Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel

Let $M$ be the tensor product of finite-dimensional polynomial evaluation Yangian $Y(gl_N)$-modules. Consider the universal difference operator $D = \sum_{k=0}^N (-1)^k T_k(u) e^{-k\partial_u}$ whose coefficients $T_k(u): M \to M$ are the XXX transfer matrices associated with $M$. We show that the difference equation $Df = 0$ for an $M$-valued function $f$ has a basis of solutions consisting of quasi-exponentials. We prove the same for the universal differential operator $D = \sum_{k=0}^N (-1)^k S_k(u) \partial_u^{N-k}$ whose coefficients $S_k(u) : M \to M$ are the Gaudin transfer matrices associated with the tensor product $M$ of finite-dimensional polynomial evaluation $gl_N[x]$-modules.


Introduction
In a quantum integrable model one constructs a collection of one-parameter families of commuting linear operators (called transfer matrices) acting on a finite-dimensional vector space. In this paper we consider the vector-valued differential or difference linear operator whose coefficients are the transfer matrices. This operator is called the universal differential or difference operator. For the XXX and Gaudin type models we show that the kernel of the universal operator is generated by quasi-exponentials or sometimes just polynomials. This statement establishes a relationship between these quantum integrable models and that part of algebraic geometry, which studies the finite-dimensional spaces of quasi-exponentials or polynomials, in particular with Schubert calculus.
We plan to develop this relationship in subsequent papers. An example of an application of this relationship see in [5].
We describe possible degrees of polynomials p(u) appearing in the kernel further in the paper. The universal differential operator has singular points at z 1 , . . . , z n . We describe behavior of elements of the kernel at these points.
The tensor product M may be naturally regarded as the tensor product of polynomial evaluation modules over the current algebra gl N [x]. Then the operators X ab (u), D K,M (u, ∂ u ) may be naturally defined in terms of the gl N Similarly, the tensor product M may be naturally regarded as the tensor product of polynomial evaluation modules over the Yangian Y (gl N ). Then one may define a linear N -th order difference operator acting on M -valued functions in u. That operator is called the universal difference operator. Its coefficients commute and are called the transfer matrices of the associated XXX model.
We prove that the kernel of the universal difference operator is generated by quasi-exponentials. We describe the quasi-exponentials entering the kernel and their behavior at singular points of the universal difference operator.
The paper has the following structure. In Section 2 we make general remarks on quasiexponentials.
In Section 3 we collect basic facts about the Yangian Y (gl N ), the fundamental difference operator and the XXX transfer matrices. In Section 3.4 we formulate Theorem 3.8 which states that the kernel of the universal difference operator is generated by quasi-exponentials. Theorem 3.8 is our first main result for the XXX type models.
In Section 4 we prove a continuity principle for difference operators with quasi-exponential kernel. Under certain conditions we show that if a family of difference operators has a limiting difference operator, and if the kernel of each operator in the family is generated by quasi-exponentials, then the kernel of the limiting difference operator is generated by quasiexponentials too.
Section 5 is devoted to the Bethe ansatz method for the XXX type models. Using the Bethe ansatz method we prove the special case of Theorem 3.8 in which M is the tensor product of vector representations of gl N . Then the functoriality properties of the fundamental difference operator and our continuity principle allow us to deduce the general case of Theorem 3.8 from the special one.
In Section 6 we give a formula comparing the kernels of universal difference operators associated respectively with the tensor products M Λ (1) ⊗ · · · ⊗ M Λ (n) and M Λ (1) (a 1 ) ⊗ · · · ⊗ M Λ (n) (an) , where a 1 , . . . , a n are non-negative integers and Λ (i) (a i ) = Λ (i) 1 + a i , . . . , Λ (i) N + a i . In Section 7 we describe the quasi-exponentials entering the kernel of the universal difference operator and the behavior of functions of the kernel at singular points of the universal difference operator. Theorems 7.1, 7.2 and 7.3 form our second main result for the XXX type models.
In Sections 8-12 we develop an analogous theory for the universal differential operator of the Gaudin type models.
