Inverse scattering method for Hitchin systems of types B n , C n , D n , and their generalizations
aa r X i v : . [ m a t h - ph ] J u l INVERSE SCATTERING METHOD FOR HITCHIN SYSTEMS OFTYPES B n , C n , D n , AND THEIR GENERALIZATIONS O.K.SHEINMAN
Abstract.
We give a solution of the Inverse Scattering Problem for integrablesystems with a finite number degrees of freedom, admitting a Lax representationwith spectral parameter on a Riemann surface. While conventional approachesdeal with the systems with GL ( n ) symmetry, we focus on the problems arising inthe case of symmetry with respect to a semi-simple group. Our main results applyto Hitchin systems of the types B n , C n , D n . Contents
1. Introduction 12. Lax integrable systems with spectral parameter on a Riemann surface 32.1. Lax equations 32.2. Hierarchies 53. Spectral transform 63.1. Spectral curve 63.2. Relation between spectra of L - and M -operators 63.3. From ( L, M )-pair to Baker–Akhiezer matrix-function 73.4. Spectral transform 84. Inverse spectral transform 94.1. Retrieving the (
L, M )-pair by spectral and Tyurin data 94.2. Theta-functional formula for the Baker–Akhieser matrix-function 134.3. Explicit expression for the exponents 14References 161.
Introduction
In [4], I.M.Krichever introduced a wide class of Lax integrable systems withspectral parameter on a Riemann surface. Basic examples are given by A n -typeHitchin and Calogero–Moser systems. Some classical integrable systems are con-tained there as well. In the above quoted work, Krichever has constructed thecorresponding integrable hierarchies and their Hamiltonian theory, gave a generalscheme of algebraic-geometrical integration of GL ( n ) Hitchin systems by means hiswell-known theory of Baker–Akhieser functions.In [6], there were constructed similar Lax operators for the classical groups SO (2 n ), SO (2 n + 1), Sp (2 n ). In the subsequent series of works summarized in [7]we generalized the hierarchies originally introduced in [4], and their Hamiltonian theory, on the Lax systems of the above type where operators of the ( L, M )-pairtake values in the corresponding simple Lie algebras. In [8] we generalized the aboveresults on the case of arbitrary semi-simple Lie algebras. It turned out to be possibledue to a new algebraic approach to the Lax operators in question developed in [9].In [10, 1], we developed explicit separation of variables technique for Hitchin systemson hyperelliptic curves with arbitrary underlying symmetry algebra (group) in theclass of complex semi-simple Lie algebras (groups, resp.).The inverse scattering method appears as the only part of the programme ini-tiated in [4], not yet done in presence of non-trivial group symmetry. It is the aimof this work to fill this gap.In the present paper we consider the case of classical simple groups SO (2 n ), SO (2 n + 1), Sp (2 n ). We set the problem as follows: to retrieve the operators ofthe Lax pair by spectral data. With this aim, we use a conventional scheme ofalgebraic-geometric integration due to Krichever [2, 3], specified for the systems inquestion in [4]. According to this scheme(1.1) L = b ΨΛ b Ψ − , M = − ∂ t b Ψ · b Ψ − where L is the Lax matrix, M is its counterpart in the Lax pair, Λ is a spectrumof L coming directly from the spectral curve, b Ψ is the matrix of Baker–Akhieserfunctions uniquely defined by spectral data.In presence of a symmetry with respect to a semi-simple group G , there is thefollowing difference with the conventional GL ( n ) case :1) b Ψ is a G -valued function;2) Conventional spectral data is not sufficient for retrieving the ( L, M )-pair. Itis necessary to give explicitly the time-dependent poles of b Ψ correspondingto the
Tyurin points of the Lax matrix while in the conventional GL ( n ) casethe Tyurin points are zeroes of the Baker–Akhieser function.3) An expression for exponents of the Baker–Akhieser function is needed.None of the problems 1, 2 arose in the GL ( n ) case, and the problem 3 had a trivialsolution. We resolve the first problem by applying a certain orthogonalization (resp.,skew-orthogonalization) process to the matrix b Ψ obtained in a conventional way.The corresponding bilinear form comes directly from the assumption of invarianceof the spectral curve with respect to a holomorphic involution (in turn, coming fromthe assumption that L is a skew-symmetric (resp., infinitesimal symplectic)) matrix.The most delicate part of the proof is to preserve the form of Tyurin poles in courseof the orthogonalization (proof of the Theorem 4.3 below).Methods of resolving the second problem are actually developed in [5]. Weadopt the equation of motion of our analogs of the Tyurin parameters from [8].To resolve the third problem, we reduce it to a calculation of gradients of in-variants of the L -matrix in terms of its spectrum. This looks like a classical problemof linear algebra, however we don’t know any reference on its solution.In Section 2 we define integrable hierarchies in question via their Lax represen-tations. In Section 3 we define the spectral data corresponding to a given Lax hierarchy.They include the spectral curve, the pole divisor and divisor of exponential singular-ities of the Baker–Akhieser function, and the form of the corresponding exponents.We stress again that these data are insufficient in the case G is semi-simple. Thereason is that in the semi-simple case the Tyurin points appear as unremovablepoles while for G = GL ( n ) they are zeroes of (determinant of) the Baker–Akhieserfunction.In Section 4 we formulate and prove main results of the paper, namely, thetheorem of existence and uniqueness of the Baker–Akhieser function possessing theabove listed special properties, and a theta-functional formula for it. We also givean explicit formula for the exponents of the Baker–Akhieser function.2. Lax integrable systems with spectral parameter on a Riemannsurface
Lax equations.
