KP Trigonometric Solitons and an Adelic Flag Manifold

0.24 MB·English

by  Luc Haine

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3(2007), 015, 15 pages
KP Trigonometric Solitons
⋆
and an Adelic Flag Manifold
Luc HAINE
Department of Mathematics, Universit´e catholique de Louvain,
Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
E-mail: [email protected]
Received November 22, 2006, in final form January 5, 2007; Published online January 27, 2007
7 Original article is available at http://www.emis.de/journals/SIGMA/2007/015/
0
0
2 Abstract. WeshowthatthetrigonometricsolitonsoftheKPhierarchyenjoyadifferential-
difference bispectral property, which becomes transparent when translated on two suitable
n
a spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen
J as a non-self-dual illustration of Wilson’s fundamental idea [Invent. Math. 133 (1998),

7 1–41] for understanding the (self-dual) bispectral property of the rational solutions of the
2 KP hierarchy. It also gives a bispectral interpretationof a (dynamical) duality between the

hyperbolic Calogero–Moser system and the rational Ruijsenaars–Schneider system, which
]
I was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127–165].
S
. Key words: Calogero–Mosertype systems; bispectral problems
n
i
l 2000 Mathematics Subject Classification: 35Q53;37K10
n
[

1
Dedicated to the memory of Vadim Kuznetsov
v
4
5
1 Introduction
0
1
0
One of the many gems I had the chance to share with Vadim Kuznetsov during his visit to
7
0 Louvain-la-Neuve in the fall semester of 2000, had to do with his work on separation of variables
/ andspectrality. Heknewaboutmyworkonbispectralproblemsandinsistedthattheseproblems
n
i were connected. He introduced me to his work with Nijhoff and Sklyanin [11], on separation
l
n of variables for the elliptic Calogero–Moser system. I would have loved to discuss the topic
:
v of the present paper with him, which deals with the (simpler) trigonometric version of this
i
X system.
One of the exciting new developments in the field of integrable systems has been the intro-
r
a
duction by George Wilson [21] of the so called Calogero–Moser spaces
C = (X,Z) gl(N,C) gl(N,C): rank([X,Z]+I) = 1 /GL(N,C), (1.1)
N
∈ ×
where gl(N,(cid:8)C) denotes the space of complex N N matrices(cid:9), 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 (Zu¨rich). 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 Gru¨nbaum, which has
played and (as we shall see) continues to play a decisive role in the subject.
⋆ThispaperisacontributiontotheVadimKuznetsovMemorialIssue“IntegrableSystemsandRelatedTopics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
2 L. Haine
In the form discussed by Wilson [20], the bispectral problem asks for the classification of
all rank 1 commutative algebras of differential operators, for which the joint eigenfunction
A
ψ(x,z) which satisfies
A(x,∂/∂x)ψ(x,z) = f (z)ψ(x,z) A , (1.2)
A
∀ ∈ A
also satisfies a (non-trivial) differential equation in the spectral variable
B(z,∂/∂z)ψ(x,z) = g(x)ψ(x,z). (1.3)
In[20], itwas foundthatall thesolutions of theproblemareparametrized byacertain subgrass-
mannianoftheSegal–Wilson GrassmannianGr,thatWilsoncalledtheadelicGrassmannianand
that he denoted by Grad. 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 Grad as the union C of the Calogero–Moser spaces introduced
N≥0 N
∪
above. The correspondence can be seen as given by the map
β: (X,Z) ψ (x,z) = exz det I (xI X)−1(zI Z)−1 , (1.4)
W
→ { − − − }
which sends a pair (X,Z) C (modulo conjugation) to the (stationary) Baker–Akhiezer
N
∈
functionofthecorrespondingspaceW = β(X,Z) Grad. From(1.4),oneseesimmediatelythat
