integrable connections related to zonal spherical functions

Invent. math. 110, 95-121 (1992)

mathematicae �9 Springer-Verlag 1992

Integrable connect ions related to zonal spherical funct ions*

Atsushi Matsuo Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan**

Oblatum 29-IV-1991 & 6-III-1992

Summary. We define a system of differential equations of first order for a function valued in the group algebra of the Weyl group associated with an arbitrary root system. This is equivalent to the system of differential equations given by Heckman and Opdam which is a deformation of the system satisfied by the zonal spherical function of the Riemannian symmetric space G / K of non-compact type. When the root system is A,-type, our equation is related to the Knizhnik-Zamolodchikov equation in conformal field theory.

Introduction

In their series of papers [HO], [H1] and [-O1, 2], Heckman and Opdam studied a deformation of the system of differential equations for zonal spherical functions. Let b be a n-dimensional complex linear space and 27 c b* a root system of rank n, not necessarily reduced. Let W be the Weyl group. We associate a complex number k~ to each ~ ~ 27, such that kw~ = k~ for any :~ ~ 27 and w ~ W. Let 2 e b* be another parameter and u a variable on b. Let c~ denote the directional differentiation with respect to each ~ e b. Then the Laplacian is defined by

i = 1 a c E ' ~ Ca(u) - - l

where {~i} is an orthonormal basis of b. The system considered by Heckman and Opdam is the following one for a scalar valued function q5 = qS(u):

(El) = e e o . Here D is a commutative algebra of differential operators containing the Laplacian and 7(P)(2) is the eigenvalue related to the parameter At b*. When e are twice the

* This paper is a revision of the preprint RIMS 750 with the title "Knizhnik-Zamolodchikov type equations and zonal spherical functions" ** Current address: Department of Mathematics, Nagoya University, Nagoya 464, Japan

96 A. Matsuo

restricted roots of the Riemannian symmetric space G/K and k~ are half the corresponding multiplicities, then D is the set of the radial parts of the invariant differential operators on G/K. This system is invariant with respect to the action of the Weyl group.

In this paper, we shall explicitly construct a system of differential equations of first order, which is equivalent to (El). Let us sketch our construction.

Let C [ W] be the group algebra of W. Let ~ be a root and s~ the corresponding reflection. We define the endomorphisms a~ and e, of C[W] by setting

a~(w) = s~w and

J" w if w - l ~ e Z +, ~(W) l - w otherwise,

for each we W. We also define er for Ceb by

er (w) = (w2, 4) w.

Consider the following system of differential equations for a C [ W]-valued function q'(u):

(E2) Or = ,~+~-(~, r - 1) + a,e, + er ~(u); ~ e b .

This equation is Weyl group invariant, integrable and with logarithmic poles. We note that some analogous equations were already introduced and studied by Cherednik [C1-5].

The main result in this paper is the following (Theorem 5.4.1):

Theorem. For any solution cb(u) = ~w~w qSw(U)W to the Eq. (E2), ~b(u) --- ~w~w ~b~(u) is a solution to the system (El). This correspondence is bijective if k~ + 2k2, + (2, ~v) 4: O for any ~ e S ~ .

Now let us give another description of the Eq. (E2) when S is the A, type root system. Let 9 be the Lie algebra ~I(n + 1, C), t the subalgebra consisting of all diagonal matrices in 9. For each i = 1 , ' - . , n + 1, let ~ : g ~ End(V3 denote a copy of the vector representation and "1r the weight subspace of 1/1 | . . . | II.+~ with weight 0. Set b* = {(P~, " " , / ~ , + x ) e C "+~;/q + . - . +/~,+~ = 0}. Take a variable z = (z~, �9 - �9 z, + z) e C" + t and regard it as a representative of u e b, where b is the dual of I)*. Then, by a certain choice of k e C and her, the Eq. (E2) is equivalent to the following equation for a ~q/~o-valued function ~g(z):

(E3) ~ ( z ) = ~ k 2 ( c o t h Z l - Z i [2 j{ . i ) \ ~ ( n i | -}- (Tzi~Tzj)(r) + 7zi(h) t/t(z);

i = l , . . . , n + l .

Here t, r are the elements of g| respectively defined by

t ---- 2 (Eij | EJ i "~ EJ i | Eij ) -}- ~ Hi | Hi , l < i < j < n + l i = 1

r = 2 (EJ i | Eij -- Eij | Eft), l<i<j<n+l

Integrable connections related to zonal spherical functions 97

where Eij denotes the matrix with only non-zero entry 1 at the (i,j)-th component and { Hi} is an orthonormal basis of t. Here we note that the Eq. (E3) is a special case of those introduced by Cherednik [C3]. Interpreting his theory, it is possible to give an integral representation of a solution to this equation hence to (El) for

= A,. Now, take the rational limit of the Eq. (E3). Then we obtain the Knizhnik-

Zamolodchikov equation:

c~ ~(z) k ~ (ni| 7J(z); i 1, n + 1. (E4) c~zi j ( , i) zl -- zj

It makes sense for a function 7'(z) valued in V~| . . . | V,+~, where V~ are representations of an arbitrary simple Lie algebra g. This equation is originated in the Wess-Zumino-Witten model in conformal field theory [KZ, TK]. We may understand the present work as a bridge between the theories of different origines: conformal field theory and harmonic analysis on symmetric spaces.

1 Preliminaries

In this section, we summarize notations and properties on a root system and the Weyl group. We refer to Bourbaki [B, Chap. 6] and Helgason [Hel, p. 455].

1.1 Root system and Weyl group

Let a be the n-dimensional Euclidean space with the inner product (,). Let a* be the dual space of a. For any c~ e a* such that ct 4= 0, we define

2 (1.1.1) cC = (e,~) e and

(1.1.2) s~# =/1 -- (#,~v)cq # ~ E .

Then s~eGL(a*) is the orthogonal reflection with respect to the hyperplane perpendicular to ct. The inner product and the reflections are uniquely extended C-linearly to t )= a| The letter u is used to denote a variable on D. We sometimes identify D* with [9 via the inner product (,):

(1.1.3) #(u) = (/t, u).

Let X c t)* be a root system of rank n. Namely, Z is a finite subset of a satisfying

0 r X and E is spanned by 27,

(~ v, fl) e Z for ~, fle Z, and

(1.1.4) s~fl~ X for ~, f l e X .

We do not assume that 27 is reduced. Let So = {cruX; :tr be the set of indivisible roots. The lattice generated by Z is called the root lattice.

The Weyl group W is the subgroup of GL(D*) generated by {s~; ~ ~ 27}. The identity element is denoted by 1 e W. Let C [ W ] = O ~ w Cw be the group algebra of Weyl group. We sometimes identify 1 e C [ IV] with the scalar 1 6 C.