Let g be a semi-simple Lie algebra over C , h be its Cartansubalgebra, and h ∈ h be such element that p i = α i ( h ) ∈ Z + for every simple root α i of g . If we denote the root lattice of g by Z ( R ) where R is the root system of g ,then h belongs to the positive chamber of the dual lattice Z ( R ) ∗ .For p ∈ Z let g p = { X ∈ g | (ad h ) X = pX } , and k = max { p | g p = 0 } . Thenthe decomposition g = k L i = − k g p gives a Z -grading on g . Call k a depth of the grading.Obviously, g p = M α ∈ Rα ( h )= p g α . Define also the following filtration on g : ˜ g p = p L q = − k g q . Then ˜ g p ⊂ ˜ g p +1 ( p ≥ − k ),˜ g − k = g − k , . . . , ˜ g k = g , ˜ g p = g , p > k .Let Σ be a complex compact Riemann surface with two given non-intersectingfinite sets of marked points: Π and Γ. Assume every γ ∈ Γ to be assigned with an h γ ∈ Z ( R ) ∗ + , and with the corresponding grading and filtration. We equip the nota-tion g p , ˜ g p with the upper γ indicating that the grading (resp. filtration) subspacecorresponds to γ . Let L be a meromorphic mapping Σ → g , holomorphic outsidethe marked points which may have poles of an arbitrary order at the points in Π,and has the expansion of the following form in a neighborhood of any γ ∈ Γ:(2.1) L ( z ) = ∞ X p = − k L p ( z − z γ ) p , L p ∈ ˜ g γp where z is a local coordinate in the neighborhood of γ , z γ is the coordinate of γ itself. For simplicity, we assume that the depth of grading k is the same all over Γ,though it would be no difference otherwise. Lemma 2.1 ([8]) . The expansion (2.1) takes place if, and only if L ( z ) = ( z − z γ ) h γ L (0) ( z )( z − z γ ) − h γ where L (0) is holomorphic in the neighborhood of z = z γ . We denote by L Π , Γ ,h a linear space of all such mappings. Since the relation (2.1)is preserved under commutator, L Π , Γ ,h is a Lie algebra called Lax operator algebra .Let M : Σ → g be a meromorphic mapping holomorphic outside Π and Γ, forevery γ ∈ Γ having a Laurent expansion(2.2) M ( z ) = ν γ h γ z − z γ + ∞ X i = − k M γi ( z − z γ ) i at γ , where M γi ∈ ˜ g γi for i < M γi ∈ g for i ≥
0, and ν γ ∈ C . We denote by M Π , Γ ,h the space of such mapping corresponding to given data Π , Γ , h (as above, h = { h γ | γ ∈ Γ } ). Obviously, L Π , Γ ,h ⊂ M Π , Γ ,h .Let D be a non-negative divisor supported at Π: D = m P + . . . + m N P N where m , . . . , m N ≥
0. Let L D Π , Γ ,h = { L ∈ L Π , Γ ,h | ( L ) + K P s =1 k s γ s + D ≥ } . Wedefine a sheaf L D formed by the spaces L D Π , Γ ,h as fibres. A point of the total spaceof the sheaf is given by a triple { Γ , h, L } where the pair { Γ , h } ( h = { h γ | γ ∈ Γ } )represents a point of the base, L ∈ L Π , Γ ,h represents a point of the fibre over { Γ , h } .We assume g to be a classical simple Lie algebra, and G be the correspondingclassical group. Then G operates on the total space of the sheaf L D . The action ofa g ∈ G is as follows: γ → γ , h γ → gh γ g − , L → gLg − .Many results below apply to a more general situation of an arbitrary semi-simple Lie algebra g and the algebraic group G corresponding to g and its faithfulrepresentation in a linear space V . Definition 2.2.
The quotient of the total space of the sheaf L D by the just defined G -action serves as a phase space of the dynamical system we are going to define.Let this space be denoted by P D . Thus P D = L D /G .We give the dynamics on the phase space by means of a Lax equation. It is ournext step to introduce it.By Lax operator we mean the map of the total space of the sheaf L D to M er (Σ → g ) given by { Γ , h, L } → L , i.e. the map forgetting the base compo-nent of a point in L D . Taking account of G -equivariance of the map, we regard toit as to a map defined on the phase space. In abuse of notation, we will denote themap (i.e. the Lax operator) by L also.We introduce the sheaf M D similarly to L D replacing L defined by (2.1) with M defined by (2.2) in the above definitions. We call the forgetting map on M D : { Γ , h, M } → M the M -operator .The equation(2.3) ˙ L = [ L, M ]where ˙ L = dL/dt , is called the Lax equation . This is a system of ordinary differ-ential equations for the dynamical variables including { ( z γ , h γ ) | γ ∈ Γ } , and theparameters of the main parts of meromorphic functions L and M at the points inΠ and Γ. Following [4], a system given by the Lax representation (2.1)–(2.3) with D = ( ω )where ω is a holomorphic differential on Σ, is called a Hitchin system .2.2.
Hierarchies.
