HYPERGEOMETRIC SOLUTIONS OF THE QUANTUM DIFFERENTIAL EQUATION OF THE COTANGENT BUNDLE OF A PARTIAL FLAG VARIETY ◦ ⋄ V.TARASOV AND A.VARCHENKO 3 ◦Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis 1 0 402 North Blackford St, Indianapolis, IN 46202-3216, USA 2 ◦St.Petersburg Branch of Steklov Mathematical Institute n a Fontanka 27, St.Petersburg, 191023, Russia J 2 ⋄Department of Mathematics, University of North Carolina at Chapel Hill 1 Chapel Hill, NC 27599-3250, USA ] G Abstract. We describe hypergeometric solutions of the quantum differential equation of A the cotangent bundle of a gln partial flag variety. These hypergeometric solutions manifest . h the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety. t a m [ Contents 1 v 1. Introduction 2 5 2. Cotangent bundles of partial flag varieties 3 0 2.1. Partial flag varieties 3 7 2.2. Equivariant cohomology 4 2 . 3. Yangian 4 1 3.1. Yangian Y(gl ) 4 0 N 3 3.2. Algebra Y(gl ) 5 N 1 3.3. Y(gl )-action on (CN)⊗n⊗C[z;h] 5 : N e v 3.4. Y(gl )-action on H∗(X ) according to [RTV] 6 i e N T n X 4. Dynamical Hamiltonians and quantum multiplication 6 e r 4.1. Dynamical Hamiltonians 6 a 4.2. Dynamical Hamiltonians Xq on H∗(T∗F ) 7 λ,i T λ 4.3. Quantum multiplication by divisors on H∗(T∗F ) 8 T λ 4.4. Dynamical Hamiltonians on (CN)⊗n⊗C[z;h] 8 4.5. qKZ difference connection 9 5. Yangian Y(gl ) weight functions 10 N 5.1. Weight functions W 10 I e 5.2. Weight functions W 10 σ,I 5.3. Stable envelope map 10 5.4. The inverse map 11 5.5. The difference connection on H∗(T∗F ) 11 T λ 6. Four more connections 12 ◦E-mail: vt@math.iupui.edu, vt@pdmi.ras.ru, supported in part by NSF grant DMS-0901616 ⋄E-mail: anv@email.unc.edu, supported in part by NSF grant DMS-1101508 2 V.TARASOVANDA.VARCHENKO 6.1. Trigonometric dynamical connection on π 12 Vλ 6.2. Rational qKZ difference connection on π 12 Vλ 6.3. Relations between flat sections 13 6.4. Trigonometric KZ connection on W 13 λ 6.5. Rational dynamical difference connection on π 14 Wλ 6.6. Equivalence of connections on π and π 14 Vλ Wλ 7. Integral representations for flat sections 15 7.1. Master function 15 7.2. gl weight functions 15 n 7.3. Hypergeometric integrals 16 7.4. Example 17 References 18 1. Introduction In [MO], D.Maulik and A.Okounkov develop a general theory connecting quantum groups and equivariant quantum cohomology of Nakajima quiver varieties, see [N1, N2]. In particular, in [MO] the operators of quantum multiplication by divisors are described. As well- known, these operators determine the equivariant quantum differential equation of a quiver variety. In this paper we apply this description to the cotangent bundles of gl partial n flag varieties and construct hypergeometric solutions of the associated equivariant quantum differential equation. In [GRTV] and [RTV], the equivariant quantum differential equation of the cotangent