Bi-Hamiltonian structure of Sutherland models coupled to two ${\mathfrak u}(n)^*$-valued spins from Poisson reduction

We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group ${\mathrm{GL}}(n,{\mathbb{C}})$, and study its Poisson reduction with respect to the action of the product group ${{\mathrm U}(n)} \times {{\mathrm U}(n)}$ arising from left- and right-multiplications. One of the pertinent Poisson structures is the canonical one, while the other is suitably transferred from the real Heisenberg double of ${\mathrm{GL}}(n,{\mathbb{C}})$. When taking the quotient of $T^*{\mathrm{GL}}(n,{\mathbb{C}})$ we focus on the dense open subset of ${\mathrm{GL}}(n,{\mathbb{C}})$ whose elements have pairwise distinct singular values. We develop a convenient description of the Poisson algebras of the ${{\mathrm U}(n)} \times {{\mathrm U}(n)}$ invariant functions, and show that one of the Hamiltonians of the reduced bi-Hamiltonian hierarchy yields a hyperbolic Sutherland model coupled to two ${\mathfrak u}(n)^*$-valued spins. Thus we obtain a new bi-Hamiltonian interpretation of this model, which represents a special case of Sutherland models coupled to two spins obtained earlier from reductions of cotangent bundles of reductive Lie groups equipped with their canonical Poisson structure. Upon setting one of the spins to zero, we recover the bi-Hamiltonian structure of the standard hyperbolic spin Sutherland model that was derived recently by a different method.


Introduction and background
The investigation of integrable many-body models of Calogero-Moser-Sutherland type has been initiated half a century ago [7,38,52], and by now an extensive body of knowledge has been amassed about such systems and their manifold applications in mathematics and physics [3,11,12,47]. Remarkably, we are still witnessing intense research in this area, regarding especially the 'relativistic' deformations [48] of the originally studied particle systems and their spin extensions [17,26,30,34]. Recent enquiries on these models are concerned, for example, with their derivation by Hamiltonian reduction [4,8,14], relations to moduli spaces of flat connections [2], harmonic analysis and special functions [10,31,46], pole dynamics in the matrix KP hierarchy [43] and quiver varieties [8,13].
Interestingly, several classical integrable systems admit a bi-Hamiltonian structure, which means that the evolution equation can be encoded by two different Hamiltonians and corresponding Poisson brackets that are compatible in the sense that their linear combinations also satisfy the Jacobi identity. This can often be used to generate a family of commuting Hamiltonians through recursion. The first example of a bi-Hamiltonian structure was found by Magri [35] for the KdV equation. Since then the bi-Hamiltonian approach has become one of the central tools in the study of soliton equations [6,36]. The bi-Hamiltonian structures of Toda lattices also received considerable attention [51], but (except for the simplest rational model [1,5,15,36]) the construction of bi-Hamiltonian structures for Calogero-Moser-Sutherland type models has been left largely unexplored in the literature. The aim of our current research is to take some steps towards filling this gap.
The present report is a continuation of our series of papers [18,19,20] devoted to the study of bi-Hamiltonian aspects of spin Sutherland models. The papers [18,19] dealt with bi-Hamiltonian structures of the hyperbolic and trigonometric real spin Sutherland models, and then in [20] we investigated the corresponding complex holomorphic system. Here, we construct a bi-Hamiltonian structure for a more general real Sutherland type model that involves two spin variables belonging to the dual u(n) * of the Lie algebra of the unitary group U(n). The bi-Hamiltonian structure of the hyperbolic spin Sutherland model will be recovered by setting one of the spins to zero. The model of our interest is a special case of models investigated previously in [16,24,29,45], but in those papers bi-Hamiltonian structures were not addressed, while this is the issue that concerns us here.
Our work fits into the Hamiltonian reduction approach to integrable many-body models. We now overview the chain of developments that motivated us and sketch the content of the present report. The first step was taken by Olshanetsky and Perelomov [41] who derived the hyperbolic Sutherland model with Hamiltonian, by reduction of geodesic motion on the Riemannian symmetric space of positive matrices. One may view the symmetric space in question as the coset space G/U(n) for G := GL(n, C) regarded as a real Lie group. The pioneering work [41] focused on solving the equations of motion defined by the Hamiltonian (1.1) via projection of suitably constrained geodesics on G/U(n). It is fruitful to interpret this as Hamiltonian reduction of the phase space T * G as follows [21]. There is a natural action of the group U(n) × U(n) on T * G (see equations (3.1) and (3.2) below), which is generated by a moment map (Φ L , Φ R ) : T * G → u(n) * ⊕ u(n) * . (1. 2) The Sutherland model lives on the reduced phase space where O KKS (named after [28]) is a coadjoint orbit of U(n) of dimension 2(n − 1) (that is, the smallest non-trivial coadjoint orbit). One can also perform the reduction in two consecutive steps, descending first to Φ −1 L (0)/U(n), which is nothing but the cotangent bundle of the symmetric space. In this construction it is useful to trivialize T * G, say by right translations, and also identify G := gl(n, C) with its own dual space by means of the pairing given by the real part of the trace form, see (2.1). Thus T * G gets identified with the manifold M := {(g, J) | g ∈ G, J ∈ G}, (1.4) and H Suth (1.1) arises from the unreduced Hamiltonian H(g, J) = 1 2 J, J . (1.5) Of course, the role of the two U(n) factors can be exchanged in (1.3).
