KP Trigonometric Solitons and an Adelic Flag Manifold

We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127-165].


Introduction
One of the many gems I had the chance to share with Vadim Kuznetsov during his visit to Louvain-la-Neuve in the fall semester of 2000, had to do with his work on separation of variables and spectrality. He knew about my work on bispectral problems and insisted that these problems were connected. He introduced me to his work with Nijhoff and Sklyanin [11], on separation of variables for the elliptic Calogero-Moser system. I would have loved to discuss the topic of the present paper with him, which deals with the (simpler) trigonometric version of this system.
One of the exciting new developments in the field of integrable systems has been the introduction by George Wilson [21] of the so called Calogero-Moser spaces where gl(N, C) denotes the space of complex N × N matrices, I is the identity matrix and the complex linear group GL(N, C) acts by simultaneous conjugation of X and Z. These spaces are at the crossroads of many areas in mathematics, connecting with such fields as non-commutative algebraic and symplectic geometry. For an introduction as well as a broad overview of the subject, I recommend Etingof's recent lectures [4] at ETH (Zürich). Since the bispectral problem is not mentioned in these lectures, it seems not inappropriate to start from this problem as introduced in the seminal paper [3] by Duistermaat and Grünbaum, which has played and (as we shall see) continues to play a decisive role in the subject.
In the form discussed by Wilson [20], the bispectral problem asks for the classification of all rank 1 commutative algebras A of differential operators, for which the joint eigenfunction ψ(x, z) which satisfies also satisfies a (non-trivial) differential equation in the spectral variable B(z, ∂/∂z)ψ(x, z) = g(x)ψ(x, z).
In [20], it was found that all the solutions of the problem are parametrized by a certain subgrassmannian of the Segal-Wilson Grassmannian Gr, that Wilson called the adelic Grassmannian and that he denoted by Gr ad . The same Grassmannian parametrizes the rational solutions in x of the Kadomtsev-Petviashvili (KP) equation (vanishing as x → ∞). The main result of [21] is to give another description of Gr ad as the union ∪ N ≥0 C N of the Calogero-Moser spaces introduced above. The correspondence can be seen as given by the map which sends a pair (X, Z) ∈ C N (modulo conjugation) to the (stationary) Baker-Akhiezer function of the corresponding space W = β(X, Z) ∈ Gr ad . From (1.4), one sees immediately that the mysterious bispectral involution b : Gr ad → Gr ad which exchanges the role of the variables x and z ψ b(W ) (x, z) = ψ W (z, x), W ∈ Gr ad , becomes transparent when expressed at the level of the Calogero-Moser spaces, as it is given by b C (X, Z) = (Z t , X t ), where X t and Z t are the transposes of X and Z. The situation is nicely summarized by the following commutative diagram, all arrows of which are bijections In [6], jointly with Plamen Iliev, we considered the following discrete-continuous version of the bispectral problem. To determine all rank 1 commutative algebras A of difference operators, for which the joint eigenfunction ψ(n, z) which satisfies also satisfies a (non-trivial) differential equation in the spectral variable B(z, ∂/∂z)ψ(n, z) = g(n)ψ(n, z). (1.7) The problem was motivated by an earlier work with Alberto Grünbaum [5], where we investigated the situation when the algebra A contains a second-order symmetric difference operator (this time, without imposing any rank condition on A). This situation extends the theory of the classical orthogonal polynomials, where the differential equation is of the second order too.
The main result of [6] was to construct from Wilson's adelic Grassmannian Gr ad an (isomorphic) adelic flag manifold Fl ad , which provides solutions of the bispectral problem raised above, and parametrizes rational solutions in n (vanishing as n → ∞) of the discrete KP hierarchy.