bundle of a gl partial flag variety was identified with the trigonometric dynamical differ- n ential equation introduced in [TV4]. By the (gl ,gl )-duality of [TV4], the trigonometric N n dynamical differential equation is identified with the trigonometric KZ differential equation. Hypergeometric solutions of the trigonometric KZ differential equation were constructed in [SV, MV]. By using this sequence of isomorphisms we obtain hypergeometric solutions of the equivariant quantum differential equation. The hypergeometric solutions have the form (1.1) I (z;q˜;h;κ) = Φ(s;z;q˜;h)h/κω(s;z;q˜;h)ds, ψ Z ψ(z;q˜;h;κ) where z = (z ,...,z ), h are equivariant parameters, q˜ = (q˜ ,...,q˜ ) quantum parameters, 1 n 1 N s = (s(i)) integration variables, κ the parameter of the differential equation, ψ(z;q˜;h;κ) j the integration cycle in the s-space, ω(s;z;q˜;h) a rational function, Φ(s;z;q˜;h) the master function, see Corollary 7.4. Studying solutions of the quantum differential equation may lead to better understanding Gromow-Witten invariants of the cotangent bundle of a partial flag variety, c.f. Givental’s study of the J-function in [G1, G2, G3]. The existence of solutions of the quantum differential equation as such oscillatory integrals manifests the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety. One may think that the logarithm of the master function is the Landau-Ginzburg potential of the mirror dual object. In particular, one may expect that the algebra of func- tionsonthecritical set ofthemaster functionΦ(·;z;q˜;h)isisomorphic tothecorresponding HYPERGEOMETRIC SOLUTIONS OF THE QUANTUM DIFFERENTIAL EQUATION 3 localizationoftheequivariant quantum cohomologyalgebraof thecotangent bundle ofa partial flag variety, see Section 7.4 and similar statements for Bethe algebras in [MTV2, MTV3]. The trigonometric KZ differential equation and trigonometric dynamical differential equation come in pairs with compatible difference equations. The trigonometric KZ differential equation is compatible with the rational dynamical difference equation introduced in [TV3]. The trigonometric dynamical differential equation is compatible with the rational qKZ difference equation, as shown in [TV4]. Under the (gl ,gl )-duality, the pair consisting of the N n trigonometric KZ differential equation and rational dynamical difference equation is identified with the pair consisting ofthe trigonometric dynamical differential equation andrational qKZ difference equation, see [TV4]. Under the identification of the trigonometric dynamical differential equation with the equivariant quantum differential equation of [BMO], the rational qKZ difference operators are identified with the shift operators of [MO], see also [BMO]. It was shown in [MV], that the hypergeometric solutions for the trigonometric KZ differential equation also satisfy the difference