Three Formulas for Eigenfunctions of Integrable Schr&Ouml

Compositio Mathematica 107: 143±175, 1997. 143

c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Three formulae for eigenfunctions of integrable SchrodingerÈ operators

GIOVANNI FELDER and ALEXANDER VARCHENKO ? Department of Mathematics, Phillips Hall, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA; e-mail: [email protected], [email protected].

Received 28 November 1995; accepted in ®nal form 15 April 1996

Abstract. We give three formulae for meromorphic eigenfunctions (scattering states) of Sutherland's

integrable N -body SchrodingerÈ operators and their generalizations. The ®rst is an explicit computation of the Etingof±Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third is a formula of Bethe ansatz type. The last two formulas are degen- erations of elliptic formulas obtained previously in connection with the Knizhnik±Zamolodchikov± Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a `Hermite±Bethe' variety, a generalization of the

spectral variety of the LameÂ operator. We also give the q -deformed version of our ®rst formula. In

the scalar sl N case, this gives common eigenfunctions of the commuting Macdonald±Ruijsenaars difference operators. Mathematics Subject Classi®cation (1991): 17B65, 82B23, 14J99 Key words: Integrable models, Ruijsenaars models, Bethe ansatz

1. Introduction

Let g be a simple complex Lie algebra, with a non-degenerate ad-invariant bilinear

; ) h h g

form ( , and a ®xed Cartan subalgebra .Let be the set of roots of .

2 e 2 g

For each let be a corresponding root vector, normalized so that

(e ;e )=

1. g

Suppose that U is a highest weight representation of with ®nite dimensional

h U [ ] zero-weight space (the space of vectors in U annihilated by ) 0 . We consider

in this paper the differential operator

X 2

H = 4 + e e ;

2 (1)

())

sin (

U [ ] 4 acting on functions on h with values in 0 . The Laplacian is the opera-

P 2 2

=@ = (b ;) @

tor in terms of coordinates for any orthonormal basis

b ;::: ;b h [ ] g = U N 1 r of . As remarked in 3 ,if sl ,and is the symmetric power

? The authors were supported in part by NSF grants DMS-9400841 and DMS-9501290 respectively.

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144 GIOVANNI FELDER AND ALEXANDER VARCHENKO

pN N N

C C U [ ] S of the de®ning representation ,then 0 is one-dimensional and this

differential operator reduces to Sutherland's integrable N -body SchrodingerÈ oper-

(p + ) ator [16] in one dimension with coupling constant p 1 . In suitable variables,

H is in this case proportional to

X 2

H = + p(p + ) V (x x );

S j

2 1 l (2)

j =

6=l 1 j

2 2

(x)= = (x) V (x)= = (x) with V 1sin (trigonometric model) or 1sinh (hyperbolic

model). We refer to these special cases as the scalar cases.

H = "

We consider in this paper eigenfunctions (functions with for

(;)

2 C f () f

some " ) of the form e ,where is meromorphic and regular as

() ! i1 8 2 , (a more precise de®nition is given below). In the language

of N -body SchrodingerÈ operators these are scattering states for the hyperbolic

E ( )

models. It turns out that for generic the space of such eigenfunctions is

[ ] ®nite dimensional and isomorphic to U 0 . We give three formulae for these eigenfunctions. The ®rst one (Theorem 3.1) is an explicit computation of an expression of Etingof and Kirillov in terms of traces of certain intertwining operators. The reason that these combinations of traces can be computed explicitly is that they turn out to be the same as for the corresponding Lie algebra without Serre relations (Proposition 9.2). This observation reduces the

computation of traces to a tractable combinatorial problem. We also give the q - deformed version of this formula (Theorem 8.1). The second formula, an integral representation (Theorem 4.1), follows from the fact that eigenfunctions can be obtained as suitable limits of solutions of the Knizhnik±Zamolodchikov±Bernard equation of conformal ®eld theory, for which we gave explicit solutions in [7]. The

third formula (Theorem 5.1) is of the Bethe ansatz (or Hermite) type: one has an

2 C explicit expression of a function depending on parameters T . This function is an eigenfunction if the parameter lie on an algebraic variety which we call

Hermite±Bethe variety. We have a regular map p from the Hermite±Bethe variety

T p

to h , sending to the corresponding value of . Conjecturally, is a covering

( ) map and a basis of E for generic is obtained by taking the eigenfunctions

( ) corresponding to the points in the ®ber p . This conjecture is proved in some cases including the scalar case (Theorem 5.3).

The three formulae hold for generic values of the spectral parameter .In

Section 6 we study trigonometric polynomial solutions of the igenvalue problem

= " H . These (Weyl antiinvariant) eigenfunctions are related to multivariable Jacobi polynomials (or Jack polynomials). Formulae for these polynomials are obtained as the spectral parameters tends to an antidominant integral weight. The construction involves the construction of the scattering matrices for the problem, which give a solution of the Yang±Baxter equation. It would be interesting to

generalize this construction to the q -deformed case, which would give formulae for Macdonald polynomials. We give such formulae in Section 8 as a conjecture.