98 A. Matsuo

1.2 Positive roots and length in Weyl group

We choose a set of positive roots 2; + c 2;. Let 2;- = - 2; + be the set of negative roots, { e ~ , . . . , ~,} the simple roots and { A ~ , . . . , A,} the fundamental weights. We put 2;o + = 2 ;+n 2; o and 2;o = X - n 2;o. Then any root is proportional to a root in 27o +, with the factor +_ 1 or + 2. The simple roots are contained in So +. We define a partial ordering on b* by

(1.2.1) 2 > / ~ i f 2 - / ~ E ~ , cj~j with cj~Z>=o. j = l

We write 2 >/~ if 2 > # and 2 . #. The set of simple reflections S = {s,,; i = 1, �9 �9 -, n} generates the Weyl group.

For each w e W, the minimal number l such that w = s~ �9 �9 �9 st for st e S is called the length of w, denoted by l(w).

Lemma 1.2.1 (i) Let ct be a simple root. Then s ~ = -- ~ S o and s~(2;~ - {~}) = -

(ii) Let ~ be a positive root. Then w - X e ~ s w - 1 ~ 2 ; +) if and only if l(s, w) < l(w)(resp, l(s,w) > l(w)).

Proo f (i) See Bourbaki [B, p. 157] or Helgason [Hel, p. 461]. (ii) See Bernstein et al. [BGG, Corollary 2.3]. Q.E.D

1.3 Exponential polynomials

Let P = {#eb*; (#, ev) EZ for any ee2;} be the weight lattice. Then P is a free Abelian group generated by the fundamental weights. A finite sum ~,~ea,e" , a, ~ C, is called an exponential polynomial. All exponential polynomials form the group algebra of the Abelian group P, denoted by C [P]. The Weyl group W acts on C [P] by w(e ~) = e wu.

Let C[P] w be the set of Weyl group invariant elements in C[P] . For each fundamental weight Ai, we set

(1.3.1) xi = 2 eWA'" weW

Then we have x~eC[P] w and

(1.3.2) C[P] w ~ " C [ - X I , " " �9 , Xn].

The Weyl denominator is defined by

(1.3.3) A = H ( e ~ - e - { - ) = f i eA' H ( 1 - e - ' ) .

2e$,~ 2eCZ

This is an exponential polynomial and anti-invariant with respect to W. Each anti-invariant exponential polynomial is divisible by A.

We set

(1.3.4) XR = {u~D; e ~u) 4= 1 for any c~Z~} .


1.4 Differential operators on b

For each ~ e b, the differentiation 0e is defined by

(1.4.1) (Ocf)(u) = -~ f (u + t~) . t = O

The set of all constant coefficient differential operators on b is denoted by ~ ). Let S = S(b) be the symmetric algebra on b, which is identified with the set C [b*] of all polynomial functions on b*- The correspondence ~--,0~ induces an algebra isomorphism

(1.4.2) 0:S(b) ~ , ~//(b).

The action of the Weyl group W is defined by extending the map ~?r ~ 0w~ to the algebra automorphism of ~//(b) for each we W.

1.5 Parameters

To each ~ e Z, we associate a complex number k, such that kw, = k, for any c~ e 22 and w e W. In particular, we have k_, = k,. The sequence (k,),~ is called the multiplicity function. If ~ r 22, then we understand k, = 0. We also use the notation

(1.5.1) e = �89 E

throughout this paper. We consider another parameter 2eb* , which is called the weight parameter.

We say 2 is regular if (~ v, 2) :t: 0 for any e e 22, or equivalently if w2:4= 2 for any w e m .

2 Zonal spherical system

We summarize some known facts about the system of differential equations satisfied by the zonal spherical functions after Heckman and Opdam. We refer to Heckman and Opdam I-HO] and Heckman [H3] for a survey.

2.1 Harish-Chandra homomorphism

Let ~ be the algebra of functions on XR generated by

1 1 The Weyl group acts on ~ by 1 - e "~-~ 1 - e w" for each we W. Let ~ | denote

the set of all differential operators with coefficients in ~. Then .r174 is an algebra with usual composition of differential operators.

100 A. Matsuo

Each element P E ~ | is expressed as

(2.1.2) P = ~, eud(P{U)), / ~ > 0

with some P~")E S(b) by expanding

1 (2.1.3) - 1 + e " + e z ' + . . .

1 - e ~

for positive roots g E Z +. The Harish-Chandra homomorphism

(2.1.4) ? : ~| --* C[b*]

is an algebra homomorphism defined by

(2.1.5) ?(e)(2) = e(O)(2 + O),

where O is given by (1.5.1). We have regarded P(~ S(b) as a polynomial function on D*.

2.2 Zonal spherical system

The Laplace-Beltrami operator L E ~ | defined by

k e ~ + l ~ 1 ~ k ~ e ~ + 1 (2.2.1) L = ~ , + y, ~eW-Z~_lC3~= c3~+~ e ~ _ ~ 0 ~ ,

i = 1 a e Z + i = 1 aeZ

where {~i} is an orthonormal basis of [~. Note that we have used the identification (1.1.3) to define a~ for :r The operator L obviously does not depend on the choice of orthonormal basis {~i} of D and of positive roots Z +. Hence L is invariant under the action of the Weyl group.

We set

(2.2.2) D = {PEN| [L, P ] = 0, w(P) = P for any we W}.

The zonal spherical system is the following system of differential equations defined on XR for a scalar valued function ~b(u):

(El) Pc~(u) = y(P)(2)~b(u); PED.

Here 2E I)* is the parameter of this system concerning the eigenvalue 7(P)(2).

Theorem 2.2.1 (Heckman and Opdam [HO; H1; O1, 2]) D is isomorphic to C[b*] w by restrictin9 the Harish-Chandra homomorphism y, and D is a commutative sub-

algebra of the Weft algebra C x l, " " �9 x,, OXl' " ' ~x, "

The following proposition is a consequence of this theorem.

P r o p o s i t i o n 2 . 2 . 2 The dimension of the space of local solutions to (El) is equal to the order of the Weft group.

Remark.When ~ are twice the restricted roots of the Riemannian symmetric space G/K of non-compact type, and k~ are half the corresponding multiplicities, then


D is the set of the radial parts of the invariant differential operators on G/K and the zonal spherical function satisfies the system (El) (cf. Helgason [He2, p. 259 and p. 339].

Note. Heckmann-Opdam [HO] called (E i) the system of hypergeometric differential equations.

2.3 Differential-difference operators

For any P e End (C [P] ), its restriction to C[P] w is denoted by Res(P):

(2.3.1) Res(P) : C[P] w --* C [ P ] .

Then Res(Pw) = Res(P) holds for any P e E n d ( C [ P ] ) and any wc W. Now, for each ~ c [, we put

1 k~ e + (2.3.2) D~ = (?~ - ~ ~- (c~, 3) ~ (s, - 1).

Then D~ acts on C[P] . This operator is called the global analogue of Dunkl's differential-difference operator, (cf. Dunkl [D] and Heckman [H2, 3]). We set

(2.3.3) De, d = ~ D~cEnd(C[P] ) . ~eW~

We have wDr w-1 = De,d, for any we W. Therefore Res(D~,d) acts on C[p jw:

(2.3.4) Res(Dr : C [P ] w ~ C[P] w.