dynamical equation. By using all of the above identifications, we conclude that the hypergeometric solutions (1.1) of the equivariant quantum differential equation give also flat sections of the difference connection defined by the shift operators of [MO] on the equivariant quantum cohomology of the cotangent bundle of a partial gl flag variety. n As shown in [TV1, TV2], the qKZ equation has solutions in the form of multidimensional q-hypergeometric integrals. These q-hypergeometric integrals are expected to satisfy the compatible dynamical differential equation. That will imply that the quantum differential equation of [MO] has solutions in the form of multidimensional q-hypergeometric integrals. See an example of such solutions in Section 8.4 of [GRTV]. We plan to develop these q- hypergeometric solutions in a separate paper. 2. Cotangent bundles of partial flag varieties 2.1. Partial flag varieties. Fix natural numbers N,n. Let λ ∈ ZN , |λ| = λ +···+λ = >0 1 N n. Consider the partial flag variety F parametrizing chains of subspaces λ 0 = F ⊂ F ⊂ ... ⊂ F = Cn 0 1 N with dimF /F = λ , i = 1,...,N. Denote by T∗F the cotangent bundle of F . Denote i i−1 i λ λ X = ∪ T∗F . n |λ|=n λ Example. If n = 1, then λ = (0,...,0,1 ,0,...,0), T∗F is a point and X is the union of i λ 1 N points. If n = 2 then λ = (0,...,0,1 ,0,...,0,1 ,0,...,0) or λ = (0,...,0,2 ,0,...,0). In the i j i first case T∗F is the cotangent bundle of projective line, in the second case T∗F is a point. λ λ Thus X istheunionofN points andN(N−1)/2 copies ofthecotangent bundle ofprojective 2 line. Let I = (I ,...,I ) be a partition of {1,...,n} into disjoint subsets I ,...,I . Denote 1 N 1 N I the set of all partitions I with |I | = λ , j = 1,...N. Denote I = ∪ I . λ j j n |λ|=n λ Let u ,...,u be the standard basis of Cn. For any I ∈ I , let x ∈ F be the point 1 n λ I λ corresponding to the coordinate flag F ⊂ ··· ⊂ F , where F is the span of the standard 1 N i 4 V.TARASOVANDA.VARCHENKO basis vectors u ∈ Cn with j ∈ I ∪...∪I . We embed F in T∗F as the zero section and j 1 i λ λ consider the points x as points of T∗F . I λ 2.2. Equivariant cohomology. Denote G = GL (C)×C×. Let A⊂ GL (C) be the torus n n of diagonal matrices. Denote T = A×C× the subgroup of G. The groups A⊂ GL act on Cn and hence on T∗F . Let the group C× act on T∗F by n λ λ multiplication in each fiber. We denote by −h its C×-weight. We consider the equivariant cohomology algebras H∗(T∗F ;C) and T λ H∗(X ) = ⊕ H∗(T∗F ;C). T n |λ|=n T λ Denote by Γ = {γ ,...,γ } the set of the Chern roots of the bundle over F with fiber i i,1 i,λi λ F /F . Let Γ = (Γ ;...;Γ ). Denote by z = {z ,...,z } the Chern roots corresponding i i−1 1 N 1 n to the factors of the torus T. Then N λi n HT∗(T∗Fλ) = C[Γ]Sλ1×···×SλN ⊗C[z]⊗C[h] (u−γi,j) = (u−za) . .DYi=1Yj=1 Ya=1 E The cohomology H∗(T∗F ) is a module over H∗(pt;C) = C[z]⊗C[h]. T