Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric $\mathrm{BC}_n$ Sutherland system

Integrable many-body systems of Ruijsenaars--Schneider--van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson-Lie group $\mathrm{SU}(2n)$. New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space $\mathbb{C}^n\simeq\mathbb{R}^{2n}$ underlies both global models, it is seen that for both systems the action variables generate the standard torus action on $\mathbb{C}^n$, and the fixed point of this action corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are found to be non-degenerate in the sense that the functional dimension of the Poisson algebra of their conserved quantities is equal to half the dimension of the phase space. The dual of the deformed Sutherland system is shown to be a limiting case of a van Diejen system.


Introduction
Integrable Hamiltonian systems have important applications in diverse fields of physics and are in the focus of intense investigation by a great variety of mathematical methods. We are interested in the family of classical many-body systems introduced in their simplest form by Calogero [2], Sutherland [41] and Ruijsenaars and Schneider [38]. The relevance of these systems to numerous areas of mathematics and physics is apparent from the reviews devoted to them [5,7,24,25,26,34,37,42]. One of their fascinating features is that several pairs of such systems enjoy a duality relation that converts the particle positions of one system into the action variables of the other system, and vice versa 1 . This intriguing phenomenon was first analyzed in the ground-breaking papers [33,36] by a direct method, while its grouptheoretic background came to light more recently [17,18,24]. The treatment of the self-dual Calogero system by Kazhdan, Kostant and Sternberg [19] served as a source of inspiration for these developments. Since this paper is devoted to the analysis of a particular dual pair, let us next outline in more precise terms the notion of duality that we use. An integrable Hamiltonian system is given by an Abelian Poisson algebra H of smooth functions on a 2n-dimensional symplectic manifold (M, ω) such that the functional dimension of H is n, and all elements of H generate complete flows. The systems of our interest possess another distinguished Abelian Poisson algebra P, which has the same properties as H and the following requirements hold: . This means that R and R −1 can be interpreted as global action-angle maps for the Liouville integrable systems (M, ω, H) and (M ,ω,Ĥ). One may also say that P represents global position-type variables for the many-body system (M ,ω,Ĥ) and global action-type variables for the system (M, ω, H), together with the analogous 'dual statement'. For further description of this curious notion and its quantum mechanical counterpart, alias the celebrated bispectral property [6], the reader may consult the reviews [34,37]. We note in passing that in some examples the λ i are globally smooth and independent, and then M o = M, while in other examples they lose their smoothness or independence outside a proper submanifold M o . This should not come as a surprise since from the dual viewpoint the λ i are action variables, which usually exhibit some singularities. Their canonical conjugates θ i may vary on the circle or on the line depending on the example.
It was realized by Gorsky and his collaborators [17,18,24], and elaborated in detail mainly by the first author and his collaborators ( [8]- [16], [27,28]), that dual pairs of integrable manybody systems can be derived by Hamiltonian reduction utilizing the following mechanism. Suppose that we have a higher dimensional 'master phase space' M that admits a symmetry group G, and two distinguished independent Abelian Poisson algebras H 1 and H 2 formed by G-invariant, smooth functions on M. Then we can apply Hamiltonian reduction to M and obtain a reduced phase space M red equipped with two Abelian Poisson algebras H 1 red and H 2 red that descend respectively from H 1 and H 2 . We need to construct two distinct models M andM of M red yielding (M, ω, H, P) and (M ,ω,Ĥ,P) in such a way that the reduction of H 1 is represented by H andP, and the reduction of H 2 is represented by P andĤ. If this is achieved, then we obtain a natural map R : M →M that corresponds to the identity map on M red and relates the Abelian Poisson algebras on M to those onM in the way stated in (1.2). A crucial, and very intricate, requirement is that the reduction must provide manybody systems: to fulfil this, one can rely only on experience and inspiration. The heart of the matter is the choice of the correct master system and its specific reduction. The examples so far treated by the mechanism just outlined include group theoretic reinterpretations of dual pairs previously constructed by direct methods as well as new dual pairs found by reduction. At the same time, there still exist such known instances of dualities as well (notably, the self-dual hyperbolic RS system [33] and the dual pair involving the relativistic Toda system [35]) that stubbornly resist treatment in the reduction framework. The goal of this paper is to present a thorough analysis of a dual pair of integrable manybody systems recently derived in [11] and [16] by reduction of the Heisenberg double of the standard Poisson-Lie group SU(2n). It is well-known [39,40] that the Heisenberg doubles are Poisson-Lie analogues (and deformations) of corresponding cotangent bundles. The relevant reduction is a direct Poisson-Lie generalization-making use of Lu's momentum map, [21]of the reduction of the cotangent bundle T * SU(2n) used for deriving the trigonometric BC n Sutherland system and its dual in [10]. Correspondingly, the reduction of the Heisenberg double leads to a deformation of this dual pair. We shall not only describe the deformed dual pair, but shall also show how the duality allows us to extract non-trivial information about the dynamics. For example, it will allow us to prove that both of the resulting integrable many-body Hamiltonians are non-degenerate since their flows densely fill the corresponding Liouville tori. Furthermore, it will be shown that all the flows of H posses a common fixed point, as do the flows ofĤ. In order to see this, we need to provide global descriptions of the dual models M andM of the reduced phase space. This problem was settled in [11] for one of the models, and will be our main task below for the other one.
Our current line of research was initiated in the paper [22], where the analogous reduction of the Heisenberg double of SU(n, n) was considered. The investigation in [10] was strongly influenced by the work of Pusztai [28], who studied a dual pair arising from reduction of T * SU(n, n). The Poisson-Lie counterpart of the SU(n, n) dual pair appears more complicated than what we report on here; its exploration is left for the future.
Before outlining the content of the paper, let us recall from [11,16] the local description of our many-body systems in duality, which arises by restricting attention to dense open submanifolds of the reduced phase space. These models have 3 real parameters, µ > 0 and u and v, whose range will be specified below. Here, we use the hatted letters to describe the model constructed in [11]. The manifoldM contains a dense open proper subsetM o parametrized by the Cartesian product where T n is an n-torus and Theλ i and the anglesθ i are Darboux coordinates , i.e., onM o we havê The main HamiltonianĤ can be written onM o aŝ (1.7) The phase space M of the 'dual model' possesses a dense open proper subset M o parametrized by It carries the Darboux form In terms of these variables, the main Hamiltonian H reads The formulae of the main HamiltoniansĤ (1.6) and H (1.11) are invariant with respect to the independent transformations µ → −µ and (u, v) → (−v, −u). Motivated by this, we assume throughout the paper that µ > 0 and at a later stage we shall also assume that |u| > |v|. (1.13) The exclusion of |u| = |v| is required for our reduction treatment, while the choice (1.13) turns out to have technical advantages. The above specified domains D + and D + emerge from the reduction, but they can also be viewed as choices made to guarantee the strict positivity of all expressions under the square roots appearing in the Hamiltonians. A few remarks are now in order. The main HamiltoniansĤ and H are reminiscent of many-body Hamiltonians introduced by van Diejen [3]. The relation regardingĤ was made precise in [11] and regarding H it will be described in this paper. The coordinatesλ i and λ i serve as position variables forĤ and H, respectively, and we shall see that they yield globally smooth (and analytic) functions on the underlying phase space. Note that the deformation parameter that brings this dual pair into the one obtained by reduction of T * SU(2n) [10] is here set to unity. The cotangent bundle limits ofĤ and H are discussed in [11] and in [16]. Now we outline the content of the paper and highlight our main results. In Section 2.1, we first recall the Heisenberg double M equipped with the Abelian Poisson algebras H 1 and H 2 , then set up the pertinent reduction. In Section 2.2, we review the global modelM of the reduced phase space. The discussion in Section 2 sheds new light on several issues. Section 3 contains the logical outline of the construction of the global model M, which is our primary task. This is summarized by Figure 2 at the end of Section 3. The elaboration of the details required new ideas and a certain amount of labour: it occupies Section 4, Section 5 and Section 6.1. Our first main result is Theorem 5.6 in Section 5. Crucially, this theorem establishes the range of the λ-variables that arises from the reduction. Building on the local results of [16], it also yields the Darboux chart (1.10) on a dense open submanifold of M red parametrized by (1.9). Our second main result is given by Theorem 6.5, which describes the symplectomorphism Ψ between (M, ω), cast as C n with its canonical symplectic structure, and (M red , ω red ). Combining Theorem 6.5 with previous developments, we explain in Section 6.2 that our reduction engenders a realization of the diagrams of Figure 1. We consider this to be our principal achievement. We also present consequences for the dynamics of the systems in duality in Section 6.2 and in Section 7. Section 7 is devoted to further discussion of the results and open problems. Finally, two appendices are included. The first one is purely technical, while in the second we clarify the connection between the Hamiltonian H (1.11) and van Diejen's five parametric integrable trigonometric Hamiltonians.