Section 8 contains basic facts about the current algebra gl N [x], the universal differential operator and the Gaudin transfer matrices. We formulate Theorem 8.4, which states that the kernel of the universal differential operator is generated by quasi-exponentials. This theorem is our first main result for the Gaudin type models. Section 9 contains the continuity principle for the differential operators with quasi-exponential kernel.
In Section 12 we describe the quasi-exponentials entering the kernel of the universal differential operator and the behavior of functions of the kernel at singular points of the universal differential operator. Theorems 12.1, 12.2 and 12.3 form our second main result for the Gaudin type models.
2 Spaces of quasi-exponentials 2.1 Quasi-exponentials 2.1.1. Define the operator τ acting on functions of u as (τ f )(u) = f (u + 1). A function f (u) will be called one-periodic if f (u + 1) = f (u). Meromorphic one-periodic functions form a field with respect to addition and multiplication.
2.1.2. Let Q be a nonzero complex number with fixed argument. Set Q u = e u ln Q . We have τ Q u = Q u Q.
Let p ∈ C[u] be a polynomial. The function Q u p will be called a (scalar) elementary quasiexponential in u. A finite sum of elementary quasi-exponentials will be called a (scalar) quasiexponential.
Let V be a complex vector space of finite dimension d. A V -valued quasi-exponential is a V -valued function of the form a f a (u)v a , where f a (u) are scalar quasi-exponentials, v a ∈ V , and the sum is finite.
We say that a quasi-exponential ab Q u a u b v ab is of degree less than k if v ab = 0 for all b k. 2.1.3. For given End (V )-valued rational functions A 0 (u), . . . , A N (u) consider the difference operator acting on V -valued functions in u.
We say that the kernel of D is generated by quasi-exponentials if there exist N d quasiexponential functions with values in V such that each of these function belongs to the kernel of D and these functions generate an N d-dimensional vector space over the field of one-periodic meromorphic functions.
The following simple observation is useful.
Lemma 2.1. Assume that a quasi-exponential ab Q u a u b v ab , with all numbers Q a being different, lies in the kernel of D defined in (2.1). Then for every a, the quasi-exponential Q u a b u b v ab lies in the kernel of D.
The lemma follows from the fact that exponential functions with different exponents are linearly independent over the field of rational functions in u.
3 Generating operator of the XXX transfer matrices 3.1.1. Let e ab , a, b = 1, . . . , N , be the standard generators of the complex Lie algebra gl N . We have gl N = n + ⊕ h ⊕ n − , where For an integral dominant gl N -weight Λ ∈ h * , denote by M Λ the irreducible finite dimensional gl N -module with highest weight Λ. For a gl N -module M and a weight µ ∈ h * , denote by M [µ] ⊂ M the vector subspace of vectors of weight µ.
3.1.2. The Yangian Y (gl N ) is the unital associative algebra with generators T {s} ab , a, b = 1, . . . , N and s = 1, 2, . . . . Let The defining relations in Y (gl N ) have the form for all a, b, c, d. The Yangian is a Hopf algebra with coproduct for all a, b.
3.1.3. We identify the elements of End (C N ) with N × N -matrices. Let E ab ∈ End (C N ) denote the matrix with the only nonzero entry 1 at the intersection of the a-th row and b-th column. 1 )). Then the defining relations for the Yangian can be written as the following equation of series in u −1 with coefficients in End ( For a monic series f (u), there is an automorphism The fixed point subalgebra in Y (gl N ) with respect to all automorphisms χ f is called the Yangian Y (sl N ). Denote by ZY (gl N ) the center of the Yangian Y (gl N ).
See [4] for a proof.
for suitable monic series c a (u). Moreover, for certain monic polynomials P a (u). The polynomials P 1 , . . . , P N −1 are called the Drinfeld polynomials of the module V . The vector v is called a highest weight vector and the series c 1 (u), . . . , c N (u) -the Yangian highest weights of the module V .
For any collection of monic polynomials P 1 , . . . , P N −1 there exists an irreducible finitedimensional Y (gl N )-module V such that the polynomials P 1 , . . . , P N −1 are the Drinfeld polynomials of V . The module V is uniquely determined up to twisting by an automorphism of the form χ f . The claim follows from Drinfeld's description of irreducible finite-dimensional Y (sl N )-modules [2] and Proposition 3.1.
Let W be the irreducible subquotient of V 1 ⊗ V 2 generated by the vector v 1 ⊗ v 2 . Then the Drinfeld polynomials of the module W equal the products of the respective Drinfeld polynomials of the modules V 1 and V 2 . .
for some monic polynomial P N (u).
For any collection of monic polynomials P 1 , . . . , P N there exists a unique polynomial irreducible finite-dimensional Y (gl N )-module V such that the polynomials P 1 , . . . , P N −1 are the Drinfeld polynomials of V and (3.2) holds.