The message of this paper is to show that the analogue of Wilson's diagram (1.5) in the context of this new bispectral problem is the following commutative diagram, with bijective arrows In this diagram, the spaces , (1.9) are trigonometric analogues of the Calogero-Moser spaces C N defined in (1.1). Gr trig is a certain subgrassmannian of linear spaces W ∈ Gr parametrizing special solitons of the KP hierarchy, that I call the "trigonometric Grassmannian", since the corresponding tau functions τ W take the form being a solution of the trigonometric Calogero-Moser-Sutherland hierarchy (as long as all x i (t 1 , t 2 , t 3 , . . .) remain distinct, see Section 3). The bispectral map b : Fl ad → Gr trig , which sends a flag V to a linear space W = b(V), and is defined by 11) trivializes when expressed at the level of the Calogero-Moser spaces, as it is now given by The result will follow easily from the expression of the (stationary) Baker-Akhiezer function ψ W (x, z) of a space W ∈ Gr trig , in terms of pairs of matrices (X, which defines the map β trig in the diagram (1.8). The definition of the map β in the same diagram follows immediately from the definition of the adelic flag manifold in terms of Wilson's adelic Grassmannian as given in [6], and will be recalled in Section 4. Some time ago, Ruijsenaars [16] (see also [17]) made a thorough study of the action-angle maps for Calogero-Moser type systems with repulsive potentials, via the study of their scattering theory. Along the way, he observed various duality relations between these systems. In particular, when the interaction between the particles in the trigonometric Calogero-Moser system is repulsive, the system is dual (in the sense of scattering theory) to the rational Ruijsenaars-Schneider system. In the last section, we show that within our picture (1.8), if τ W (t 1 , t 2 , t 3 , . . .) is the tau function of a space W ∈ Gr trig as in (1.10), the tau function of the flag b −1 (W ) is with λ i (t 1 , t 2 , . . .) solving now the rational Ruijsenaars-Schneider hierarchy, thus representing Ruijsenaars' duality as a bispectral map. The likelihood of this last statement was formulated previously by Kasman [8], on the basis of a similar relationship between the quantum versions of these systems, as studied by Chalykh [1] (see also [2]). It is also implicitly suggested by a recent work of Iliev [7], which relates the polynomial tau functions in n of the discrete KP hierarchy with the rational Ruijsenaars-Schneider hierarchy.

Trigonometric solitons of the KP hierarchy
In this section, we construct a class of special solitons of the KP hierarchy. Their relation with the trigonometric version of the Calogero-Moser hierarchy will be explained in the next section, justifying the appellation "trigonometric solitons". We first need to recall briefly the definition of the Segal-Wilson Grassmannian Gr and its subgrassmannian Gr rat , from which solitonic solutions of the KP hierarchy can be constructed, see [18] for details. Let S 1 ⊂ C be the unit circle, with center the origin, and let H denote the Hilbert space L 2 (S 1 , C). We split H as the orthogonal direct sum H = H + ⊕ H − , where H + (resp. H − ) consists of the functions whose Fourier series involves only non-negative (resp. only negative) powers of z. Then Gr is the Grassmannian of all closed subspaces W of H such that (i) the projection W → H + is a Fredholm operator of index zero (hence generically an isomorphism); (ii) the projection exp −1 (t, z)W belongs to Gr and, for almost any t, it is isomorphic to H + , so that there is a unique function in itψ W (t, z) which projects onto 1. The function ψ W (t, z) = exp(t, z)ψ W (t, z) is called the Baker-Akhiezer function of the space W andψ W (t, z) is called the reduced Baker-Akhiezer function. A fundamental result of Sato asserts that there is a unique (up to multiplication by a constant) function τ W (t 1 , t 2 , t 3 , . . .), the celebrated tau function, such that An element of the form k≤s a k z k , a s = 0, is called an element of finite order s. For W ∈ Gr, W alg denotes the subspace of elements of finite order of W . It is a dense subspace of W . We also need the ring The rational Grassmannian Gr rat is the subset of Gr, for which Spec(A W ) is a rational curve. In this case, A W is a subset of the ring C[z] of polynomials in z and the map C → Spec(A W ) induced by the inclusion is a birational isomorphism, sending ∞ to a smooth point completing the curve. Let us fix N distinct complex numbers λ 1 , . . . , λ N inside of S 1 , and another N non-zero complex numbers µ 1 , . . . , µ N . We assume that λ i − λ j = 1, ∀ i = j. We define W alg λ,µ as the space of rational functions f (z) such that (i) f is regular except for (at most) simple poles at λ 1 , . . . , λ N and poles of any order at infinity; (ii) f satisfies the N conditions The Baker-Akhiezer function ψ W of a space W = W λ,µ has the form for some functions b j (t) determined by (2.3). By a simple computation, we obtain We introduce the N × N matrices X and Z with entries with δ ij the usual Kronecker symbol so that Z is diagonal, and we put With these notations, the system of equations (2.5) determining the functions b i (t) is written as where b(t) and µ denote column vectors of length N , with entries b i (t) and µ i respectively. From (2.4) and (2.8), we get that the reduced Baker-Akhiezer functionψ can be written as with µ t the row vector obtained by transposing µ.
The following commutation relation will be crucial For short set Considering that if T is a matrix of rank 1 then 1 + tr T = det(I + T ) (with tr denoting the trace), from (2.7), (2.9), (2.10) and (2.11), we find Using the fact that the determinant of a product of matrices does not depend on the orders of the factors, we get In particular, setting t 2 = t 3 = · · · = 0 and t 1 = x, we find that the so-called stationary Baker-Akhiezer function of W admits the following form By Cauchy's determinant formula applied to X in (2.6) which is non-zero by the assumptions we made on the λ j 's and µ j 's. Thus X is invertible and it follows from (2.10) that rank(XZX −1 − Z + I) = 1, as announced in the Introduction, see (1.9) and (1.13). It is now easy to deduce the following proposition.
Proposition 1. The tau function τ W (t 1 , t 2 , t 3 , . . .) of a space W = W λ,µ is given by with X and Z defined as in (2.6).
Proof . Denoting for short by exp{· · · } the expression that appears inside the exponential in (2.15), one computes withX andZ defined as in (2.7) and (2.11). This shows that the reduced Baker-Akhiezer function obtained in (2.12) satisfies Sato's formula (2.1) with τ W as in (2.15). Since this formula determines the tau function up to a constant, the proof is complete.
Remark 1. 2 Kasman and Gekhtman [9] (see Corollary 3.2 in their paper) have established that for any triple (X, Y, Z) of N × N matrices such that rank (XZ − Y X) = 1, the function is a tau function of the KP hierarchy, associated with some W ∈ Gr rat , by showing that it satisfies the Hirota equation in Miwa form. The special choice leads to a N -soliton solution of the KP hierarchy, and Proposition 1 can be obtained by picking Thanks to Remark 1, it makes sense to introduce the following definition: Definition 1. The trigonometric Grassmannian Gr trig is defined to be the following subgrassmannian of Gr rat with C trig N defined as in (1.9) and with τ (X,Z−I,Z) (t 1 , t 2 , . . .) defined as in (2.16). The corresponding solutions of the KP hierarchy will be called trigonometric solitons. Example 1. As an example of a non-generic W ∈ Gr trig (i.e. not of the form W λ,µ as above), let us define W ∈ Gr rat to be the closure in L 2 (S 1 , C) of the space W alg formed with the rational functions f (z) such that (i) f is regular except for (at most) a double pole at z = 0 and a pole of any order at ∞; (ii) f satisfies the two conditions A simple computation shows that the corresponding stationary Baker-Akhiezer function is which can be put into the form (2.13) with which forms a trigonometric Calogero-Moser pair as defined in (1.9), with a non-diagonalizable Z. The corresponding tau function is

Gr trig and the Calogero-Moser-Sutherland hierarchy
In this section, we relate the Grassmannian Gr trig introduced in Definition 1 with the Calogero-Moser-Sutherland system, also referred as the trigonometric (or hyperbolic) Calogero-Moser system. This is a system of N particles on the line whose motion is governed by the Hamiltonian . (3.1) We allow the particles to move in the complex plane. The original Moser-Sutherland system [12] is recovered if we suppose our particles to be confined to the imaginary axis, since the hyperbolic sine becomes then the trigonometric one. When the motion of the particles is confined to the real axis, the potential is attractive if velocities are real, and repulsive if velocities (and time) are purely imaginary. In this last case, the particles ultimately behave like free particles. But for most initial conditions in the complex plane, some collisions will take place after a finite time.
Moser [12] proved that the system (3.1) is completely integrable, showing that it describes an isospectral deformation of the N × N matrix L(x, y) with entries where δ ij is the usual Kronecker symbol. More precisely, the quantities F k (x, y) = (1/k) tr L k (x, y) (with tr denoting the trace), k = 1, 2, . . . , N , are N independent first integrals in involution for the system. In particular, F 2 (x, y) gives back the original Hamiltonian H(x, y).
In order to relate the system to the KP trigonometric solitons introduced in Definition 1, we need the following lemma which can be extracted from Ruijsenaars [16], and was motivated by his study of the scattering theory of the system (3.1) when the interaction between the particles is repulsive (which, with our conventions, amounts to pick velocities and time imaginary).

L. Haine
Lemma 1. Let (X, Z) ∈ C trig N , N ≥ 1, as defined in (1.9). If X is diagonalizable, there is a conjugation with K(x) a diagonal matrix of the form
Proof . Since det(X) = 0, when diagonalizing X, we can always assume K to have the form (3.4). Denoting by L the result of the conjugation of Z by the same matrix U , since by the definition of C trig N the rank of [X, Z] + X is 1, we have with α and β two (non-zero) column vectors of length N . Writing (3.5) componentwise, we get which, by putting i = j, shows that α i β i = e x i = 0, ∀ i. Thus, by multiplying U to the right by an appropriate diagonal matrix, we can always arrange that α i = β i = e x i /2 . With this choice, one sees from (3.6) that necessarily e x i = e x j , ∀ i = j, and while the diagonal entries L ii are free. Denoting L ii = y i establishes the lemma.
The explicit integration of the system (3.1) was performed by Olshanetsky and Perelomov [13]. An interpretation in terms of Hamiltonian reduction was given by Kazhdan, Kostant and Sternberg [10], leading to the following result whose proof can be found in the nice treatise [19], by Suris. We identify gl(N, C) with its dual via the trace form X, Y = tr(XY ), and we define accordingly the gradient of a smooth function ϕ : gl(N, C) → C by dϕ(X)(Y ) = ∇ϕ(X), Y . [19,Theorem 27.6] for a proof.) Let ϕ : gl(N, C) → C be an Ad-invariant function, and let H(x, y) = ϕ(L(x, y)), with L(x, y) as in (3.2). Let (x i (t), y i (t)) be the solution of Hamilton's equationṡ

Proposition 2. (See
Then, the quantities e x i (t) are the eigenvalues of the matrix with K 0 = K(x 0 ) as in (3.4), L 0 = L(x 0 , y 0 ) and ∇ϕ the gradient of ϕ. Moreover, the matrix V (t) which diagonalizes K 2 0 exp t∇ϕ(L 0 ) , so that K 2 (x(t)) = V (t)K 2 0 exp t∇ϕ(L 0 ) V (t) −1 , and which is normalized by the condition V (t)K 0 e = K(x(t))e, e = (1, . . . , 1) t , is such that Remark 2. As explained at the beginning of this section, to describe a repulsive interaction in the trigonometric Calogero-Moser system with Hamiltonian H(x, y) = ϕ L(x, y) , ϕ(L) = (1/2) tr L 2 as in (3.1) and (3.2), we have to pick the x i 's real, the y i 's imaginary and t imaginary also. In this case, t∇ϕ(L 0 ) = tL 0 , t ∈ √ −1 R, is hermitian and thus K 2 0 exp(tL 0 ) in (3.7) is always diagonalizable. In the general case, this matrix can become non-diagonalizable for some values of t, which leads to collisions in the system.
Let us now consider the following Hamiltonians H k (x, y) = ϕ k (L(x, y)), with ϕ k (L(x, y) where x i (t 1 , t 2 , t 3 , . . .) denotes the solution of the trigonometric Calogero-Moser hierarchy with initial condition (x 0 , y 0 ) ∈ C 2N specified by (3.3) as in Lemma 1 above.
Proof . Let us consider the solution x i (t) of the trigonometric Calogero-Moser hierarchy with initial condition (x 0 , y 0 ) ∈ C 2N specified by (3.3), i.e. U −1 XU = K 2 (x 0 ) ≡ K 2 0 , U −1 ZU = L(x 0 , y 0 ) ≡ L 0 (remember from the proof of Lemma 1 that necessarily x i (0) = x j (0), ∀ i = j). From Proposition 2 and the definition of ϕ k in (3.8), it follows easily that the quantities e x i (t) are the eigenvalues of the matrix

Since the determinant of a matrix is invariant under conjugation, we obtain
With account of (2.16), this establishes (3.9), up to the inessential (in the sense that it leads to the same solution of the KP hierarchy) exponential factor e . The proof is complete.

The bispectral property of the KP trigonometric solitons
The adelic Grassmannian Gr ad consists of the spaces W ∈ Gr rat for which the curve Spec(A W ) (with A W as in (2.2)) is unicursal, that is the birational isomorphism C → Spec(A W ) corresponding to the inclusion A W ⊂ C[z] is bijective. As established in [20], Gr ad parametrizes all commutative rank 1 algebras A of bispectral differential operators, in the sense of (1.2) and (1.3).
The corresponding algebras A are isomorphic to A W . The joint eigenfunction of the operators in A is given by the stationary Baker-Akhiezer function ψ W (x, z) of W , and it can be written in terms of a pair of matrices (X, Z) satisfying rank([X, Z] + I) = 1 as in (1.4). Since in Section 2 we have chosen in the definition of Gr the circle S 1 to be of radius 1, all the eigenvalues of Z (which correspond to the singular points of Spec(A W ) under the map C → Spec(A W )) will be inside of S 1 , see [21]. We shall thus assume without loss of generality that any pair (X, Z) ∈ C N as defined in (1.1) satisfies the condition that the spectrum of Z is inside the unit circle 3 . Following [6], starting from any W ∈ Gr ad and its corresponding tau function τ W (t 1 , t 2 , . . .), we build a function ψ(n, t, z) = ψ(n, t 1 , t 2 , . . . , z) via the formula ψ(n, t, z) = (1+z) n exp(t, z) τ W (t 1 +n−1/z, t 2 −n/2−1/(2z 2 ), t 3 +n/3−1/(3z 3 ), . . .) τ W (t 1 + n, t 2 − n/2, t 3 + n/3, . . .) . (4.1) We define a corresponding flag of subspaces in L 2 (S 1 , C) with V n the closure in L 2 (S 1 , C) of the space V alg n = span of {ψ(n, 0, z), ψ(n + 1, 0, z), ψ(n + 2, 0, z), . . .}.
The set of these flags was called the adelic flag manifold in [6], and we shall denote it by Fl ad . In order to formulate the main result of [6], we need to introduce the following algebra with poles only at z = −1 and z = ∞, It can be shown that the curve Spec(A V ) is also unicursal, and that there is a bijective birational isomorphism C \ {−1} → Spec(A V ) which sends −1 and ∞ to two smooth points completing the curve (see [6,Theorem 4.4]).
Theorem 2. (see ). Any V ∈ Fl ad gives rise to a rank one bispectral commutative algebra of difference operators A as in (1.6) and (1.7), isomorphic to A V as defined in (4.2).
The function ψ V (n, z) ≡ ψ(n, 0, z) is the joint eigenfuntion of the operators in A, and is called the (stationary) Baker-Akhiezer function of the flag. From the definition of Fl ad and the result of [21] which establishes a bijection between the union of the Calogero-Moser spaces C N , N ≥ 0, introduced in (1.1) and Gr ad , it is easy to deduce the following lemma. Proof . From Sato's formula (2.1) and formula (1.4) for the (stationary) Baker-Akhiezer function of a space W ∈ Gr ad , one deduces easily that the corresponding tau function is given by Using the fact that the spectrum of Z is inside the unit circle, one finds From (4.1), a simple computation shows then that the (stationary) Baker-Akhiezer function ψ V (n, z) of the corresponding adelic flag is given by (4.3), which establishes the lemma.
We can now formulate the main result of our paper.
Theorem 3. The stationary Baker-Akhiezer function ψ W (x, z) of a space W ∈ Gr trig as defined in (2.17), besides being an eigenfunction of a commutative algebra of differential operators in x is also an eigenfunction of a commutative algebra of difference operators in the spectral variable z built from an adelic flag V, i.e.
Proof . Any stationary Baker-Akhiezer function associated with a space W in the Segal-Wilson Grassmannian, for which Spec(A W ) is an irreducible affine algebraic curve (which completes by adding one non-singular point at infinity), satisfies (4.5) for a commutative algebra of differential operators A f , f ∈ A W . This is a reformulation of the classical Burchnall-Chaundy-Krichever theory as explained in Section 6 of [18]. In particular, it applies to any W ∈ Gr trig ⊂ Gr rat . From the definition of Gr trig (2.17), the same proof as in Proposition 1 shows that the stationary Baker-Akhiezer function ψ W (x, z) of W ∈ Gr trig is given by (2.13). Let us define ψ b (n, z) = ψ W (log(1 + z), n) (4.7) We claim that ψ b (n, z) is the stationary Baker-Akhiezer function of an adelic flag. In view of the characterization given in (4.3) of the stationary Baker-Akhiezer function of an adelic flag, using the fact that the determinants of a matrix and its transpose are equal, it is enough to find two matricesX andZ satisfying rank([Z,X] + I) = 1, so that ψ b (n, z) can be put into the form It is easy to see that (4.8) agrees with (4.9) by picking X = ZX −1 andZ = X − I. Since ψ b (n, z) is the stationary Baker-Akhiezer function of an adelic flag V, by Theorem 2, there exists a commutative algebra of difference operators {B g , g ∈ A V }, so that Remembering the definition (4.7) of ψ b (n, z) in terms of ψ W (x, z), we obtain which establishes (4.6) and completes the proof of the theorem. Proof . Looking at the proof of Theorem 3 and more specifically at equations (4.7), (4.10) and (4.11), we have in fact constructed a map, the inverse of the map b C in (1.12) with the property that This establishes the commutativity of the diagram (1.8).