Preparations and review of known results
In this section, mostly based on [11], we recall the reduction of our interest and review the modelM of the reduced phase space found earlier. All manifolds in this article are viewed as real. Hence the expression "analytic" must always be understood to mean "real-analytic".
We shall focus on the C ∞ character of the manifolds and maps of our concern, but shall often also indicate their analytic nature by parenthetical remarks.

The master system and its reduction
We shall reduce the master phase space M := SL(2n, C). Here, SL(2n, C) is viewed as a real Lie group, and we also need its subgroups K := SU(2n), B := SB(2n), (2.1) where the latter is formed by upper triangular complex matrices with positive entries along the diagonal. Every element g ∈ M admits the alternative Iwasawa decompositions By using these, M is equipped with the Alekseev-Malkin [1] symplectic form To display the corresponding Poisson bracket, for any F ∈ C ∞ (M, R) we introduce the sl(2n, C)-valued left-and right-derivatives ∇F and ∇ ′ F by We prepare the linear operator on sl(2n, C), utilizing the projectors associated with the real vector space decomposition sl(2n, C) = K + B, (2.6) where K and B are the Lie algebras of K and B, respectively. The Poisson bracket reads (2.7) The structure described above is known [39,40] as the Heisenberg double of the standard Poisson-Lie group SU(2n). The Abelian Poisson algebra H 2 is defined as follows. Let P denote the space of positive definite Hermitian matrices of size 2n and determinant 1. Consider the ring C ∞ (P) K of smooth real function on P that are invariant with respect to the natural action of K on P given by conjugation of a Hermitian matrix by a unitary one. We set i.e., H 2 is the pull-back of C ∞ (P) K by the map M ∋ g → bb † ∈ P. A generating setĤ j for H 2 M is provided by the functionsĤ j having the form H j (g) =ĥ j (bb † ) withĥ j (bb † ) := 1 2 tr (bb † ) j for j = 1, . . . , 2n − 1. (2.9) The Hamiltonian vector field and the corresponding (complete) flow can be written down explicitly for anyĤ ∈ H 2 . After our reduction the n Hamiltonians descending from the functionsĤ 1 ,Ĥ 2 , . . . ,Ĥ n remain independent, and the many-body Hamiltonian displayed in (1.6) results fromĤ 1 .
To present the other Abelian Poisson algebra of interest, H 1 , we define the matrix where 1 n is the n × n unit matrix, and introduce the subgroup Let C ∞ (K) K + ×K + denote those functions on K that are invariant with respect to both leftand right-multiplications by elements of K + . Then, referring to the Iwasawa decomposition (2.2), we define A generating set is furnished by the functions H j given by We recall that C ∞ (K) carries a natural Poisson bracket associated with (2.7), for which the map g → k by (2.2) is a Poisson map. Explicitly, (2.14) Here the B-valued left-and right-derivatives, Df and D ′ f , of any f ∈ C ∞ (K) are defined analogously to (2.4). It is well-known that K is a Poisson-Lie group and K + < K is a Poisson-Lie subgroup of K with respect to this Poisson structure. The following lemma implies that H 1 is an Abelian Poisson algebra. Proof. Let us start by noting that every k ∈ K may be written in the form k = κ 1 ∆κ 2 , for κ 1 , κ 2 ∈ K + , and ∆ = Γ iΣ iΣ Γ , (2.15) where Γ = diag(cos q 1 , . . . , cos q n ), Σ = diag(sin q 1 , . . . , sin q n ) (2.16) If h 1 and h 2 are two (K + × K + )-invariant smooth functions on K, then their Poisson bracket is also (K + × K + )-invariant. Therefore it is enough to show that {h 1 , h 2 } vanishes at any point of the form ∆ given in (2.15). The (K + × K + )-invariance of h ∈ C ∞ (K) means that the B-valued left-and rightderivatives Dh, D ′ h have the form where we use the obvious 2 × 2 block-structure defined by I (2.10). On account of the identity we must also have (2.20) Applying this at k = ∆, we obtain where the dependence of A andÃ on ∆ is suppressed. This gives us the conditions (the first two from skew symmetry of the diagonal blocks, and the third-after a calculation-from comparison of the off-diagonal blocks) Let h 1 and h 2 be two (K + × K + )-invariant functions, and use A 1 ,Ã 1 and A 2 ,Ã 2 as in (2.18) for their derivatives. By substitution into the Poisson bracket (2.14) on K we get (2.22). Utilizing alternatively (ii) and (i) then gives The combination of (i) and (ii) yields Γ 2 AΣ 2 = Σ 2 AΓ 2 , and thence [Σ 2 , A] = 0. Applying this to the two expressions in (2.24) and then adding them, we have which completes the proof.
The Hamiltonian vector fields corresponding to the collective Hamiltonians H ∈ H 1 (2.12) are all complete. Actually the completeness is valid for any H ∈ C ∞ (M) given by H(g) = h(k) using the Iwasawa decomposition g = kb (2.2) and any h ∈ C ∞ (K). In this case the derivatives of H are related to the derivatives of h according to This implies that the integral curves g(t) = k(t)b(t) of the Hamiltonian vector field of H on M are determined by the 'decoupled' differential equationṡ The vector field on K occurring in the first equation is complete due to compactness of K.
After substituting a solution k(t) into the second equation, b(t) can be found (in principle) by performing a finite number of integrations: this is because of the triangular structure of the group B. Now, with the master phase space M and its two distinguished Abelian Poisson algebras H 1 and H 2 at our disposal, we summarize the reduction procedure that concerns us. The basic steps of defining a reduction are the specifying of the symmetry and of the constraints to be used. As our symmetry group, we take the direct product K + × K + and let it act on the phase space by the map This is a Poisson action if K + is endowed with its natural multiplicative Poisson structure inherited from (2.14) [39,40]. The momentum map generating this action sends g to the pair of matrices given by the block-diagonal parts of b L and b (2.2). The constraints restrict the value of the momentum map to a suitable constant. To define the constraints, we fix a positive number µ and a vectorv ∈ C n , and let σ denote the unique upper triangular matrix with positive diagonal entries that verifies (2.29) Then we impose the 'left-handed' momentum map constraint forcing b L to have the form and also impose the 'right-handed' momentum map constraint by requiring that with real parameters u and v subject to |u| = |v|. We use a 2 × 2 block-matrix notation corresponding to I (2.10), and thus χ L , χ are n × n complex matrices. The submanifold M 0 of M defined by these momentum constraints, is stable under the action of the 'gauge group' K + (σ) × K + , where According to general principles, the reduced phase space M red is the quotient It was shown in [11] that the 'effective gauge group' acts freely on M 0 , where Z diag 2n is the subgroup that acts trivially In other words, π 0 : M 0 → M red is a principal fibre bundle with structure group (2.35). It follows that M red is a smooth (and analytic) symplectic manifold, and we let ω red denote its symplectic form that descends from ω M . It is readily seen that all elements of H 1 and H 2 are invariant with respect to the group action Φ (2.28) on M, and thus they give rise to two Abelian Poisson algebras H 1 red and H 2 red on the symplectic manifold (M red , ω red ). Referring to equations (2.9), (2.13) and using the embedding ι 0 : M 0 → M as in Figure 1, the defining relations of the reduced Hamiltonians of our principal interest are and of course π * 0 (ω red ) = ι ⋆ 0 (ω M ). In the spirit of the general scheme outlined in the Introduction, our task now is to construct a suitable pair of models of M red . One model was already found before, and next we briefly recall it.