3.1.8.
There is a one-parameter family of automorphisms where in the right hand side, (u − z) −1 has to be expanded as a power series in u −1 . The Yangian Y (gl N ) contains the universal enveloping algebra U (gl N ) as a Hopf subalgebra. The embedding is given by the formula e ab → T {1} ba for all a, b. We identify U (gl N ) with its image.
The evaluation homomorphism ǫ : Y (gl N ) → U (gl N ) is defined by the rule: T is an irreducible Y (gl N )-module and the corresponding highest weight series c 1 (u), . . . , c N (u) have the form The corresponding Drinfeld polynomials are N ∈ Z 0 for all i. Then the polynomial P N (u) has the form 3.1.9. Consider C N as the gl N -module with highest weight (1, 0, . . . , 0).
For any complex numbers z 1 , . . . , z n , all irreducible subquotients of the Y (gl N )-module C N (z 1 ) ⊗ · · · ⊗ C N (z n ) are polynomial Y (gl N )-modules. Moreover, for any polynomial irreducible finite-dimensional Y (gl N )-module V , there exist complex numbers z 1 , . . . , z n such that V is isomorphic to a subquotient of the Y (gl N )-module C N (z 1 ) ⊗ · · · ⊗ C N (z n ).
The numbers z 1 , . . . , z n are determined by the formula This formula follows from consideration of the action of the center of the Yangian Y (gl N ) in the module C N (z 1 ) ⊗ · · · ⊗ C N (z n ).
3.1.10. A finite-dimensional Y (gl N )-module will be called polynomial if it is the direct sum of tensor products of polynomial irreducible finite-dimensional Y (gl N )-modules. If V is a polynomial finite-dimensional Y (gl N )-module, then for any a, b = 1, . . . , N , the series T ab (u)| V converges to an End (V )-valued rational function in u.
3.1.11. Let π : U (gl N ) → End (C N ) be the representation homomorphism for the gl N -module C N . Clearly, for any x ∈ U (gl N ) we have For a non-degenerate matrix A ∈ End (C N ), define an automorphism ν A of Y (gl N ) by the formula Let V be a finite-dimensional Yangian module with the representation µ : Y (gl N ) → End (V ) andμ : GL N → End (V ) the corresponding representation of the group GL N . The automorphism ν A induces a new Yangian module structure V A on the same vector space with the representation µ A = µ • ν A . Formula (3.4) yields that for any x ∈ Y (gl N ),

Universal difference operator
If V is a polynomial finite-dimensional Y (gl N )-module, then X ab (u, τ ) acts on V -valued functions in u, Following [13], introduce the difference operator where the sum is over all permutations σ of {1, . . . , N }. The operator D(u, τ ) will be called the universal difference operator associated with the matrix Q.
If π is not bijective, then The statement is Proposition 4.10 in [6].

