Tarasov v, Varchenko a - Identities Between Q-Hypergeometric and Hyper Geometric Integrals of...

8/9/2019 Tarasov v, Varchenko a - Identities Between Q-Hypergeometric and Hyper Geometric Integrals of Different Dimensi

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http://www.elsevier.com/locate/aim

Advances in Mathematics 191 (2005) 2945

Identities between q-hypergeometric and

hypergeometric integrals of different dimensions

V. Tarasova,c,1 and A. Varchenkob,,2

a

St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, RussiabDepartment of Mathematics, University of North Carolina at Chapel Hill,

Chapel Hill, NC 27599-3250, USAcDepartment of Mathematical Sciences, Indiana University Purdue University at Indianapolis,

Indianapolis, IN, 46202-3216, USA

Received 26 September 2003; accepted 24 January 2004

communicated by P.Etingof

Abstract

Given complex numbers m1; l1 and nonnegative integers m2; l2; such that m1 m2 l1 l2;for any a; b 0;y; minm2; l2 we define an l2-dimensional Barnes type q-hypergeometricintegral Ia;bz;m; m1; m2; l1; l2 and an l2-dimensional hypergeometric integralJa;bz; m; m1; m2; l1; l2: The integrals depend on complex parameters z and m: We showthat Ia;bz; m; m1; m2; l1; l2 equals Ja;be

m; z; l1; l2; m1; m2 up to an explicit factor, thusestablishing an equality of l2-dimensional q-hypergeometric and m2-dimensional hypergeo-

metric integrals. The identity is based on the glk; gln duality for the qKZ and dynamicaldifference equations.

r 2004 Elsevier Inc. All rights reserved.

Keywords: Hypergeometric integrals; q-hypergeometric integrals; KnizhnikZamolodchikov equations;(glk, gln) duality

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Corresponding author.

E-mail addresses:

[email protected], [email protected] (V. Tarasov), [email protected],[email protected] (A. Varchenko).1Supported in part by RFFI Grant 02-01-00085a and CRDF Grant RM1-2334-MO-02.2Supported in part by NSF Grant DMS-0244579.

0001-8708/$ - see front matterr 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.aim.2004.01.007


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1. Introduction

1.1. q-Hypergeometric integrals

Let k be a positive number. Let m1; l1 be complex numbers and m2; l2 nonnegativeintegers such that

m1 m2 l1 l2:

We say that an integer a is admissible with respect to m2; l2 if

0papminm2; l2:

For a pair of admissible numbers a; b we define a function Ia;bz; m; m1; m2; l1; l2 ofcomplex variables z; m: The function is defined as an l2-dimensional Barnes typeq-hypergeometric integral:

Ia;bz; m; m1; m2; l1; l2

Zdl2 z;m1;m2 Fl2 t; z; m; m1; m2wl2a;at; z; m1; m2Wl2b;bt; z; m1; m2 dtl2 : 1:1

Here t t1;y; tl2 and dtl2 dt1ydtl2 : The functions Fl2 t; z; m; m1; m2;

wl2a;at; z; m1; m2 and Wl2b;bt; z; m1; m2 are defined below. The l2-dimensional

integration contour dl2 z; m1; m2 lies in Cl2 and is also defined below.

The q-master function Fl is defined by the formula

Flt1;y; tl; z; m; m1; m2 exp m Xl

u1

tu=k !Yl

u1

Gtu=kGtu z=k

Gtu

m1

=kGtu

z m2

=k

Y

1puovpl

Gtu tv 1=k

Gtu tv 1=k:

The rational weight function wla;a is defined by the formula

wla;at1;y; tl; z; m1; m2

Y1puovpltu tv

tu tv 1 Yl

u1

1

tu m1

SymYl

ula1

tu

tu z m2

Y1puovpl

tu tv 1

tu tv

" #;

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V. Tarasov, A. Varchenko / Advances in Mathematics 191 (2005) 294530


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where Sym ft1;y; tl P

sASlfts1 ;y; tsl: The trigonometric weight function

Wlb;b is defined by the formula

Wlb;bt1;y; tl; z; m1; m2 Y

1puovpl

sinptu tv=k

sinptu tv 1=k

Ylu1

epitu=k

sinptu m1=k

SymYl

ulb1

epiz=k sinptu=k

sinptu z m2=k

"