A generalization [23] of the construction just outlined is to consider the reduction to Φ −1 L (0)/(U(n) × U(n)). (1.6) After restriction to a dense open subset, this gives rise to the spin Sutherland model with Hamiltonian where the 'spin variable' ξ ∈ u(n) * ≃ u(n) has zero diagonal part. The spin ξ can be restricted to a coadjoint orbit by imposing a corresponding constraint on Φ R , as was done in [23], but this will not be advantageous for our present purpose. The most general reduction of M based on the group U(n) × U(n) consists in descending to the quotient M/(U(n) × U(n)), (1.8) which is naturally a Poisson space. Taking the quotient of a Poisson manifold by a group action, having the key property that the Poisson bracket closes on the invariant functions, is usually called Poisson reduction. The smooth functions on the quotient stem from the smooth invariant functions upstairs. In the paper [24] the closely related method of symplectic reduction was applied, which means that we considered with two arbitrary coadjoint orbits of U(n). By general principles [42], these quotients represent Poisson subspaces of the (singular) Poisson space (1.8). They are stable under the projections of the Hamiltonian flows of all the U(n) × U(n) invariant functions. The fact that the quotient space (1.8) is not a smooth manifold does not cause any serious difficulty. Still, it is convenient to focus on the dense open subset M reg ⊂ M, where the singular value decomposition of g ∈ G has the form g = Ae q B −1 with q = diag(q 1 , q 2 , . . . , q n ), q i ∈ R, q 1 > q 2 > · · · > q n , (1.10) and A, B ∈ U(n). Note that q is uniquely determined by g, but A and B are unique only up to the transformations with T n < U(n) denoting the diagonal subgroup. Introduce the space where q is restricted as in (1.10) and J ∈ G. Then consider the mapping from M reg onto M reg 0 . This induces a bijective mapping of the dense subset M reg /(U(n)×U(n)) of the quotient (1.8) onto M reg 0 /T n . The quotient by T n expresses that J matters up to the gauge transformations J → τ J τ −1 , as inherited from (1.11). According to earlier results (see equation (2.25) in [24], and see also [29,45]), one can parametrize the matrix J in terms of two u(n) * -valued spin variables ξ l and ξ r as follows: (1.14) The spin variables Poisson commute with the canonical pairs q i , p i . We may decompose any n × n complex matrix X into anti-Hermitian part X + and Hermitian part X − , and then we have ξ l = −J + and ξ r = (e −q J e q ) + . The two spins are coupled by the constraint that the diagonal part of (ξ l + ξ r ) vanishes, and the T n gauge transformations act on them by simultaneous conjugations. The reduction of the Hamiltonian (1.5) takes the form Upon setting ξ l = 0, J simplifies to the Hermitian Lax matrix of the spin Sutherland model and H spin−2 turns into the spin Sutherland Hamiltonian (1.7). The essential new contribution of the present paper is this. We first exhibit an interesting bi-Hamiltonian structure on the cotangent bundle T * G. This consists of a quadratic Poisson bracket (in terms of natural coordinate functions) and the canonical one. The quadratic Poisson bracket comes from the theory of Poisson-Lie groups [49], and its suitable Lie derivative is the canonical Poisson bracket. The Hamiltonians given by the real and imaginary parts of tr (J k ), for k ∈ N, generate a hierarchy of bi-Hamiltonian evolution equations on M. The Poisson reduction with respect to U(n) × U(n) gives rise to a bi-Hamiltonian structure on the quotient space (1.8), which we will realize in the form of compatible Poisson brackets on the space of T n invariant functions C ∞ (M reg 0 ) T n . These functions represent the ring of smooth functions on the regular part of the quotient (1.8). In practice, this means that we present the reduced Poisson brackets in terms of T n invariant functions of the pair (q, J ). By constraining the anti-Hermitian part of J to zero, we reach a joint Poisson subspace of the compatible Poisson brackets, and on this subspace we recover the bi-Hamiltonian structure of the spin Sutherland model (1.7) that was derived originally by a rather ad hoc method [18]. In the general case, we shall explain that in terms of the variables (q, p, ξ l , ξ r ) the first reduced Poisson structure takes the form that underlies the interpretation of the Hamiltonian (1.15) as a Sutherland model coupled to two spins.
The article is organized as follows. In Section 2, we describe the compatible Poisson brackets and a family of bi-Hamiltonian evolution equations on M. In Section 3, we develop the reduction of these structures. Section 4 is devoted to the spin Sutherland interpretation of the reduced system. Our conclusions are given in Section 5. There are also three appendices. Appendix A provides an explanation of the origin of the second Poisson bracket on M, Appendix B expounds some technical details, and Appendix C contains the proof of the key property of the change of variables behind the spin Sutherland interpretation.
Finally, let us state explicitly what we consider as our new results and specify their location in the text. First, we think that the unreduced bi-Hamiltonian hierarchy that we present in Section 2 has not been described before. We studied the analogous holomorphic system in [20]. Our principal results are contained in Section 3: Theorems 3.4 and 3.5 give the reduced Poisson brackets in terms of T n invariant functions on M reg 0 , Propositions 3.6 and 3.8 characterize the reduced bi-Hamiltonian hierarchy of evolution equations. In the general case, the interpretation as a spin Sutherland model with two spins is explained by Lemma 4.6 and Proposition 4.7, which were motivated by previous results [24]. Theorem 4.3 represents an important new result, since it clarifies the origin of the bi-Hamiltonian structure of the hyperbolic spin Sutherland model having the 'main Hamiltonian' (1.7).