Let us observe

Res(D~) -- Res(Dr D~- 1)

k~ e ~ + 1 = r Res(D~- ') - ~ ~-(cq ~) e - ; ~ - 1 Res((s~ - 1)D~ -1)

k~ e ~ + 1 Res(D~or t _ D~-'). (2.3.5) = c~r -1) - ~ ~-(ct, ~)e~ _ 1 ~ +

We define the differential opera to r /9 t a) for ~ ~ b and d c Z ~ o inductively by

1 k, e + 5~,_ 1)), (2.3.6) 5~ a) = ~r " - ' ) - ~ ~-(a, ~)e~_~_ 1 (5,a:~ ~ -

and/5~~ = 1. We set

(2.3.7) /5~, ,= Z /~d). neW~

Then, by cons t ruc t ion , / )~a)e~ | and we have

5~,a = Res(n~,.) on C [ P ] w.

102 A. Matsuo

Theorem 2.3.1 (Heckman EH3]) Dr are differential operators in the Weyl algebra

�9 . o ~ C xx, �9 ' ' , x . , ~x~ ' communte with each other and generate D.

3 Construction of integrable connection

In this section, we define a C[W]-valued differential equation of first order associated with a root system. This equation is Weyl group invariant and integrable.

3.1 Operators on C [ W ]

Let C [ W ] be the group algebra of the Weyl group and v : W--, G L ( C [ W]) the left regular representation�9

First we set a , = v(s~) for any eE2;, namely we define

(3.1.1) try(w) = s~w.

Then, for any w E W and e E S, we have

(3�9149 v(w)~r~v(w) ~ 1 = aw~ and a_~ = cry.

Next we define e, E E n d ( C [ W ] ) for c~2; by setting

(3.1.3) e,(w) = { if W - 1~ E z~ + ' W

- w if w - l ~ E S -.

Then we have

(3.1.4) v(w)e,v(w) -1 = ew, and e_~ + e, = 0.

Note that e2, -- e~ if both c~ and 2c~ belong to L'. Finally, for any 2eb* and Ceb, we define ee(2)EEnd(C[W]) by

(3�9149 er = (w2, ~)w.

Then we have

(3.1.6) v(w)e~(Z)v(w) -1 = ewe(2) for any we W.

3.2 C [ W]-valued differential equation

Let us consider the following system of differential equations defined on XR for a C I-W]-valued function ~(u):

(E2) 8~q~(u)= 2 + 1 ) + ~ r ~ +e~(2) ~b(u); tED.


We set

(3.2.1)

Proposition 3.2.1

k~ ( e ~ ) + 1 ) ar = ~e+ y~ ~-(:4 ~) \e~.,~ _ 1 (a~ - 1) + ~r~ + ed;0

k. ( c~(u) , ) = ~ ~ (~ ,4 ) c o t h y t % - l ) + a ~ e ~ +er

~EX ~

The system (E2) is Weyl group invariant, i.e.

v(w) Ar 1 u) v(w)- 1 = A~(u), .for any w ~ W.

v(w)Ar ~ u) v(w)- { k~ ( (~, w-~u)

= v ( w ) ,~+ ~-(c~, ~) coth 2

- yProof

( a , - 1 ) + a~a~)+ er v(w) -~

k, ( (wc4 u) ) = ~ ~-(~,4) c o t h ~ ( a w , - 1 ) + a w , Sw, +ewe(2)

.-,.~x~ 2 c o t h ~ ( %

= ~ ~-( , w~) coth (a~ - 1) + %s~ + e~r = A,~r Q.E.D ~ E ~ t

Corollary 3.2.2 I f eb(u) is a solution to the Eq. (E2), then v(w) ~(w- ~ u), w e W, is also a solution to (E2).

3.3 h~tegrability

Proposition 3.3.1 The Eq. (E2) is integrabIe, i.e.

[~r - A~, 0, - A,] = 0, for any 4, ~l E b.

Proof Since 0~A, = 0nAr obviously holds, it suffices to prove

(3.3.1) [A o A,] = 0.

The terms in [Ar A,] concerning ee(2) and e,(2) are calculated as

k~ ~(u) -~ coth Y {(c~, ~) [cry, %(2)] -- (c~, t/)[a~, er + [er e,(2)]

k~ ~(u) �9 ~z ~ coth ~ - {(cq 4)(c~ v r/)a~e~(2) - (~, t/)(e v, ~)a~e~(2)} = 0.

The other terms in [A o A,] are written as $1 + $2 + $3 + $4 where

1 0 4 A . M a t s u o

~(u) fl(u) S l = ~ B ( ~ , f l ) c o t h - T c O t h - T a ~ a ~,

a , B e Z +

$2 = ~ B(e, f l )coth~-~ a~a~e B, e, fleY. +

. . . . fl(u) $ 3 = ~ B(e,p)com--~-a~e~o'B, at, r i t z +

S~= ~ B ( ~ , f l ) ~ B. ~,BeZ +

k~ ka Here B(~, fl) is defined by B(~, fl) = T {(~' ~)(fl' t/) - (a, t/)(fl, ~)}. Note that

B(a, fl) = - B ( f l , a) and B(s~a,s~fl)= B(fl, e) for any reflection s~ with 7e Z + c~ (Ra + Rfl), (cf. Dunkl [D]). Now $3 is equal to

S(fl, ~) coth a(u) or ~,Bez+ T aBeBa~ = ~,Bez+ ~ S(fl, a) coth -Ta~a~,Be~.B

, . , ~ ( u )

= ~ B(s, fl, a l c o m - T a , a~eB = - ~ B(a, fl) coth ~(u) a, BeF. + a, BeZ + T ~Ta~TB ~'B '

which cancels with $2. Since S~ written as

rt,B~-r + a, B e z + ~ , B e z +

it suffices to show

(3.3.2) ~ B(a, B) coth ~(u) fl(u) ~ B(a, fl)e~e B ~,B~+ T c ~ 2 - ~,Bez + s ~ s o = w s ~ s ~ = w

for each w e W. We note that each side of (3.3.2) does not depend on the choice of positive roots. If ~ and fl are parallel, then B(c~, B) = 0 and (3.3.2) trivialy holds. Otherwise, {ct, f leZ+; s,s B = w} is contained in some two dimensional subspace b* ~ b* and w is a rotation with axis perpendicular to b*. Since the reflection hyperplane of s~ with y ~ 2 + ~ b * contains the axis of rotation w, we have s~ 1 wsr = w-1. Let W~ denote the subgroup of W generated by {s~; 7 e Z ~- c~ b*}. Now let I~ be the left-hand side of (3.3.2), then it is invariant under the action of Ww. In fact, for any sr with 7 e Z d- c~ I9 *, we have

s~(I~) = s~c( s ~ f l

B(a, fl) coth ~ - coth �9 , BE,~ + s , s ~ = w

= ~ B(s,a,s,fl) coth a fl �9 , Pez + ~ coth

Ss~SsrB = w

= ~ B(~, B) coth ~ coth ~

s~$# = w - 1

= 2 B(~,fl) coth a �9 ,Bez + ~coth = Iw.

