On a Poisson-Lie deformation of the BC(n) Sutherland system

A deformation of the classical trigonometric BC(n) Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson-Lie analogue of the Kazhdan-Kostant-Sternberg type reduction of the free particle on SU(2n) that leads to the BC(n) Sutherland system. We prove that this yields a Liouville integrable Hamiltonian system and construct a globally valid model of the smooth reduced phase space wherein the commuting flows are complete. We point out that the reduced system, which contains 3 independent coupling constants besides the deformation parameter, can be recovered (at least on a dense submanifold) as a singular limit of the standard 5-coupling deformation due to van Diejen. Our findings complement and further develop those obtained recently by Marshall on the hyperbolic case by reduction of the Heisenberg double of SU(n,n).


Introduction
Models amenable to exact treatment provide key paradigms for our understanding of natural phenomena and form a fertile field of research crossing the border of physics and mathematics. The study of integrable Hamiltonian systems is a very active subfield with particularly strong ties to group theory and symplectic geometry. For reviews, see e.g. [9,22,30,5,8]. One of the time-honoured approaches to such systems consists in viewing them as 'shadows' of natural free systems enjoying high symmetries. This is alternatively known as the projection method or as Hamiltonian reduction [24,25]. The list of the free 'master systems' is monotonically expanding in time. To name a few, it includes free particles on Lie groups together with their Poisson-Lie symmetric deformations and quasi-Hamiltonian analogues. For example, it was shown in the pioneering paper [17] that the integrable many-body system of Sutherland [34], which describes particles on the circle interacting via a pair potential given by the inverse square of the chord-distance, is a reduction of the free particle on the unitary group U(n). Various deformations of the Sutherland system due to Ruijsenaars and Schneider [31,29] were derived [11,12] from Poisson-Lie symmetric free motion on U(n), whose phase space is the Heisenberg double [33] of the Poisson-Lie group U(n), and from the internally fused quasi-Hamiltonian double [2] of U(n), which arose from Chern-Simons field theory.
The projection method was enriched by an interesting recent contribution of Marshall [20], who obtained an integrable Ruijsenaars-Schneider (RS) type system by reducing the Heisenberg double of SU(n, n), which directly motivated our present work 1 . Here, we shall deal with a reduction of the Heisenberg double of SU(2n) and derive a Liouville integrable Hamiltonian system related to Marshall's one in a way similar to the connection between the original trigonometric Sutherland system and its hyperbolic variant. Although this is essentially analytic continuation, it should be noted that the resulting systems are qualitatively different in their dynamical characteristics and global features. In addition, what we hope makes our work worthwhile is that our treatment is different from the one in [20] in several respects and we go considerably further regarding the global characterization of the reduced phase space and the completeness of the relevant Hamiltonian flows.
The main Hamiltonian of the system that we obtain can be displayed as follows Here u, v and x are real coupling parameters that will be assumed to satisfy u < v, v = −u and x = 0. (1. 2) The components ofq parametrize the torus T n by e iq andp belongs to the domain C x := {p ∈ R n | 0 >p 1 ,p k −p k+1 > |x|/2 (k = 1, . . . , n − 1)}. (1.3) The dynamics is then defined via the symplectic form ω = n j=1 dq j ∧ dp j . (1.4) It will be shown that this system results by restricting a reduced free system on a dense open submanifold of the pertinent reduced phase space. The Hamiltonian flow is complete on the full reduced phase space, but it can leave the submanifold parametrized by C x × T n . By glancing at the form of the Hamiltonian, one may say that it represents an RS type system coupled to external fields. Since differences of the 'position variables'p k appear, one feels that this Hamiltonian somehow corresponds to an A-type root system. To better understand the nature of this model, let us now introduce new Darboux variables q k , p k following essentially [20] as exp(p k ) = sin(q k ) andq k = p k tan(q k ). (1.5) In terms of these variables H(p,q; x, u, v) = H 1 (q, p; x, u, v) with the 'new Hamiltonian' H 1 (q, p; x, u, v) = e −2u + e 2v 2 n j=1 1 sin 2 (q j ) − n j=1 cos(p j tan(q j )) 1 − 1 + e 2(v−u) sin 2 (q j ) + 4e 2(v−u) 4 sin 2 (q j ) − sin 2 (2q j ) 1 2 × n k=1 (k =j) 1 − 2 sinh 2 x 2 sin 2 (q j ) sin 2 (q k ) sin 2 (q j − q k ) sin 2 (q j + q k ) 1 2 . (1. 6) Remarkably, only such combinations of the new 'position variables' q k appear that are naturally associated with the BC n root system and the Hamiltonian H 1 enjoys symmetry under the corresponding Weyl group. Thus now one may wish to attach the Hamiltonian H 1 to the BC n root system. Indeed, this interpretation is preferable for the following reason. Introduce the scale parameter (corresponding to the inverse of the velocity of light in the original RS system) β > 0 and make the substitutions u → βu, v → βv, x → βx, p → βp,ω → βω. (1.7) Then consider the deformed Hamiltonian H β (q, p; x, u, v) := H 1 (q, βp; βx, βu, βv). (1.8) The point is that one can then verify the following relation: H β (q, p; x, u, v) − n β 2 = H Suth BCn (q, p; γ, γ 1 , γ 2 ), (1.9) where H Suth BCn = 1 2 n j=1 p 2 j + 1≤j<k≤n γ sin 2 (q j − q k ) + γ sin 2 (q j + q k ) + n j=1 γ 1 sin 2 (q j ) + n j=1 γ 2 sin 2 (2q j ) (1.10) is the standard trigonometric BC n Sutherland Hamiltonian with coupling constants γ = x 2 4 , γ 1 = 2uv, γ 2 = 2(v − u) 2 . (1.11) Note that the domain of the variablesq,p, and correspondingly that of q, p also depends on β, and in the β → 0 limit it is easily seen that we recover the usual BC n domain π 2 > q 1 > q 2 > · · · > q n > 0, p ∈ R n .
(1. 12) In conclusion, we see that H in its equivalent form H β is a 1-parameter deformation of the trigonometric BC n Sutherland Hamiltonian. We remark in passing that the conditions (1.2) imply that γ 2 > 0 and 4γ 1 + γ 2 > 0, which guarantee that the flows of H Suth BCn are complete on the domain (1.12).
Marshall [20] obtained similar results for an analogous deformation of the hyperbolic BC n Sutherland Hamiltonian. His deformed Hamiltonian differs from (1.1) above in some important signs and in the relevant domain of the 'position variables'p. Although in our impression the completeness of the reduced Hamiltonian flows was not treated in a satisfactory way in [20], the completeness proof that we shall present can be adapted to Marshall's case as well.
It is natural to ask how the system studied in the present paper (and its cousin in [20]) is related to van Diejen's [35] 5-coupling trigonometric BC n system? It was shown already in [35] that the 5-coupling trigonometric system is a deformation of the BC n Sutherland system, and later [36] several other integrable systems were also derived as its ('Inozemtsev type' [16]) limits. Motivated by this, we can show that the Hamiltonian (1.1) is a singular 2 limit of van Diejen's general Hamiltonian. Incidentally, a Hamiltonian of Schneider [32] can be viewed as a subsequent singular limit of the Hamiltonian (1.1). Schneider's system was mentioned in [20], too, but the relation to van Diejen's system was not described.
The original idea behind the present work and [20] was that a natural Poisson-Lie analogue of the Hamiltonian reduction treatment [13] of the BC n Sutherland system should lead to a deformation of this system. It was expected that a special case of van Diejen's standard 5-coupling deformation will arise. The expectation has now been confirmed, although it came as a surprise that a singular limit is involved in the connection.
The outline of the paper is as follows. We start in Section 2 by defining the reduction of interest. In Section 3 we observe that several technical results of [11] can be applied for analyzing the reduction at hand, and solve the momentum map constraints by taking advantage of this observation. The heart of the paper is Section 4, where we characterize the reduced system. In Subsection 4.1 we prove that the reduced phase space is smooth, as formulated in Theorem 4.4. Then in Subsection 4.2 we focus on a dense open submanifold on which the Hamiltonian (1.1) lives. The demonstration of the Liouville integrability of the reduced free flows is given in Subsection 4.3. In particular, we prove the integrability of the completion of the system (1.1) carried by the full reduced phase space. Our main result is Theorem 4.9 (proved in Subsection 4.4), which establishes a globally valid model of the reduced phase space. We stress that the global structure of the phase space on which the flow of (1.1) is complete was not considered previously at all, and will be clarified as a result of our group theoretic interpretation. Section 5 contains our conclusions, further comments on the related paper by Marshall [20] and a discussion of open problems. The main text is complemented by four appendices. Appendix A deals with the connection to van Diejen's system; the other 3 appendices contain important details relegated from the main text.