The modelM of M red and its consequences
The material presented in this subsection is extracted from [11], where proofs can be found.
As the first main step, a parametrization of a dense open submanifold of the reduced phase space by the domain D + × T n (1.4) was constructed, where the variablesλ i are related to the invariant ∆ (2.15) formed from k in g = kb ∈ M 0 by setting sin q i = exp(λ i ), (2.38) using that q n > 0 for g ∈ M 0 . It proves useful to combine theλ i ∈ R <0 and their canonical conjugatesθ i ∈ R/2πZ into complex variables by defining The variable Z is naturally extended ro run over the whole of C n , equipped with the symplectic form The main result of [11] says that is a model of the full reduced phase space (M red , ω red ) (2.34). In fact, one can construct a symplectomorphismΨ : The n-tuples (λ 1 , . . . ,λ n ) and (|Z 1 | 2 , . . . , |Z n | 2 ) yield analytic maps fromM to R n , which are related by an affine GL(n, Z) transformation. Explicitly, we havê The functions |Z j | 2 generate the obvious Hamiltonian action of the torus T n onM = C n . Namely, the flows of |Z 1 | 2 , . . . , |Z n | 2 with time parameters t 1 , . . . , t n act by the map The origin Z 1 = · · · = Z n = 0 is the unique fixed point of this action. Applying (2.15) and (2.38), the reduced Hamiltonians H red j ∈ C ∞ (M red ) that descend from the functions H j (2.13) are found to take the following form in terms of the modelM : where P j is the polynomial determined by the relations cos(2jq a ) = P j (exp(λ a )), exp(λ a ) = sin q a for 0 < q a ≤ π 2 . (2.47) That is, where {T m (x) | m = 0, 1, 2, . . . } are Chebyshev polynomials of the first kind, characterized by T m (cos ϕ) = cos(mϕ). Altogether, we see that theλ j , or equivalently the |Z j | 2 , are action variables for the Liouville integrable system defined by the reduced Hamiltonians H red 1 , . . . , H red n . The subset ofM on which n j=1 Z j = 0 is mapped by (2.44) onto the boundary of the closure of D + , with Z = 0 corresponding to the vertex The point Z = 0 is a common equilibrium for the Hamiltonians H red j •Ψ −1 . Moreover, H red 1 •Ψ −1 reaches its global minimum onM at Z = 0. This follows from the fact that cos(2q a ) is monotonically decreasing for 0 < q a ≤ π 2 and the joint maxima of the q a for a = 1, . . . , n is reached at the vertex (2.49) corresponding to Z = 0.
On the dense open subset parametrized by D + × T n , the flow generated by H red where {U m (x) | m = 0, 1, 2, . . . } are Chebyshev polynomials of the second kind, characterized by U m (cos ϕ) = sin((m + 1)ϕ)/sin(ϕ). It is obvious that for genericλ and any fixed j the frequenciesΩ are independent over the field of rational numbers, and therefore the flow of H red j •Ψ −1 densely fills the generic Liouville tori. This implies that every element in the commutant of H red On the full phase spaceM, the flow generated by the function H red j •Ψ −1 has the following form: where hereλ is evaluated on the initial value Z(0). As for the reduced HamiltoniansĤ j :=Ĥ red j •Ψ −1 descending fromĤ j (2.9);Ĥ ≡Ĥ 1 takes the Ruijsenaars-Schneider-van Diejen (RSvD) type many-body form (1.6) in terms of the variables (λ,θ). This Hamiltonian as well as all members of its commuting family yield analytic functions on the full reduced phase space modelled byM . Explicit formulae can be obtained following the lines of [11]. By using its analyticity and the asymptotic behavior where the particles are far apart, it can be shown that the determinant det dθĤ 1 , dθĤ 2 , . . . , dθĤ n is non-zero on a dense open subset of D + × T n . This not only implies the Liouville integrability of the HamiltoniansĤ j , but it shows also that the 2n functionsλ j ∈P andĤ j ∈Ĥ, for j = 1, . . . , n, are functionally independent. In particular, the Hamiltonian vector fields of the elements ofP andĤ together span the tangent space T mM at generic points m ∈M . As a consequence of the formula (2.46), {λ j } n j=1 and {H red j •Ψ −1 } n j=1 represent alternative generating sets for the algebraP of the global position variables.
Remark 2.1. In [11] the modelM was obtained under the assumptions that v > u and |u| = |v|, but now we find that essentially nothing changes if only |u| = |v| is assumed. The conditionλ 1 ≤ s = min(0, v − u) arises from the requirement that all entries of the n × n diagonal matrix given by [11] must be non-negative. Another difference is that [11] defined z 1 , . . . , z n−1 in the same way as (2.39), but introduced a variable z n instead of Z n (2.40) by This z n varies in the open disc D r of radius r = e s/2 , and is related to Z n ∈ C by an analytic diffeomorphism, which one can write down explicitly, for any s ≤ 0, if desired.

Constructing the model M of M red : general outline
The modelM of M red was obtained by explicitly constructing a global cross-section of the gauge orbits in M 0 . The construction of the new model M that we achieve in this paper is somewhat more complicated. We here collect the main concepts that will appear in the construction, hoping that this will enhance readability. The reader is recommended to keep an eye on Figure 2, placed at the end of the section. We shall describe the quotient M red (2.34) by exhibiting a new set of unique representatives for each orbit of the 'gauge group' K + (σ) × K + acting on M 0 . We now display M red as emphasising that (η L , η R ) ∈ K + (σ)×K + acts by left-and by right-multiplication, respectively. We shall arrange taking the quotient into convenient consecutive steps, using in addition to the obvious direct product structure of the gauge group also the fact that K + (σ) itself can be decomposed as the direct product where andv ∈ C n is the fixed vector in (2.29). It is easy to check that every element of K + (σ) can be written as a product of these two disjunct, mutually commuting subgroups.
As in [16], and define the subset M 1 of the 'constraint surface' M 0 ⊂ M by The 'left-handed' gauge transformations by K + (σ) map M 1 to itself and by using this we introduce the quotient It will be useful to identify N with the image of the map and thus operate on the constituent χ (2.31) of b according to where we employ the block-matrix notation Recalling the singular value decomposition of n×n matrices, we observe from (3.10) that every element g ∈ M 0 can be gauge transformed into M 1 , and the components β i of the resulting element of M 1 are uniquely determined by g. To proceed further, we restrict ourselves to the 'regular part' defined by the strict inequalities We call such β and the corresponding quasi-diagonal b regular, and apply the notations and M reg red = K + (σ)\M reg 0 /K + . Later it will emerge that in fact M 0 = M reg 0 . If β is regular, then the corresponding b in (3.5) is fixed by the following Abelian subgroup, T n−1 , of K + : We shall also use the subgroup of T 1 × T n−1 given bỹ where Z 2n denotes the (2n) th roots of unity and we employ the notation (3.3). Defining which is provided by the map using the above parametrizations of the elements of T 1 (3.3), T n−1 (3.13) and T n (3.15). After these preparations, we come to the main points. First, we let δ ∈ T n−1 act on and also let η R ∈ K + act by Then we introduce the identification which is invariant with respect to the action (3.18) of T n−1 . Since the actions of T n−1 and where on the right-end we refer to the action of T n−1 on M reg 1 given by M reg 1 ∋ kb → kbδ −1 = kδ −1 b. We continue by applying the decomposition (3.2) of K + (σ) to deduce the identification where we have taken into account that N reg = K + (ŵ)\M reg 1 (see (3.7)). The action of T 1 × T n−1 on N reg factors through the homomorphism (3.17). The induced action of T n (3.15) on N reg is given, in terms of the triples (w, L, β) in (3.8) representing the elements of N reg , by the formula One sees this from the definitions in (3.8) and in (3.17) The final outcome is the following identification: The action (3.24) of T n on N reg is actually a free action. This is a consequence of the fact [11] that the action of (K + (σ) × K + )/Z diag 2n on M 0 is free. Remark 3.1. Every element of M 0 can be mapped into M 1 by a gauge transformation, which is unique up to residual gauge transformations acting on M 1 . It is a useful fact that locally, in a neighbourhood of any fixed element of M reg 0 , a well-defined map f 0 can be chosen, in such a manner that the gauge transformed matrix g 1 depends analytically on the local coordinates on the manifold M reg 0 . We next explain this statement.
