q -Hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma theorem
aa r X i v : . [ m a t h . AG ] A ug q -HYPERGEOMETRIC SOLUTIONS OF QUANTUM DIFFERENTIALEQUATIONS, QUANTUM PIERI RULES, AND GAMMA THEOREM VITALY TARASOV ◦ AND ALEXANDER VARCHENKO ⋆ ⋆ Department of Mathematics, University of North Carolina at Chapel HillChapel Hill, NC 27599-3250, USA ⋆ Faculty of Mathematics and Mechanics, Lomonosov Moscow State UniversityLeninskiye Gory 1, 119991 Moscow GSP-1, Russia ◦ Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis402 North Blackford St, Indianapolis, IN 46202-3216, USA ◦ St. Petersburg Branch of Steklov Mathematical InstituteFontanka 27, St. Petersburg, 191023, Russia
Key words : Flag varieties, quantum differential equation, dynamical connection, q -hyper-geometric solutions : 82B23, 17B80, 14N15, 14N35 Abstract.
We describe q -hypergeometric solutions of the equivariant quantum differentialequations and associated qKZ difference equations for the cotangent bundle T ∗ F λ of apartial flag variety F λ . These q -hypergeometric solutions manifest a Landau-Ginzburgmirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantumequivariant cohomology of the cotangent bundle. Our Gamma theorem for T ∗ F λ says thatthe leading term of the asymptotics of the q -hypergeometric solutions can be written asthe equivariant Gamma class of the tangent bundle of T ∗ F λ multiplied by the exponentialsof the equivariant first Chern classes of the associated vector bundles. That statementis analogous to the statement of the gamma conjecture by B. Dubrovin and by S. Galkin,V. Golyshev, and H. Iritani, see also the Gamma theorem for F λ in Appendix B. In memory of Victor Lomonosov (1946 – 2018)
Contents
1. Introduction 32. Dynamical and qKZ equations 42.1. Notations 42.2. Dynamical differential equations 42.3. Difference qKZ equations 53. Weight functions 53.1. Weight functions ˇ W I W σ,I ◦ E -mail : [email protected] , [email protected] ⋆ E -mail : [email protected] , supported in part by NSF grants DMS-1362924, DMS-1665239
VITALY TARASOV AND ALEXANDER VARCHENKO W I ( t ; z ) 73.5. Modification of the three-term relation 73.6. Shuffle properties 73.7. Factorization 83.8. Useful identities 84. Master function and discrete differentials 94.1. Master function 94.2. Definition of discrete differentials 104.3. Special discrete differentials 104.4. First key formula 114.5. Second key formula 125. Proof of Theorem 4.3 126. Proof of Theorem 4.4 146.1. Proof of Theorem 4.4 for N = 2, λ = ( n, I = ( { , . . . , n } , ∅ ) 146.2. Proof of Theorem 4.4 for N = 2 and I = I max i = 1, arbitrary N , and I = ( { , . . . , n } , ∅ , . . . , ∅ ) 166.4. Proof of Theorem 4.4 for i = 1, arbitrary N , and I = I max i >
1, arbitrary N , and I = ( { , . . . , n } , ∅ , . . . , ∅ ) 176.6. Proof of Theorem 4.4 for i >
1, arbitrary λ , and I = I max H ∗ T ( T ∗ F λ )-valued weight function 299.5. Quantum multiplication by divisors on H ∗ T ( T ∗ F λ ) 3010. Quantum Pieri rules 3310.1. Quantum equivariant cohomology algebra H ˜ qT ( T ∗ F λ ) 3310.2. Quantum equivariant Pieri rules 3410.3. Bethe ansatz equations 3510.4. Proof of Theorem 10.2 3510.5. Limit ˜ q i / ˜ q i +1 → i = 1 , . . . , N − K -theory 3611.1. Solutions and equivariant K -theory 3611.2. End of proof of Lemma 11.2 3811.3. The homogeneous case z = 0 3911.4. The limit h → ∞ UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 3 Introduction
In [MO], D. Maulik and A. Okounkov develop a general theory connecting quantum groupsand equivariant quantum cohomology of Nakajima quiver varieties, see [N1, N2]. In partic-ular, in [MO] the operators of quantum multiplication by divisors are described. As itis well-known, these operators determine the equivariant quantum differential equations ofa quiver variety. In this paper we apply this description to the cotangent bundles T ∗ F λ of the gl n N -step partial flag varieties and construct q -hypergeometric solutions of theassociated equivariant quantum differential equations and qKZ difference equations. The q -hypergeometric solutions are constructed in the form of Jackson integrals.Studying solutions of the equivariant quantum differential equations may lead to betterunderstanding Gromov-Witten invariants of the cotangent bundle, cf. Givental’s study ofthe J -function in [Gi1, Gi2, Gi3].The presentation of solutions of the equivariant quantum differential equations as q -hypergeometric integrals manifests a version of the Landau-Ginzburg mirror symmetry forthe cotangent bundle.In [MO] the equivariant quantum differential equations come together with a compatiblesystem of difference equations called the qKZ equations. In [GRTV, RTV1] the equivari-ant quantum differential equations and qKZ difference equations were identified with thedynamical differential equations and qKZ difference equations with values in the tensorproduct ( C N ) ⊗ n of vector representations of gl N . The q -hypergeometric solutions of the( C N ) ⊗ n -valued qKZ difference equations were constructed long time ago in [TV1], see also[TV2]-[TV4]. It was expected that those q -hypergeometric solutions are also solutions ofthe compatible dynamical differential equations. That fact is proved in this paper and is thefirst main result of the paper. The proof is based on some new rather nontrivial identitiesfor the integrand of the Jackson integral. The integrand is the product of the scalar masterfunction and a vector-valued function, whose coordinates are called weight functions. In[RTV1] it was shown that the weight functions are nothing else but the stable envelopesof [MO] for the cotangent bundle of the partial flag varieties. Our new identities can beinterpreted as new identities for stable envelopes. We interpret these new identities as Pierirules in quantum equivariant cohomology of the cotangent bundle of the partial flag variety.That is our second main result.Our Gamma theorem for T ∗ F λ (Theorem B.1) says that the leading term of the asymp-totics of the q -hypergeometric solutions for T ∗ F λ is the product of the equivariant gammaclass of the tangent bundle of T ∗ F λ and the exponentials of the equivariant first Chern classesof the associated vector bundles. That statement is analogous to the statement of the gammaconjecture by B. Dubrovin and by S. Galkin, V. Golyshev, and H. Iritani, see Appendix B.See also the Gamma theorem for F λ (Theorem B.2).The paper is organized as follows. In Section 2 we introduce the ( C N ) ⊗ n -valued dynamicaland qKZ equations. In Section 3 we define the weight functions and list their basic properties.In Section 4 we introduce the master function and describe the discrete differentials — thequantities with zero Jackson integrals. We also formulate there two key identities for theweight functions — Theorems 4.3 and 4.4. We prove Theorem 4.3 in Section 5 and Theorem4.4 in Section 6. In Section 7, we summarize Theorems 4.3 and 4.4 as a statement about theintegrand of the main Jackson integral. In Section 8 we construct integral representations VITALY TARASOV AND ALEXANDER VARCHENKO for solutions of the ( C N ) ⊗ n -valued dynamical equations. In Section 9 we introduce theequivariant quantum differential equations and explain how their q -hypergeometric solutionsare obtained from solutions of the ( C N ) ⊗ n -valued dynamical equations. In Section 10 weformulate and prove Pieri rules. In Section 11 we show that the space of solutions of thequantum differential equation can be identified with the vector space of the equivariant K-theory algebra. We also discuss two limiting cases of the quantum differential equation. InAppendix A we discuss the basic properties of Schubert polynomials, and in Appendix B weformulate our Gamma theorems.The authors thank G. Cotti, V. Golyshev, and R. Rimanyi for useful discussions. Thesecond author thanks the Hausdorff Institute for Mathematics in Bonn for hospitality inMarch 2018, when the Gamma theorem was discovered. The second author also thanks theMax Planck Institute for Mathematics in Bonn for hospitality in May–June 2018.2. Dynamical and qKZ equations
Notations.
Fix
N, n ∈ Z > and h, κ ∈ C × . Let λ ∈ Z N > , | λ | = λ + . . . + λ N = n . Let I = ( I , . . . , I N ) be a partition of { , . . . , n } into disjoint subsets I , . . . , I N . Denote I λ theset of all partitions I with | I j | = λ j , j = 1 , . . . , N .Consider C N with basis v i = (0 , . . . , , i , , . . . , i = 1 , . . . , N , and the tensor product( C N ) ⊗ n with basis v I = v i ⊗ · · · ⊗ v i n , where the index I is a partition ( I , . . . , I N ) of { , . . . , n } into disjoint subsets I , . . . , I N and i j = m if j ∈ I m .The space ( C N ) ⊗ n is a module over the Lie algebra gl N with basis e i,j , i, j = 1 , . . . , N .The gl N -module ( C N ) ⊗ n has weight decomposition ( C N ) ⊗ n = P | λ | = n ( C N ) ⊗ n λ , where ( C N ) ⊗ n λ is the subspace with basis ( v I ) I ∈I λ .2.2. Dynamical differential equations.
Define the linear operators X , . . . , X n actingon ( C N ) ⊗ n -valued functions of z = ( z , . . . , z n ) , q = ( q , . . . , q N ) and called the dynamicalHamiltonians: X i ( z ; h ; q ) = n X a =1 z a e ( a ) i,i − h (cid:16) ˜ e i,i (1 − ˜ e i,i )2 + X a
Define the R -matrices acting on ( C N ) ⊗ n , R ( i,j ) ( u ) = u − hP ( i,j ) u − h , i, j = 1 , . . . , n , i = j . Define the qKZ operators K , . . . , K n acting on ( C N ) ⊗ n : K i ( z ; h ; q ; κ ) = R ( i,i − ( z i − z i − + κ ) . . . R ( i, ( z i − z + κ ) ×× q e ( i )1 , . . . q e ( i ) N,N N R ( i,n ) ( z i − z n ) . . . R ( i,i +1) ( z i − z i +1 ) . The qKZ operators preserve the weight decomposition of ( C N ) ⊗ n and form a discrete flatconnection, K i ( z , . . . , z j + κ, . . . , z n ; q ; κ ) K j ( z ; h ; q ; κ ) = K j ( z , . . . , z i + κ, . . . , z n ; q ; κ ) K i ( z ; h ; q ; κ )for all i, j , see [FR]. The system of difference equations with step κ ,(2.4) f ( z , . . . , z i + κ, . . . , z n ; q ) = K i ( z ; h ; q ; κ ) f ( z , . . . , z n ; q ) , i = 1 , . . . , N , on a ( C N ) ⊗ n -valued function f ( z , q ) is called the qKZ equations. Theorem 2.1 ([TV2]) . The systems of dynamical and qKZ equations are compatible. (cid:3) Weight functions
Weight functions ˇ W I . For I ∈ I λ , we define the weight functions ˇ W I ( t ; z ), cf. [TV1,TV4, RTV1]. The functions ˇ W I ( t ; z ) here coincide with the functions W I ( t ; z ; h ) definedin [RTV1, Section 3.1].Recall λ = ( λ , . . . , λ N ). Denote λ ( i ) = λ + . . . + λ i , i = 1 , . . . , N − λ ( N ) = n , and λ { } = P N − i =1 λ ( i ) = P N − i =1 ( N − i ) λ i . Recall I = ( I , . . . , I N ). Set S jk =1 I k = { i ( j )1 < . . . i ( j ) a ( t ( j ) a − t ( j +1) d ) λ ( j ) Y b = a +1 t ( j ) a − t ( j ) b − ht ( j ) a − t ( j ) b (cid:19) . VITALY TARASOV AND ALEXANDER VARCHENKO
In these formulas for a function f ( t , . . . , t k ) of some variables, we denoteSym t ,...,t k f ( t , . . . , t k ) = X σ ∈ S k f ( t σ , . . . , t σ k ) . Example.
Let N = 2, n = 2, λ = (1 , I = ( { } , { } ), J = ( { } , { } ). Thenˇ W I ( t ; z ) = − h ( t (1)1 − z ) , ˇ W J ( t ; z ) = − h ( t (1)1 − z − h ) . Example.
Let N = 2, n = 3, λ = (1 , I = ( { } , { , } ). Thenˇ W I ( t ; z ) = − h ( t (1)1 − z − h ) ( t (1)1 − z ) . Example.
Let N = 2, n = 3, λ = (2 , I = ( { , } , { } ). Thenˇ W I ( t ; z ) = ( − h ) (cid:16) ( t (1)1 − z ) ( t (1)1 − z ) ( t (1)2 − z − h )( t (1)2 − z − h ) t (1)1 − t (1)2 − ht (1)1 − t (1)2 ++ ( t (1)2 − z ) ( t (1)2 − z ) ( t (1)1 − z − h )( t (1)1 − z − h ) t (1)2 − t (1)1 − ht (1)2 − t (1)1 (cid:17) . For a subset A = { a , . . . , a j } ⊂ { , . . . , n } , denote z A = ( z a , . . . , z a j ). For I ∈ I λ , denote z I = ( z I , . . . , z I N ). For f ( t (1) , . . . , t ( N ) ) ∈ C [ t (1) , . . . , t ( N ) ] S λ (1) × ... × S λ ( N ) , we define f ( z I ) bysubstituting t ( j ) = ( z I , . . . , z I j ) , j = 1 , . . . , N .3.2. Weight functions ˇ W σ,I . For σ ∈ S n and I ∈ I λ , we define(3.2) ˇ W σ,I ( t ; z ) = ˇ W σ − ( I ) ( t ; z σ (1) , . . . , z σ ( n ) ) , ˇ U σ,I ( t ; z ) = ˇ U σ − ( I ) ( t ; z σ (1) , . . . , z σ ( n ) ) , where σ − ( I ) = ( σ − ( I ) , . . . , σ − ( I N )). Example.
Let N = 2, n = 2, λ = (1 , I = ( { } , { } ), J = ( { } , { } ). Thenˇ W id , I ( t ; z ) = − h ( t (1)1 − z ) , ˇ W id , J ( t ; z ) = − h ( t (1)1 − z − h ) , ˇ W s, I ( t ; z ) = − h ( t (1)1 − z − h ) , ˇ W s, J ( t ; z ) = − h ( t (1)1 − z ) , where s is the transposition.3.3. Three-term relation.Lemma 3.1 ([RTV1, Lemma 3.6]) . For any σ ∈ S n , I ∈ I λ , i = 1 , . . . , n − , we have (3.3) ˇ W σs i,i +1 ,I = z σ ( i ) − z σ ( i +1) z σ ( i ) − z σ ( i +1) + h ˇ W σ,I + hz σ ( i ) − z σ ( i +1) + h ˇ W σ,s σ ( i ) ,σ ( i +1) ( I ) , where s i,j ∈ S n is the transposition of i and j . (cid:3) UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 7
Weight functions W I ( t ; z ) . Let σ ∈ S n be the longest permutation, σ ( i ) = n +1 − i , i = 1 , . . . , n . For I ∈ I λ , denote(3.4) W I ( t ; z ) = ( − h ) − λ { } ˇ W σ ,I ( t ; z ) , U I ( t ; z ) = ˇ U σ ,I ( t ; z ) . In other words, we have(3.5) W I ( t ; z ) = Sym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − U I ( t ; z ) ,U I ( t ; z ) =(3.6) = N − Y j =1 λ ( j ) Y a =1 (cid:18) λ ( j +1) Y c =1 i ( j +1) c i ( j ) a ( t ( j ) a − t ( j +1) d − h ) λ ( j ) Y b = a +1 t ( j ) b − t ( j ) a − ht ( j ) b − t ( j ) a (cid:19) . Example.
Let N = 2, n = 2, λ = (1 , I = ( { } , { } ), J = ( { } , { } ). Then W I ( t ; z ) = t (1)1 − z − h , W J ( t ; z ) = t (1)1 − z . Modification of the three-term relation.