∈
the mysterious bispectral involution b: Grad Grad which exchanges the role of the variables x
→
and z
ψ (x,z) = ψ (z,x), W Grad,
b(W) W
∈
becomes transparent when expressed at the level of the Calogero–Moser spaces, as it is given by
bC(X,Z) = (Zt,Xt), where Xt and Zt are the transposes of X and Z. The situation is nicely
summarized by the following commutative diagram, all arrows of which are bijections
C β Grad
N≥0 N
∪ −−−−→
bC b (1.5)
C β Grad
∪N≥y0 N −−−−→ y
In [6], jointly with Plamen Iliev, we considered the following discrete-continuous version of
thebispectralproblem. Todetermineallrank1commutative algebras of differenceoperators,
A
for which the joint eigenfunction ψ(n,z) which satisfies
Aψ(n,z) a (n)ψ(n+j,z) = f (z)ψ(n,z) A , (1.6)
j A
≡ ∀ ∈ A
finitely manyj∈Z
X
also satisfies a (non-trivial) differential equation in the spectral variable
B(z,∂/∂z)ψ(n,z) = g(n)ψ(n,z). (1.7)
TheproblemwasmotivatedbyanearlierworkwithAlbertoGru¨nbaum[5],whereweinvestigated
the situation when the algebra contains a second-order symmetric difference operator (this
A
time, without imposing any rank condition on ). This situation extends the theory of the
A
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 Grad an (isomor-
phic) adelic flag manifold Flad, which provides solutions of the bispectral problem raised above,
and parametrizes rational solutions in n (vanishing as n ) of the discrete KP hierarchy.
→ ∞
KP Trigonometric Solitons and an Adelic Flag Manifold 3
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
C β Grad Flad
N≥0 N
∪ −−−−→ ≡
bC b (1.8)
Ctrig βtrig Grtrig
∪N≥y0 N −−−−→ y
In this diagram, the spaces
Ctrig = (X,Z) GL(N,C) gl(N,C): rank (XZX−1 Z +I) = 1 /GL(N,C), (1.9)
N ∈ × −
aretrigonome(cid:8)tricanalogues oftheCalogero–Moser spacesC definedin(1.(cid:9)1). Grtrig isacertain
N
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 take
W
the form
N
x x (t ,t ,t ,...)
i 1 2 3
τ (x+t ,t ,t ,...) = 2sinh − , (1.10)
W 1 2 3
2
i=1 (cid:0) (cid:1)
Y
withx (t ,t ,t ,...)beingasolution of thetrigonometric Calogero–Moser–Sutherland hierarchy
i 1 2 3
(as long as all x (t ,t ,t ,...) remain distinct, see Section 3). The bispectral map b: Flad
i 1 2 3
→
Grtrig, which sends a flag to a linear space W = b( ), and is defined by
V V
ψ (x,z) =ψ (z,ex 1), (1.11)
b(V) V
−
trivializes when expressed at the level of the Calogero–Moser spaces, as it is now given by1
bC(X,Z) = I +Zt,Xt(I +Zt) . (1.12)
The result will fo(cid:0)llow easily from the(cid:1) expression of the (stationary) Baker–Akhiezer function
ψ (x,z) of a space W Grtrig, in terms of pairs of matrices (X,Z) Ctrig
W ∈ ∈ N
ψ (x,z) = exzdet I X(exI X)−1(zI Z)−1 , (1.13)
W
{ − − − }
which defines the map βtrig 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
mapsforCalogero–Moser typesystemswithrepulsivepotentials,viathestudyoftheirscattering
theory. Along the way, he observed various duality relations between these systems. In partic-
ular, 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 (t ,t ,t ,...)
W 1 2 3
is the tau function of a space W Grtrig as in (1.10), the tau function of the flag b−1(W) is
∈
N
τb−1(W)(n,t1,t2,...) = (n λi(t1,t2,...)), n Z,
− ∈
i=1
Y
with λ (t ,t ,...) solving now the rational Ruijsenaars–Schneider hierarchy, thus representing
i 1 2
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 thequantum 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.
1The additional condition det(I +Z) 6= 0 needed for this definition to make sense, follows from fixing the
radius of thecircle used in thedefinition of Gr tobe 1, which can always be assumed, see Section 4.
