential Ω(Z) ∈ M (CZ), with poles p jk(Z) ∈ CZ, ( j,k) ∈ I (Λ), such that ! 1 I H (Z) = √ Ω(Z) , (5.35) 2π −1 c jk ( j,k)∈I (Λ)

where c jk is a small loop around the point p jk(Z) ∈ CZ, for every ( j,k) ∈ I (Λ). 2 Proof. For every Z ∈ O(Λ), consider CZ : ζ − PΛ(w;Z) = 0, such that PΛ(w;Z) = rΛ(w)det(w1r − L(Z)), and deﬁne MΛ := {CZ | Z ∈ O(Λ)}. In order to prove item 1), we observe that

PΛ(w;Z) = µΛ(w)FΛ(w;Z), (5.36) 1 n−1 1 see Lemma 5.2 and Lemma 5.3. Since det(w r − L(Z)) = ∏k=1 det w k − Lk(Z) , we can set Pk(w;Z) = gcd FΛ(w;Z),det(w1k − Lk(Z)) , i = 1,...,n − 1. (5.37)

From this we obtain the decomposition PΛ(w;Z) = µΛ(w)P1(w;Z)···Pn−1(w;Z). For item 2), we observe the −1 0 −1 following. Given u ∈ ∆Λ, since H (Z) = u, ∀Z ∈ H (u), we have CZ = CZ0 , ∀Z,Z ∈ H (u). Indeed, (k) denoting u = (u j )( j,k)∈I (Λ), it follows from Eq. 5.29 that

2 2 (k) CZ = (w,ζ) ∈ C ζ − µΛ(w) ∏( j,k)∈I (λ)(w − u j ) = 0 , (5.38) −1 −1 for all Z ∈ H (u). The description above shows that CZ is smooth, for some Z ∈ H (u), if and only if the components of u are pairwise distinct and gcd FΛ(w;Z), µΛ(w) = 1, i.e., if and only if u ∈ int(∆Λ). Thus, from Corollary 5.1, we obtain item 2). The proof of item 3) follows from the following fact. Given a d-dimensional face F of ∆Λ, if u ∈ int(F ), from the Gelfand-Tsetlin patterns (Eq. 5.4) we have that at least one of the following equalities is satisﬁed by the components of u:

22 (k) (k) (k+1) (k) (k+1) u j = λ, u j = u j or u j = u j+1 , for some eigenvalue λ ∈ C of Λ. Therefore, if u ∈ int(F ), for some d-dimensional face F of ∆Λ, we have −1 have that PΛ(w;Z) has at least one root with multiplicity ≥ 2, ∀Z ∈ H (u), which implies that CZ is singular for all Z ∈ H −1(u), so we have item 3). Now, given u ∈ int(F ), for some d-dimensional face F , if we have as stated in item 4) that P0(w;Z) 0 ≤ Res k ;α ≤ 1, (5.39) Pk(w;Z) for all (α,0) ∈ Sing(CZ) and for all 1 ≤ k ≤ n − 1, it follows that

Pk(α;Z) = 0 if and only if multα (Pk(w;Z)) = 1, −1 d for all (α,0) ∈ Sing(CZ) and for all 1 ≤ k ≤ n − 1. On the other hand, since H (u) =∼ T ×Y(u) (Theorem

5.1), if dimR(Y(u)) > 0, it follows that at some stage 1 ≤ k0 ≤ n − 1 of the iterated bundle given in item (i) of 2`−1 Remark 5.2 we have a S -factor in Sk0 (` > 1). This factor corresponds to the pattern  (n−1) (n−1) λ j > u = λ j+1 = ··· = λ j+` = u > λ j+`+1, if k0 = n − 1,  j j+` or  (k0+1) (k0) (k0+1) (k0+1) (k0) (k0+1) u j > u j = u j+1 = ··· = u j+` = u j+` > u j+`+1, if 1 ≤ k0 ≤ n − 2,