Let P reg denote the manifold of n × n Hermitian matrices having distinct, positive eigenvalues, and G reg denote the open subset of GL(n, C) diffeomorphic to P reg ×U(n) by the polar decomposition, presented as Here, p(χ) and u(χ) depend analytically on χ. Let D reg ⊂ P reg denote the manifold of real diagonal matrices β = diag(β 1 , . . . , β n ) satisfying β 1 > · · · > β n > 0. We recall that P reg is diffeomorphic to D reg × (U(n)/T(n)) by the correspondence with the standard maximal torus T(n) < U(n). Invoking the fact [20] that U(n) is a locally trivial bundle over the coset space U(n)/T(n), we see that ξ(p) ∈ U(n) in (3.28) can be locally chosen to be a well-defined, smooth function of p. Now introduce ζ(χ) := (det(ξ(p(χ)) −2 u(χ))) 1 2n , choosing it so as to give a smooth function locally around a fixed χ at hand. As the final outcome, a locally well-defined map f 0 (3.26) is obtained as follows: Since χ depends analytically on g, the local choices guarantee that g 1 (g) depends analytically on the coordinates on M reg 0 . We remark in passing that β 2 n resulting from (3.10) is the smallest eigenvalue of χχ † , and therefore β n is not a smooth function on M 0 at those points where it vanishes. As we shall see later (from equation (4.12) and Theorem 5.6), the assumption (1.13) excludes this eventuality.
In the above we established the various identifications only at the set-theoretic level. Although we shall not rely on it technically, we wish to note that all above identifications hold in the category of smooth (and analytic) manifolds as well. We next prove a lemma, which implies that M reg 1 is an embedded submanifold of M 0 ; itself known-from [11]-to be an embedded submanifold of M. Utilizing the assumption (1.13), it will be shown later that M reg 1 = M 1 . Then it follows that M 1 ⊂ M 0 represents a reduction of the structure group (2.35) of the principal fibre bundle M 0 over M red to the subgroup (K + (σ) × T n−1 )/Z diag 2n , and N is a principal fibre bundle over M red with structure group T n , in the standard sense [20]. where ξ is any n × n complex matrix with real diagonal, and we use χ in (2.31). LetM reg 1 ⊂ M 1 consist of the elements g = kb with b of the form (3.5), but β ∈ R n now restricted by β i = β j for i = j and n i=1 β i = 0. Then the exterior derivatives of the functions φ ξ are linearly independent at each point ofM reg 1 . Proof. Take an arbitrary g ∈M reg 1 and note that the infinitesimal gauge transformations by the elements of Lie(K + ) generate a (2n − 1)n dimensional subspace of the tangent space T g M 0 , which coincides with the dimension of the real linear space of the matrices ξ. A general element of Lie(K + ) is a matrix of the form diag(X, Y ), X, Y ∈ u(n), tr(X + Y ) = 0, (3.31) and denoting the induced tangent vector by V (X,Y ) (g), we find the derivative One can easily check that this derivative vanishes for every (X, Y ) if and only if ξ = 0. This means that the exterior derivatives dφ ξ (g) span a (2n − 1)n dimensional subspace of T * g M 0 at each g ∈M reg 1 , which establishes the claim. The statement of the lemma is non-trivial only ifM reg 1 is non-empty, which turns out to hold. We then also have the non-empty open subsetM reg 0 , which is defined by the condition that the real parts of the diagonal entries of χ are pairwise distinct and non-zero. The lemma implies directly thatM reg 1 is an embedded submanifold ofM reg 0 , and hence it is an embedded submanifold of M 0 , too. Finally, we see that M reg 1 (specified by (3.12)) is itself an open subset ofM reg 1 . It turns out non-empty, and is therefore also an embedded submanifold of M 0 .
Eventually, we shall obtain the desired model M of M red as an explicit global crosssection for the action of T n on N = N reg . We shall use Remark 3.1 to show the analyticity of the natural map from M 0 onto this cross-section. This will enable us to prove that the construction gives a model of the symplectic manifold (M red , ω red ). The procedure is summarized in the following commutative diagram: . More precisely, we shall proceed with the help of new variables (w, Q, λ) equivalent to (w, L, β). The usefulness of this characterization lies in the fact that we will be able to describe all solution of the constraint equation (4.14) explicitly, and shall rely on this to construct the desired model M of M red (3.25). We start by recalling a lemma from [16]. whereŵ ∈ C 2n is the fixed vector introduced in (3.4).
Proof. Irrespective of the constraints, b L in g = b L k R ∈ M can be written as and the left-handed constraint requires that b 2 = y1 n and b 1 = y −1 σ. By simply spelling it out for g = b L k R , the matrix on the L.H.S. of (4.1) reads explicitly as The equality of the bottom-right blocks on the two sides of (4.1) is equivalent to b 2 = y1 n . Then the off-diagonal blocks on both sides are zero, while (using (2.29)) the top-left block boils down to the equality b 1 b † 1 = y −2 σσ † , which implies the statement. From now on we work on M 1 ⊂ M 0 (3.6). Taking any quasi-diagonal b, it will prove useful to diagonalize the positive definite matrix Let us introduce the real functions s(t) and c(t) by the formulae 2 which imply the identity c 2 (t) + s 2 (t) = 1. Then consider λ ∈ R n subject to the condition λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ |v|, C(λ) = diag(c(λ 1 ), . . . , c(λ n )), S(λ) = diag(s(λ 1 ), . . . , s(λ n )), (4.8) and the matrix Notice that ρ(λ) is real, symmetric and orthogonal, Here and throughout the paper, the suffix star on matrices and vectors denotes complex conjugate, and dagger denotes Hermitian adjoint.
Lemma 4.2 [16]. For any quasi-diagonal b given by (3.5), bb † can be written as where β is related to λ according to the one-to-one correspondence Now we are ready to formulate the main result of this section.
Proposition 4.3. Take an arbitrary g = kb ∈ M 1 (3.6) for which β and λ are connected by (4.12), and (using L and w from (3.8)) define Q ∈ U(2n) andw ∈ C 2n by Then the matrix Q and the vectorw satisfy the constraint equation and the relationsw †w = α 2 (α −2n − 1), Qw =w. Conversely, pick λ ∈ R n verifying (4.6) and suppose that a matrix Q ∈ U(2n) and a vector w ∈ C 2n satisfy (4.14) as well as the relations (4.15) and the condition that Q is conjugate to I (2.10). Then there exists g = kb ∈ M 1 from which Q andw can be constructed according to (4.13), and such g is unique up to left-multiplication by the elements of the subgroup K + (ŵ) (3.4) of the left-handed gauge group K + (σ).