4 L. Haine
2 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 Grrat, from which
solitonic solutions of the KP hierarchy can be constructed, see [18] for details. Let S1 C be
⊂
the unit circle, with center the origin, and let H denote the Hilbert space L2(S1,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
∞
W H is a compact operator. For t = (t ,t ,...), let exp(t,z) = exp( t zk). For all t,
− 1 2 k
→
k=1
exp−1(t,z)W belongstoGrand,foralmostanyt,itisisomorphictoH ,sothPatthereisaunique
+
function in it ψ˜ (t,z) which projects onto 1. Thefunction ψ (t,z) = exp(t,z)ψ˜ (t,z) is called
W W W
the Baker–Akhiezer function of the space W and ψ˜ (t,z) is called the reduced Baker–Akhiezer
W
function. A fundamental result of Sato asserts that there is a unique (up to multiplication by
a constant) function τ (t ,t ,t ,...), the celebrated tau function, such that
W 1 2 3
τ (t 1/z,t 1/(2z2),t 1/(3z3),...)
W 1 2 3
ψ (t,z) = exp(t,z) − − − . (2.1)
W
τ (t ,t ,t ,...)
W 1 2 3
An element of the form a zk, a = 0, is called an element of finite order s. For W Gr,
k s
6 ∈
k≤s
Walg denotes the subspacPe of elements of finite order of W. It is a dense subspace of W. We
also need the ring
A = f analytic in a neighborhood of S1: f.Walg Walg . (2.2)
W
{ ⊂ }
The rational Grassmannian Grrat is the subset of Gr, for which Spec(A ) is a rational curve.
W
In this case, A is a subset of the ring C[z] of polynomials in z and the map C Spec(A )
W 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 ,...,λ inside of S1, and another N non-zero
1 N
alg
complex numbers µ ,...,µ . We assume that λ λ = 1, i= j. We define W as the space
1 N i− j 6 ∀ 6 λ,µ
of rational functions f(z) such that
(i) f is regular except for (at most) simple poles at λ ,...,λ and poles of any order at
1 N
infinity;
(ii) f satisfies the N conditions
res f(z)+µ2f(λ 1) = 0, 1 i N, (2.3)
λi i i− ≤ ≤
where res f(z) is the residue of f(z) at λ . The closure of Walg in L2(S1,C) defines a space
λi i λ,µ
W Grrat.
λ,µ
∈
The Baker–Akhiezer function ψ of a space W = W has the form
W λ,µ
N
µ b (t)
j j
ψ (t,z) = exp(t,z) 1 , (2.4)
W
− z λ
( j )
j=1 −
X
for some functions b (t) determined by (2.3). By a simple computation, we obtain
j
N
µ b (t)
j j
exp(t,λ )b (t)+µ exp(t,λ 1) 1+ = 0. (2.5)
i i i i
− − 1+λ λ
( j i)
j=1 −
X
KP Trigonometric Solitons and an Adelic Flag Manifold 5
We introduce the N N matrices X and Z with entries
×
µ µ
i j
X = , Z = λ δ , (2.6)
ij ij i ij
1+λ λ
j i
−
with δ the usual Kronecker symbol so that Z is diagonal, and we put
ij
∞
X˜ = exp t (Zk (Z I)k) X. (2.7)
k
− − −
( )
k=1
X
With these notations, the system of equations (2.5) determining the functions b (t) is written as
i
X˜b(t) = µ, (2.8)
whereb(t)and µ denotecolumn vectors of length N, with entries b (t) andµ respectively. From
i i
(2.4)and (2.8), we get that thereduced Baker–Akhiezer function ψ˜ (t,z) = exp−1(t,z)ψ (t,z)
W W
can be written as
ψ˜ (t,z) = 1 µt zI Z −1X˜−1µ, (2.9)
W
− −
with µt the row vector (cid:0)obtained(cid:1)by transposing µ.