s~s# = w


Let d,~ be the Weyl denominator of the root system Z c~ b*. Then I,,Aw, which is an exponential polynomial, is anti-invariant with respect to the action of W,~. There- fore it is divisible by A~ and Iw itself is an exponential polynomial. Since the degree of lw is zero, it is constant. Let us fix a u0~b such that (7, Uo)> 0 for any 7 e Z § c~ l) *w. Substitute tUo for u in lw and let t > 0 tend to infinity. Then we obtain

lw = ~ B(a, fl). On the other hand, put X~ = {x ~ W; l(w'x) > l(x) for any

w'~ W~}. Any element in W is decomposed as w'x by some w'~ Ww and some xeXw. Now, for any 7 e Z ~ c~b*, w' e Ww and x e X ~ ,

B(~, fl)e~e~(syw' x) = ~ B(~, fl) e.,e~a~(w' x) 6 , f l ~Y . + a , 0 ~ ' s + s~s~ = w s i s ~ = w

= ~ B(o~, fl)are~e~p(w'x) ot, f l e Z § s ~ s o = w

= ~ B(srcr srfl) aTe, e~(w'x ) .

B(fl, ~)are, e~(w' x)

= ~ B(~, f l )a~e~(w'x) .

By the decomposition given above and by induction on the length of w' together with e~(x) = x for ~ Z § c~ b*, we see that the right-hand side of(3.3.2) is a scalar,

whose value is equal to ~ B(~, fl). This shows (3.3.2) and the proof is

completed. Q.E.D.

Corollary 3.3.2 The dimension of the space of local solutions of the Eq. (3.2.1) is equal to the order of the Weyl group.

Remark. The Eq. (E2) defines an integrable connection on the trivial bundle XR • C [W] ~ XR. Moreover, it is written by only in terms of logarithmic forms:

d - ~ k ~ { ( a ~ - l ) d l ~ 1 7 6 1 7 6 �9 ~ =

Here {~* } is the dual basis of {cq).

Notes. (i) Some of techniques in the proof of Proposition 3.3.1 are taken from the proof of commutativity of the infinitesimal differential-difference operators given by Dunkl [D], (cf. Heckman [HI]). (ii) Propositions 3.2.1 and 3.3.1 mean that the End(C[W])-valued functions r~(u) defined by

r,(u) = T ( f ~cotn ~ - (a~ -- 1) + a ~ a e Z

106 A. Matsuo

form a set of Weyl group invariant classical r-matrices in the sense of Cherednik [C1, 2], in which the equation of the following type is studied:

O'a Or = ~ k~(a, 4) ~-~ cb(u); 4 e b.

Cherednik [C4, 5] also constructed a class of Weyl group invariant differential equations with trigonometric coefficients.

4 Correspondence of solutions

We construct a homomorphism from the solution space of the Eq. (E2) to that of (El). We also study fundamental solutions of (E2) given by series expansions, from which we see that the homomorphism is bijective for general values of the parameters.

4.1 Correspondence of solutions

We define the linear map Sym : C [ W] ~ C by

c"

for any ~,w~wCwW~C[W]. Then Sym(v(w)0 ) = Sym(0) holds for any ~,eC[W] and w e W.

We set

k~ (4.1.2) C~ = ~, ~-(c~, 4)a~s~ + e~(2).

Then the Eq. (E2) is written as

{ ~ k~ e~'+l(a~_l)+Cr Or ~ + ~(~' 4)e ~_~

We have v(w)Ccv(w) -1 = Cwr for any we W.

Lemma 4.1.1 If �9 = ~(u) is a solution to the Eq. (E2), then

(4.1.3) /5~ d) Sym (~) = Sym (Cff q0.

Proof By the definition (2.3.6), we have

(4.1.4) 1

/Ska)Sym(~b) = Or -- ~ ~-(a, w e-UZ~_ 1 ~ ~,r -- D~ d-a)) Sym((b).

We assume that (4.1.3) holds for d - 1. Then the right-hand side of (4.1.4) becomes

2-k" e'e" -----1+ 1 Sym ((C~ -~ - Cg- ~) 4~).


The first term is equal to

1 k~ e + Sym( Ca- 1(6~r (~b)) = 2 2- (0~, ~) ~ Sym ((C~- ~(a~ - 1))4)) + Sym(C a - ' Cr 4))

k~ e: + 1 _ = ~ ~-(a, r e ~ _ 1 Sym ((C~d~ t - C~-')rb) + Sym(C~q~).

~ff~+

Thus we have

/~d) Sym@) = Sym(C~ r

so that (4.1.3) is proved by induction on d. Q.E.D.

Lemma 4.1.2 Let ~ : ~'| ~ C[b*] be the Harish-Chandra homomorphism defined by (2.1.5). Then

(4.1.5) 7(/5~d))(2) = Sym(C~(1)).

Proof. We assume that (4.1.5) holds for d - 1. Then we can calculate as follows.

(~d)) ()~) = (~ + ~), ~) ~ ( ~ d - 1)) (,~) _[_ ~t~Z' ' ' 2 ( , ~) ~ (~d~-I) __ j ~ d - l ) ) ()~1

-- ka 0~ = (2 + O, ~) Sym(C~- 1 (1)) + ~ . ~ - ( , ~) Sym(C~a~ t (1))

- ~+~- ( , ~ ) S y m ( ( ~ - 1(1))

= (2, ~) Sym(C~ -~ (1)) + ~ ~-( , ~) Sym (C~-~g~(1))

2 k~ (e, ~)Sym(C~_ ~ a~ e~(l)) = Sym(C~- ~ er (1)) + ~r + 2

= Sym(C~(1)).

Therefore (4.1.5) is proved by induction on d. Q.E.D.

Proposition 4.1.3 I f q) = Cb(u) is a solution to the Eq. (E2), then ~b(u) = Sym@(u)) is a solution to the system (El).

Proof. Combining Lemma 4.1.1 and Lemma 4.1.2, we obtain

/)r Sym(qb) = 2 /)~d)Sym(~) qeW~

= ~ Sym(C, a 4)t ~eW~

= 2 Sym(C,a(1))Sym@)

= ~ 7(D;n))(2)Sym(tb) rt~W~

= 7(D+,d) (2) Sym (<b).

Since /5~,d generate D, this completes the proof. Q.E.D.

108 A. Matsuo

Thus we obtain a homomorphism

(4.1.6) Sym : Sol(E2) ~ Sol(E1)

of the solution sheaves of the Eqs, (E2) and (El).

4.2 Local solutions

We put e ~ .... )=y~, which we take as local coordinates on b. Then we have a 1

- - = -- 0~. Here {~*} is the dual basis of {~}. Hence the Eq. (E2) is written as: 0Yi Yi (A,0, ) (4.2.1) Oy~ = Y~' + Regular at = 0 4~,

where

k~ (4.2.2) A(r ~ = ~ ~ (~, r a~(e~ - 1) + e~(2) + (e, ~).

Notice that A~ ~ commute with each other thanks to the commutativity of A~.

Lemma 4.2.1 Let ~ be a positive root. Then the length of w ~ C [ W ] is shortened by a , (e~- 1).