(k0) (k0) for some j, see for instance [58, Lemma 5.14]. Thus, denoting α = u j = ··· = u j+`, from above we have 0 Pk(w;Z) (α,0) ∈ Sing(CZ), such that multα (Pk (w;Z)) > 1. Therefore, if 0 ≤ Res ;α ≤ 1, for all (α,0) ∈ 0 Pk(w;Z) −1 ∼ d Sing(CZ) and for all 1 ≤ k ≤ n − 1, we must have that dimR(Y(u)) = 0, i.e., H (u) = T , which concludes the proof of item 4). The proof of item 5) is a consequence of the previous argument, that is, if u ∈ int(F ), −1 ∼ d for some d-dimensional face F of ∆Λ, and H (u) = T ×Y(u), with dimR(Y(u)) > 0, then there exists −1 (α,0) ∈ Sing(CZ), for some Z ∈ H (u), such that multα (Pk0 (w;Z)) > 1, for some 1 ≤ k0 ≤ n − 1. H −1 H In order to prove item 6) we observe that, since O(Λ) = H (int(∆Λ)), for every Z ∈ O(Λ) we have that the roots of PΛ(w;Z) are pairwise distinct, which implies that the roots of FΛ(w;Z) are also pairwise (k) distinct, notice that the roots of FΛ(w;Z) are the components of H (Z) = (λ j (Z))( j,k)∈I (Λ). Since in this case 1 CZ is a hyperelliptic curve, we can take Ω(Z) ∈ M (CZ), such that

1 wF0 (w;Z) 1 wdw (Z) := Λ dw = . (5.40) Ω ∑ (k) 2 FΛ(w;Z) 2 ( j,k)∈I (Λ) w − λ j (Z)

(k) The poles of Ω(Z) on CZ are given by the branch points p jk(Z) = (λ j (Z),0) ∈ CZ, where ( j,k) ∈ I (Λ). (k) From a straightforward computation as in Example 5.1, it follows that λ (Z) = √1 H Ω(Z), for some j 2π −1 c jk small loop c jk around the point p jk(Z) ∈ CZ, for all ( j,k) ∈ I (Λ). From above we obtain item 6) and conclude the proof.

5.2.2 Examples and ﬁnal comments

In this subsection, we discuss by means of some concrete examples how the results of Theorem 5.3 can be applied in the study of the Liouville foliation defined by the Gelfand-Tsetlin integrable systems. The main purpose is to provide a classification for the Gelfand-Tsetlin fibers in terms of elementary tools of the theory of algebraic curves and investigate their relationship with vanishing cycles.

23 Example 5.2 (Warm-up example). Consider the Hamiltonian U(2)-space (O(Λ),ωO(Λ),U(2),Φ), such that Λ = 1 1 , > O( ) ∼ ( , 1 , ) diag λ1 λ2 , where λ1 λ2. In this case we have Λ = CP and the Gelfand-Tsetlin system CP ωCP H can be obtained as in Theorem 4.6 through the spectral equation 1 (1) det(w 1 − L) = (w − λ1 ) = 0, (5.41) (1) observe that L = L1. Thus, we have H (Z) = λ1 (Z), for all Z ∈ O(Λ). By applying Theorem 5.3, we obtain a 1 family of algebraic curves MΛ = {CZ | Z ∈ CP }, such that 2 CZ : ζ − µΛ(w)FΛ(w;Z) = 0, (5.42)

1 (1) for all Z ∈ CP , where µΛ(w) = (w − λ1)(w − λ2) and FΛ(w;Z) = (w − λ1 (Z)). In this case, the Gelfand- Tsetlin polytope ∆Λ is deﬁned by the pattern (1) λ2 ≤ u1 ≤ λ1, (5.43)

(1) −1 1 −1 and for every u = u1 ∈ ∆Λ, we have H (u) = T , if u ∈ Int(∆Λ), and H (u) = point, if u ∈ ∂∆Λ. Given −1 u ∈ Int(∆Λ), for all Z ∈ H (u) we have that CZ ∈ MΛ is a smooth algebraic curve deﬁned by the afﬁne part of a genus 1 Riemann surface Σ1 (Fig. 1).