To be more precise about the related literature [16,24,29,45], we note that [16] and [45] considered generalized spin Sutherland models on symplectic leaves of quotient spaces of the form K 1 \T * G/K 2 , where K 1 and K 2 are the fixed point subgroups of two involutions of a semisimple or reductive Lie group G, while in [24,29] K 1 = K 2 was taken to be the maximal compact subgroup. Our construction of the bi-Hamiltonian master system on T * G is specific to G = GL(n, C), since it uses that GL(n, C) is an open subset of its Lie algebra, which contains the unit matrix. One may also apply a similar construction to GL(n, R), but no other generalizations are apparent to us.
2 Master system on the cotangent bundle of GL(n, C) We first fix some notations. Let us consider G := GL(n, C), regarded as a real Lie group, and endow its Lie algebra G := gl(n, C) with the trace form X, Y := ℜtr (XY ), ∀X, Y ∈ G.
(2.1) Any X ∈ G admits the unique decomposition into strictly upper triangular part X > , diagonal part X 0 , and strictly lower triangular part X < . Correspondingly, one obtains the vector space direct sum This decomposition gives rise to the standard solution of the modified classical Yang-Baxter equation on G, r ∈ End(G). Specifically, we define and also introduce r ± := r ± 1 2 id, (2.5) where id(X) := X. Note that They yield another direct sum decomposition where the two subspaces are orthogonal to each other with respect to the bilinear form (2.1). Thus we may decompose any X ∈ G as X = X + + X − with X ± ∈ G ± . We shall use that and, in consequence of (2.4), Our immediate aim is to present two Poisson structures on the smooth, real manifold We view M as a model of the cotangent bundle of T * G. Here, the cotangent bundle is trivialized by means of right translations, and G * is identified with G using the trace form.
For any F ∈ C ∞ (M, R), introduce the G-valued derivatives ∇ 1 F , ∇ ′ 1 F and d 2 F by the defining relations and where t is a real parameter and X ∈ G is arbitrary. In addition, it will be convenient to define the G-valued functions ∇ 2 F and ∇ ′ 2 F by 14) It is worth remarking that ∇ 1 F (g, J) = g (∇ ′ 1 F (g, J)) g −1 .
Theorem 2.1. For F, H ∈ C ∞ (M, R), the following formulae define two Poisson brackets:

15)
and where the derivatives are evaluated at (g, J), and we put rX for r(X).
Proof. The first Poisson bracket is easily recognized to be the canonical one carried by the cotangent bundle T * G, presented by means of right trivialization. The second one is the real analogue of the holomorphic Poisson bracket of M, then regarded as a complex manifold, which has been constructed in [20]. The formula (2.16) has the same form as the second Poisson bracket in [20], except that now we use smooth real functions and the real part of the trace for the pairing , . We give a terse outline of the origin of this Poisson bracket in Appendix A. Of course, its Jacobi identity can be verified directly as well.
Let us recall [35,36] that two Poisson brackets are called compatible if their arbitrary linear combination is also a Poisson bracket. A bi-Hamiltonian manifold is a manifold equipped with a pair of compatible Poisson brackets. The non-trivial condition is posed by the Jacobi identity for the linear combination of the Poisson brackets. In several examples it is guaranteed by the following mechanism. Suppose that D is a derivation of the functions, and { , } is a Poisson bracket. Then the formula where D[F ] is the derivative of the function F , provides an anti-symmetric bi-derivation. This is called the Lie derivative bracket (since its underlying bi-vector is the Lie derivative of the Poisson tensor of { , } along the corresponding vector field). It is well-known [50] that if { , } D satisfies the Jacobi identity, then this holds also for arbitrary linear combinations of { , } and { , } D . Now consider the vector field on M whose flow through (g, J) is where 1 n denotes the n×n unit matrix, and let D be the corresponding derivation of C ∞ (M, R).
As coordinate functions on M, we may employ 'evaluation functions' that on (g, J) return the real and imaginary parts of the matrix elements of g, g r αβ := ℜg αβ , g i αβ := ℑg αβ , (2.19) and the components J k := T k , J with respect to an arbitrary basis T k (k = 1, . . . , 2n 2 ) of the real vector space G. Their derivatives are where we extended the derivation to complex functions. Consequently, the two Poisson brackets of Theorem 2.1 are compatible.
Proof. Since both { , } 1 and { , } D 2 are bi-derivations, it is enough to verify the equality (2.21) for coordinate functions. Hence, we inspect the cases for which F and H are taken from the functions J k and g i αβ , g r αβ . Since D[g αβ ] = 0 and the second Poisson bracket closes on the functions that depend only on g, we see that the equality (2.21) holds if both F and H depend only on g, for in these cases both sides of (2.21) vanish. Next, we note that 22) and this leads to which equals {J k , J l } 1 . To continue, we introduce the n × n matrix {g, J k } a , for a = 1, 2, by defining its αβ entry to be The formulae of Theorem 2.1 give From here it is easily checked that whereby the proof is complete.