3.2.2.
Introduce the coefficients T 0 (u), . . . , T N (u) of D(u, τ ): The coefficients T k (u) are called the transfer matrices of the XXX type model associated with Q.
The main properties of the transfer matrices: see [13,6].

3.2.4.
If V is a polynomial finite-dimensional Y (gl N )-module, then the universal operator D(u, τ ) induces a difference operator acting on V -valued functions in u. This operator will be called the universal difference operator associated with Q and V and denoted by D Q,V (u, τ ). The linear operators T k (u)| V ∈ End (V ) will be called the transfer matrices associated with Q and V and denoted by T k,Q,V (u). They are rational functions in u.
Consider the algebra U (gl N ) ⊗n . For a, b = 1, . . . , N and i = 1, . . . , n, define Lemma 3.5. If D Q (u, τ ) is the universal difference operator associated with the matrix Q and is the automorphism defined in Section 3.1.11, then is the universal difference operator associated with the matrix AQA −1 .
Proof . Consider matrices Q = (Q ab ), T = (T ab (u)), and X = (X ab (u, τ )). Then formula (3.6), defining X ab , may be read as X = 1−QT τ −1 . Formula (3.7) for the universal difference operator may be understood as the row determinant of the matrix X , D(u, τ ) = det X . The definition of the automorphism ν A reads as (id ⊗ ν A )(T ) = A −1 T A. Then we have Now formula (3.8) will be proved if we were able to write that last determinant as the product: The last formula may be proved the same way as the standard formula det M N = det M det N in ordinary linear algebra, using two observations. The first is that the entries of A are numbers and commute with the entries of X . The second observation is Lemma 3.2 describing the transformations of the row determinant of X with respect to row replacements.

More properties of transfer matrices
As we know T 0,Q,M Λ (z)[m] (u) = 1, and we have Theorem 3.6. The followings statements hold.
(i) The operators T 10 , T 20 , . . . , T N 0 and T 11 , T 21 , . . . , T N 1 are scalar operators. Moreover, the following relations hold: Proof . Part (i) follows from Proposition B.1 in [6]. Part (ii) follows from the definition of the universal difference operator and the fact that the coefficients of the series T N (u) belong to the center of Y (gl N ).

3.3.1.
Assume that Q is the identity matrix. Then the associated transfer matrices preserve Sing M Λ (z)[m] and we may consider the universal difference operator D Q=1, for suitable coefficients S k (u). Note that the operators S k (u) coincide with the action in Sing M Λ (z)[m] of the modified transfer matrices S k (u) from formula (10.4) of [6].
Theorem 3.7. The following three statements hold.
Since Q is the identity matrix, formula (3.6) reads now as follows Then part (ii) is straightforward from formulas (3.9) and (3.7).
Let v be any vector in Sing M Λ (z)[m] and d any number. To prove part (iii) we apply the difference operators in formula (3.9) to the function vu d . The expansion at infinity of result of the application of the right side is So it remains to show that To prove (3.11), observe that according to formula (3.10) we have Applying this remark to formula (3.7) we get Indeed only the identity permutation contributes nontrivially to the sum in the left side. This proves part (iii) of Theorem 3.7.

First main result
Theorem 3.8. Let V be a polynomial finite-dimensional Y (gl N )-module and Q ∈ GL N . Consider the universal difference operator D Q,V (u, τ ) associated with Q and V . Then the kernel of D Q,V (u, τ ) is generated by quasi-exponentials.
The theorem will be proved in Section 5.