The following commutation relation will be crucial
[X,Z] =µµt X. (2.10)
−
For short set
Z˜ = zI Z. (2.11)
−
Considering that if T is a matrix of rank 1 then 1+trT = det(I +T) (with tr denoting the
trace), from (2.7), (2.9), (2.10) and (2.11), we find
ψ˜ (t,z) = 1 tr X˜−1µµtZ˜−1 = det I X˜−1([X,Z]+X)Z˜−1
W
− −
= det I(cid:8) X˜−1([X˜,Z˜(cid:9)]+X)Z˜(cid:8)−1 = det X˜−1(Z˜X˜ X)(cid:9)Z˜−1 .
− −
(cid:8) (cid:9) (cid:8) (cid:9)
Using the fact that the determinant of a product of matrices does not depend on the orders of
the factors, we get
ψ˜ (t,z) = det I XX˜−1Z˜−1 . (2.12)
W
−
(cid:8) (cid:9)
In particular, setting t = t = = 0 and t = x, we find that the so-called stationary
2 3 1
···
Baker–Akhiezer function of W admits the following form
ψ (x,z) = exzdet I X(exI X)−1(zI Z)−1 . (2.13)
W
− − −
(cid:8) (cid:9)
By Cauchy’s determinant formula applied to X in (2.6)
N
1
det(X) = µ2 1 , (2.14)
i − 1 (λ λ )2
i=1 1≤i<j≤N(cid:18) − i− j (cid:19)
Y Y
which is non-zero by the assumptions we made on the λ ’s and µ ’s. Thus X is invertible and it
j j
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.
6 L. Haine
Proposition 1. The tau function τ (t ,t ,t ,...) of a space W = W is given by
W 1 2 3 λ,µ
∞
τ (t ,t ,t ,...) = det I Xexp t (Z I)k Zk , (2.15)
W 1 2 3 k
− − −
( ( ))
k=1
X (cid:0) (cid:1)
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
1 1 1
τ t ,t ,t ,... =det I Xexp (zI (Z I))(zI Z)−1
W 1− z 2− 2z2 3− 3z3 { − {···} − − − }
(cid:18) (cid:19)
= det I Xexp Xexp (zI Z)−1 ,
{ − {···}− {···} − }
from which it follows that
τ (t 1/z,t 1/(2z2),t 1/(3z3),...)
W 1− 2 − 3− = det I X(exp−1 ... X)−1(zI Z)−1
τ (t ,t ,t ,...) { − { }− − }
W 1 2 3
= det I XX˜−1Z˜−1 ,
{ − }
with X˜ and Z˜ 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 as in (2.15). Sincethis formula
W
determines the tau function up to a constant, the proof is complete. (cid:4)
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 YX) = 1, the function
× −
∞ ∞
τ (t ,t ,...)= det I Xexp t Zk exp t Yk , (2.16)
(X,Y,Z) 1 2 k k
− −
( ( ) ( ))
k=1 k=1
X X
is a tau function of the KP hierarchy, associated with some W Grrat, by showing that it
∈
satisfies the Hirota equation in Miwa form. The special choice
µ µ
i j
X = , Y = ν δ , Z = λ δ , with ν = λ , i,j,
ij ij i ij ij i ij i j
λ ν 6 ∀
j i
−
leads to a N-soliton solution of the KP hierarchy, and Proposition 1 can be obtained by picking
Y = Z I.
−
Thanks to Remark 1, it makes sense to introduce the following definition:
Definition 1. The trigonometric Grassmannian Grtrig is defined to be the following subgrass-
mannian of Grrat
Grtrig= W Grrat: τ (t ,t ,...)=τ (t ,t ,...),for (X,Z) Ctrig , (2.17)
∈ W 1 2 (X,Z−I,Z) 1 2 ∈ N≥0 N
(cid:8) S (cid:9)
trig
with C defined as in (1.9) and with τ (t ,t ,...) defined as in (2.16). The corre-
N (X,Z−I,Z) 1 2
sponding solutions of the KP hierarchy will be called trigonometric solitons.
2I am thankfulto thetwo referees both of whom made this important observation.
KP Trigonometric Solitons and an Adelic Flag Manifold 7
Example 1. As an example of a non-generic W Grtrig (i.e. not of the form W as above),
λ,µ
∈
let usdefineW Grrat tobetheclosurein L2(S1,C)ofthespace Walg formedwiththerational
∈
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
res zf(z)+f( 1) = 0,
0
−
res f(z)+f′( 1) = 0.