Proof From Lemma 1.2.1, we have

J" 0 if l(s, w) > l(w), ff ct(sa 1)(w)

- 2 s , w i f l(s, w) < l(w).

This proves the lemma. Q.E.D.

Proposition 4.2.2 Suppose that w2 - 2 does not belon9 to the root lattice for any w ~ W. Then,for each w ~ W, the Eq. (E2) has the unique solution ofthefollowinaform:

l l

where 0 ~ e C [ W ] and O~ ~ = ~ ' ~ w c~,~, w', c~,~,eC, is determined by

(4.2.4) A t ~ 0 ~) = (w2 + Q, ~) ~ ~ and c~, ~ = 1.

k~ Proof. Order the basis {w~C[ W]} according to the length. The ~ , ~ - ( e , ~ )~

(e~- 1) is nilpotent and triangular by Lemma 4.2.1. Hence A~ ~ are triangular matrices with diagonal parts er + (Q, ~). Because of the assumption, the common eigenvalues w2 + 0 do not differ by integers. Therefore the proof follows immediately from the local theory of linear differential equations. Q.E.D.

Remark. If the exponents differ by integers, then logarithmic terms occur in the local expression of solutions. However, if 2 is regular, then the common eigenvalues of A~ ~ are distinct and the Eq. (4.2.4) for all ~ e b has the unique solution 0 ~ ) for each w.

I n t e g r a b l e c o n n e c t i o n s r e l a t ed to z o n a l spher ica l func t ions

Corollary 4.2.3 Le t 2 be regular. I f l(w') > l(w) and w' 4= w then c . . . . , = O, i.e.

= w + Z w'eW.l(w')<ttw)

f o r some cw, ~, ~ C.

109

4.3 Analysis o f leading coefficients

Let c~,~,(2)= Cw,~, defined as in Proposition 4.2.2 for a given 2eb*, which is supposed to be regular. We understand k, = 0 if ~ r X.

Lemma 4.3.1 (i)Ifo~ is a simple root, then

k~ +2k2~ (4.3.1) c~ ,a(2) - (2, a~)

(ii) I f ~ is a simple root and l(s, w) > l(w), then

(4.3.2) c . . . . . ,(2) = c . . . . . ,(2) + Cw. ~,(2) c~,, ~ (wZ).

P r o o f (i) Substituting ~, ~) = ~w, ,W c~, ~, (2)w' in (4.2.4) and equating coefficients ! to w, we have

(4.3.3) C~.w,(2)(w'2 - w2) = ~ c . . . . ~,(2)(kafl + 2k2pfl), fleY.~;l(s~w')>l(w')

for any w, w' ~ W. In particular, we have

G,,~(2)(2 -- s~2) = (k~ + 2k2~)~.

By the definition of reflections, this implies

e~,~ (2)(2 - ~" ) = (k~ + 2kz~)ct.

which proves (4.3.1). (ii) By Lemma 1.2.1, (4.3.3) is rewritten as

(4.3.4) Cw,~,,(2)(w'2 - w2) = ~ c . . . . . ,(2)(ka + 2k2r #ES,~ n w'2:~

In particular, we have

(4.3.5) c . . . . . , (2)(s~w'2 - w2) = ~ Cw.,,s,w,(2)(kp + 2k2a)fl, #~Z~ ns,w'z;

(4.3.6) c~,, x (wZ)(w2 - s~wZ) = (k~ + 2k2~)c~.

Then multiplied by s, , (4.3.5) is equivalent to

(4.3.7) c . . . . . , (2 ) (w '2 -- s~w2) = ~" c . . . . . . . ,(2)(ka + 2k2a)s~fl. #Ez6 ~ n s,w'Zg

110 A. Matsuo

N o w let ~ . . . . . , (2) denote the right hand side of (4.3.2). We have

~ . , . , (2) (w'2 - s~ w2) = { c . . . . . , (2) + c . , w, (2) c~, ~ (w2)} (w '2 - s~ w~)

= c . . . . . ,(2) (w '2 - s~,w2) + c . , w, (2) c~, ~ ( w 2 ) ( w 2 - s~w2)

+ c~,,l(w2)Cw, w,(2)(w'2 - w2).

By (4.3.4), (4.3.5) and (4.3.7), this is writ ten as

Z

(4.3.8) +

Cw, sps, w,(J.)(k# + 2k2#)s~fl q.- cw, w,(2)(ka + 2ka,)Ct

~, c,~,,~,w'(2)c~,,dw2)(k s + 2k2s)fl. #eZ~ n w'_r~

Replacing fl with s,fl, we rewrite the first term in (4.3.8) as

c . . . . . ~w, (2) (k s + 2k2a) ft.

Thus the first and the second terms in (4.3.8) are written in a unified manner:

y, C . . . . . . . ' ( 2 ) (ks q- 2k2s) f l �9

Hence we obtain

. . . . . ,(2)(w'2 - w2) = ~ c . . . . . ,w,(2)(k s + 2kzs)fl

+ ~ c . . . . . ,(2)c~,l(w2)(k~ + 2k2s)fi SeZ,~ c~ w's

= Y~ es . . . . , w , ( 2 ) ( k s + 2 k 2 p Z . SeG c~ w',~

This equality says that

Since c . . . . . ( 2 ) = 0 follows f rom the assumpt ion l(s~w)> l(w), we have G . . . . . . (2) = Cw,~(2) + c . . . . . (2)c~,a (w2) = 1. Hence we obta in ?~ . . . . ,(2) = c . . . . . ,(2) by the uniqueness of the solution to {4.2.4). Q.E.D.

Proposit ion 4.3.2 Let ~h ~) be defined by (4.2.4). Then

(4.3.9) Sym(f f~ )) = 1--[ ks + 2k2s + (2, fl ") s ~ ; ~ - - ~ ; (2, p ~)

Proof. We shall prove (4.3.9) by induction on the length of w �9 IV. Since ~h (1 ~ = l, (4.3.9) is obvious for w = 1. Let a be a simple root and suppose that w �9 W satisfies l(s~w) > l(w) = I. By L e m m a 4.3.1, we have

Integrable connect ions related to zonal spherical funct ions 111

= 2 c . . . . . .(~) + Y~ c~,,,,.(;OCs,.,(~O w ' ~ W w 'EW

= Sym@[O)) + Sym(O~>) k: + 2kz: (w2, ~ v)

k, + 2k2~ + (2, w - t a v ) Sym@ ~)) �9 ( L w - ~ v)

Assuming that (4.3.9) holds for w ~ W with l(w) < l, we rewrite this as

k, + 2k2~ + (2, w - ~ ~ v) l-I kr + 2k2a + (2,/~ v )

(L w - ~ " ~ ) e ~ ~ ~ - ' z c (L/~v )

By L e m m a 1.2.1, w-~ e ~ Zo + follows from the assumpt ion t(s~ w) > I(w). Therefore Z~- ~ (s, w)- 1 20- = (Zo + c~ w- 1 Xo-) w {w- 1 cr Hence we obtain

S 'm "~'l~ k~ + 2k2t~ + (2,/~") y w~,~1 = 1-I ( ~ , / ~ ) ,

#e,~g c~ (s,,~,)- aIu

which completes the induction. Q.E.D.