The evolution equation of a bi-Hamiltonian system can be written in Hamiltonian form with respect to two compatible, linearly independent, Poisson brackets, and respective Hamiltonians. We now present interesting bi-Hamiltonian systems on M.
All these Hamiltonians are in involution, and they define bi-Hamiltonian systems according to the relations, and corresponding flows, listed as follows: Proof. We can calculate that ) both vanish, and we obtain Taking an arbitrary F ∈ C ∞ (M, R), the formulae of Theorem 2.1 then yield and This means that the Poisson bracket relations (2.28) and (2.29) hold. It follows also immediately that the corresponding Hamiltonian flows through the initial value (g(0), J(0)) have the simple form as claimed. In particular, J and all its functions are constant along these flows, which implies the involutivity of the underlying Hamiltonians.
Thus the functional dimension of the ring of functions, A, depending on J andJ equals 4n 2 − 2n. This ring is closed under both Poisson brackets, since in addition to (2.22) one has {J k ,J l } i = 0 (i = 1, 2) and The elements of A Poisson commute with the elements of the ring of functions, H ⊂ A, generated by the Hamiltonians (2.27). It is clear that the functional dimension of H is 2n. Let us recall [39] that a degenerate integrable system on a symplectic manifold of dimension N is given by an Abelian Poisson algebra of functional dimension N 1 < N/2, whose element admit N − N 1 functionally independent joint constants of motion 1 . We can directly apply this definition to the first Poisson structure, which is symplectic. Therefore we see that the Hamiltonians (2.27) define a degenerate integrable system, at least with respect the first Poisson bracket. The origin of the second Poisson bracket from Poisson-Lie groups (see Appendix A) shows that it is also symplectic on a dense open subset of M, and therefore we may say that the Hamiltonians of the 'bi-Hamiltonian hierarchy' of Proposition 2.3 form a degenerate integrable system with respect to both Poisson brackets.

Poisson reduction with respect to U(n) × U(n)
We are going to reduce with respect to an action of U(n) × U(n), which is engendered by two commuting actions of U(n), called right-handed and left-handed actions. Regarding the first Poisson structure, it will be obvious that the Poisson bracket closes on the invariant functions. For the second Poisson bracket, this needs a proof, and we start by providing it. We define the right-handed action of U(n) on M by associating with any η ∈ U(n) the diffeomorphism R η of M that acts according to The left-handed action is given by L η , These are commuting U(n) actions on M, for we have We shall consider the respective sets of invariant functions, and

7)
and it belongs to C ∞ (M) 1 Degenerate integrable systems are also called superintegrable or integrable in the non-commutative sense [32,37,44]. Liouville integrability corresponds to the limiting case On account of (2.10) and the orthogonality of the subspaces (2.8), we see that for right- Thus we obtain (3.7) from (2.16). The closure of the Poisson bracket on C ∞ (M) is then a consequence of the relations which follow directly from the definitions.

12)
and it belongs to C ∞ (M) Proof. The function F (and similarly H) enjoys the invariance property Taking the derivative at t = 0 we obtain (3.14) Here and below, we apply the decomposition Y = Y + + Y − for any Y ∈ G, as defined by the direct sum (2.8). We now look at the terms that appear in the formula (2.16). By using that r maps G ± into G ∓ (2.10) and taking (3.14) into account, we derive the identity In a similar manner, we obtain Next, we can write In these derivations we used the anti-symmetry of r (2.6). Observe that the terms given by the last three equations cancel altogether. In other words, the terms that contain r in r ± (2.5) all cancel from (2.16). Noticing the elementary identity we then obtain (3.12) from (2.16). The closure of the Poisson bracket on the left U(n) invariant functions follows from (3.12) and the transformation rules of the derivatives, which hold for all η ∈ U(n).
We have seen that the U(n) invariant functions form Poisson subalgebras with respect to the second Poisson bracket (2.16). Of course, the same is true regarding the first Poisson bracket (2.15). This can be seen directly from (2.15), and also follows from well known results about cotangent lifts of actions on a configuration space.
According to the singular value decomposition, also called Cartan (KAK) decomposition, every element g ∈ GL(n, C) can be decomposed as Here, q is uniquely determined by g, and if q 1 > q 2 > · · · > q n then η L and η R are unique up to the freedom with the maximal torus T n < U(n). In this paper, we call GL(n, C) reg the dense open subset of GL(n, C) whose elements obey the strict inequalities, and we also introduce It is easily seen that every invariant function F ∈ C ∞ (M) U(n)×U(n) can be recovered from its restriction to the following submanifold of M: With the tautological embedding ι : M reg 0 → M, the restriction of F reads Obviously, f is invariant under the T n action on M reg 0 given by the diffeomorphisms A τ , That is to say, f = F • ι belongs to C ∞ (M reg 0 ) T n . Next, we introduce the reduced ring of functions, This space of functions naturally inherits a pair of compatible Poisson brackets, called reduced Poisson brackets, which are defined as follows: The reduced Poisson brackets are well-defined, since the original Poisson brackets close on the U(n) × U(n) invariant functions. They contain all information about the Poisson algebra carried by the (singular) quotient space . Now our goal is to establish intrinsic formulae of the reduced Poisson brackets, which contain only derivatives with respect to the variables on M reg 0 . To this end, we need to express the derivatives of F at (e q , J) ∈ M reg 0 in terms of the derivatives of the restricted function f = F • ι. We shall use the decompositions where G ± 0 ⊂ G ± contain the respective diagonal matrices, and G ± ⊥ contain the off-diagonal ones. Thus we can write and we define ∇ 2 f and ∇ ′ 2 f similarly to (2.14). The definition (3.30) makes sense since for small enough t the components of (q + tX 0 ) satisfy the same ordering condition as those of q. Plainly, we have Let us introduce the linear operator R(q) ∈ End(G) by letting it act on an arbitrary matrix X ∈ G according to Notice that R(q) maps G ± ⊥ onto G ∓ ⊥ (3.29), respectively, in an invertible manner, and it satisfies Due to the following result, R(q) will appear in the expressions of the reduced Poisson brackets. where the superscripts refer to the decomposition (2.7).