0
−
A simple computation shows that the corresponding stationary Baker–Akhiezer function is
2 1 ex
ψ (x,z) = exz 1+ + − ,
W (1+e2x)z (1+e2x)z2
(cid:26) (cid:27)
which can be put into the form (2.13) with
0 1 0 1
X = − , Z = ,
1 0 0 0
(cid:18) (cid:19) (cid:18) (cid:19)
which forms a trigonometric Calogero–Moser pair as defined in (1.9), with a non-diagonalizab-
le Z. The corresponding tau function is
∞ ∞ ∞
τ (x,t ,t ,...)=e−2x exp 2 ( 1)kt exp ( 1)kt ( 1)k+1kt ex+e2x .
W 2 3 k k k
− − − −
( ! ! ! )
k=2 k=2 k=2
X X X
3 Grtrig and the Calogero–Moser–Sutherland hierarchy
In this section, we relate the Grassmannian Grtrig introduced in Definition 1 with the Calogero–
Moser–Sutherland system, also referred as the trigonometric (or hyperbolic) Calogero–Moser
system. Thisis asystemof N particles on thelinewhosemotion is governed bytheHamiltonian
N
1 1
H(x,y) = y2 . (3.1)
2 i − 4sinh2((1/2)(x x ))
i=1 1≤i<j≤N i − j
X X
Weallow theparticles tomoveinthecomplex plane. TheoriginalMoser–Sutherlandsystem[12]
isrecovered ifwesupposeourparticles tobeconfinedtotheimaginary axis, sincethehyperbolic
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
×
1
L (x,y) = δ y +(1 δ ) , (3.2)
ij ij i ij
− 2sinh((1/2)(x x ))
i j
−
where δ is the usual Kronecker symbol. More precisely, the quantities F (x,y) =
ij k
(1/k)trLk(x,y) (with tr denoting the trace), k = 1,2,...,N, are N independent first integrals
in involution for the system. In particular, F (x,y) gives back the original Hamiltonian H(x,y).
2
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).
8 L. Haine
Lemma 1. Let (X,Z) Ctrig,N 1, as defined in (1.9). If X is diagonalizable, there is
∈ N ≥
a conjugation
U−1XU = K2(x), U−1ZU =L(x,y), (3.3)
with K(x) a diagonal matrix of the form
K(x) = diag ex1/2,...,exN/2 , (3.4)
and L(x,y) as in ((cid:0)3.2). (cid:1)
Proof. Since det(X) = 0, when diagonalizing X, we can always assume K to have the form
6
(3.4). Denoting by L the result of the conjugation of Z by the same matrix U, since by the
trig
definition of C the rank of [X,Z]+X is 1, we have
N
[K2,L] = αβt K2, (3.5)
−
with α and β two (non-zero) column vectors of length N. Writing (3.5) componentwise, we get
(exi exj)L = α β δ exi, (3.6)
ij i j ij
− −
which, by putting i= j, shows that α β = exi = 0, i. Thus, by multiplying U to the right by
i i
6 ∀
an appropriate diagonal matrix, we can always arrange that α = β = exi/2. With this choice,
i i
one sees from (3.6) that necessarily exi = exj, i= j, and
6 ∀ 6
1
L = , for i= j,
ij
2sinh (1/2)(x x ) 6
i j
−
while the diagonal(cid:0)entries L are(cid:1)free. Denoting L = y establishes the lemma. (cid:4)
ii ii i
Theexplicitintegrationofthesystem(3.1)wasperformedbyOlshanetskyandPerelomov[13].
An interpretation in terms of Hamiltonian reduction was given by Kazhdan, Kostant and Stern-
berg [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
h i
accordingly the gradient of a smooth function ϕ: gl(N,C) C by dϕ(X)(Y)= ϕ(X),Y .