Figure 1: On the right-hand side we have the Riemann surface Σ1 which compactiﬁes CZ.

−1 If u ∈ ∂∆Λ, we have two possibilities, i.e., u = λ2 or u = λ1. In both cases, for every Z ∈ H (u), we have that CZ ∈ MΛ is a singular algebraic curve (Fig. 2a and Fig. 2b).

(a) : 2 − (w − )(w − )2 = 0. 2 2 CZ ζ λ1 λ2 (b) CZ : ζ − (w − λ1) (w − λ2) = 0.

−1 Figure 2: On the left-hand side we have a pinched torus associated to the singular ﬁber H (λ2). On the −1 right-hand side we have a horn torus associated to the singular ﬁber H (λ1).

As it can be seen above, the singular ﬁbers of the Gelfand-Tsetlin system correspond to singular curves obtained from the vanish of certain homology cycles of a generic smooth element in MΛ (Fig. 3).

Figure 3: Vanishing cycles which characterize the singular ﬁbers of the GT-system.

−1 For every u ∈ ∆Λ = [λ2,λ1], and every Z ∈ H (u), we have Ordp(CZ) = 1 or 2, ∀p ∈ CZ. Moreover, it is straightforward to show that:

24 −1 1 1. Ordp(CZ) = 1, ∀p ∈ CZ, if and only if H (u) =∼ T ;

−1 2. Ordp(CZ) = 2, for some p ∈ CZ, if and only if H (u) = point.

From above we have a characterization of the Gelfand-Tsetlin ﬁbers purely in terms of the elements in MΛ.

Example 5.3 (Generic orbits in u(3)). Consider the Hamiltonian U(3)-space (O(Λ),ωO(Λ),U(3),Φ), where ∼ Λ = diag λ1,λ2.λ3 , such that λ1 > λ2 > λ3. It is straightforward to see that O(Λ) = Fl(3), and from Theorem 4.6 we have the Gelfand-Tsetlin system (Fl(3),ωFl(3),H ) given by the solutions of the spectral equation

1 (1) (2) (2) det(w 3 − L) = (w − λ1 )(w − λ1 )(w − λ2 ) = 0, (5.44)

(2) (2) (1) i.e., H = (λ1 ,λ2 ,λ1 ). Applying Theorem 5.3 we get MΛ = {CZ | Z ∈ Fl(3)}, such that

2 CZ : ζ − µΛ(w)FΛ(w;Z) = 0, (5.45) for every Z ∈ Fl(3), where µΛ(w) = (w − λ1)(w − λ2)(w − λ3) and FΛ(w;Z) = det(w13 − L(Z)). It is worth mentioning that also from Theorem 5.3 we can describe the Guillemin and Sternberg’s action coordinates (2) (2) (1) H {λ1 ,λ2 ,λ1 } on Fl(3) purely in terms of hyperelliptic integrals. In fact, since in this case FΛ(w;−) = H 10 det(w13 − L), by following Eq. 5.40, for every Z ∈ Fl(3) , we have I (k) 1 1 1 −1 λ j (Z) = √ tr ( 3 − w L(Z)) dw, ∀1 ≤ j ≤ k ≤ 2. (5.46) 4π −1 c jk

Now, as in the previous example, let us analyze the relationship between singular Gelfand-Tsetlin ﬁbers and vanishing cycles. In this case, the associated Gelfand-Tsetlin polytope is deﬁned by the following patterns:

λ1 λ2 λ3 ≥ ≥ ≥ ≥ (2) (2) u1 u2 (5.47) ≥ ≥ (1) u1

−1 ∼ 3 From above, given u ∈ Int(∆Λ), for all Z ∈ H (u) = T , we have that CZ ∈ MΛ is a smooth algebraic curve deﬁned by the afﬁne part of a genus 2 Riemann surface Σ2 (Fig. 4a and Fig. 4b).

(1) (1) (a) u1 ≥ λ2. (b) λ2 ≥ u1 .

−1 Figure 4: Riemann surface which compactiﬁes CZ, for all Z ∈ H (u), such that u ∈ Int(∆Λ).

As in the previous example, we can distinguish the singular ﬁbers of the Gelfand-Tsetlin integrable system by means of the vanish of certain homology cycles (Fig. 5) of a generic smooth element in MΛ.

10 −1 H Observe that ∂w det(w13 − L(Z)) = det(w13 − L(Z))tr[(w13 − L(Z)) ], for every Z ∈ Fl(3) . Thus, after some suitable rear- rangement in Eq. 5.46, one can also express Guillemin and Sternberg’s action coordinates in terms of the resolvent R(w;L(Z)) := −1 (w13 − L(Z)) , w ∈ C\σ(L(Z)).

25 Figure 5: The homology cycles for Σ2.

(a) 0-dimensional face; Associated ﬁber H −1(u) = point. (b) 0-dimensional face; Associated ﬁber H −1(u) =∼ S3.

Figure 6: Examples of the correspondence between the faces of ∆Λ and vanishing cycles. On the right-hand side of Fig. 6a-6d we have the vanishing cycles represented by the blue and red colored circles.

By looking at the face structure of the underlying GT-polytope ∆Λ described in terms of certain subgraphs −1 ∼ −1 of the ladder diagram corresponding to Λ (e.g. [2]), and observing that H (u1) = H (u2) if u1 and u2 are contained in the relative interior of the same face of ∆Λ, e.g. [58], one has the precise description of the vanishing cycles which defines the singular curve corresponding to each singular fiber of the Gelfand-Tsetlin 11 −1 ∼ 3 system (Fig. 6a-6d). From the definition of MΛ, the unique non-torus Lagrangian fiber H (u) = S (Fig. 6b) corresponds to the singular algebraic curve

2 4 −1 CZ : ζ − (w − λ1)(w − λ2) (w − λ3) = 0, ∀Z ∈ H (u). (5.48)

−1 It is straightforward to see from the associated GT-patterns (Eq. 5.47) that, ∀u ∈ ∂∆Λ, H (u) is Lagrangian −1 if and only if there exists p ∈ CZ, for some Z ∈ H (u), such that Ordp(CZ) = 4. Joining this last fact with the −1 results of Theorem 5.3 we conclude that, for every u ∈ ∆Λ and every Z ∈ H (u), we have 1 ≤ Ordp(CZ) ≤ 4, ∀p ∈ CZ. Moreover, we obtain the following characterization: −1 3 1. (Lagrangian torus) Ordp(CZ) = 1, ∀p ∈ CZ, if and only if H (u) =∼ T ;

−1 2. (Isotropic non-Lagrangian) 1 < Ordp(CZ) < 4, for some p ∈ CZ, if and only if H (u) = point or H −1(u) =∼ T d, d = 1,2;

−1 3 3. (Non-torus Lagrangian) Ordp(CZ) = 4, for some p ∈ CZ, if and only if H (u) =∼ S .

Therefore, we have a classiﬁcation for the leaves of the underlying Liouville foliation in terms of MΛ.

Example 5.4 (Non-regular orbits). Consider now the Hamiltonian U(4)-space (O(Λ),ωO(Λ),U(4),Φ), such that Λ = diag λ1,λ2,λ3,λ4 , ∼ where λ1 = λ2 > λ3 = λ4. In this particular case we have O(Λ) = Gr(2,4), and from Theorem 4.6 we obtain the Gelfand-Tsetlin integrable system (Gr(2,4),ωGr(2,4),H ) by means of the solutions of the spectral equation

1 (3) (2) (2) (1) det w 6 − L = w − λ1 w − λ2 w − λ3 w − λ1 w − λ2 w − λ1 = 0. (5.49) 11See for instance [46, Theorem 1.1].