Proof. We know from (3.9) that (∇ ′ 1 F ) + = 0 holds, because F ∈ C ∞ (M) . By using that we can write (3.38) at any (e q , J), where ad q (X) := [q, X]. This implies that The proof is completed by noting that where the derivatives are taken at (e q , J) and we use R(q) (3.34). If h is the restriction of any of the Hamiltonians H k orH k (2.27), then this formula can be written as Proof. The first and second lines of (3.43) represent the contributions of the second and third terms of (3.36) obtained upon substitution in the formula (2.15). Regarding the third line, it arises from the first term of (3.36) by taking the sum of To get (3.44) from (3.43), we use that for any Hamiltonian h proportional with the real or imaginary parts of tr (J k ) one has [d 2 h, J] = 0. In this case, the second line of (3.43) gives 47) and the third line becomes where the derivatives are taken at (e q , J) and R(q) is given by (3.34). If h is the restriction of any of the Hamiltonians H k orH k (2.27), then this formula can be recast in the form Proof. We see by combining (3.10) (obtained in the proof of Lemma 3.1) and Lemma 3.2 that we substitute the identity (3.36) both for ∇ 1 F and ∇ 1 H. After that, we spell out the first two terms of (3.51) and the contributions coming from the first terms of ∇ 1 F and ∇ 1 H. The formula (3.49) is then obtained by collecting terms. Turning to the proof of (3.50), instead of (3.49) it is shorter to go back to (3.51). We observe from (2.30) The verification is finished by noting that and (3.55) The following statement summarizes the outcome of our construction.

56)
give rise to bi-Hamiltonian evolution equations since they satisfy (3.57) Proof. This is obvious from our construction. In particular, the compatibility of the reduced Poisson brackets follows from the compatibility of the original Poisson brackets on C ∞ (M, R) by applying the definition (3.27). The properties (3.57) are consequences of (3.44) and (3.50) taking into account that which are implied by (2.30).
In the next remark, we explain that the compatible Poisson brackets can be defined also on the ring of function C ∞ (M reg 0 ) T n .
Remark 3.7. The ring of functions C ∞ (M) red (3.26) is contained in C ∞ (M reg 0 ) T n , but is not equal to it. For example, the components of q give elements of C ∞ (M reg 0 ) T n , and can be extended to unique, U(n)×U(n) invariant continuous functions on M, but these functions lose their differentiability at the locus where q i = q i+1 for some i. This holds since the e 2q i (3.20) are the ordered eigenvalues of gg † and, as is well known, the differentiability of eigenvalues is in general lost where they coincide. Nevertheless, the formulae (3.43) and (3.49) define compatible Poisson brackets on the whole of C ∞ (M reg 0 ) T n . In order to see this, consider M red (3.22), which is the set of elements of M that can be transformed into M reg 0 by the action of U(n) × U(n). This is a dense open subset and the pull-back by ι : M reg 0 → M reg yields an injective and surjective map from C ∞ (M reg ) U(n)×U(n) onto C ∞ (M reg 0 ) T n , that is, with some function T : M reg 0 → G + 0 . Note that G + 0 is just the Lie algebra of T n . This ambiguity drops out after projection to the quotient space M reg 0 /T n . For definiteness, we shall fix this ambiguity of the vector field E i h by by imposing the condition where {f, h} red i is understood to be given by the formulae (3.43) and (3.49). These formulae define anti-symmetric bi-derivations on C ∞ (M reg 0 ), but the Jacobi identity holds only for the T n invariant functions. By some abuse of terminology, we call the vector field E i h the Hamiltonian vector field associated with h by means of the bracket { , } red i .

Proposition 3.8. Consider the vector fields
with the Hamiltonians (3.56). Using R(q) (3.34), these vector fields have the explicit form (3.65) All these vector fields are tangent to the submanifold defined by imposing the constraint J + = 0, as well as to the submanifold

68)
while the restriction ofẼ i k vanishes identically on M reg 0,− for all k and i. The vector fields V i k reproduce the evolutional vector fields of the spin Sutherland hierarchy described in [18]. Denoting the restrictions of E i k andẼ i k on M reg 0,+ by U i k andŨ i k , we obtain  [18] shows that the bi-Hamiltonian vector fields (3.68) reproduce the spin Sutherland hierarchy studied earlier.