In order to prove the converse, which gives the reconstruction of g ∈ M 1 from λ, Q andw, we start by noting that if Q ∈ U(2n) is conjugate to I (2.10), then we can a find an element κ ∈ K for which ρQρ −1 = κ † Iκ, where ρ := ρ(λ). (4.16) Next, we observe that the auxiliary condition Qw =w is equivalent to By using (4.17) and the property thatw †w = α 2 (α −2n −1), we see that there exists an element k + ∈ K + for which k + κρw =ŵ. (4.18) Let us now define g = kb by using k := k + κ (4. 19) together with the quasi-diagonal b associated to λ via (4.12). Then routine manipulations show that equation (4.14) implies for g the left-handed momentum map constraint (4.1). Now let us inspect the ambiguity in the above constructed k, and thus in g. If κ ′ and k ′ + represent another choice in the above equalities, then we have and thus Therefore and hence k ′ + η + =η L k + for someη L ∈ K + (ŵ). (4.23) This entails that that is, k and g are unique up to left-multiplication by the isotropy subgroup of the vectorŵ in K + .
Definition 4.4. Let us call a triple (w, Q, λ) ∈ C 2n ×C 2n×2n ×R n admissible if λ satisfies (4.6), the constraint equation (4.14) holds, Q is unitary, conjugate to I (2.10), and the auxiliary conditions (4.15) are met. Denote by N (λ) the set of admissible triples associated with fixed λ, and let M 1 (λ) stand for the subset of M 1 corresponding, by Proposition 4.3, to the admissible triples with fixed λ. Moreover, denote by M 0 (λ) the subset of M 0 whose elements can be gauge transformed into M 1 (λ). Finally, denote by D(u, v, µ) the set of the admissible λ variables, i.e., those that appear in admissible triples.
It is clear from the relations (4.12) and (4.13) that the triple (w, Q, λ) is equivalent to the triple (w, L, β) (in the obvious sense that one can be expressed in terms of the other). By using this equivalence, and Proposition 4.3, we identify N (λ) as defined above with the image of the map (3.8), with β taking the value (4.12).
In order to construct the desired model of M red , we need to describe all admissible triples (w, Q, λ). A crucial part of this problem is to find the admissible λ, which parametrize the eigenvalues of bb † for g = kb ∈ M 0 . These eigenvalues, and thus also the components of λ, can be viewed as continuous functions on M 0 , and we are looking for the range of the corresponding map, L, D(u, v, µ) = L(M 0 ) with L : g → λ. (4.30) In the following section, we shall describe D(u, v, µ) and the corresponding solutions of (4.14) explicitly. See Theorem 5.6 for the result.
We can explain at this point why an open subset of the reduced phase space can be parametrized by the λ i together with n angular variables; which appear in (1.8). To this end, let us take an arbitrary element e iξ ≡ diag(e iξ 1 , . . . , e iξ 2n ) (4.31) from the torus T 2n , and notice that if (w, Q, λ) is admissible, then so is (e iξw , e iξ Qe −iξ , λ), ∀e iξ ∈ T 2n . (4.32) Indeed, the conditions described in Proposition 4.3 are respected by these transformations. In addition to the gauge transformations by τ ∈ T n in (4.29), these T 2n transformations involve n arbitrary angles, which parametrize T 2n /T n . It is clear that, for generic λ, equation (4.14) permits the expression of Q in terms of λ andw. Moreover, we shall see shortly that the |w a | can be expressed in terms of λ, and generically none of them vanish. This implies that generically the elements of N (λ)/T n can indeed be parametrized by n-angles.
Remark 4.5. We know that the T n action on N (λ) is free, and shall also confirm this explicitly later. Moreover, it will turn out that the T 2n action, sending (w, Q, λ) to (4.32), is transitive on N (λ); and is also free except for a certain lower dimensional subset of the admissible λ values.

Solution of the constraints
Locally, the general solution of the constraint equation (4.14) was already found in [16]. Here, 'locally' means that the form of the domain of the λ-variables was not established. In this section, we shall prove that D(u, v, µ) (4.30) is the closure of D + in (1.9), as was anticipated in [16]. Moreover, we shall describe all admissible triples forming N (3.7) explicitly. When combined with the local results of [16], this yields a model of the reduced system coming from the Abelian Poisson algebra H 1 (2.12) restricted to a dense open submanifold, and will permit us to derive the desired global model M of M red in Section 6. For technical reasons that will become clear shortly, we initially work on a certain dense open subset of M 0 . To define this subset, let us consider the following symmetric polynomials in 2n indeterminates: and Since M 0 (2.32) is a joint level surface of independent analytic functions on M, it is an analytic submanifold of M, and thus we obtain analytic functions on M 0 if we substitute the eigenvalues Λ k (g) of gg † = kbb † k −1 into the above polynomials. This follows since, being symmetric polynomials in the eigenvalues, the p i (Λ(g)) can be expressed as polynomials in the coefficients of the characteristic polynomial of gg † . We know that M 0 is connected and, as explained in Remark 5.1, can also conclude that p(g) := p 1 (Λ(g))p 2 (Λ(g)) (5.3) does not vanish identically on M 0 . By analyticity, this implies that is a dense open subset of M 0 . We call its elements strongly regular. We shall apply the same adjective to the λ-values for which (using (4.7)) p(Λ(λ)) = 0, and call also strongly regular the corresponding admissible triples (w, Q, λ), whose set is denoted N sreg . The admissible strongly regular λ-values form the dense subset Focusing on N sreg , we introduce the 2n × 2n diagonal matrices W := diag(w 1 , . . . ,w 2n ), and the Cauchy-like matrix C, The denominators do not vanish since λ is strongly regular. The constraint equation (4.14) leads to the following formula for the matrix Q: Since Q is conjugate to I (2.10), Q 2 = 1 2n holds, and this translates into Let us observe that the matrix W is invertible whenever λ is strongly regular. Indeed, if some componentw a = 0, then (5.11) yields D 2 a = 1, which is excluded by strong regularity. Next, we substitute (5.10) into the equation Qw =w in (4.15), which gives Dividing byw j produces the formula (5.13) where C −1 is the inverse of the matrix C (5.9) and we took into account (4.7). This expresses the moduli |w j | as functions of λ.
Using the parameter µ instead of α = e −µ , define the 2n functions (5.14) as well as the functions and depend only on λ that parametrizes the eigenvalues of bb † according to (4.7) and (4.11). Explicitly, these functions are given by the relation with the functions F j (5.15). The component Q in any admissible strongly regular triple (w, Q, λ) ∈ N sreg can be written as (5.10), where the phases of the entries ofw ∈ C 2n can be chosen arbitrarily.
Proof. For a strongly regular admissible λ, the formula (5.16) is a reformulation [16] of (5.13). It remains valid on the whole of M reg 1 , since the functions on the two sides of (5. in M 0 ). In the strongly regular case, the formula (5.10) for Q was derived above. The phases ofw j can take arbitrary values, because one can use arbitrary e iξ ∈ T 2n in equation (4.32).
The definitions guarantee the positivity of |w j |(λ) for every λ ∈ D(u, v, µ) sreg (see below (5.11)). Thus, the explicit formula (5.16) leads to a necessary condition on λ to belong to the (still unknown) set D(u, v, µ) sreg . Indeed, our aim below is to identify the 'maximal domain' on which the functions F j as given by the formula (5.15) are positive. More precisely, we are interested in the set We stress that in this definition λ is not assumed to be admissible or strongly regular; the formula (5.15) is used to define F j (λ) for the λ that occur. Next, we shall give the elements of D + (u, v, µ) explicitly. After that, we shall prove that D(u, v, µ) (4.30) is the closure of D + (u, v, µ). Our notation anticipates that the definition (5.17) turns out to give the set (1.9).
We now observe that if λ i − λ i+1 > µ for all i, then F n+a (λ) > 0 is valid for all a = 1, . . . , n as well. Therefore the positivity of F 2n (λ) requires that (e −2u − e −2λn ) > 0, (5.27) which in the case u > 0 enforces that λ n > |u|. At this stage, the proof is complete whenever (5.20) is guaranteed. Therefore, it only remains to show that λ n > |u| must hold also when |u| > |v| and u < 0. This follows from Lemma 5.4 below.