26 From Theorem 5.3 we get MΛ = {CZ | Z ∈ Gr(2,4)}, such that

2 CZ : ζ − µΛ(w)FΛ(w;Z) = 0, (5.50) for every Z ∈ Gr(2,4), where µΛ(w) = (w − λ1)(w − λ3) and (1) (2) (2) (3) FΛ(w;Z) = w − λ1 (Z) w − λ1 (Z) w − λ2 (Z) w − λ2 (Z) . The associated Gelfand-Tsetlin patterns in this case are given by:

λ1 λ1 λ3 λ3 = ≥ = = ≥ = (3) λ1 u2 λ3 ≥ ≥ ≥ ≥ (5.51) (2) (2) u1 u2 ≥ ≥ (1) u1 −1 ∼ 4 Given u ∈ Int(∆Λ), as in the previous example, for all Z ∈ H (u) = T , we have that CZ ∈ MΛ is a smooth algebraic curve deﬁned by the afﬁne part of a genus 2 Riemann surface Σ2 (Fig. 7a and Fig. 7b).

(1) (3) (3) (1) (a) u1 ≥ u2 . (b) u2 ≥ u1 .

H Figure 7: Riemann surface which compactiﬁes CZ, for all Z ∈ O(Λ) .

4 (3) (2) (2) (1) (3) Under the projection R → R, H = λ2 ,λ2 ,λ1 ,λ1 7→ λ2 , we have that ∆Λ is fibered over [λ3,λ1] by Gelfand-Tsetlin polytopes defined by patterns as in Eq. 5.47, and the fiber shrinks to a two dimensional (3) (3) (2) (2) (1) triangle on the boundaries λ2 = λ1,λ3 (Fig. 8). By taking u2 = u2 = u1 = u1 = s, such that s ∈ [λ3,λ1], −1 ∼ 3 1 −1 we have H (u) = S × S , if λ1 > s > λ3, and H (u) = point, if s = λ1 or λ3. More precisely, for every −1 u ∈ ∂∆Λ, H (u) is a Lagrangian fiber if and only if u = (s,s,s,s), such that λ1 > s > λ3. Fibers over other boundary faces are lower dimensional isotropic torus, see for instance [46, Theorem 1.2].

Figure 8: The Gelfand-Tsetlin polytope for Gr(2,4).

From item 1) of Theorem 5.3, for every Z ∈ Gr(2,4), we have FΛ(w;Z) = P1(w;Z)P2(w;Z)P3(w;Z), such that

27 (1) (2) (2) (3) P1(w;Z) = w − λ1 (Z) , P2(w;Z) = w − λ1 (Z) w − λ2 (Z) and P3(w;Z) = w − λ2 (Z) .

Hence, for all Z ∈ Gr(2,4), such that Sing(CZ) 6= /0,we have

P0(w;Z) 0 ≤ Res k ;α ≤ 1, Pk(w;Z) for k = 1,3 and for all (α,0) ∈ Sing(CZ). In the above setting, note that 1 ≤ Ordp(CZ) ≤ 5, ∀p ∈ CZ. Combining these last facts with item 4) and item 5) of Theorem 5.3, we conclude from the Gelfand-Tsetlin patterns (Eq. −1 5.51) that, for every u ∈ ∆Λ and every Z ∈ H (u), the following holds:

−1 4 1. (Lagrangian torus) Ordp(CZ) = 1, ∀p ∈ CZ, if and only if H (u) =∼ T ; 0 Pk(w;Z) 2. (Isotropic non-Lagrangian) Ordp(CZ) = 5, for some p ∈ Sing(CZ), or 0 ≤ Res ;α ≤ 1, for all Pk(w;Z) −1 −1 d (α,0) ∈ Sing(CZ), and for all 1 ≤ k ≤ 3, if and only if H (u) = point or H (u) =∼ T , d = 1,2,3;

3. (Non-torus Lagrangian) Ordp(CZ) = 4 and multα (P2(w;Z)) > 1, for some p = (α,0) ∈ Sing(CZ), if and only if H −1(u) =∼ S3 × S1.

As in the previous example, we have a classiﬁcation for the leaves of the Liouville foliation deﬁned by the Gelfand-Tsetlin integrable system (Gr(2,4),ωGr(2,4),H ) in terms of MΛ.

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