Interpretation as spin Sutherland models
We below develop the physical interpretation of the reduced bi-Hamiltonian system. This interpretation will be reached via a suitable parametrization of the variable J for (e q , J) ∈ M reg 0 . In this parametrization the first reduced Poisson structure and one of the reduced Hamiltonians take the form characteristic of spin Sutherland models. We first present the particular case corresponding to the Hamiltonian (1.7) and deal with the general case (1.15) subsequently. We will not spell out the expression of the second Poisson bracket in terms of the new variables, since those formulae are complicated and do not enhance our understanding. We have seen in Proposition 3.8 that the evolutional vector fields of the spin Sutherland hierarchy (given by (3.68)) result from our construction by restriction to the submanifold M reg 0,− . Now we show that the bi-Hamiltonian structure found in [18] also results from this restriction. We start by demonstrating that the derivative of J + vanishes along all Hamiltonian vector fields E i h at those point where J + = 0, i.e., on the submanifold M reg 0,− (3.66). To see this, we rearrange the Poisson brackets in the form 1) and show that the functions vanish for both Poisson brackets upon imposing the constraint J + = 0.
The following lemmas are obtained by straightforward, somewhat tedious, calculations. We sketch only the proof of the second lemma, which is the more complicated one.
It follows that E 2 h is tangent to both submanifolds M reg 0,− (3.66) and M reg 0,+ (3.67). Proof. We shall use that, for any X ∈ G, X − = 1 2 (X + X * ) and X + = 1 2 (X − X * ), together with obvious properties of the trace form (2.1), like X * , Y * = X, Y . By directly spelling it out, we find Thus, the first term of {f, h} red 2 (3.49) gives (4.6), and the second term of (3.49) gives the last terms in both lines (4.7) and (4.8). In order to confirm the latter statement, note that Next, we inspect the terms of (3.49) that contain R(q) (3.34). We use the anti-symmetry of R(q) (3.35), the property (R(q)X) * = −R(q)X * , and that it maps G ± into G ∓ , respectively. Then we can write Similarly, we obtain (4.12) By adding these two expressions, and taking into account the factor 2 on the left-hand side of (3.49), we get the terms containing R(q) in (4.7) and (4.8). The first terms on the right-hand sides of (4.7) and (4.8) result by expanding and collecting all terms coming from the last line of (3.49), which is laborious but fully straightforward. Some further details are given in Appendix B.
Let us remember [42] where the derivatives are evaluated at (e q , J − ). These formulae reproduce the compatible Poisson brackets of the hyperbolic spin Sutherland hierarchy described earlier in [18].
Proof. Suppose that F , H are the restrictions of f, h ∈ C ∞ (M reg 0 ) T n . Then, according to the definition of the restricted brackets, where the formulae of the preceding two lemmas are applied, at J + = 0. Here, we substitute (d 2 f ) − (e q , J − ) = d 2 F (e q , J − ) and analogous relations for the other derivatives. This readily leads to the above forms of the restricted Poisson brackets. For example, to obtain (4.15), we also use that This holds by virtue of the identities Taking into account some obvious differences of notation, one sees by direct comparison that the Poisson brackets in (4.14) and (4.15) coincide with those in Theorem 1 of [18].
Remark 4.4. The Poisson bracket (4.15) was obtained in [18] by suitably rewriting the Poisson structure of an example of models derived by Li [33] applying a rather complicated method based on dynamical Poisson-Lie groupoids. Then it was directly shown to be compatible with the first Poisson bracket (4.14) extracted from [23]. The present derivation is simpler and it highlights that both Poisson brackets originate from a single reduction in a unified manner. If we parametrize the Hermitian matrix J = J − in the form where the p i are arbitrary real numbers and ξ in an off-diagonal anti-Hermitian matrix, then the reduced Hamiltonian 1 2 tr (J 2 ) reproduces (1.7). The spin ξ matters up to the gauge transformations ξ → τ ξτ −1 , ∀τ ∈ T n . Under the reduced first Poisson bracket (4.14), the T n invariant functions of ξ are those arising from the u(n) Lie-Poisson bracket reduced by the first class constraints ξ kk = 0 for all k. The q i , p i (i = 1, . . . , n) form canonical pairs with respect to the reduced first Poisson bracket, and they Poisson commute with the T n invariant functions of ξ. These statements are proved in [18,23], and will be generalized in the next subsection.
The combined message of Theorem 4.3 and Proposition 3.8 is summarized in the next corollary. Finally, let us observe that J + and J − appear rather symmetrically in the formula of Lemma 4.2. In particular, E 2 h [J − ] = 0 holds after restriction to the submanifold M reg 0,+ (3.67). Therefore we can restrict the second reduced Poisson bracket on this submanifold. Moreover, n out of the 2n commuting Hamiltonians (3.56) survives this restriction, in correspondence to the vector fields in (3.70). It can be verified that the substitution J + = iJ − , with the new variable J − , converts the restricted Poisson bracket on C ∞ (M reg 0,+ ) T n into the restricted second Poisson bracket on C ∞ (M reg 0,− ) T n . This means that we do not obtain anything new from this restriction, and hence we omit its more detailed description.
Proof. By decomposing J as J = J + + J − using (2.7), we can write Since the first two terms give the anti-Hermitian part (e −q Je q ) + , we obtain from (4.20) This relation can be solved for J − ij as and J − ii = p i by definition. Thus we have derived (4.22) and its easy to see that this yields a smooth bijection between (e q , J) ∈ M reg 0 and the set of quadruplets (q, p, ξ l , ξ r ) satisfying the conditions stated by the lemma. The transformation rule (4.23) is equivalent to (3.25). The forms of ℜtr (J) and 1 2 ℜtr (J 2 ) then follow by straightforward evaluation. To obtain the formula (1.15), one applies the identity 2 cosh x/ sinh 2 x = 1/ sinh 2 (x/2) − 2/ sinh 2 x.