Corollary 4.3.3 Suppose that w 2 - 2 does not belon9 to the root lattice. I f k, + 2k2~ + (2, c~ ~ ) 4: O for any ~ ~ Z ~, then any S y m @ ~ )) does not vanish and the homomorphism Sym : Sol(E2) -+ Sol(E1) is bijective.

5 Bijectivity condition

In this section, we prove the equivalence of the systems (El) and (E2) under a certain condition. Roughly speaking, it is established by analysing the determi- nan t of the correspondence introduced in the previous section.

5.1 Harmonic polynomials and Poincard series

We first summarize some propert ies of ha rmonic polynomials associated with the Weyl group. We refer to [He2, p. 356] for detail.

For the symmetr ic algebra S on b, let S w be the set of all W-invariant elements in S. Let S* be the symmetr ic algebra on [9*.

Definition 5.1.1 An element q ~ S * is harmonic if ~ (P)q = 0 for any P ~ S w such that P(O) = O.

The set of all harmonics is denoted by ~ . Using the inner product ( , ) on b, S* is identified with S, Therefore W is regarded as a supspace of S. Then H has a natural gradat ion

(5.1.1) ~ , = @ ~f(a> d>O

112 A. Matsuo

with respect to the degree. Its Poincar~ series is, by definition,

P(~Cf; t ) = ~ dim ~f td) t n. d > O

This is described as follows. Let d l , " ' , d, be the exponents of W, namely the uniquely determined degrees of homogenious generators of invariant polynomials. They have the following properties:

(5.1.2) fi d, = I WI and ~ (d,.- 1) = IZffl- i = 1 i = 1

Then the Poincar6 series of ~ is given by

(5.1.3) e ( ~ ; t) = I~I - - i = 1

Lemma 5.1.2

1 - - t d~

1 - - t

d dim ocg ~a) = �89 wI I Zo + I. d > O

Proof Differentiate (5.1.3) with respect to t and let t ~ 1. Then the lemma is proved by (5.1.2). Q.E.D.

Remark. Furthermore dim ~ d ) isequal to the number of we W with length d (cf. Helgason [He2, p. 383]).

Lemma 5.1.3 Let {hi; i = 1 , . . - , L WI} be a homogenious basis of oW and {2i; i = 1, �9 �9 �9 I W]} a sequence in b*. Then the determinant det(hi(2~ +/~)) does not depend on # ~ b*.

Proof For any/~ ~ b*, we have

det(h~(~ + t/~ = ~ det E~, t = O k = l

where the (i, j)-th component of Ek is given by

h/(2j) if i ~e k,

~ hi (2j + t~) if i = k. t = O

Then [ d hi(2J + t#)l,=o =(c'~uh/)(2i) 'SinceO"h/isaharm~

lower than deg h/, it is a linear combination of hz, l + i. Hence det Ek = 0 for any k, which completes the proof. Q.E.D.

5.2 Discrirninant of field extension with Galob group W

Consider the natural representation of the Weyl group on ~uf. Then this is equivalent to the left regular representation of W. An important point is that


a basis {hi, i = 1, �9 - ", [W[} of a~ is a base of the Galois extension C(S)/C(S w) whose Galois group is isomorphic to 14/, where C(S) is the fractional field of the algebra S and C(S w) is that of invariants S w (cf. Chevalley [Ch] and Helgason [He2, p. 383]). Now we set {wl; i = 1, �9 . . , f W[ } = W and take a homogenious basis {hi;i = 1,' " " , {WI } of o~. Consider the matrix whose (i, j)-th component is w~-~(hfl. Then its determinant D is the discriminant of the extension C(S)/C(SW).

L e m m a 5 . Z 1 det (h i (wj ,~ + #)) is a non-zero polynomial on ;'.e~* of degree �89 WI I,~,~ [for any/z~ll*.

Pro@ By the Lemma 5.I.3, det(hi(wj)~ +/~)) does not depend on/x. Therefore it is equal to the value of discriminant D(2). It is not identically zero, since the extension C(S)/C(S w) is separable. Q.E.D.

5.3 Determinant formula

Since Ar defined by (3.2.2) commute with each other, the map ~ ~ Ar extends to an algebra homomorphism S ~ E n d ( C [ W ] ) | h~,Ah . For each h e ~ and O e C [ W ] , we define J h ' C [ W ] ~ Gl by

(5.3.1) Jh(O) = Sym(Ah (0)) ~ Jl.

We consider the series expansion of Ar = A~(u):

(5.3.2) Ar = A(~ ) + 2 eU(")A~ ") # > 0

where A ~)e End(C [ W]) and A ~o)is given by (4.2.2). This expansion is convergent ~ + o) (o) W -o C b m {u~ll; Rea(u)< 0 for any ~ Z o }. We define A~ and Jh C[ ] y

substituting A~ ~ for Ar in the definition of An and Jh. Then we have

(5.3.3) Jh = J~h ~ + ~ J~")e"(") t~>O

for some J~")" C [ W] --, C. Now let us take h i e c f and wieC[W] as in Sect. 5.2. Consider the matrix

whose (i,j)-th component is Jh,(W~). Then its constant term with respect to the expansion (5.3.2) is equal to J~~

Our fundamental formula is the following:

Proposition 5.3.1 There exists a non-zero constant C, which does not depend on the parameters, such that

det(d~~ = C 1-I (k~ + 2kz~ + 0-, c~v)) ~.

Proof. Suppose that 2 ell* is regular. Then, as in Proposition 4.2.2, A~ ~ have the unique common eigenvector for each w e W of the following form:

(5.3.4) ~ ) = w + ~ C~,w, w', w'; l(w~)<l(w)

114 A. M a t s u o

where c~.~, are some complex numbers. Since the base change (5.3.4) is given by a triangular matrix with diagonal part being the identity, we have

det(J(n~ det~1~o~,l,(o)~ = t'Jh~ 1,Wwj l )

(5.3.5) = det (Sym (Aht~ (ff ~w~

Because ~O~ ~ is a common eigenvector of A[ ~ with eigenvalue (w2 + ~9, ~), we have

", .t.(O) ~h,~(~ J = h~(wi2 + @)qJw �9

Hence (5.3.5) becomes

det (J~hO)(wj)) = det(h,(wj2 + p)Sym(O~wm))

= IF[ 1-I k, + 2k2~, + (2, e't)

(5.3.6)

~,~w ,,~z~ ~,,,z~ (2, ~ ) det(h~(wj2 + p))

= det(h~(wj2 + ~9)) I-I (k, + 2k2, + (2, ~v)) ~ 1-L~(~,~ v),~ ~

Here we have used the evaluation of SymtqJ ~ , given by Proposition 4.3.2. Now, by Lemma 5.2.1 applied for/~ = Q, the demoninator and the numerator in the right of (5.3.6) must cancel out and the resulting form must be

c I ] (k, + 2k2, + (,~, ~v))~

for some non-zero constant C which depend neither on 2 �9 b* nor on k,. Thus the proposition is proved for a regular 2 �9 b*. Now the result is an identity of polynomials so is continued to arbitrary 2 �9 b*. Q.E.D

Note. The method to calculate the determinant given above is analogous to the discussion of Kato [Ka], where a certain representation of the affine Hecke algebras is studied. This kind of representation is closely related to the monodromy of the system (El), (see Sect. 7).