In the above it was assumed that n > 2, but the arguments are easily adapted to cover the n = 2 case, too.
The only other source of potential singularity of Q lm (5.38) is the vanishing of the denominators of D 2n,2n (5.8) and C 2n,2n (5.9) as λ n tends to µ/2. This may be excluded by the form of D + (u, v, µ), but when it is not excluded then one can check easily that these poles cancel against each other. The continuity of the resulting functions on D + (u, v, µ) × T 2n and their analyticity on the interior also follow immediately from their explicit formulae. The statements regarding ρ(λ) and β(λ) are plainly true.
The following theorem summarizes one of our main results.
This yields a symplectomorphism between M + red equipped with the restriction of ω red and the product manifold D + (u, v, µ) × T n equipped with the symplectic form n j=1 dθ j ∧ dλ j . Proof. In what follows, we first show that all triples given by Lemma 5.5 are admissible, that is, they represent elements on N . In particular 3 , D defined in (4.30) is the closure of D + in (5.18). Then we apply a density argument to demonstrate that the admissible triples of Lemma 5.5 exhaust N . Finally, we explain the statement about the model of the subset M + red of M red . We have seen that for any λ ∈ D sreg ⊂ D every admissible triple (w, Q, λ) is of the form (5.38), and we also know that D sreg is a non-empty open subset of D + . By noting that the triple (4.32) is admissible whenever (w, Q, λ) is admissible, we conclude that the conditions on admissible triples formulated in Definition 4.4 are satisfied by the triples given by (5.38) with (λ, e iξ ) taken from the open subset D sreg × T 2n ⊂ D + × T 2n . Because these conditions require the vanishing of analytic functions, they must then hold on the connected open set D + × T 2n , and by continuity on its closure as well. Thus, we have proved that all triples given by Lemma 5.5 are admissible. On account of (5.36), this implies that D = D + .
We now show that Lemma 5.5 gives all admissible triples. To this end, let us choose an admissible triple, denoted (w ♯ , Q ♯ , λ ♯ ), for which λ ♯ ∈ (D \ D sreg ). This corresponds by equation (4.13) to some element g 1♯ ∈ M 1 , which is obtained by a right-handed gauge transformation from some element g ♯ ∈ M 0 . We fix g 1♯ and g ♯ . We can find a sequence g(j) ∈ M sreg 0 that converges to g ♯ , because M sreg 0 is a dense subset of M 0 . It is easy to see that the sequence g(j) can be gauge transformed into a sequence g 1 (j) ∈ M 1 (3.6) that converges to g 1♯ . (This follows from the continuous dependence on g of the eigenvalues β 2 i of χχ † , where χ is the top-right block of b from g = kb ∈ M 0 .) The convergent sequence g 1 (j) ∈ M 1 corresponds by equation (4.13) to a sequence (w(j), Q(j), λ(j)) of strongly regular admissible triples that converges to (w ♯ , Q ♯ , λ ♯ ). Then, as for any λ ∈ D sreg every admissible triple is of the form (5.38) , we obtain a sequence (λ(j), e iξ (j)) ∈ D sreg × T 2n that obeys lim j→∞ w λ(j), e iξ (j) , Q λ(j), e iξ (j) , λ(j) = (w ♯ , Q ♯ , λ ♯ ) . (5.43) By the compactness of T 2n , possibly going to a subsequence, we can assume that e iξ (j) converges to some e iξ ♯ . By the continuous dependence of the triple in Lemma 5.5 on (λ, e iξ ), it finally follows that i.e., every admissible triple is given by Lemma 5.5. It remains to establish the symplectomorphism between M + red in (5.41) and D + × T n . Before going into this, we need some preparation. We first note M + red is open subset of M red since D + is an open subset of R n and L : M 0 → R n defined in (4.30) is a continuous, gauge invariant map, which descends to a continuous map from M red to R n . As a consequence of (5.35), M + red is dense in M red . It is also true that L is an analytic map, because its components are logarithms of eigenvalues of gg † , and (5.36) ensures that the eigenvalues of gg † are pairwise distinct positive numbers for any g ∈ M 0 . Let us define M + 0 := L −1 (D + ), and introduce also M + 1 := M 1 ∩M + 0 , as well as the subset N + ⊂ N consisting of the admissible triples (w, Q, λ) for which λ ∈ D + . Finally, let S + ⊂ N + stand for the set of admissible triples parametrized by D + × T n using (5.38) with λ ∈ D + and the phases e iξa ofw a satisfying (5.42).
Any admissible triple (w, Q, λ) ∈ N + is gauge equivalent to a unique admissible triple in S + , parametrized by (λ, e iθ ) ∈ D + × T n with e iθ j =w jw * j+n |w jwn+j | , j = 1, . . . , n. By this formula, we can view e iθ as a gauge invariant function on N + , and we also obtain the identification N + /T n ≃ S + with respect to the gauge action in (4.29). Now we define a map by composing a gauge transformation f 0 : M + 0 → M + 1 with the map π 1 : M + 1 → N + given by equation (4.13), and with the map N + → S + operating according to (5.45). (The notations are borrowed from Figure 2. See also Remark 3.1.) Since the λ-values belonging to D + are regular, the map ψ + is smooth (even analytic). It is obviously gauge invariant, surjective and maps different gauge orbits to different points. Therefore ψ + descends to a one-to-one smooth map Ψ + : M + red → D + × T n . It was shown in [16] (without explicitly specifying the domain D + in the calculation) that Ψ + satisfies with the restriction ω + red of the reduced symplectic form on M + red ⊂ M red . In particular, the Jacobian determinant of Ψ + is everywhere non-degenerate, and therefore the inverse map is also smooth (and analytic).
We finish this section with a few remarks. The strong regularity condition was employed to ensure that we never divide by zero in the course of the analysis. The non-vanishing of p 1 (5.1) and the first factor of p 2 (5.2) prevents zero denominators in (5.8), (5.9) and (5.14). The non-vanishing of the second factor of p 2 was used in the argument below (5.11). The last two factors of p 2 exclude the vanishing of the functions F k (5.15) or of a component of ρ (4.9), which are not differentiable at those excluded values of λ on account of some square roots becoming zero.
Notice from (5.45) that (because of vanishing denominators) the variables e iθ j cannot all be well-defined at such points where λ belongs to the boundary of D.
Up to this point in the paper, we have not used the assumption (1.13). We shall utilize it in the following section, where we introduce new variables that cover also the part of M red associated with the boundary of D. Imposing |u| > |v| ensures, by virtue of D = D + (5.37), that the regularity condition (3.12) holds globally, since λ n > |v| is equivalent to β n > 0. This in turn ensures, by the arguments developed in Section 3 and Section 4 (see (3.25) and (4.26)), that we have the identification If |v| > |u|, then β n = 0 corresponding to λ n = |v| is allowed for elements of M 1 . As mentioned after equation (3.29), this would complicate the arguments. Also, if β n = 0, then the corresponding isotropy group K + (λ) that appears in (4.26) is larger then T n−1 in (3.13). The desire to avoid these complications, together with the symmetry mentioned above (1.13), motivates adopting this assumption in Section 6. Finally, we recall from [16] that the reduction of H 1 (2.13) gives the RSvD type Hamiltonian (1.11) in terms of the Darboux variables (λ, e iθ ).

The global model M of M red and consequences
We construct the global model M by bringing every admissible triple (w, Q, λ) ∈ N to a convenient normal form. We then present consequences for our pair of integrable systems.

Construction of the model M of M red
Adopting the assumption (1.13), we start with the observation that most (but not all) functions |w a |(λ) contain a factor of the form λ j − λ j+1 − µ, j = 1, . . . , n − 1, λ n − |u|, (6.1) multiplied by a function of λ which is strictly positive and analytic in an open neighbourhood of D ≡ D(u, v, µ). On account of the formula (5.38), the moduli of the components of Q depend only on λ, and for certain indices they are strictly positive, analytic functions. The precise way in which this happens depends on the sign of u, and now we assume for concreteness that |u| > |v| and u < 0. (6.2) We shall comment on the modifications necessary when this does not hold.