It turns out that under the reduced first Poisson bracket (3.43) the q i , p i form canonically conjugate pairs, the Poisson brackets of the T n invariant functions of ξ l , ξ r are governed by the Lie-Poisson bracket of u(n) ⊕ u(n) reduced by the constraints (4.21), and these two sets of variables decouple under { , } red 1 . This result can be obtained as a consequence of the symplectic reduction approach adopted in [24]. Alternatively, we can directly perform the required change of variables in the formula (3.43). In order to make this paper self-contained, we present the second method, but relegate all computational details to Appendix C. Incidentally, we have verified that the two methods give the same result, which provides an excellent check on our considerations.
Thus, by using (4.22), we parametrize M reg 0 by the quintets of variables For any smooth, real function F of these variables, we have the 'partial gradients' 28) which are defined in the natural manner using the restriction of the pairing (2.1) to G + ⊥ and to G + 0 (remember that G + = u(n)). For arbitrary T n invariant functions f, h of the old variables (e q , J) we write f (e q , J) = F (q, p, ξ l ⊥ , ξ r ⊥ , ξ 0 ) and h(e q , J) = H(q, p, ξ l ⊥ , ξ r ⊥ , ξ 0 ), (4.29) and calculate {F, H} red 1 from the identity Proposition 4.7. Let F, H be T n invariant smooth functions on M reg 0 , parametrized by the variables (4.27) that transform according to In terms of these variables, the first reduced Poisson bracket (3.43) can be written as By using this identity, and its counterpart with F and H exchanged, one may write the terms containing d ξ 0 in the second line of (4.32) in many alternative ways. Upon imposing the constraint ξ l = 0, and putting ξ := ξ r ⊥ , J in (4.22) reproduces J − in (4.19). Setting also all derivatives with respect to ξ 0 and ξ l ⊥ to zero, the Poisson bracket (4.32) reproduces the first reduced Poisson bracket of the spin Sutherland model (1.7) described in Remark 4.4. Of course, by setting ξ r = 0 instead of ξ l = 0 one reaches the same model. Remark 4.9. Let us recall from Remark 2.4 that the master system behind the spin Sutherland model (1.15) is a degenerate integrable system. It is known [27,53] that degenerate integrability is generically preserved under Hamiltonian reduction. The degenerate integrability of the model (1.15) on generic symplectic leaves of the first Poisson structure follows as a special case of results of [45]. It would be nice to enhance these results by explicitly exhibiting the required number of independent constants of motion. We here restrict ourselves to displaying a large number of U(n) × U(n) invariant elements of the ring A of unreduced constants of motion, discussed in Remark 2.4, which descend to constants of motion of the reduced system. Namely, let P be an arbitrary product of non-negative powers of J + and J − . That is, P has the form with non-negative integers k 1 , k 2 , k 3 , k 4 etc. On account of the transformation rules (3.1) and (3.2), the real and imaginary parts of tr P are U(n) × U(n) invariant elements of A. Similarly, using (2.34), one obtains U(n) × U(n) invariant unreduced constants of motion by taking the trace of an arbitrary product of powers ofJ + andJ − . Degenerate integrable systems are also known to be Liouville integrable under very general conditions [27]. Considering the restriction of the model to generic symplectic leaves, a construction of sufficient number of constant of motion in involution can be found in [29].

Conclusion
We here introduced a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group GL(n, C) and analyzed its quotient with respect to the symmetry group U(n) × U(n). We described the form of the compatible reduced Poisson brackets (Theorems 3.4 and 3.5) as well as the bi-Hamiltonian vector fields generated by the commuting reduced Hamiltonians (Propositions 3.6 and 3.8). We found that the restriction of the reduced bi-Hamiltonian hierarchy to a joint Poisson subspace of its two Poisson brackets reproduces the hyperbolic spin Sutherland hierarchy associated with the Hamiltonian (1.7). In the general case, the reduced system was identified as a Sutherland model coupled to two spin variables according to the Hamiltonian (1.15). Thus the commuting flows of the model that are generated by the spectral invariants of the Lax matrix (4.22) all admit bi-Hamiltonian description, which may be considered as our main result. The spin Sutherland interpretation arose from using the variables (q, p, ξ l , ξ r ) instead of (e q , J), in which the first reduced Poisson bracket takes the form displayed in Proposition 4.7. It is in principle possible to present also the second Poisson bracket in these variables, but the resulting formulae are not expected to have a transparent structure. At least on the subspace where J is Hermitian and positive definite, it should be possible to construct an alternative parametrization that would allow tr (J) to be interpreted as a spin Ruijsenaars-Schneider Hamiltonian. In the corresponding trigonometric case, such a change of variables is known [19], and it permits one to recover the spinless trigonometric Ruijsenaars-Schneider model on a special symplectic leaf of the reduced second Poisson bracket. Thus we suspect that the spinless hyperbolic Ruijsenaars-Schneider model should be found on a symplectic leaf of the second Poisson structure described in Theorem 4.3. However, we do not know how to characterize the symplectic leaves of this Poisson structure. We encountered difficulties when trying to find them by 'analytic continuation' from the trigonometric to the hyperbolic case. This poses a very interesting open problem for future work. It is worth mentioning that, at least to our knowledge, no derivation of the the real, hyperbolic Ruijsenaars-Schneider model by Hamiltonian reduction is known at present, as opposed to the real trigonometric model and its complex holomorphic counterpart, for which several reduction treatments are available [3,8,22,25,40]. It would be important to construct such a derivation, and finding the symplectic leaves of the Poisson bracket (4.15) could help to resolve this long-standing conundrum.
Acknowledgements. I wish to thank Maxime Fairon for several useful remarks on the manuscript. This work was supported in part by the NKFIH research grant K134946.

A An explanation of the second Poisson bracket
We below outline how the second Poisson bracket (2.16) arises by a change of variables from Semenov-Tian-Shansky's Poisson bracket [49] on the Heisenberg double G × G. We will be brief since this explanation closely follows the appendix in [20]. However, note that in [20] we considered holomorphic complex functions, while now we deal with real smooth functions.
We begin by introducing a non-degenerate, invariant bilinear form on the real Lie algebra where (X 1 , X 2 ) and (Y 1 , Y 2 ) are from G ⊕ G, and (2.1) is applied. It is not difficult to see that G ⊕ G is the vector space direct sum of the subalgebras Recall that r ± are defined in (2.5), i.e., r + (X) = X > + 1 2 X 0 and r − (X) = −X < − 1 2 X 0 for X written as in (2.2) These are isotropic subalgebras, meaning that the bilinear form (A.1) vanishes on them separately. After identifying G with G δ , we can use the bilinear form to take G * (A.3) as a model of the dual space of G, which explains the notation.
Let us define the linear operator ρ on G ⊕ G by using the projections P G δ onto G δ and P G * onto G * associated with the vector space direct sum G ⊕ G = G δ + G * . It features in two well-known [49] Poisson brackets on C ∞ (G × G, R). For , R), the G ⊕ G-valued left-and right-derivatives are determined by DF (g 1 , g 2 ), (X 1 , X 2 ) 2 := d dt t=0 F (e tX 1 g 1 , e tX 2 g 2 ), where t ∈ R, (X 1 , X 2 ) runs over G ⊕ G and (g 1 , g 2 ) ∈ G × G. With these notations, the two Poisson brackets read The minus bracket is called the Drinfeld double bracket, and the plus one the Heisenberg double bracket. The former makes G × G into a Poisson-Lie group, and the latter is symplectic in a neighbourhood of the identity [49]. Now we introduce new variables in a neighbourhood of (1 n , 1 n ) ∈ G × G. We need the connected subgroups of G corresponding to the subalgebras in (2.3). These are denoted G > , G < and G 0 , respectively, where G > contains the upper triangular complex matrices in G having 1 in their diagonal entries, and G 0 contains the diagonal matrices in G. Then the connected Lie subgroups of G × G associated with the subalgebras G δ (A.2) and G * (A.3) are G δ := {g δ | g δ := (g, g), g ∈ G} (A.7) and Since G ⊕ G = G δ + G * , there exist open neighbourhoods of the identity in G × G whose elements can be factorized uniquely as where (g δL , η * R ) and (g δR , η * L ) vary in corresponding open sets around the identity in G δ × G * . Let us write g δR = (g, g) and η * R = η > η 0 , (η 0 η < ) −1 , (A. 10) and use the factorizations (A.9) to define the map ψ : (g 1 , g 2 ) → (g, J) with J := η > η 2 0 η < . (A.11) By suitably choosing its domain, ψ gives a diffeomorphism between open subsets of G × G containing (1 n , 1 n ), but it is advantageous for us to regard its image as a subset of G × G.
The following statement can be verified by essentially the same calculations that were presented in [20]. The proposition guarantees that the Jacobi identity holds for the restriction of the second Poisson bracket (2.16) to an open set around (1 n , 1 n ). This implies that (2.16) gives a Poisson bracket on C ∞ (M, R), too. Indeed, as determined by (2.16), the Poisson brackets of the coordinate functions on M provided by the matrix elements (2.19) and the components of J are real-analytic functions on M, and thus the Jacobi identity holds for them globally since we know from the proposition that it holds on an open set.
Remark A.2. Incidentally, by writing ∇ 1 F (g, J) = gd 1 F (g, J), where d 1 F is defined similarly to (2.13) using that G is an open subset of G, one may extend both Poisson brackets (2.15) and (2.16) to C ∞ (G × G, R) as well.
Remark A.3. For the aficionados of Poisson-Lie groups, we remark that u(n) δ < G δ has the property that its annihilator inside G * (A.3) with respect to the pairing (A.1) is a Lie subalgebra of G * . By applying the general theory [49], one can trace back the closure statements of Lemma 3.1 and Lemma 3.2 to this property. We gave direct proofs of these lemmas, and thus there is no need to elaborate this point.

B A remark on the proof of Lemma 4.2
The goal of this appendix is to help those readers who wish to go through the details of the last step of the proof of Lemma 4.2, which requires the demonstration of the following identity: where E(f, h) stands for the 2 terms obtained by exchanging f and h. We note that X as defined above can be expressed in the alternative form Finally, the claimed formula (B.1) follows by combining the statements displayed above.