For a function f ( z , . . . , z n ) and i = 1 ,. . . , n −
1, define the operator S i,i +1 by the formula(3.7) S i,i +1 f ( z , . . . , z n ) = z i − z i +1 − hz i − z i +1 f ( z , . . . , z i +1 , z i , . . . , z n ) + hz i − z i +1 f ( z , . . . , z n ) . Lemma 3.1 can be reformulated as follows.
Lemma 3.2.
For any I ∈ I λ , i = 1 , . . . , n − , we have W s i,i +1 ( I ) ( t ; z ) = ( S i,i +1 W I )( t ; z )(3.8) = z i − z i +1 − hz i − z i +1 W I ( t ; z , . . . , z i +1 , z i , . . . , z n ) + hz i − z i +1 W I ( t ; z , . . . , z n ) . (cid:3) Shuffle properties.
Let n, n , n ∈ Z > , n = n + n . Let λ , λ , λ ∈ Z N > , | λ | = n , | λ | = n , λ = λ + λ . Let I = ( I , . . . , I N ) be a decomposition of the set { , . . . , n } intosubsets such that | I j | = λ j . Let I = ( I , . . . , I N ) be a decomposition of the set { n + 1 , . . . ,n } into subsets such that | I j | = λ j . Define the decomposition I = ( I , . . . , I N ) of the set { ,. . . , n } by the rule: I j = I j ∪ I j .Consider the weight function W I of variables ( t (1) , . . . , t ( N ) ), where t ( j ) = ( t ( j )1 , . . . , t ( j ) λ ( j ) ), λ ( j ) = λ + . . . + λ j , j = 1 , . . . , N −
1, and t ( N ) = ( z , . . . , z n ).Consider the weight function W I of variables (˜ t (1) , . . . , ˜ t ( N ) ), where ˜ t ( j ) = ( t ( j )1 , . . . , t ( j )( λ ) (1) ),( λ ) ( j ) = λ + . . . + λ j , j = 1 , . . . , N −
1, and ˜ t ( N ) = ( z , . . . , z n ). Consider the weight function W I of variables ˇ t = (ˇ t (1) , . . . , ˇ t ( N ) ), where ˇ t ( j ) = ( t ( j )( λ ) ( j ) +1 , . . . , t ( j ) λ ( j ) ) for j = 1 , . . . , N − t ( N ) = ( z n +1 , . . . , z n ). Denote ( λ ) ( j ) = λ + . . . + λ j , j = 1 , . . . , N − VITALY TARASOV AND ALEXANDER VARCHENKO
Define the connection coefficient C λ , λ ( t ; z ) = N − Y j =1 h(cid:16) ( λ ) ( j ) Y a =1 λ ( j ) Y b =( λ ) ( j ) +1 t ( j ) b − t ( j ) a − ht ( j ) b − t ( j ) a (cid:17) × (3.9) × (cid:16) ( λ ) ( j ) Y a =1 λ ( j +1) Y c =( λ ) ( j +1) +1 ( t ( j ) a − t ( j ) c − h ) (cid:17)(cid:16) λ ( j ) Y a =( λ ) ( j ) +1 ( λ ) ( j +1) Y c =1 ( t ( j ) a − t ( j ) c ) (cid:17)i . Lemma 3.3.
We have (3.10) W I ( t ; z ) = Sym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − e W I ,I ( t ; z ) Q N − j =1 (( λ ) ( j ) )!(( λ ) ( j ) )! , where e W I ,I ( t ; z ) = C λ , λ ( t ; z ) W I (˜ t (1) , . . . , ˜ t ( N − ; z , . . . , z n ) × W I (ˇ t (1) , . . . , ˇ t ( N − ; z n +1 , . . . , z n ) . (cid:3) Factorization.
Consider the gl N weight function W { ,...,n } , ∅ ,..., ∅ of variables ( t (1) , . . . , t ( N ) ), where t ( j ) = ( t ( j )1 , . . . , t ( j ) n ) for j = 1 , . . . , N −
1, and t ( N ) = ( z , . . . , z n ). Lemma 3.4.
We have (3.11) W { ,...,n } , ∅ ,..., ∅ ( t (1) , . . . , t ( N − ; t ( N ) ) = N − Y j =1 W gl { ,...,n } , ∅ ( t ( j ) ; t ( j +1) ) , where W gl { ,...,n } , ∅ ( t ( j ) ; t ( j +1) ) is the gl weight function assigned to the partition of the set { ,. . . , n } into two subsets { , . . . , n } and ∅ .Proof. The function W gl { ,...,n } , ∅ ( t ( j ) ; t ( j +1) ) is symmetric in variables ( t ( j +1)1 , . . . , t ( j +1) n ) due tothe gl three-term relations of Lemma 3.1. That symmetry and formula (3.1) imply formula(3.11). (cid:3) Useful identities.Theorem 3.5.
Given k ∈ Z > , consider variables t (0)1 , t ( k +1)1 and t ( i )1 , t ( i )2 for i = 1 , . . . , k .Set F = ( t (0)1 − t (1)1 ) ( t ( k )1 − t ( k +1)1 − h ) − ( t (0)1 − t (1)2 − h ) ( t ( k )2 − t ( k +1)1 ) ,G = ( t (0)1 − t (1)2 − h ) ( t ( k )1 − t ( k +1)1 − h ) − ( t (0)1 − t (1)1 ) ( t ( k )2 − t ( k +1)1 ) , and H = k − Y i =1 (cid:0) ( t ( i )1 − t ( i +1)2 − h ) ( t ( i )2 − t ( i +1)1 ) (cid:1) k Y i =1 t ( i )2 − t ( i )1 − ht ( i )2 − t ( i )1 . Then (3.12) Sym t (1)1 , t (1)2 . . . Sym t ( k )1 , t ( k )2 ( F H ) = 0
UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 9 and (3.13) Sym t (1)1 , t (1)2 . . . Sym t ( k )1 , t ( k )2 ( GH ) = 0 . Proof.
Formulae (3.12), (3.13) are equivalent toSym t (1)1 , t (1)2 . . . Sym t ( k )1 , t ( k )2 (cid:0) ( F ± G ) H (cid:1) = 0 . Observe that F + G = (2 t (0)1 − t (1)1 − t (1)2 − h ) ( t ( k )1 − t ( k )2 − h ) ,F − G = ( t (1)1 − t (1)2 − h ) ( t ( k )1 + t ( k )2 − t ( k +1)1 − h ) , andSym s ,s (cid:16) ( s − u − h ) ( s − u ) s − s − hs − s (cid:17) = Sym u ,u (cid:16) ( s − u − h ) ( s − u ) u − u − hs − s (cid:17) . Hence the function W gl { , } , ∅ ( s , s , u , u ) = Sym s ,s (cid:16) ( s − u − h ) ( s − u ) s − s − hs − s (cid:17) is symmetric both in s , s and u , u . Therefore,Sym t (1)1 , t (1)2 . . . Sym t ( k )1 , t ( k )2 (cid:0) ( F + G ) H (cid:1) = (2 t (0)1 − t (1)1 − t (1)2 − h ) ×× Sym t ( k )1 , t ( k )2 (cid:16) ( t ( k )1 − t ( k )2 − h ) t ( k )2 − t ( k )1 − ht ( k )2 − t ( k )1 (cid:17) k − Y i =1 W gl { , } , ∅ ( t ( i )1 , t ( i )2 , t ( i +1)1 , t ( i +1)2 ) = 0 . and Sym t (1)1 , t (1)2 . . . Sym t ( k )1 , t ( k )2 (cid:0) ( F − G ) H (cid:1) = ( t ( k )1 + t ( k )2 − t ( k +1)1 − h ) ×× Sym t (1)1 , t (1)2 (cid:16) ( t (1)1 − t (1)2 − h ) t (1)2 − t (1)1 − ht (1)2 − t (1)1 (cid:17) k − Y i =1 W gl { , } , ∅ ( t ( i )1 , t ( i )2 , t ( i +1)1 , t ( i +1)2 ) = 0 . Theorem 3.5 is proved. (cid:3) Master function and discrete differentials
Master function.
Let φ ( x ) = Γ( x/κ ) Γ (cid:0) ( h − x ) /κ (cid:1) . Define the master function:Φ λ ( t ; z ; h ; q ) = ( e π √− n − λ N ) q N ) P na =1 z a /κ N − Y i =1 (cid:16) e π √− λ i +1 − λ i ) q i q i +1 (cid:17) P λ ( i ) j =1 t ( i ) j /κ × (4.1) × N − Y i =1 λ ( i ) Y a =1 (cid:18) λ ( i ) Y b =1 b = a t ( i ) a − t ( i ) b − h ) φ ( t ( i ) a − t ( i ) b ) λ ( i +1) Y c =1 φ ( t ( i ) a − t ( i +1) c ) (cid:19) . It is a symmetric function of variables in each of the groups t ( i ) , i = 1 , . . . , N − Definition of discrete differentials.
Consider the space S of functions of the formΦ λ ( t ; z ; h ; q ) f ( t ; z ; h ; q ) where f ( t ; z ; h ; q ) is a rational function. Consider the lattice κ Z λ { } whose coordinates are labeled by variables t ( i ) j ∈ t . The shifts t ( i ) j t ( i ) j + κ of any of the t -variables preserve the space S and extend to an action of the lattice κ Z λ { } on S . A discretedifferential is a finite sum of rational functions of the form(4.2) Φ λ ( t + w ; z ; h ; q )Φ λ ( t ; z ; h ; q ) f ( t + w ; z ; h ; q ) − f ( t ; z ; h ; q ) , where w ∈ κ Z λ { } .4.3. Special discrete differentials.
For integers 1 α < β N , split the variables t =( t (1) , . . . , t ( N − ) , t ( i ) = ( t ( i )1 , . . . , t ( i ) λ ( i ) ) , into two groups t { α, β } and t { α, β } as follows: t { α, β } = ( t ( α ) λ ( α ) , t ( α +1) λ ( α +1) , . . . , t ( β − λ ( β − )and t { α, β } = ( t (1) { α, β } , . . . , t ( N − { α, β } ) , where t ( i ) { α, β } = ( t ( i )1 , . . . , t ( i ) λ ( i ) − ) if α i < β and t ( i ) { α, β } = t ( i ) , otherwise.For a rational function g of t { α, β } , z , q , denote d t { α,β } g := g ( t { α, β } ; z ; h ; q ) q α − q β (cid:16) q β ( t ( α − λ ( α − − t ( α ) λ ( α ) ) β − Y i = α λ ( i − − Y a =1 ( t ( i − a − t ( i ) λ ( i ) ) × (4.3) × ( t ( β − λ ( β − − t ( β ) λ ( β ) − h ) β − Y i = α λ ( i +1) − Y a =1 ( t ( i ) λ ( i ) − t ( i +1) a − h ) β − Y i = α λ ( i ) − Y a =1 t ( i ) a − t ( i ) λ ( i ) − ht ( i ) a − t ( i ) λ ( i ) −− q α ( t ( α − λ ( α − − t ( α ) λ ( α ) − h ) β − Y i = α λ ( i − − Y a =1 ( t ( i − a − t ( i ) λ ( i ) − h ) ×× ( t ( β − λ ( β − − t ( β ) λ ( β ) (cid:1) β − Y i = α λ ( i +1) − Y a =1 ( t ( i ) λ ( i ) − t ( i +1) a ) β − Y i = α λ ( i ) − Y a =1 t ( i ) λ ( i ) − t ( i ) a − ht ( i ) λ ( i ) − t ( i ) a (cid:17) . Lemma 4.1.
The function d t { α,β } g is a discrete differential. (cid:3) Proof.
Formula (4.3) is an example of formula (4.2), where f ( t ; z ; h ; q ) = g ( t { α, β } ; z ; h ; q ) ( t ( α − λ ( α − − t ( α ) λ ( α ) ) β − Y i = α λ ( i − − Y a =1 ( t ( i − a − t ( i ) λ ( i ) ) ×× ( t ( β − λ ( β − − t ( β ) λ ( β ) − h ) β − Y i = α λ ( i +1) − Y a =1 ( t ( i ) λ ( i ) − t ( i +1) a − h ) β − Y i = α λ ( i ) − Y a =1 t ( i ) a − t ( i ) λ ( i ) − ht ( i ) a − t ( i ) λ ( i ) and w has coordinates t ( α ) λ ( α ) , t ( α +1) λ ( α +1) , . . . , t ( β − λ ( β − equal to κ and other coordinates equal tozero. (cid:3) UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 11
We rewrite d t { α,β } g as d t { α,β } g = q β q α − q β ˜ d t { α,β } g − ˇ d t { α,β } g, where ˜ d t { α,β } g := g ( t { α, β } ; z ; h ; q ) (cid:16) ( t ( α − λ ( α − − t ( α ) λ ( α ) ) β − Y i = α λ ( i − − Y a =1 ( t ( i − a − t ( i ) λ ( i ) ) × (4.4) × ( t ( β − λ ( β − − t ( β ) λ ( β ) − h ) β − Y i = α λ ( i +1) − Y a =1 ( t ( i ) λ ( i ) − t ( i +1) a − h ) β − Y i = α λ ( i ) − Y a =1 t ( i ) a − t ( i ) λ ( i ) − ht ( i ) a − t ( i ) λ ( i ) −− ( t ( α − λ ( α − − t ( α ) λ ( α ) − h ) β − Y i = α λ ( i − − Y a =1 ( t ( i − a − t ( i ) λ ( i ) − h ) ×× ( t ( β − λ ( β − − t ( β ) λ ( β ) (cid:1) β − Y i = α λ ( i +1) − Y a =1 ( t ( i ) λ ( i ) − t ( i +1) a ) β − Y i = α λ ( i ) − Y a =1 t ( i ) λ ( i ) − t ( i ) a − ht ( i ) λ ( i ) − t ( i ) a (cid:17) and ˇ d t { α,β } g := g ( t { α, β } ; z ; h ; q ) ( t ( α − λ ( α − − t ( α ) λ ( α ) − h ) β − Y i = α λ ( i − − Y a =1 ( t ( i − a − t ( i ) λ ( i ) − h ) × (4.5) × ( t ( β − λ ( β − − t ( β ) λ ( β ) (cid:1) β − Y i = α λ ( i +1) − Y a =1 ( t ( i ) λ ( i ) − t ( i +1) a ) β − Y i = α λ ( i ) − Y a =1 t ( i ) λ ( i ) − t ( i ) a − ht ( i ) λ ( i ) − t ( i ) a . Denote(4.6) d { α, β } g := Sym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − d t { α,β } g, (4.7) ˜ d { α, β } g := Sym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − ˜ d t { α,β } g, (4.8) ˇ d { α, β } g := Sym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − ˇ d t { α,β } g . Then(4.9) d { α, β } g = q β q α − q β ˜ d { α, β } g − ˇ d { α, β } g. Corollary 4.2.
The function d { α, β } g is a discrete differential. (cid:3) First key formula.
Let λ ∈ Z N > , | λ | = n . For α, β = 1 , . . . , N , α = β , denote(4.10) λ α, β = ( λ , . . . , λ α − , . . . , λ β + 1 , . . . , λ N ) . Notice that | λ α, β | = | λ | .Let I = ( I , . . . , I N ) ∈ I λ , I k = ( ℓ k, , . . . , ℓ k,λ k ), k = 1 , . . . , N . For α = β , a = 1 , . . . ,λ α , b = 1 , . . . , λ β , denote(4.11) ( I ) bβ, α = ( I , . . . , I α ∪ { ℓ β,b } , . . . , I β − { ℓ β,b } , . . . , I N ) ∈ I λ α,β . For J ∈ I λ α,β and b = 1 , . . . , λ β + 1, we have ( J ) bβ, α ∈ I λ . The function U J defined byformula (3.5) is a function of variables t α, β , z . Theorem 4.3.
We have (4.12) ( ˜ d { α, β } U J )( t ; z ) = − h λ β +1 X b =1 W ( J ) bβ,α ( t ; z ) . Theorem 4.3 is proved in Section 5.4.5.
Second key formula.