λ T Example. If n = 1, then H∗(X ) = ⊕N H∗(T∗F ) T 1 i=1 T (0,...,0,1i,0,...,0) is naturally isomorphic to CN ⊗C[z ;h] with basis v = (0,...,0,1 ,0,...,0), i = 1,...,N. 1 i i For i = 1,...,N, denote λ(i) = λ + ··· + λ . Denote Θ = {θ ,...,θ } the Chern 1 i i i,1 i,λ(i) roots of the bundle F over F with fiber F . Let Θ = (Θ ,...,Θ ). The relations i λ i 1 N λ(i) i λi (u−θ ) = (u−γ ), i = 1,...,N, i,j i,j Yj=1 Yℓ=1Yj=1 define the homomorphism C[Θ]Sλ(1)×···×Sλ(N) ⊗C[z]⊗C[h] → HT∗(T∗Fλ). 3. Yangian 3.1. Yangian Y(gl ). The Yangian Y(gl ) is the unital associative algebra with generators N N T{s} for i,j = 1,...,N, s ∈ Z , subject to relations i,j >0 (u−v) T (u),T (v) = T (v)T (u)−T (u)T (v), i,j,k,l = 1,...,N , i,j k,l k,j i,l k,j i,l (cid:2) (cid:3) where T (u) = δ + ∞ T{s}u−s. The Yangian Y(gl ) is a Hopf algebra with the co- i,j i,j s=1 i,j N product ∆ : Y(gl ) →PY(gl )⊗Y(gl ) given by ∆ T (u) = N T (u)⊗T (u) for N N N i,j k=1 k,j i,k i,j = 1,...,N. The Yangian Y(gl ) contains, as a H(cid:0)opf sub(cid:1)algebPra, the universal envelop- N ing algebra U(gl ) of the Lie algebra gl . The embedding is given by e 7→ T{1}, where N N i,j j,i e are standard standard generators of gl . i,j N Notice that T{1},T{s} = δ T{s} − δ T{s} for i,j,k,l = 1,...,N, s ∈ Z , which i,j k,l i,l k,j j,k i,l >0 implies that the(cid:2)Yangian Y(cid:3)(gl ) is generated by the elements T{1} , T{1} , i = 1,...,N−1, N i,i+1 i+1,i {s} and T , s > 0. 1,1 HYPERGEOMETRIC SOLUTIONS OF THE QUANTUM DIFFERENTIAL EQUATION 5 3.2. Algebra Y(gl ). In this section we follow [GRTV, Section 3.3]. In formulas of that N Section 3.3 we replace h with −h. e Let Y(gl ) bethesubalgebra of Y(gl )⊗C[h] generatedover C by C[h] andthe elements N N (−h)s−1T{s} for i,j = 1,...,N, s > 0. Equivalently, the subalgebra Y(gl ) is generated e i,j N over C by C[h] and the elements T{1} , T{1} , i = 1,...,N −1, and (−h)s−1T{s}, s > 0. i,i+1 i+1,i e 1,1 For p = 1,...,N, i = {1 6 i < ··· < i 6 N}, j = {1 6 j < ··· < j 6 N}, define 1 p 1 p M (u) = (−1)σT (u)...T (u−p+1). i,j i1,jσ(1) ip,jσ(p) X σ∈Sp Introduce the series A (u),...,A (u), E (u),...,E (u), F (u),...,F (u): 1 N 1 N−1 1 N−1 ∞ (3.1) A (u) = M (−u/h) = 1+ (−h)sA u−s, p i,i p,s Xs=1 ∞ (3.2) E (u) = −h−1M (−u/h) M (−u/h) −1 = (−h)s−1E u−s, p j,i i,i p,s (cid:0) (cid:1) Xs=1 ∞ F (u) = −h−1 M (−u/h) −1M (−u/h) = (−h)s−1F u−s, p i,i i,j p,s (cid:0) (cid:1) Xs=1 where in formulas (3.1) and (3.2) we have i = {1,...,p}, j = {1,...,p−1,p+1}. Observe {1} {1} {s} that E = T , F = T and A = T , so the coefficients of the series E (u), p,1 p+1,p p,1 p,p+1 1,s 1,1 p F (u) and h−1(A (u) − 1 together with C[h] generate Y(gl ). In what follows we will p p N describe actions of the alg(cid:1)ebra Y(gl ) by using series (3.1), (3.2). N e e 3.3. Y(gl )-action on (CN)⊗n ⊗C[z;h]. Set N e L(u) = (u−z −hP(0,n))...