In order that the system (2.3) is closed, it is necessary to give M as a function of L . Here we will do it following the lines of [4, 8, 7].Let a denote a triple of the form { χ, P ∈ Π , m > − m P } where χ is an invariantpolynomial on the Lie algebra g , m P is the mulniplicity of P in the divisor D .We define the gradient ∇ χ ( L ) ∈ g of the polynomial χ at the point L ∈ g bymeans of the relation(2.4) dχ ( L ) = h∇ χ ( L ) , dL i , where dχ is the differential of χ as of a function on g , h· , ·i is the Cartan–Killingform on g . If L ∈ L Π , Γ ,h , i.e. we regard to L as to a meromorphic g -valued functionon Σ, then ∇ χ ( L ) will be such function too. If it is considered as a function of alocal coordinate w on Σ, we write ∇ χ ( w ). Lemma 2.3 ([4, 8]) . For every triple a = { χ, P, m } , and every L ∈ L D Π , Γ ,h , thereis a unique appropriately normalized M a ∈ M D Π , Γ ,h having a unique pole outside Γ ,namely, at P , and such that in a neighborhood of P (with a local parameter ω ) (2.5) M a ( w ) = w − m ∇ χ ( L ( w )) + O (1) , and the equation ˙ L = [ L, M a ] is well-defined (i.e. ([ L, M a ]) is a tangent vector to L D Π , Γ ,h . Lemma 2.3 gives M a as a function of L which we denote by M a ( L ). We referto [8] for the details on the normalization mentioned in the theorem.A point P ∈ Σ is called a regular point of L if L ( P ) is well-defined, and isa regular element of the Lie algebra g . It is called irregular if this evaluation iswell-defined but irregular in g . The set of all irregular points of a given L ∈ L D Π , Γ ,h is finite. This implies that the set of Lax operators for which the set of irregularpoints has an empty intersection with Π, is open. Theorem 2.4 ([4, 8]) . Relations (2.6) ∂ a L = [ L, M a ] where M a = M a ( L ) , give a family of commuting vector fields (parameterized by a )on the open subset in P D given by Lax operators such that their sets of irregularpoints have empty intersections with Π . Corollary 2.5.
Lax equations ∂ a L = [ L, M a ] give commuting flows on P D . In [8], Theorem 2.4 has been proven under conditions [8, (3.21)] which read asfollows in our notation(2.7) ∂ a z γ = ν a,γ , ∂ a h γ = [ h γ , M γa, ]( ν a,γ coming from (2.2)). The same relations appear as the conditions of holomorphyof spectra of M -operators along trajectories of (2.3) [8, Section 4.5]. In our approach,equations (2.7) play the same role as the equations of motion of Tyurin parameters in [5, 4]. Theorem 2.6.
For any a the vector field ∂ a is Hamiltonian with the Hamiltonian H a = res P a w − m a a χ ( L ( w a )) ω ( w a ) with respect to the Krichever–Phong symplectic structure. We refer to [8] for the version of the Krichever–Phong symplectic structureneeded here. For the origin of this notion we refer to [4] and references therein, andto [7]. 3.
Spectral transform
It is the aim of the section, to assign a certain spectral data to the aboveconstructed hierarchy of Lax equations.3.1.
Spectral curve.
By the spectral curve of an L ∈ P D we mean a curve Σ L given by the equation(3.1) det( L ( q ) − λ ) = 0 , q ∈ Σ . It is a d -fold branch covering of Σ where d = dim V , V is the space of the standardrepresentation of G .The spectral curve possesses the following simple properties. Lemma 3.1. ◦ Represent (3.1) in the form (3.2) R ( q, λ ) = λ d + d X j =1 r j ( q ) λ d − j = 0 . Then r j ∈ O (Σ , − jD ) . ◦ Assume g is one of the Lie algebras so (2 n ) , so (2 n + 1) , sp (2 n ) . Then thespectral curve is invariant with respect to the involution λ → − λ .Proof. ◦ According to [4, 8, 7] the spectra of Lax operators are holomorphic atTyurin points. Hence polynomials r j ( q ) are holomorphic there too. Next, ( L ) ≥ − D .Since r j is a polynomial of degree j in L , we obtain ( r j ) ≥ − jD .2 ◦ Let σ be a matrix of the scalar product in V in the cases g = so (2 n ) , so (2 n +1), and a matrix of the symplectic form in the case g = sp (2 n ). Then σL T σ − = − L ( L T denotes the matrix transposed to L ), and (3.1) is obviously invariant withrespect to the replacement of L with σL T σ − . But this replacement results in theequation det( L ( q ) + λ ) = 0. (cid:3) Let D L be a pull-back of the divisor D with respect of the covering Σ L → Σ.3.2.