Definition of the Hamiltonian reduction
We below introduce the 'free' Hamiltonians and define their reduction. We restrict the presentation of this background material to a minimum necessary for understanding our work. The conventions follow [11], which also contains more details. As a general reference, we recommend [7].

The unreduced free Hamiltonians
We fix a natural number 3 n ≥ 2 and consider the Lie group SU(2n) equipped with its standard quadratic Poisson bracket defined by the compact form of the Drinfeld-Jimbo classical rmatrix, where E αβ is the elementary matrix of size 2n having a single non-zero entry 1 at the αβ position. In particular, the Poisson brackets of the matrix elements of g ∈ SU(2n) obey Sklyanin's formula Thus SU(2n) becomes a Poisson-Lie group, i.e., the multiplication SU(2n) × SU(2n) → SU(2n) is a Poisson map. The cotangent bundle T * SU(2n) possesses a natural Poisson-Lie analogue, the so-called Heisenberg double [33], which is provided by the real Lie group SL(2n, C) endowed with a certain symplectic form [1], ω. To describe ω, we use the Iwasawa decomposition and factorize every element K ∈ SL(2n, C) in two alternative ways with uniquely determined Here SB(2n) stands for the subgroup of SL(2n, C) consisting of upper triangular matrices with positive diagonal entries. The symplectic form ω reads Before specifying free Hamiltonians on the phase space SL(2n, C), note that any smooth function h on SB(2n) corresponds to a functionh on the space of positive definite Hermitian matrices of determinant 1 by the relatioñ Then introduce the invariant functions (2.7) These in turn give rise to the following ring of functions on SL(2n, C): where we utilized the decomposition (2.3). An important point is that H forms an Abelian algebra with respect to the Poisson bracket associated with ω (2.5). The flows of the 'free' Hamiltonians contained in H can be obtained effortlessly. To describe the result, define the derivative d R f ∈ C ∞ (SB(2n), su(2n)) of any real function f ∈ C ∞ (SB(2n)) by requiring The Hamiltonian flow generated by H ∈ H through the initial value K(0) = g L (0)b R (0) −1 is in fact given by where H and h are related according to (2.8). This means that g L (t) follows the orbit of a one-parameter subgroup, while b R (t) remains constant. Actually, g R (t) also varies along a similar orbit, and b L (t) is constant. The constants of motion b L and b R generate a Poisson-Lie symmetry, which allows one to define Marsden-Weinstein type [19] reductions.