(u−z −hP(0,1)), n 1 where the factors of CN⊗ (CN)⊗n are labeled by 0,1,...,n and P(i,j) is the permutation of the i-th and j-th factors. The operator L(u) is a polynomial in u,z,h with values in End(CN⊗(CN)⊗n). We consider L(u) as an N×N matrix with End(V)⊗C[u;z;h]-valued entries L (u). i,j Proposition 3.1 (Proposition 4.1 in [GRTV]). The assignment n (3.3) φ T (−u/h) = L (u) (u−z )−1 i,j i,j a (cid:0) (cid:1) aY=1 defines the action of the algebra Y(gl ) on (CN)⊗n ⊗C[z;h]. Here the right-hand side of N (3.3) is a series in u−1 with coefficients in End((CN)⊗n)⊗C[z;h]). e Under this action, the subalgebra U(gl ) ⊂ Y(gl ) acts on (CN)⊗n ⊗ C[z;h] in the N N standard way: any element x ∈ gl acts as x(1) +...+x(n). The action φ was denoted in N e [GRTV] by φ+. 6 V.TARASOVANDA.VARCHENKO 3.4. Y(gl )-action on H∗(X ) according to [RTV]. We define the Y(gl )-action ρ on N T n N H∗(X ) by formulas (3.4), (3.6), (3.7) below. We define ρ A (u) : H∗(T∗F ) → H∗(T∗F ) T en p T e λ T λ by (cid:0) (cid:1) p λp h (3.4) ρ A (u) : [f] 7→ f(Γ;z;h) 1− , p u−γ (cid:0) (cid:1) h aY=1 Yi=1 (cid:16) p,i(cid:17)i for p = 1,...,N. In particular, (3.5) ρ(X∞) : [f] 7→ [(γ +...+γ )f(Γ;z;h)], i = 1,...,N . i i,1 i,λi Let α ,...,α be simple roots, α = (0,...,0,1,−1,0,...,0), with p−1 first zeros. We 1 N−1 p define ρ E (u) : H∗(T∗F ) 7→ H∗(T∗F ), p T λ−αp T λ (cid:0) (cid:1) λp f(Γ′i;z;h) λp 1 λp+1 (3.6) ρ E (u) : [f] 7→ (γ −γ −h) , p p,i p+1,k (cid:20) u−γ γ −γ (cid:21) (cid:0) (cid:1) Xi=1 p,i Yj=1 p,j p,i kY=1 j6=i ρ F (u) : H∗(T∗F ) 7→ H∗(T∗F ), p T λ+αp T λ (cid:0) (cid:1) λp+1 f(Γi′;z;h) λp+1 1 λp (3.7) ρ F (u) : [f] 7→ (γ −γ −h) , p p,k p+1,i (cid:20) u−γ γ −γ (cid:21) (cid:0) (cid:1) Xi=1 p+1,i Yj=1 p+1,i p+1,j kY=1 j6=i where Γ′i = (Γ ;...;Γ ;Γ −{γ };Γ ∪{γ };Γ ;...;Γ ), 1 p−1 p p,i p+1 p,i p+2 N Γi′ = (Γ ;...;Γ ;Γ ∪{γ };Γ −{γ };Γ ;...;Γ ). 1 p−1 p p+1,i p+1 p+1,i p+2 N Theorem 3.2 (Theorem 5.10 in [GRTV]). These formulas define a Y(gl )-module structure N on H∗(X ). T n e This Y(gl )-module structure was denoted in [GRTV] by ρ− and h in [GRTV] is replaced N with −h. The topological interpretation of this Y(gl )-action see in [GRTV, Theorem 5.16]. e N In [MO], a Yangian module structure on H∗(X ) was introduced. T en Theorem 3.3 (Corollary6.4in[RTV]). The Y(gl )-module structure ρ on H∗(X ) coincides N T n with the Yangian module structure on H∗(X ) introduced in [MO]. T ne 4. Dynamical Hamiltonians and quantum multiplication 4.1. Dynamical Hamiltonians. Assume that q ,...,q are distinct numbers. Define the 1 N elements Xq,...,Xq ∈ Y(gl ) by the rule 1 N N e N h q Xq = −hT{2} + e e −1 − h j G , i i,i 2 i,i i,i q −q i,j (cid:0) (cid:1) Xj=1 i j j6=i HYPERGEOMETRIC SOLUTIONS OF THE QUANTUM DIFFERENTIAL EQUATION 7 where G = e e − e = e e − e . By taking the limit q /q → 0 for all i = 1, i,j i,j j,i i,i j,i i,j j,j i+1 i ...,N −1, we define the elements X∞,...,X∞ ∈ Y(gl ), 1 N N X∞ = −hT{2} + h e e −1 +he(G +...