Continuity principle for dif ference operators
with quasi-exponential kernel 4

.1 Independent quasi-exponentials
Let V be a complex vector space of dimension d. Let p ∈ C[u] be a monic polynomial of degree k.
Consider the differential equation for a V -valued function f (u). Denote by W p the complex vector space of its solutions. The map assigning to a solution its initial condition at u = 0, is an isomorphism.
Let λ 1 , . . . , λ l be all distinct roots of the polynomial p of multiplicities k 1 , . . . , k l , respectively. Let v 1 , . . . , v d be a basis of V . Then the quasi-exponentials form a basis in W p .
Then the kd functions listed in (4.1) are linear independent over the field of one-periodic functions.

Admissible difference operators
Assume that each of these functions has limit as u → ∞ and A 0 (u) = 1 in End (V ). For every k, let be the Laurent expansion at infinity. Consider the algebraic equation with respect to variable x and the difference operator   Assume that D is admissible at infinity and a nonzero V -valued quasi-exponential Then Q is a root of the characteristic equation (4.2).

Continuity principle
• for every ǫ ∈ (0, 1) the kernel of D ǫ is generated by quasi-exponentials, • there exists a natural number m such that for every ǫ ∈ (0, 1) all quasi-exponentials generating the kernel of D ǫ are of degree less than m.
Theorem 4.4. Under these conditions the kernel of the difference operator D ǫ=0 is generated by quasi-exponentials.
Proof . For every ǫ ∈ [0, 1), the characteristic equation for D ǫ has N d roots counted with multiplicities. As ǫ tends to 0 the roots of the characteristic equation of D ǫ tend to the roots of the characteristic equation of D ǫ=0 . All these roots are nonzero numbers. For small positive ǫ the set of multiplicities of roots does not depend on ǫ.
The following lemma is evident.
Lemma 4.5. There exist • a numberǭ with 0 <ǭ 1, • for any ǫ, 0 < ǫ <ǭ, a way to order the roots of the characteristic equation of D ǫ (we denote the ordered roots by Q ǫ 1 , . . . , Q ǫ N d ), • a way to assign the logarithm q ǫ j to every root Q ǫ j such that for every j the number q ǫ j continuously depends on ǫ and q ǫ j = q ǫ l whenever Q ǫ j = Q ǫ l . Let m be the number described in Section 4.3. For every ǫ, 0 < ǫ <ǭ, we define p ǫ ∈ C[u] to be the monic polynomial of degree k = mN d, whose set of roots consists of m copies of each of the numbers q ǫ 1 , . . . , q ǫ N d . Let W pǫ be the kd-dimensional vector space of quasi-exponentials assigned to the polynomial p ǫ in Section 4.1. By assumptions of Theorem 4.1, for every ǫ, 0 < ǫ <ǭ, the space W pǫ contains an N d-dimensional subspace U ǫ generating the kernel of D ǫ . This subspace determines a point in the Grassmannian The points ρ pǫ (U ǫ ) all lie in the compact manifold Gr(V ⊕k , N d) and the set of all such points has an accumulation pointŨ ∈ Gr(V ⊕k , N d) as ǫ tends to zero. Then Using Lemma 4.1, we conclude that the space ρ −1 p ǫ=0 (Ũ ) generates the kernel of D ǫ=0 .

Bethe ansatz equations associated with a weight subspace
Consider a nonzero weight subspace M (z)[m 1 , . . . , m N ]. Introduce l = (l 1 , . . . , l N −1 ) with l j = m j+1 + · · · + m N . We have n l 1 · · · l N −1 0. Set l 0 = l N = 0 and l = l 1 + · · · + l N −1 . We shall consider functions of l variables The following system of l algebraic equations with respect to l variables t is called the Bethe ansatz equations associated with M (z)[m 1 , . . . , m N ] and Q, Here the equations of the first group are labeled by j = 1, . . . , l 1 , the equations of the second group are labeled by a = 2, . . . , N − 2, j = 1, . . . , l a , the equations of the third group are labeled by j = 1, . . . , l N −1 . A solutiont of system (5.1) will be called off-diagonal ift  The statement follows from Theorem 6.1 in [6]. For k = 1, the result is established in [3].

Weight function and Bethe ansatz theorem
Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel 15 The eigenvalues of the Bethe vector are given by the following construction. Set for a = 2, . . . , N . Define the functions λ k (u, t, z) by the formula for k = 0, . . . , N , see Theorem 6.1 in [6].

Difference operator associated with an off-diagonal solution
Lett be an off-diagonal solution of the Bethe ansatz equations. The scalar difference operator will be called the associated fundamental difference operator.
Theorem 5.2. The kernel of Dt(u, τ ) is generated by quasi-exponentials of degree bounded from above by a function in n and N . This is Proposition 7.6 in [9], which is a generalization of Proposition 4.8 in [8].   A vector a = (a 1 , . . . , a n ) with coordinates a i from the set {0, 2, 3, . . . , N } will be called admissible if for any j = 1, . . . , N − 1 we have l j = #{ a i | i = 1, . . . , n, and a i > j } In other words, a is admissible if m j = #{ a i | i = 1, . . . , n, and a i = j}.

5.6.4.
For q = 0 the right hand sides of equations (5.3) equal zero and to solve the equations for q = 0 one needs to find common zeros of the left hand sides.
Let a = (a 1 , . . . , a N ) be an admissible index. Definet(a, 0) = (t  Hence for every small nonzero q, there exists a unique pointt(a, q) = (t (i) j (a, q)), such that •t(a, q) is an off-diagonal solution of system (5.3) with the same q, •t(a, q) holomorphically depends on q and tends tot(a, 0) as q tends to zero.
Therefore, for all i, j, we havẽ Lemma 5.6. As q tends to zero, the Bethe vector ω(t(a, q), z) has the following asymptotics: where C a is a nonzero number and the dots denote a linear combination of basis vectors e a ′ v with indices a ′ lexicographically greater than a.

Comparison of kernels
Let V be a finite-dimensional polynomial Y (gl N )-module. Consider the universal difference operators D Q,V (u, τ ) and D Q,V f (u, τ ). Let C(u) be a function satisfying the equation Then a V -valued function g(u) belongs to the kernel of D Q,V (u, τ ) if and only if the function C(u)g(u) belongs to the kernel of D Q,V f (u, τ ).  = (a 1 , . . . , a n ) be a collection on complex numbers. Introduce the new collection Λ(a) = (Λ (1) (a 1 ), . . . , Λ (n) (a n )).
at the points of S i . (ii) If v N = v N −1 = · · · = v j = 0 for some j > 1, then f (z i − k) = 0 for k = 1, 2, . . . , Λ Proof . Consider the polynomial difference operator . Now we may continue on this reasoning up to the equation and so on. Repeating this reasoning we prove parts (i) and (ii) of the theorem.
The same reasoning shows that if f (u) = 0 for u ∈ S i , then the quasi-exponential f (u) identically equals zero. Since the kernel ofD is generated by quasi-exponentials, we obtain part (iii) of the theorem.