Remark 3. In [8] (see also [14] for KdV solitons), it was shown that for any W ∈ Gr rat , the Baker-Akhiezer function ψ W (x, z), besides satisfying (4.5), also satisfies for appropriate λ j ∈ C, with b j (z, ∂/∂z) ordinary differential operators in z. As shown in Theorem 3, the KP trigonometric solitons are distinguished by the property that this equation can be taken to be a difference equation in z, with the b j 's functions of z and the λ j 's integers.

Ruijsenaars' duality revisited
A simple computation, using the commutativity of the diagram (1.8), the definition of (b C ) −1 in (4.12) and formula (4.4) for the tau function of an adelic flag, shows that for (X, Z) ∈ C trig To make contact with the rational Ruijsenaars-Schneider system, we assume that Z is diagonalizable, with no two eigenvalues differing by a unit. Then, by an argument similar to the one used in Lemma 1, there is a conjugation which diagonalizes Z in such a way that the rational Ruijsenaars-Schneider system [15] is the Hamiltonian systeṁ with Hamiltonian H(λ, θ) = tr L RS (λ, θ). It is a completely integrable system, whose solution (as well as the solution of the flows commuting with it) is described by the next proposition.
As before, we refer to λ i (t 1 , t 2 , . . .) as the solution of the rational Ruijsenaars-Schneider hierarchy with initial condition (λ 0 , θ 0 ) ∈ C 2N . Our final theorem, when combined with Theorem 1, reveals a duality between the trigonometric Calogero-Moser hierarchy and the rational Ruijsenaars-Schneider hierarchy, in terms of the bispectral map b defined in (1.11). This duality was first discovered by Ruijsenaars in [16], by studying the scattering theory of these systems.
Proof . Let L RS (λ 0 , θ 0 ) ≡ L RS 0 and K RS (λ 0 ) ≡ K RS 0 be defined from X and Z as in (5.2), (5.3) and (5.4). Since a determinant is invariant under conjugation, using the definition of ϕ k (5.6), from (5.1) we find From Proposition 3, it follows that the functions λ i (t 1 , t 2 , . . .) which solve the rational Ruijsenaars-Schneider hierarchy with initial condition (λ 0 , θ 0 ) are the eigenvalues of the matrix which establishes the assertion (5.7) and completes the proof of the theorem.

Remark 4.
A trigonometric Calogero-Moser pair (X, Z) for which both X and Z are diagonalizable with no two eigenvalues of Z differing by a unit, can be represented (modulo conjugation) by K 2 (x 0 ), L(x 0 , y 0 ) as in (3.3), or L RS (λ 0 , θ 0 ), K RS (λ 0 ) as in (5.2). Let W = β trig (X, Z) be the corresponding space in Gr trig . By Theorem 1, the zeros of τ W (x + t 1 , t 2 , . . .) as a function of x solve the trigonometric Calogero-Moser hierarchy with initial condition (x 0 , y 0 ) and, by Theorem 4, the zeros as a function of n of τ b −1 (W ) (n, t 1 , t 2 , . . .) solve the rational Ruijsenaars-Schneider hierarchy with initial condition (λ 0 , θ 0 ). To describe a repulsive interaction in the trigonometric Calogero-Moser system (3.1), one chooses the x i 's real with x 1 < · · · < x N , the y i 's imaginary and time imaginary too. In this case L(x, y) is anti-hermitian and thus diagonalizable with imaginary eigenvalues. As shown in [16,Section 2C], the matrix T in (5.2) which conjugates the pair K 2 (x), L(x, y) to L RS (λ, θ), K RS (λ) , is then uniquely determined by requiring it to be unitary and such that √ −1 λ 1 < · · · < √ −1 λ N and µ i > 0. The real solution of the hyperbolic Calogero-Moser system is then given by x i ( √ −1 t), √ −1 y i ( √ −1 t) , t ∈ R, and the map √ −1 y i , x i → √ −1 λ i , θ i , is the scattering map, providing the action-angle variables for the system. Vice versa, the map θ i , √ −1 λ i → x i , √ −1 y i gives the action-angle variables for the rational Ruijsenaars-Schneider system.