ANNALS OF MATHEMATICS
The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
By Evgeny Mukhin, Vitaly Tarasov, and Alexander Varchenko
SECOND SERIES, VOL. 170, NO. 2 September, 2009
anmaah
Annals of Mathematics, 170 (2009), 863–881
The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
By EVGENY MUKHIN, VITALY TARASOV, and ALEXANDER VARCHENKO
Abstract
We prove the B. and M. Shapiro conjecture that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This, in particular, implies the following result: If all ramification points of a parametrized rational curve ރސ1 ރސr lie W ! on a circle in the Riemann sphere ރސ1, then maps this circle into a suitable real subspace ޒސr ރސr . The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has real spectrum. In Appendix A, we discuss properties of differential operators associated with Bethe vectors in the Gaudin model. In particular, we prove a statement, which may be useful in complex algebraic geometry; it claims that certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of the corresponding Gaudin Hamiltonians is simple. In Appendix B, we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with Lie algebras of types Ar , Br and Cr .
1. The B. and M. Shapiro conjecture
1.1. Statement of the result. Fix a natural number r > 1. Let V ރŒx be a vector subspace of dimension r 1. The space V is called real if it has a basis C consisting of polynomials in ޒŒx.
Tarasov is supported in part by RFFI grant 05-01-00922. Varchenko is supported in part by NSF grant DMS-0244579.
863 864 EVGENY MUKHIN, VITALY TARASOV, and ALEXANDER VARCHENKO
For a given V , there exists a unique linear differential operator r 1 r d C d d D r 1 1.x/ r r .x/ r 1.x/; D dx C C dx C C dx C C whose kernel is V . This operator is called the fundamental differential operator of V . The coefficients of the operator are rational functions in x. The space V is real if and only if all coefficients of the fundamental operator are real rational functions. The Wronskian of functions f1; : : : ; fi in x is the determinant 0 .1/ .i 1/1 f1 f1 f1 .1/ .i 1/ Bf2 f2 f2 C Wr.f ; : : : ; f / det B C : 1 i B : : : C D @ : : : A . / . / f f 1 f i 1 i i i Let f1; : : : ; fr 1 be a basis of V . The Wronskian of the basis does not depend on the choice of theC basis up to multiplication by a number. The monic representative is called the Wronskian of V and denoted by WrV .
THEOREM 1.1. If all roots of the polynomial WrV are real, then the space V is real.
This statement is the B. and M. Shapiro conjecture formulated in 1993. The conjecture is proved in [EG02b] for r 1; see a more elementary proof also for D r 1 in [EG05]. The conjecture, its supporting evidence, and applications are D discussed in [EG02b], [EG02a], [EG05], [EGSV06], [ESS06], [KS03], [RSSS06], [Sot97a], [Sot97b], [Sot99], [Sot00b], [Sot03], [Sot00a] and [Ver00].
1.2. Parametrized rational curves with real ramification points. For a projec- r tive coordinate system .v1 vr 1/ on the complex projective space ރސ , the W W C subset of points with real coordinates is called the real projective subspace and is denoted by ޒސr . 1 r Let ރސ ރސ be a parametrized rational curve. If .u1 u2/ are projective W !1 W r coordinates on ރސ and .v1 vr 1/ are projective coordinates on ރސ , then W W C is given by the formula
.u1 u2/ . 1.u1; u2/ r 1.u1; u2//; W W 7! W W C where i are homogeneous polynomials of the same degree. We assume that at 1 any point of ރސ at least one component i is nonzero. Choose the local affine 1 coordinate u u1=u2 on ރސ and local affine coordinates v1=vr 1; : : : ; vr =vr 1 D C C on ރސr . In these coordinates, the map takes the form  à f1.u/ fr .u/ (1.1) f u ;:::; ; where fi .u/ i .u; 1/. W 7! fr 1.u/ fr 1.u/ D C C THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 865
The map is said to be ramified at a point of ރސ1 if its first r derivatives at this point do not span ރސr [KS03]. More precisely, a point u is a ramification point if the vectors f .1/.u/; : : : ; f .r/.u/ are linearly dependent. We assume that a generic point of ރސ1 is not a ramification point.