Y

1puovpl

sinptu tv 1=k

sinptu tv=k

#:

Integral (1.1) is defined for 0oIm mo2p: With respect to other parameters wedefine integral (1.1) by analytic continuation from the region where m1; m2 arecomplex numbers with negative real parts and Re z 0: In that case we put

dlz; m1; m2 ft1;y; tlACl j Re tu e; u 1;y; lg;

where e is a positive number less than minRe m1; Re m2: In the consideredregion of parameters the integrand in (1.1) is well defined on dl2 z; m1; m2 for anya; b; the integral is convergent and gives a meromorphic function of z; m1; m2; see[TV1]. It is also known that Ia;bz; m; m1; m2; l1; l2 can be analytically continued to a

nonnegative integer value of m2; if a; b are admissible with respect to m2; l2 at thatpoint, and the analytic continuation is given by the integral over a suitable

deformation of the imaginary plane ft1;y; tl2 ACl2 j Re tu 0; u 1;y; l2g;

see [MuV].

Remark. There is an alternative way to describe the integrand of integral (1.1), again

writing it down as a product of three factors. Namely, consider the functions

Xlt1;y; tl; z; m; m1; m2

pkll3=2 exp m piXlu1

tu=k !

Ylu1

Gtu=kGm1 tu=kGtu z=kGz m2 tu=k

Y

1puovpl

tu tv sinptu tv=kGtu tv 1=kGtv tu 1=k;

pla;at1;y; tl; z; m2 SymYlau1

tu z m2Yl

ula1

tuY

1puovpl

tu tv 1

tu tv

" #;

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V. Tarasov, A. Varchenko / Advances in Mathematics 191 (2005) 2945 31


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Plb;bt1ytl; z; m2 exppibz=k Sym

Ylb

u1

sinptu z m2=k

"

Yl

ulb1

sinptu=kY

1puovpl

sinptu tv 1=k

sinptu tv=k

#:

Then

Fl2 t; z; m; m1; m2wl2a;at; z; m1; m2Wl2b;bt; z; m1; m2

Xl2 t; z; m; m1; m2pl2a;at; z; m2Pl2b;bt; z; m2:

In this description of the integrand the function Xl is such that it contains all thepoles of the integrand and has no zeros anywhere but on the shifted diagonals:

tu tvAkZ; the functions pla;a are polynomials in t1;y; tl; z; m2; and the functionsPlb;b are trigonometric polynomials in t1;y; tl; z; m2:

1.2. Hypergeometric integrals

Let k be a positive number. Let m1; l1 be complex numbers and m2; l2 nonnegativeintegers such that

m1 m2 l1 l2:

For a pair of admissible numbers a; b we define a function Ja;bz; m; m1; m2; l1; l2 ofcomplex variables z; m: The function is defined as an l2-dimensional hypergeometricintegral

Ja;bz; m; m1; m2; l1; l2

Zgl2 b;b

z

Cl2 t; z; m; m1; m2gl2a;at; z dtl2 : 1:2

Here t t1;y; tl2 ; dtl2 dt1;y; dtl2 : The functions Cl2 t; z; m; m1; m2 and

gl2a;at; z are defined below. The l2-dimensional integration contour gl2b;bz lies

in Cl2 and is also defined below.The master function Cl is defined by the formula

Clt1;y; tl; z; m; m1; m2

Ylu1

tmm1m22l1=ku 1 tum1=kz tu

m2=kY

1puovpl

tu tv2=k:

The weight function gla;a is defined by the formula

gla;at1;y; tl; z SymYlau1

1

1 tu

Ylula1

1

z tu

" #:

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We define the integral in (1.2) by analytic continuation from the region

za0; 0oarg zo2p; Re m{0: 1:3

In that region the integration contour glb;bz is shown in Fig. 1. It has the form

glb;bz ft1;y; tlACl j tuACu; u 1;y; lg: Here Cu; u 1;y; l; are noninter-

secting oriented loops in C: The first b loops start at infinity in the direction of z; goaround z

;and return to infinity in the same direction. For 1puovpb

;the loop C

ulies inside the loop Cv: The last l b loops start at infinity in the real positivedirection, go around 1; and return to infinity in the same direction. For b 1puovpl; the loop Cu lies inside the loop Cv:

Ifz; m are in region (1.3), we fix a univalued branch of the master function Cl overglb;bz by fixing the arguments of all its factors. Namely, we assume that at the

point ofglb;bz where all numbers t1=z;y; tb=z; tb1;y; tl belong to 0; 1 we have

arg tuA0; 2p; arg 1 tuAp; p; argz tuA0; 2p; argtu tvA0; 2p;

for u

1;y

;l;

v

u

1;y

;l:

Integral (1.2) is convergent in region (1.3).