All components of ρ(λ) (4.9) are also analytic functions in a neighbourhood of D.
It is straightforward to write explicit formulae for the functions f i and f j+1,n+j . We shall not use them, but for completeness present some of them in Appendix A. Here, we note only that, as was pointed out in the proof of Lemma 5.5, the vanishing denominators of C j+1,n+j in Q j+1,n+j are cancelled by a zero ofw j+1w * n+j , for any j. Analogous formulae can be written for all matrix elements of Q. The only non-displayed matrix element of Q that never vanishes is Q 1,2n .
The factors (6.1) lose their smoothness when they become zero, which happens at the boundary of D. This is analogous to the failure of the function f : C → R given by f (z) = |z| to be differentiable at the origin in C. Our globally valid new variables will be n complex numbers running over C, whose moduli are the factors (6.1). Before presenting this, let us remark that in terms of a complex variable the standard symplectic form on R 2 ≃ C can be written (up to a constant) as idz ∧ dz * , and the equality idz ∧ dz * = dr 2 ∧ dφ with z = re iφ (6.5) holds on C * = C \ {0}. This may motivate one to introduce new Darboux coordinates on D + × T n like in the next lemma. The symplectic form that appears in (5.47) satisfies Extending the definition (6.6) to D × T n , the boundary of D corresponds to the subset of C n on which n i=1 ζ i = 0. Since we know that the boundary of D is part of the admissible λ values, it is already rather clear that ζ i as defined above extend to global coordinates on M red . Nevertheless, this requires a proof. The proof will enlighten the origin of the complex variables ζ i .
It is clear from Lemma 6.1 that for any (w, Q, λ) ∈ N there exists a unique gauge transformation 4 (4.29) by τ = τ (w, Q, λ) ∈ T n (3.15) such that for the gauge transformed triple the first and last components of τw are real and positive and the components (τ Qτ −1 ) j+1,j+n are real and negative for all j = 2, . . . , n − 2. (The choice of negative sign stems from (5.39).) This map can be calculated explicitly. By using this, we are able to obtain an analytic, gauge invariant map from M 0 onto C n , which gives rise to a symplectomorphism between M red and C n . Below, we elaborate this statement. Definition 6.3. Let S ⊂ N be the set of admissible triples, denoted (w S , Q S , λ), satisfying the following gauge fixing conditions: As in the proof Theorem 5.6, let S + ⊂ N + denote the set of admissible triples parametrized by D + × T n using (5.38) with λ ∈ D + and the phases e iξa ofw a satisfying (5.42).
We know that S + defines a unique normal form for the elements of N + ⊂ N , and S defines a unique normal form for the whole of N . For any (w, Q, λ) ∈ N , we define the n phases X 1 , X n , X j+1,n+j ∈ U(1) by writing for every j = 1, . . . , n − 2. The map (w, Q, λ) → (w S , Q S , λ) sends any admissible triple to the intersection of its T n orbit (defined by (4.29)) with S, which is given by ,n+i (6.10) for j = 2, . . . , n − 1. This yieldsw S and Q S as gauge invariant functions on N , and by using them we can define the C n valued gauge invariant map π N : (w, Q, λ) → ζ on N as follows: ζ j (w, Q, λ) :=w S n+j /f n+j (λ), j = 1, . . . , n − 1, ζ n (w, Q, λ) := (Q S n,2n−1 ) * /f n,2n−1 (λ). (6.11) For the remaining components of the functionw S given by (6.10), we find with the functions of λ in (6.3), and of coursew S 1 = f 1 (λ) andw S 2n = f 2n (λ). The function Q S (6.10) is given by substitutingw S forw in the formula (5.38).
Equation (6.12) can be checked by writing every (w, Q, λ) in terms of (λ, e iξ ) ∈ D × T 2n as in (5.38), cf. Lemma 5.5. By applying this, we obtain, for j = 1, . . . , n − 1, e −iξ l e iξ n+l and ζ n = λ n − |u| n l=1 e −iξ l e iξ n+l . (6.13) This shows manifestly that the range of ζ covers the whole of C n . If we restrict this formula to S + , parametrized by D + × T n using (5.42), then we recover our previous formulae (6.6). We now summarize these claims.
Proposition 6.4. The T n gauge invariant map π N : (w, Q, λ) → ζ exhibited in (6.11) induces a bijection between N /T n and C n . The restriction of the component functions ζ i to S + ⊂ N is given by the formula (6.6). The inverse map from C n to S ≃ N /T n can be written down explicitly by first expressing λ in terms of ζ as λ j = |u| + (n − j)µ + n l=j |ζ l | 2 , j = 1, . . . , n, (6.14) then expressingw S by means of ζ using (6.11) and (6.12), and finally obtaining Q S as a function of ζ via substitution ofw S (ζ) forw in the formula (5.38).
Proof. The surjectivity onto C n was explained above, and the injectivity is clear because we can explicitly write down the inverse from C n onto the global cross-section S of the T n action on N .
Our main theorem says that the construction just presented gives a global model of M red : Theorem 6.5. Take an arbitrary element g 0 ∈ M 0 and pick g(g 0 ) to be an element of M 1 which is gauge equivalent to g 0 . Then define the map ψ : M 0 → C n by the rule ψ : g 0 → ζ (w(g(g 0 )), Q(g(g 0 )), λ(g(g 0 ))) , (6.16) combining (6.11) with the map M 1 ∋ g → (w, Q, λ) ∈ N given by equations (4.12) and (4.13). The map ψ is analytic, gauge invariant and it descends to a diffeomorphism Ψ : M red → C n having the symplectic property Ψ * (ω can ) = ω red . (6.17) Proof. Since it does not depend on the choice for g(g 0 ), the analyticity of ψ follows from the possibility of an analytic local choice (see Remark 3.1) and the explicit formulae involved in the definition (6.16). Its bijective character is a direct consequence of Proposition 6.4. The symplectic property follows from Theorem 5.6 and a density argument. Namely, on M + red we can convert Ψ + satisfying (5.47) into Ψ by means of the map (λ, e iθ ) → ζ as given by (6.6). This and Lemma 6.2 imply the equality (6.17) for the restriction of Ψ on M + red , and then the equality extends to the whole space by the smoothness of Ψ, ω can and ω red . As a consequence of (6.17), the inverse map is smooth as well.
Remark 6.6. The formulae of the complex variables used in Section 2.2 can be converted into those applied in this section by introducing new 'tilded variables' as λ j := −λ n+1−j + c,θ j := −θ n+1−j ,Z k := Z n−k ,Z n = Z n , (6.18) for j = 1, . . . , n and k = 1, . . . , n − 1. ThenZ depends onλ,θ by the same formula (6.6) whereby ζ depends on λ, θ. By choosing the constant c appropriately, the domain ofλ also becomes identical to the domain of λ. Remark 6.7. As promised, we now comment on the modification of the construction for the cases when (6.2) does not hold. If instead we have |u| > |v| and u > 0, then the definition (6.6) is still applicable, but (5.15) implies that the factor λ n − |u| is contained in |w 2n | instead of |w n |, and thus |Q n,2n−1 | does not contain this factor (cf. (6.3)). Then one may proceed by defining a global cross-section S ⊂ N with the help of the gauge fixing conditions w S 1 > 0 and Q S j+1,n+j < 0 for all j = 1, . . . , n − 1 (cf. (6.8)). The construction works quite similarly to the above one, and all consequences described in the next subsection remain true. As was discussed in the Introduction, we can impose (1.13) without loss of generality. Nevertheless, it could be a good exercise to detail the construction of the counterpart of our model M when (1.13) does not hold. We only note that one must then define ζ n in such a way that |ζ n | = λ n − |v| and use that, on account of (4.5), this factor is contained in a matrix element of ρ(λ) (4.9).