THEOREM 1.2. If all ramification points of the parametrized rational curve lie on a circle in the Riemann sphere ރސ1, then maps this circle into a suitable real subspace ޒސr ރސr . A maximally inflected curve is, by definition [KS03], a parametrized real rational curve whose ramification points are all real. Theorem 1.2 implies the existence of maximally inflected curves for every placement of the ramification points. Theorem 1.2 follows from Theorem 1.1. Indeed, if all ramification points lie on a circle, then after a linear change of coordinates .u1 u2/, we may assume that W the ramification points lie on the real line ޒސ1 and that the point .0 1/ is not a W r ramification point. After a linear change of coordinates .v1 vr 1/ on ރސ , W W C we may assume that r 1 is not zero at any of the ramification points. Let us use C the affine coordinates u u1=u2 and v1=vr 1; : : : ; vr =vr 1, and formula (1.1). D C . / C . / Then the determinant of coordinates of the vectors f 1 .u/; : : : ; f r .u/ is equal to  à f1 fr 1 Wr ;:::; ; 1 .u/ r 1 Wr.f1; : : : ; fr ; fr 1/.u/: fr 1 fr 1 D .fr 1/ C C C C C Hence the vectors f .1/.u/; : : : ; f .r/.u/ are linearly dependent if and only if the 1 Wronskian of f1; : : : ; fr 1 at u is zero. Since not all points of ރސ are ramification C points, the complex span V of polynomials f1; : : : ; fr 1 is an .r 1/-dimensional C C space. By assumptions of Theorem 1.2, all zeros of the Wronskian of V are real. By Theorem 1.1, the space V is real. This means that there exist projective coordinates r on ރސ in which all polynomials f1; : : : ; fr 1 are real. Theorem 1.2 is deduced from Theorem 1.1. C
1.3. Reduction of Theorem 1.1 to a special case.
THEOREM 1.3. If all roots of the Wronskian are real and simple, then V is real.
We deduce Theorem 1.1 from Theorem 1.3. Indeed, let V0 be an .r 1/- C dimensional space of polynomials whose Wronskian has real roots only. Let d be the degree of a generic polynomial in V0. Let ރ Œx be the space of polynomials of degree not greater than d. d Let G.r 1; d/ be the Grassmannian of .r 1/-dimensional vector subspaces C C in ރd Œx. 866 EVGENY MUKHIN, VITALY TARASOV, and ALEXANDER VARCHENKO
Let ސ.ރ.r 1/.d r/Œx/ be the projective space associated with the vector space C ރ.r 1/.d r/Œx. C The varieties G.r 1; d/ and ސ.ރ.r 1/.d r/Œx/ have the same dimension. The C C assignment V WrV defines a finite morphism 7! G.r 1; d/ ސ.ރ.r 1/.d r/Œx/ W C ! C I see for example [Sot97b] and [EG02b]. The space V0 is a point of G.r 1; d/. C Since is finite and V0 has Wronskian with real roots only, there exists a continuous curve V G.r 1; d/ for Œ0; 1/ such that the Wronskian of V 7! 2 C 2 for > 0 has simple real roots only. By Theorem 1.3, the space V is real for > 0. Hence, the fundamental differential operator of V has real coefficients. Therefore, the fundamental differential operator of V0 has real coefficients and the space V0 is real. Theorem 1.1 is deduced from Theorem 1.3.
1.4. The upper bound for the number of complex vector spaces with the same exponents at infinity and the same Wronskian. Let f1; : : : ; fr 1 be a basis of V C such that deg fi di for some sequence d d1 < < dr 1 . We say that V has D D f C g exponents d at infinity. If V has exponents d at infinity, then deg WrV n, where Pr 1 Qn D n i C1 .di i 1/. Let T s 1.x zs/ be a polynomial in x with simple D D C D D (complex) roots z1: : : : ; zn. Then the upper bound for the number of complex vector spaces V with exponents d at infinity and Wronskian T is given by the number N.d/ defined as follows. Consider the Lie algebra slr 1 with Cartan decomposition C slr 1 n h n C D ˚ ˚ C and simple roots ˛1; : : : ; ˛r h . Fix the invariant inner product on h by the 2 condition .˛i ; ˛i / 2. For any integral dominant weight ƒ h , denote by Lƒ D 2 the irreducible slr 1-module with highest weight ƒ. Let !r h be the last C 2 fundamental weight. For i 1; : : : ; r, introduce the numbers D Pi li j 1.dj j 1/ D D C and the integral dominant weight Pr (1.2) ƒ.d/ n!r i 1 li ˛i : D D Set N.d/ to be the multiplicity of the module Lƒ.d/ in the n-fold tensor product n L L! L! : ˝!r D r ˝ ˝ r According to Schubert calculus, the number of complex .r 1/-dimensional C vector spaces V with exponents d at infinity and Wronskian T is not greater than the THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 867 number N.d/. This is a standard statement of Schubert calculus; see for example [MV04, 5]. Thus, in order to prove Theorem 1.3, it is enough to prove this:
THEOREM 1.4. For generic real z1; : : : ; zn, there exist exactly N.d/ distinct Qn real vector spaces V with exponents d at infinity and with Wronskian T s 1.x D D zs/.
1.5. Structure of the paper. In Section 2, for generic complex z1; : : : ; zn, we construct exactly N.d/ distinct complex vector spaces V with exponents d at infinity and Wronskian T . In Section 3, we show that all of these vector spaces are real if z1; : : : ; zn are real. This proves Theorem 1.4. The constructions of Sections 2 and 3 are the Bethe ansatz constructions for n the Gaudin model on L!˝r . In Appendix A, we discuss properties of differential operators associated with the Bethe vectors in the Gaudin model and give applications of the Bethe ansatz constructions of Section 3. In particular, we prove Corollary A.3, which may be useful in complex algebraic geometry; it claims that certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of the corresponding Gaudin Hamiltonians is simple; cf. [EH83] and [MV04]. In Appendix B, we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with Lie algebras of types Ar , Br , Cr .