Generalized Marsden-Weinstein reduction
The free Hamiltonians in H are invariant with respect to the action of SU(2n) × SU(2n) on SL(2n, C) given by left-and right-multiplications. This is a Poisson-Lie symmetry, which means that the corresponding action map is a Poisson map. In (2.11) the product Poisson structure is taken using the Sklyanin bracket on SU(2n) and the Poisson structure on SL(2n, C) associated with the symplectic form ω (2.5). This Poisson-Lie symmetry admits a momentum map in the sense of Lu [18], given explicitly by Φ : The key property of the momentum map is represented by the identity where f ∈ C ∞ (SL(2n, C)) is an arbitrary real function and the Poisson bracket is the one corresponding to ω (2.5). The map Φ enjoys an equivariance property and one can [18] perform Marsden-Weinstein type reduction in the same way as for usual Hamiltonian actions (for which the symmetry group has vanishing Poisson structure). To put it in a nutshell, any H ∈ H gives rise to a reduced Hamiltonian system by fixing the value of Φ and subsequently taking quotient with respect to the corresponding isotropy group. The reduced flows can be obtained by the standard restriction-projection algorithm, and under favorable circumstances the reduced phase space is a smooth symplectic manifold. Now, consider the block-diagonal subgroup G + := S(U(n) × U(n)) < SU(2n). (2.15) Since G + is also a Poisson submanifold of SU(2n), the restriction of (2.12) yields a Poisson-Lie action G + × G + × SL(2n, C) → SL(2n, C) (2. 16) of G + × G + . The momentum map for this action is provided by projecting the original momentum map Φ as follows. Let us write every element b ∈ SB(2n) in the block-form and define G * + < SB(2n) to be the subgroup for which b(12) = 0 n . If π : SB(2n) → G * + denotes the projection then the momentum map Φ + : SL(2n, C) → G * + × G * + is furnished by Indeed, it is readily checked that the analogue of (2.14) holds with X, Y taken from the blockdiagonal subalgebra of su(2n) and b L , b R replaced by their projections. The equivariance property of this momentum map means that in correspondence to We briefly mention here that, as the notation suggests, G * + is itself a Poisson-Lie group that can serve as a Poisson dual of G + . The relevant Poisson structure can be obtained by identifying the block-diagonal subgroup of SB(2n) with the factor group SB(2n)/L, where L is the block-upper-triangular normal subgroup. This factor group inherits a Poisson structure from SB(2n), since L is a so-called coisotropic (or 'admissible') subgroup of SB(2n) equipped with its standard Poisson structure. The projected momentum map Φ + is a Poisson map with respect to this Poisson structure on the two factors G * + in (2.19). The details are not indispensable for us. The interested reader may find them e.g. in [6].
Inspired by the papers [13,11,20], we wish to study the particular Marsden-Weinstein reduction defined by imposing the following momentum map constraint: with some real constants u, v and x. Here, ν(x) ∈ SB(n) is the n × n upper triangular matrix defined by whose main property is that ν(x)ν(x) † has the largest possible non-trivial isotropy group under conjugation by the elements of SU(n). Our principal task is to characterize the reduced phase space where Φ −1 + (µ) = {K ∈ SL(2n, C) | Φ + (K) = µ} and is the isotropy group of µ inside G + × G + . Concretely, G + (µ L ) is the subgroup of G + consisting of the special unitary matrices of the form where η L (2) is arbitrary and In words, η L (1) belongs to the little group of ν(x)ν(x) † in U(n). We shall see that Φ −1 + (µ) and M are smooth manifolds for which the canonical projection

Solution of the momentum map constraints
The description of the reduced phase space requires us to solve the momentum map constraints, i.e., we have to find all elements K ∈ Φ −1 + (µ). Of course, it is enough to do this up to the gauge transformations provided by the isotropy group G µ (2.25). The solution of this problem will rely on the auxiliary equation (3.11) below, which is essentially equivalent to the momentum map constraint, Φ + (K) = µ, and coincides with an equation studied previously in great detail in [11]. Thus we start in the next subsection by deriving this equation.

A crucial equation implied by the constraints
We begin by recalling (e.g. [21]) that any g ∈ SU(2n) can be decomposed as where g + , h + ∈ G + and q = diag(q 1 , . . . , q n ) ∈ R n satisfies The vector q is uniquely determined by g, while g + and h + suffer from controlled ambiguities. First, apply the above decomposition to g L in K = g L b −1 R ∈ Φ −1 + (µ) and use the righthanded momentum constraint π(b R ) = µ R . It is then easily seen that up to gauge transformations every element of Φ −1 + (µ) can be represented in the following form: Here ρ ∈ SU(n) and α is an n × n complex matrix. By using obvious block-matrix notation, we introduce Ω := K 22 and record from (3.3) that For later purpose we introduce also the polar decomposition of the matrix Ω, where T ∈ U(n) and the Hermitian, positive semi-definite factor Λ is uniquely determined by the relation ΩΩ † = Λ 2 . Second, by writing with an n × n matrix χ. Now we inspect the components of the 2 × 2 block-matrix identity which results by substituting K from (3.3). We find that the (22) component of this identity is equivalent to On account of the condition (1.2), this uniquely determines Λ in terms of q, and shows also that Λ is invertible. A further important consequence is that we must have q n > 0, (3.9) and therefore sin q is an invertible diagonal matrix. Indeed, if q n = 0, then from (3.4) and (3.8) we would get (ΩΩ † ) nn = e 2v = e −2u , which is excluded by (1.2). Next, one can check that in the presence of the relations already established, the (12) and the (21) components of the identity (3.7) are equivalent to the equation (3.10) Observe that K uniquely determines q, T and ρ, and conversely K is uniquely defined by the above relations once q, T and ρ are found. Now one can straightforwardly check by using the above relations that the (11) component of the identity (3.7) translates into the following equation: This is to be satisfied by q subject to (3.2), (3.9) and T ∈ U(n), ρ ∈ SU(n). What makes our job relatively easy is that this is the same as equation (5.7) in the paper [11] by Klimčík and one of us. In fact, this equation was analyzed in detail in [11], since it played a crucial role in that work, too. The correspondence with the symbols used in [11] is This motivates to introduce the variablep ∈ R n in our case, by setting sin q k = ep k , k = 1, . . . , n.
(3.13) Notice from (3.2) and (3.9) that we have 0 ≥p 1 ≥ · · · ≥p n > −∞. (3.14) If the components ofp are all different, then we can directly rely on [11] to establish both the allowed range ofp and the explicit form of ρ and T . The statement thatp j =p k holds for j = k can be proved by adopting arguments given in [11,12]. This proof requires combining techniques of [11] and [12], whose extraction from [11,12] is rather involved. We present it in Appendix B, otherwise in the next subsection we proceed by simply stating relevant applications of results from [11].
Remark 3.1. In the context of [11] the components ofp are not restricted to the half-line and both k L and k R vary in U(n). These slight differences do not pose any obstacle to using the results and techniques of [11,12]. We note that essentially the same equation (3.11) surfaced in [20] as well, but the author of that paper refrained from taking advantage of the previous analyses of this equation. In fact, some statements of [20] are not fully correct. This will be specified (and corrected) in Section 5.

Consequences of equation (3.11)
We start by pointing out the foundation of the whole analysis. For this, we first display the identity which holds with a certain n-component vectorv =v(x). By introducing and settingp ≡ diag(p 1 , . . . ,p n ), we rewrite equation (3.11) as The equality of the characteristic polynomials of the matrices on the two sides of (3.17) gives a polynomial equation that containsp, the absolute values |w j | 2 and a complex indeterminate. Utilizing the requirement that |w j | 2 ≥ 0 must hold, one obtains the following result.
Proposition 3.2 can be proved by merging the proofs of Lemma 5.2 of [11] and Theorem 2 of [12]. This is presented in Appendix B.
The above-mentioned polynomial equality permits to find the possible vectors w (3.16) as well. Ifp and w are given, then T is determined by equation (3.17) up to left-multiplication by a diagonal matrix and ρ is determined by (3.16) up to left-multiplication by elements from the little group ofv(x). Following this line of reasoning and controlling the ambiguities in the same way as in [11], one can find the explicit form of the most general ρ and T at any fixedp ∈C x . In particular, it turns out that the range of the vectorp equalsC x .
Before presenting the result, we need to prepare some notations. First of all, we pick an arbitraryp ∈C x and define the n × n matrix θ(x,p) as follows: (3.19) and (3.20) All expressions under square root are non-negative and non-negative square roots are taken. Note that θ(x,p) is a real orthogonal matrix of determinant 1 for which θ(x,p) −1 = θ(−x,p) holds, too. Next, define the real vector r(x,p) ∈ R n with non-negative components 21) and the real n × n matrix ζ(x,p), Finally, define the n × n matrix κ(x) as where, again, a = n if x > 0 and a = 1 if x < 0. It can be shown that both κ(x) and ζ(x,p) are orthogonal matrices of determinant 1 for anyp ∈C x . Now we can state the main result of this section, whose proof is omitted since it is a direct application of the analysis of the solutions of (3.11) presented in Section 5 of [11]. Proposition 3.3. Take anyp ∈C x and any diagonal unitary matrix e iq ∈ T n . By using the preceding notations define K ∈ SL(2n, C) (3.3) by setting and also applying the equations (3.4), (3.5), (3.8) and (3.13). Then the element K belongs to the constraint surface Φ −1 + (µ), and every orbit of the gauge group G µ (2.25) in Φ −1 + (µ) intersects the set of elements K just constructed.
Remark 3.4. It is worth spelling out the expression of the element K given by Proposition 3.3. Indeed, we have using the above definitions and Remark 3.5. Let us call S the set of the elements K(p, e iq ) constructed above, and observe that this set is homeomorphic toC by its very definition. This is not a smooth manifold, because of the presence of the boundary ofC x . However, this does not indicate any 'trouble' since it is not true (at the boundary of C x ) that S intersects every gauge orbit in Φ −1 + (µ) in a single point. Indeed, it is instructive to verify that ifp is the special vertex ofC x for whichp k = (1 − k)|x|/2 for k = 1, . . . , n, then all points K(p, e iq ) lie on a single gauge orbit. This, and further inspection, can lead to the idea that the variablesq j should be identified with arguments of complex numbers, which lose their meaning at the origin that should correspond to the boundary ofC x . Our Theorem 4.9 will show that this idea is correct. It is proper to stress that we arrived at such idea under the supporting influence of previous works [29,11].

Characterization of the reduced system
The smoothness of the reduced phase space and the completeness of the reduced free flows follows immediately if we can show that the gauge group G µ acts in such a way on Φ −1 + (µ) that the isotropy group of every point is just the finite center of the symmetry group. In Subsection 4.1, we prove that the factor of G µ by the center acts freely on Φ −1 + (µ). Then in Subsection 4.2 we explain that C x × T n provides a model of a dense open subset of the reduced phase space by means of the corresponding subset of Φ −1 + (µ) defined by Proposition 3.3. Adopting a key calculation from [20], it turns out that (p, e iq ) ∈ C x × T n are Darboux coordinates on this dense open subset. In Subsection 4.3, we demonstrate that the reduction of the Abelian Poisson algebra of free Hamiltonians (2.8) yields an integrable system. Finally, in Subsection 4.4, we present a model of the full reduced phase space, which is our main result.

Smoothness of the reduced phase space
It is clear that the normal subgroup of the full symmetry group G + × G + consisting of matrices of the form acts trivially on the phase space. This subgroup is contained in G µ (2.25). The corresponding factor group of G µ is called 'effective gauge group' and is denoted byḠ µ . We wish to show thatḠ µ acts freely on the constraint surface Φ −1 + (µ). We need the following elementary lemmas. with g + , h + , g ′ + , h ′ + ∈ G + and q = diag(q 1 , . . . , q n ) subject to π 2 ≥ q 1 > · · · > q n > 0. Then there exist diagonal matrices m 1 , m 2 ∈ T n having the form If (4.3) holds with strict inequality π 2 > q 1 , then m 1 = m 2 , i.e., a = b. Lemma 4.2. Pick anyp ∈C x and consider the matrix θ(x,p) given by (3.19) and (3.20). Then the entries θ n,1 (x,p) and θ j,j+1 (x,p) are all non-zero if x > 0 and the entries θ 1,n (x,p) and θ j+1,j (x,p) are all non-zero if x < 0.
For convenience, we present the proof of Lemma 4.1 in Appendix C. The property recorded in Lemma 4.2 is known [29,11], and is easily checked by inspection. Proposition 4.3. The effective gauge groupḠ µ acts freely on Φ −1 + (µ). Proof. Since every gauge orbit intersects the set S specified by Proposition 3.3, it is enough to show that if (η L , η R ) ∈ G µ maps K ∈ S (3.27) to itself, then (η L , η R ) equals some element (η, η) given in (4.1). For K of the form (3.3), we can spell out K ′ ≡ η L Kη −1 R as The equality K ′ = K implies by the uniqueness of the Iwasawa decomposition and Lemma 4.1 that we must have where K(p, e iq ) stands for the expression (3.27). Note that S o is in bijection with C x × T n . The next lemma says that no two different point of S o are gauge equivalent. with some (η L , η R ) ∈ G µ . By spelling out the gauge transformation as in (4.6), using the shorthand sin q = ep, we observe thatp ′ =p since q in (3.1) does not change under the action of G + ×G + . Since now we have π 2 > q 1 (which is equivalent to 0 >p 1 ), the arguments applied in the proof of Proposition 4.3 permit to translate the equality (4.11) into the relations complemented with the condition which is equivalent to (4.14) We stress that m ∈ T n and notice from (3.20) that forp ∈ C x all the diagonal entries θ(−x,p) jj are non-zero. Therefore we conclude from (4.14) that e iq ′ = e iq . This finishes the proof, but of course we can also confirm that m = z1 n , consistently with Proposition 4.3.
Now we introduce the map P : SL(2n, C) → R n by defined by writing g L in the form (3.1) with sin q = ep. The map P gives rise to a map P : M → R n verifyingP (π µ (K)) = P(K), ∀K ∈ Φ −1 + (µ), (4.16) where π µ is the canonical projection (2.28). We notice that, since the 'eigenvalue parameters' p j (j = 1, . . . , n) are pairwise different for any K ∈ Φ −1 + (µ),P is a smooth map. The continuity ofP implies that is an open subset. The second equality is a direct consequence of our foregoing results about S and S o . Note thatP −1 (C x ) = π µ (S) = M. Since π µ is continuous (actually smooth) and any point of S is the limit of a sequence in

Liouville integrability of the reduced free Hamiltonians
The Before settling the above question, let us focus on the Hamiltonian H ∈ H defined by Using the formula of K(p, e iq ) in Remark 3.4, it is readily verified that H(K(p, e iq )) = H(p,q; x, u, v), ∀(p, e iq ) ∈ C x × T n , Turning to the demonstration of Liouville integrability, consider the n functions The restriction of the corresponding elements of H M on M o ≃ C x × T n gives the functions where α has the form (3.28). These are real-analytic functions on C x × T n . It is enough to show that their exterior derivatives are linearly independent on a dense open subset of C x × T n . This follows if we show that the function f (p,q) = det dqH 1 , dqH 2 , . . . , dqH n (4.23) is not identically zero on C x × T n . Indeed, since f is an analytic function and C x × T n is connected, if f is not identically zero then its zero set cannot contain any accumulation point. This, in turn, implies that f is non-zero on a dense open subset of C x × T n ≃ M o , which is also dense and open in the full reduced phase space M. In other words, the reductions of H k (k = 1, . . . , n) possess the property of Liouville integrability. It is rather obvious that the function f is not identically zero, since H k involves dependence onq through e ±ikq and lower powers of e ±iq . It is not difficult to inspect the function f (p,q) in the 'asymptotic domain' where all differences |p j −p m | (m = j) tend to infinity, since in this domain α becomes close to a diagonal matrix. We omit the details of this inspection, whereby we checked that f is indeed not identically zero. The above arguments prove the Liouville integrability of the reduced free Hamiltonians, i.e., the elements of H M . Presumably, there exists a dual set of integrable many-body Hamiltonians that live on the space of action-angle variables of the Hamiltonians in H M . The construction of such dual Hamiltonians is an interesting task for the future, which will be further commented upon in Section 5.

The global structure of the reduced phase space
We here construct a global cross-section of the gauge orbits in the constraint surface Φ −1 + (µ). This engenders a symplectic diffeomorphism between the reduced phase space (M, ω M ) and the manifold (M c ,ω c ) below. It is worth noting that (M c ,ω c ) is symplectomorphic to R 2n carrying the standard Darboux 2-form, and one can easily find an explicit symplectomorphism if desired. Our construction was inspired by the previous papers [29,11], but detailed inspection of the specific example was also required for finding the final result given by Theorem 4.9. After a cursory glance, the reader is advised to go directly to this theorem and follow the definitions backwards as becomes necessary. See also Remark 4.10 for the rationale behind the subsequent definitions.
To begin, consider the product manifold  It verifies which means that Z x is a symplectic embedding of (C . . , n − 1), e iqn (z) = z n−1 |z n−1 | .

(4.30)
It is important to remark that thep k (z) (k = 1, . . . , n) given above yield smooth functions on the whole ofM c , while the anglesq k are of course not well-defined on the complementary locus ofM o c . Our construction of the global cross-section will rely on the building blocks collected in the following long definition. Definition 4.7. For any (z 1 , . . . , z n−1 ) ∈ C n−1 consider the smooth functions and recalling (3.21), introduce the n × n matrixζ(x, z) by the formulaê where a = n if x > 0 and a = 1 if x < 0. Then introduce the matrixθ(x, z) for x > 0 aŝ Although the variable z n appears only inγ 1 , we can regard all objects defined above as smooth functions onM c , and we shall do so below.
The key properties of the matricesζ,θ,α andγ are given by the following lemma, which can be verified by straightforward inspection. The role of these identities and their origin will be enlightened by Theorem 4.9. Then the following identities hold for all (p, e iq ) ∈C x × T n : Here we use Definition 4.7 and the functions onC x × T n introduced in Subsection 3.2.
For the verification of the above identities, we remark that the vector r (3.21) can be expressed as a smooth function of the complex variables as Q jk (x, z), j = 1, . . . , n.   Proof. The proof is based upon the identitŷ (4.47) which is readily seen to be equivalent to the set of identities displayed in Lemma 4.8. It means thatK(z(p, e iq )) is a gauge transform of K(p, e iq ) in (3.27). Indeed, the above transformation of K(p, e iq ) has the form (2.20) with where c is a harmless scalar inserted to ensure det(η L ) = det(η R ) = 1. Using (3.25) and is gauge equivalent to S o in (4.10), we obtain the equalitŷ K * (ω) =ω (4.51) by using Theorem 4.6 and equation (4.29). More precisely, we here also utilized thatK * (ω) is (obviously) smooth andM o c is dense inM c . The only statements that remain to be proved are that the intersection ofŜ with any gauge orbit consists of a single point and thatK is injective. (These are already clear for S o ⊂Ŝ and forK|M o c .) Now suppose that for some gauge transformation and z, z ′ ∈M c . Let us observe from the definitions that we can write where sin q(z) = ep (z) , with π 2 ≥ q 1 > · · · > q n > 0, and D(z) is a diagonal unitary matrix of the form D(z) = diag(d 1 , 1 n−1 ,d 1 , 1 n−1 ). Then the uniqueness properties of the Iwasawa decomposition of SL(2n, C) and the generalized Cartan decomposition (3.1) of SU(2n) allow to establish the following consequences of (4.52). First, (4.54) Second, using Lemma 4.1, for some diagonal unitary matrices of the form (4.4). Third, we havê For definiteness, let us focus on the case x > 0. Then we see from the definitions that the componentsα k+1,k andα 1,n depend only onp(z) and are non-zero. By using this, we find from (4.56) that m 1 = m 2 = C1 n with a scalar C, and thereforê α(z ′ ) =α(z). (4.57) Inspection of the components (1, 2), . . . , (1, n − 1) of this matrix equality and (4.54) permit to conclude that z ′ 2 = z 2 , . . . , z ′ n−1 = z n−1 , respectively. Then, the equality of the (2, n) entries in (4.57) gives z ′ 1 = z 1 which used in the (1, 1) position implies z ′ n = z n . Thus we see that z ′ = z and the proof is complete. (Everything written below (4.56) is quite similar for x < 0.) Remark 4.10. Let us hint at the way the global structure was found. The first point to notice was that all or some of the phases e iq j cannot encode gauge invariant quantities if p belongs to the boundary ofC x , as was already mentioned in Remark 3.5. Motivated by [11], then we searched for complex variables by requiring that a suitable gauge transform of K(p, e iq ) in (3.27) should be expressible as a smooth function of those variables. Given the similarities to [11], only the definition of z n was a true open question. After trial and error, the idea came in a flash that the gauge transformation at issue should be constructed from a transformation that appears in Lemma C.1. Then it was not difficult to find the correct result.
Remark 4.11. Let us elaborate on how the trajectoriesp(t) corresponding to the flows of the reduced free Hamiltonians, arising from H k (4.21) for k = 1, . . . , n, can be obtained. Recall that for k = 1 the reduction of H 1 completes the main Hamiltonian H (1.1). Since Thus the curve g L (t) (2.10) has the form (4.58) The reduced flow results by the usual projection algorithm. This starts by picking an initial value z(0) ∈M c and setting K(0) =K(z(0)) by applying (4.46), which directly determines g L (0) and b R (0) as well. Then the map P (4.15) gives rise top(t) via the decomposition of g L (t) ∈ SU(2n) as displayed in (3.1), that iŝ p(t) = P(K(t)). from whichp j (t) can be obtained using (3.13). In particular, the 'particle positions' evolve according to an 'eigenvalue dynamics' similarly to other many-body systems. This involves the one-parameter group e −itL(0) k , where L(0) is the initial value of the Lax matrix (cf. (4.22)) where we suppressed the dependence ofα (4.37) on the parameters x, u, v. A more detailed characterization of the dynamics will be provided elsewhere.