Let I = ( I , . . . , I N ) ∈ I λ , I k = ( ℓ k, , . . . , ℓ k,λ k ) . For k , k =1 , . . . , N , k = k , and m = 1 , . . . , λ k , m = 1 , . . . , λ k , define the element I k ,k ; m ,m =( ˜ I , . . . , ˜ I N ) ∈ I λ such that ˜ I k = I k if k = k , k , and(4.13) ˜ I k = I k ∪ { ℓ k ,m } − { ℓ k ,m } , ˜ I k = I k ∪ { ℓ k ,m } − { ℓ k ,m } . Theorem 4.4.
For I ∈ I λ and i = 1 , . . . , N − , we have (cid:16) λ ( i ) X j =1 t ( i ) j − λ ( i − X j =1 t ( i − j − X a ∈ I i z a (cid:17) W I =(4.14) = h i − X j =1 λ i X m =1 λ j X m =1 ℓ i,m >ℓ j,m W I i,j ; m ,m − h N X j = i +1 λ i X m =1 λ j X m =1 ℓ i,m <ℓ j,m W I i,j ; m ,m ++ N X j = i +1 λ i X a =1 ˇ d { i,j } U ( I ) ai,j − i − X j =1 λ j X a =1 ˇ d { j,i } U ( I ) aj,i . Theorem 4.4 is proved in Section 6.5.
Proof of Theorem 4.3
For n = 1 , Theorem 4.3 is the following statement. Lemma 5.1.
Let n = 1 . For γ n , let J γ = ( J , . . . , J N ) be the decomposition of theone-element set { } , such that J γ = { } and J j = ∅ for j = γ . Let α < β N .Then ˜ d { α, β } U J γ = − hW J α , β = γ , (5.1) ˜ d { α, β } U J γ = 0 , β = γ . (5.2) Proof.
For any γ , we have W J γ = U J γ , and U J γ is the function of t ( γ )1 , . . . , t ( N − , t ( N )1 = z ,which is identically equal to 1, see (3.5).If β = γ , then˜ d { α, β } U J γ = ˜ d { α,γ } U J γ = ( t ( γ − − t ( γ )1 − h ) − ( t ( γ − − t ( γ )1 ) = − h = − h W J α , which proves (5.1) . UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 13
The proof of (5.2) is by cases. If β < γ , then ˜ d { α, β } U J γ = (1 − · γ < α < β ,then ˜ d { α, β } U J γ = 0 by identity (3.12). If α < γ < β , then ˜ d { α, β } U J γ = 0 by identity (3.13).If α = γ < β , then ˜ d { α, β } U J γ = 0 by the degeneration of identity (3.12) as t (0)1 → ∞ . (cid:3) For arbitrary n , Theorem 4.3 follows by induction on n from the shuffle properties ofweight functions in Lemma 3.3. To avoid writing numerous indices we illustrate the reasoningby an example.Let N = 3, n = 2, J = ( ∅ , { , } , ∅ ), α = 1, β = 2. Then formula (4.12) reads(5.3) ˜ d { , } U J = − hW ( { } , { } , ∅ ) − hW ( { } , { } , ∅ ) . Indeed, we have˜ d { , } U J = Sym t (2)1 ,t (2)2 (cid:16)(cid:0) ( t (1)1 − t (2)1 − h ) ( t (1)1 − t (2)2 − h ) − ( t (1)1 − t (2)1 ) ( t (1)1 − t (2)2 ) (cid:1) ×× ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 (cid:17) == Sym t (2)1 ,t (2)2 (cid:16)(cid:0) ( t (1)1 − t (2)1 − h ) ( t (1)1 − t (2)2 − h ) − ( t (1)1 − t (2)1 ) ( t (1)1 − t (2)2 − h ) ++ ( t (1)1 − t (2)1 ) ( t (1)1 − t (2)2 − h ) − ( t (1)1 − t (2)1 )( t (1)1 − t (2)2 ) (cid:1) ×× ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 (cid:17) . This is the sum of four terms. The first two areSym t (2)1 ,t (2)2 (cid:16)(cid:0) ( t (1)1 − t (2)1 − h ) ( t (1)1 − t (2)2 − h ) − ( t (1)1 − t (2)1 ) ( t (1)1 − t (2)2 − h ) (cid:1) ×× ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 (cid:17) == − h Sym t (2)1 ,t (2)2 (cid:16) ( t (1)1 − t (2)2 − h ) ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 (cid:17) = − h W { } , { } , ∅ , the last two areSym t (2)1 ,t (2)2 (cid:16)(cid:0) ( t (1)1 − t (2)1 ) ( t (1)1 − t (2)2 − h ) − ( t (1)1 − t (2)1 ) ( t (1)1 − t (2)2 ) (cid:1) ×× ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 (cid:17) == − h Sym t (2)1 ,t (2)2 (cid:16) ( t (1)1 − t (2)2 ) ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 (cid:17) = − h W { } , { } , ∅ , and we get (5.3).The treatment of these four terms is an inductive step from n = 1 to n = 2. The analysisof the first two terms is the application of Theorem 4.3 for n = 1 at the first point z . Namely, the factor (cid:0) ( t (1)1 − t (2)1 − h ) − ( t (1)1 − t (2)2 ) (cid:1) corresponds to ˜ d { , } at z and the product( t (1)1 − t (2)2 − h ) ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 is the connection coefficient between W { } , ∅ , ∅ sitting at z and W ∅ , { } , ∅ sitting at z , seeLemma 3.3. And the analysis of the last two terms is the application of Theorem 4.3 for n = 1 at the second point z . Namely, the factor (cid:0) ( t (1)1 − t (2)1 − h ) − ( t (1)1 − t (2)2 ) (cid:1) correspondsto ˜ d { , } at z and the product( t (1)1 − t (2)2 ) ( t (2)1 − z − h ) ( t (2)2 − z ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 is the connection coefficient between W { } , ∅ , ∅ sitting at z and W ∅ , { } , ∅ sitting at z , seeLemma 3.3. 6. Proof of Theorem 4.4
Proof of Theorem 4.4 for N = 2 , λ = ( n, , I = ( { , . . . , n } , ∅ ) .Lemma 6.1. We have (6.1) n X l =1 ( t (1) l − z l ) W gl { ,...,n } , ∅ = n X a =1 ˇ d { , } U gl { ,...,a − ,a +1 ,...,n } , { a } . Proof.
We will prove formula (6.1) by induction on n . Denote t ′ = ( t (1)1 , . . . , t (1) n − ) , t ′′ = ( t (1)1 , . . . , t (1) n − ) , z ′ = ( z , . . . , z n − ) ,A n ( t , z ′ ) = n − Y b =1 (cid:16) ( t (1) n − z b ) t (1) n − t (1) b − ht (1) n − t (1) b (cid:17) , B ( t ′′ , z ) = n − Y a =1 ( t (1) a − z − h ) . By formulae (4.5), (4.8), equality (6.1) reads as follows:Sym t (1)1 ,...,t (1) n n X a =1 (cid:16) ( t (1) a − z a ) U gl { ,...,n } , ∅ ( t , z ) − (6.2) − ( t (1) n − z n ) A n ( t , z ′ ) U gl { ,...,a − ,a +1 ,...,n } , { a } ( t ′ , z ) (cid:17) = 0 . For n = 1 , formula (6.2) is clearly true. For the induction step, we explore formula (3.6). Itimplies that the summation term with a = n in formula (6.2) vanishes,(6.3) U gl { ,...,n } , ∅ ( t , z ) = A n ( t , z ) U gl { ,...,n − } , ∅ ( t ′ , z ′ ) B ( t ′′ , z n ) ( t (1) n − − z n − h ) , and for a < n , U gl { ,...,a − ,a +1 ,...,n } , { a } ( t ′ , z ) =(6.4) = ( t (1) n − − z n − ) A n − ( t ′ , z ′ ) U gl { ,...,a − ,a +1 ,...,n − } , { a } ( t ′′ , z ′ ) B ( t ′′ , z n ) . UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 15
The last formula and the identitySym t ( n ) n − , t ( n ) n ( t ( n ) n − z n ) t ( n ) n − t ( n ) n − − ht ( n ) n − t ( n ) n − = Sym t ( n ) n − , t ( n ) n ( t ( n ) n − − z n − h ) t ( n ) n − t ( n ) n − − ht ( n ) n − t ( n ) n − , yield Sym t (1)1 ,...,t (1) n ( t (1) n − z n ) A n ( t , z ′ ) U gl { ,...,a − ,a +1 ,...,n } , { a } ( t ′ , z ) =(6.5) = Sym t (1)1 ,...,t (1) n ( t (1) n − − z n − ) A n ( t , z ′ ) A n − ( t ′ , z ′ ) ×× U gl { ,...,a − ,a +1 ,...,n − } , { a } ( t ′′ , z ′ ) B ( t ′′ , z n ) ( t (1) n − − z n − h ) . Summarizing all observations, we see that formula (6.2) follows from the equalitySym t (1)1 ,...,t (1) n − A n ( t , z ′ ) B ( t ′′ , z n ) ( t (1) n − − z n − h ) × (6.6) × n − X a =1 (cid:16) ( t (1) a − z a ) U gl { ,...,n − } , ∅ ( t ′ , z ′ ) −− ( t (1) n − − z n − ) A n − ( t ′ , z ′ ) U gl { ,...,a − ,a +1 ,...,n − } , { a } ( t ′′ , z ′ ) (cid:17) = 0 , with t (1) n not involved in the symmetrization. Since the product A n ( t , z ′ ) B ( t ′′ , z n ) ( t (1) n − − z n − h )is symmetric in t (1)1 , . . . , t (1) n − , formula (6.6) follows from the induction assumptionSym t (1)1 ,...,t (1) n − n − X a =1 (cid:16) ( t (1) a − z a ) U gl { ,...,n − } , ∅ ( t ′ , z ′ ) − (6.7) − ( t (1) n − − z n − ) A n − ( t ′ , z ′ ) U gl { ,...,a − ,a +1 ,...,n − } , { a } ( t ′′ , z ′ ) (cid:17) = 0 . Lemma 6.1 is proved. (cid:3)
Proof of Theorem 4.4 for N = 2 and I = I max . For N = 2, λ = ( k, n − k ) , wedenote I max = (cid:0) { n − k + 1 , . . . , n } , { , . . . , n − k } (cid:1) . Then formula (4.14) becomes formula(6.8) below. Lemma 6.2.
We have k X l =1 ( t (1) l − z n − k + l ) W gl { n − k +1 ,...,n } , { ,...,n − k } (6.8) = n X a = n − k +1 ˇ d { , } U gl { n − k +1 ,...,a − ,a +1 ,...,n } , { ,...,n − k,a } , Proof.
Dividing both sides of the equation by Q ki =1 Q n − ka =1 ( t (1) l − z a ) , turns formula (6.8) intoformula (6.1). (cid:3) Proof of Theorem 4.4 for i = 1 , arbitrary N , and I = ( { , . . . , n } , ∅ , . . . , ∅ ) .Proposition 6.3. For I = ( { , . . . , n } , ∅ , . . . , ∅ ) , we have (6.9) (cid:16) n X l =1 t (1) l − n X a =1 z a (cid:17) W I = n X j =2 n X a =1 ˇ d { ,j } U ( I ) ′ a ,j . Proof.
Formula (6.9) is equivalent to the formula n X j =2 n X l =1 ( t ( j − l − t ( j ) l ) W I = n X j =2 n X a =1 ˇ d { ,j } U ( I ) ′ a ,j which follows from the next lemma. Lemma 6.4.
For j = 2 , . . . , N we have (6.10) n X l =1 ( t ( j − l − t ( j ) l ) W I = n X a =1 ˇ d { ,j } U ( I ) ′ a ,j Proof.
By Lemma 3.4, the left-hand side of (6.10) equals(6.11) (cid:16) n X l =1 ( t ( j − l − t ( j ) l ) W gl { ,...,n } , ∅ ( t ( j − ; t ( j ) ) (cid:17) N Y i =2 i = j W gl { ,...,n } , ∅ ( t ( i − ; t ( i ) ) . It is easy to see that the right hand side equals(6.12) (cid:16) n X a =1 ˇ d { j − ,j } U ( { ,...,a − ,a +1 ,...,n } , { a } ) ( t ( j − ; t ( j ) ) (cid:17) N Y i =2 i = j W gl { ,...,n } , ∅ ( t ( i − ; t ( i ) ) . Hence, Lemma 6.4 follows from formula (6.1). (cid:3)
Proposition 6.3 is proved. (cid:3)
Proof of Theorem 4.4 for i = 1 , arbitrary N , and I = I max . For λ = ( λ , . . . ,λ N ) , we denote I max = (cid:0) { n − λ + 1 , . . . , n } , . . . , { , . . . , λ N } (cid:1) . Then formula (4.14) takesthe form(6.13) N X j =2 λ X l =1 ( t ( j − l + λ ( j − − λ − t ( j ) l + λ ( j ) − λ ) W I max = N X j =2 λ X a =1 ˇ d { ,j } W ( I max ) ′ a ,j . The following lemma implies formula (6.13).
Lemma 6.5.
For j = 2 , . . . , N , we have (6.14) λ X l =1 ( t ( j − l + λ ( j − − λ − t ( j ) l + λ ( j ) − λ ) W I max = λ X a =1 ˇ d { ,j } W ( I max ) ′ a ,j . UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 17
Proof.
The left-hand side of formula (6.14) equalsSym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − (cid:16) λ X l =1 ( t ( j − l + λ ( j − − λ − t ( j ) l + λ ( j ) − λ )(6.15) × N Y m =2 U gl { λ m +1 ,...,λ ( m ) } , { ,...,λ m } ( t ( m − ; t ( m ) ) (cid:17) , while the right-hand side of (6.14) equals by definitionSym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − (cid:16) λ ( j ) Y l =1 ( t ( j − λ ( j − − t ( j ) l ) λ ( j − − Y l =1 t ( j − λ ( j − − t ( j − l − ht ( j − λ ( j − − t ( j − l (6.16) × λ ( j ) X b =1+ λ ( j ) − λ U gl { λ j +1 ,...,λ ( j ) − λ ,...,b − ,b +1 ,...,λ ( j ) } , { ,...,λ j ,b } ( t ( j − \ { t ( j − λ ( j − } , t ( j ) ) × N Y m =2 m = j U gl { λ m +1 ,...,λ ( m ) } , { ,...,λ m } ( t ( m − , t ( m ) ) (cid:17) . The equality of (6.15) and (6.16) follows from the following case of formula (6.1) : λ X l =1 ( t ( j − l + λ ( j − − λ − t ( j ) l + λ ( j ) − λ ) W gl { ,...,λ } , ∅ ( t ( j − λ ( j − − λ , . . . , t ( j − λ ( j − ; t ( j )1+ λ ( j ) − λ , . . . , t ( j ) λ ( j ) )= λ X a =1 ˇ d { , } U gl { ,...,a − ,a +1 ,...,λ } , { a } ( t ( j − λ ( j − − λ , . . . , t ( j − λ ( j − − ; t ( j )1+ λ ( j ) − λ , . . . , t ( j ) λ ( j ) ) . (cid:3) Proof of Theorem 4.4 for i > , arbitrary N , and I = ( { , . . . , n } , ∅ , . . . , ∅ ) . For i = 2 , . . . , N − I = ( { , . . . , n } , ∅ , . . . , ∅ ) , Theorem 4.4 says that n X l =1 ( t ( i ) l − t ( i − l ) W I = − n X a =1 ˇ d { ,i } U ( I ) ′ a ,j , which is formula (6.10).6.6. Proof of Theorem 4.4 for i > , arbitrary λ , and I = I max . To prove this caseof Theorem 4.4, we introduce a partition I max , j = ( I max , j , . . . , I max , jj ) of the set (1 , . . . , λ ( j ) )by the rule(6.17) I max , ja = { i | λ ( j ) − λ ( a ) < i λ ( j ) − λ ( a − } , so that | I max , ja | = λ a . For example, I max , N = I max . Formula (4.14) for I = I max can be written as j − X l =1 (cid:16)(cid:16) X i ∈ I max ,jl t ( j ) i − X i ∈ I max ,j − l t ( j − i (cid:17) W I max + λ l X a =1 ˇ d { l,j } U ( I max ) ′ al,j (cid:17) (6.18) + N X l = j +1 (cid:16)(cid:16) X i ∈ I max ,l − j t ( l − j − X j ∈ I max ,li t ( l ) i (cid:17) W I max − λ l X a =1 ˇ d { j,l } U ( I max ) ′ aj,l (cid:17) = 0 . Formula (6.18) follows from the next Proposition.