Relation between spectra of L - and M -operators. Since L and ∂ a + M a commute by virtue of the Lax equations, there is a meromorphic vector-valuedfunction ψ on Σ L such that for a general Q = ( q, λ ) ∈ Σ L (3.3) L ( q ) ψ ( Q ) = λ ( Q ) ψ ( Q ) , ( ∂ a + M a ( q )) ψ ( Q ) = f a ( Q ) ψ ( Q )where λ ( Q ) is a solution of (3.1). For a given q ∈ Σ, let Λ( q ) be a diagonal matrixwith all solutions of (3.1) on the diagonal. To define it, we fix an arbitrary orderof the sheets of the covering Σ L → Σ, that is an order of the roots of the equation (3.1). As well, we consider a diagonal matrix F a ( q ) with the eigenvalues f a ( q ) onthe diagonal, placed in the same order.Let Λ( q ) = diag ( λ ( q ) , . . . , λ d ( q )). Placing the vectors ψ ( q, λ ) , . . . , ψ ( q, λ d ) inthe corresponding order, we obtain a matrix Ψ( q ) which we will call the matrix ofeigenvectors . The following relations are equivalent to (3.3)(3.4) L Ψ = ΨΛ , ( ∂ a + M a )Ψ = Ψ F a . The two matrices Λ and F a represent the spectra of the operators L and M a ,resp. The relation between the spectra in a neighborhood of q = P a is given by(3.5) F a ( w a ) = w − m a a ∇ χ a (Λ( w a )) + O (1)where w a is a local parameter in the neighborhood. It has been proven in [8].Indeed, in a neighborhood of P a we have, first, F a = Ψ − · ∂ a Ψ + Ψ − M a Ψ =Ψ − M a Ψ+ O (1) by holomorphy of Ψ and Ψ − at the points in Π (which is a conditionof a general position), and second, M a = w − m ∇ χ ( L ( w )) + O (1) by Lemma 2.3.By invariance of χ , its gradient ∇ χ is equivariant, thus we have Ψ ∇ χ ( L )Ψ − = ∇ χ (Ψ L Ψ − ) = ∇ χ (Λ) (the first equality follows from [8, Eq. (3.29)]). Therefore, F a = ∇ χ (Λ) + O (1).3.3. From ( L, M ) -pair to Baker–Akhiezer matrix-function. Given a pair (
L, M a )defined as above, consider the following matrix-valued function on Σ:(3.6) b Ψ = Ψ exp − t a Z F a dt a . where t a is the time parameter of the flow given by M a . Theorem 3.2. (A0) ( ∂ a + M a ) b Ψ = 0 . (A1) Columns of b Ψ are orthogonal if g = so (2 n ) , so (2 n + 1) , and skew-orthogonalif g = sp (2 n ) . (A2) b Ψ is meromorphic except at P a where it has an essential singularity of theform b Ψ( w a ) = Ψ a ( w a )exp (cid:0) − t a w − m a a ∇ χ a (Λ( w a )) (cid:1) ,w a is a local parameter in the neighborhood, Ψ a is a local matrix-valued func-tion, holomorphic and invertible, Λ( w a ) is a local notation for Λ( q ) . (A3) b Ψ is meromorphic outside D . Apart from Tyurin points, the divisor D p ofits poles is time independent. (A4) At the Tyurin points (3.7) b Ψ = ( z − z γ ) h γ Ψ where Ψ is holomorphic and holomorphically invertible in a neighborhood of z γ .Proof. (A0) We have( ∂ a + M a ) b Ψ = ∂ a Ψ · e − ta R F a dt a − Ψ F a e − ta R F a dt a + M a Ψ e − ta R F a dt a = ( ∂ a + M a )Ψ e − ta R F a dt a − Ψ F a e − ta R F a dt a . By (3.4), ( ∂ a + M a )Ψ = Ψ F a , which completes the proof of the statement.(A1) holds for the reason that L is skew-symmetric in the case g = so (2 n ) , so (2 n +1), and infinitesimal symplectic in the case g = sp (2 n ).(A2-A3) We refer to [3] for the proof of those properties which are conventional.The only point which is not standard is the form of exponent which immediatelyfollows from (3.5).(A4) By Lemma 2.1, there takes place the following relation for the Lax oper-ator: L ( z ) = ( z − z γ ) h γ L (0) ( z )( z − z γ ) − h γ where L (0) ( z ) is holomorphic. Let Ψ ( z ) be the matrix formed by eigenvectors of L (0) ( z ). In a general position, it is holomorphic, and holomorphically invertiblein a neighborhood of z = z γ . We have L (0) ( z )Ψ ( z ) = Ψ ( z )Λ. Together withthe previous relation it implies L ( z )( z − z γ ) h γ Ψ ( z ) = ( z − z γ ) h γ Ψ ( z )Λ, henceΨ( z ) = ( z − z γ ) h γ Ψ ( z ) (up to normalization of eigenvectors which does not affectthe proof). (cid:3) By the property (A2) b Ψ is a conventional
Baker–Akhieser matrix , as introducedin [3, 4]. However, by the property (A4), the Tyurin points appear as time-dependentpoles. Indeed, tr h γ = 0, hence ( z − z γ ) h γ has both zeroes and poles on the diagonalat z = z γ (while for GL ( n ) ( z − z γ ) h γ is conjugated to diag ( z − z γ , , . . . ,
1) [8]). Thismakes a substantial difference with [4], though in general such situation is not new.It had been considered in [5] for the Kadomtsev–Petviashvili equation. Anotherdifference with the conventional situation makes the property (A1).3.4.
Spectral transform. By spectral transform we mean the correspondence { L, { M a }} −→ { Σ L , D, { ( γ, h γ ) } , {∇ χ a (Λ) }} . In other words, the above hierarchy is assigned with the spectral data consistingof the spectral curve, the divisors D (of exponential points of the Baker–Akhieserfunction) and D p (its pole divisor) on it, gradients of invariant polynomials, andthe set of Tyurin points together with associated grading elements. We would likecomment on the the last two items (Tyurin data and {∇ χ a (Λ) } ). Integration ofthe Tyurin data in spectral data is not conventional. In the precedential work [4],due to a special form of the elements h γ for g = gl ( n ), the Tyurin points turn outto be zeroes of eigenfunctions. The situation for semisimple algebras consideredhere is completely different: there are always poles at Tyurin points. As for thegradients of invariant polynomials, they are included as polynomials giving a formof the exponents which is conventional [3].It is the aim of the next section to retrieve the hierarchy by the spectral data,in other words to construct the inverse spectral transform. Inspite the above introduced b Ψ is not included to spectral data, it gives a hint atwhich properties the Baker–Akhieser function giving the inverse spectral transformshould possess. 4.