Discussion and outlook on open problems
In this paper we derived a deformation of the trigonometric BC n Sutherland system by means of Hamiltonian reduction of a free system on the Heisenberg double of SU(2n). Our main result is the global characterization of the reduced phase space given by Theorem 4.9. The Liouville integrability of our system holds on this phase space, wherein the reduced free flows are complete. These flows can be obtained by the usual projection method applied to the original free flows described in Section 2.
The local form of our reduced 'main Hamiltonian' (1.1) is similar to the Hamiltonian derived in [20], which deforms the hyperbolic BC n Sutherland system. However, besides a sign difference corresponding to the difference of the undeformed Hamiltonians, the local domain of our system, C x × T n in (1.3), is different from the local domain appearing in [20], which in effect has the form C ′ x × T n with the open polyhedron 5 C ′ x := {p ∈ R n |p n > 0,p k −p k+1 > |x|/2 (k = 1, . . . , n − 1)}. (5.1) We here wish to point out that C ′ x × T n is not the full reduced phase space that arises from the reduction considered in [20]. In fact, similarly to our case, the constraint surface contains a submanifold of the formC ′ x × T n in the case of [20], whereC ′ x is the closure of C ′ x . Then a global model of the reduced phase space can be constructed by introducing complex variables suitably accommodating the procedure that we utilized in Subsection 4.4. Erroneously, in [20] the full phase space was claimed to be C ′ x × T n ; the details of the correct description will be presented elsewhere.
Throughout the text we assumed that n > 1, but we now note that the reduced system can be specialized to n = 1 and the reduction procedure works in this case as well. The assumption was made merely to save words. The formalism actually simplifies for n = 1 since the Poisson structure on G + = S(U(1) × U(1)) < SU(2) is trivial.
As explained in Appendix A, the Hamiltonian (1.1) is a singular limit of a specialization of the trigonometric van Diejen Hamiltonian [35], which (in addition to the deformation parameter) contains 5 coupling constants. As a result, at least classically, van Diejen's system can be degenerated into the trigonometric BC n Sutherland system either directly, as described in [35], or in a roundabout way, going through our system. Of course, a similar statement holds in relation to hyperbolic BC n Sutherland and the system of [20].
Except in the rational limit [26], no Lax matrix is known that would generate van Diejen's commuting Hamiltonians. In the reduction approach a Lax matrix arises automatically, in our case it features in equations (4.22) and (4.61). This might be helpful in searching for a Lax matrix behind van Diejen's 5-coupling Hamiltonian. The search would be easy if one could derive van Diejen's system by Hamiltonian reduction. It is a long standing open problem to find such derivation. Perhaps one should consider some 'classical analogue' of the quantum group interpretation of the Koornwinder (BC n Macdonald) polynomials found in [23], since those polynomials diagonalize van Diejen's quantized Hamiltonians [37].
Another open problem is to construct action-angle duals of the deformed BC n Sutherland systems. Duality relations are not only intriguing on their own right, but are also very useful for extracting information about the dynamics [28,29,30,27]. The duality was used in [4,10] to show that the hyperbolic BC n Sutherland system is maximally superintegrable, the trigonometric BC n Sutherland system has precisely n constants of motion, and the relevant dual systems are maximally superintegrable in both cases. These studies, which were heavily influenced by Pusztai's paper [26] (see also [14]), may provide inspiration for a future investigation of the dualities for the deformed BC n Sutherland systems. We here only remark that deformed dual systems should arise from considering the reduction of alternative sets of commuting free Hamiltonians on the pertinent Heisenberg doubles.
After we finished our work, there appeared a preprint [38] dealing with the quantum mechanics of a lattice version of a 4-parameter Inozemtsev type limit of van Diejen's trigonometric/hyperbolic system. The systems studied in [20] and in our paper correspond to further limits of specializations of this one. The statements about quantum mechanical dualities contained in [38] and its references should be related to classical dualities.
We hope be able to return to some of these questions in the future.