→ h∇ i
Proposition 2. (See [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 (t),y (t)) be the solution
i i
of Hamilton’s equations
∂H ∂H
x˙ = , y˙ = ,
i i
∂y −∂x
i i
with initial conditions (x ,y ) = x (0),...,x (0),y (0),...,y (0) C2N, such that x (0) =
0 0 1 N 1 N i
∈ 6
x (0), i= j. Then, the quantities exi(t) are the eigenvalues of the matrix
j ∀ 6 (cid:0) (cid:1)
K2exp t ϕ(L ) , (3.7)
0 ∇ 0
with K = K(cid:0)(x ) as in(cid:1) (3.4), L = L(x ,y ) and ϕ the gradient of ϕ. Moreover, the mat-
0 0 0 0 0
∇
rix V(t) which diagonalizes K2exp t ϕ(L ) , so that
0 ∇ 0
K2(x(t)) = V(t)K2exp t ϕ((cid:0)L ) V(t)−(cid:1)1,
0 ∇ 0
and which is normalized by th(cid:0)e conditio(cid:1)n
V(t)K e= K(x(t))e, e= (1,...,1)t,
0
is such that
L(x(t),y(t)) = V(t)L V(t)−1.
0
KP Trigonometric Solitons and an Adelic Flag Manifold 9
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)trL2 as in (3.1) and (3.2), we have to pick the x ’s real, the y ’s imaginary and t imaginary
i i
(cid:0) (cid:1)
also. In this case, t ϕ(L ) = tL ,t √ 1 R, is hermitian and thus K2exp(tL ) in (3.7) is
∇ 0 0 ∈ − 0 0
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 (x,y) = ϕ (L(x,y)), with
k k
1
ϕ (L(x,y)) = tr (L(x,y) I)k+1 Lk+1(x,y) , k = 1,2,..., (3.8)
k
k+1 { − − }
and let us denote by x (t) x (t ,t ,t ,...) the solution obtained by flowing along the Hamil-
i i 1 2 3
≡
tonian vector field X during a time t , X during a time t etc., starting from some initial
H1 1 H2 2
condition (x ,y ) C2N. For short, we shall refer to x (t) x (t ,t ,t ,...) as the solution of
0 0 i i 1 2 3
∈ ≡
the trigonometric Calogero–Moser hierarchy with initial condition (x ,y ) C2N.
0 0
∈
Theorem 1. The tau function τ (x+t ,t ,t ,...)as givenin (2.16), with (X,Z) Ctrig
(X,Z−I,Z) 1 2 3 ∈ N
and X diagonalizable, is (up to an inessential exponential factor) a trigonometric polynomial in
the variable x
N
(x x (t ,t ,t ,...))
i 1 2 3
τ (x+t ,t ,t ,...)= 2sinh − , (3.9)
(X,Z−I,Z) 1 2 3
2
i=1
Y
where x (t ,t ,t ,...) denotes the solution of the trigonometric Calogero–Moser hierarchy with
i 1 2 3
initial condition (x ,y ) C2N specified by (3.3) as in Lemma 1 above.
0 0
∈
Proof. Let us consider the solution x (t) of the trigonometric Calogero–Moser hierarchy with
i
initial condition (x ,y ) C2N specified by (3.3), i.e. U−1XU = K2(x ) K2,U−1ZU =
0 0 ∈ 0 ≡ 0
L(x ,y ) L (remember from the proof of Lemma 1 that necessarily x (0) = x (0), i = j).
0 0 0 i j
≡ 6 ∀ 6
From Proposition 2 and the definition of ϕ in (3.8), it follows easily that the quantities exi(t)
k
are the eigenvalues of the matrix
∞ ∞
K2exp t ϕ (L ) = K2exp t (L I)k Lk .
0 k∇ k 0 0 k 0− − 0
( ) ( )
k=1 k=1
X X (cid:0) (cid:1)
From this, we deduce
N N
ex+x2i(t) 2sinh(x−xi(t)) = (ex exi(t))
2 −
i=1 i=1
Y Y
∞
= det exI K2exp t ((L I)k Lk)
− 0 k 0 − − 0
( ( ))
k=1
X
∞
= eNxdet I K2exp (t +x)I + t ((L I)k Lk) .