Kernel of the fundamental difference operator associated with an eigenvector
is an eigenvector of all transfer matrices, Then the scalar difference operator will be called the fundamental dif ference operator associated with the eigenvector v. Theorems 7.1, 7.2 and 7.3 give us information on the kernel of the fundamental difference operator.
(i) Assume that Q ∈ GL N is diagonal with distinct diagonal entries. Then the kernel of D v (u, τ u ) is generated by quasi-exponentials Q u 1 p 1 (u), . . . , Q u N p N (u), where for every i the polynomial p i (u) ∈ C[u] is of degree m i . (ii) Assume that Q is the identity matrix and the eigenvector v belongs to Sing M Λ (z) [m].
Corollary 7.5. Assume the index i is such that z i − z j / ∈ Z for any j = i. Let f (u) be a quasiexponential lying in the kernel of D v (u, τ u ). Then (iii) For any numbers v N , v N −1 , . . . , v 1 there exists a quasi-exponential f (u) which lies in the kernel of D v (u, τ u ) and takes these values at S i .
If v is a Bethe eigenvector, then these two corollaries were proved in [8] and [9]. For any a, b, the series L ab (u) acts on M Λ (z) by the formula be the decomposition of the tensor product of gl N -modules into the direct sum of irreducible gl N -modules. Then for any z ∈ C, is the decomposition of the tensor product of evaluation gl N [x]-modules into the direct sum of irreducible gl N [x]-modules. The previous remarks show that for any z 1 , . . . , z n ∈ C, all irreducible submodules of the gl N [x]-module C N (z 1 ) ⊗ · · · ⊗ C N (z n ) are tensor products of polynomial evaluation gl N -module which is the tensor product of polynomial evaluation gl N [x]-modules, then there exist z 1 , . . . , z n such that V is isomorphic to a submodule of the gl N [x]-module C N (z 1 ) ⊗ · · · ⊗ C N (z n ).
that is, the gl N [x]-modules V and V A are isomorphic. In particular, if V is the tensor product of polynomial evaluation gl N [x]-modules, then V A is the tensor product of polynomial evaluation gl N [x]-modules too.

Fundamental differential operator
Let K = (K ab ) be an N × N matrix with complex entries. For a, b = 1, . . . , N , define the differential operator Following [13], introduce the differential operator where the sum is over all permutations σ of {1, . . . , N }. The operator D(u, ∂ u ) will be called the universal differential operator associated with the matrix K.

8.2.2.
If V is the tensor product of evaluation finite-dimensional gl N [x]-modules, then the universal operator D(u, ∂ u ) induces a differential operator acting on V -valued functions in u. This operator will be called the universal differential operator associated with K and V and denoted by D K,V (u, ∂ u ). The linear operators S k (u)| V ∈ End (V ) will be called the transfer matrices associated with K and V and denoted by S k,K,V (u). They are rational functions in u.

8.2.3.
If D K (u, τ ) is the universal differential operator associated with the matrix K and is the automorphism defined in Section 8.1.4. Then Lemma 8.1 implies that is the universal differential operator associated with the matrix AKA −1 , cf. Lemma 3.5. Let V be the tensor product of finite-dimensional evaluation gl N [x]-modules, andμ : G N → GL(V ) the associated GL N -representation. Then Consider the universal differential operator Theorem 8.2. The following statements hold.
(i) The operators S 10 , S 20 , . . . , S N 0 and S 11 , S 21 , . . . , S N 1 are scalar operators. Moreover, the following relations hold: (ii) For k = 1, . . . , N − 1, we have whereS k (u) is a polynomial in u of degree nk. Moreover, the operators Proof . Part (i) follows from Proposition B.1 in [6]. The existence of presentation (8.2) follows from the definition of the universal differential operator. To prove equation (8.3) it is enough to notice that the leading singular term of the universal differential operator at u = z r is equal to the leading singular term of the universal differential operator associated with one evaluation module M Λ (r) (z r ), which in its turn expresses via the quantum determinant.
where the operators S 1,0 , . . . , S N,0 are scalar operators. Moreover, The proof of Theorem 8.3 is similar to the proof of Theorem 3.7. 9 Continuity principle for dif ferential operators with quasi-exponential kernel 9.1 Quasi-exponentials Let V be a complex vector space of dimension d. Let A 0 (u), . . . , A N (u) be End (V )-valued rational functions in u. Assume that each of these functions has limit as u → ∞ and A 0 (u) = 1 in End (V ). Then the differential operator

First main result in the Gaudin case
acting on V -valued functions in u, will be called admissible at infinity. For every k, let A k (u) = A ∞ k,0 + A ∞ k,1 u −1 + A ∞ k,2 u −2 + · · · be the Laurent expansion at infinity. Consider the algebraic equation det x N + x N −1 A 1,0 + · · · + xA N −1,0 + A N,0 = 0 (9.1) with respect to variable x.
Lemma 9.1. If a nonzero V -valued quasi-exponential e λu (u d v d + u d−1 v d−1 + · · · + v 0 ) lies in the kernel of an admissible at infinity differential operator D, then λ is a root of equation (9.1).