+ G ), i i,i 2 i,i i,i i,1 i,i−1 (cid:0) (cid:1) see [GRTV]. The elements Xq, X∞, i = 1,...,N, are called the dynamical Hamiltonians. i i Observe that i−1 n q q Xq = X∞ − h i G − h j G . i i q −q i,j q −q i,j Xj=1 i j jX=i+1 i j Given λ = (λ ,...,λ ), set G = e e for λ > λ and G = e e for λ < λ . 1 N λ,i,j j,i i,j i j λ,i,j i,j j,i i j We define the elements Xq ,...,Xq ∈ Y(gl ), λ,1 λ,N N i−1 e n q q Xq = X∞ − h i G − h j G . λ,i i q −q λ,i,j q −q λ,i,j Xj=1 i j jX=i+1 i j Let κ ∈ C×. The formal differential operators ∂ (4.1) ∇ = κq − Xq, i = 1,...,N, q,κ,i i∂q i i pairwise commute and, hence, define a flat connection ∇ for any Y(gl )-module, see q,κ N [GRTV]. e Lemma 4.1 (Lemma 3.5 in [GRTV]). The connection ∇ defined by λ,q,κ ∂ (4.2) ∇ = κq − Xq , λ,q,κ,i i∂q λ,i i i = 1,...,N, is flat for any κ. Proof. The connection ∇ is gauge equivalent to connection ∇ , λ,q,κ κ (4.3) ∇ = (Υ )−1 ∇ Υ , Υ = (1−q /q )hελ,i,j/κ, λ,q,κ,i λ q,κ,i λ λ j i 16iY λ , and ε = e for λ < λ . (cid:3) λ,i,j j,j i j λ,i,j i,i i j Connection (4.1) was introduced in [TV4], see also Appendix B in [MTV1], and is called the trigonometric dynamical connection. Later the definition was extended from sl to other N simple Lie algebras in [TL] under the name of the trigonometric Casimir connection. The trigonometric dynamical connection is defined over CN with coordinates q ,...,q , 1 N it has singularities at the union of the diagonals q = q . i j 4.2. Dynamical Hamiltonians Xq on H∗(T∗F ). Recall the Y(gl )-module structure λ,i T λ N ρ defined on H∗(X ) = ⊕ H∗(T∗F ) in Section 3.4. For any µ = (µ ,...,µ ) ∈ ZN , T n |λ|=n T λ e 1 N >0 |µ| = n, the action of the dynamical Hamiltonians Xq preserve each of H∗(T∗F ). µ,i T λ 8 V.TARASOVANDA.VARCHENKO Lemma 4.2 (Lemma 7.6 in [RTV]). For any λ and i = 1,...,n, the restriction of ρ(Xq ) λ,i to H∗(T∗F ) has the form: T λ (4.4) i−1 n q q ρ(Xq ) = (γ +···+γ )− h i ρ(G ) − h j ρ(G ) = λ,i i,1 i,λi q −q λ,i,j q −q λ,i,j Xj=1 i j jX=i+1 i j i−1 n q q i j = (γ +···+γ )− h ρ(e e ) − h ρ(e e ) +C, i,1 i,λi q −q j,i i,j q −q i,j j,i Xj=1 i j jX=i+1 i j where (γ +···+γ ) denotes the operator of multiplication by the cohomology class γ + i,1 i,λi i,1 ···+γ , the operator C is a scalar operator on H∗(T∗F ), and for any i 6= j the element i,λi T λ ρ(G ) annihilates the identity element 1 ∈ H∗(T∗F ). λ,i,j λ T λ 4.3. Quantum multiplication by divisors on H∗(T∗F ). In [MO], the quantum multi- T λ plication by divisors on H∗(T∗F ) is described. The fundamental equivariant cohomology T λ classes of divisors on T∗F are linear combinations of D = γ + ···+ γ , i = 1,...,N. λ i i,1 i,λi The quantum multiplication D ∗ depends on parameters q˜ = (q˜ ,...,q˜ ) ∈ (C×)N. i q˜ 1 N Theorem 4.3 (Theorem 10.2.1 in [MO]). For i = 1,...,N, the quantum multiplication by D is given by the formula: i (4.5) i−1 n q˜/q˜ q˜/q˜ j i i j D ∗ = (γ +···+γ )+ h ρ(e e ) − h ρ(e e ) +C, i q˜ i,1 i,λi 1−q˜/q˜ j,i i,j 1−q˜/q˜ i,j j,i Xj=1 j i jX=i+1 i J where C is a scalar operator on H∗(T∗F ) fixed by the requirement that the purely quantum T λ part of D ∗ annihilates the identity 1 . i q˜ λ Corollary 4.4 (Corollary 7.8 in [RTV]). For i = 1,...,N, the operator D ∗ of quantum i q˜ multiplication by D on H∗(T∗F ) equals the action ρ(Xq ) on H∗(T∗F ) of the dynamical i T λ λ,i T λ Hamiltonian Xq if we put (q ,...,q ) = (q˜−1,...,q˜−1). λ,i 1 N 1 N The quantum connection ∇ on H∗(T∗F ) is defined by the formula quant,λ,q˜,κ T λ ∂ (4.6) ∇ = κq˜ − D ∗ , i = 1,...,N, quant,λ,q˜,κ,i i i q˜ ∂q˜ i where κ ∈ C× is a parameter of the connection, see [BMO]. By Corollary 4.4, we have (4.7) ∇ = ρ(∇ ), i = 1,...,N. quant,λ,q˜,κ,i λ,q˜−1,...,q˜−1,−κ 1 N 4.4. Dynamical Hamiltonians on (CN)⊗n ⊗ C[z;h]. Recall that e , i,j = 1,...,N, i,j denote standard generators of gl . A vector v of a gl -module M has weight λ = (λ ,..., N N 1 λ ) ∈ CN if e v = λ v for i = 1,...,N. We denote by M ⊂ M the weight subspace of N i,i i λ weight λ. We consider CN as the standard vector representation of gl with basis v ,...,v such N 1 N that e v = δ v for all i,j,k. Denote V = (CN)⊗n. For I = (I ,...,I ) ∈ I , we define i,j k j,k i 1 N n HYPERGEOMETRIC SOLUTIONS OF THE QUANTUM DIFFERENTIAL EQUATION 9 v ∈ V by the formula v = v ⊗···⊗v , where i = i if i ∈ I . Let I I i1 in j j i V = V λ M |λ|=n be the weight decomposition. The vectors (v ) form a basis of V . I I∈Iλ λ (a) As always, we denote by e the action of e on the a-th tensor factor of V and denote i,j i,j e = n e(a). i,j a=1 i,j RecPall the Y(gl )-module structure φ on V ⊗ C[z;h] and the dynamical Hamiltonians N Xq introduced in Section 4.1. The dynamical Hamiltonians preserve each of the weight suibspaces V ⊗e C[z;h]. λ Lemma 4.5 (Lemma 4.17 in [GRTV]). For i = 1,...,n, we have n N N h q (4.8) φ(Xq) = z e(a) + (e2 −e )− h e(a)e(b) − h j G . i a i,i 2 i,i i,i i,j j,i q −q i,j Xa=1 Xj=1 16Xai(aj) In these formulas for a function f(t ,...,t ) of some variables we denote 1 k Sym f(t ,...,t ) = f(t ,...,t ). t1,...,tk 1 k σ1 σk X σ∈Sk Example. Let N = 2, n = 2, λ = (1,1), I = ({1},{2}), J = ({2},{1}). Then W (t;z;h) = −h(t(1)−z ), W (t;z;h) = −h(t(1)−z −h). I 1 2 J 1 1 5.2. Weight functions W . For σ ∈ S and I ∈ I , we define σ,I n λ W (t;z;h) = W (t;z ,...,z ;h), σ,I σ−1(I) σ(1) σ(n) where σ−1(I) = (σ−1(I ),...,σ−1(I )). 1 N For a subset A ⊂ {1,...,n}, denote z = (z ) . For I ∈ I , denote z = (z ,...,z ). A a a∈A λ I I1 IN For f(t(1),...,t(N)) ∈ C[t(1),...,t(N)]Sλ(1)×···×Sλ(N), we define f(zI) by replacing t(j) with ∪j z . Denote k=1 Ik N−1 c (z ) = (z −z −h), λ I i j Ya=1 i,j∈Y∪ab=1Ib R(z ) = (z −z ), Q(z ) = (z −z −h). I i j I i j 16aY

Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety

Similar Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety