Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum
aa r X i v : . [ m a t h . QA ] M a y BETHE ALGEBRA OF HOMOGENEOUS
XXX
HEISENBERG MODELHAS SIMPLE SPECTRUM
E. MUKHIN ∗ , , V. TARASOV ∗ ,⋆, , AND A. VARCHENKO ∗∗ , ∗ Department of Mathematical Sciences, Indiana University – Purdue University,Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA ⋆ St. Petersburg Branch of Steklov Mathematical InstituteFontanka 27, St. Petersburg, 191023, Russia ∗∗ Department of Mathematics, University of North Carolina at Chapel Hill,Chapel Hill, NC 27599-3250, USA
Abstract.
We show that the algebra of commuting Hamiltonians of the homogeneous
XXX
Heisenberg model has simple spectrum on the subspace of singular vectors of thetensor product of two-dimensional gl -modules. As a byproduct we show that there existexactly (cid:0) nl (cid:1) − (cid:0) nl − (cid:1) two-dimensional vector subspaces V ⊂ C [ u ] with a basis f, g ∈ V such that deg f = l, deg g = n − l + 1 and f ( u ) g ( u − − f ( u − g ( u ) = ( u + 1) n . Introduction
XXX
Heisenberg model.
Consider the vector space ( C ) ⊗ n andthe linear operator H XXX = − n X j =1 ( σ ( j )1 σ ( j +1)1 + σ ( j )2 σ ( j +1)2 + σ ( j )3 σ ( j +1)3 ) , where σ ( k ) a = 1 ⊗ ( k − ⊗ σ a ⊗ ( n − k ) , σ ( n +1) a = σ (1) a , and σ , σ , σ are the Pauli matrices, σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . The operator H XXX is the Hamiltonian of the celebrated
XXX
Heisenberg model, alsocalled the homogeneous
XXX model, and the problem is to find eigenvalues and eigen-vectors of the Hamiltonian.This problem was first addressed in the pioneering work [Be] by H. Bethe, who lookedfor eigenvectors of H XXX in a certain special form. His method and its further extensions Supported in part by NSF grant DMS-0601005. Supported in part by RFFI grant 08-01-00638. Supported in part by NSF grant DMS-0555327.
E. MUKHIN, V. TARASOV, AND A. VARCHENKO are traditionally called the Bethe ansatz. The current literature on the
XXX model andits generalizations,
XXZ and
XYZ models, as well as their counterparts in statistical me-chanics, the six- and eight-vertex models, is enormous. We limit ourselves to mentioningjust two books, [B1] and [KBI]. However, even numerous references therein hardly covera half of the bibliography on the subject.The Hamiltonian H XXX can be included into a one -parameter family of commutinglinear operators called the transfer matrix, see [B1], [FT], [KBI]. We call a commutativeunital subalgebra of linear operators on ( C ) ⊗ n generated by the transfer matrix the Bethealgebra. The actual problem is to construct eigenvalues and eigenvectors for the Bethealgebra.The elements of the Bethe algebra commute with the natural gl -action on ( C ) ⊗ n .Therefore, the eigenspaces of the Bethe algebra are representations of gl , and it sufficesto construct highest weight vectors of those representations.The Bethe ansatz method associates to every admissible solution ( λ , . . . λ l ) of thesystem of equations (cid:18) λ j + i λ j − i (cid:19) n = l Y k =1 k = j λ j − λ k + iλ j − λ k − i , j = 1 , . . . , l , (1.1)a vector in ( C ) ⊗ n , called the corresponding Bethe vector, see [FT]. A solution ( λ , . . . , λ l )is called admissible if all λ , . . . , λ l are distinct, and all factors in (1.1) are nonzero. Anonzero Bethe vector is a highest weight vector of an ( n − l + 1)-dimensional irreduciblerepresentation of gl , and all vectors in that representation are eigenvectors of each elementof the Bethe algebra sharing the same eigenvalue.It is an important question whether the Bethe ansatz method produces all eigenvectorsof the Bethe algebra. This question is referred to as the question of completeness of theBethe ansatz for finite chains. It was discussed by H. Bethe himself in [Be] and manytimes since then by other authors. For instance, see a recent discussion in [B2]. However,no rigorous proof is available even for the so-called inhomogeneous models. Moreover,as one can see from the results of this paper, Sklyanin’s separation of variables does notprove completeness of the Bethe ansatz to the very end, though it is indeed an importantstep towards the proof.To be more precise, there are certain quantum integrable models for which the com-pleteness of the Bethe ansatz has been proved. For example, see [YY] and Theorem 1.2.2in [KBI]. The proofs for those models are based on a variational principle and convexityof some auxiliary action. However, for the the XXX model, the corresponding action isnot convex, and that technique fails.In this paper we establish the completeness of the Bethe ansatz method for the homo-geneous
XXX model provided the method is improved in a certain way, see below in theintroduction. We show that the spectrum of the Bethe algebra of the homogeneous
XXX model is simple, that is, all eigenspaces of the Bethe algebra are irreducible gl -modules. ETHE ALGEBRA OF HOMOGENEOUS
XXX
MODEL HAS SIMPLE SPECTRUM 3
We also show that eigenvalues of the Bethe algebra are in a one -to -one correspondencewith certain second-order linear difference equations with two linearly independent poly-nomial solutions. We prove similar results for inhomogeneous higher spin
XXX models.To continue with an introduction and match the notation in the main part of the paper,we change the variables in system (1.1), λ j = i t j + 1) , j = 1 , . . . , l , and write the system in the polynomial form( t j + 2) n l Y k =1 k = j ( t j − t k −
1) = ( t j + 1) n l Y k =1 k = j ( t j − t k + 1) , j = 1 , . . . , l . (1.2)We call system (1.2) the system of the Bethe ansatz equations. The system is invariantwith respect to permutations of t , . . . , t l , so the symmetric group S l acts on solutions tothe Bethe ansatz equations.We denote by ω ( t , . . . , t l ) the Bethe vector corresponding to an admissible solution( t , . . . , t l ) of the Bethe ansatz equations. The Bethe vectors corresponding to admissiblesolutions with permuted coordinates are equal. The number of Bethe vectors ω ( t , . . . , t l )is equal to the number of S l -orbits of admissible solutions to system (1.2).Since each element of the Bethe algebra commutes with the natural gl -action on ( C ) ⊗ n ,it is enough to diagonalize the action of the Bethe algebra on each subspace of gl -singularvectors of given weight,Sing ( C ) ⊗ n [ l ] = { v ∈ ( C ) ⊗ n | e v = 0 , e v = ( n − l ) v, e v = l v } , with 2 l n . For every admissible solution ( t , . . . , t l ) of the Bethe ansatz equations, theBethe vector ω ( t , . . . , t l ) belongs to the subspace Sing ( C ) ⊗ n [ l ] .To illustrate the problem with completeness of the Bethe ansatz in the standard formand the way it can be resolved, let us consider an example.Let n = 4 and l = 2. Then dim Sing ( C ) ⊗ [ 2 ] = 2 , the operator H XXX restricted toSing ( C ) ⊗ [ 2 ] has eigenvalues 5 and − t + 2) ( t − t −
1) = ( t + 1) ( t − t + 1) , (1.3)( t + 2) ( t − t −
1) = ( t + 1) ( t − t + 1) , and there is only one orbit of admissible solutions: t = −
32 + 12 r − , t = − − r − . (1.4)The Bethe vector ω ( t , t ) is an eigenvector of H XXX with eigenvalue 5.
E. MUKHIN, V. TARASOV, AND A. VARCHENKO
The results of this paper say that each eigenspace of H XXX acting on Sing ( C ) ⊗ [ 2 ]corresponds to a difference equation u f ( u ) − B ( u ) f ( u −
1) + ( u + 1) f ( u −
2) = 0 , (1.5)where B ( u ) is a polynomial, and the difference equation has polynomial solutions of degree2 and 3. The corresponding eigenvalue of H XXX equals 1 − B ′ (0) / B (0) .Indeed, there are exactly two such difference equations. The first one has B ( u ) =2 u + 4 u − u + 1, and solutions u + 3 u + and u + 6 u + 11 u + . The roots of thequadratic polynomial are numbers t and t given by (1.4).The second difference equation (1.5) with polynomial solutions of degree 2 and 3 has B ( u ) = 2 u + 4 u − u −
1, and solutions ( u + 1) ( u + 2) and u + 6 u + 10 u + . Theroots of the quadratic polynomial, t = − t = −
2, form a nonadmissible solutionto system (1.3) , and the Bethe vector ω ( t , t ) for t = − t = − n and l such that 2 l n , the results of this paper for the homogeneous XXX model say that eigenspaces of the Bethe algebra acting on Sing ( C ) ⊗ n [ l ] are one-dimensional. They are in a one -to -one correspondence with difference equations u n f ( u ) − B ( u ) f ( u −
1) + ( u + 1) n f ( u −
2) = 0 , (1.6)where B ( u ) is a polynomial, and those difference equations have polynomial solutions ofdegree l and n − l + 1. The corresponding eigenvalues of elements of the Bethe algebraare described by the polynomial B ( u ). In particular, the eigenvalue of H XXX equals1 − B ′ (0) / B (0) . The roots t , . . . t l of the polynomial solution of equation (1.6) of degree l form a solution of system (1.2). The Bethe vector ω ( t , . . . t l ) is nonzero if and only ifthe solution ( t , . . . t l ) is admissible.To obtain an eigenvector of the Bethe algebra corresponding to a difference equa-tion (1.6) with two polynomial solutions, we use the following construction. The space( C ) ⊗ n has a structure of a module over the Yangian Y ( gl ), and the Bethe algebra ofthe homogeneous XXX model is the image of a commutative subalgebra B ⊂ Y ( gl ),called the Bethe subalgebra. We take another Y ( gl )-module W a,d , described in Sec-tion 2.5, which is the holomorphic representation of Y ( gl ) associated with the polyno-mials a ( u ) = ( u + 1) n and d ( u ) = u n . There is a natural epimorphism σ : W a,d → ( C ) ⊗ n of Y ( gl )-modules.Using the roots t , . . . , t l of the polynomial solution of equation (1.6) of degree l and Sklyanin’s procedure of separation of variables [Sk], we define a nonzero vector˜ ω ( t , . . . , t l ) in W a,d , which is an eigenvector of B acting on W a,d . We consider themaximal B -invariant subspace V ⊂ W a,d that contains ˜ ω ( t , . . . , t l ) and does not containother linearly independent eigenvectors of B . We show that the image σ ( V ) ⊂ ( C ) ⊗ n is a one-dimensional subspace of Sing ( C ) ⊗ n [ l ]. Since σ is an homomorphism of Y ( gl )-modules, σ ( V ) is an eigenspace of the Bethe algebra acting on Sing ( C ) ⊗ n [ l ] with thesame eigenvalues as the eigenvalues of ˜ ω ( t , . . . , t l ) with respect to the action of B on ETHE ALGEBRA OF HOMOGENEOUS
XXX
MODEL HAS SIMPLE SPECTRUM 5 W a,d . The subspace σ ( V ) ⊂ Sing ( C ) ⊗ n [ l ] is that one-dimensional subspace of eigenvec-tors which we assigned to difference equation (1.6) with two polynomial solutions.If ( t , . . . , t l ) is an admissible solution, then the subspace V ⊂ W a,d is one-dimensional,and the subspace σ ( V ) is spanned by the Bethe vector ω ( t , . . . , t l ).The construction described above provides a generalization of the Bethe ansatz methodin which the solutions to the Bethe ansatz equations are replaced by difference equa-tion (1.6) with two polynomial solutions, and the Bethe vectors in Sing ( C ) ⊗ n [ l ] arereplaced by the subspaces σ ( V ). Our result says that the generalized Bethe vectors forma basis in Sing ( C ) ⊗ n [ l ] and, moreover, the spectrum of the Bethe algebra is simple.As a remark, we would like to indicate another way to obtain the eigenspace of theBethe algebra acting on Sing ( C ) ⊗ n [ l ] corresponding to the difference equation (1.6). Wemay consider the inhomogeneous XXX model depending on parameters z , . . . , z n . Thecorresponding system of the Bethe ansatz equations are n Y s =1 ( t j − z s + 2) l Y k =1 k = j ( t j − t k −
1) = n Y s =1 ( t j − z s + 1) l Y k =1 k = j ( t j − t k + 1) , (1.7) j = 1 , . . . , l . It follows from the results of this paper that if f ( u ) = Q lj =1 ( u − t j ) is asolution of the difference equation (1.6), then for generic z = ( z , . . . , z n ) there exists anadmissible solution t ( z ) = ( t ( z ) , . . . , t l ( z )) of system (1.7) such that t ( z ) → ( t , . . . t l )as z →
0. The Bethe vectors ω ( t ( z ); z ) are nonzero for generic z , and the eigenspaces C ω ( t ( z ); z ) have a one-dimensional limit as z →
0, which is the eigenspace of the Bethealgebra of the homogeneous
XXX model. A similar approach for the gl N Gaudin modelis developed in [MTV5].The correspondence between the eigenvectors of the Bethe algebra and second-orderlinear difference equation with two polynomial solutions is in the spirit of the geometricLanglands correspondence in which eigenfunctions of commuting differential operatorscorrespond to connections on curves.Equation (1.6) is known in the physical literature as Baxter’s equation. Its connectionwith the Bethe ansatz equations has been studied in many papers. The fact that the rootsof a polynomial solution of Baxter’s equation give a solution of the Bethe ansatz equations(provided the roots are distinct) is known as Manakov’s principle and the analytic Betheansatz. An important observation about the existence of a second polynomial solution ofBaxter’s equation has been done in [PS]. A similar observation in a much more generalcontext has been made independently in [MV2], [MV3].