5.4 The main theorem

Our main theorem is the following:

Theorem 5.4.1 For any solution ~(u)=~w~w49w(U)W of the equation (E2), 49 (u) = ~w~w 49w(U) is a solution to the system (El). This correspondence is bijective if k~ + 2k2~ + (2, ~ v ) 4:0 for any ~ �9 Z~'.

Proof. The first part is already proved as Proposition 4.1.3. Let q~ = ~(u) be a solution of (E2): 0r = Ar and assume that Sym(q~) = 0. Then we have

Jh(Cb) = Sym(Ah(~))

(5.4.1) = Sym(t3(h)q0 = O(h)Sym(~)= 0,

for any h � 9 Now suppose that k, + 2k2~ + (2, ~t v) 4:0 for any e � 9 Then det(Jh,(Wj)) ~: 0 so that we have ~ = 0. Therefore the map Sym : Sol(E2) ~ Sol(E1) is injective. Thus we conclude that it is bijective because dim Sol(E1)= dim Sol(E2) = [WI. Q.E.D.

Integrable connections related to zonal spherical functions t 15

Remark. Suppose that one of k, + 2k2~ + ()~, c~ ~) vanishes. Then, by Proposi t ion 4.3.2, the leading term in some of Sym(4h,,(u)) vanishes. On the other hand, a fundamenta l system of solutions to the system (El) is given in the following form under certain general conditions:

c~(u) = e("a+v.~) {1 + ~ a~,ue~U'~)}, ~ > 0

(cf. [HO, p. 341]). Therefore, Sym : Sol(E2) ~ Sol(E1) is not injective in this case. Along this line, we can prove that it is bijective if and only ifk, + 2k2, + (2, e ~) 4= 0 for any c~eZg.

6 Knizhnik-Zamolodchikov equations

In this section, we only deal with the case of A,- type root system. We discuss the relation of the Eq. (E2) and the Kn izhn ik -Zamolodch ikov equat ion originated in conformal field theory.

6.1 Trigonometric Knizhnik-Zamolodchikov equation

Let g be a simple Lie algebra over C of rank n and (I) the Car tan-Ki l l ing form. Choose a Caf tan subalgebra t. We consider the root system A = t* and the set A + of positive roots. Then we have the root space decomposi t ion:

(6.1.1) .q = @ g, O t O @ g-~ .

For any ~ A + we choose E=~9=, E_=~g_= such that (E, JE_=)= 1. Let {H~; i = 1, - �9 ', n} be an o r thonorma l basis of t. We define

(6.1.2)

t= E (E.| + E_.| H. |

r = ~ (E_~| - E~|

Fix a positive integer N. For each i = 1 , . . - , N , we take a g-module ~i : g ~ End( V 3. Let k be a complex paramete r and h ~ t. We introduce the following differential equat ion for a V I | �9 " �9 | VN-valued function ~U(z) = I / I ( Z 1 , - . - , ZN):

~k ~ (cothZi-Zj ) } (E3) ~z/Y'(z) = ~ (~zi| + (~i| + ~/(h) ~(z) .

We note that, by taking the rat ional limit of (E3), we obtain the ordinary K n i z h n i k - Z a m o l o d c h i k o v equation:

0 7j(z ) = k ~ (~' | ~j) (t) ~U(z), (E4) Oz--~ zi - z i (j , i)

116 A. Matsuo

Notes. (i) Equation (E3) was studied by Cherednik [C3]. The integrability of this equation comes from the fact that the 9| function r(x) = coth(x)t + r satisfies the classical Yang-Baxter equation. (ii) In the Wess-Zumino-Witten model in conformal field theory, the N-point functions on P1 satisfy the Knizhnik-Zamolodchikov Eq. (E4) for a certain value of k~C, (cf. Knizhnik and Zamolodchikov [KZ] and Tsuchiya and Kanie [TK]).

6.2 Vector representation of ~l (n + 1, C)

Let 9 be the Lie algebra ~l(n + 1, C) with the Cartan subalgebra t consisting of all diagonal matrices in g. The Killing form is given by (x ly )= t rxy for any x, y ~ g. Let Eij denote the matrix with only non-zero entry 1 at the (i,j)-th component. Let ~ t * be the simple root corresponding to E , - E~+li+l. Set c q j = ~ + . - - + ~ j _ l for any I < i 4 = j < n + 1 and define E~, s = E u. The set of positive roots is A + = {7~s; 1 < i < j < n + 1}. We define Aiet* by (~ilAs) = 6~s where 6~j denote the Kronecker delta. Then {At} form the set of fundamental weights.

A vector space V = C "§ equipped with the natural action of ~l(n + 1, C) is called the vector representation of 9, denoted by n : g ~ End(V). Consider the vector e~ with only non-zero entry 1 in the i-th component. Then ei is a weight vector with weight A1 - (~1 + . . . q- ~-1). In particular, e~ is the highest weight vector.

Now let us consider the trigonometric Knizhnik-Zamolodchikov equation in the sense of the last subsection for g = ~l(n + 1, C) and V~ . . . . = VN = V being the vector representation.

Lemma 6.2.1 In the notations of 6.1, we have

1 (6.2.1) (n|174 = (et| -:--7(ek| and

n + 1

(6.2.2)

e l @ e k

(r~| (r)(ek| = -- e,|

0

i f k < l ,

i f l < k ,

otherwise.

Proof. Because E~ s ek = (~jk ei, we have

(Eij| Esi + Esi| Eij)(ek| { ; lQek 1 < i < j < n + l

and

if k . l otherwise,

E 1 < i < j < n + l

(Es~ | Eis - Eij | Esi) (ek | ez) =

e l @ e k

- - e l @ e k

0

if k < / ,

if l < k ,

otherwise.

I n t e g r a b l e c o n n e c t i o n s r e l a t ed to z o n a l s p h e r i c a l f u n c t i o n s 117

The lat ter proves (6.2.2). Since the weight of el is equal to A1 - (cq + �9 �9 . + ~i-a), we have

~ (H i |174 = (AI - (al + " " " + c ~ - l ) I A l - (~l + " " " + c~z- ~))ek| i = 1

{1 t = { ( A L I A 1 ) - 1 + 6kt}e~ | = n + 1 + 6kt e ~ |

Thus (6.2.1) is also proved. Q.E.D

6.3 Identification with C [ ~ , + 1J-valued case

W e keep the settings in 6.2. Let us restrict ourselves to the case where N = n + 1. W e consider the weight subspace of V1 | �9 . - | V,+~ with the weight 0:

(6.3.1) ~4/~o = V e V 1 | " ' " | V n + l ; 7ri(h)v = 0 f o r a n y h e t . / = l

Then we have

L e m m a 6.3.1 "/C o is given by

~/C0 = @ Cew-,(1) | " " | ew-,~,+l) W E a n + 1

where an element o f the symmetric group ~ , + 1 is regarded as a permutation o f letters { 1 , . " , n + l } .