1.3. Main result

Theorem 1. Let k be a generic positive number. Let m1; l1 be complex numbers andm2; l2 nonnegative integers, such that m1 m2 l1 l2: Let 0oIm mo2p: Then forany a; b 1;y; minm2; l2 we have

Cbm1; m2; l1; l2Ia;bz; m; m1; m2; l1; l2

Dbm1; m2; l1; l2Ebl1; l2Xz; m1; m2

Ym; m1; m2; l1; l2Ja;bem; z; l1; l2; m1; m2; 1:4

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Fig. 1. Integration contour glb;bz:



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where

Cbm1; m2; l1; l2 2pil2

l2!l2 b!b! Yl2b1

j0sinpm1 j=k Yb1

j0sinpm2 j=k

Yl21j0

G1 1=kG1 m1 j=k

G1 j 1=k

Dbm1; m2; l1; l2 2im2

Ym2b1j0

1

sinp j 1=k

Yb1j0

1

sinp j 1=k

Ym21j0

G1 l1 j=kG1=kG1 j 1=k

;

Ebl1; l2 exppib2 b l2l1 l2 l2l2 1=2=k;

Xt; m1; m2 Ym21

j0

G j m1 t=k

G j 1 t=k; 1:5

Ym; m1; m2; l1; l2 eml2l22m11=2k1 eml11l2=k; arg1 emAp; p: 1:6

Remark. There is a similar theorem establishing an equality of suitable hypergeo-

metric integrals of different dimensions, see [TV4]. The factor Dbm1; m2; l1; l2 in thepresent paper corresponds to the normalization factor Cbl1; l2; m1; m2 in [TV4] andcontains the same product of sines and gamma-functions.

Example. Let m2 l2 1: In this case formula (1.4) becomes the classical equalityof two integral representations of the Gauss hypergeometric function 2F1: Forinstance, if a b 0; then taking a m1=k; b z m1=k; g 1 z m1=k; after simple transformations one gets

1

2pi

ZiNeiNe

esmpiGsGs aGs b

Gs gds

1 emgabGb

Gg a ZN

1

tbt 1a1t embg dt

Gb

Gg a

Z10

ua11 uga11 uemb du GaGb

Gg2F1a; b; g; e

m:

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Here it is assumed that Re g4Re a40; Re b40 and 0oeominRe a; Re b:The second equality is obtained by the change of integration variable

u t 1=t em:

Theorem 1 claims that an l2-dimensional q-hypergeometric integral equals an

m2-dimensional hypergeometric integral up to an explicit factor. Note that in the first

integral the numbers m1; m2 are shifts of arguments of the gamma-functions enteringits q-master function, while in the second integral m2 is its dimension and m1 is not

present explicitly.

It is well known that studying asymptotics of integrals with respect to their

dimension is an interesting problem appearing, for instance, in the theory of

orthogonal polynomials and in matrix models. The duality of the theorem allows us

to study asymptotics of integrals with respect to their dimension. Namely, assume

that in the 4-tuple m1; m2; l1; l2 the nonnegative integer l2 tends to infinity while thenumbers m1 and m2 remain fixed. Then Ia;bz; m; m1; m2; l1; l2 is a q-hypergeometricintegral of growing dimension l2; whose master function has fixed shifts m1; m2: Atthe same time, Ja;be

m; z; l1; l2; m1; m2 is a hypergeometric integral of the fixeddimension m2; whose master function has growing exponents l1; l2: The asymptoticsof Ja;be

m; z; l1; l2; m1; m2 can be calculated using the steepest descent method. Anexample of such calculation is given in [TV4].

Similarly, one can assume that in the 4-tuple m1; m2; l1; l2 the nonnegative integerm2 tends to infinity while the numbers l1 and l2 are fixed. Then Ja;be

m; z; l1; l2; m1; m2

is a hypergeometric integral of growing dimension m2; whose master function hasfixed exponents l1; l2: On the other hand, Ia;bz; m; m1; m2; l1; l2 is a q-hypergeometricintegral of the fixed dimension l2; whose master function has growing shifts m1; m2:The asymptotics of Ia;bz; m; m1; m2; l1; l2 in principle can be calculated using theStirling formula for asymptotics of the gamma-function.

To prove Theorem 1 we show that the matrices

Ia;bz; m; m1; m2; l1; l20pa;bpminm2;l2 1:7

and

Xz; m1; m2Ja;bem; z; l1; l2; m1; m20pa;bpminm2;l2

satisfy the same system of first-order linear difference equations with respect to z; seeCorollary 4 and Theorems 5 and 7. Studying asymptotics of those matrices as

Re z-N allows us to compute the connection matrix, thus proving Theorem 1.The fact that both matrices satisfy the same system of difference equations is based

on the duality of the qKZ and dynamical equations for glk and gln [TV3], which is a

generalization of the duality between the rational differential KZ and dynamical

equations observed in [TL]. Namely, in [TV2] a system of dynamical difference

equations was introduced. It is proved there that the dynamical difference equationsare compatible with the trigonometric KZ differential equations. In [MV]

hypergeometric solutions of the trigonometric KZ and dynamical difference

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equations were presented. On the other hand, q-hypergeometric solutions of the

rational qKZ difference equations were constructed in [TV1]. It was shown in [TV3]

that the system of dynamical difference equations for glk and the system of rational

qKZ equations for gln are naturally transformed into each other under the glk; glnduality. In this way one gets two sets of solutions of the same system of equations,

and in principle, these two sets of solutions can be identified. Theorem 1 is a

realization of this idea for the case of k n 2: We will discuss the case of anarbitrary pair k; n in a separate paper.

Let us make an additional remark. In [TV3] it was introduced a system of

dynamical differential equations which is transformed under the glk; gln duality tothe trigonometric KZdifferential equations. It is proved in [TV5] that the dynamical

differential equations are compatible with the rational qKZdifference equations. It is

shown in [TV6] that the q-hypergeometric solutions of the qKZ equations satisfy the

dynamical differential equations. In the present case of k n 2 this means thatmatrices (1.7) and

Ym; m1; m2; l1; l2Ja;bem; z; l1; l2; m1; m20pa;bpminm2;l2 1:8

satisfy the same system of first order linear differential equations with respect to m:Therefore, the connection matrix of (1.7) and (1.8) does not depend on m; whichagrees with formula (1.4). Other factors in (1.4) also have natural explanation in

terms of the q-deformation of the glk; gln duality.

The rest of the paper is as follows. In Section 2, we discuss the gl2; gl2 duality forthe qKZ and dynamical equations. In Section 3, we describe their hypergeometric

solutions and prove Theorem 1. In Section 4, we give some facts about the dynamical

differential equations.

2. The gl2; gl2 duality for KZ and dynamical equations

In this section we follow the exposition in [TV3].

2.1. The trigonometric KZ and associated dynamical difference equations

Let Eij; i;j 1; 2; be the standard generators of the Lie algebra gl2: Let h C E11"C E22 be the Cartan subalgebra, while E12 and E21 are, respectively, thepositive and negative root vectors.

The trigonometric r-matrix rzAgl#22 is defined by the formula

rz 1

z 1z 1E11#E11 E22#E22=2 zE12#E21 E21#E12:

Fix a nonzero complex number k: The trigonometric KnizhnikZamolodchikovKZ operators r1;y; rn are the following differential operators with coefficients in

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Ugl2#n

acting on functions of complex variables z1;y; zn; l1; l2:

raz1;y; zn; l1; l2 kza@

@za X2

i1li

1

2 Xnb1

Ebii !Eaii Xn

b1baa

rza=zbab

: 2:1

Here Ebii 1#?#Eii

bth

#?#1; and the meaning of rza=zbab

is similar. It is

known that the operators r1;y; rn pairwise commute.Let V1;y; Vn be gl2-modules. The trigonometric KZ equations for a function

Uz1;y; zn; l1; l2 with values in V1#?#Vn are

raz1;y; zn; l1; l2Uz1;y; zn; l1; l2 0; a 1;y; n: 2:2

Let Tu be the difference operator acting on functions fu by the formula

Tu fu fu k:

Introduce a series Bt depending on a complex variable t

Bt 1 XNs1

Es21Es12

Ysj1

1

jt E11 E22 j:

For any gl2-module V with a locally nilpotent action of E12 and finite-dimensionalweight subspaces the series Bt has a well-defined action in any weight subspaceVmCV as a rational EndVm-valued function of t:

Let V1;y; Vn be gl2-modules as above. Introduce the dynamical differenceoperators Q1; Q2 acting on V1#?#Vn-valued functions of complex variablesz1;y; zn; l1; l2 by the formulae

Q1z1;y; zn; l1; l2 Bl1 l21Yna1

zE

a11

a Tl1 ;

Q2z1;y; zn; l1; l2 Yna1

zEa22a Bl1 l2 kTl2 : 2:3

One can see that the operators Q1; Q2 commute.

Proposition 2 (Tarasov and Varchenko [TV2]). One has ra; Qi 0 for alla 1;y; n and i 1; 2:

The dynamical difference equations associated with the trigonometric KZ

equations for a function Uz1;y; zn; l1; l2 with values in V1#?#Vn are

Qiz1;y; zn; l1; l2Uz1;y; zn; l1; l2 Uz1;y; zn; l1; l2; i 1; 2: 2:4

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The trigonometric KZ and dynamical difference operators preserve the weight

decomposition of V1#?#Vn: Thus the KZ and dynamical equations can beconsidered as equations for a function Uz1;y; zn; l1; l2 taking values in a given

weight subspace of V1#?#Vn:

2.2. The qKZ equations

Let V; W be irreducible highest weight gl2-modules with highest weight vectorsv; w; respectively. There is a unique rational function RVWt taking values inEndV#W such that

RVWt; g#1 1#g 0 for any gAgl2;

RVWtE21#E11 E22#E21 tE21#1

E11#E21 E21#E22 tE21#1RVWt;

RVWtv#w v#w:

The function RVWt is called the rational R-matrix for the tensor product V#W: Itcomes from the representation theory of the Yangian Ygl2:

Let V1;y; Vn be irreducible highest weight gl2-modules. Introduce the qKZ

difference operators Z1;y; Zn acting on V1#?#Vn-valued functions of complexvariables z1;y; zn; l1; l2 by the formula

Zaz1yzn; l1; l2 Ranza znyRa;a1za za11l

Ea11

1 lE

a22

2

R1az1 za kyRa1;aza1 za kTza : 2:5

These operators are called the qKZ operators. It is known that they pairwise

commute [FR]. The difference equations

Zaz1;y; zn; l1; l2Uz1;y; zn; l1; l2 Uz1;y; zn; l1; l2; a 1;y; n; 2:6

for a V1#?#Vn-valued function Uz1;y; zn; l1; l2 are called the qKZ equations.The qKZ operators preserve the weight decomposition of V1#?#Vn: Thus the

qKZ and dynamical differential equations can be considered as equations for a

function Uz1;y; zn; l1; l2 taking values in a given weight subspace ofV1#?#Vn:

2.3. The duality

For a complex number m; denote Mm the Verma module over gl2 with highestweight m; 0 and highest weight vector vm: The vectors E

d2;1vm; dAZX0; form a basis

in Mm:

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For a nonnegative integer m; denote Lm the irreducible gl2-module with highest weight

m; 0 and highest weight vector vm: The vectors Ed2;1vm; d 0;y; m; form a basis in Lm:

Let m1; l1 be complex numbers and m2; l2 nonnegative integers such thatm1 m2 l1 l2: Consider the weight subspace Mm1#Lm2 l1; l2 of the tensorproduct Mm1#Lm2 : The weight subspace has a basis

Fam1; m2; l1; l2 1

l2 a!a!El2a2;1 vm1#E

a2;1vm2 ; a 0;y; minm2; l2: 2:7

There is a linear isomorphism

j : Mm1#Lm2 l1; l2-Ml1#Ll2 m1; m2;

Fam1; m2; l1; l2/Fal1; l2; m1; m2: 2:8

Theorem 3 (Tarasov and Varchenko [TV3]). The isomorphism j transforms the qKZ

operators acting in Mm1#Lm2 l1; l2 into the dynamical difference operators acting inMl1#Ll2 m1; m2: More precisely, we have

jZ1z1; z2; l1; l2 Gz1 z2; m1; m21

Q1l1; l2; z1; z2j;

jZ2z1; z2; l1; l2 Gz1 z2 k; m1; m2Q2l1; l2; z1; z2j;

where

Gt; m1; m2 Ym21j0

t j m1t j 1

:

Let St be any solution of the equation

St k Gt; m1; m2St:

For instance, one of solutions is given by St Xt; m1; m2; cf. (1.5).

Corollary 4. Let an Ml1#Ll2 m1; m2-valued function Uz1; z2; l1; l2 solve thedynamical difference equations:

Qaz1; z2; l1; l2Uz1; z2; l1; l2 Uz1; z2; l1; l2; a 1; 2:

Then the Mm1#Lm2 l1; l2-valued function

eUUz1; z2; l1; l2 Sz1 z2j

1Ul1; l2; z1; z2

solves the qKZ equations

Zaz1; z2; l1; l2 eUUz1; z2; l1; l2 eUUz1; z2; l1; l2; a 1; 2:

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More facts on the gl2; gl2 duality for KZ and dynamical equations are given inSection 4.

3. Hypergeometric solutions

3.1. Hypergeometric solutions of the qKZ equations

For any b 0;y; minm2; l2 define an Mm1#Lm2 l1; l2-valued function

%Ibz; m; m1; m2; l1; l2 Xl2a0

Ia;bz; m; m1; m2; l1; l21

l2 a!a!El2a2;1 vm1#E

a2;1vm2 :

Here the actual range of summation is until a minm2; l2; since Ea2;1vm2 0

for a4m2:

Theorem 5. Let li emi; i 1; 2: For any b 0;y; minm2; l2 the function

Ubz1; z2; l1; l2 em1m1z1m2z2m

21

m22

=2m1m2l2z1l2=2=k

1 em2m1 l2=k %Ibz2 z1; m2 m1; m1; m2; l1; l2 3:1

is a solution of the qKZ equations (2.6) with values in Mm1#L

m2l

1; l

2: Moreover, if

l1=l2 is not real, than any solution of that qKZ equations is a linear combination offunctions Ubz1; z2; l1; l2 with coefficients being k-periodic functions of z1; z2:

The theorem is a direct corollary of the construction ofq-hypergeometric solutions

of the qKZ equations given in [TV1,MuV].

We describe asymptotics of the integral Ia;bz; m; m1; m2; l1; l2 as Re z-N andm is fixed. For a positive number k; complex numbers m; m; Im mA0; 2p; and anonnegative integer l consider the Selberg-type integral

Alm; m Zdlm

exp m pi Xlu1

su !Ylu1

GsuGsu m=k

Yl

u;v1uav

Gsu sv 1=k

Gsu svdsl: 3:2

The integral is defined by analytic continuation from the region where Re m is

negative. In that case

dlm fs1;y; slACl j Re su Re m=2; u 1;y; lg:

In the considered region of parameters the integrand in (3.2) is well defined on

dlm and the integral is convergent, see [TV1]. The formula for Alm is

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well known,

Alm; m 2pilempil12ml=2k 1 emlml1=k

Yl1j0

G1 j 1=k

G1 1=kG j m=k;

where arg1 emAp; p; see, for example, [TV1].

Remark. Other versions of Selberg-type integrals see in [FSV,TV7].

Lemma 6. Let Re z-N andm is fixed. Then

Ia;bz; m; m1; m2; l1; l2 l2!l2 b!b!pl2 expmzb=k

z=k2b2bm1m22l2m2l2=k

Al2bm; m1Abm; m2dab Oz1:

The lemma follows from [TV1].

3.2. Hypergeometric solutions of the trigonometric KZ and difference dynamical

equations

For b 0;y; minm2; l2; define an Ml1#Ll2 m1; m2-valued function

%Jbz; m; l1; l2; m1; m2 Xm2a0

Ja;bz; m; l1; l2; m1; m21

m2 a!a!Em2a2;1 vl1#E

a2;1vl2 :

Here the actual range of summation is until a minm2; l2; since Ea2;1vl2 0 for

a4l2:

Theorem 7. For any b 0;y; minm2; l2 the function

Ubz1; z2; l1; l2 zl1l1m2l2m2m

22

m1l1l21=2=k

1 zl2l1m1l2=2=k2

z1 z2l1l2=k %Jbz2=z1; l2 l1; l1; l2; m1; m2

is a solution of the KZ equations (2.2) and difference dynamical equations (2.4) with

values in Ml1#Ll2 m1; m2:

The theorem is a direct corollary of [MV].We describe asymptotics of the integral Ja;be

m; z; l1; l2; m1; m2 as Re z-N andm is fixed. For a positive number k; a complex number l and a nonnegative integer m

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consider the Selberg-type integral

Bml

Zgm

ePm

u1su=k

Ymu1

su1l=k

Y1puovpm

su sv2=k

dsm:

The integration contour gm has the form

gm fs1;y; smACm j suACu; u 1;y; mg;

see Fig. 2. Here Cu; u 1;y; m; are nonintersecting oriented loops in C: The loopsstart at N; go around 0, and return to N: For uov; the loop Cu lies inside theloop Cv: We fix a univalued branch of the integrand by assuming that at the point ofgm where all numbers s1;y; sm are negative we have argsu 0 for u 1;y; m;and argsu sv 0 for 1puovpm:

The formula for Bml is well known

Bml 2pimkmm1l=k

Ym1j0

G1 1=k

G1 l j=kG1 j 1=k; 3:3

for example, cf. [TV2,MTV].

Lemma 8. Let Re z-N andm is fixed. Then

Ja;bem; z; l1; l2; m1; m2 m2 b!b!e

pim2b2bl2=k

embzl12m2b=k1 em2b2bl1l22m2m2l2=k

z2b2bl1l22m2m2m2l11=k

Bm2bl1Bbl2dab Oz1:

The proof is straightforward.

3.3. Proof of Theorem 1

Theorems 5 and 7, and Corollary 4 imply that for any b 0;y; minm2; l2 the

functions %Ibz; m; m1; m2; l1; l2 and Xz; m1; m2j1 %Jbe

m; z; l1; l2; m1; m2 satisfy the

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Fig. 2. The contour gm:



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same first-order difference equation with respect to z with step k: Hence, for anya 0;y; minm2; l2 one has

Xz; m1; m2Ja;bem; z; l1; l2; m1; m2

Xminm2;l2c0

Ia;cz; m; m1; m2; l1; l2Gb;cz; m; m1; m2; l1; l2; 3:4

the connection coefficients Gb;cz; m; m1; m2; l1; l2 being k-periodic functions ofz andholomorphic functions of m in the strip 0oIm mo2p: Taking into accountasymptotics of the integrals Ia;cz; m; m1; m2; l1; l2 and Ja;be

m; z; l1; l2; m1; m2 asRe z-N and m is fixed, see Lemmas 6 and 8, one can compute the connection

coefficients and obtain formula (1.4). Theorem 1 is proved. &

Remark. One can see from formula (1.4) that all connection coefficients

Gb;cz; m; m1; m2; l1; l2 in (3.4) as functions of m are proportional to the samefunction Ym; m1; m2; l1; l2: This fact, which is pure computational in the given proofof Theorem 1, can be observed independently in advance, because Theorems 7

and 12, and Corollary 11 imply that the functions %Ibz; m; m1; m2; l1; l2 and

Ym; m1; m2; l1; l2j1 %Jbe

m; z; l1; l2; m1; m2 satisfy the same first-order differentialequation with respect to m:

4. Differential dynamical operators

Introduce the dynamical differential operators D1; D2 with coefficients in Ugl2#n

acting on functions of complex variables z1;y; zn; l1; l2 by the formula

Diz1;y; zn; l1; l2 kli@

@li

eEE2ii2

Xn

a1

zaEaii

X2j1

X1paobpn

Eaij E

bji

li0

li li0 eEE21 eEE12 eEE22:

Here eEEkl Pna1 Eakl ; and i0 is supplementary to i; that is, fi; i0g f1; 2g:Proposition 9 (Tarasov and Varchenko [TV5]). One has Za; Di 0 for alla 1;y; n and i 1; 2; where Z1;y; Zn are the qKZ operators (2.5).