Continuity principle
Let A 0 (u, ǫ), . . . , A N (u, ǫ) be End (V )-valued rational functions in u analytically depending on ǫ ∈ [0, 1). Assume that is admissible at infinity, • for every ǫ ∈ (0, 1) the kernel of D ǫ is generated by quasi-exponentials, • there exists a natural number m such that for every ǫ ∈ (0, 1) all quasi-exponentials generating the kernel of D ǫ are of degree less than m.
Theorem 9.2. Under these conditions the kernel of the differential operator D ǫ=0 is generated by quasi-exponentials.
The proof is similar to the proof of Theorem 4.4.
10 Bethe ansatz in the Gaudin case 10.

Preliminaries
Consider C N as the gl N -module with highest weight (1, 0, . . . , 0). For complex numbers z 1 , . . . , z n , denote which is the tensor product of polynomial gl N [x]-modules. Let be its gl N -weight decomposition with respect to the Cartan subalgebra of diagonal matrices.
Here m = (m 1 , . . . , m N ). Assume that K = diag (K 1 , . . . , K N ) is a diagonal N × N -matrix with distinct coordinates and consider the universal differential operator associated with M(z) and K. Acting on M(z)-valued functions, the operator D K, M(z) (u, ∂ u ) preserves the weight decomposition.
In this section we shall study the kernel of this operator restricted to M(z)[m]-valued functions.

Weight function and Bethe ansatz theorem
We denote by ω(t, z) the universal weight function of the Gaudin type associated with the weight subspace M(z) [m]. The universal weight function of the Gaudin type is defined in [12]. A convenient formula for ω(t, z) is given in Appendix in [11] and Theorems 6.3 and 6.5 in [11].
Ift is a solution of the Bethe ansatz equations, then the vector ω(t, z) ∈ M(z)[m] is called the Bethe vector associated witht. The statement follows from Theorem 9.2 in [6]. For k = 1, the result is established in [10].
The eigenvalues of the Bethe vector are given by the following construction. Set for a = 2, . . . , N . Define the functions λ k (u, t, z) by the formula Then for k = 0, . . . , N ,

Differential operator associated with a solution of the Bethe ansatz equations
Lett be a solution of system (10.1). The scalar difference operator will be called the associated fundamental differential operator.
Theorem 10.2. The kernel of Dt(u, τ ) is generated by quasi-exponentials of degree bounded from above by a function in n and N .
This is Proposition 6.4 in [9], which is a generalization of Lemma 5.6 in [7].

Completeness of the Bethe ansatz
Lemma 10.5. For any admissible index a = (a 1 , . . . , a N ) and small nonzero q, there exists a solutiont(a, q) = (t for every i, j. Lemma 10.6. As q tends to zero, the Bethe vector ω(t(a, q), z) has the following asymptotics: ω(t(a, q), z) = C a q −l e a v + O q −l+1 , where C a is a nonzero number.
Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel 29 The lemma follows from formula (10.3) and the formula for the universal weight function in [11]. Lemma 10.6 implies Theorem 10.3.
Theorem 11.2. For any matrix K, an M Λ (z)-valued function g(u) belongs to the kernel of the differential operator D K,M Λ (z) (u, ∂ u ) if and only if the function C(u)g(u) belongs to the kernel of the differential operator D K,M Λ(a) (z) (u, ∂ u ), where The proof easily follows from part (ii) of Theorem 8.2. Then the scalar differential operator (−1) k λ k,v (u) ∂ N −k u will be called the fundamental differential operator associated with the eigenvector v. Theorems 12.1, 12.2 and 12.3 give us information on the kernel of the fundamental differential operator.