Proof. Consider a pure tensor % @ - �9 �9 | ei.~,. This is a weight vector with the weight

n + I

(6.3.2) (n + 1)A1 - ~ (~i, + " " + ~i~_,). m - - I

It is 0 if and only if i l , " ", i,+1 are distinct. Q.E.D.

Hence the correspondence w ~ , e~- ,~) @ �9 �9 �9 | ew- ,~,+ ~ defines an identifica- t ion

(6.3.3) C [ ~ , + l ] ~ , ~/Co

which commutes with the na tu ra l ac t ion of ~ , + x. We next consider a complex vector space C "+ 1 and let (C "+ 1), be its dual space.

Take a basis {e~; i = 1 , . . . , n + 1} of (C"+1) * and set z i = e~(z) for a var iable

z~C n+l. We define b* = {ZT+t 1Pie ie(C"+l)*; /~1 + " " + P.+1 = 0} and b = C " + I / N where N = {veC"+X; (l)*, v) = 0}, so that b and b* are dual to each other. Define

Z = { e ~ - ~ j ; 1 < i : l : j < n + 1},

then Z c D* is a roo t system of A, type with C-bi l inear form ( , ) defined by (ei, ej) = 6u. We also identify t* with b* via ~ij~--,e~ - e~.

118 A. Matsuo

Lemma 6.3.2 Under the identifications above, we have

1 (7~ i ( ~ 7~j) (t) = f f i j - - - - and (~ i ( ~ 7r j ) (r) : o i j F.ij ,

n + l where aij = a~,j and eij = e~,j are defined by (3.1.1) and (3.1.3) respectively.

Proof Immediate from Lemma 6.2.1. Q.E.D.

We define, for each h s t,

n + l

(6.3.4) 2 = ~ (A1 - (~1 + "' + c~,-0)(h)~i~b*. i = 1

Then the action of a diagonal element is given as follows:

Lemma 6.3.3

gi(h)ew-,~l~ | " ' " Q ew-~,+ll = (w2, ei)ew-,r | " ' " | e,v ,~,+1).

Proof By (6.3.4), we have n ( h ) e i = ( A l - ( ~ l + ' " + a i - 1 ) ) ( h ) e i = ( A , ei)ei, from which the lemma immediately follows. Q.E.D.

Let us regard z ~ C" + 1 as a representative of u ~ b. Notice that the Eq. (E3) is invariant with respect to the translation ( z l , ' " , z,+l)F--*(z~ + x , . . . , z,+~ + x) for any x ~ C. Therefore it is well-defined on XR ~ b.

Now let k, = k for any ~ ~ Z and 2 as (6.3.4). Then, combining these lemmas, we obtain the main proposition:

Proposition 6.3.4 By the correspondence

. _ . -k.~-~ ~b(u) = I- I sinh ~ ) ~g (z),

i < j \

the restriction o f the Eq. (E4) to ~1/~o is equivalent to the Eq. (E2) associated with the root system Z, = A. .

7 Remarks

We finally make two comments on the equations (E1)-(E4) according to some advances related to conformal field theory and the theory of hypergeometric functions.

7.1 Monodromy representations

If Va . . . . = VN then the Knizhnik-Zamolodchikov Eq. (E4) has an invariance with respect to the action of symmetric group ~N in certain sense. When V~ are the vector representations of ~1,,, the corresponding braiding monodromy, which is a priori a representation of the braid group, gives a representation of Iwahori's Hecke algebra [TK, Ko]. It would then be obvious that the counterpart for the Eq. (E3) is a representation of affine Hecke algebra of A,-type.

Cherednik [C 1, 2, 5] gave some analogous results for C [ W]-valued case, where the (affine) Hecke algebra corresponding to arbitrary root system 2; took place. On

lntegrable connections related to zonal spherical functions 119

the other hand, it was already shown by Heckman and Opdam [HO] that the monodromy of the system (El) also gives a representation of affine Hecke algebra of type 2;. It coincides with what is called the principal series representation or the induced representation (see [H3]) of affine Hecke algebras. In [Ka], [H3] and [C4], irreducibility of such representations is studied, and their results agree with our consequence (Remark after Theorem 5.4.1) that the monodromy representation of the Eq. (El) or (E2) is reducible when (2, ~v) + k~ + 2kz~ = 0 for some ~ e Z d .

These phenomena support our main theorem.

7.2 In t egra l represen ta t ions o f solut ions

By the author [M], Schechtman and Varchenko [SV1] and others, it is known that solutions to the Knizhnik-Zamolodchikov Eq. (E4) are represented by certain integrals which generalize an integral representation of Gauss' hypergeometric function. Analogous theory for the equation (E3) was constructed by Cherednik [C3]. It is worth a mention that in view of contours of the integral solutions some geometric structure behind the monodromy of (E4) was studied by [FW, L, SV2] etc.

Now these integral solutions give some special class (sometimes called the generalized Selberg type integrals) in the category of generalized hypergeometric functions studied by Aomoto [A1, 2-1 and Gelfand and others [VGZ]. Having combined with Cherednik's work [C3-1 via the results of the present paper (Prop- osition 4.1.3 and Proposition 6.3.4), zonal spherical function of A,-type is expressed as a sum of generalized hypergeometric functions mentioned above. Let us describe it briefly.

We shall follow the context of Sect. 6. First take integration variables indexed as { @ ; j = 1, �9 �9 �9 n, l = j , �9 �9 �9 n}. Take a total ordering < in the set of the indices. Consider the following multivalued function

2(A,~/) t I (D(Z, t) e~(,t, Al)z, = ~ x e I I

( j , l )~l 1 <- i<-n+l

( H sinh 1-I 1 < i < n + l ( j , l ) < ( j ' , l ' )

l <_l<n

sinh tlj) - tlJ")) k~'~'~''

Then the integral

r = ~ o9(z, t)q~(z, t )d t ] 1~ " " dt~, "~, F

for a certain choice of a function q~(z, t) and a contour F, is a solution to the zonal spherical system (El) for Z --- A, in the context of Sect. 6. Here q~(z, t) is a p o l y -

t ? - - t l ! nomial of coth ~ and coth- ; 2 written in a complicated manner (see

[C3] for detail). Note that the meromorphic function ~0(z, t) does not contribute the monodromy of ~b(u).

120 A. Matsuo

Acknowledgements. The author would like to express his gratitude to G. Heckman for kind explanations on his literatures, and to I. Cherednik for discussion, encouragement and useful comments. He is also grateful to J. Kaneko and M. Okado for information. His special thanks are due to M. Kashiwara for suggestions on the proofs.

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[HI]

[H2]

[H3]

[H4]

[Hel]

[He2] [K]

[Ka]

[KZ]

[Ko]

[L]

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