− 0 − 1 k 0− − 0
( ( ))
k=2
X
Since the determinant of a matrix is invariant under conjugation, we obtain
N N ∞
exi(2t)−x 2sinh (x−xi(t))=det I Xexp (t1+x)I+ tk((Z I)k Zk) .
2 − − − −
( ( ))
i=1 i=1 k=2
Y Y X
With account of (2.16), this establishes (3.9), up to the inessential (in the sense that it leads to
PN xi(t)−x
thesame solution of theKP hierarchy) exponential factor ei=1 2 . Theproofis complete. (cid:4)
10 L. Haine
4 The bispectral property of the KP trigonometric solitons
The adelic Grassmannian Grad consists of the spaces W Grrat for which the curve Spec(A )
W
∈
(with A as in (2.2)) is unicursal, that is the birational isomorphism C Spec(A ) corre-
W W
→
sponding to the inclusion A C[z] is bijective. As established in [20], Grad parametrizes all
W
⊂
commutativerank1algebras ofbispectraldifferentialoperators,inthesenseof (1.2)and(1.3).
A
The corresponding algebras are isomorphic to A . The joint eigenfunction of the operators
W
A
in is given by the stationary Baker–Akhiezer function ψ (x,z) of W, and it can bewritten in
W
A
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 S1 to be of radius 1, all the eigenvalues of Z
(which correspond to the singular points of Spec(A ) under the map C Spec(A )) will be
W W
→
inside of S1, 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 circle3.
Following [6], starting from any W Grad and its corresponding tau function τ (t ,t ,...),
W 1 2
∈
we build a function ψ(n,t,z) = ψ(n,t ,t ,...,z) via the formula
1 2
τ (t +n 1/z,t n/2 1/(2z2),t +n/3 1/(3z3),...)
ψ(n,t,z)=(1+z)nexp(t,z) W 1 − 2− − 3 − . (4.1)
τ (t +n,t n/2,t +n/3,...)
W 1 2 3
−
We define a corresponding flag of subspaces in L2(S1,C)
: V V V
n+1 n n−1
V ··· ⊂ ⊂ ⊂ ⊂ ···
with V the closure in L2(S1,C) of the space
n
Valg = span of ψ(n,0,z),ψ(n+1,0,z),ψ(n+2,0,z),... .
n { }
The set of these flags was called the adelic flag manifold in [6], and we shall denote it by Flad.
In order to formulate the main result of [6], we need to introduce the following algebra
A = rational functionsf(z) with poles only at z = 1and z = ,
V
{ − ∞
such that k Z, for which f(z).V V , n . (4.2)
n n+k
∃ ∈ ⊂ ∀ }
Itcan beshownthatthecurveSpec(A )is also unicursal,andthatthereis abijective birational
V
isomorphism C 1 Spec(A ) which sends 1 and to two smooth points completing
V
\{− } → − ∞
the curve (see [6, Theorem 4.4]).
Theorem 2. (seeHaine–Iliev [6]). Any Flad gives rise to a rank one bispectral commutative
V ∈
algebra of difference operators as in (1.6) and (1.7), isomorphic to A as defined in (4.2).
V
A
Thefunctionψ (n,z) ψ(n,0,z) isthejointeigenfuntion of theoperatorsin , andiscalled
V
≡ A
the (stationary) Baker–Akhiezer function of the flag. From the definition of Flad and the result
of [21]which establishes a bijection between the unionof theCalogero–Moser spaces C ,N 0,
N
≥
introduced in (1.1) and Grad, it is easy to deduce the following lemma.
Lemma 2. There is a bijection β: C Flad given by the map
N≥0 N
∪ →
(X,Z) ψ (n,z) = (1+z)ndet I +(X n(I +Z)−1)−1(zI Z)−1 . (4.3)
V
→ { − − }
3In[20,21],theradiusofS1 isallowedtovaryinthedefinitionofGrad. Asexplainedin[18],thereisnolossof
generalityinfixingtheradiustobe1,sincethescalingtransformationsψRλW(x,z)=ψW(λx,λ−1z), 0<|λ|≤1,
act on Gr as defined in Section 2.