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

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by  V. Tarasov

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: [email protected], [email protected], supported in part by NSF grant DMS-0901616
⋄E-mail: [email protected], 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 partic-
ular, 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 par-
tial flag variety, see Section 7.4 and similar statements for Bethe algebras in [MTV2, MTV3].
The trigonometric KZ differential equation and trigonometric dynamical differential equa-
tion 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 dif-
ference 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 identi-
fied with the pair consisting ofthe trigonometric dynamical differential equation andrational
qKZ difference equation, see [TV4]. Under the identification of the trigonometric dynam-
ical 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 dif-
ferential 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 quan-
tum 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<j6N
where ε = e for λ > 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 16Xa<b6n Xj=1 i j
j6=i
To obtain the lemma we replace h with −h in Lemma 4.17 of [GRTV].
4.5. qKZ difference connection. Recall the Y(gl )-action φ on (CN)⊗n ⊗C[z;h] intro-
N
duced in Section 3.3. Let
e
u−hP(i,j)
R(i,j)(u) = , i,j = 1,...,n, i 6= j.
u−h
For κ ∈ C×, define operators K ,...,K ∈ End((CN)⊗n)⊗C[z;h],
1 n
K (z;q;h;κ) = R(i+1,i)(z −z ) ... R(n,i)(z −z ) ×
i i+1 i n i
× q−e1(i,)1... q−e(Ni),N R(1,i)(z −z −κ) ... R(i−1,i)(z −z −κ).
1 N 1 i i−1 i
Considerthedifferenceoperators K ,...,K actingon (CN)⊗n-valuedfunctionsof z,q,h,
κ,1 κ,n
b b
(4.9) K F(z;q;h) = K (z;q;h;κ)F(z ,...,z ,z +κ,z ,...,z ;q;h).
z,q,h,κ,i i 1 i−1 i i+1 n
Theorem 4b.6 ([FR]). The operators K ,...,K pairwise commute.
z,q,h,κ,1 z,q,h,κ,n
Theorem 4.7 ([TV4]). The operatorsbK ,...b,K , φ(∇ ),...,φ(∇ )
z,q,h,κ,1 z,q,h,κ,n λ,q,κ,1 λ,q,κ,N
pairwise commute.
b b
The commuting difference operators K ,...,K define the rational qKZ dif-
z,q,h,κ,1 z,q,h,κ,n
ference connection. We say that a (CN)⊗n-valued function F(z;q;h) is a flat section of the
b b
difference connection if
K F(z;q;h) = F(z;q;h), i = 1,...,n.
z,q,h,κ,i
Theorem 4.7 says tbhat the qKZ difference connection commutes with the trigonometric
dynamical connection φ(∇ ) .
λ,q,κ
10 V.TARASOVANDA.VARCHENKO
5. Yangian Y(gl ) weight functions
N
5.1. Weight functions W . For I ∈eI , we define the weight functions W (t;z;h), c.f.
I λ I
[TV1, TV5].
Recall λ = (λ ,...,λ ). Denote λ(i) = λ + ... + λ and λ{1} = N−1λ(i) =
1 N 1 i i=1
N−1(N − i)λ . Recall I = (I ,...,I ). Set j I = {i(j) < ... < i(j)P}. Consider
i=1 i 1 N k=1 k 1 λ(j)
tPhe variables t(j), j = 1,...,N, a = 1,...,λ(j),Swhere t(N) = z , a = 1,...,n. Denote
a a a
t(j) = (t(j)) and t = (t(1),...,t(N−1)).
k k6λ(j)
The weight functions are
(5.1) W (t;z;h) = (−h)λ{1} Sym ... Sym U (t;z;h),
I t(1),...,t(1) t(N−1),...,t(N−1) I
1 λ(1) 1 λ(N−1)
N−1 λ(j) λ(j+1) λ(j+1) λ(j) (j) (j)
t −t −h
U (t;z;h) = (t(j)−t(j+1) −h) (t(j)−t(j+1)) a b .
I (cid:18) a c a d t(j)−t(j) (cid:19)
Yj=1 Ya=1 Yc=1 Yd=1 b=Ya+1 a b
i(cj+1)<i(aj) i(dj+1)>i(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<b6N iY∈Ia jY∈Ib 16aY<b6N iY∈Ia jY∈Ib
5.3. Stable envelope map. Following [RTV], we define the weight function map
[W ] : V ⊗C[z;h] → H∗(X ), v 7→ [W (Θ;z;h)],
id T n I id,I
where W (Θ;z;h) is the polynomial W (t;z;h) in which variables t(j) are replaced with
id,I id,I i
θ and [W (Θ;z;h)] is the cohomology class represented by W (Θ;z;h).
j,i id,I id,I