2. Construction of spaces of polynomials
2.1. Construction of (not necessarily real) spaces with exponents d at infinity Qn and with Wronskian T s 1.x zs/ having simple roots. Write z .z1; : : : ; zn/. D D D Introduce a function of l1 lr variables C C t .t .1/; : : : ; t .1/; : : : ; t .r/; : : : ; t .r// D 1 l1 1 lr by the formula
lr n r Y Y .r/ 1 Y Y .i / .i / 2 (2.1) ˆd.t z/ .t zs/ .t t / I D j j s j 1 s 1 i 1 16j The master functions arise in the hypergeometric solutions of the KZ equations [SV91], [Var95] and in the Bethe ansatz method for the Gaudin model [RV95], [SV03], [MV00], [MV04], [MV05], [Var06]. For more general master functions, see Appendix B. In particular, the master function (2.1) corresponds to the collection .!r ;:::;!r / of integral dominant slr 1 weights and the integral dominant weight ƒ.d/; see (1.2). C The product of symmetric groups †l † † acts on the variables t D l1 lr by permuting the coordinates with the same upper index. The master function is †l -invariant. We call a point t with complex coordinates a critical point of ˆd . z/ if I  à 1 @ˆd ˆd .i / .t z/ 0 for i 1; : : : ; r and j 1; : : : ; li : @tj I D D D In other words, a point t will be called a critical point if the system l1 l2 X 2 X 1 (2.2) 0 ; D t .1/ t .1/ C t .1/ t .2/ s 1;s j j s s 1 j s D ¤ D li li 1 li 1 X 2 X 1 XC 1 0 ; D t .i / t .i / C t .i / t .i 1/ C t .i / t .i 1/ s 1;s j j s s 1 j s s 1 j s C D ¤ D D n lr lr 1 X 1 X 2 X 1 0 .r/ .r/ .r/ .r/ .r 1/ D t zs t t C t t s 1 j s 1;s j j s s 1 j s D D ¤ D of l1 lr equations is satisfied, where j 1; : : : ; l1 in the first group of C C D equations, i 2; : : : ; r 1 and j 1; : : : ; li in the second group of equations, and D D j 1; : : : ; lr in the last group of equations. We require that all denominators in D these equations are not equal to zero. In the Gaudin model, the equations (2.2) are called the Bethe ansatz equations. The set of critical points of ˆd . z/ is †l -invariant. I t For a critical point t, define the tuple y .y1; : : : ; yr / of polynomials in D variable x by li Y .i / (2.3) yi .x/ .x t / for i 1; : : : ; r: D j D j 1 D Consider the .r 1/-st order linear differential operator C d T ÁÁ d yr ÁÁ d y2 ÁÁ d Á Dt ln0 ln0 ln0 ln0.y1/ ; D dx yr dx yr 1 dx y1 dx where ln0.f / denotes .df =dx/=f for any f . Denote by Vt the kernel of Dt . THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 869 Call Dt the fundamental operator of the critical point t, and call Vt the fundamental space of the critical point t. THEOREM 2.1 [MV04, 5]. The fundamental space Vt is an .r 1/-dimen- C sional space of polynomials with exponents d at infinity and Wronskian T . The tuple t y can be recovered from the fundamental space Vt as follows. Let f1; : : : ;fr 1 be C a basis of Vt consisting of polynomials with deg fi di for all i. Then y1; : : : ; yr D are respective scalar multiples of the polynomials f1; Wr.f1; f2/; Wr.f1; f2; f3/; : : : ; Wr.f1; : : : ; fr /: Thus distinct orbits of critical points define distinct .r 1/-dimensional spaces C V with exponents d at infinity and Wronskian T . THEOREM 2.2 [MV05, Th. 6.1]. For generic complex z1; : : : ; zn, the master function ˆd . z/ has N.d/ distinct orbits of critical points. I Therefore, by Theorems 2.1 and 2.2, we constructed N.d/ distinct spaces of polynomials with Wronskian T . All these spaces are fundamental spaces of critical points of the master function ˆd . z/. I 3. Bethe vectors 3.1. Generators. Let Ei;j for i; j 1; : : : ; r 1 be the standard generators D C of glr 1. The elements Ei;j for i j and Hi Ei;i Ei 1;i 1 for i 1; : : : ; r C ¤ D C C D are the standard generators of slr 1. We have slr 1 n h n , where C C D C ˚ ˚ r M M M n ރ Ei;j ; h ރ Hi ; n ރ Ei;j : C D D D i E v E 1 1 E 1 1 v E n n E n n v I i 1;i i 1;i !r i1 1;i1 ik 1;ik !r D 1 C 1 k1 C k1 ˝ ˝ C n C n n in L Œ and scalar functions !I; labeled by .1; : : : ; r / †l , where !˝r D 2 !I; !I;;1.z1/:::!I;;n.zn/; D 1 !I;;j .zj / j j j j j : D .tI . / tI . // .tI . / tI . // .tI . / zj / 1 I 2 I kj 1I kj I kj I We set X X (3.1) !.t z/ !I; EI v: I D I P.l;n/ †l 2 2 The universal weight function is invariant with respect to the †l -action on vari- .i / ables tj . The universal weight function was introduced in [Mat90] and [SV91] to solve the KZ equations. The other formulas for the universal weight function can be found in [RSV05]. Examples. If n 2 and l .1; 1; 0; : : : ; 0/, then D D E2;1E3;2v! v! E3;2E2;1v! v! !.t z/ r ˝ r r ˝ r I D .t .1/ t .2//.t .2/ z / C .t .2/ t .1//.t .1/ z / 1 1 1 1 1 1 1 1 E2;1v! E3;2v! E3;2v! E2;1v! r ˝ r r ˝ r C .t .1/ z /.t .2/ z / C .t .2/ z /.t .1/ z / 1 1 1 2 1 1 1 2 v! E2;1E3;2v! v! E3;2E2;1v! r ˝ r r ˝ r : C .t .1/ t .2//.t .2/ z / C .t .2/ t .1//.t .1/ z / 1 1 1 2 1 1 1 2 If l .2; 0; : : : ; 0/, then D 1 1 Á 2 !.t z/ E v! v! I D .t .1/ t .1//.t .1/ z / C .t .1/ t .1//.t .1/ z / 2;1 r ˝ r 1 2 2 1 2 1 1 1 1 1 Á E2;1v! E2;1v! C .t .1/ z /.t .1/ z / C .t .1/ z /.t .1/ z / r ˝ r 1 1 2 2 2 1 1 2 1 1 Á 2 v! E v! : C .t .1/ t .1//.t .1/ z / C .t .1/ t .1//.t .1/ z / r ˝ 2;1 r 1 2 2 2 2 1 1 2 THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 871 The values of the universal weight function at the critical points of the master function are called the Bethe vectors. The Bethe vectors of critical points of the same †l -orbit coincide, since both the critical point equations and the universal weight function are †l -invariant. n The universal weight function takes values in L!˝r Œ. But if t is a critical point of the master function, then the Bethe vector !.t z/ belongs to the subspace I of singular vectors Sing L nŒ L nŒ; see [RV95] and comments on this fact !˝r !˝r in [MV05, 2]. By Theorem 2.2, the master function ˆd . z/ has N.d/ distinct orbits of I critical points for generic z. Form a list t1;:::; tN.d/ of representatives in each of the orbits. These critical points define a collection of Bethe vectors !.t1 z/; : : : ; I !.tN.d/ z/ belonging to Sing L nŒ. The space Sing L nŒ has dimension I !˝r !˝r N.d/. THEOREM 3.1 [MV05, Th. 6.1]. For generic z, the Bethe vectors form a basis n in Sing L!˝r Œ. n 3.3. The Gaudin model. The Gaudin Hamiltonians on Sing L!˝r Œ are certain n linear operators acting on Sing L!˝r Œ and (rationally) depending on a complex parameter x. We use the construction of the Gaudin Hamiltonians suggested in [Tal04] and [CT04]; see also [MTV06]. We consider the slr 1-module L!r as the gl -module of highest weight .0; : : : ; 0; 1/. C r 1 C To define the Gaudin Hamiltonians consider the differential operators n .s/ d X Ej;i Xi;j .x/ ıi;j for all i; j 1; : : : ; r 1, D dx x z D C s 1 s D .s/ .s 1/ .n s/ where ıi;j is the Kronecker symbol and Ej;i 1˝ Ej;i 1˝ . These n D ˝ ˝ differential operators act on L!˝r -valued functions in x. The order of Xij is one if i j and is zero otherwise. D Set X (3.2) M . 1/ X1; .1/.x/X2; .2/.x/ : : : Xr 1;.r 1/.x/; D C C †r 1 2 C where . 1/ denotes the sign of the permutation. The operator M is the row- determinant of the matrix .Xij /. For example, for r 1, we have D n .s/ n .s/ n .s/ n .s/ d X E1;1 Á d X E2;2 Á X E2;1 ÁX E1;2 Á M : D dx x z dx x z x z x z s 1 s s 1 s s 1 s s 1 s D D D D 872 EVGENY MUKHIN, VITALY TARASOV, and ALEXANDER VARCHENKO Write r 1 r d C d M r 1 M1.x/ r Mr 1.x/; D dx C C dx C C C n n where Mi .x/ L L are linear operators depending on x. The coefficients W !˝r ! !˝r M1.x/,..., Mr 1.x/ are called the Gaudin Hamiltonians. C LEMMA 3.2. The Gaudin Hamiltonians commute: ŒMi .u/; Mj .v/ 0 for all D n i, j , u and v. The Gaudin Hamiltonians commute with the glr 1-action on L!˝r ; n C in particular, they preserve Sing L!˝r Œ. The first statement is seen, for example, in [KS82], [Tal04], [CT04], and [MTV06, Prop. 8.2]. The second statement is seen, for example, in [KS82] and in [MTV06, Prop. 8.3]. THEOREM 3.3 [MTV06, Th. 9.2]. For any critical point t of the master function ˆd . z/, the Bethe vector !.t z/ is an eigenvector of Mi .x/ for i I I D 1; : : : ; r 1. The corresponding eigenvalues i .x/ are given by the formula C r 1 r d C d r 1 1.x/ r r 1.x/ dx C C dx C C C D d Á d y2 ÁÁ d yr ÁÁ d T ÁÁ ln0.y1/ ln0 ln0 ln0 : dx C dx C y1 dx C yr 1 dx C yr Set r 1 r d C d r 1 K r 1 r M1.x/ . 1/ C Mr 1.x/ D dx C dx C C C r 1 r d C d r 1 K1.x/ r Kr 1.x/: D dx C C dx C C C This is the differential operator that is formally adjoint to the differential op- r 1 n n erator . 1/ M. The coefficients Ki .x/ L L are linear operators C W !˝r ! !˝r depending on x. These coefficients can be expressed as differential polynomials in M1.x/; : : : ; Mr 1.x/. For instance, C d K1.x/ M1.x/; K2.x/ M2.x/ r M1.x/; D D dx and so on. Similarly, the operators M1.x/; : : : ; Mr 1.x/ can be expressed as C differential polynomials in K1.x/; : : : ; Kr 1.x/. C By Lemma 3.2, the operators K1.x/; : : : ; Kr 1.x/ pairwise commute, that is, C ŒKi .u/; Kj .v/ 0 for all i, j , u and v, and they commute with the glr 1-action n D C on L!˝r . The operators K1.x/; : : : ; Kr 1.x/ will be called the Gaudin Hamiltonians, C just like the operators M1.x/; : : : ; Mr 1.x/. C For any critical point t of the master function ˆd . z/, the Bethe vector I !.t z/ is an eigenvector of the Gaudin Hamiltonians Ki .x/ for i 1; : : : ; r 1 by I D C THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 873 Theorem 3.3 and Lemma 3.2. The corresponding eigenvalues i .x/ are given by the formula r 1 r d C d r 1 1.x/ r r 1.x/ du C C dx C C C d T ÁÁ d yr ÁÁ d y2 ÁÁ d Á ln0 ln0 ln0 ln0.y1/ : D dx yr dx yr 1 dx y1 dx Note that this is the fundamental differential operator Dt of the critical point t. COROLLARY 3.4. For generic z, the Bethe vectors form an eigenbasis of the Gaudin Hamiltonians Ki .x/ for i 1; : : : ; r 1, D C the operators K1.x/; : : : ; Kr 1.x/ have simple joint spectrum, that is, their eigenvalues separate the basisC Bethe eigenvectors. The first statement follows from Theorem 3.1 and Theorem 3.3. Let us prove the second statement. If two Bethe vectors have the same eigen- values, then they have the same fundamental operators, hence the same fundamental spaces. The fundamental space of a critical point uniquely determines the orbit of the critical point by Theorem 2.1. Hence the two Bethe vectors correspond to the same orbit of critical points and hence are equal. 3.4. The Shapovalov form and real z. Define an anti-involution glr 1 glr 1;Ei;j Ej;i for all i; j . W C ! C 7! Let W be a highest weight glr 1-module with highest weight vector w. The Shapovalov form on W is the uniqueC symmetric bilinear form S defined by the conditions S.w; w/ 1 and S.gu; v/ S.u; .g/v/ D D for all u; v W and g gl ; see [Kac90]. The Shapovalov form is nondegenerate 2 2 r 1 on an irreducible module WCand is positive definite on the real part of the irreducible module W . Let Lƒ1 Lƒn be the tensor product of irreducible highest weight glr 1- ˝ ˝ C modules. Let vƒ Lƒ be a highest weight vector and Si the corresponding i 2 i Shapovalov form on Lƒi . Define the symmetric bilinear form on the tensor product by the formula S S1 Sn: The form S is called the tensor Shapovalov D ˝ ˝ form. THEOREM 3.5 [MTV06, Prop. 9.1]. The Gaudin Hamiltonians Ki .x/ for i 1; : : : ; r 1 are symmetric with respect to the tensor Shapovalov form S: D C S.Ki .x/u; v/ S.u; Ki .x/v/ for all i; x; u; v: D 874 EVGENY MUKHIN, VITALY TARASOV, and ALEXANDER VARCHENKO COROLLARY 3.6. If all of z1; : : : ; zn; x are real numbers, then the Gaudin Hamiltonians Ki .x/ for i 1; : : : ; r 1 are real linear operators on the real part D C of the tensor product Lƒ Lƒ . These operators are symmetric with respect 1 ˝ ˝ n to the positive definite tensor Shapovalov form. Hence they are simultaneously diagonalizable and have real spectrum. 3.5. Proof of Theorem 1.4. If z1; : : : ; zn; x are real, then all of the Gaudin n Hamiltonians on Sing L!˝r Œ have real spectrum, since they are symmetric opera- tors on a Euclidean space. If t is a critical point of ˆd . z/, then the eigenvalues I 1.x/; : : : ; r 1.x/ of the corresponding Bethe vector !.t z/ are real rational C I functions. Hence the fundamental differential operator Dt has real coefficients. Therefore, the fundamental vector space of polynomials Vt is real. Thus for generic real z1; : : : ; zn we have N.d/ distinct real spaces of polynomials with exponents Qn d at infinity and Wronskian s 1.x zs/. Thus Theorem 1.4 is proved. D Appendix A A.1. The differential operator K has polynomial solutions only. Suppose z1; : : : ; zn ރ. Let ƒ1; : : : ; ƒn; ƒ h be dominant integral weights. Assume 2 1 2 that the irreducible slr 1-module Lƒ is a submodule of the tensor product C 1 Lƒ1 Lƒn . ˝ ˝ Pi For any s 1; : : : ; n; , and i 1; : : : ; r, set ms;i .ƒs; j 1 ˛j / and D 1 D D D r n 1 X X Á l ms;i m ;i : D r 1 1 C i 1 s 1 D D For any s 1; : : : ; n, we will consider the slr 1-module Lƒs as the glr 1- D C C module of highest weight .0; ms;1; ms;2;:::; ms;r /. Considered as a sub- module of the glr 1-module Lƒ1 Lƒn , the slr 1-module Lƒ has the C ˝ ˝ C 1 glr 1-highest weight C . l; l m ;1; l m ;2;:::; l m ;r /: 1 1 1 THEOREM A.1. Consider the operator K as a differential operator acting on Lƒ Lƒ -valued functions in x. 1 ˝ ˝ n (i) Then all singular points of the operator K are regular and lie in the set z1; : : : ; zn; . f 1g (ii) Let u.x/ be any germ of an Lƒ Lƒ -valued function such that Ku 0. 1˝ ˝ n D Then u is the germ of an Lƒ Lƒ -valued polynomial in x. 1 ˝ ˝ n (iii) Let w Sing.Lƒ1 Lƒn /Œƒ be an eigenvector of the operators 2 ˝ ˝ 1 K1.x/; : : : ; Kr 1.x/ with the eigenvalues 1.x/; : : : ; r 1.x/, respectively. C C THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 875 Consider the scalar differential operator r 1 r d C d Dw r 1 1.x/ r r 1.x/: D dx C C dx C C C Then the exponents of the differential operator Dw at are 1 l; m ;1 1 l; : : : ; m ;r r l: 1 1 (iv) If z1; : : : ; zn are distinct, then for any s 1; : : : ; n, the exponents of the differ- D ential operator Dw at zs are 0; ms;1 1; : : : ; ms;r r. C C (v) The kernel of the differential operator Dw is an .r 1/-dimensional space of C polynomials. Proof. Part (i) is a direct corollary of the definition of the operator K. We first prove part (ii) in the special case of ƒ1 ƒn !r and generic D D D z1; : : : ; zn. By construction, the operator K commutes with the glr 1-action on n C L!˝r . This fact and Theorem 3.1 imply that K has an eigenbasis consisting of the Bethe vectors and their images under the glr 1-action. Then by Theorems 3.3 and 2.1, all solutions of the differential equationCKu 0 are polynomials. D The proof of part (ii) for arbitrary ƒ1; : : : ; ƒn and z1; : : : ; zn clearly follows from the special case and the following remarks: The operator K is well defined for any z1; : : : ; zn, not necessarily distinct, and rationally depends on z1; : : : ; zn. If for generic z1; : : : ; zn, all solutions of the differential equation Ku 0 are D polynomial, then for any z1; : : : ; zn, all solutions of the differential equation Ku 0 are polynomial. D Assume that some of z1; : : : ; zn coincide. Partition the set z1; : : : ; zn into f g several groups of coinciding points of sizes n1; : : : ; nk, whose sum is n. Denote the representatives in the groups by u1; : : : ; uk ރ, where u1; : : : ; uk are ns 2 distinct. Denote Ws L!˝ for s 1; : : : ; k. Choose an irreducible module D r D L Ws for every s. Then the operator K defined for those z1; : : : ; zn on s W1 W preserves the space of functions with values in the submodule ˝ ˝ k L L . If we restrict K to the space of functions with values in 1 ˝ ˝ k L L , then this restriction coincides with the operator K defined 1 ˝ ˝ k for the tensor product L L and u1; : : : ; u . 1 ˝ ˝ k k Any highest weight irreducible finite dimensional glr 1-module with highest C weight .m0; : : : ; mr /, where m0 ޚ60, is a submodule of a suitable tensor 2 power of L! (considered as the gl -module with highest weight .0; : : : ; 0; r r 1 1/). C Part (ii) is proved. 876 EVGENY MUKHIN, VITALY TARASOV, and ALEXANDER VARCHENKO To calculate the exponents of the operator Dw at singular points, we calculate the exponents of its formal adjoint operator. Namely, we consider the operator r 1 r d C d r 1 Dw r 1 r 1.x/ . 1/ C r 1.x/ D dx C dx C C C r 1 r d C d r 1 1.x/ r r 1.x/: D dx C C dx C C C The vector w is an eigenvector of the operators M1.x/; : : : ; Mr 1.x/, with eigen- C values 1.x/; : : : ; r 1.x/, respectively. C LEMMA A.2. Let the exponents of Dw at a point x z be p1; : : : ; pr 1. Then D C the exponents of Dw at the point x z are r pr 1; : : : ; r p1. D C Consider the U.glr 1/-valued polynomial C X (A.1) A.x/ . 1/ .x r/ı1; .1/ E .1/;1 D †r 1 2 C .x 1/ır; .r/ E .r/;r xır 1; .r 1/ E .r 1/;r 1 : C C C C It is known that the coefficients of this polynomial are central elements in U.glr 1/; C see for example [MNO96, Remark 2.11]. If v is a singular vector of a glr 1-weight C .p1; : : : ; pr 1/, then formula (A.1) yields C r 1 YC A.x/v .x r 1 i pi /v: D C i 1 D Hence, the operator A.x/ acts on Lƒs as the identity operator multiplied by r Y s.x/ .x r i ms;i /: D C C i 0 D Let s 1; : : : ; n. It follows from (3.2) that the indicial polynomial of Dw at D .s 1/ .n s/ the singular point zs is the eigenvalue of the operator 1 A.x/ 1 ˝ ˝ ˝ ˝ acting on the vector w, that is, s.x/. Similarly, the indicial polynomial of Dw at infinity is the eigenvalue of A. x/ acting on the vector w that belongs to the submodule Lƒ of the glr 1-module Lƒ1 Lƒn , that is, 1 C ˝ ˝ r Y .x/ . x r i l m ;i /: 1 D C C C 1 i 0 D Hence, by Lemma A.2, the exponents of the operator Dw are as required. This proves parts (iii) and (iv). Part (v) follows from parts (i)–(iv). THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 877 COROLLARY A.3. Assume that the operators Ki .x/; : : : ; Kr 1.x/ acting on C the subspace of weight singular vectors Sing.Lƒ1 Lƒn /Œƒ are diagonal- ˝ ˝ 1 izable and have simple joint spectrum. Then there exist dim Sing.Lƒ1 Lƒn /Œƒ ˝ ˝ 1 distinct polynomial .r 1/-dimensional spaces V with the following properties. If C D is the fundamental differential operator of such a space, then D has singular points at z1; : : : ; zn; only, with the exponents 1 0; ms;1 1; : : : ; ms;r r at zs for any s; C C l; m ;1 1 l; : : : ; m ;r r l at : 1 1 1 Consider all .r 1/-dimensional polynomial spaces V , whose fundamental C operator has exponents at z1; : : : ; zn; as indicated in Corollary A.3. Schubert 1 calculus says that the number of such spaces is not greater than the dimension of Sing.Lƒ1 Lƒn /Œƒ ; see for example [MV04]. Thus, according to ˝ ˝ 1 Corollary A.3, the simplicity of the spectrum of the Gaudin Hamiltonians on Sing.Lƒ1 Lƒn /Œƒ implies the transversality of the Schubert cycles ˝ ˝ 1 corresponding to these exponents at z1; : : : ; zn; ; cf. [MV04] and [EH83]. 1 The operators K1.x/; : : : ; Kr 1.x/ acting on Sing.Lƒ1 Lƒn /Œƒ C ˝ ˝ 1 are diagonalizable if z1; : : : ; zn are real; see Section 3.4. Remark A.4. It was conjectured in [CT04] that the monodromy of the dif- ferential operator M, acting on Lƒ Lƒ -valued functions in x, is trivial. 1 ˝ ˝ n However, the proof of this statement in [CT04] is not satisfactory. On the other hand, Theorem A.1 implies that the monodromy of the differential operator K, acting on Lƒ Lƒ -valued functions in x, is trivial. Together with Theorem 1˝ ˝ n 3.5, this implies that the monodromy of the operator M is trivial as well. A.2. Bethe vectors in Sing.Lƒ1 Lƒn /Œƒ . Let z .z1; : : : ; zn/ be n ˝ ˝ 1 D a point in ރ with distinct coordinates. Let ƒ1; : : : ; ƒn; ƒ h be dominant 1 2 integral weights. Assume that the irreducible slr 1-module Lƒ is a submodule C 1 of the tensor product Lƒ1 Lƒn . ˝ ˝ Pn Pr Introduce l .l1; : : : ; lr / by the formula ƒ s 1 ƒs i 1 li ˛i . Set D 1 D D C l l1 lr . Consider the associated master function D C C r li n r Y Y Y .i / .ƒs ;˛i / Y Y .i / .i / 2 ˆ.t z/ .t zs/ .t t / I D j j s i 1j 1 s 1 i 1 16j l n Consider the universal weight function ! ރ ރ .Lƒ1 Lƒn /Œƒ W ! ˝ ˝ 1 defined by the formulas of Section 3.2. The value !.t z/ of the universal weight I function at a critical point t of the master function ˆ. z/ is called a Bethe vector I (see [RV95] and [MV04]), and belongs to Sing.Lƒ1 Lƒn /Œƒ ; see [RV95]. t ˝ ˝ 1 For a critical point t, define the tuple y .y1; : : : ; yr / of polynomials in D variable x by formulas of Section 2.1. Define polynomials T1;:::;Tr in x by the formula n Y .ƒs ;˛i / Ti .x/ .x zs/ : D s 1 D Consider the linear differential operator of order r 1 given by C d T1 Tr ÁÁ d yr T1 Tr 1 ÁÁ Dt ln0 ln0 D dx yr dx yr 1 d y2T1 ÁÁ d Á ln0 ln0.y1/ : dx y1 dx All singular points of Dt are regular and lie in z1; : : : ; zn; . The exponents f 1g of Dt at zs are 0; ms;1 1; : : : ; ms;r r for any s, and the exponents of Dt C C at are l; m ;1 1 l; : : : ; m ;r r l. The kernel Vt of Dt is an 1 1 1 .r 1/-dimensional space of polynomials; see [MV04]. C t The tuple y can be recovered from Vt as follows. Let f1; : : : ; fr 1 be a C basis of Vt consisting of monic polynomials of strictly increasing degree. Then y1; : : : ; yr are respective scalar multiples of the polynomials Wr.f1; f2/ Wr.f1; f2; f3/ Wr.f1; : : : ; fr / f1; ; 2 ;:::; 2 r 1 T1 T2T1 Tr 1Tr 2 :::T1 I see [MV04]. THEOREM A.5 [MTV06, Th. 9.2]. For any critical point t of the master function ˆ. z/, the Bethe vector !.t z/ is an eigenvector of Ki .x/ for i I I D 1; : : : ; r 1 and the corresponding eigenvalues 1.x/; : : : ; r 1.x/ are given by C C the formula r 1 r d C d r 1 1.x/ r r 1.x/ Dt : du C C dx C C C D COROLLARY A.6. Any two distinct nonzero Bethe vectors cannot have the same eigenvalues for all Gaudin Hamiltonians. The proof of the corollary is similar to the proof of the second statement of Corollary 3.4. Appendix B Let g be a simple Lie algebra, let h its Cartan subalgebra, let ˛i h for 2 i 1; : : : ; r be simple roots, and let . ; / be the standard invariant scalar product D THEB.ANDM.SHAPIROCONJECTUREANDTHEBETHEANSATZ 879 on g. Let ƒ .ƒ1; : : : ; ƒn/ be integral dominant weights of g. Let l .l1; : : : ; lr / D D be nonnegative integers such that the weight Pn Pr ƒ s 1 ƒs i 1 li ˛i 1 D D D is dominant integral. Let z .z1; : : : ; zn/ be distinct complex numbers. Introduce D the associated master function depending on variables t .t .1/; : : : ; t .1/; : : : ; t .r/; : : : ; t .r// D 1 l1 1 lr by the formula r li n r Y Y Y .i / .ƒs ;˛i / Y Y .i / .i / .˛i ;˛i / ˆg;ƒ;l .t z/ .t zs/ .t t / I D j j s i 1j 1 s 1 i 1 16j Acknowledgments We thank A. Eremenko and A. Gabrielov for useful discussions. References [CT04] A. V. 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MR 2007m:32016 Zbl 05184506 [Ver00] J.VERSCHELDE, Numerical evidence for a conjecture in real algebraic geometry, Experi- ment. Math. 9 (2000), 183–196. MR 2001i:65062 Zbl 1054.14080 (Received May 8, 2006) (Revised August 27, 2007) E-mail address: [email protected] DEPARTMENT OF MATHEMATICAL SCIENCES,INDIANA UNIVERSITY–PURDUE UNIVERSITY, 402 NORTH BLACKFORD ST.,INDIANAPOLIS, IN 46202-3216, UNITED STATES E-mail address: [email protected] E-mail address: [email protected] ST.PETERSBURG BRANCHOF STEKLOV MATHEMATICAL INSTITUTE,FONTANKA 27, ST.PE- TERSBURG, 191023, RUSSIA and DEPARTMENT OF MATHEMATICAL SCIENCES,INDIANA UNIVERSITY–PURDUE UNIVERSITY, 402 NORTH BLACKFORD ST.,INDIANAPOLIS, IN 46202-3216, UNITED STATES E-mail address: [email protected] DEPARTMENT OF MATHEMATICS,UNIVERSITYOF NORTH CAROLINA AT CHAPEL HILL,CHAPEL HILL, NC 27599-3250, UNITED STATES