The differential equations

Diz1;y; zn; l1; l2Uz1;y; zn; l1; l2 0; i 1; 2; 4:1

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for a V1#?#Vn-valued function Uz1;y; zn; l1; l2 are called the dynamicaldifferential equations associated with the qKZ equations.

Theorem 10 (Tarasov and Varchenko [TV3]). The isomorphism j; see (2.8),transforms the dynamical differential operators acting in Mm1#Lm2 l1; l2 into thetrigonometric KZ operators (2.1) acting in Ml1#Ll2 m1; m2:

jDaz1; z2; l1; l2 ral1; l2; z1; z2j; a 1; 2:

Corollary 11. Let an Ml1#Ll2 m1; m2-valued function Uz1; z2; l1; l2 solve thetrigonometric KZ equations:

raz1; z2; l1; l2Uz1; z2; l1; l2 0; a 1; 2:

Then the Mm1#Lm2 l1; l2-valued function

eUUz1; z2; l1; l2 j1Ul1; l2; z1; z2solves the system of the dynamical differential equations (4.1):

Daz1; z2; l1; l2

eUUz1; z2; l1; l2 0; a 1; 2:

The next statement describes q-hypergeometric solutions of the dynamicaldifferential equations.

Theorem 12. For any b 0;y; minm2; l2 the function Ubz1; z2; l1; l2 defined by(3.1) is a solution of Eq. (4.1) with values in Mm1#Lm2 l1; l2:

The theorem follows from [TV6].

Acknowledgments

The authors thank Y. Markov for useful discussions.

References

[FR] I.B. Frenkel, N.Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations,

Comm. Math. Phys. 146 (1992) 160.

[FSV] G. Felder, L. Stevens, A. Varchenko, Elliptic Selberg integrals and conformal blocks, preprint,

2002, 113; math.QA/0210040.

[MTV] Y. Markov, V. Tarasov, A. Varchenko, The determinant of a hypergeometric period matrix,

Houston J. Math. 24 (2) (1998) 197220.

[MV] Y. Markov, A. Varchenko, Hypergeometric solutions of trigonometric KZ equations satisfy

dynamical difference equations, Adv. Math. 166 (1) (2002) 100147.

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[MuV] E. Mukhin, A. Varchenko, The quantized KnizhnikZamolodchikov equation in tensor

products of irreducible sl2-modules, CalogeroMoserSutherland models, Montreal, QC,1997, CRM Series Math. Phys., Springer, New York, 2000.

[TL] V. Toledano-Laredo, A KohnoDrinfeld theorem for quantum Weyl groups, Duke Math.J. 112 (3) (2002) 421451.

[TV1] V. Tarasov, A. Varchenko, Geometry of q-hypergeometric functions as a bridge between

Yangians and quantum affine algebras, Invent. Math. 128 (1997) 501588.

[TV2] V. Tarasov, A. Varchenko, Difference equations compatible with trigonometric KZdifferential

equations, Internat. Math. Res. Notices (15) (2000) 801829.

[TV3] V. Tarasov, A. Varchenko, Duality for KnizhnikZamolodchikov and dynamical equations,

Acta Appl. Math. 73 (12) (2002) 141154.

[TV4] V. Tarasov, A. Varchenko, Identities for hypergeometric integrals of different dimensions,

preprint, 2003, pp. 19; math.QA/0305224.

[TV5] V. Tarasov, A. Varchenko, Dynamical differential equations compatible with rational qKZ

equations, preprint, 2004, pp. 17; math. QA/0403416.

[TV6] V. Tarasov, A. Varchenko, Hypergeometric solutions of rational qKZ equations satisfydynamical differential equations, in preparation.

[TV7] V. Tarasov, A. Varchenko, Selberg-type integrals associated with sl3; Lett. Math. Phys. 65 (3)(2003) 173185.

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