Inverse spectral transform
It is the aim of the section to retrieve the L - and M -operators by the spec-tral data and Tyurin data. We formulate and prove the theorem of existence anduniqueness of the Baker–Akhieser matrix possessing the above formulated properties(A1)–(A4) (Theorem 3.2).4.1. Retrieving the ( L, M ) -pair by spectral and Tyurin data. Given a genus g curve Σ, and a positive divisor D on it, let b Σ be a Riemann surface of the algebraicfunction λ given by the equation(4.1) R ( q, λ ) = λ n + n X j =1 r j ( q ) λ n − j = 0where r j ∈ O (Σ , − jD ) (the left equality in (4.1) is just a notation). b Σ is a branchcovering of Σ. It possesses a holomorphic involution σ : λ → − λ . Assume that b Σhas no more than nodal singularities. In the case b Σ is singular, let b Σ n denote itsnormalization. Let π : b Σ → Σ ( π : b Σ n → Σ, resp.) be the covering map.Let λ ( Q ) be a function on b Σ defined by λ ( q, λ j ) = λ j where Q = ( q, λ j ) is anarbitrary point of b Σ. Let b D be a pull-back of the divisor D . Then b D is a positivedivisor on b Σ, obviously invariant with respect to σ . Let Λ( q ) be a diagonal matrix–valued function on Σ: Λ( q ) ij = δ ij λ j where λ j is the j -th root of the spectral curveequation over the point q ∈ Σ for an arbitrary (but fixed) ordering of the roots. Bydefinition(4.2) (Λ) + D ≥ . The diagonal matrix-valued function Λ is defined up to a preserving the involutionenumeration of the roots of the characteristic equation, i.e. up to the action of theWeyl group on h . Similarly we can assign a diagonal matrix-valued function onΣ to any (scalar-valued) function on b Σ. Once the enumeration is fixed, there is aversion of the direct image construction which takes a vector-valued function on b Σto a matrix-valued function on Σ. The evaluation of the last at q ∈ Σ is formedby the evaluations of the first at the preimages of q , as by columns, taken in thecorresponding order.Let D = P P ∈ Σ m P P where m P ∈ Z , m P >
0. Let the time t a correspond to apoint P a ∈ supp D , integer m a ≥ − m P , and an invariant polynomial χ a . Theorem 4.1.
Given a non-special divisor D p on b Σ , such that deg D p = b g + d − ( d is the dimension of the standard representation of g ), there exists the uniqueBaker–Akhieser vector-function b ψ , and the corresponding matrix-valued function b Ψ satisfying the following conditions: (BA1) b Ψ is a G -valued function on Σ . (BA2) For every P ∈ supp D , b Ψ( w P ) = Ψ P ( w P ) · exp − X a : P a = P,m a ≤ m P t a w − m a P ∇ χ a (Λ( w P )) in a punctured neighborhood of the point P , where w P is a local parameterin the neighborhood, Ψ P is a local matrix-valued function, holomorphic andinvertible, Λ( w P ) is a local notation for Λ( q ) . (BA3) Outside D ( b D , resp.), b Ψ ( b ψ , resp.) is meromorphic. Apart from Tyurinpoints, ( b ψ ) + D p ≥ (hence, the poles of b ψ are time independent). (BA4) At the Tyurin points (4.3) b Ψ = ( z − z γ ) h γ Ψ where Ψ is holomorphic and holomorphically invertible in a neighborhood of z γ (including z γ ).Proof. Let p = { ( Q, p Q ) | Q ∈ b Σ , p Q ∈ C [ w − Q ] } be finite ( w Q denotes a localparameter in a neighborhood of Q ), D p be a positive divisor of degree b g + d − b Σ. Relying on the conventional theory of Baker–Akhieser functions [3], we willconsider as proven the existence of a d -dimensional space B ( D p , p ) of scalar-valuedBaker–Akhieser functions on b Σ with an arbitrary p as a set of essential singularities,and a pole divisor D p , where for a ( Q, p Q ) ∈ p we consider Q as a point of essentialsingularity, and p Q as a main part of the exponent at Q .For an arbitrary time t a we can take Q ∈ π − ( P a ), and p Q ( w a ) = w − m a a ∇ χ a (Λ( w a )),and apply the above statement to the set p obtained this way. Then functions inthe space B ( D p , p ) have the exponents as claimed in (BA2).However, we claim existence and uniqueness of the Baker–Akhieser functionwith an additional requirement (BA4). We will show that the dimension of thespace of modified this way Baker–Akhieser functions does not change.Let V ≃ C d and V i = { v ∈ V | h γ v = iv } . Obviously,(4.4) V = m M i = − m V i . Define a flag F : { } ⊆ F − m ⊆ . . . ⊆ F m = V by setting(4.5) F j = j M i = − m V i . Then F m = V . We set F i = V also for i > m . It has been shown in course ofthe proof of [11, Lemma 3.4] that for a (locally) holomorphic vector function ψ thefollowing expansion takes place: z h γ ψ = ∞ X i = − m ψ i z i where ψ i ∈ F i for all i ≥ − m . On the one hand, by Riemann–Roch theorem, thecontribution of the pole at any γ to the dimension of such functions should be equalto m dim V . On the other hand, the total codimension of the relations ψ i ∈ F i is also equal to m dim V which easy follows from the equality dim V i = dim V − i .Therefore, the contribution is actually equal to zero. Applying this argument to anycolumn of the matrix-function z h γ Ψ we obtain that the contribution of this matrix-function into the dimension of the space of Baker–Akhieser functions is equal to0. Therefore adding the condition (BA4) leaves the conventional dimension of thespace of Baker–Akhieser functions invariant.By the above procedure of generation of vector-valued and matrix-valued Baker–Akhieser functions, given an arbitrary base ψ , . . . , ψ d in B ( D p , a ) we can form avector-function ψ ( Q ) = ( ψ ( Q ) , . . . , ψ d ( Q )) T on b Σ. The upper T denotes transposi-tion here. Further on, given a point q ∈ Σ, we consider the matrix b Ψ( q ) formed bythe vector-functions ψ ( Q j ) as columns where j = 1 , . . . , d , Q j = ( q, λ j ), λ , . . . , λ d are the roots of the spectral curve equation over q . b Ψ depends on the order ofroots of the spectral equation, but apart that, it is unique up to an appropriatenormalization.It is our next modification that we take b Ψ( q ) ∈ G for all q ∈ Σ. With this aim,we first define a bilinear form on the space B ( b D, p ). For g = so (2 n ) , so (2 n + 1),and ψ , ψ ∈ B ( D p , p ), we set(4.6) ( ψ , ψ )( q ) = 12 X Q : π ( Q )= q ψ ( Q ) ψ ( Q σ )where π : b Σ → Σ is the covering map, q ∈ Σ, Q ∈ b Σ, and σ : ( q, λ ) → ( q, − λ ) isthe involution (where λ , − λ is a pair of opposite roots of the characteristic equation R ( λ, q ) = 0). Let P be an exponential point of b Ψ (see (BA2)), and Q is in a smallenough neighborhood of P . Then Λ( Q ) = − Λ( Q σ ) by definition of σ . Since χ is σ -invariant, ∇ χ is σ -antiinvariant, i.e. ∇ χ (Λ( Q )) = −∇ χ (Λ( Q σ )). Hence everysingularity of exponential type turns to its inverse under the involution σ . The sameholds for the Tyurin-type poles, because the sum of every two diagonal elements of h γ which are permuted by σ , vanishes, by definition of a Cartan subalgebra. Hencethe just defined point-wise scalar product of Baker–Akhieser vector-functions is ameromorphic function (holomorphic at the just listed points). Choose the abovebase ψ ( Q ) , . . . , ψ d ( Q ) to be orthogonal with respect to the scalar product (4.6).Then the columns of the corresponding matrix b Ψ will be orthogonal with respect tothe scalar product ( x, y ) = xσy T where x, y ∈ C d are considered as rows, T denotestransposition, and, in abuse of notation, σ here denotes an operator in C d permutingthe standard basis e , . . . , e d ∈ C d according to the involution σ on the sheets of thecovering b Σ → Σ, namely e , . . . , e d → e σ (1) , . . . , e σ ( d ) . Observe that L is consideredto be skew-symmetric matrix exactly with respect to this scalar product. Finally,we normalize ψ , . . . , ψ d so that the matrix b Ψ ∈ SO ( d ). The argument is the same in the symplectic case except that the scalar product(4.6) is replaced with the symplectic form :(4.7) h ψ , ψ i ( q ) = 12 X Q : π ( Q )= q ( ψ ( Q ) ψ ( Q σ ) − ψ ( Q σ ) ψ ( Q )) . (cid:3) Theorem 4.2.
Let b Ψ be given by Theorem 4.1, and h γ ( t ) = g ( t, z ) hg ( t, z ) − where h ∈ h is integral, g , g − and ˙ g are holomorphic, g ( t ) ∈ G for all t . Then ◦ . L and M a given by (1.1) with t = t a provide a solution of the Lax equationwith the relation (2.5) between L and M a . ◦ . L ∈ L D Π , Γ ,h , and M a ∈ M D Π , Γ ,h , as defined in Section 2.1, in particular (cid:16) L (cid:12)(cid:12) Σ D (cid:17) + D ≥ .Proof. By (BA2) Λ is a h -valued function where h is a diagonal (Cartan) subalgebrain g . Since by (BA1) b Ψ( q ) ∈ G · D , and D commutes with h , L = b ΨΛ b Ψ − is a g -valued function on Σ. The relation M = − ∂ t b Ψ · b Ψ − obviously implies the samefor M . By t we denote any of the times t a .By differentiating the both sides of the relation L = b ΨΛ b Ψ − , taking account oftime independence of Λ, we obtain˙ L = ∂ b Ψ · Λ b Ψ − − b ΨΛ b Ψ − ∂ b Ψ · b Ψ − = [ΨΛΨ − , − ∂ Ψ · Ψ − ] = [ L, M ] . At a Turin point z γ , by (4.3) and L = b ΨΛ b Ψ − , we have L = ( z − z γ ) h γ (Ψ ΛΨ − )( z − z γ ) − h γ where Ψ ΛΨ − is holomorphic. By Lemma 2.1, and for the reason that Ψ ΛΨ − isholomorphic, L has a Laurent expansion (2.1) at z = z γ .Since the exponential factor in (BA2) commutes with Λ, we have L = Ψ P ΛΨ − P on Σ D where Ψ P is holomorphic and holomorphically invertible. Hence, takingaccount of (4.2) we obtain (cid:16) L (cid:12)(cid:12) Σ D (cid:17) = (cid:16) Λ (cid:12)(cid:12) Σ D (cid:17) ≥ − D. Next, consider the expression for the M -operator at a Tyurin point γ . By differen-tiation of both parts of (BA4) we obtain ∂ t b Ψ · b Ψ − = ∂ t ( z − z γ ) h γ · ( z − z γ ) − h γ + ( z − z γ ) h γ ( ∂ t Ψ · Ψ − )( z − z γ ) − h γ . To calculate ∂ t ( z − z γ ) h γ · ( z − z γ ) − h γ we plug ( z − z γ ) h γ = g ( z − z γ ) h g − where h istime-independent. Then ∂ t ( z − z γ ) h γ · ( z − z γ ) − h γ = ˙ gg − + νh γ z − z γ − ( z − z γ ) h γ ˙ gg − ( z − z γ ) − h γ where ν = ∂ t z γ . We obtain(4.8) M = ˙ gg − + νh γ z − z γ − ( z − z γ ) h γ ( ˙ gg − + ∂ t Ψ · Ψ − )( z − z γ ) − h γ . By assumption, ˙ gg − + ∂ t Ψ · Ψ − is holomorphic. Therefore by Lemma 2.1, thethird summand in (4.8) has an L -type Laurent expansion. The first occurence of˙ gg − in (4.8) we replace with O (1). We conclude that the the right hand side of(4.8) has an expansion of the type (2.2).3 ◦ . In the neighborhood of an exponential point P , plugging (BA2) to the relation M a = − ∂ t a b Ψ · b Ψ − we obtain M a = − ∂ t Ψ P · Ψ P − + Ψ P ( w − m ∇ χ (Λ) + O (1))Ψ P − Since Ψ P is holomorphic at P , we have M a = Ψ P ( w − m ∇ χ (Λ))Ψ P − + O (1) . By Ad-invariance of the polynomial χ it follows that Ψ( w − m ∇ χ (Λ))Ψ − = w − m ∇ χ ( L )which implies the relation (2.5) between L and M a . (cid:3) Theta-functional formula for the Baker–Akhieser matrix-function.
The aim of the section is to appropriately modify a conventional Krichever-typetheta-functional formula for the Baker–Akhieser function (for a Bloch functionssuch kind of relation has been introduced by A.Its). By (4.3), it follows that at anyTyurin point γ , all entries of the i th row of the matrix b Ψ have the same order z h iγ where h iγ is the i th diagonal element of h γ . We assign these orders to the i th basiselement of B ( D p , p ) by modifying the 1-form involved in such kind of formulas. Thelast will be no longer the same for all rows.For any P ∈ supp D let { P j | j = 1 , . . . , d } be its full preimage in b Σ. For anyinvariant polynomial χ a let µ ja ( w P j ) denote the j th diagonal element of ∇ χ a (Λ( w P ))(which is nothing but the j th eigenvalue of M a ), where w P j is a local coordinate inthe neighborhood of P j ( w P j = w P if P j is not a branch point of the spectral curve,i.e. in a general position). Theorem 4.3.
Let Ω i be the unique 1-form on Σ such that ◦ . In a neighborhood of any P j Ω i ( w P j ) = − X a : P a = P,m a ≤ m P t a w − m a P j µ ja ( w P j ) + O (1) dw P j ;2 ◦ . In a neighborhood of any Tyrin point γ Ω i ( z ) = (cid:18) h iγ z − z γ + O (1) (cid:19) dz. ◦ . Ω i is holomorphic outside the above points, and its a -periods are equal tozero.Then the Baker–Akhieser matrix (4.9) b Ψ ij ( q ) = exp ( q,λ j ) Z Ω i θ (A( Q ) + Z ( D i ) + U ) θ (A( Q ) + Z ( D i )) , i = 0 , . . . , n − , subjected to the further orthogonalization (skew-orthogonalization, resp.) procedureintroduced above, satisfies the conditions of Theorem 4.1.Proof. It is conventional that the g -valued functions given by the relations (1.1),(4.9) are well-defined on Σ whatever be the differential Ω satisfying the conditions1 ◦ , 3 ◦ . It can be proven exactly as in the case G = GL ( n ) [3].In presence of involution, since the divisor D is invariant with respect to it,the procedure of orthogonalization (skew-orthogonalization, resp.) can be applied.The problem is to fulfill the conditions (BA2) and, especially, (BA4) in the process.For instance, the condition (BA4) means that the i th row of b Ψ has the order h iγ at γ where h γ = { h γ , . . . , h dγ } . We will show that these orders are preserved in theprocess of orthogonalization (skew-orthogonalization, resp.), hence (BA4) stays tobe fulfilled.Indeed, let vector-functions e , . . . , e k , e k +1 have the orders h , . . . , h k , h k +1 at apoint, say, z = 0. Thus, locally, e i ∼ z h i , i = 1 , . . . , k + 1. On the ( k + 1)th step ofthe orthogonalization process we are looking for λ , . . . , λ k satisfying the equations( λ e + . . . + λ k e k + e k +1 , e j ) = 0 , j = 1 , . . . , k + 1 . Then λ s = ( − k − s det[( e i , e j )] j =1 ,..., b s,...,k,k +1 i =1 ,...,k / det[( e i , e j )] j =1 ,...,ki =1 ,...,k . Obviously,det[( e i , e j )] j =1 ,...,ki =1 ,...,k ∼ z h + ... + h k ) , det[( e i , e j )] j =1 ,..., b s,...,k,k +1 i =1 ,...,k ∼ z h + ... + h k ) − h s + h k +1 , hence λ s ∼ z h k +1 − h s . Since e s ∼ z h s , all terms of the linear combination λ e + . . . + λ k e k + e k +1 are of the same order z h k +1 .As for (BA2), it will be preserved for the reason that the columns stay untouchedin the process of orthogonalization of rows.The resulting Baker–Akhieser matrix satisfies the conditions of Theorem 4.2. (cid:3) Explicit expression for the exponents.
Here we would like to give an ex-plicit expression (the relation (4.12) below) for µ ja ( w P j ) in Theorem 4.3.Recall that µ ja ( w P j ) is the j th diagonal element of ∇ χ a (Λ( w P )), as introducedin Section 4.2. It gives the main part of the differential Ω i at the point P j , asformulated in Theorem 4.3,1 ◦ . Thus, it will be our first step to give an explicitexpression for ∇ χ (Λ) where χ is a basis invariant of the Lie algebra g . Example . Let χ ( L ) = tr L n +1 . Then dχ ( L ) = tr dL n +1 = ( n + 1)tr L n dL , hence ∇ χ ( L ) = ( n + 1) L n .This example reproduces the expression for the main part of the M -operator in [4]. Example . Let χ ( L ) = det L . Then ( ∇ χ )( L ) = (det L ) L − . Proof 1.
By differentiating determinant as a polylinear functional of its columns,we obtain: d det L = P ij A ij dL ij where A ij is a cofactor of the ( i, j )th element ofthe matrix L . Let A = ( A ij ). Then the previous relation can be written down as d det L = tr( A T dL ) ( A T being the transposed matrix for A ). It is conventional that A T = (det L ) L − , which implies the statement. (cid:3) Proof 2.
We start with det L = exp tr log L . Then d det L = (exp tr log L ) d (tr log L ) = (det L )tr d log L = (det L )tr ( L − dL ) . Finally, d det L = tr ((det L ) L − · dL ). (cid:3) Example 4.5 possesses a certain universality, namely it suggests the way tocalculate gradients of all coefficients of the characteristic polynomial. To be moreprecise, we will calculate their generating function. On the one hand, such generatingfunction is given by the gradient of the characteristic polynomial itself: ∇ det( λ − L ) = l X i =1 λ n − i ∇ r i ( L ) . On the other hand ∇ det( λ − L ) = det( λ − L )( λ − L ) − . Consider the characteristic equation in a most general form R ( q, λ ) = λ d + n X j =1 λ d − d j r j ( q ) = 0where d j is a degree of r j as of a polynomial in L .Assuming r ≡
1, det L = 0, and taking L − off, we obtaindet( λ − L )( λ − L ) − = − n X j =0 λ n − d j r j ( L ) ! (1 − λL − ) − L − = − n X j =0 λ n − d j r j ( L ) ! ∞ X p =0 ( − p λ p L − p ! L − . Finally, we obtaindet( λ − L )( λ − L ) − = n X j =0 ∞ X p =0 ( − p +1 λ n − d j + p r j ( L ) L − p − , which implies ∇ r i ( L ) = X d j − p = d i j ≤ n, p ≥ ( − p +1 r j ( L ) L − p − . For a simple Lie algebra this relation writes as ∇ r i ( L ) = X d j − p = d i j ≤ n, p ≥ ( − p +1 r j ( L ) L − p − . Here, p = d j − d i , and p ≥ j ≥ i . Therefore, changing the summationlimits, we write down the expression for the gradient as follows:(4.10) ∇ r i ( L ) = X i ≤ j ≤ n ( − d j − d i +1 r j ( L ) L d i − d j − . The last is a relation between diagonal matrix-valued functions on Σ. For theinduced scalar-valued functions on the spectral curve we have(4.11) ∇ r i ( q, λ ) = X i ≤ j ≤ n ( − d j − d i +1 r j ( q ) λ d i − d j − . This implies the following expression for µ ja :(4.12) µ ja ( w P j ) = X i ≤ j ≤ n ( − d j − d i +1 r j ( w P j ) λ d i − d j − j . In the case g = so (2 n ), the r n ( L ) = det L is not a basis spectral invariant, and whatwe need is a gradient of the Pfaffian. For the last, from ∇ Pf( L ) = (det L ) L − wederive ∇ Pf( L ) = 12 Pf( L ) L − which implies µ ja ( w P j ) = 12 Pf( L ( w P j )) λ − j for the times corresponding to the Hamiltonians related to the Pfaffian.In the case g = so (2 n + 1) L is always degenerate. However, the above calcu-lation can be carried out after excluding the characteristic root λ = 0 (dividing theboth parts of the spectral equation by λ ) which means that we consider only thenon-trivial irreducible component of the spectral curve. Remark . In the case the base curve is hyperelliptic (given in the form y = P g +1 ( x )), and P j runs over the preimages of ∞ , the asimptotic behavior of x , y , λ at every P j , as well as explicit expressions for r i ( x, y ) (for all i ), have been explicitlycalculated in [10, 1]. This enables one to explicitly express Ω i in terms of the basisholomorphic Prym differentials which are also known. References [1] Borisova, P.I., Sheinman, O.K.
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