Proposition 6.6.
For i = 1 , , . . . , N − , and j = i, i + 1 , . . . , N − , we have (6.19) (cid:16) X l ∈ I max ,ji t ( j ) l − X l ∈ I max ,ji t ( j +1) l (cid:17) W I max = λ i X a =1 ˇ d { i,j +1 } U ( I max ) ′ ai,j +1 . Proof.
For i = 1 , formula (6.19) follows from Lemma 6.5. For i >
1, we prove formula(6.19) by induction on λ ( i − , see Lemmas 6.7 and 6.8 below. If λ ( i − = 0 , that is, λ j = 0for all j = 1 , . . . , i − I max , I max ,l on λ : I max λ = ( I max λ , , . . . , I max λ ,N ) , I max ,l λ = ( I max ,l λ , , . . . , I max ,l λ ,l ) . We fix i, j until the end of the proof of Proposition 6.6, and omit the condition | λ | = n . Lemma 6.7.
Assume that formula (6.19) holds for λ = (0 , . . . , , λ k , . . . , λ N ) with k i .Then formula (6.19) holds for ˜ λ = (0 , . . . , , , λ k , . . . , λ N ) .Proof. Formula (6.19) for λ has the form (cid:16) X l ∈ I max ,j λ ,i t ( j ) l − X l ∈ I max ,j +1 λ ,i t ( j +1) l (cid:17) Sym t ( k ) . . . Sym t ( N − (cid:0) U I max λ ( t ) (cid:1) (6.20) = Sym t ( k ) . . . Sym t ( N − (cid:16) C λ ,i,j +1 λ i X a =1 U ( I max λ ) ′ ai,j +1 ( t ) (cid:17) , where C λ , i,j +1 is the factor in the second and third lines of definition (4.5).In addition to the variables t = ( t ( k ) , . . . , t ( N ) ) appearing in formula (6.20), formula (6.19)for ˜ λ contains the new variables t new = (cid:0) t ( k − , t ( k )1+ λ ( k ) , t ( k +1)1+ λ ( k +1) , . . . , t ( N − λ ( N − , t ( N )1+ λ ( N ) (cid:1) ,and has the form (cid:16) X l ∈ I max ,j ˜ λ ,i t ( j ) l − X l ∈ I max ,j +1˜ λ ,i t ( j +1) l (cid:17) Sym ˜ t ( k ) . . . Sym ˜ t ( N − (cid:0) U I max˜ λ (˜ t ) (cid:1) (6.21) = Sym ˜ t ( k ) . . . Sym ˜ t ( N − (cid:16) C ˜ λ ,i,j +1 λ i X a =1 U ( I max˜ λ ) ′ ai,j +1 (˜ t ) (cid:17) , UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 19 where ˜ t = t ∪ t new = (˜ t ( k − , ˜ t ( k ) , . . . , ˜ t ( N ) ) . It is easy to see from definition (3.6) that(6.22) U I max˜ λ (˜ t ) = U I max λ ( t ) F (˜ t ) , where F (˜ t ) is the product of all factors appearing in (3.6) involving the interrelation of twovariables at least one of those being from t new . Moreover, F (˜ t ) is symmetric in the variables t ( l ) for each l = k, . . . , N . Furthermore, since I max ,j λ ,i = I max ,j ˜ λ ,i and I max ,j +1 λ ,i = I max ,j +1˜ λ ,i , thefirst factors in the left-hand sides of formulas (6.20) and (6.21) coincide.By all these observations, to get formula (6.21) from (6.20), we need to verify that(6.23) Sym ˜ t ( k ) . . . Sym ˜ t ( N − (cid:16) C ˜ λ , i,j +1 λ i X a =1 U ( I max˜ λ ) ′ ai,j +1 − F C λ , i,j +1 λ i X a =1 U ( I max λ ) ′ ai,j +1 (cid:17) = 0 . This equality follows from identity (3.12) for the variables t ( i − λ ( i − , t ( i ) λ ( i ) , t ( i )1+ λ ( i ) , . . . , t ( j ) λ ( j ) ,t ( j )1+ λ ( j ) , t ( j +1)1+ λ ( j +1) . Lemma 6.7 is proved. (cid:3) Example.
Let N = 5 , λ = (0 , , , ,
0) , ˜ λ = (1 , , , ,
0) . For i = 3 , j = 3 , formulas(6.20) and (6.23) take the form t (3)1 − t (4)1 = t (3)1 − t (4)1 andSym t (3)1 , t (3)2 Sym t (4)1 , t (4)2 (cid:16) ( t (3)1 − t (4)1 ) ( t (2)1 − t (3)1 ) ( t (3)1 − t (4)2 − h ) ( t (3)2 − t (4)1 ) × ( t (4)1 − t (5)2 − h ) ( t (4)2 − t (5)1 ) t (3)2 − t (3)1 − ht (3)2 − t (3)1 t (4)2 − t (4)1 − ht (4)2 − t (4)1 (cid:17) = Sym t (3)1 , t (3)2 Sym t (4)1 , t (4)2 (cid:16) ( t (3)1 − t (4)2 − h ) ( t (2)2 − t (3)1 ) ( t (3)2 − t (4)2 ) ( t (3)1 − t (4)1 ) × ( t (4)1 − t (5)2 − h ) ( t (4)2 − t (5)1 ) t (3)2 − t (3)1 − ht (3)2 − t (3)1 t (4)2 − t (4)1 − ht (4)2 − t (4)1 (cid:17) , respectively. The last equality follows from identity (3.12) for the variables t (2)1 , t (3)1 , t (3)2 ,t (4)2 .For i = 3 , j = 4 , formulas (6.20) and (6.23) take the form t (4)1 − t (5)1 = t (4)1 − t (5)1 andSym t (3)1 , t (3)2 Sym t (4)1 , t (4)2 (cid:16) ( t (4)1 − t (5)1 ) ( t (2)1 − t (3)1 ) ( t (3)1 − t (4)2 − h ) ( t (3)2 − t (4)1 ) × ( t (4)1 − t (5)2 − h ) ( t (4)2 − t (5)1 ) t (3)2 − t (3)1 − ht (3)2 − t (3)1 t (4)2 − t (4)1 − ht (4)2 − t (4)1 (cid:17) = Sym t (3)1 , t (3)2 Sym t (4)1 , t (4)2 (cid:16) ( t (2)1 − t (3)2 − h ) ( t (2)2 − t (3)1 ) ( t (3)1 − t (4)2 − h ) ( t (4)2 − t (5)1 ) × ( t (4)2 − t (5)2 ) ( t (4)1 − t (5)1 ) t (3)2 − t (3)1 − ht (3)2 − t (3)1 t (4)2 − t (4)1 − ht (4)2 − t (4)1 (cid:17) , respectively. The last equality follows from identity (3.12) for the variables t (2)1 , t (3)1 , t (3)2 ,t (4)1 , t (4)2 , t (5)2 . Lemma 6.8.
Assume that formula (6.19) holds for λ = (0 , . . . , , λ k , . . . , λ N ) with k < i and λ k > . Then formula (6.19) holds for ˜ λ = (0 , . . . , , , λ k + 1 , . . . , λ N ) .Proof. The proof is completely similar to that of Lemma 6.7. The only change is that thenew variables are t new = (cid:0) t ( k )1+ λ ( k ) , t ( k +1)1+ λ ( k +1) , . . . , t ( N − λ ( N − , t ( N )1+ λ ( N ) (cid:1) . (cid:3) Example.
Let N = 3 , λ = (1 , ,
0) , ˜ λ = (2 , ,
0) . For i = 2 , j = 2 , formula (6.23) prooffollows from identity (3.12) for the variables t (1)2 , t (2)2 , t (2)3 , t (3)3 .Lemmas 6.7 and 6.8 yield Proposition 6.6. (cid:3) Theorem 4.4 for i > N , λ , and I = I max is proved.6.7. Modification of the three-term relation.
For integers α, β , 1 α < β N , and λ ∈ Z N > , | λ | = n , recall the notations t { α, β } , t { α, β } , λ α, β , in Sections 4.3 and 4.4. Lemma 6.9.
For any α < β N and I ∈ I λ α,β , we have ˇ d { α, β } U I = c α, β ˇ d { α, β } W I ,where c α, β = Q β − i = α λ ( i ) Q N − i =1 λ ( i ) ! .Proof. Let G α, β ( t , z ) = ( t ( α − λ ( α − − t ( α ) λ ( α ) − h ) β − Y i = α λ ( i − − Y a =1 ( t ( i − a − t ( i ) λ ( i ) − h ) × (6.24) × ( t ( β − λ ( β − − t ( β ) λ ( β ) (cid:1) β − Y i = α λ ( i +1) − Y a =1 ( t ( i ) λ ( i ) − t ( i +1) a ) β − Y i = α λ ( i ) − Y a =1 t ( i ) λ ( i ) − t ( i ) a − ht ( i ) λ ( i ) − t ( i ) a be the product in the right-hand side of formula (4.5). Since G α, β ( t , z ) is symmetric inthe variables t ( i ) { α, β } for every i = 1 , . . . , N − U I ( t { α, β } , z ) and divide the result by the order of the relevant product of thesymmetric groups before doing the overall symmetrization in formula (4.8) for ˇ d { α, β } U I .This results in replacing U I ( t { α, β } , z ) by c α, β W I ( t { α, β } , z ) , see formula (3.5). (cid:3) Recall the operator S i,i +1 acting on functions of z , . . . , z n given by formula (3.7). Lemma 6.10.
For any i = 1 , . . . , n − , α < β N , and I ∈ I λ α,β , we have (6.25) S i,i +1 ( ˇ d { α, β } U I ) = ˇ d { α, β } U s i,i +1 ( I ) . Proof.
The product G α, β ( t , z ) , see (6.24), is symmetric in z , . . . , z n . Hence S i,i +1 ( ˇ d { α, β } U I ) = c α, β S i,i +1 ( ˇ d { α, β } W I ) = c α, β ˇ d { α, β } (cid:0) S i,i +1 ( W I ) (cid:1) = c α, β ˇ d { α, β } W s i,i +1 ( I ) = ˇ d { α, β } U s i,i +1 ( I ) . by Lemmas 6.9 and 3.2. (cid:3) UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 21
The end of the proof of Theorem 4.4.
Given l , 1 l N − i = 1 , . . . , l . The result is (cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ I i z a (cid:17) W I + h l X i =1 N X j = l +1 λ i X m =1 λ j X m =1 ℓ i,m <ℓ j,m W I i,j ; m ,m (6.26) = l X i =1 N X j = l +1 λ i X a =1 ˇ d { i,j } U ( I ) ai,j , To finish the proof of Theorem 4.4, we need to prove formula (6.26) for any I and any i = 1 ,. . . , N − σ , denote by | σ | the length of σ . For any J, J ′ ∈ I λ , define thepermutation σ J,J ′ as follows: if J m = { j m, < . . . < j m, λ m } , J ′ m = { j ′ m, < . . . < j ′ m, λ m } ,then σ J,J ′ ( j ′ m, l ) = j m, l . Set σ J = σ J,I max . The permutation σ J has the minimal lengthamongst all permutations σ such that σ ( I max ) = J . Lemma 6.11.
Assume that for J ∈ I λ and a transposition s i,i +1 , we have | s i,i +1 σ J | < | σ J | .Then s i,i +1 σ J = σ s i,i +1 ( J ) . (cid:3) We will prove formula (6.26) by induction with respect to the length of σ I . For the baseof induction I = I max , formula (6.26) is proved already.Fix I ∈ I λ and find m such that | s m,m +1 σ I | < | σ I | . Let p, r be such that m ∈ I p and m + 1 ∈ I r . Since | s m,m +1 σ I | < | σ I | , we have p < r .Denote ˜ I = s m,m +1 ( I ) . Then ˜ I p = I p − { m } ∪ { m + 1 } , ˜ I r = I r − { m + 1 } ∪ { m } , and˜ I c = I c , otherwise. And clearly, I = s m,m +1 ( ˜ I ) .Write formula (6.26) for ˜ I : (cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ ˜ I i z a (cid:17) W ˜ I + h l X i =1 N X j = l +1 λ i X m =1 λ j X m =1˜ ℓ i,m < ˜ ℓ j,m W ˜ I i,j ; m ,m (6.27) = l X i =1 N X j = l +1 λ i X a =1 ˇ d { i,j } U (˜ I ) ai,j , where ˜ I c = (˜ ℓ c, , . . . , ˜ ℓ c, λ c ). We will show that applying the operator S m,m +1 to both sidesof formula (6.27) transforms it to formula (6.26) for I .To compare the right-hand sides, observe that s m,m +1 (cid:0) ( ˜ I ) ai,j (cid:1) = ( I ) ai,j . Hence, Lemma 6.10yields S m,m +1 (cid:0) ˇ d { i,j } U (˜ I ) ai,j (cid:1) = ˇ d { i,j } U ( I ) ai,j , that proves the desired assertion.To compare the left-hand sides, observe first that s m,m +1 ( ˜ I i,j ; m ,m ) = I i,j ; s m,m +1 ( m ) , s m,m +1 ( m ) and S m,m +1 ( W ˜ I i,j ; m ,m ) = W I i,j ; sm,m +1( m ,sm,m +1( m by Lemma 3.2. This proves the desired transformation of the second sum in the left-handside of (6.27) term by term provided p > l or r l . If p l < r , the matching between theterms of the second sums in (6.27) and (6.26) is not perfect and the sum in (6.26) containsone more term h W I p,r ; m,m +1 .If p > l or r l , the sum P li =1 P a ∈ ˜ I i z a in formula (6.27) is symmetric in z m , z m +1 and equals the sum P li =1 P a ∈ I i z a in formula (6.26). Thus S m,m +1 (cid:16)(cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ ˜ I i z a (cid:17) W ˜ I (cid:17) = (cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ I i z a (cid:17) S m,m +1 ( W ˜ I )= (cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ I i z a (cid:17) W I by Lemma 3.2. If p l < r , then we have S m,m +1 (cid:16)(cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ ˜ I i z a (cid:17) W ˜ I (cid:17) = (cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ I i z a (cid:17) S m,m +1 ( W ˜ I ) + h W ˜ I = (cid:16) λ ( l ) X j =1 t ( l ) j − l X i =1 X a ∈ I i z a (cid:17) W I + h W I p,r ; m,m +1 , since ˜ I = I p,r ; m,m +1 . This shows that the operator S m,m +1 transforms formula (6.27) toformula (6.26). This completes the induction step. Theorem 4.4 is proved. Example.
Let N = 2 , n = 3 , λ = (2 ,
1) , I = ( { , } , { } ) , I max = ( { , } , { } ) , σ I = s , . Formula (6.26) is(6.28) ( t (1)1 + t (1)2 − z − z ) W { , } , { } + h W { , }{ } = ˇ d { , } ( U { } , { , } + U { } , { , } ) , formula (6.27) is(6.29) ( t (1)1 + t (1)2 − z − z ) W { , } , { } = ˇ d { , } ( U { } , { , } + U { } , { , } ) . and the operator S , transforms formula (6.29) to formula (6.28).7. Corollary of Theorems 4.3 and 4.4
Let λ ∈ Z N > , | λ | = n , and I ∈ I λ . Recall the notations ( I ) aα, β , I i,j ; m ,m , see (4.11),(4.13), and the discrete differentials d { α, β } g , see (4.6). Define the discrete differential(7.1) D I,i = N X j = i +1 λ j X a =1 d { j,i } U ( I ) aj,i − i − X j =1 λ i X a =1 d { i,j } U ( I ) ai,j . UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 23
Corollary 7.1.
We have (cid:18) λ ( i ) X j =1 t ( i ) j − λ ( i − X j =1 t ( i − j − X a ∈ I i z a (cid:19) W I + h (cid:18) i − X j =1 q i λ j q i − q j + N X j = i +1 q j λ i q i − q j (cid:19) W I +(7.2) + h N X j =1 j = i λ i X m =1 (cid:18) λ j X m =1 ℓ i,m <ℓ j,m W I i,j ; m ,m + q j q i − q j λ j X m =1 W I i,j ; m ,m (cid:19) = D I,i . Proof.
Theorems 4.3 and 4.4, and formula (4.9) imply that (cid:16) λ ( i ) X j =1 t ( i ) j − λ ( i − X j =1 t ( i − j − X a ∈ I i z a (cid:17) W I − h i − X j =1 λ i X m =1 λ j X m =1 ℓ i,m >ℓ j,m W I i,j ; m ,m + h N X j = i +1 λ i X m =1 λ j X m =1 ℓ i,m <ℓ j,m W I i,j ; m ,m + h i − X j =1 q i q i − q j (cid:16) λ j W I + λ i X m =1 λ j X m =1 W I i,j ; m ,m (cid:17) + h N X j = i +1 q j q i − q j (cid:16) λ i W I + λ i X m =1 λ j X m =1 W I i,j ; m ,m (cid:17) = D I,i . Formula (7.2) is obtained now by rearranging the terms in the left-hand side of this equality. (cid:3)
Recall the scalar master function Φ λ ( t ; z ; h ; q ) given by (4.1). Define(7.3) Ω λ ( q ) = N − Y i =1 N Y j = i +1 (1 − q j /q i ) hλ i /κ . Introduce the ( C N ) ⊗ n λ -valued weight function(7.4) W λ ( t ; z ; h ) = X I ∈I λ W I ( t ; z ; h ) v I . Recall the dynamical Hamiltonians X i ( z ; h ; q ) defined in (2.1). Theorem 7.2.
For every i = 1 , . . . , N , we have (cid:16) κ q i ∂∂q i − X i ( z ; h ; q ) (cid:17) Ω λ ( q ) Φ λ ( t ; z ; h ; q ) W λ ( t ; z ) =(7.5) = Ω λ ( q ) Φ λ ( t ; z ; h ; q ) X I ∈I λ D I,i ( t ; z ; h ; q ) v I . Proof.
The statement is equivalent to Corollary 7.1. (cid:3) Integral representations for solutions of dynamical equations
Formal integrals.
Let λ ∈ Z N > , | λ | = n , and κ ∈ C × . Consider the space of func-tions of the form Φ λ ( t ; z ; h ; q ) f ( t ; z ; h ; q ) , where Φ λ ( t ; z ; h ; q ) is the master function (4.1),and f ( t ; z ; h ; q ) is a polynomial in t and holomorphic function of z , h, q on some domain L ⊂ C n × C × C N . Assume that we have a map M assigning to a function Φ λ f a function M (Φ λ f ) of variables z , h, q , holomorphic on L , such that:(i) The map M is linear over the field of meromorphic on L functions in z , q ,h ,(8.1) M (cid:0) Φ λ ( g f + g f ) (cid:1) = g M (Φ λ f ) + g M (Φ λ f )for any meromorphic functions g , g of z , h, q , such that g f and g f are holo-morphic on L .(ii) For any i = 1 , . . . , N , we have(8.2) ∂∂q i M (Φ λ f ) = M (cid:18) ∂∂q i (Φ λ f ) (cid:19) . (iii) If f is a discrete differential of a polynomial in t , then(8.3) M (Φ λ f ) = 0 . A map M is called a formal integral. We have the following corollary of Theorem 7.2. Lemma 8.1. If M is a formal integral, then the ( C N ) ⊗ n λ -valued function F M ( z ; h ; q ) := Ω λ M (Φ λ W λ ) = Ω λ X I ∈I λ M (Φ λ W I ) v I holomorphic on L , is a solution of the dynamical differential equations (2.3) . (cid:3) Jackson integral.
Consider the space C λ { } × C n × C × C N with coordinates t , z , h, q .The lattice κ Z λ { } naturally acts on this space by shifting the t -coordinates.Let J = ( J , . . . , J N ) ∈ I λ . Recall the notation S ki =1 J i = { j ( k )1 < . . . < j ( k ) λ ( k ) } . DefineΣ J ⊂ C λ { } × C n × C × C N by the equations:(8.4) t ( k ) i = z j ( k ) i , k = 1 , . . . , N − , i = 1 , . . . , λ ( k ) , and call it a discrete cycle .For a function of t and a point s ∈ C λ { } , define Res t = s to be the iterated residue,Res t = s = Res t (1)1 = s (1)1 . . . Res t (1) λ (1) = s (1) λ (1) . . . Res t ( N − = s ( N − . . . Res t ( N − λ ( N − = s ( N − λ ( N − . Let L ′ be the complement in C n × C of the union of the hyperplanes(8.5) h = mκ , z a − z b = mκ , z a − z b + h = mκ , for all a, b = 1 , . . . , n , a = b , and all m ∈ Z . Let L ′′ ⊂ C N be the domain(8.6) | q i +1 /q i | < , i = 1 , . . . , N − , UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 25 with additional cuts fixing a branch of log q i for all i = 1 , . . . , N . Set L = L ′ × L ′′ ⊂ C n × C × C N .Let f ( t ; z ; h ; q ) be a polynomial in t and a holomorphic function of z ; h ; q on L . For( z ; h ; q ) ∈ L , define(8.7) M J (Φ λ f )( z ; h ; q ) = X r ∈ Z λ { } Res t =Σ J + r κ (cid:0) Φ λ ( t ; z ; h ; q ) f ( t ; z ; h ; q ) (cid:1) . This sum is called the
Jackson integral over the discrete cycle Σ J . Lemma 8.2.
The map M J is a formal integral.Proof. Each term of the sum in formula (8.7) is a holomorphic function on L ′ . Moreover,Res t =Σ J + r κ (cid:0) Φ λ ( t ; z ; h ; q ) f ( t ; z ; h ; q ) = 0 if r Z λ { } . Hence, the sum over Z λ { } reducesto the sum over Z λ { } . The result is similar to a multidimensional hypergeometric series mul-tiplied by some fractional powers of q , . . . , q N . The obtained sum converges if | q i +1 /q i | < i = 1 , . . . , N − L .Properties (8.1) – (8.3) for the map M J are clear. Lemma 8.2 is proved. (cid:3) Lemma 8.3.
The function M J (Φ λ f ) analytically continues to the hyperplanes h = mκ for m ∈ Z > .Proof. By the proof of Lemma 8.2,(8.8) M J (Φ λ f )( z ; h ; q ) = X r ∈ Z λ { } Res t =Σ J + r κ (cid:0) Φ λ ( t ; z ; h ; q ) f ( t ; z ; h ; q ) (cid:1) . for ( z ; h ; q ) ∈ L . By inspection, if h → mκ , m ∈ Z > , and r ∈ Z λ { } , thenRes t =Σ J + r κ (cid:0) Φ λ ( t ; z ; h ; q ) f ( t ; z ; h ; q ) (cid:1) → Res t =Σ J + r κ (cid:0) Φ λ ( t ; z ; mκ ; q ) f ( t ; z ; mκ ; q ) (cid:1) . Hence(8.9) M J (Φ λ f )( z ; h ; q ) → X r ∈ Z λ { } Res t =Σ J + r κ (cid:0) Φ λ ( t ; z ; mκ ; q ) f ( t ; z ; mκ ; q ) (cid:1) , since the sum in the right-hand side converges if | q i +1 /q i | < i = 1 , . . . , N − (cid:3) Remark.
For m ∈ Z > , the sum P r ∈ Z λ { } Res t =Σ J + r κ (cid:0) Φ λ ( t ; z ; mκ ; q ) f ( t ; z ; mκ ; q ) (cid:1) di-verges, and the function M J (Φ λ f )( z ; mκ ; q ) is not given by formula (8.7). Example.
Let N = 2 , λ = (1 , n −
1) , J = ( { } , { , , . . . , n } ) . ThenΦ λ ( t ; z ; h ; q ) = ( e π √− q ) P na =1 z a /κ (cid:16) e π √− n − q q (cid:17) t (1)1 /κ n Y a =1 Γ (cid:16) t (1)1 − z a κ (cid:17) Γ (cid:16) z a − t (1)1 + hκ (cid:17) , and(8.10) M ( { } , { , ,...,n } ) (Φ λ f ) = X r ∈ Z Res t (1)1 = z + rκ (Φ λ f ) . Nonzero contributions to the sum in the right-hand side of (8.10) come from the poles ofΓ (cid:0) ( t (1)1 − z ) /κ (cid:1) . Explicitly, the answer is M ( { } , { , ,...,n } ) (Φ λ f ) = ( e π √− n − q ) z /κ ( e π √− q ) P na =2 z a /κ ×× κ Γ (cid:16) hκ (cid:17) n Y a =2 Γ (cid:16) z − z a κ (cid:17) Γ (cid:16) z a − z + hκ (cid:17) ×× ∞ X l =0 f ( z − lκ ; z ; h ; q ) (cid:16) q q (cid:17) l l − Y j =0 (cid:18) h + j κκ + j κ n Y a =2 z a − z + h + j κz a − z + κ + j κ (cid:19) , and the series converges if | q /q | < Solutions of dynamical equations.
Recall the ( C N ) ⊗ n λ -valued weight function W λ ( t ; z ),given by (7.4). For J ∈ I λ , define(8.11) Ψ J ( z ; h ; q ) = Ω λ ( q ) M J (Φ λ W λ )( z ; h ; q ) = Ω λ ( q ) X I ∈I λ M J (Φ λ W I )( z ; h ; q ) v I . Theorem 8.4.
The function Ψ J ( z ; h ; q ) is a holomorphic ( C N ) ⊗ n λ -valued function of z , h, q on the domain L ⊂ C n × C × C N such that h κ Z , z a − z b κ Z , z a − z b + h κ Z , for all a, b = 1 , . . . , n , a = b , | q i +1 /q i | < , i = 1 , . . . , N − , and a branch of log q i is fixed for each i = 1 , . . . , N . Furthermore, Ψ J ( z ; h ; q ) is a solutionof the dynamical differential equations (2.3) .Proof. The weight functions W I ( t ; z ; h ) are polynomials in t , z , h and do not depend on q .Hence, Theorem 8.4 follows from Lemmas 8.2, 8.3, and 8.1. (cid:3) Theorem 8.5.
Under conditions of Theorem 8.4, the collection of ( C N ) ⊗ n λ -valued functions (cid:0) Ψ J ( z ; h ; q ) (cid:1) J ∈I λ is a basis of solutions of the dynamical equations (2.3) .Proof. By formulas (8.8), (8.11), if | q i +1 /q i | → i = 1 , . . . , N − J ( z ; h ; q ) ≃ Ω λ ( q ) Res t =Σ J (cid:0) Φ λ ( t ; z ; h ; q ) (cid:1) W λ (Σ J ; z ; h ) =(8.12) = Ω λ ( q ) Res t =Σ J (cid:0) Φ λ ( t ; z ; h ; q ) (cid:1) X I ∈I λ W I (Σ J ; z ; h ) v I . By [RTV1, Lemma 3.1], the matrix (cid:0) W I (Σ J ; z ; h ) (cid:1) I,J ∈I λ is triangular and the diagonal entries W I (Σ I ; z ; h ) are nonzero if h = 0 and z a − z b = 0 , z a − z b + h = 0 , for all a, b = 1 , . . . ,n , a = b . Hence the vectors W λ (Σ J ) , J ∈ I λ , form a basis of ( C N ) ⊗ n λ and the collection (cid:0) Ψ J ( z ; h ; q ) (cid:1) J ∈I λ is a basis of solutions of the dynamical equations (2.3). (cid:3) UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 27
The functions Ψ J ( z ; h ; q ) were considered in [TV1]. It follows from [TV1, Theorem 1.5.2],cf. [TV4], that for every J ∈ I λ , the function Ψ J ( z ; h ; q ) is a solution of the qKZ equations(2.4). Corollary 8.6.
The collection of ( C N ) ⊗ n λ -valued functions (cid:0) Ψ J ( z ; h ; q ) (cid:1) J ∈I λ is a basis ofsolutions of both the dynamical and qKZ equations, see (2.3) , (2.4) , with values in ( C N ) ⊗ n λ . Remark.
The functions Ψ J ( z ; h ; q ) are called the multidimensional q -hypergeometric solu-tions of the dynamical equations. In [TV5], we constructed another type of solutions of thedynamical equations called the multidimensional hypergeometric solutions .9. Equivariant quantum differential equations
Partial flag varieties.
Let λ ∈ Z N > , | λ | = n . Consider the partial flag variety F λ parametrizing chains of subspaces0 = F ⊂ F ⊂ . . . ⊂ F N = C n with dim F i /F i − = λ i , i = 1 , . . . , N . Denote by T ∗ F λ the cotangent bundle of F λ and X n = [ | λ | = n T ∗ F λ . Let u , . . . , u n be the standard basis of C n . For I ∈ I λ , let x I ∈ F λ be the pointcorresponding to the coordinate flag F ⊂ . . . ⊂ F N , where F i is the span of the standardbasis vectors u j ∈ C n with j ∈ I ∪ . . . ∪ I i . We embed F λ in T ∗ F λ as the zero sectionand consider the points x I as points of T ∗ F λ .9.2. Equivariant cohomology.
Let A ⊂ GL n ( C ) be the torus of diagonal matrices and T = A × C × . The group A acts on C n and hence on T ∗ F λ . Let the group C × act on T ∗ F λ by multiplication in each fiber. We denote by − h its C × -weight.We consider the equivariant cohomology algebras H ∗ T ( T ∗ F λ ; C ) and H ∗ T ( X n ) = M | λ | = n H ∗ T ( T ∗ F λ ; C ) . Denote by Γ i = { γ i, , . . . , γ i,λ i } the set of the Chern roots of the bundle over F λ with fiber F i /F i − . Let Γ = ( Γ ; . . . ; Γ N ) . Denote by z = { z , . . . , z n } the Chern roots correspondingto the factors of the torus A . Then(9.1) H ∗ T ( T ∗ F λ ) = C [ Γ ] S λ × ... × S λN ⊗ C [ z ] ⊗ C [ h ] .D N Y i =1 λ i Y j =1 ( u − γ i,j ) = n Y a =1 ( u − z a ) E . The cohomology H ∗ T ( T ∗ F λ ) is a module over H ∗ T ( pt ; C ) = C [ z ] ⊗ C [ h ] .Notice that(9.2) N − Y i =1 N Y j = i +1 λ i Y a =1 λ j Y b =1 ( γ j,b − γ i,a ) ( γ i,a − γ j,b − h ) is the equivariant total Chern class of the tangent bundle of T ∗ F λ and(9.3) c ( E i ) = λ i X a =1 γ i,a , i = 1 , . . . , N , is the equivariant first Chern class of the vector bundle E i over T ∗ F λ with fiber F i /F i − .For i = 1 , . . . , N , denote Θ i = { θ i, , . . . , θ i,λ ( i ) } the Chern roots of the bundle F i over F λ with fiber F i . Let Θ = (Θ , . . . , Θ N ) . The relations(9.4) λ ( i ) Y a =1 ( u − θ i,a ) = i Y j =1 λ j Y k =1 ( u − γ j,k ) , i = 1 , . . . , N , define the homomorphism C [ Θ ] S λ (1) × ... × S λ ( N ) ⊗ C [ z ] ⊗ C [ h ] → H ∗ T ( T ∗ F λ ) . Stable envelope map.
Recall the weight functions ˇ W I defined in Sections 3.1. Letˇ W I ( Θ ; z ) ∈ H ∗ T ( T ∗ F λ ) be the cohomology class represented by the polynomial ˇ W id ,I ( t ; z )with the variables t ( i ) a replaced by θ i,a for all i = 1 , . . . , N − a = 1 , . . . , λ ( i ) . Denote c λ ( Θ ) = N − Y i =1 λ ( i ) Y a =1 λ ( i ) Y b =1 ( θ i,a − θ i,b − h ) ∈ H ∗ T ( T ∗ F λ ) . Observe that c λ ( Θ ) is the equivariant Euler class of the bundle L N − a =1 Hom( F a , F a ) if wemake C × act on it with weight − h . Theorem 9.1 ([RTV1, Theorem 4.1]) . For any λ and any I ∈ I λ , the cohomology class ˇ W I ( Θ ; z ) ∈ H ∗ T ( T ∗ F λ ) is divisible by c λ ( Θ ) , that is, there exists a unique element Stab I ∈ H ∗ T ( T ∗ F λ ) such that (9.5) [ ˇ W I ( Θ ; z )] = c λ ( Θ ) · Stab I . (cid:3) Define the stable envelope map by the rule(9.6) Stab : ( C N ) ⊗ n ⊗ C [ z ] ⊗ C [ h ] → H ∗ T ( X n ) , v I Stab I . Remark.
Stable envelope maps for Nakajima quiver varieties were introduced in [MO].They were defined there geometrically in terms of the associated torus action. The mapStab given by formula (9.6) is the stable envelope map of [MO] for the Nakajima quivervariety X n , described in terms of the Chern roots Θ , z , h , see [RTV1]. Remark.
After the substitution h = 1 the classes Stab I ∈ H ∗ T ( T ∗ F λ ) can be consideredas elements of the equivariant cohomology H ∗ ( C × ) n ( F λ ) of the partial flag variety F λ (andnot of the cotangent bundle T ∗ F λ ). These new classes are the equivariant Chern-Schwartz-MacPherson classes (CSM classes) of the corresponding Schubert cells, see [RV]. UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 29
Let C ( z ; h ) be the algebra of rational functions in z , h . The map(9.7) Stab : ( C N ) ⊗ n ⊗ C ( z ; h ) → H ∗ T ( X n ) ⊗ C ( z ; h ) , v I Stab I , is an isomorphism of C ( z ; h )-modules by [RTV1, Lemma 6.7].9.4. H ∗ T ( T ∗ F λ ) -valued weight function. Define the H ∗ T ( T ∗ F λ )-valued function b W ( t ; Γ )as follows:(9.8) b W ( t ; Γ ) = Sym t (1)1 ,..., t (1) λ (1) . . . Sym t ( N − ,..., t ( N − λ ( N − b U ( t ; Γ ) , b U ( t ; Γ ) = N − Y j =1 λ ( j ) Y a =1 (cid:18) a − Y c =1 ( t ( j ) a − t ( j +1) c − h ) λ ( j +1) Y d = a +1 ( t ( j ) a − t ( j +1) d ) λ ( j ) Y b = a +1 t ( j ) a − t ( j ) b − ht ( j ) a − t ( j ) b (cid:19) . where ( t ( N )1 , . . . , t ( N ) n ) = ( γ , , . . . , γ , λ , γ , , . . . , γ , λ , . . . , γ N, , . . . , γ N, λ N ) , cf. formula (3.1)for I = I min = (cid:0) { , . . . , λ } , . . . , { n − λ N + 1 , . . . , n } (cid:1) . Example.
Let N = 2, λ = (1 , n − b W ( t ; Γ ) = Q n − a =1 ( t (1)1 − γ ,a ) .Let N = 3, λ = (1 , , b W ( t ; Γ ) = ( t (1)1 − t (2)2 ) ( t (2)1 − γ , ) ( t (2)1 − γ , ) ( t (2)2 − γ , − h ) ( t (2)2 − γ , ) t (2)1 − t (2)2 − ht (2)1 − t (2)2 ++ ( t (1)1 − t (2)1 ) ( t (2)2 − γ , ) ( t (2)2 − γ , ) ( t (2)1 − γ , − h ) ( t (2)1 − γ , ) t (2)2 − t (2)1 − ht (2)2 − t (2)1 . Define(9.9) Q ( Γ ) = N − Y i =1 N Y j = i +1 λ i Y a =1 λ j Y b =1 ( γ i,a − γ j,b − h ) ∈ H ∗ T ( T ∗ F λ ) . The image of the ( C N ) ⊗ n λ -valued weight function W λ ( t ; z ) , see (7.4), is given by the nextproposition. Proposition 9.2.
We have (9.10) X I ∈I λ W I ( t ; z ) Stab id , I = Q ( Γ ) b W ( t ; Γ ) . Proof.
Recall that W I ( t ; z ) = ( − h ) − λ { } ˇ W σ , I ( t ; z ) , see (3.4). Recall the discrete cycle Σ J given by (8.4). Let σ I ∈ S n be a permutation such that σ I ( I min ) = I . Then formula (9.10)is equivalent to the following equality(9.11) X I ∈I λ ˇ W σ , I ( t ; z ) ˇ W I (Σ J ; z ) = c λ (Σ J ) Q ( z J ) ˇ W σ J , J ( t ; z ) . For the proof of formula (9.11), consider the function Z ( t ; ˜ t ; z ) = X I ∈I λ ˇ W σ , I ( t ; z ) ˇ W I (˜ t ; z ) . Here ˜ t is an additional set of variables similar to t . Then formula (9.11) reads(9.12) Z ( t ; Σ J ; z ) = c λ (Σ J ) Q ( z J ) ˇ W σ J , J ( t ; z ) . Three-term relations (3.3) imply that for any σ ∈ S n , we have(9.13) Z ( t ; ˜ t ; z ) = X I ∈I λ ˇ W σ, I ( t ; z ) ˇ W σσ , I (˜ t ; z ) . By [RTV1, Lemma 3.2], we have ˇ W σ J σ , I (Σ J , z ) = c λ ( z J ) Q ( z J ) δ I,J . Thus taking σ = σ J ,˜ t = Σ J in formula (9.13) , we get equality (9.12). Proposition 9.2 is proved. (cid:3) Define the cohomology classes(9.14) R ( Γ ) = N − Y i =1 N Y j = i +1 λ i Y a =1 λ j Y b =1 ( γ i,a − γ j,b )and(9.15) R I ( Γ ; z ) = N − Y i =1 N Y j = i +1 λ i Y a =1 Y b ∈ I j ( γ i,a − z b ) , I ∈ I λ . Notice that R I ( z J ; z ) = R ( z J ) δ I,J . Proposition 9.3.
For any K ∈ I λ , we have (9.16) X I ∈I λ W I (Σ K ; z ) Stab id , I = ( − h ) − λ { } c λ ( Θ ) R K ( Γ ; z ) Q ( Γ ) . Proof.
Formula (9.16) is equivalent to the equality(9.17) X I ∈I λ ˇ W σ , I (Σ K ; z ) ˇ W I (Σ J ; z ) = (cid:0) c λ (Σ J ) (cid:1) R ( z J ) Q ( z J ) δ J,K . By [RTV1, Lemma 3.2], we have ˇ W σ J , J (Σ K , z ; h ) = c λ ( z J ) R ( z J ) δ J,K . Thus taking ˜ t = Σ K in formula (9.11) , we get equality (9.17).Formula (9.17) also follows from [RTV1, Lemma 3.4]. Proposition 9.3 is proved. (cid:3) Quantum multiplication by divisors on H ∗ T ( T ∗ F λ ) . The quantum multiplicationby divisors on H ∗ T ( T ∗ F λ ) is described in [MO]. The fundamental equivariant cohomologyclasses of divisors on T ∗ F λ are linear combinations of D i = γ i, + . . . + γ i,λ i , i = 1 , . . . , N .The quantum multiplication D i ∗ ˜ q : H ∗ T ( T ∗ F λ ) → H ∗ T ( T ∗ F λ ) by the divisor D i depends onparameters ˜ q = (˜ q , . . . , ˜ q N ) ∈ ( C × ) N and is given in [MO, Theorem 10.2.1].The quantum connection ∇ quant λ , ˜ q , ˜ κ on H ∗ T ( T ∗ F λ ) is defined by the formula ∇ quant λ , ˜ q , ˜ κ,i = ˜ κ ˜ q i ∂∂ ˜ q i − D i ∗ ˜ q , i = 1 , . . . , N , UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 31 where ˜ κ ∈ C × is the parameter of the connection, see [BMO]. The system of equationsfor flat sections of the quantum connection is called the system of the equivariant quantumdifferential equations .The isomorphism Stab allows us to compare the operators ∇ λ , q ,κ,i := ∇ q ,κ,i (cid:12)(cid:12) ( C N ) ⊗ n λ ofthe dynamical connection on ( C N ) ⊗ n λ , see (2.2), and the operators ∇ quant λ , ˜ q , ˜ κ,i of the quantumconnection on H ∗ T ( T ∗ F λ ) .Recall the dynamical Hamiltonians X i ( z ; h ; q ) , see (2.1). Define the modified dynamicalHamiltonians X − λ ,i ( z ; h ; q ) =(9.18) = X i ( z ; h ; q ) (cid:12)(cid:12) ( C N ) ⊗ n λ − h i − X j =1 q i q i − q j min( λ i , λ j ) − h N X j = i +1 q j q i − q j min( λ i , λ j ) . The modified dynamical connection on ( C N ) ⊗ n λ is(9.19) ∇ − λ , q ,κ,i = κ q i ∂∂q i − X − λ ,i ( z ; h ; q ) , i = 1 , . . . , N . see [GRTV, Section 3.4]. Recall that h GRTV = − h . Theorem 9.4 ([RTV1, Corollary 7.6]) . The isomorphism
Stab identifies the operators D i ∗ ˜ q of quantum multiplication by D i on H ∗ T ( T ∗ F λ ) with the action of the modified dynamicalHamiltonians X − λ ,i ( z ; h ; ˜ q − ) on ( C N ) ⊗ n λ , where ˜ q − = (˜ q − , . . . , ˜ q − N ) . Consequently, thedifferential operators ∇ quant λ , ˜ q , ˜ κ,i are identified with the differential operators ∇ λ , ˜ q − , − ˜ κ,i . (cid:3) See also [RTV1, Theorem 7.5].Set(9.20) b Ω λ ( ˜ q ; ˜ κ ) = N − Y i =1 N Y j = i +1 (1 − ˜ q i / ˜ q j ) h min(0 , λ j − λ i ) / ˜ κ . Set λ { } = P i
Let N = 2 , n = 2 , λ = (1 ,
1) . Recall the Gauss hypergeometric series F ( a, b ; c ; x ) = ∞ X m =0 ( a ) m ( b ) m ( c ) m x m m ! , where ( u ) m = u ( u − . . . ( u − m + 1) . Set F ( z , z ; h ; ˜ κ ; x ) = F (cid:16) − h ˜ κ , z − z − h ˜ κ ; 1 + z − z ˜ κ ; x (cid:17) and F ′ ( z , z ; h ; ˜ κ ; x ) = ∂F∂x ( z , z ; h ; ˜ κ ; x ) . Then b Ψ ( { } , { } ) ( z , z ; h ; ˜ q , ˜ q ; ˜ κ ) == ˜ κ − ( e − π √− ˜ q ) z / ˜ κ ( e − π √− ˜ q ) z / ˜ κ Γ (cid:16) z − z ˜ κ (cid:17) Γ (cid:16) z − z − h ˜ κ (cid:17) × ( γ , − γ , − h ) (cid:0) ( γ , − z ) F ( z , z ; h ; ˜ κ ; ˜ q / ˜ q ) − ˜ κ (˜ q / ˜ q ) F ′ ( z , z ; h ; ˜ κ ; ˜ q / ˜ q ) (cid:1) and b Ψ ( { } , { } ) ( z , z ; h ; ˜ q , ˜ q ; ˜ κ ) = b Ψ ( { } , { } ) ( z , z ; h ; ˜ q , ˜ q ; ˜ κ ) . Theorem 9.5.
The collection of functions (cid:0) b Ψ I ( z ; h ; ˜ q ; ˜ κ ) (cid:1) I ∈I λ is a basis of solutions of boththe quantum differential equations ∇ quant λ , ˜ q , ˜ κ,i f = 0 , i = 1 , . . . , N , and the associated qKZ difference equations.Proof. The statement follows from Theorems 8.4, 8.5, 9.4, and Proposition 9.2, see Corollary8.6. (cid:3)
Remark.
The integral representations for solutions of the equivariant quantum differentialequations is a manifestation of a version of mirror symmetry. The basis of solutions givenby Theorem 9.5 is an analog of Givental’s J -function.For ˜ q i / ˜ q i +1 → i = 1 , . . . , N − b Ψ I ( z ; h ; ˜ q ; ˜ κ ) is given by taking the residue at t = Σ I . Theorem 9.6.
Assume that ˜ q i / ˜ q i +1 → for all i = 1 , . . . , N − . Then b Ψ I ( z ; h ; ˜ q ; ˜ κ ) = N Y i =1 (cid:0) e π √− λ i − n ) ˜ q i (cid:1) P a ∈ Ii z a / ˜ κ (9.23) × N − Y i =1 N Y j = i +1 Y a ∈ I i Y b ∈ I j Γ (cid:16) z b − z a ˜ κ (cid:17) Γ (cid:16) z a − z b − h ˜ κ (cid:17) × (cid:18) ∆ I + X m ∈ Z N − > m =0 b Ψ I, m ( z ; h ; ˜ κ ) N − Y i =1 (cid:16) ˜ q i ˜ q i +1 (cid:17) m i (cid:19) , UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 33 where ∆ I ( Γ , z ) = R I ( Γ ; z ) /R ( z I ) is the cohomology class such that ∆ I ( z J ; z ) = δ I,J , andthe classes b Ψ I, m ( z ; h ; ˜ κ ) are rational functions in z , h, ˜ κ , regular on the domain h ˜ κ Z > , z a − z b ˜ κ Z , z a − z b + h ˜ κ Z , for all a, b = 1 , . . . , n , a = b .Proof. The statement follows from formula (9.21) and Propositions 9.2, 9.3. (cid:3)
Example.
Let N = 2 , n = 2 , λ = (1 ,
1) . As ˜ q / ˜ q → b Ψ ( { } , { } ) ( z , z ; h ; ˜ q , ˜ q ; ˜ κ ) is the cohomology class( e − π √− ˜ q ) z / ˜ κ ( e − π √− ˜ q ) z / ˜ κ Γ (cid:16) z − z ˜ κ (cid:17) Γ (cid:16) z − z − h ˜ κ (cid:17) ∆ ( { } , { } ) and the leading term of the solution b Ψ ( { } , { } ) ( z , z ; h ; ˜ q , ˜ q ; ˜ κ ) is the cohomology class( e − π √− ˜ q ) z / ˜ κ ( e − π √− ˜ q ) z / ˜ κ Γ (cid:16) z − z ˜ κ (cid:17) Γ (cid:16) z − z − h ˜ κ (cid:17) ∆ ( { } , { } ) . Quantum Pieri rules
Quantum equivariant cohomology algebra H ˜ qT ( T ∗ F λ ) . Let ˜ q = (˜ q , . . . , ˜ q N ) ∈ ( C × ) N have distinct coordinates. The quantum equivariant cohomology algebra H ˜ qT ( T ∗ F λ )is the algebra generated by the operators D i ∗ ˜ q : H ∗ T ( T ∗ F λ ) → H ∗ T ( T ∗ F λ ) of quantummultiplication by the divisors D i , i = 1 , . . . , N , see details in [MO, GRTV]. The algebracan be defined by generators and relations as follows.Introduce the variables ˜ γ i, , . . . , ˜ γ i,λ i for i = 1 , . . . , N . Set(10.1) W ˜ q ( u ) = det (cid:18) ˜ q j − i λ i Y k =1 (cid:0) u − ˜ γ i,k − h ( i − j ) (cid:1)(cid:19) Ni,j =1 . Theorem 10.1.
The quantum equivariant cohomology algebra H ˜ qT ( T ∗ F λ ) is isomorphic tothe algebra (10.2) C [ e Γ ] S λ × ... × S λN ⊗ C [ z ] ⊗ C [ h ] .D W ˜ q ( u ) = Y i Let N = 2 , n = 2 , λ = (1 , 1) . Then D i ∗ ˜ q ˜ γ i, , i = 1 , γ , + ˜ γ , = z + z , ˜ γ , ˜ γ , − h ˜ q ˜ q − ˜ q ( ˜ γ , − ˜ γ , − h ) = z z . It is easy to see that the algebra H ˜ qT ( T ∗ F λ ) does not change if all ˜ q , . . . , ˜ q N are multipliedby the same number. In the limit ˜ q i / ˜ q i +1 → i = 1 , . . . , N − 1, the relations in H ˜ qT ( T ∗ F λ )turn into the relations in H ∗ T ( T ∗ F λ ) , see formula (9.1).10.2. Quantum equivariant Pieri rules. Recall the weight functions W I ( t ; z ) , see (3.5).Introduce the variables e Θ i = { ˜ θ i, , . . . , ˜ θ i,λ ( i ) } , e Θ = ( e Θ , . . . , e Θ N ) . Let W I ( e Θ ; z ) be thepolynomial W I ( t ; z ) with the variables t ( i ) a replaced by ˜ θ i,a for all i = 1 , . . . , N − a = 1 ,. . . , λ ( i ) . For any m = 1 , . . . , N − (cid:18) ˜ q j − i λ i Y k =1 (cid:0) u − ˜ γ i,k − h ( i − j ) (cid:1)(cid:19) mi,j =1 = Y i For any i = 1 , . . . , N and I ∈ I λ , the following relation in H ˜ qT ( T ∗ F λ ) holds: (cid:18) λ i X k =1 ˜ γ i,k − X a ∈ I i z a (cid:19) { W I } = h λ i (cid:18) i − X j =1 ˜ q j ˜ q i − ˜ q j + N X j = i +1 ˜ q i ˜ q i − ˜ q j (cid:19) { W I } +(10.6) − h N X j =1 j = i λ i X m =1 (cid:18) λ j X m =1 ℓ i,m <ℓ j,m { W I i,j ; m ,m } − ˜ q i ˜ q i − ˜ q j λ j X m =1 { W I i,j ; m ,m } (cid:19) , where ℓ i,m , ℓ j,m , I i,j ; m ,m are defined in Section 4.5. Theorem 10.2 is proved in Section 10.4. Example. Let N = 2 , n = 2 , λ = (1 , 1) . Then { W ( { } , { } ) } = ˜ γ , − z − h , { W ( { } , { } ) } = ˜ γ , − z , and the quantum Pieri rules take the form( ˜ γ , − z ) { W ( { } , { } ) } = h ˜ q ˜ q − ˜ q (cid:0) { W ( { } , { } ) } + { W ( { } , { } ) } (cid:1) − h { W ( { } , { } ) } , (10.7) ( ˜ γ , − z ) { W ( { } , { } ) } = h ˜ q ˜ q − ˜ q (cid:0) { W ( { } , { } ) } + { W ( { } , { } ) } (cid:1) . UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 35 These are the same relations as in formula (10.3).10.3. Bethe ansatz equations. The Bethe ansatz equations is the following system ofalgebraic equations with respect to the variables t :(10.8) λ ( i − Y k =1 t ( i − k − t ( i ) j − ht ( i − k − t ( i ) j λ ( i +1) Y k =1 t ( i ) j − t ( i +1) k t ( i ) j − t ( i +1) k − h λ ( i ) Y k =1 k = j t ( i ) j − t ( i ) k − ht ( i ) j − t ( i ) k + h = q i +1 q i , for i = 1 , . . . , N − j = 1 , . . . , λ ( i ) . This system can be reformulated as the system ofequations:(10.9) lim κ → Φ λ ( . . . , t ( i ) j + κ, . . . ; z ; h ; q )Φ λ ( t ; z ; h ; q ) = 1 , i = 1 , . . . , N − , j = 1 , . . . , λ ( i ) , see [TV1, MTV1]. Lemma 10.3. For I ∈ I λ and i = 1 , . . . , N − , let D I,i ( t ; z ; h ; q ) be the function definedin (7.1) . Let ˇ t be a solution of the Bethe ansatz equations (10.8) . Then D I,i (ˇ t ; z ; h ; q ) = 0 and the right-hand side of formula (7.2) equals zero at t = ˇ t .Proof. If ˇ t is a solution of equations (10.8), then the second of the two factors in the right-hand side of formula (4.3) equals zero at t = ˇ t . (cid:3) Proof of Theorem 10.2. We have the following theorem. Theorem 10.4. Let ˇ t be a solution of the Bethe ansatz equations (10.8) . Then there existunique polynomials Q λ i k =1 ( u − ˇ γ i,k ) ∈ C [ u ] , i = 1 , . . . , N , such that (10.10) det (cid:18) q m − ji λ i Y k =1 (cid:0) u − ˇ γ i,k − h ( i − j ) (cid:1)(cid:19) mi,j =1 = Y i Proof of Theorem 10.2. Formula (10.6) is obtained from formula (7.2) by several substi-tutions. First take q i = ˜ q − i for all i = 1 , . . . , N , substitute the variables t ( i ) j by ˜ θ i,j ,and replace the term D I,i by zero. Then write symmetric functions in the variables e Θ m via symmetric functions in the variables e Γ i , i = 1 , . . . , m . As a result, the expression P λ ( i ) j =1 t ( i ) j − P λ ( i − j =1 t ( i − j becomes P λ i j =1 ˜ γ i,j − h P i − j =1 ( λ i − λ j ) ˜ q j / (˜ q i − ˜ q j ) according to for-mula (10.5).Lemma 10.3 and Theorem 10.4 mean that formula (10.6) holds for those values of e Γ , . . . , e Γ N − that come from solutions ˇ t of the Bethe ansatz equations (10.8). By [MTV2, Theorem e Γ , . . . , e Γ N − form a Zariski opensubset of all values of e Γ , . . . , e Γ N − satisfying defining relations of the algebra H ˜ qT ( T ∗ F λ ) ,see (10.2). This proves Theorem 10.2. (cid:3) Limit ˜ q i / ˜ q i +1 → , i = 1 , . . . , N − , and CSM classes of Schubert cells. In thelimit ˜ q i / ˜ q i +1 → i = 1 , . . . , N − H ˜ qT ( T ∗ F λ ) turns into the algebra H ∗ T ( T ∗ F λ ) and the classes { W I } ∈ H ˜ qT ( T ∗ F λ ) become the classes [ W I ] ∈ H ∗ T ( T ∗ F λ ) . Thenformula (10.6) takes the form (cid:16) λ i X k =1 γ i,k − X a ∈ I i z a (cid:17) [ W I ] =(10.12) = h i − X j =1 λ i X m =1 λ j X m =1 ℓ i,m >ℓ j,m [ W I i,j ; m ,m ] − h N X j = i +1 λ i X m =1 λ j X m =1 ℓ i,m <ℓ j,m [ W I i,j ; m ,m ] . In particular, identities in (10.7) turn into the identities(10.13) ( γ , − z ) [ W ( { } , { } ) ] = − h [ W ( { } , { } ) ] , ( γ , − z ) [ W ( { } , { } ) ] = 0 . Remark. After the substitution h = 1 , the classes [ W I ] ∈ H ∗ T ( T ∗ F λ ) can be considered aselements of the equivariant cohomology H ∗ ( C × ) n ( F λ ) . By [RV] these new classes [ W I ] h =1 areproportional to the CSM classes κ I of the corresponding Schubert cells with the coefficient ofproportionality independent of the index I . Hence formula (10.12) induces the equivariantPieri rules for the equivariant CSM classes: (cid:16) λ i X k =1 γ i,k − X a ∈ I i z a (cid:17) κ I =(10.14) = h i − X j =1 λ i X m =1 λ j X m =1 ℓ i,m >ℓ j,m κ I i,j ; m ,m − h N X j = i +1 λ i X m =1 λ j X m =1 ℓ i,m <ℓ j,m κ I i,j ; m ,m , see detailed definitions of the CSM classes in [RV].11. Solutions of quantum differential equations and equivariant K -theory Solutions and equivariant K -theory. Introduce more variables: y = e π √− h/ ˜ κ ,´ t ( i ) j = e π √− t ( i ) j / ˜ κ , ´ z i = e π √− z i / ˜ κ , ´ γ i,j = e π √− γ i,j / ˜ κ , etc. We will use the acute superscriptalso for the corresponding collections of those variables like ´ Γ , ´ t , ´ z . We will write ´ Γ ± , ´ t ± , ´ z ± for the collections extended by the inverse variables, for instance, ´ z ± = (´ z ± , . . . , ´ z ± n ) .Let P be a Laurent polynomial in the variables ´ t , ´ z , y , symmetric in ´ t ( i )1 , . . . , ´ t ( i ) λ ( i ) foreach i = 1 , . . . , N − b Ψ P ( z ; h ; ˜ q ; ˜ κ ) = X I ∈I λ P ( ´Σ I , ´ z , y ) b Ψ I ( z ; h ; ˜ q ; ˜ κ ) , UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 37 where b Ψ I ( z ; h ; ˜ q ; ˜ κ ) are given by (9.21). Lemma 11.1. The function b Ψ P ( z ; h ; ˜ q ; ˜ κ ) is a solution of both the quantum differentialequations ∇ quant λ , ˜ q , ˜ κ,i f = 0 , i = 1 , . . . , N , and the associated qKZ difference equations.Proof. The statement follows from Theorem 9.5. (cid:3) Lemma 11.2. For a Laurent polynomial P in ´ t , ´ z , y symmetric in ´ t ( i )1 , . . . , ´ t ( i ) λ ( i ) for each i = 1 , . . . , N − , the function b Ψ P ( z ; h ; ˜ q ; ˜ κ ) is holomorphic in ˜ q on the domain L ′′ ⊂ C N such that | ˜ q i / ˜ q i +1 | < , i = 1 , . . . , N − , and a branch of log ˜ q i is fixed for each i = 1 ,. . . , N , and b Ψ P ( z ; h ; ˜ q ; ˜ κ ) is holomorphic in z , h on the domain L ′′′ ⊂ C n × C such that (11.2) h ˜ κ Z > , z a − z b + h ˜ κ Z , a, b = 1 , . . . , n , a = b . Proof. By the properties of b Ψ I ( z ; h ; ˜ q ; ˜ κ ) , see (9.22), we need only to show that the function b Ψ P ( z ; h ; ˜ q ; ˜ κ ) is regular at the hyperplanes z a − z b ∈ ˜ κ Z . This will be done in Section 11.2below. (cid:3) Consider the equivariant K -theory algebra, see [RTV2, Section 2.3], [RTV3, Section 4.4],(11.3) K T ( T ∗ F λ ) = C [ ´ Γ ± ] S λ × ... × S λN ⊗ C [ ´ z ± ] ⊗ C [ y ± ] .D N Y i =1 λ i Y j =1 ( u − ´ γ i,j ) = n Y a =1 ( u − ´ z a ) E , cf. (9.1). Introduce the variables ´ θ i,λ ( i ) , i = 1 , . . . , N . The relations(11.4) λ ( i ) Y a =1 ( u − ´ θ i,a ) = i Y j =1 λ j Y k =1 ( u − ´ γ j,k ) , i = 1 , . . . , N , define the epimorphism C [ ´ Θ ± ] S λ (1) × ... × S λ ( N ) ⊗ C [ ´ z ± ] ⊗ C [ y ± ] → K T ( T ∗ F λ ) . Thus theassignment P b Ψ P defines a map from K T ( T ∗ F λ ) to the space of solutions of the quantumdifferential equations and the associated qKZ difference equations with values in H ∗ T ( T ∗ F λ )extended by functions in z , h, ˜ q holomorphic in the domain L ′′′ × L ′′ . We evaluate belowthe determinant of this map.The cohomology algebra H ∗ T ( T ∗ F λ ) is a free module over H ∗ T ( pt ; C ) = C [ z ] ⊗ C [ h ] , witha basis given by the classes of Schubert polynomials(11.5) Y I ( Γ ) = A σ I ( γ , , . . . , γ , λ , γ , , . . . , γ , λ , . . . , γ N, , . . . , γ N, λ N ) , I ∈ I λ . Similarly, the algebra K T ( T ∗ F λ ) is a free module over K T ( pt ; C ) = C [ ´ z ± ] ⊗ C [ y ± ] , witha basis given by the classes of Schubert polynomials(11.6) b Y I ( ´ Γ ) = A σ I ( ´ γ , , . . . , ´ γ , λ , ´ γ , , . . . , ´ γ , λ , . . . , ´ γ N, , . . . , ´ γ N, λ N ) , I ∈ I λ . Both assertions are clear from Proposition A.7. Expand solutions of the quantum differential equation using those Schubert bases:(11.7) b Ψ b Y I ( Γ ; z ; h ; ˜ q ; ˜ κ ) = X J ∈I λ b Ψ I,J ( z ; h ; ˜ q ; ˜ κ ) Y J ( Γ ) . Theorem 11.3. Let n > . We have det (cid:0) b Ψ I,J ( z ; h ; ˜ q ; ˜ κ ) (cid:1) I,J ∈I λ = N − Y i =1 N Y j = i +1 (1 − ˜ q i / ˜ q j ) h min( λ i ,λ j ) / ˜ κ N Y i =1 ˜ q d (1) λ ,i P na =1 z a / ˜ κi × (11.8) × n Y a =1 n Y b =1 b = a (cid:16) π √− (cid:16) z a − z b − h ˜ κ (cid:17)(cid:17) d (2) λ , where (11.9) d (1) λ ,i = λ i ( n − λ ! . . . λ N ! , d (2) λ = 2 ( n − λ ! . . . λ N ! N − X i =1 N X j = i +1 λ i λ j . Proof. By Lemma 11.1, the left-hand side of (11.8) solves the differential equations (cid:16) ˜ κ ˜ q i ∂∂ ˜ q i − tr (cid:0) X − λ ,i ( z ; − h ; ˜ q − ) (cid:1)(cid:17) det (cid:0) b Ψ I,J ( z ; h ; ˜ q ; ˜ κ ) (cid:1) I,J ∈I λ = 0 , i = 1 , . . . , N , where X − λ ,i are the modified dynamical Hamiltonians (9.18). Thus det( b Ψ I,J ) equals the firsttwo products in the right-hand side of (11.8) multiplied by a factor that does not depend on ˜ q . The remaining factor is found by taking the limit ˜ q i / ˜ q i +1 → i = 1 , . . . , N − (cid:3) Corollary 11.4. The collection of functions (cid:0) b Ψ b Y I ( z ; h ; ˜ q ; ˜ κ ) (cid:1) I ∈I λ is a basis of solutions ofboth the quantum differential equations ∇ quant λ , ˜ q , ˜ κ,i f = 0 , i = 1 , . . . , N , and the associated qKZ difference equations. (cid:3) End of proof of Lemma 11.2. It is enough to show the regularity of b Ψ P ( z ; h ; ˜ q ; ˜ κ )at the hyperplanes z a − z b ∈ ˜ κ Z assuming that h/ ˜ κ is real negative and sufficiently large.For a number A , let C ( A ) ⊂ C be a parabola with the following parametrization:(11.10) C ( A ) = { ˜ κ (cid:0) A + s − s √− (cid:1) | s ∈ R } . Given z , ˜ κ , take A such that all the points z , . . . , z n are inside C ( A + N − 2) . Suppose h/ ˜ κ is a sufficiently large negative real so that all the points z + h, . . . , z n + h are outside C ( A ) . Set(11.11) C λ ( z ) = (cid:0) C ( A ) (cid:1) × λ (1) × . . . × (cid:0) C ( A + N − (cid:1) × λ ( N − indicating the dependence on λ and z explicitly. The integral (11.12) below does notdepend on a particular choice of A . UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 39 Lemma 11.5. For a Laurent polynomial P in ´ t , ´ z , y symmetric in ´ t ( i )1 , . . . , ´ t ( i ) λ ( i ) for each i = 1 , . . . , N − , we have (11.12) b Ψ P ( z ; h ; ˜ q ; ˜ κ ) = ˜ κ − λ { } − λ { } ( − λ { } + λ { } (cid:0) π √− − h/ ˜ κ ) (cid:1) − λ { } e Ψ P ( z ; h ; ˜ q ; ˜ κ ) , e Ψ P ( z ; h ; ˜ q ; ˜ κ ) = b Ω λ ( ˜ q ; ˜ κ ) λ (1) ! . . . λ ( N − ! Z C λ ( z ) P (´ t ; ´ z ; y ) Φ λ ( t ; z ; ˜ q − ; − ˜ κ ) Q ( Γ ) b W ( t ; Γ ) d λ { } t . Proof. The integral converges provided | ˜ q i / ˜ q i +1 | < i = 1 , . . . , N − q i is fixed for each i = 1 , . . . , N . Evaluate the integral by residues in the followingway: replace C λ ( z ) by (cid:0) C ( A + B ) (cid:1) × λ (1) × . . . × (cid:0) C ( A + B + N − (cid:1) × λ ( N − , where B ∈ R > ,and send B to infinity. Then by (9.21), the resulting series yields formula (11.1). (cid:3) The integrand in formula (11.12) is regular at the hyperplanes z a − z b ∈ ˜ κ Z , and so doesthe function b Ψ P ( z ; h ; ˜ q ; ˜ κ ) . Lemma 11.2 is proved. (cid:3) The homogeneous case z = 0 . The quantum differential equations ∇ quant λ , ˜ q , ˜ κ,i f = 0depend on z as a parameter and are well defined at z = 0 .For any Laurent polynomial P in ´ t , y , symmetric in ´ t ( i )1 , . . . , ´ t ( i ) λ ( i ) for each i = 1 ,. . . , N − b Ψ P (0; h ; ˜ q ; ˜ κ ) is a solution of the quantum differential equations ∇ quant λ , ˜ q , ˜ κ,i, z =0 f = 0 , i = 1 , . . . , N , see Lemma 11.1. Lemma 11.6. The function b Ψ P (0; h ; ˜ q ; ˜ κ ) is holomorphic in ˜ q , h provided | ˜ q i / ˜ q i +1 | < , i = 1 , . . . , N − , a branch of log ˜ q i is fixed for each i = 1 , . . . , N , and h ˜ κ Z > .Proof. By Lemma 11.2, we need only to show that b Ψ P (0; h ; ˜ q ; ˜ κ ) is regular if h ∈ ˜ κ Z < .We will prove that b Ψ P (0; h ; ˜ q ; ˜ κ ) is regular if h/ ˜ κ ∈ R < .If h/ ˜ κ is a sufficiently large negative real, write b Ψ P (0; h ; ˜ q ; ˜ κ ) by formula (11.12). Thenone can replace the integration contour C λ (0) by the contour C ′ λ ( h, ˜ κ ) = (cid:0) C (cid:0) ( N − ε (cid:1)(cid:1) × λ (1) × (cid:0) C (cid:0) ( N − ε (cid:1)(cid:1) × λ (2) × . . . × (cid:0) C ( ε ) (cid:1) × λ ( N − , where ε = h/ ( N ˜ κ ) , without changing the integral. With the integration over C ′ λ ( h, ˜ κ ) , itis clear that b Ψ P (0; h ; ˜ q ; ˜ κ ) continues to a function regular for all negative real h/ ˜ κ . (cid:3) Consider the algebras H ∗ C × ( T ∗ F λ ) = H ∗ T ( T ∗ F λ ) / h z = 0 i , K C × ( T ∗ F λ ) = K T ( T ∗ F λ ) / h ´ z = (1 , . . . , i . The algebra H ∗ C × ( T ∗ F λ ) is a free module over C [ h ] and the algebra K C × ( T ∗ F λ ) is a freemodule over C [ y ± ] , with bases given by the respective classes of Schubert polynomials, see(11.5), (11.6).Expand solutions of the quantum differential equation at z = 0 using those Schubertbases:(11.13) b Ψ b Y I ( Γ ; 0; h ; ˜ q ; ˜ κ ) = X J ∈I λ b Ψ I,J (0; h ; ˜ q ; ˜ κ ) Y J ( Γ ) . Let d λ = n ! / ( λ ! . . . λ N !) . Formula (11.8) at z = 0 takes the formdet (cid:0) b Ψ I,J (0; h ; ˜ q ; ˜ κ ) (cid:1) I,J ∈I λ =(11.14) = (cid:16) π √− (cid:16) − h ˜ κ (cid:17)(cid:17) d λ P i The collection of functions (cid:0) b Ψ b Y I (0; h ; ˜ q ; ˜ κ ) (cid:1) I ∈I λ is a basis of solutions ofthe quantum differential equations ∇ quant λ , ˜ q , ˜ κ,i, z =0 f = 0 , i = 1 , . . . , N . (cid:3) The limit h → ∞ . Suppose that ˜ q i / ˜ q i +1 = ( − h ) − λ i − λ i +1 p i /p i +1 , i = 1 , . . . , N − q N = p N , where p , . . . , p N are new variables. The limit h → ∞ keeping p , . . . , p N fixed corresponds to replacing the cotangent bundle T ∗ F λ by the partial flag variety F λ itself, the algebras H ∗ T ( T ∗ F λ ) , K T ( T ∗ F λ ) by the respective algebras H ∗ A ( F λ ) , K A ( F λ ) ,where A ⊂ GL n ( C ) is the torus of diagonal matrices, and the equivariant quantum differ-ential equations for T ∗ F λ by the analogous equations for F λ . We will discuss this limit indetail in a separate paper making here only a few remarks.We identify H ∗ A ( F λ ) with the subalgebra in H ∗ T ( T ∗ F λ ) of h -independent elements, and K A ( F λ ) with the subalgebra in K T ( T ∗ F λ ) of y -independent elements.The discussion of the limit h → ∞ is based on Stirling’s formula(11.15) Γ( α − h/ ˜ κ )Γ( β − h/ ˜ κ ) ∼ ( − h/ ˜ κ ) α − β , h → ∞ . For F λ , we have the following counterparts of the master functionΦ ◦ λ ( t ; z ; p ; ˜ κ ) = ( e π √− λ N − n ) p N ) P na =1 z a / ˜ κ N − Y i =1 (cid:16) e π √− λ i − λ i +1 ) ˜ κ λ i + λ i +1 p i p i +1 (cid:17) P λ ( i ) j =1 t ( i ) j / ˜ κ × (11.16) × N − Y i =1 λ ( i ) Y a =1 (cid:18) λ ( i ) Y b =1 b = a (cid:0) ( t ( i ) b − t ( i ) a ) / ˜ κ (cid:1) λ ( i +1) Y c =1 Γ (cid:0) ( t ( i +1) c − t ( i ) a ) / ˜ κ (cid:1)(cid:19) , the weight function(11.17) b W ◦ ( t , Γ ) = N − Y i =1 N Y j = i +1 λ ( i ) Y a =1 λ ( i +1) Y b = λ ( i ) +1 ( t ( i ) a − γ j,b ) , and solutions of the quantum differential equations(11.18) b Ψ ◦ I ( z ; p ; ˜ κ ) = ( − ˜ κ ) − λ { } − λ { } ˜ κ P N − i =1 P a ∈ Ii ( λ i + λ i +1 ) z a / ˜ κ e Ψ ◦ I ( z ; p ; ˜ κ ) , e Ψ ◦ I ( z ; p ; ˜ κ ) = X r ∈ Z λ { } > Res t =Σ I + r ˜ κ (cid:0) Φ ◦ λ ( t ; z ; p ; ˜ κ ) (cid:1) b W ◦ (Σ I + r ˜ κ ; Γ ) , UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 41 where I i = S ij =1 I j and λ { } = P i For references regarding Schubert polynomials, see for example [L, M].Let D , . . . , D n − be the divided difference operators acting on functions of x , . . . , x n : D i f ( x , . . . , x n ) = f ( x , . . . , x n ) − f ( x , . . . , x i +1 , x i , . . . , x n ) x i − x i +1 , cf. (3.7). They satisfy the nil-Coxeter algebra relations,(A.1) ( D i ) = 0 , D i D i +1 D i = D i +1 D i D i +1 , D i D j = D j D i , | i − j | > . Given σ ∈ S n with a reduced decomposition σ = s i ,i +1 . . . s i j ,i j +1 , define D σ = D i . . . D i j .For instance, D id is the identity operator and D s i,i +1 = D i . Due to relations (A.1), theoperator D σ does not depend on the choice of a reduced decomposition. Moreover, D σ D τ = D στ , if | σ | + | τ | = | στ | , D σ D τ = 0 , otherwise . Here | σ | is the length of σ . Denote x σ = ( x σ (1) , . . . , x σ ( n ) ) . Let σ be the longest permu-tation, σ ( i ) = n + 1 − i , i = 1 , . . . , n . Then D σ f ( x ) = Y i For any σ, τ ∈ S n , (A.3) D σ (cid:0) A σ ( x ) A τσ ( x σ ) (cid:1) = ( − σσ δ σ,τ . (cid:3) UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 43 Proposition A.2. Cauchy formula holds, (A.4) X σ ∈ S n ( − σ A σ ( x ) A σσ ( y ) = n − Y i =1 n − i Y j =1 ( y i − x j ) . (cid:3) For any f ∈ C [ x ] and σ ∈ S n , define f σ ∈ C [ x ] S n by the rule(A.5) f σ ( x ) = ( − σσ D σ (cid:0) f ( x ) A σσ ( x σ ) (cid:1) . Proposition A.3. For any f ∈ C [ x ] , (A.6) f ( x ) = X σ ∈ S n f σ ( x ) A σ ( x ) . (cid:3) Thus C [ x ] is a free module over C [ x ] S n of rank n ! with a basis given by Schubert poly-nomials.Recall the notation from Section 2.1, and I min , I max ∈ I λ , I min = (cid:0) { , . . . , λ } , . . . , { n − λ N + 1 , . . . , n } (cid:1) ,I max = (cid:0) { n − λ + 1 , . . . , n } , . . . , { , . . . , λ N } (cid:1) . For I = ( I , . . . , I N ) ∈ I λ , I j = { i j, < . . . < i j,λ j } , define the permutations σ I , σ I ( k ) = i j,k − λ ( j − , k ∈ I min j , j = 1 , . . . , N , and σ I = σ I ( σ I max ) − . Then σ I ( I min ) = σ I ( I max ) = I .Let S λ × . . . × S λ N ⊂ S n be the isotropy subgroup of I min . Lemma A.4. For any I ∈ I λ , we have A σ I ( x ) ∈ C [ x ] S λ × ... × S λN . (cid:3) For example, A σ I max ( x ) = Q N − a =1 Q i ∈ I min a x N − ai . Proposition A.5. For any I, J ∈ I λ , (A.7) D σ I max (cid:0) A σ I ( x ) A σ J ( x σ ) (cid:1) = ( − σ I δ I,J . (cid:3) Proposition A.6. We have, (A.8) X I ∈I λ ( − σ I A σ I ( x ) A σ I ( y σ ) = Y a
For any f ∈ C [ x ] S λ × ... × S λN , we have (A.9) f ( x ) = X I ∈I λ f σ I ( x ) A σ I ( x ) , that is, in formula (A.6) , f σ = 0 unless σ = σ I for some I ∈ I λ , and (A.10) f σ I ( x ) = ( − σ I D σ I max (cid:0) f ( x ) A σ I ( x σ ) (cid:1) . (cid:3) Define R λ ( x ) = Y a
For any f ∈ C [ x ] S λ × ... × S λN , we have (A.11) D σ I max f ( x ) = X I ∈I λ f ( x σ I ) R λ ( x σ I ) . (cid:3) Proposition A.9. Let n > . Then (A.12) det (cid:0) A σ I ( x σ J ) (cid:1) I,J ∈I λ = Y i The formula (9.23) for the asymptotics of solutions (cid:0) b Ψ I ( z ; h ; ˜ q ; ˜ κ ) (cid:1) I ∈I λ to the joint systemof the quantum differential equations and associated qKZ difference equations reminds thestatement of the gamma conjecture, see [D1, D2, KKP, GGI, GI, GZ].The gamma conjecture [D2, GGI] is a conjecture relating the quantum cohomology of aFano manifold X with its topology. The quantum cohomology of X defines a flat quantumconnection over C × in the direction of first Chern class c ( X ). The connection has a regularsingular point at t = 0 and an irregular singular point at t = ∞ . The connection has adistinguished (multivalued) flat section J X ( t ) defined by Givental in [Gi1] and called theJ-function. Under certain assumptions, the limit of the J-function: A X := lim t →∞ J X ( t ) h [pt] , J X ( t ) i ∈ H ∗ ( x )exists and defines the principal asymptotic class A X of X . The gamma conjecture says that A X equals the gamma class ˆΓ X of the tangent bundle of X .The gamma class of a holomorphic vector bundle E over a topological space X is themultiplicative characteristic class, in the sense of Hirzebruch, associated to the power seriesexpansion Γ(1 + x ) = 1 − γx + γ + ζ (2)2 x + . . . of the gamma function at 1, where γ is theEuler constant and ζ (2) is the value at 2 of the zeta function. In other words, the gammaclass is the function that associates to a holomorphic bundle E over X the cohomologyclass ˆΓ( E ) = Q i Γ(1 + τ i ) ∈ H ∗ ( X ; R ), where the total Chern class of E has the formalfactorization c ( E ) = Q i (1 + τ i ) with the Chern roots τ i of degree 2. If E is the tangentbundle of X , we write ˆΓ X for ˆΓ( E ). Its terms of degree E ) = 1 − γc + (cid:0) − ζ (2) c + ζ (2) + γ c (cid:1) (B.1) + (cid:0) − ζ (3) c + ( ζ (3) + γζ (2)) c c − ζ (3) + 3 γζ (2) + γ c (cid:1) + . . . , see [GZ]. UANTUM DIFFERENTIAL EQUATIONS, PIERI RULES, AND GAMMA-THEOREM 45 Consider the equivariant gamma class of T ∗ F λ , b Γ T ∗ F λ = N − Y i =1 N Y j = i +1 λ i Y a =1 λ j Y b =1 Γ(1 + γ j,b − γ i,a ) Γ(1 + γ i,a − γ j,b − h ) . cf. (9.2), and the equivariant first Chern classes c ( E i ) = P λ i a =1 γ i,a , i = 1 , . . . , N , ofthe vector bundles E i over T ∗ F λ with fibers F i /F i − , see (9.3). Theorem 9.6 can bereformulated as follows. Theorem B.1 (Gamma theorem for T ∗ F λ ) . For κ = 1 , the leading term of the asymptoticsof the q -hypergeometric solutions (cid:0) b Ψ I ( z ; h ; ˜ q ; ˜ κ ) (cid:1) I ∈I λ in (9.23) is the product of the equi-variant gamma class of T ∗ F λ and the exponentials of the equivariant first Chern classes ofthe associated vector bundles E , . . . , E N : (B.2) ˆΓ T ∗ F λ N Y l =1 (cid:0) e π √− λ i − n ) ˜ q i (cid:1) c ( E i ) . (cid:3) Similarly formula (11.19) can be reformulated as follows. Theorem B.2 (Gamma theorem for F λ ) . For κ = 1 , the leading term of the asymptotics ofthe q -hypergeometric solutions (cid:0) b Ψ ◦ I ( z ; p ; ˜ κ ) (cid:1) I ∈I λ in (11.18) is the product of the equivariantgamma class of T ∗ F λ and the exponentials of the equivariant first Chern classes of theassociated vector bundles E , . . . , E N : (B.3) ˆΓ F λ N Y l =1 (cid:0) e π √− λ i − n ) p i (cid:1) c ( E i ) . (cid:3) Example. Let N = 2 , n = 2 , λ = (1 , 1) . 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