Consequences of the model of M and the duality map
cosh(2jλ l ). (6.23) Thus, the functions λ l : M → R are action variables for the (completed) integrable manybody system (1.6) onM. The actions λ l are related by a GL(n, Z) transformation combined with a constant shift to the distinguished action variables defined by the functions |ζ i | 2 on M = C n . These latter action variables generate the standard T n action on M = C n . The origin ζ = 0 is a fixed point for the torus action, and it represents the unique joint minimum of the Hamiltonians (6.23). Moreover, this is the only equilibrium point that any single Hamiltonian of the form (6.23) possesses. It follows from the above that R(0) ∈M is a joint equilibrium point for the Hamiltonianŝ H red j •Ψ −1 . It also follows that each HamiltonianĤ red j •Ψ −1 is non-degenerate (has no extra conserved quantities), because this property of the equivalent Hamiltonians (6.23) is easily seen. Of course, one can write down the analogues of equations (2.50) -(2.53) for the flows of the Hamiltonians (6.23) on M. For any fixed j, the counterparts of the n frequencies (2.52) are given by Ω j,a (λ) = 2j sinh (2jλ a ), which generically are independent over the field of rational numbers. The existence of an equilibrium point forĤ ( A special feature of the dual pair at hand is that the action-angle phase spaces (M, ω) and (M ,ω) are also the same in an obvious manner, namely, both are equal to (C n , ω can ). Distinguished action variables of both systems generate the standard torus action on C n ≃ R 2n equipped with its canonical symplectic form. It is by no means true that every Liouville integrable system corresponds to a globally well-defined Hamiltonian torus action, and for global torus actions there could be several inequivalent possibilities. Integrable many-body systems in action-angle duality live on symplectomorphic phase spaces, but their respective action variables cannot in general be intertwined by a symplectomorphism. Apart from the current example and self-dual systems, such an action-intertwining symplectomorphism was previously found only for dual pairs of purely scattering systems, such as the hyperbolic Sutherland system and its Ruijsenaars dual [33], and the analogous BC n systems [28].
It may be worth stressing that the duality map R (6.20) is just the identity map on M red written in terms of two distinct models. On the other hand, the map M →M given by ζ → Z = ζ encodes a non-trivial map on M red , for which Ψ −1 (ζ) →Ψ −1 (ζ), ∀ζ ∈ C n .
We end by remarking that one can perform semiclassical quantization for both systems using their respective action variables. Even more, one can quantize any action variable of the form |ζ j | 2 ∈ C ∞ (C) by the replacement where the hatted letters stand for annihilation and creation operators on the standard Fock space. In this manner, one obtains that the spectrum of each action variable |ζ j | 2 consists of all non-negative integers. This then gives immediately the (semi-classical) spectra of the corresponding integrable Hamiltonians. Regarding the Hamiltonians (6.23), one simply expresses {λ i } in terns of {|ζ j | 2 }. One can deal with the Hamiltonians H red j •Ψ −1 (2.45) in the same spirit.

Discussion and outlook
We have presented a thorough description of the models M andM of the reduced phase space M red (2.34) and gained a detailed understanding of how these models are equipped with a pair of integrable many-body systems in action-angle duality. Our principal result is that we have established the validity of Figure 1 of the Introduction for the case at hand. In particular, we have seen that λ : M → R n yields via the duality map R the momentum map for the torus action associated with the integrable HamiltoniansĤ red j •Ψ −1 that containĤ (1.6) and at the same time it provides global position variables for the Hamiltonians H red j •Ψ −1 that contain H (1.11). This and the analogous dual interpretations for the mapλ :M → R n are explained in Section 2.2 and Section 6.2. To put it slightly differently, we have seen that {λ j } and {Ĥ red j • Ψ −1 } (6.23) are alternative generating sets for the Abelian Poisson algebra P on (M, ω), while {λ j } and {H red j •Ψ −1 } (2.46) provide alternative generating sets forP on (M ,ω).
The main technical achievement of this paper is the construction of the model M, which is summarized by Figure 2 and Theorem 6.5. The constructions of the maps ψ andψ that feature in the two figures rely respectively on the singular value decomposition and on the generalized Cartan decomposition of certain matrices, and other algebraic operations. These maps, and especially the duality map R, cannot be presented explicitly, basically since the eigenvalues of higher rank matrices cannot be given in closed form. Nevertheless, the duality proves very useful for understanding the qualitative features of the respective systems.
Our study gives rise to the first example of systems in duality for which the two systems are different (not a self-dual case) and both have quasi-periodic motions on compact Liouville tori. The duality map R allowed us to demonstrate that in our case each one of the two systems (M, ω, H, P, H) and (M ,ω,Ĥ,P,Ĥ) has a unique equilibrium position, which corresponds to the origin in C n used to represent both M andM . We also pointed out that each reduced Hamiltonian H red j andĤ red j possesses Abelian commutants in the Poisson algebra C ∞ (M red ). As another spin-off, let us now explain that the particle positions evaluated along any fixed phase space trajectory of our Hamiltonians stay in a compact set, i.e., all motions are bounded. Indeed, any trajectory of H red j • Ψ −1 is contained in a set (λ • R) −1 (λ 0 ) for someλ 0 ∈ R n , which is compact, since-being equivalent to the standard T n momentum map on (C n , ω can ) (2.42)-the mapλ :M → R n is proper. This compact subset of M is sent by λ onto a compact subset of R n , simply because λ : M → R n is continuous. A similar argument can be applied to the trajectories generated by the HamiltoniansĤ red j •Ψ −1 as well. We remark that in principle we can derive Lax pairs for our systems, since we know the 'unreduced Lax matrices' (see (2.9) and (2.13)) that generate the Abelian Poisson algebras H 1 and H 2 on M, and those unreduced Lax matrices satisfy Lax equations already before reduction [16,23]. The specific formulae should be worked out and compared with the Lax matrices obtained recently in [30].
We have seen that the complex 'oscillator variables' provide an easy way for finding the semiclassical spectra of the actions, by (6.24), and thus also the spectra of the many-body Hamiltonians. It is an interesting problem for future work to compare this 'action-angle quantization' with a 'Schrödinger quantization' of the RSvD type many-body Hamiltonians (1.6) and (1.11) built on analytic difference operators. For this, the recent paper by van Diejen and Emsiz [4] should serve as a good starting point.
Another promising project is to explore reductions of the Heisenberg double of SU(2n) at generic values of the momentum map. This is expected to produce extensions with internal degrees of freedom of the many-body systems (1.6) and (1.11). A suitably generalized version of action-angle duality should hold also for such systems, analogously to the systems investigated by Reshetikhin [31,32].
Finally, we wish to draw attention to our supplementary new result presented in Appendix B, where we show how the Hamiltonian H (1.11) can be recovered as a scaling limit of van Diejen's 5-parametric integrable Hamiltonians [3]. We stress that our reduced Hamiltonians automatically have complete flows on M red , while the completeness of the flow for general real forms of van Diejen's systems has not yet been studied. However, see [30], and also [29] for a detailed study of classical scattering in a 2-parameter hyperbolic case. The most intriguing open problem in this area is to find a Hamiltonian reduction treatment for van Diejen's 5-parametric systems. This would enhance their group theoretic understanding, and would also help to explore their classical dynamics. B The relation of H (1.11) to van Diejen's Hamiltonian Our starting point is the following real form of van Diejen's Hamiltonian [3], with real parameters a, b, c, d, µ, ±j and V (1,2) ±j given by ±j (λ) = n k =j sinh µ ± (λ j + λ k ) sinh (µ ± (λ j − λ k ) sinh(λ j + λ k ) sinh(λ j − λ k ) .

(B.2)
For convenience, we shall refer to the two terms in the formula for H vD (λ, θ) as "kinetic" and "potential". We will prove the following result.