A Links to systems of van Diejen and Schneider
Recall that the trigonometric BC n van Diejen system [35] has the Hamiltonian and v, w denoting the trigonometric potentials v(z) = sin(µ + z) sin(z) and w(z) = sin(µ 0 + z) sin(z) where µ, µ 0 , µ 1 , µ ′ 0 , µ ′ 1 are arbitrary parameters. By making the substitutions the potentials become hyperbolic functions and their R → ∞ limit exists, namely In the 1-particle case we have V ± (λ) = w(±λ), thus H vD takes the following form By utilizing (A.6) one obtains .
Equating the R → ∞ limit of H vD (λ, θ) (A.7) with the Hamiltonian H(p,q; x, u, v) (1.1) yields a system of linear equations involving g 0 , g 1 , g ′ 0 , g ′ 1 as unknowns and u, v as parameters. Actually, four sets of linear equations can be constructed, each with infinitely many solutions depending on one (real) parameter, but these sets are 'equivalent' under the exchanges: Therefore it is sufficient to give only one set of solutions, e.g. i.e., the Hamiltonian H (1.1) is recovered as a singular limit of H vD (A.1). Consider now the function H(p,q; x, u, v) and introduce the real parameter σ through the substitutions and apply the canonical transformation (A.14) Then we have lim with Schneider's [32] Hamiltonian Remark A.1. (i ) In (A.4) only two of the four external field couplings µ 0 , µ ′ 0 , µ 1 , µ ′ 1 are scaled with R. However, scaling all four of these parameters also leads to an integrable Ruijsenaars-Schneider type system with a more general 4-parameter external field. For details, see Section II.B of [36]. (ii ) The connection to Schneider's Hamiltonian was mentioned in Remark 7.1 of [20] as well, where a singular limit, similar to (A.15) was taken.

B Proof of a key result
In this appendix we prove Proposition 3.2 which states that the range of the 'position variable' p is contained in the closed thick-walled Weyl chamberC x (3.18).
The above proof is a straightforward adaptation of the proofs of Lemma 5.2 of [11] and Theorem 2 of [12]. We presented it since it could be awkward to extract the arguments from those lengthy papers, and also our notations and the ranges of our variables are different.

C Proof of an elementary lemma
We here prove the following equivalent formulation of Lemma 4.1.

D Auxiliary material on Poisson-Lie symmetry
The statements presented here are direct analogues of well-known results [3,15] about Hamiltonian group actions with zero Poisson bracket on the symmetry group. They are surely familiar to experts, although we could not find them in a reference. Let us consider a Poisson-Lie group G with dual group G * and a symplectic manifold P equipped with a left Poisson action of G. Essentially following Lu [18] (cf. Remark D.4), we say that the G-action admits the momentum map ψ : P → G * if for any X ∈ G, the Lie algebra of G, and any f ∈ C ∞ (P ) we have (L X P f )(p) = X, {f, ψ}(p)ψ(p) −1 , ∀p ∈ P, (D. 1) where X P is the vector field on P corresponding to X, ., . stands for the canonical pairing between the Lie algebras of G and G * , and the notation pretends that G * is a matrix group. Using the Hamiltonian vector field V f defined by L V f h = −{f, h} (∀h ∈ C ∞ (P )), we can spell out equation (D.1) equivalently as (L X P f )(p) = − X, D ψ(p) R ψ(p) −1 (D p ψ)(V f (p)) , ∀p ∈ P, (D. 2) where D p ψ : T p P → T ψ(p) G * is the derivative, and R ψ(p) −1 denotes the right-translation on G * by ψ(p) −1 . Since the vectors of the form V f (p) span T p P , we obtain the following characterization of the Lie algebra of the isotropy subgroup G p < G of p ∈ P .
Lemma D.1. With the above notations, we have This directly leads to the next statement.
Corollary D.
Let us further suppose that ψ : P → G * is G-equivariant, with respect to the appropriate dressing action of G on G * . Then we have G p < G µ , ∀p ∈ ψ −1 (µ). (D.4) Here G p and G µ refer to the respective actions of G on P and on G * . Corollary D.2 and equation (D.4) together imply the following useful result.
We finish by a clarifying remark concerning the momentum map.
Remark D.4. Let B be the Poisson tensor on P , for which {f, h} = B(df, dh) = L V h f . We can write V h = B ♯ (dh) with the corresponding bundle map B ♯ : T * P → T P . Any X ∈ G = T e G = (T e ′ G * ) * extends to a unique right-invariant 1-form ϑ X on G * (e ∈ G and e ′ ∈ G * are the unit elements). With this at hand, equation (D.1) can be reformulated as which is a slight variation of the defining equation of the momentum map found in [18].