The results of this paper for the
XXX model are discreteanalogues of the results of [MTV3] for the Gaudin model.In Section 2 we discuss the Yangian Y ( gl ), the Bethe subalgebra B ⊂ Y ( gl ), andYangian modules. In particular, we describe the holomorphic representation W a,d of the E. MUKHIN, V. TARASOV, AND A. VARCHENKO
Yangian Y ( gl ). The module W a,d is associated with two monic polynomials a ( u ) = n Y i =1 ( u − z i + m i ) and d ( u ) = n Y i =1 ( u − z i )and is isomorphic to C [ x , . . . , x n ] as a vector space.We introduce a collection (cid:0) ( m , , . . . , ( m n , (cid:1) of gl -weights and say that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating if P ni =1 m i − l + 1 + s = 0 for all s = 1 , . . . , l .In Sections 3 – 7, we study the algebras A W and A D , and relations between them.Eventually, we show that the algebras A W and A D are isomorphic, see Theorem 7.3.1.The algebra A W is the image of the Bethe subalgebra B acting on the subspaceSing W a,d [ l ] ⊂ W a,d of gl -singular vectors. We consider a polynomial B ( u, H ) = 2 u n + H u n − + · · · + H n , whose coefficients H k ∈ End (Sing W a,d [ l ]) are generators of A W , andintroduce the universal difference operator D Sing W a,d [ l ] = d ( u ) − B ( u, H ) ϑ − + a ( u ) ϑ − acting on Sing W a,d [ l ]-valued functions in u . Here ϑ : f ( u ) f ( u + 1).The algebra A D is defined in Section 4. We consider the space C l + n with coordinates a = ( a , . . . , a l ) and h = ( h , . . . , h n ), polynomials B ( u, h ) = 2 u n + h u n − + · · · + h n and p ( u, a ) = u l + a u l − + · · · + a l , and the difference operator D h = d ( u ) − B ( u, h ) ϑ − + a ( u ) ϑ − We define the scheme C D of points p ∈ C l + n such that the polynomial p ( u, a ( p )) lies inthe kernel of the difference operator D h ( p ) . The algebra A D is the algebra of functions on C D . There is a natural epimorphism ψ DW : A D → A W such that ψ DW ( h k ) = H k , seeTheorem 4.3.3.Using the Bethe ansatz method, we prove that if z , . . . , z n are generic and the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating, then the scheme C D considered as a set has atleast dim Sing W a,d [ l ] distinct points, see Section 5.In Section 6, we review Sklyanin’s procedure of separation of variables in the XXX model and construct the universal weight function. Theorem 6.3.2 connects the algebras A D , A W and the universal weight function.The algebra A D acts on itself by multiplication operators. We denote by L f the operatorof multiplication by an element f ∈ A D . The algebra A D acts on its dual space A ∗ D by op-erators L ∗ f , dual to multiplication operators. Using the universal weight function we definea linear map τ : A ∗ D → Sing W a,d [ l ] and prove that if the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l isseparating, then τ is an isomorphism that intertwines the action of operators L ∗ f , f ∈ A D ,with the action of operators ψ DW ( f ) ∈ End(Sing W a,d [ l ]), see Theorem 7.3.1. Therefore,we prove that ψ DW : A D → A W is an algebra isomorphism. Theorem 7.3.1 is our firstmain result. ETHE ALGEBRA OF HOMOGENEOUS
XXX
MODEL HAS SIMPLE SPECTRUM 7
Using the Grothendieck residue, we define an isomorphism φ : A D → A ∗ D of A D -modules, see Section 7.4. Therefore, if the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating, thecomposition τ φ : A D → Sing W a,d [ l ] is a linear isomorphism which intertwines the actionof the algebra A D on itself by multiplication operators and the action of the Bethe algebra A W on Sing W a,d [ l ].In Sections 8 through 11, we impose more conditions on m , . . . , m n and z , . . . , z n .We assume that m , . . . , m n are natural numbers. We keep the assumption that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating, that takes the form 2 l P ns =1 m s . We also assumethat z i − z j / ∈ Z if i = j .In Sections 8 – 11, we study three more algebras A G , A P and A L , and relations betweenthem. The algebra A G is defined in Section 8. We consider the subspace C d [ u ] ⊂ C [ u ] ofall polynomials of degree d for a suitably large number d , and the Grassmannian ofall two-dimensional subspaces of C d [ u ]. Using the numbers z , . . . , z n and m , . . . , m n wedefine n + 1 Schubert cycles C F ( z ) , Λ (1) , . . . , C F ( z n ) , Λ ( n ) , C F ( ∞ ) , Λ ( ∞ ) in the Grassmannian.The algebra A G is the algebra of functions on the intersection of the Schubert cycles.The algebra A P is defined in Section 9.1. Let ˜ l = P ns =1 m s + 1 − l , ˜ a = (˜ a , . . . , ˜ a ˜ l − l − , ˜ a ˜ l − l +1 , . . . , ˜ a ˜ l ) , ˜ p ( u, ˜ a ) = u ˜ l + ˜ a u ˜ l − + · · · + ˜ a ˜ l − l − u l +1 + ˜ a ˜ l − l +1 u l − + · · · + ˜ a ˜ l , andconsider the space C ˜ l + l + n − with coordinates ˜ a , a , h . We define the scheme C P as thescheme of points p ∈ C ˜ l + l + n − such that the polynomials p ( u, a ( p )) and ˜ p ( u, ˜ a ( p )) lie inthe kernel of the difference operator D h ( p ) . The algebra A P is the algebra of functionson C P . The map ( p ( u, a ( p )) , ˜ p ( u, ˜ a ( p )) , D h ( p ) ) ( p ( u, a ( p )) , D h ( p ) ) defines a naturalepimorphism ψ DP : A D → A P . We also show that the algebras A G and A P are naturallyisomorphic.To define the algebra A L , see Section 9.3, we consider the tensor product L Λ ( z ) = L Λ (1) ( z ) ⊗ · · · ⊗ L Λ ( n ) ( z n )of evaluation Yangian modules, where L Λ ( i ) is the irreducible gl -module of highest weightΛ ( i ) = ( m i ,
0) . The algebra A L is the image of the Bethe subalgebra B ⊂ Y ( gl ) actingon the subspace Sing L Λ [ l ] ⊂ L Λ ( z ) of gl -singular vectors. The Y ( gl )-module L Λ ( z ) isisomorphic to the quotient module W a,d /K , where K ⊂ W a,d is the kernel of the YangianShapovalov form on W a,d . We denote by σ : Sing W a,d [ l ] → Sing L Λ [ l ] the epimorphismof vector spaces corresponding to the epimorphism W a,d → L Λ ( z ) of Y ( gl )-modules. Theepimorphism σ induces the algebra epimorphism ψ W L : A W → A L .We denote by ξ : A D → Sing L Λ [ l ] the composition of maps στ φ , and by ψ DL : A D → A L the composition of maps ψ W L ψ DW . We show that the kernels of the maps ξ , ψ DL and ψ DP coincide. This allows us to obtain an algebra isomorphism ψ P L : A P → A L and a linear isomorphism ζ : A P → Sing L Λ [ l ] intertwining the action of A P on itself bymultiplication operators and the action of the Bethe algebra A L on Sing L Λ [ l ]. This isour second main result, see Theorem 10.3.1. E. MUKHIN, V. TARASOV, AND A. VARCHENKO
In Section 11, we use the Yangian Shapovalov form on L Λ ( z ) and the map ζ to obtain alinear isomorphism θ : A ∗ P → Sing L Λ [ l ] intertwining the action of operators L ∗ f , f ∈ A P ,with the action of operators ψ P L ( f ) ∈ End(Sing L Λ [ l ]), see Theorem 11.2.1. Using theisomorphism, we show that eigenvectors of the action of the algebra A L on Sing L Λ [ l ] arein a one -to -one correspondence with certain second-order linear difference equations withtwo polynomial solutions of degrees l and n − l + 1, see Corollary 11.2.3.Section 12 contains the analogues of the previous results for the homogeneous XXX
Heisenberg model.We recapitulate the main results of this paper as three commutative diagrams. Thehorizontal arrows of the diagrams are isomorphisms, the downward vertical arrows areepimorphisms, and the upward vertical arrow is an embedding.The first diagram shows the algebras of functions A D , A P on difference operators withrespectively one or two polynomials in the kernels, the algebra A G of functions on the in-tersection of Schubert cycles, the Bethe algebras A W and A L , associated with Sing W a,d [ l ]and Sing L Λ [ l ], respectively, and their homomorphisms: A D ψ DW −−−→ A Wψ DP y y ψ WL A G −−−→ ψ GP A P −−−→ ψ PL A L The other two diagram show the vector spaces involved: A ∗ D τ −−−→
Sing W a,d [ l ] ( ψ DP ) ∗ x y σ A ∗ P −−−→ θ Sing L Λ [ l ] A D τφ −−−→
Sing W a,d [ l ] ψ DP y y σ A P −−−→ ζ Sing L Λ [ l ]Each vector space on these diagrams is a module over the corresponding algebra on thefirst diagram, and all linear maps are consistent with the algebra homomorphisms. Acknowledgments.
The authors thank referees for helpful comments.2.
Yangian Y ( gl ) and Yangian modules gl . Let e ab , a, b = 1 ,
2, be the standard generators of the complex Liealgebra gl . We have gl = n + ⊕ h ⊕ n − where n + = C · e , h = C · e ⊕ C · e , n − = C · e . For a gl -weight Λ ∈ h ∗ , we denote by M Λ the Verma gl -module with highest weightΛ and by L Λ the irreducible gl -module with highest weight Λ. ETHE ALGEBRA OF HOMOGENEOUS
XXX
MODEL HAS SIMPLE SPECTRUM 9
Let Λ = (Λ (1) , . . . , Λ ( n ) ) be a collection of gl -weights, where Λ ( i ) = (Λ ( i )1 , Λ ( i )2 )for i = 1 , . . . , n . Let l be a nonnegative integer. The pair Λ , l will be called separating if P ni =1 (Λ ( i )1 − Λ ( i )2 ) − l + 1 + s = 0 for all s = 1 , . . . , l , cf. [MV1], [MV2], [MTV3].In the following, we need the next lemma. Let m be a complex number and l a nonnegative integer. Let V bea gl -module with weight decomposition V = L ∞ k =0 V [ k ] , where V [ k ] ⊂ V is a weightsubspace of weight ( m − k, k ) . Assume that m − l + 1 + s = 0 for all s = 1 , . . . , l . Thenthe map e e : V [ l − → V [ l − is an isomorphism of vector spaces.Proof. Let U k = ker (cid:0) e l − k | V [ l − (cid:1) . Clearly, V [ l −
1] = U ⊃ U · · · ⊃ U l − ⊃ U l = { } .Let C = e ( e + 1) − e e . Set P ( x ) = Q l − k =0 ( x − c k ), where c k = k ( m − k + 1),and Q ( x ) = P ( x ) − P ( c l ) x − c l . We have e e | V [ l − = ( c l − C ) | V [ l − . Since C is a central element, we have ( C − c k ) U k ⊂ U k +1 . Therefore, P ( C ) | V [ l − = 0,and ( c l − C ) | V [ l − Q ( C ) | V [ l − = P ( c l ) . The assumption on m and l implies that P ( c l ) = 0. Hence, the operator ( c l − C ) | V [ l − is invertible. (cid:3) The Yangian Y ( gl ) is the unital associative algebra with generators T { s } ab , a, b = 1 , s = 1 , , . . . . Let T ab ( u ) = δ ab + ∞ X s =1 T { s } ab u − s , a, b = 1 , . Then the defining relations in Y ( gl ) have the form( u − v ) (cid:0) T ab ( u ) T cd ( v ) − T cd ( v ) T ab ( u ) (cid:1) = T cb ( v ) T ad ( u ) − T cb ( u ) T ad ( v ) , (2.1)for all a, b, c, d . The Yangian is a Hopf algebra with coproduct∆ : T ab ( u ) X c =1 T cb ( u ) ⊗ T ac ( u ) (2.2)for all a, b . [KBI]. The following relations hold : T ( u ) T ( u ) . . . T ( u k ) = k Y i =1 u − u i − u − u i T ( u ) . . . T ( u k ) T ( u ) ++ 1( k − T ( u ) X σ ∈ S k (cid:18) u − u σ k Y i =2 u σ − u σ i − u σ − u σ i T ( u σ ) . . . T ( u σ k ) T ( u σ ) (cid:19) ,T ( u ) T ( u ) . . . T ( u k ) = k Y i =1 u − u i + 1 u − u i T ( u ) . . . T ( u k ) T ( u ) ++ 1( k − T ( u ) X σ ∈ S k (cid:18) u − u σ k Y i =2 u σ − u σ i + 1 u σ − u σ i T ( u σ ) . . . T ( u σ k ) T ( u σ ) (cid:19) . A series f ( u ) in u − is called monic if f ( u ) = 1 + O ( u − ). For a monic series f ( u ), there is an automorphism ϕ f : Y ( gl ) → Y ( gl ) , T ab ( u ) f ( u ) T ab ( u ) . There is a one-parameter family of automorphisms ρ z : Y ( gl ) → Y ( gl ) T ab ( u ) T ab ( u − z ) , where in the right-hand side, ( u − z ) − has to be expanded as a power series in u − .The Yangian Y ( gl ) contains the universal enveloping algebra U ( gl ) as a Hopf subal-gebra. The embedding is given by the formula e ab T { } ba for all a, b . We identify U ( gl )with its image.The evaluation homomorphism ǫ : Y ( gl ) → U ( gl ) is defined by the rule: T { } ab e ba for all a, b , and T { s } ab a, b and all s > + : Y ( gl ) → Y ( gl ) the antiinvolution defined by (cid:0) T ab ( u ) (cid:1) + = T ba ( u ) . (2.3) The seriesqdet T ( u ) = T ( u ) T ( u − − T ( u ) T ( u −
1) (2.4)is called the quantum determinant . The coefficients of the series qdet T ( u ) belong to thecenter of the Yangian Y ( gl ) [IK].The series T ( u ) + T ( u ) is called the transfer matrix . It is known that the coefficientsof the series T ( u ) + T ( u ) commute [FT].We call the unital subalgebra B ⊂ Y ( gl ) generated by coefficients of the seriesqdet T ( u ) and T ( u ) + T ( u ) the Bethe subalgebra . The Bethe subalgebra is commuta-tive. Elements of the Bethe subalgebra commute with elements of the subalgebra U ( gl )and are invariant under the antiinvolution (2.3). ETHE ALGEBRA OF HOMOGENEOUS
XXX
MODEL HAS SIMPLE SPECTRUM 11 [T].
Let V be an irreducible finite-dimensional Y ( gl ) -module. Thereexists a unique up to proportionality vector v ∈ V , monic series c ( u ) , c ( u ) , and a monicpolynomial P ( u ) such that T ( u ) v = 0 ,T aa ( u ) v = c a ( u ) v , a = 1 , , and c ( u ) c ( u ) = P ( u + 1) P ( u ) . (2.5)The vector v is called a highest weight vector , the series c ( u ) , c ( u ) — the Yangianhighest weights , and the polynomial P ( u ) — the Drinfeld polynomial of the module V . [T]. For any monic series c ( u ) , c ( u ) and a monic polynomial P ( u ) obeying relation (2.5) , there exists a unique irreducible finite-dimensional Y ( gl ) -module V such that c ( u ) , c ( u ) are the Yangian highest weights of the module V . Let V , V be irreducible finite-dimensional Y ( gl )-modules with respective high-est weight vectors v , v . Then for the Y ( gl )-module V ⊗ V , we have T ( u ) v ⊗ v = 0 ,T aa ( u ) v ⊗ v = c (1) a ( u ) c (2) a ( u ) v ⊗ v , a = 1 , . Let W be the irreducible subquotient of V ⊗ V generated by the vector v ⊗ v . Thenthe Drinfeld polynomial of the module W equals the products of the Drinfeld polynomialsof the modules V and V . For a gl -module V , let the Y ( gl )-module V ( z ) be the pullback of V throughthe homomorphism ǫ ◦ ρ z ; that is, the series T ab ( u ) acts on V ( z ) as 1 + ( u − z ) − e ba . Themodule V ( z ) is called the evaluation module with evaluation point z . Let Λ = (Λ (1) , . . . , Λ ( n ) ) be a collection of integral dominant gl -weights, whereΛ ( i ) = (Λ ( i )1 , Λ ( i )2 ), Λ ( i )1 > Λ ( i )2 , for i = 1 , . . . , n . For generic complex numbers z , . . . , z n ,the tensor product of evaluation modules L Λ ( z ) = L Λ (1) ( z ) ⊗ · · · ⊗ L Λ ( n ) ( z n )is an irreducible finite-dimensional Y ( gl )-module and the corresponding highest weightseries c ( u ) , c ( u ) have the form c a ( u ) = n Y i =1 u − z i + Λ ( i ) a u − z i . (2.6) The corresponding Drinfeld polynomial equals P ( u ) = n Y i =1 Λ ( i )1 − Y s =Λ ( i )2 ( u − z i + s ) . The results of this section go back to [T].Choose monic polynomials a ( u ) , d ( u ) ∈ C [ u ] of positive degree n , a ( u ) = n Y i =1 ( u − z i + m i ) , d ( u ) = n Y i =1 ( u − z i ) . (2.7) There exists a unique Y ( gl ) -action on the vector space C [ x , . . . , x n ] such that (cid:0) T ( u ) · p (cid:1) ( x ) = 1 d ( u ) n X i =1 u n − i x i p ( x , . . . , x n ) (2.8)= (cid:18) x u + x − x P ni =1 z i u + . . . (cid:19) p ( x , . . . , x n ) for any polynomial p ∈ C [ x , . . . , x n ] , and T ( u ) · a ( u ) d ( u ) · , T ( u ) · , T ( u ) · , (2.9) where stands for the constant polynomial equal to as an element of C [ x , . . . , x n ] . We denote by W a,d the Y ( gl )-module defined by formulae (2.8), (2.9) and call it theholomorphic representation of Y ( gl ), associated with the polynomials a ( u ) , d ( u ).The Yangian module W a,d is cyclic: every element of W a,d can be obtained from 1 bythe action of a suitable polynomial in T { } , T { } , . . . . Formulae (2.9) mean that 1 isan eigenvector of the operators T { s } , T { s } and 1 is annihilated by the operators T { s } with s = 1 , , . . . . Then the Yangian commutation relations (2.1) allow us to determine theaction of T { s } , T { s } , T { s } on all elements of W a,d .Since the coefficients of the series qdet T ( u ) are central, and the module W a,d is gener-ated by the polynomial 1, we haveqdet T ( u ) (cid:12)(cid:12) W a,d = a ( u ) d ( u ) . (2.10)For every i, j = 1 ,
2, we have T ij ( u ) (cid:12)(cid:12) W a,d = ˜ T ij ( u ) d ( u ) , (2.11)where ˜ T ij ( u ) is an End ( W a,d )-valued polynomial in u of degree n for i = j , and of degree n − i = j . ETHE ALGEBRA OF HOMOGENEOUS
XXX
MODEL HAS SIMPLE SPECTRUM 13
The embedding U ( gl ) ֒ → Y ( gl ) defines a gl -module structure on W a,d . The gl -weight decomposition of W a,d is the degree decomposition W a,d = ⊕ ∞ l =0 W a,d [ l ] intosubspaces of homogeneous polynomials. The subspace W a,d [ l ] of homogeneous polynomi-als of degree l has gl -weight (cid:0) P ni =1 m i − l, l (cid:1) . Let
Sing W a,d [ l ] = { p ∈ W a,d [ l ] | e p = 0 } be the subspace of gl -sinular vectors. Assume that the pair (( m , , . . . , ( m n , , l is separating. Then dim Sing W a,d [ l ] = dim W a,d [ l ] − dim W a,d [ l − . Proof.
The map e e : W a,d [ l − → W a,d [ l − m , , . . . , ( m n , l is separating, see Lemma 2.1.2. The fact that e e is an isomorphism implies the lemma. (cid:3) Denote by + : Y ( gl ) → Y ( gl ) the antiinvolution defined by T + ij ( u ) = T ji ( u ).Denote by φ : W a,d → C the linear function p ( x , . . . , x n ) p (0 , . . . , The YangianShapovalov form on W a,d is the unique symmetric bilinear form S on W a,d defined by theformula S ( x · , y ·
1) = φ ( x + y ·
1) for all x, y ∈ Y ( gl ).Different gl -weight subspaces of W a,d are orthogonal with respect to the form S , anddet S | W a,d [ l ] = const n Y i,j =1 l − Y s =0 ( z i − z j + m j − s )( n + l − s − n − ) , where the constant does not depend on z , . . . , z n , m , . . . , m n . The kernel of the Yangian Shapovalov form K ⊂ W a,d is a Y ( gl )-submodule. The Y ( gl )-module W a,d /K is irreducible. The Yangian Shapovalov form on W a,d induces anondegenerate symmetric bilinear form on W a,d /K called the Yangian Shapovalov form of the module W a,d /K . [T]. For generic z , . . . , z n , m , . . . , m n , the Y ( gl ) -module W a,d is ir-reducible and isomorphic to the tensor product of evaluation Verma modules M ( m , ( z ) ⊗· · · ⊗ M ( m n , ( z n ) . Any such an isomorphism sends to a scalar multiple of the tensorproduct v ( m , ⊗· · ·⊗ v ( m n , of highest weight vectors of the corresponding Verma modules. [T]. Let m i ∈ Z > for i = 1 , . . . , n , and m m · · · m n . As-sume that z i − z j + m j − s = 0 and z i − z j − − s = 0 for all i < j and s = 0 , , . . . , m i − .Then for any permutation σ ∈ S n , the irreducible Y ( gl ) -module W a,d /K is isomorphicto the tensor product of evaluation irreducible modules L ( m σ , ( z σ ) ⊗ · · · ⊗ L ( m σn , ( z σ n ) .Any such an isomorphism sends the element corresponding to to a scalar multiple ofthe tensor product v ( m σ , ⊗ · · · ⊗ v ( m σn , of highest weight vectors of the correspondingirreducible modules. For a proof of this theorem see also [CP].
The assumption of Theorem 2.6.2 can be formulated geometrically as the assump-tion that for i < j the sets Z i = { z i , z i − , . . . , z i − m i } and Z j = { z j , z j − , . . . ,z j − m j } either do not intersect, or the smaller set Z i is a subset of the larger set Z j (since we assumed that m i m j ).3. Algebra A W and universal difference operator Let V be a Y ( gl )-module. We call the image of the Bethe algebra B ⊂ Y ( gl ) in End ( V ) the Bethe algebra associated with V . If U ⊂ V is a vectorsubspace preserved by elements of the Bethe algebra B V , then their restrictions to U define a commutative unital subalgebra B U ⊂ End ( U ) called the Bethe algebra associatedwith U . Define the operator ϑ acting on functions of u as ( ϑf )( u ) = f ( u + 1).Let V be a Y ( gl )-module such that for all a, b the series T ab ( u ) | V sum up to End ( V )-valued rational functions in u . Let U ⊂ V be a vector subspace preserved by the Bethealgebra B V . The universal difference operator D U acting on U -valued functions in u isdefined by the formula D U = 1 − (cid:0) T ( u ) + T ( u ) (cid:1)(cid:12)(cid:12) U ϑ − + qdet T ( u ) (cid:12)(cid:12) U ϑ − , see [Tal], [MTV1, (4.16) ], [MTV2]. The operator D U is a linear second-order differenceoperator. A W . Operator D Sing W a,d [ l ] Consider the Bethe algebra B W a,d associatedwith the Y ( gl )-module W a,d . Recall that (cid:0) qdet T ( u ) (cid:1)(cid:12)(cid:12) W a,d = a ( u ) d ( u ) , see (2.10), and (cid:0) T ( u ) + T ( u ) (cid:1)(cid:12)(cid:12) W a,d = B ( u, ˜ H ) d ( u )where B ( u, ˜ H ) = ˜ H u n + ˜ H u n − + · · · + ˜ H n (3.1)for suitable coefficients ˜ H k ∈ End (cid:0) W a,d (cid:1) , see 2.11. It follows from Proposition (2.2.1)that the coefficients ˜ H , ˜ H are scalar operators, ˜ H = 2 , ˜ H = P ni =1 ( m i − z i ).The elements ˜ H k are called the XXX
Hamiltonians associated with W a,d . The Hamiltonians ˜ H k preserve the subspace Sing W a,d [ l ] defined in Section 2.5.3.Set H k = ˜ H k | Sing W a,d [ l ] ∈ End (Sing W a,d [ l ])and B ( u, H ) = H u n + H u n − + · · · + H n . ETHE ALGEBRA OF HOMOGENEOUS
XXX
MODEL HAS SIMPLE SPECTRUM 15
The coefficients H , H , H are scalar operators, H = 2 , H = n X i =1 ( m i − z i ) ,H = l (cid:0) l − − n X i =1 m i (cid:1) + X i Hamiltonians associated with Sing W a,d [ l ]. The operators of the algebra A W are symmetric with respect to the YangianShapovalov form on W a,d , S ( f v, w ) = S ( v, f w )for all f ∈ A W and v, w ∈ W a,d , see [MTV1]. D Sing W a,d [ l ] . Consider the universal difference operator D Sing W a,d [ l ] actingon Sing W a,d [ l ]-valued functions, D Sing W a,d [ l ] = 1 − B ( u, H ) d ( u ) ϑ − + a ( u ) d ( u ) ϑ − , The modified universal difference operator D Sing W a,d [ l ] is defined by the formula D Sing W a,d [ l ] = d ( u ) D Sing W a,d [ l ] . Then D Sing W a,d [ l ] = d ( u ) − B ( u, H ) ϑ − + a ( u ) ϑ − . Assume that the pair (( m , , . . . , ( m n , , l is separating. Then forany v ∈ Sing W a,d [ l ] there exist unique v , . . . , v l ∈ Sing W a,d [ l ] such that the function w ( u ) = v u l + v u l − + . . . + v l is a solution of the difference equation D Sing W a,d [ l ] w ( u ) = 0 .Proof. By Lemma 2.5.3 the dimension of Sing W a,d [ l ] does not depend on z , . . . , z n , m , . . . , m n , if the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Because of that, we mayconsider the difference equation D Sing W a,d [ l ] v ( u ) = 0 as a difference equation on a fixedvector space with coefficients of the difference equation algebraically depending on pa-rameters z , . . . , z n , m , . . . , m n .Given a vector v ∈ Sing W a,d [ l ], we look for a solution of the difference equation D Sing W a,d [ l ] v ( u ) = 0 in the form v u l + P ∞ j =1 v j u l − j . Substituting this expression intothe equation, we can calculate all of the coefficients v j recursively, and they are algebraicfunctions of z , . . . , z n , m , . . . , m n . For generic z , . . . , z n and large positive integral m , . . . , m n , the coefficients v j areequal to zero for all j > l by Theorem 2.6.2 and [MTV3, Theorem 7.3]. Hence, thesame coefficients are equal to zero for all z , . . . , z n , m , . . . , m n such that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. (cid:3) Algebra A D From now on until the end of Section 11 we fix complex numbers z , . . . , z n , m , . . . , m n , and a nonnegative integer l . We always assume that the polyno-mials a ( u ) and d ( u ) are given by formulae (2.7).Let a = ( a , . . . , a l ) and h = ( h , . . . , h n ). Consider the space C l + n with coordinates a , h . Let D be the affine subspace of C l + n defined by equations q ( h ) = 0 , q ( h ) = 0,where q ( h ) = h − n X i =1 ( m i − z i ) ,q ( h ) = h − l ( l − − n X i =1 m i ) − X i XXX MODEL HAS SIMPLE SPECTRUM 17 Proof. It suffices to prove two facts:(i) For any z , there are no algebraic curves over C lying in C D ( z ).(ii) Let a sequence z ( i ) , i = 1 , , . . . , tend to a finite limit z = ( z , . . . , z n ). Let p ( i ) ∈ C D ( z ( i ) ) , i = 1 , , . . . , be a sequence of points. Then all coordinates (cid:0) a ( p ( i ) ) , h ( p ( i ) (cid:1) remain bounded as i tends to infinity.By fact (i), the dimension of A D ( z ) is finite for any z , whereas fact (ii) implies thatdim A D ( z ) does not depend on z , . . . , z n .For a point p in C D ( z ), the operator D h ( p ) has the form d ( u ) − (2 u n + h ( p ) u n − + h ( p ) u n − + h ( p ) u n − + · · · + h n ( p )) ϑ − + a ( u ) ϑ − , where the coefficients h ( p ) , h ( p ) are determined by the equations q ( h ) = 0 and q ( h ) = 0.Assume that (i) is not true. Since any affine algebraic curve over C is unbounded, thereexists a sequence of points p ( i ) ∈ C D ( z ), i = 1 , , . . . , which tends to infinity as i tends toinfinity. Then it is easy to see that h ( p ( i ) ) cannot tend to infinity since it would contradictthe fact that D h ( p ( i ) ) (cid:0) p ( u, a ( p ( i ) )) (cid:1) = 0. Choosing a subsequence, we may assume that h ( p ( i ) ) has a finite limit as i tends to infinity. Then a ( p ( i ) ) cannot tend to infinity sinceit would mean that the limiting difference equation has a polynomial solution of degreeless than l , and this is impossible.This reasoning implies that p ( i ) ∈ C D ( z ) cannot tend to infinity. Thus we get a con-tradiction and statement (i) is proved.The proof of statement (ii) is similar. (cid:3) A D and epimorphism ψ DW : A D → A W .4.3.1. Theorem. Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Assumethat h satisfies equations q ( h ) = 0 and q ( h ) = 0 . Consider the system q i ( a , h ) = 0 , i = 3 , . . . , l + 2 , (4.2) as a system of linear equations with respect to a , . . . , a l . Then this system has a uniquesolution a i = a i ( h ) , i = 1 , . . . , l , where a i ( h ) are polynomials in h . (cid:3) Proof. The claim follows from the fact that q i ( a , h ) = i (cid:16) n X s =1 m s − l + i + 1 (cid:17) a i + i − X j =1 q ij ( h ) a j for i = 1 , . . . , l . Here q ij are some linear functions of h . The coefficient of a i does notvanish since the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. (cid:3) Denote by I ′ D be the ideal in C [ h ] generated by the polynomials q ( h ) , q ( h ) , q j ( a ( h ) , h ) , j = l + 3 , . . . , l + n . The ideal I ′ D defines a scheme C ′ D in the space C n withcoordinates h = ( h , . . . , h n ). The scheme C ′ D is the scheme of points r ∈ C n such thatthe difference equation D h ( r ) w ( u ) = 0 has a polynomial solution of degree l .Theorem 4.3.1 implies that A D ∼ = C [ h ] /I ′ D . (4.3)Let H , . . . , H n be the operators introduced in Section 3.2.1. Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Thenthe assignment h s H s , s = 1 , . . . , n , determines an algebra epimorphism ψ DW : A D → A W .Proof. We use description (4.3) of the algebra A D . The equations defining the scheme C ′ D are the equations of existence of a polynomial solution of degree l to the polynomialdifference equation D h w ( u ) = 0. The operators H , . . . , H n satisfy the defining equationsfor C ′ D by Theorem 3.3.1. (cid:3) Bethe ansatz equations The Bethe ansatz equations is the following system ofequations with respect to complex numbers t = ( t , . . . , t l ) : n Y s =1 ( t j − z s + 1 + m s ) Y k = j ( t j − t k − 1) = n Y s =1 ( t j − z s + 1) Y k = j ( t j − t k + 1) , (5.1) j = 1 , . . . , l . A solution t is called admissible if all t , . . . , t l are distinct, and all factors in (5.1) arenonzero.The permutation group S l acts on admissible solutions. If t = ( t , . . . , t l ) is an admis-sible solution, then any permutation of these numbers is an admissible solution too. Weshall consider S l -orbits of admissible solutions.The following lemma is well-known, see for example Lemma 2.2 in [MV2]. Let t be an admissible solution of system (5.1) . Let p ( u ) = l Y i =1 ( u − t i ) , B ( u ) = d ( u ) p ( u ) + a ( u ) p ( u − p ( u − . Then B ( u ) is a polynomial of degree n and p ( u ) is annihilated by the difference operator d ( u ) − B ( u ) ϑ − + a ( u ) ϑ − . (cid:3) Any S l -orbit of admissible solutions of the Bethe ansatz equationsgives a point of the scheme C D considered as a set. Moreover, different S l -orbits givedifferent points. (cid:3) ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 19 Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Thenfor generic z , . . . , z n the Bethe ansatz equations have at least dim Sing W a,d [ l ] distinct S l -orbits of admissible solutions. Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Thenfor generic z , . . . , z n the scheme C D considered as a set has at least dim Sing W a,d [ l ] distinct points. (cid:3) Proof of Theorem 5.1.3. Make the change of variables: z s = ˆ z s /ε , s = 1 , . . . , n , and t i = ˆ t i /ε , i = 1 , . . . , l . Then equations (5.1) take the form n Y s =1 ˆ t j − ˆ z s + ε + m s ε ˆ t j − ˆ z s + ε Y k = j ˆ t j − ˆ t k − ε ˆ t j − ˆ t k + ε = 1 , j = 1 , . . . , l . (5.2)As ε tends to zero, equations (5.2) take the form n X s =1 m s ˆ t j − ˆ z s − X k = j t j − ˆ t k = O ( ε ) , j = 1 , . . . , l , and in the limit we obtain n X s =1 m s ˆ t j − ˆ z s − X k = j t j − ˆ t k = 0 , j = 1 , . . . , l . (5.3)The last system is the system of the Bethe ansatz equations for the Gaudin model. Itwas proved in [RV] that if the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating and ˆ z , . . . , ˆ z n aregeneric, then system (5.3) has at least dim Sing W a,d [ l ] distinct S l -orbits of admissiblesolutions. This proves Theorem 5.1.3. (cid:3) Separation of variables For a nonnegative integer l let C l [ y , . . . , y n − ] Sym be thevector space of symmetric polynomials in y , . . . , y n − of degree not greater than l withrespect to each variable. Let W a,d [ l ] = y l C l [ y , . . . , y n − ] Sym ⊂ C [ y , y , . . . , y n − ]and set W a,d = ⊕ ∞ l =0 W a,d [ l ]. Define an isomorphism of vector spaces W a,d ∼ = W a,d (6.1)using the formula n X i =1 x i u n − i = y n − Y j =1 ( u − y j ) , that is, by setting x i = ( − i − y σ i − ( y , . . . , y n − ) , where σ i − is the ( i − n = 2 wehave x u + x = y ( u − y ) and x = y , x = − y y .We will identify the spaces W a,d and W a,d using isomorphism (6.1). In particular, thisdefines a Y ( gl )-module structure on W a,d . We denote by Sing W a,d [ l ] ⊂ W a,d [ l ] thesubspace of gl -singular vectors.Isomorphism (6.1) defines on W a,d and its subspaces the operators which were previouslydefined on W a,d and its subspaces. Those new operators will be denoted by the samesymbols. In particular, we shall consider the action of operators ˜ T ij ( u ) and ˜ H , . . . , ˜ H n on W a,d . [Sk] . The action of e , e , ˜ T ( u ) , ˜ T ( u ) on W a,d is given by thefollowing formulae: e = n X i =1 m i − y ∂∂y , e = y ∂∂y , (6.2)˜ T ( u ) = (cid:16) u + e − n X i =1 z i + n − X j =1 y j (cid:17) n − Y j =1 ( u − y j ) + n − X j =1 a ( y j ) Y j ′ = j u − y j ′ y j − y j ′ ϑ − y j , (6.3)˜ T ( u ) = (cid:16) u + e − n X i =1 z i + n − X j =1 y j (cid:17) n − Y j =1 ( u − y j ) + n − X j =1 d ( y j ) Y j ′ = j u − y j ′ y j − y j ′ ϑ y j , (6.4) where ϑ y j : f ( y , . . . , y n − ) f ( y , . . . , y j + 1 , . . . , y n − ) .Proof. The proofs of formulae (6.2) are straightforward.The proofs of formulae (6.3) and (6.4) are similar. We will prove formula (6.4). Clearly,the weight subspace W a,d [ l ] is spanned by vectors of the form˜ T ( u ) . . . ˜ T ( u l ) · y l l Y i =1 n − Y j =1 ( u i − y j ) (6.5)with various u , . . . , u l . So, it suffices to verify formula (6.4) on such vectors.Both the expression ˜ T ( u ) ˜ T ( u ) . . . ˜ T ( u l ) · T ( u ) . . . ˜ T ( u l ) · u of degree n . Therefore, theyare uniquely determined by their coefficients at u n and u n − , and the values at n − y , . . . , y n − .Proposition 2.2.1 and formulae (2.9), (2.11), (6.5) yield that˜ T ( u ) ˜ T ( u ) . . . ˜ T ( u l ) · (cid:16) u n + (cid:0) l − n X i =1 z i (cid:1) u n − (cid:17) y l l Y i =1 n − Y j =1 ( u i − y j ) + O ( u n − ) ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 21 as u → ∞ , and (cid:0) ˜ T ( u ) ˜ T ( u ) . . . ˜ T ( u l ) · (cid:1)(cid:12)(cid:12) u = y j = d ( y j ) y l l Y i =1 (cid:16) ( u i − y j + 1) Y j ′ = j ( u i − y j ) (cid:17) , which proves the theorem. (cid:3) We have B ( u, ˜ H ) = ˜ T ( u ) + ˜ T ( u ) = (2 u + n X i =1 ( m i − z i ) + 2 n − X j =1 y j ) n − Y j =1 ( u − y j ) ++ n − X j =1 (cid:18) Y j ′ = j u − y j ′ y j − y j ′ (cid:19) (cid:16) a ( y j ) ϑ − y j + d ( y j ) ϑ y j (cid:17) . Let y = ( y , . . . , y n − ). Recall that a = ( a , . . . , a l ) , h = ( h , . . . , h n ) and p ( x, a ) = x l + a x l − + · · · + a l . Let ω ( y , a ) = y l n − Y j =1 p ( y j − , a ) . This element of W a,d [ l ] ⊗ C [ a ] ⊂ W a,d [ l ] ⊗ C [ a , h ] is called the universal weight function. A trivial but important property of the universal weight function is given by the fol-lowing lemma. Consider C l + n with coordinates a , h . Then for every p ∈ C l + n , thevector ω ( y , a ( p )) is a nonzero vector of W a,d [ l ]. (cid:3) Denote by ω D the projection of the universal weight function ω ( y , a ) to W a,d [ l ] ⊗ A D = W a,d [ l ] ⊗ A D . Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Thenfor s = 1 , . . . , n , we have ˜ H s ω D = h s ω D (6.6) in W a,d [ l ] ⊗ A D . Moreover, we have ω D ∈ Sing W a,d [ l ] ⊗ A D ⊂ W a,d [ l ] ⊗ A D . (6.7) Let p be a point of the scheme C D considered as a set. Then ω ( y , a ( p )) ∈ Sing W a,d [ l ] . (6.8) Moreover, for s = 1 , . . . , n , we have H s ω ( y , a ( p )) = h s ( p ) ω ( y , a ( p )) . (6.9) Proof of Corollary 6.3.3. Let π : C [ a , h ] → A D be the canonical projection. A point p ∈ C D determines uniquely an algebra homomorphism b p : A D → C , such that f ( p ) = b p (cid:0) π ( f ) (cid:1) for any f ∈ C [ a , h ]. In particular, ω ( y , a ( p )) = (id ⊗ b p )( ω D ) . (6.10)Therefore, formulae (6.8) and (6.9) follow from formulae (6.7) and (6.6), respectively. (cid:3) Let p , . . . p d be distinct points of the scheme C D considered as aset. Then the vectors ω ( y , a ( p )) , . . . , ω ( y , a ( p d )) are linearly independent.Proof of Corollary 6.3.4. The vector ω ( y , a ( p j )) is nonzero by Lemma 6.3.1 and is aneigenvector of the operator H s with eigenvalue h s ( p j ) by formula (6.9). Moreover, thecollections of eigenvalues h ( p ) , . . . , h ( p d ) are distinct, because a point p ∈ C D is uniquelydetermined by its coordinates h ( p ) by Theorem 4.3.1. The corollary is proved. (cid:3) Proof of Theorem 6.3.2. To prove formula (6.6) it is enough to show that the polynomial (cid:0) B ( u, ˜ H ) − B ( u, h ) (cid:1) ω ( y , a ) projects to zero in C [ u ] ⊗ W a,d [ l ] ⊗ A D . Let B ( u, y , . . . , y n − , h ) = n − X j =1 B ( y j , h ) Y j ′ = j u − y j ′ y j − y j ′ . For j = 1 , . . . , n , we have B ( y j , y , . . . , y n − , h ) = B ( y j , h ) and B ( u, y , . . . , y n − , h ) is apolynomial in u of degree n − 2. Hence B ( u, h ) − B ( u, y , . . . , y n − , h ) = (cid:16) u + h + 2 n − X j =1 y j (cid:17) n − Y j =1 ( u − y j ) . We have (cid:0) B ( u, ˜ H ) − B ( u, h ) + B ( u, y , . . . , y n − , h ) − B ( u, y , . . . , y n − , h )) ω ( y , a (cid:1) = (cid:18)(cid:16) − h + n X i =1 ( m i − z i ) (cid:17) n − Y j =1 ( u − y j ) (cid:19) ω ( y , a ) + n − X j =1 y l (cid:18) Y j ′ = j u − y j ′ y j − y j ′ p ( y j ′ − , a ) (cid:19)(cid:16) a ( y j ) ϑ − y j − B ( y j , h ) ϑ − y j + d ( y j ) (cid:17) p ( y j , a ) . Clearly all terms in the right-hand side of this formula project to zero in C [ u ] ⊗ W a,d [ l ] ⊗ A D . Hence, formula (6.6) is proved.The proof of formula (6.7) is based on the following lemma. Lemma. We have e e ω D = 0 .Proof. From the formula for the quantum determinant we have˜ T ( u ) ˜ T ( u − ω ( y , a ) = (cid:0) ˜ T ( u ) ˜ T ( u − − a ( u ) d ( u − (cid:1) ω ( y , a ) , (6.11) ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 23 where ˜ T ( u ) ˜ T ( u − 1) = e e u n − + O ( u n − ). Therefore, our goal is to calculate thecoefficient of u n − in the right-hand side. We have T ( u ) T ( u − 1) = 1 + e + e u + e ( e + 1) + T { } + T { } u + O ( u − ) . Hence T ( u ) T ( u − − T ( u ) − T ( u ) + 1 = e ( e + 1) u + O ( u − )and˜ T ( u ) ˜ T ( u − − B ( u, ˜ H ) d ( u − 1) + d ( u ) d ( u − 1) = e ( e + 1) u n − + O ( u n − ) . Thus the right-hand side of (6.11) equals (cid:0) B ( u, ˜ H ) − a ( u ) − d ( u ) (cid:1) d ( u − ω ( y , a ) + e ( e + 1) u n − ω ( y , a ) + O ( u n − ) . Here e ( e + 1) ω ( y , a ) = l (cid:16) n X i =1 m i − l + 1 (cid:17) ω ( y , a ) ,B ( u, ˜ H ) ω ( y , a ) = B ( u, h ) ω ( y , a ) ,a ( u ) + d ( u ) = 2 u n − n X s =1 (2 z s − m s ) u n − + X i XXX MODEL HAS SIMPLE SPECTRUM 25 τ : A ∗ D → Sing W a,d [ l ] . Let f , . . . , f µ be a basis of A D considered asa vector space over C . Write ω D = X i v i ⊗ f i with v i ∈ Sing W a,d [ l ] = Sing W a,d [ l ] . (7.1)Denote by V ⊂ Sing W a,d [ l ] the vector subspace spanned by v , . . . , v µ . Define the linearmap τ : A ∗ D → Sing W a,d [ l ] , g g ( ω D ) = X i g ( f i ) v i . (7.2)Clearly, V is the image of τ . Let p be a point of C D considered as a set. Let ω ( y , a ( p )) ∈ W a,d [ l ] = W a,d [ l ] be the value of the universal weight function at p . Then the vector ω ( y , a ( p )) belongs tothe image of τ .Proof. The statement follows from formula (6.10). (cid:3) Let ψ DW : A D → A W be the epimorphism defined in Theorem 4.3.3. Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Then forany f ∈ A D and g ∈ A ∗ D , we have τ ( L ∗ f ( g )) = ψ DW ( f )( τ ( g )) . In other words, the map τ intertwines the action of the algebra of multiplication oper-ators L ∗ f on A ∗ D and the action on the Bethe algebra on Sing W a,d [ l ]. Proof. The algebra A D is generated by h , . . . , h n . It is enough to prove that for any s we have τ ( L ∗ h s ( g )) = H s ( τ ( g )). But τ ( L ∗ h s ( g )) = P i g ( h s f i ) v i = g (cid:0)P i v i ⊗ h s f i (cid:1) = g (cid:0)P i H s v i ⊗ f i (cid:1) = H s ( τ ( g )). (cid:3) The vector subspace V ⊂ Sing W a,d [ l ] is invariant with respect to theaction of the Bethe algebra A W and the kernel of τ is a subspace of A ∗ D , invariant withrespect to multiplication operators L ∗ f , f ∈ A D . Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Thenthe image of τ is Sing W a,d [ l ] and the kernel of τ is zero. The map τ identifies the action of operators L ∗ f , f ∈ A D , on A ∗ D andthe action of the Bethe algebra on Sing W a,d [ l ] . Hence the epimorphism ψ DW : A D → A W is an isomorphism. Proof of Theorem 7.3.1. First we will show that τ is an epimorphism for generic z .Let d l = dim Sing W a,d [ l ]. Corollary 5.1.4 says that for generic z there exists d l distinctpoints p , . . . , p d l in C D . By Corollary (6.3.4), the vectors ω ( y , a ( p )), . . . , ω ( y , a ( p d l ))are linearly independent and hence form a basis for Sing W a,d [ l ]. Therefore, τ is anepimorphism for generic z by Lemma 7.2.1.By Theorem 4.2.1 and Lemma 2.5.3, dimensions of A D and Sing W a,d [ l ] do not dependon z . Hence dim A D > dim Sing W a,d [ l ] for all z , . . . , z n . Therefore, to prove Theorem7.3.1 it remains to prove that τ has zero kernel.Denote the kernel of τ by K . Let A D = ⊕ p A p ,D be the decomposition into the directsum of local algebras. Since K is invariant with respect to multiplication operators, wehave that K = ⊕ p K ∩ A ∗ p ,D , and for every p , the vector subspace K ∩ A ∗ p ,D is invariantwith respect to multiplication operators. By Lemma 7.1.1, if K ∩ A ∗ p ,D is nonzero, then K ∩ A ∗ p ,D contains the one-dimensional subspace m ⊥ p .Let { f a,b } be the basis of A p ,D constructed in the proof of Lemma 7.1.1, and let { f a,b } be the dual basis of A ∗ p ,D . Then the vector f , generates m ⊥ p . By definition of τ , thevector τ ( f , ) is equal to the value of the universal weight function at p . By Lemma 6.3.1,this value is nonzero and that contradicts the assumption that f , lies in the kernel of τ . (cid:3) A D . Realize the algebra A D as C [ h ] /I ′ D , where I ′ D is the ideal generated by n polynomials q ( h ) , q ( h ) , q j ( a ( h ) , h ) , j = l + 3 , . . . , l + n ,see (4.3).Let ̺ : A D → C , be the Grothendieck residue, f πi ) n Res C D fq ( h ) q ( h ) Q l + nj = l +3 q j ( a ( h ) , h ) . Let ( , ) D be the Grothendieck symmetric bilinear form on A D defined by the rule( f, g ) D = ̺ ( f g ) . The Grothendieck bilinear form is nondegenerate.The form ( , ) D determines a linear isomorphism φ : A D → A ∗ D , f ( f, · ) D . The isomorphism φ intertwines the operators L f and L ∗ f for any f ∈ A D .Proof. For g ∈ A D we have φ ( L f ( g )) = φ ( f g ) = ( f g, · ) D = ( g, f · ) D = L ∗ f (( g, · ) D ) = L ∗ f φ ( g ). (cid:3) Assume that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating. Thenthe composition τ φ : A D → Sing W a,d [ l ] is a linear isomorphism which intertwines thealgebra of multiplication operators on A D and the action of the Bethe algebra A W on Sing W a,d [ l ] . ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 27 Algebra A G ( m , , . . . , ( m n , , l . In the remainder of the paper weassume that Λ = (Λ (1) , . . . , Λ ( n ) ) = (cid:0) ( m , , . . . , ( m n , (cid:1) is a collection of dominant integral gl -weights, that is, m s ∈ Z > for s = 1 , . . . , n .We assume that l ∈ Z > is such that the weight (cid:0) P ns =1 m s − l, l (cid:1) is dominant integral,that is, P ns =1 m s − l > l . This assumption implies that the pair (cid:0) ( m , , . . . , ( m n , (cid:1) , l is separating.Let ˜ l = P ns =1 m s + 1 − l . We have ˜ l > l . The (discrete) Wronskian of polynomials f, g ∈ C [ u ] is the polynomialWr ( f ( u ) , g ( u )) = f ( u ) g ( u − − f ( u − g ( u ) . Let f, g, B ∈ C [ u ] . Assume that f, g are monic polynomials of degrees l, ˜ l , respectively, that lie in the kernel of the difference operator d ( u ) − B ( u ) ϑ − + a ( u ) ϑ − . Then Wr ( f ( u ) , g ( u )) = ( l − ˜ l ) n Y s =1 m s Y j =1 ( u − z s + j ) . Proof. Let C ( u ) = Wr ( f ( u ) , g ( u )). Then the top coefficient of C ( u ) equals l − ˜ l , and C ( u ) C ( u − 1) = a ( u ) d ( u ) , which determines the polynomial C ( u ) uniquely. (cid:3) Let f, g ∈ C [ u ] , z ∈ C , m ∈ Z > . Assume that f ( z − j ) = 0 for j = 1 , . . . , m + 1 . Then the polynomial Wr ( f ( u ) , g ( u )) is equal to zero at u = z − j , j = 1 , . . . , m , and the polynomial f ( u ) g ( u − − f ( u − g ( u ) is equal to zero at u = z − j , j = 1 , . . . , m − . (cid:3) Let f, g, C ∈ C [ u ] , z ∈ C , and Wr ( f ( u ) , g ( u )) = C ( u ) . (i) If C ( z ) = 0 and f ( z − 1) = 0 , then g ( z − = 0 . (ii) If C ( z ) = 0 and f ( z ) = 0 , then f ( z − = 0 . (cid:3) Let f, g ∈ C [ u ] , z ∈ C . Then Wr (( u − z ) f ( u ) , ( u − z ) g ( u )) = ( u − z )( u − z − 1) Wr ( f ( u ) , g ( u )) . (cid:3) C G . Let d be a sufficiently large natural numberwith respect to the numbers m , . . . , m n considered in Section 8.1. Let C d [ u ] be the vectorsubspace in C [ u ] of polynomials of degree not greater than d . Denote by G the Grassmannian of all two-dimensional vector subspaces in C d [ u ].Let F = { F d +1 ⊂ F d ⊂ · · · ⊂ F ⊂ F = C d [ u ] } be a complete flag and Λ = ( a, b )a gl dominant integral weight such that d > a > b > a, b ∈ Z . Define a Schubertcell C o F , Λ ⊂ G to be the set of all two-dimensional subspaces V ⊂ C d [ u ] having a basis f, g such that f ∈ F a +1 − F a +2 and g ∈ F b − F b +1 . Define a Schubert cycle C F , Λ ⊂ G as the closure of the Schubert cell C o F , Λ .For z ∈ Z and i ∈ Z > , set ϕ i ( u, z ) = i Y j =1 ( u − z + j ) . Introduce a complete flag in C d [ u ] : F ( z ) = { F d +1 ( z ) ⊂ F d ( z ) ⊂ · · · ⊂ F ( z ) ⊂ F ( z ) = C d [ u ] } , where F i ( z ) consists of all polynomials divisible by ϕ i ( u, z ).Introduce the complete flag in C d [ u ] associated with infinity: F ( ∞ ) = { F d +1 ( ∞ ) ⊂ F d ( ∞ ) ⊂ · · · ⊂ F ( ∞ ) ⊂ F ( ∞ ) = C d [ u ] } , where F i ( ∞ ) consists of all polynomials of degree d − i .We consider the Schubert cells C o F ( z s ) , Λ ( s ) ⊂ G , s = 1 , . . . , n , where Λ ( s ) = ( m s , C o F ( ∞ ) , Λ ( ∞ ) ⊂ G , where Λ ( ∞ ) = ( d − l, d − ˜ l − C o F ( z s ) , Λ ( s ) isthe set of all two-dimensional subspaces V ⊂ C d [ u ] having a basis f, g such that g ( z s − = 0 , f ( z s − m s − = 0 , f ( z s − j ) = 0 for j = 1 , . . . , m s + 1 , and the cell C o F ( ∞ ) , Λ ( ∞ ) is the set of all two-dimensional subspaces V ⊂ C d [ x ] having abasis f, g such that deg f = l and deg g = ˜ l .Consider the (scheme-theoretic) intersection C G = C F ( ∞ ) , Λ ( ∞ ) T (cid:0) ∩ ns =1 C F ( z s ) , Λ ( s ) (cid:1) (8.1)of the corresponding Schubert cycles. Denote by A G the algebra of functions on C G . Let z i − z j / ∈ Z for i = j . Then C G = C o F ( ∞ ) , Λ ( ∞ ) T (cid:0) ∩ ns =1 C F ( z s ) , Λ ( s ) (cid:1) as sets. Proof. Let V be a point of C F ( ∞ ) , Λ ( ∞ ) T (cid:0) ∩ ns =1 C F ( z s ) , Λ ( s ) (cid:1) . Let f, g be a monic basisof V , such that deg f l and deg g ˜ l . Then deg Wr ( f ( u ) , g ( u )) l + ˜ l − 1. Onthe other hand, the polynomial Wr ( f ( u ) , g ( u )) is divisible by Q ns =1 Q m s j =1 ( u − z s + j ) byLemma 8.2.2. Since P ns =1 m s = ˜ l + l − 1, we conclude that deg f = l , deg g = ˜ l , V is a ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 29 point of C o F ( ∞ ) , Λ ( ∞ ) , andWr ( f ( u ) , g ( u )) = ( l − ˜ l ) n Y s =1 m s Y j =1 ( u − z s + j ) . Since a suitable linear combination of f and g is divisible by ϕ m s +1 ( u, z s ), the subspace V is a point of C o F ( z s ) , Λ ( s ) by Lemma 8.2.3. (cid:3) Let V be a point of C G considered as a set. Then there exists a unique basis f, g of V such that f ( u ) = u l + f u l − + · · · + f l ,g ( u ) = u ˜ l + g u ˜ l − + · · · + g ˜ l − l − u l +1 + g ˜ l − l +1 u l − + · · · + g ˜ l for suitable complex numbers f , . . . , f l , g , . . . , g ˜ l − l − , g ˜ l − l +1 , . . . , g ˜ l . Let z i − z j / ∈ Z for i = j . Then all polynomials of the subspace V areannihilated by the difference operator D V = d ( u ) − B V ( u ) ϑ − + a ( u ) ϑ − , where B V ( u ) = 1˜ l − l (cid:0) g ( u ) f ( u − − g ( u − f ( u ) (cid:1) n Y s =1 m s − Y j =1 ( u − z s + j ) − is a polynomial of degree n .Proof. Let W ( u ) = Wr (cid:0) f ( u ) , g ( u ) (cid:1) . It is straightforward to see that all polynomials ofthe subspace V are annihilated by the difference operator W ( u − − (cid:0) g ( u ) f ( u − − g ( u − f ( u ) (cid:1) ϑ − + W ( u ) ϑ − . (8.2)Since W ( u ) = ( l − ˜ l ) n Y s =1 m s Y j =1 ( u − z s + j ) , see the proof of Lemma 8.3.2, and all coefficients of the difference operator (8.2) aredivisible by Q ns =1 Q m s − j =1 ( u − z s + j ) by Lemma 8.2.2, the statement follows. (cid:3) Write B V ( u ) = 2 u n + h u n − + · · · + h n . Recall that the scheme C D is defined in Section 4.1. Consider the schemes C D and C G as sets. Thenthe assignment V ( f , . . . , f l , h , . . . , h n ) ∈ C l + n defines an injective map of sets C G → C D . Let z i − z j / ∈ Z for i = j . Assume that V ∈ G has a basis f, g suchthat deg f = l and deg g = ˜ l , and V is annihilated by a difference operator of the form d ( u ) − B ( u ) ϑ − + a ( u ) ϑ − , where B ( u ) is a polynomial. Then V is a point of C G . The proof is similar to the proof of Theorem 7.2 in [MTV2]. (cid:3) A G .8.4.1. Lemma. Let z i − z j / ∈ Z for i = j . Then A G considered as a vector space isfinite-dimensional. Moreover, this dimension does not depend on z .Proof. The claim follows from Corollary 8.3.5 and the reasoning similar to the proof ofTheorem 4.2.1. (cid:3) Under conditions of Lemma 8.4.1, the dimension of A G as a vector space is givenby Schubert calculus. Namely, let Λ = (Λ (1) , . . . , Λ ( n ) ) be the collection of gl -highestweights, where Λ ( s ) = ( m s , L Λ = L Λ (1) ⊗ · · · ⊗ L Λ ( n ) the tensor product of irreducible gl -modules with highest weights Λ (1) , . . . , Λ ( n ) , respec-tively. Let Sing L Λ [ l ] be the subspace of L Λ of gl -singular vectors of weight ( P ns =1 m s − l, l ). Then by Schubert calculus,dim A G = dim Sing L Λ [ l ] , (8.3)see [Fu]. A G . If z i − z j / ∈ Z for i = j , we shall use the followingpresentation of the algebra A G .Let ˜ a = (˜ a , . . . , ˜ a ˜ l − l − , ˜ a ˜ l − l +1 , . . . , ˜ a ˜ l ) . Consider the space C ˜ l + l + n − with coordinates ˜ a , a , h , cf. Section 4.1.Denote by ˜ p ( u, ˜ a ) the following polynomial in u depending on parameters ˜ a ,˜ p ( u, ˜ a ) = u ˜ l + ˜ a u ˜ l − + · · · + ˜ a ˜ l − l − u l +1 + ˜ a ˜ l − l +1 u l − + · · · + ˜ a ˜ l . Recall that p ( u, a ) = u l + a u l − + · · · + a l and B ( u, h ) = 2 u n + h u n − + · · · + h n .Let us writeWr (˜ p ( u, ˜ a ) , p ( u, a )) = (˜ l − l ) u ˜ l + l − + w (˜ a , a ) u ˜ l + l − + · · · + w ˜ l + l − (˜ a , a ) , ˜ p ( u, ˜ a ) p ( u − , a ) − ˜ p ( u − , ˜ a ) p ( u, a ) =2(˜ l − l ) u ˜ l + l − + ˆ w (˜ a , a ) u ˜ l + l − + · · · + ˆ w ˜ l + l − (˜ a , a ) ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 31 for suitable polynomials w , . . . , w ˜ l + l − , ˆ w , . . . , ˆ w ˜ l + l − in variables ˜ a , a . Let us write(˜ l − l ) n Y s =1 m s Y j =1 ( u − z s + j ) = (˜ l − l ) u ˜ l + l − + c u ˜ l + l − + · · · + c ˜ l + l − , (˜ l − l ) B ( u, h ) n Y s =1 m s − Y j =1 ( u − z s + j ) = 2(˜ l − l ) u ˜ l + l − + ˆ c ( h ) u ˜ l + l − + · · · + ˆ c ˜ l + l − ( h ) , for suitable numbers c , . . . , c ˜ l + l − and polynomials ˆ c , . . . , ˆ c ˜ l + l − in variables h .Denote by I G the ideal in C [˜ a , a , h ] generated by 2(˜ l + l − 1) polynomials w i (˜ a , a ) − c i , ˆ w i (˜ a , a ) − ˆ c i ( h ) , i = 1 , . . . , ˜ l + l − . (8.4) Let z i − z j / ∈ Z for i = j . Then A G = C [˜ a , a , h ] /I G . Proof. The scheme defined by the ideal I G consists of points p such thatWr (˜ p ( u, ˜ a ( p )) , p ( u, a ( p ))) = (˜ l − l ) n Y s =1 m s Y j =1 ( u − z s + j ) , ˜ p ( u, ˜ a ) p ( u − , a ) − ˜ p ( u − , ˜ a ) p ( u, a ) = (˜ l − l ) B ( u, h ( p )) n Y s =1 m s − Y j =1 ( u − z s + j ) . Hence, the polynomials ˜ p ( u, ˜ a ( p )) , p ( u, a ( p )) span a vector subspace V lying in theintersection C G , see Theorem 8.3.6. Conversely, if V is a point of C G , then V has a basis f, g like in Lemma 8.3.4. Then by Lemma 8.3.4 we haveWr( g ( u ) , f ( u )) = (˜ l − l ) n Y s =1 m s Y j =1 ( u − z s + j ) ,g ( u ) f ( u − − g ( u − f ( u ) = (˜ l − l ) B ( u ) n Y s =1 m s − Y j =1 ( u − z s + j )for a suitable polynomial B ( u ). Hence, the triple g, f, B determines a point p , whosecoordinates satisfy equations (8.4). (cid:3) Algebras A P and A L A P . Consider the space C ˜ l + l + n − with coordinates ˜ a , a , h . Let D h = d ( u ) − B ( u, h ) ϑ − + a ( u ) ϑ − be the difference operator defined in (4.1). If h satisfies equations q ( h ) = 0 and q ( h ) = 0,then the polynomial D h (˜ p ( u, ˜ a )) is a polynomial in u of degree ˜ l + n − D h (˜ p ( u, ˜ a )) = ˜ q (˜ a , h ) u ˜ l + n − + . . . + ˜ q ˜ l + n (˜ a , h ) . The coefficients ˜ q i (˜ a , h ) are functions linear in ˜ a and linear in h .Recall that if p ( u, a ) = u l + a u l − + · · · + a l , and h satisfies equations q ( h ) = 0 and q ( h ) = 0, then the polynomial D h ( p ( u, a )) is a polynomial in u of degree l + n − D h ( p ( u, a )) = q ( a , h ) u l + n − + . . . + q l + n ( a , h ) . Denote by I P the ideal in C [˜ a , a , h ] generated by polynomials q , q , q , . . . , q l + n , ˜ q , . . . , ˜ q ˜ l + n . The ideal I P defines a scheme C P ⊂ C ˜ l + l + n − . The algebra A P = C [˜ a , a , h ] /I P is the algebra of functions on C P .The scheme C P is the scheme of points p ∈ C ˜ l + l + n − such that the difference equation D h ( p ) w ( u ) = 0 has two polynomial solutions ˜ p ( u, ˜ a ( p )) and p ( u, a ( p )). ψ GP : A G → A P .9.2.1. Theorem. If z i − z j / ∈ Z for i = j , then the identity map C ˜ l + l + n − → C ˜ l + l + n − induces an algebra isomorphism ψ GP : A G → A P .Proof. If p is a point of C P , then polynomials ˜ p ( u, ˜ a ( p )), p ( u, a ( p )) are annihilated bythe difference operator d ( u ) − B ( u, h ( p )) ϑ − + a ( u ) ϑ − . If z i − z j / ∈ Z for i = j , then thespan V of polynomials ˜ p ( u, ˜ a ( p )), p ( u, a ( p )) is a point of C G by Theorem 8.3.6. Thisreasoning defines an algebra homomorphism ψ GP : A G → A P .Conversely, if p is a point of C G , then the triple ˜ p ( u, ˜ a ( p )), p ( u, a ( p )), B ( u, h ( p ))satisfies equationsWr (˜ p ( u, ˜ a ( p )) , p ( u, a ( p ))) = (˜ l − l ) n Y s =1 m s Y j =1 ( u − z s + j ) , ˜ p ( u, ˜ a ) p ( u − , a ) − ˜ p ( u − , ˜ a ) p ( u, a ) = (˜ l − l ) B ( u, h ( p )) n Y s =1 m s − Y j =1 ( u − z s + j ) . Hence the polynomials ˜ p ( u, ˜ a ( p )), p ( u, a ( p )) are annihilated by the difference operator d ( u ) − B ( u, h ( p )) ϑ − + a ( u ) ϑ − . Therefore, p is a point of C P . (cid:3) A L . Assume that m , . . . , m n , l satisfy conditions of Section 8.1. Let Λ =(Λ (1) , . . . , Λ ( n ) ) be the collection of gl -highest weights with Λ ( s ) = ( m s , L Λ = L Λ (1) ⊗ · · · ⊗ L Λ ( n ) ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 33 be the tensor product of irreducible gl -modules with highest weights Λ (1) , . . . , Λ ( n ) , re-spectively, and v Λ = v ( m , ⊗ · · · ⊗ v ( m n , the tensor product of the corresponding highestweight vectors. Denote by L Λ ( z ) = L Λ (1) ( z ) ⊗ · · · ⊗ L Λ ( n ) ( z n )the tensor product of evaluation modules.Let Sing L Λ [ l ] ⊂ L Λ ( z ) be the subspace of gl -singular vectors of weight ( P ni =1 m i − l, l ).The algebra A L is the Bethe algebra associated with Sing L Λ [ l ].Assume that m i ∈ Z > for i = 1 , . . . , n , and m m · · · m n . Assume that z i − z j + m j − s = 0 and z i − z j − − s = 0 for all i < j and s = 0 , , . . . , m i − 1. Thenby Theorem 2.6.2, there is a natural isomorphism W a,d /K → L Λ ( z ) such that 1 v Λ .Here K ⊂ W a,d is the kernel of the Yangian Shapovalov form on W a,d .The Yangian Shapovalov form on W a,d induces the Yangian Shapovalov form S on L Λ ( z ) such that S ( v Λ , v Λ ) = 1 and S ( x · v, w ) = S ( v, x + · w ) for all x ∈ Y ( gl ) and v, w ∈ L Λ ( z ). The form S is nondegenerate and symmetric.We have the composition of linear maps W a,d → W a,d /K → L Λ ( z ) . Restricting this composition to Sing W a,d we get a linear epimorphism σ : Sing W a,d [ l ] → Sing L Λ [ l ] . The Bethe algebra A W preserves the kernel of σ and induces a commutative subalgebrain End (Sing L Λ [ l ]). The induced subalgebra coincides with the Bethe algebra A L . Wedenote by ψ W L : A W → A L the corresponding epimorphism.The operators of the algebra A L are symmetric with respect to the Yangian Shapovalovform on L Λ ( z ). Denote by D L = d ( u ) − (2 u n + ψ W L ( H ) u n − + · · · + ψ W L ( H n )) ϑ − + a ( u ) ϑ − the universal difference operator associated with the subspace Sing L Λ [ l ] and collection z . Assume that the pair Λ , l satisfies conditions of Section 8.1. Thenfor any v ∈ Sing L Λ [ l ] there exist v , . . . , v ˜ l ∈ Sing L Λ [ l ] such that the function w ( u ) = v u ˜ l + v u ˜ l − + . . . + v ˜ l is a solution of the difference equation D L w ( u ) = 0 . This theorem is a particular case of Theorem 7.3 in [MTV2]. Homomorphisms of algebras A D , A P and A L ψ DP : A D → A P . A point p of C P determines the differenceequation D h ( p ) w ( u ) = 0 and two solutions ˜ p ( u, ˜ a ( p )) and p ( u, a ( p )). Then the pair,consisting of the difference operator D h ( p ) and the solution p ( u, a ( p )) of the smallerdegree, determines a point of C D , see Section 4.1. This correspondence defines a naturalalgebra epimorphism ψ DP : A D → A P . ξ : A D → Sing L Λ [ l ] . Assume that z , . . . , z n , m , . . . , m n satisfy theassumptions of Theorem 2.6.2. Then we have the composition of linear maps A D φ −→ A ∗ D τ −→ Sing W a,d [ l ] σ −→ Sing L Λ [ l ] . Denote this composition by ξ : A D → Sing L Λ [ l ]. By Theorem 7.3.1, ξ is a linearepimorphism.Let ψ DL : A D → A L be the algebra epimorphism defined as the composition ψ W L ψ DW . If z , . . . , z n , m , . . . , m n satisfy the assumptions of Theorem 2.6.2,then the linear map ξ intertwines the action of the multiplication operators L f , f ∈ A D ,on A D and the action of the Bethe algebra A L on Sing L Λ [ l ] , that is, for any f, g ∈ A D we have ξ ( L f ( g )) = ψ DL ( f )( ξ ( g )) . The lemma follows from Corollary 7.4.2. If z , . . . , z n , m , . . . , m n satisfy the assumptions of Theorem 2.6.2,then the kernel of ξ coincides with the kernel of ψ DL .Proof. If ψ DL ( f ) = 0, then ξ ( f ) = ξ ( L f (1)) = ψ DL ( f )( ξ (1)) = 0. On the other hand, if ξ ( f ) = 0, then for any g ∈ A D we have ψ DL ( f )( ξ ( g )) = ξ ( L f ( g )) = ξ ( f g ) = ξ ( L g ( f )) = ψ DL ( g )( ξ ( f )) = 0. Since ξ is an epimorphism, this means that ψ DL ( f ) = 0. (cid:3) If z i − z j / ∈ Z for i = j , then the kernel of ξ coincides with the kernelof ψ DP .Proof. If z i − z j / ∈ Z for i = j , then the assumptions of Theorem 2.6.2 are satisfied and ξ is defined.By Schubert calculus, dim Sing L Λ [ l ] = dim A G . By Theorem 9.2.1 dim A G =dim A P if z i − z j / ∈ Z for i = j . Hence it suffices to show that the kernel of ξ con-tains the kernel of ψ DP . But this follows from Theorems 3.3.1 and 9.3.2.Indeed the defining relations in A P = A D / (ker ψ DP ) are the conditions on the operator D h to have two linear independent polynomials in the kernel. Theorems 3.3.1 and 9.3.2guarantee these relations for elements of the Bethe algebra A L . Hence, the kernel of ψ DL contains the kernel of ψ DP . By Lemma 10.2.2, the kernel of ξ coincides with the kernelof ψ DL . Therefore, the kernel of ξ contains the kernel of ψ DP . (cid:3) ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 35 Let z i − z j / ∈ Z for all i = j . Then the algebras A P , A L and A G are isomorphic.Proof. Since the algebra epimorphisms ψ DP and ψ DL have the same kernels, the algebras A P and A L are isomorphic. Then A L and A G are isomorphic by Theorem 9.2.1. (cid:3) Let z i − z j / ∈ Z for all i = j . Denote by ψ P L : A P → A L the isomorphism induced by ψ DL and ψ DP . Lemmas 10.2.1 – 10.2.3 imply the followingtheorem. If z i − z j / ∈ Z for all i = j , then the linear map ξ induces a linearisomorphism ζ : A P → Sing L Λ [ l ] which intertwines the multiplication operators L f , f ∈ A P , on A P and the action ofthe Bethe algebra A L on Sing L Λ [ l ] , that is, for any f, g ∈ A P we have ζ ( L f ( g )) = ψ P L ( f )( ζ ( g )) . (cid:3) Let z i − z j / ∈ Z for all i = j . Assume that every operator f ∈ A L isdiagonalizable. Then the algebra A L has simple spectrum and all points of the intersectionof Schubert cycles C G = C F ( ∞ ) , Λ ( ∞ ) T ( ∩ ni =1 C F ( z i ) , Λ ( i ) ) are of multiplicity one.Proof. The algebras A L , A P and A G are isomorphic. We have A P = ⊕ p A p ,P wherethe sum is over the points of the scheme C P considered as a set and A p ,P is the localalgebra associated with a point p . The algebra A p ,P has nonzero nilpotent elements ifdim A p ,P > 1. If every element f ∈ A P is diagonalizable, then the algebra A P is thedirect sum of one-dimensional local algebras. Hence A P has simple spectrum as well asthe algebras A L and A G . (cid:3) Corollary 10.3.2 has the following application. Assume that z , . . . , z n are real, z i − z j / ∈ Z and | z i − z j | ≫ forall i = j . Then all points of the intersection of Schubert cycles C G = C ∞ , Λ ( ∞ ) T ( ∩ ni =1 C z i , Λ ( i ) ) are of multiplicity one.Proof. If z , . . . , z n are real and | z i − z j | ≫ i = j , then the Yangian Shapo-valov form, restricted to the real part of Sing L Λ [ l ], is positive definite, see Appendix Cin [MTV1]. The Hamiltonians ψ W L ( H ) , . . . , ψ W L ( H ), restricted to the real part ofSing L Λ [ l ], are real symmetric operators operators with respect to the Yangian Shapo-valov form, see [MTV1]. Hence, all elements of the Bethe algebra A L are diagonalizableoperators. Therefore, the spectrum of A G is simple and all points of C G are of multiplicityone. (cid:3) Corollary 10.3.3 is related to Theorem 1 from [EGSV] and Theorem 2.1 from [MTV4]concerning the real Schubert calculus. Let n = 3, Λ ( s ) = (1 , 0) , s = 1 , , 3, Λ ( ∞ ) = (2 , R = 4 ( z + z + z − z z − z z − z z ) − . If R = 0, then every element of A L is diagonalizable and the algebra A L is isomorphic tothe direct sum C ⊕ C . If R = 0, then the algebra A L contains a nonzero nilpotent matrixand is isomorphic to C [ b ] / h b i .11. Operators with polynomial kernel and Bethe algebra A L θ : A ∗ P → Sing L Λ [ l ] . Let z i − z j / ∈ Z for all i = j . Definethe symmetric bilinear form on A P by the formula( f, g ) P = S (cid:0) ζ ( f ) , ζ ( g ) (cid:1) for all f, g ∈ A P , where S ( , ) denotes the Yangian Shapovalov form on Sing L Λ [ l ]. The form ( , ) P is nondegenerate. The lemma follows from the fact that the Yangian Shapovalov form on Sing L Λ [ l ] isnondegenerate and the fact that ζ is an isomorphism. We have ( f g, h ) P = ( g, f h ) P for all f, g, h ∈ A P . The lemma follows from the fact the elements of the Bethe algebra are symmetricoperators with respect to the Yangian Shapovalov form, see Section 3.2.2.The form ( , ) P defines a linear isomorphism π : A P → A ∗ P , f ( f , · ) P . Let z i − z j / ∈ Z for all i = j . Then the map π intertwines themultiplication operators L f , f ∈ A P , on A P and the dual operators L ∗ f , f ∈ A P , on A ∗ P . Summarizing Theorem 10.3.1 and Corollary 11.1.3 weobtain the following theorem. Let z i − z j / ∈ Z for all i = j . Then the composition θ = ζ π − is alinear isomorphism from A ∗ P to Sing L Λ [ l ] which intertwines the multiplication operators L ∗ f , f ∈ A P , on A ∗ P and the action of the Bethe algebra A L on Sing L Λ [ l ] , that is, for any f ∈ A P and g ∈ A ∗ P we have θ ( L ∗ f ( g )) = ψ P L ( f )( θ ( g )) . (cid:3) Let z i − z j / ∈ Z for all i = j . Assume that v ∈ Sing L Λ [ l ] is an eigenvector ofthe Bethe algebra A L , that is, ψ W L ( H s ) v = λ s v for suitable λ s ∈ C and s = 1 , . . . , n .Then, by Corollary 7.4 in [MTV2], the difference equation (cid:0) d ( u ) − (2 u n + λ u n − + · · · + λ n ) ϑ − + a ( u ) ϑ − (cid:1) w ( u ) = 0has two linearly independent polynomial solutions, one of degree l and the other of degree˜ l . The following corollary of Theorem 11.2.1 gives the converse statement. ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 37 Let z i − z j / ∈ Z for all i = j . Assume that ( λ , . . . , λ n ) ∈ C n is a point such that λ = n X i =1 ( m i − z i ) , λ = l (cid:16) l − − n X i =1 m i (cid:17) + X i Indeed, such a point ( λ , . . . , λ n ) defines a linear function η : A P → C , h s λ s , for s = 1 , . . . , n . Moreover, η ( f g ) = η ( f ) η ( g ) for all f, g ∈ A P .Hence η ∈ A ∗ P is an eigenvector of operators L ∗ f acting on A ∗ P . By Theorem 11.2.1, thevector v = θ ( η ) ∈ Sing L Λ [ l ] is an eigenvector of the action of the Bethe algebra A L witheigenvalues prescribed in Corollary 11.2.3.Let v ′ ∈ Sing L Λ [ l ] satisfy (11.2), then η ′ = θ − ( v ) ∈ A ∗ P satisfies η ′ ( f g ) = η ( f ) η ′ ( g ) forall f, g ∈ A P . Hence, for g = 1 we have η ′ ( f ) = η ( f ) η ′ (1). Therefore, η ′ is proportionalto η , and v ′ is proportional to v . (cid:3) Assume that ( λ , . . . , λ n ) ∈ C n is a point satisfying the assumptions of Corol-lary 11.2.3. We describe how to find the eigenvector v ∈ Sing L Λ [ l ], indicated in Corol-lary 11.2.3.Let f ( u ) be the monic polynomial of degree l which is a solution of the differenceequation (11.1). Consider the polynomial ω ( y ) = y l n − Y j =1 f ( y j − W a,d , see Section 6.3. By Theorem 6.3.2 this vector lies in Sing W a,d [ l ]and ω ( y ) is an eigenvector of the Bethe algebra A W with eigenvalues prescribed in Corol-lary 11.2.3. Consider a maximal subspace V ⊂ Sing W a,d [ l ] with three properties:i) V contains ω ( y ),ii) V does not contain other eigenvectors of the Bethe algebra A W ,iii) V is invariant with respect to the Bethe algebra A W .Such a maximal subspace does exist and is unique. Let σ ( V ) ⊂ Sing L Λ [ l ] be the imageof V under the epimorphism σ . Then by Corollary 11.2.3, the subspace σ ( V ) containsa unique one-dimensional subspace of eigenvectors of the Bethe algebra A L . Any suchan eigenvector may serve as an eigenvector of the Bethe algebra A L indicated in Corol-lary 11.2.3. Homogeneous XXX Heisenberg model In Sections 8–11, in most of the assertions we assumedthat z , . . . , z n ∈ C are such that z i − z j / ∈ Z for i = j , and m , . . . , m n are naturalnumbers. In this section we assume that z = · · · = z n = 0 and m = · · · = m n = 1 . (12.1)This special case is called the homogeneous XXX Heisenberg model .In other words, in this section we consider the Y ( gl )-module L ( ) = L (1 , (0) ⊗ · · · ⊗ L (1 , (0) , which is the tensor product of n copies of the two-dimensional evaluation module, andthe subspace of gl -singular vectors of weight ( n − l, l ),Sing L [ l ] = { p ∈ L ( ) | e p = 0 , e p = lp } . The subspace Sing L [ l ] is not empty if and only if 2 l n , that is, if and only if the pair (cid:0) (1 , , . . . , (1 , (cid:1) , l is separating. In that casedim Sing L [ l ] = (cid:18) nl (cid:19) − (cid:18) nl − (cid:19) . The algebra A L is the Bethe algebra associated with the subspace Sing L [ l ]. It isgenerated by the coefficients of the series (cid:0) T ( u ) + T ( u ) (cid:1)(cid:12)(cid:12) Sing L [ l ] .The main result of this section is the following theorem. For the homogeneous XXX Heisenberg model, the Bethe algebra A L has simple spectrum. The theorem will be proved in Section 12.7. Let ˜ l = n + 1 − l . We have ˜ l + l − n and ˜ l > l . Denote by f, g twopolynomials in C [ u ] of the form: f ( u ) = u l + f u l − + · · · + f l , (12.2) g ( u ) = u ˜ l + g u ˜ l − + · · · + g ˜ l − l − u l +1 + g ˜ l − l +1 u l − + · · · + g ˜ l . As a byproduct of the proof of Theorem 12.1.1 we prove the following theorem. Theorem. There exist exactly (cid:0) nl (cid:1) − (cid:0) nl − (cid:1) distinct pairs of polynomials f, g of the form (12.2) , such that f ( u ) g ( u − − f ( u − g ( u ) = ( l − ˜ l ) ( u + 1) n . Theorem 12.1.2 will be proved in Section 12.8. ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 39 A L for the homogeneous XXX model. Consider the Yangian module W a,d corresponding to the polynomials a ( u ) = ( u + 1) n , d ( u ) = u n . The numbers (12.1) satisfy the assumptions of Theorem 2.6.2. Therefore the Y ( gl )-module L ( ) is irreducible, and there is a natural epimorphism W a,d → L ( ) of Y ( gl )-modules. Restricting this epimorphism to Sing W a,d [ l ], we obtain a linear epimorphism σ : Sing W a,d [ l ] → Sing L Λ [ l ] . The Bethe algebra A W preserves the kernel of σ and induces a commutative subalge-bra in End (Sing L Λ [ l ]). The induced subalgebra coincides with the Bethe algebra A L ,see Section 9.3.Denote by ψ W L : A W → A L the corresponding epimorphism. We have (cid:0) T ( u ) + T ( u ) (cid:1)(cid:12)(cid:12) Sing L [ l ] = 2 + ψ W L ( H ) u − + · · · + ψ W L ( H n ) u − n , where ψ W L ( H ) = n , ψ W L ( H ) = l ( l − − n ) + n ( n − , see Section 3.2.1. Thus the Bethe algebra A L is generated by elements ψ W L ( H ) , . . . ,ψ W L ( H n ). A P for the homogeneous XXX model. Consider the space C n withcoordinates ˜ a , a , h , as in Section 8.5, and polynomials ˜ p ( u, ˜ a ), p ( u, a ), B ( u, h ).Given the polynomials a ( u ) = ( u + 1) n and d ( u ) = u n , we define the ideal I P , thealgebra A P , and the scheme C P as in Section 9.1. The scheme C P is the scheme of points p ∈ C n such that the difference equation( u n − B ( u, h ) ϑ − + ( u + 1) n ϑ − ) w ( u ) = 0has two polynomial solutions ˜ p ( u, ˜ a ( p )) and p ( u, a ( p )). A G for the homogeneous XXX model. Consider the space C n withcoordinates ˜ a , a , h , and polynomials ˜ p ( u, ˜ a ), p ( u, a ), B ( u, h ). Let us writeWr (˜ p ( u, ˜ a ) , p ( u, a )) = (˜ l − l ) u n + w (˜ a , a ) u n − + · · · + w n (˜ a , a ) , ˜ p ( u, ˜ a ) p ( u − , a ) − ˜ p ( u − , ˜ a ) p ( u, a ) =2(˜ l − l ) u n + ˆ w (˜ a , a ) u n − + · · · + ˆ w n (˜ a , a )for suitable polynomials w , . . . , w n , ˆ w , . . . , ˆ w n in variables ˜ a , a .Denote by I G the ideal in C [˜ a , a , h ] generated by 2 n polynomials w i (˜ a , a ) − (˜ l − l ) (cid:18) ni (cid:19) , ˆ w i (˜ a , a ) − (˜ l − l ) h i , i = 1 , . . . , n . (12.3) The ideal I G defines a scheme C G ⊂ C n . Then A G = C [˜ a , a , h ] /I G is the algebra of functions on C G .The scheme C G is the scheme of points p ∈ C n such thatWr (˜ p ( u, ˜ a ( p ) , p ( u, a ( p )) = (˜ l − l ) ( u + 1) n , (12.4)˜ p ( u, ˜ a ) p ( u − , a ) − ˜ p ( u − , ˜ a ) p ( u, a ) = (˜ l − l ) B ( u, h ( p )) . The identity map C n → C n induces an algebra isomorphism ψ GP : A G → A P .Proof. The proof is similar to the proof of Theorem 9.2.1. (cid:3) The dimension of A G considered as a vector space is equal to dim Sing L [ l ] = (cid:18) nl (cid:19) − (cid:18) nl − (cid:19) . Proof. Consider the ideal I G ( z ) defined by (8.4) for m = · · · = m n = 1 and arbitrary z , . . . , z n . Consider the algebra A G ( z ) = C [˜ a , a , h ] /I G ( z ). By Lemma 8.5.1, if z , . . . , z n are distinct and close to zero, then A G ( z ) is the algebra of functions on the intersectionof Schubert cells C G ( z ), see (8.1), and by (8.3) we havedim A G ( z ) = (cid:18) nl (cid:19) − (cid:18) nl − (cid:19) . To complete the proof of Lemma 12.4.2, it suffices to verify two facts:(i) There are no algebraic curves over C lying in the scheme C G ( ), defined by theideal (12.3).(ii) Let a sequence z ( i ) , i = 1 , , . . . , tend to . Let p ( i ) ∈ C G ( z ( i ) ) , i = 1 , , . . . , be asequence of points. Then all coordinates (cid:0) ˜ a ( p ( i ) ) , a ( p ( i ) ) , h ( p ( i ) (cid:1) remain boundedas i tends to infinity.By Theorem 9.2.1, the schemes C G ( z ) and C P ( z ) are isomorphic if z , . . . , z n are distinctand close to zero. By Theorem 12.4.1, the schemes C G ( ) and C P ( ) are isomorphic aswell. Claims (i) and (ii) hold for the scheme C P ( z ) by Theorem 4.2.1 because C P ( z ) is asubscheme of the scheme C D ( z ). (cid:3) XXX model. In Sec-tions 10.1 and 10.2, we define an algebra epimorphism ψ DP : A D → A P , a linear epimor-phism ξ : A D → Sing L [ l ] as the composition of linear maps A D φ −→ A ∗ D τ −→ Sing W a,d [ l ] σ −→ Sing L [ l ] . and an algebra epimorphism ψ DL : A D → A L as the composition ψ W L ψ DW . ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 41 For the homogeneous XXX model, we have Lemmas 10.2.1 and 10.2.2 and the followinganalogue of Lemma 10.2.3. For the homogeneous XXX model, the kernel of ξ coincides with thekernel of ψ DP .Proof. The proof is similar to the proof of Lemma 10.2.3 with Theorem 12.4.1 replacingTheorem 9.2.1. (cid:3) For the homogeneous XXX model, the algebras A P , A L and A G areisomorphic. Denote by ψ P L : A P → A L the isomorphism induced by ψ DL and ψ DP . We have thefollowing analogue of Theorem 10.3.1. For the homogeneous XXX model, the linear map ξ induces a linearisomorphism ζ : A P → Sing L [ l ] which intertwines the multiplication operators L f , f ∈ A P , on A P and the action ofthe Bethe algebra A L on Sing L [ l ] , that is, for any f, g ∈ A P we have ζ ( L f ( g )) = ψ P L ( f )( ζ ( g )) . (cid:3) A L of the homogeneous XXX model is diagonalizable.12.6.1. Theorem. For the homogeneous XXX model, all elements of A L are diagonal-izable operators.Proof. Let v + be a highest gl -weight vector of L (1 , and v − = e v + . Then v + , v − forma basis of L (1 , . Consider the Hermitian form on L ( ) for which the vectors v i ⊗ · · · ⊗ v i n with i j ∈ { + , −} generate an orthonormal basis of L ( ). For any X ∈ End (cid:0) L ( ) (cid:1) , denote by X † theHermitian conjugate operator with respect to this Hermitian form. It is clear that (cid:0) (1 ⊗ ( j − ⊗ e ab ⊗ ⊗ ( n − j ) ) | L ( ) (cid:1) † = (1 ⊗ ( j − ⊗ e ba ⊗ ⊗ ( n − j ) ) | L ( ) . Using the fact that ( e + e ) | L (1 , = 1 and the definition of the coproduct (2.2), it isstraightforward to verify by induction on n that (cid:0) T ab ( u ) | L ( ) (cid:1) † = ( − a + b + n T − a, − b ( − ¯ u − | L ( ) , where ¯ u is the complex conjugate of u . Therefore, (cid:0) ( T ( u ) + T ( u )) | L ( ) (cid:1) † = − ( T ( − ¯ u − 1) + T ( − ¯ u − | L ( ) . This means that for any X ∈ A L , the Hermitian conjugate operator X † lies in A L . Hence,any element of A L commutes with its Hermitian conjugate and, therefore, is diagonaliz-able. (cid:3) The proof is similar to the proof of Corollary 10.3.2,because every element of A L is diagonalizable by Theorem 12.6.1. (cid:3) The algebras A G and A L are isomorphic. So, byTheorem 12.6.1 every element f ∈ A G is diagonalizable. Therefore, the algebra A G is thedirect sum of one-dimensional local algebras. Hence C G considered as a set consists ofdim A G (cid:0) nl (cid:1) − (cid:0) nl − (cid:1) distinct points, see Lemma 12.4.2. Theorem 12.1.2 is proved. Assume that v ∈ Sing L [ l ] is an eigenvector of the Bethe algebra A L , that is, ψ W L ( H s ) v = λ s v for suitable λ s ∈ C and s = 1 , . . . , n . Then by Corollary 7.4 in [MTV2],the difference equation (cid:0) u n − (2 u n + λ u n − + · · · + λ n ) ϑ − + ( u + 1) n ϑ − (cid:1) w ( u ) = 0has two linearly independent polynomial solutions, one of degree l and the other of degree n − l + 1. The following corollary of Theorem 12.1.1 gives the converse statement. Assume that ( λ , . . . , λ n ) ∈ C n is a pointsuch that λ = n , λ = l ( l − − n ) − n ( n − , and the difference equation (cid:0) u n − (2 u n + λ u n − + · · · + λ n ) ϑ − + ( u + 1) n ϑ − (cid:1) w ( u ) = 0 has two linearly independent polynomial solutions. Then there exists a unique up to nor-malization eigenvector v ∈ Sing L [ l ] of the action of the Bethe algebra A L of the homo-geneous XXX model such that for every s = 1 , . . . , n we have ψ W L ( H s ) v = λ s v . The proof of Corollary 12.8.2] is similar to the proof of Corollary 11.2.3. Assume that ( λ , . . . , λ n ) ∈ C n is a point satisfying the assumptions of Corol-lary 12.8.2. In order to find the eigenvector v ∈ Sing L [ l ], indicated in Corollary 12.8.2,one needs to apply the procedure described in Section 11.2.4. ETHE ALGEBRA OF HOMOGENEOUS XXX MODEL HAS SIMPLE SPECTRUM 43 References [B1] R. Baxter, Exactly solved models in statistical mechanics , Academic Press, Inc.,London, 1982[B2] R. 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