This conjecture can be proved by the same method as in the classical case for sl2 and sl3, but the regularity property of Weyl antiinvariant eigenfunction appears to be more dif®cult to prove in the general case. The fact that we have three different formulae for the same thing can be explained

in informal terms as follows. Our second formula is an integral depending on a

complex parameter 0. Its form is

() =

2i 1

! (T; ): ()= (T )

()

D =(C f ; g)

The function 0 is a many-valued holomorphic function on n 01

[ fT j T = T g ! (T; ) U [ ] n

i j i

D () (T )

n .Thecycle has coef®cients in the local system determined by 0 .It

() H turns out that, for all , , is an eigenfunction of with the same eigenvalue

(; )

4 ,andthat(if is generic) for any ®xed generic all eigenfunctions can

()

be obtained by suitable . One can thus consider suitable families of cycles

! !1 parametrized by in the limits 0, . In the former case the homology

reduces to ordinary homology, and the answer is given in terms of residues of ! (our ®rst formula). In the latter limit, the integral can be evaluated by the saddle

point method and the eigenfunctions are given by evaluating ! at the critical points

of 0 (our third formula). This reasoning turns out to be useful to write down the formulas, (and to `understand' them) but it is then easier to prove them by other methods. In Section 7, we generalize our results on Bethe ansatz eigenfunctions (the third formula) to the elliptic case. The SchrodingerÈ operator in this case has the

Weierstrass }-function as a potential:

}( ()) e e + H = 4 + :

e const (3)

pN N

g = U = S C V = }

In the scalar case slN , , we obtain (2) with .Bethe H

ansatz eigenfunctions of e were given in [7]. After reviewing this construction, we give a result on completeness of Bethe states (Theorem 7.3) parallel to the

trigonometric case, in the scalar case and the case of the adjoint representation of

(U [ ])

slN : for generic values of the spectral parameter there exist dim 0 solutions of the Bethe ansatz equations corresponding to linearly independent quasi-periodic

eigenfunctions with multiplier given by . The formulae discussed here in the trigonometric case were obtained by con- sidering the trigonometric limit of solutions of the KZB equations for an elliptic curve with one marked point. The same construction could be done in the case of

N marked points. It turns out that in the trigonometric limit one of the equations U always reduces to an eigenvalue equation for H . The new feature is that is the

tensor product of N representations. The remaining commuting operators de®ne

differential equations in the space of eigenfunctions of H with a ®xed eigenvalue.

In the q -deformed case, we could only generalize the ®rst formula. It would be

interesting to ®nd the second and third formula in the q -deformed case. The organization of the paper is as follows: in Section 2 we de®ne and describe the space of eigenfunctions we consider in the trigonometric case. Three formulae for these eigenfunctions are given in Section 3 in terms of coordinates of singular vectors, in Section 4 as integrals, and in Section 5 by a Bethe ansatz. In Section 6, the action of the Weyl group is discussed along with the relation to multivariable Jacobi polynomials at special values of the spectral parameter. In Section 7 we discuss the Bethe ansatz in the elliptic case. Section 8 gives a generalization of the

®rst formula to the q -deformed case. The proof of the ®rst formula is contained in Section 9.

We conclude this introduction by ®xing some notations and conventions. The

( ; )

Cartan subalgebra h will be often identi®ed with its dual via .We®xasetof

;::: ; 2 Q Z

i i

simple roots 1 r , and write for the root lattice . Its positive

N N = f ; ;::: ;g Q

i +

part i , with 0 1 , will be denoted by . We denote by half the

0 0

> 2 Q

sum of the positive roots. We will use the partial ordering iff +

0 0 0

h > > 6= =\Q 2 Q + on and write if but .Weset + .If ,we

j =( ; ) jAj

write j . We will also use the notation to denote the cardinality of

A n S

aset . The group of permutations of letters is denoted by n .

2. The functions

We introduce a space of meromorphic functions on h that is preserved by the

2 h

differential operator H .For ,let

i ()

X ()=

e(4)

X = X = A h

i i

and write if .Let be the algebra of functions on which can be

(X ;::: ;X ) 2 C

represented as meromorphic functions of 1 r with poles belonging

[ fX = g

to set 1 . For instance, if is a positive root, the functions

X + X

1 4 1

; ( ()) = i ;

2 = 2 cot (5)

( ()) ( X ) X

sin 1 1

A A A = C [[ X ;::: ;X ]]

belong to . The algebra is a subalgebra of 1 r

^ ^

A A( ) 2 h A A( )

If , we introduce the -module and the -module of functions

i )f f 2 A f 2 A

of the form exp(2 where ,resp. . These modules are preserved

H A( ) U [ ]

by derivatives with respect to . Therefore preserves the spaces 0 ,

( ) U [ ]

A 0 .

2 A( ) U [ ]

Thus any 0 has the form

()= X () ;

2 U [ ] 62 Q + with 0 vanishing if .

THREE FORMULAE FOR EIGENFUNCTIONS 147

2 h u 2 U [ ]

THEOREM 2.1. For generic , and any non-zero 0 , there exists a

= X 2A() U[ ] H = " " 2 C

unique 0, such that ,forsome , and

= u " =( ) (; ) 2 A( ) U [ ] such that 0 . Moreover, 2, and 0 .

The existence and uniqueness of as a formal power series was proved in the scalar case by Heckman and Opdam (see Section 3 of [8]), who generalized a classical construction of Harish-Chandra (see [9], IV.5). The fact that the series converges to a meromorphic function seems to be new.

We next prove this theorem except for the statement that the formal power series

A( ) U [ ] actually belongs to 0 , which will follow from the explicit formulas

for given below.

= "

Proof. The idea is that the eigenvalue equation H is a recursion relation

for the coef®cients . We use the fact that and give the same contribution

H 4X ()=( i) ( ;

to the sum in , to apply (5). With the formula 2

)X () H = "

, we see that is equivalent to the recursion relation

X X

e e ; [( ; ) ( ) "] = j

2 2 (6)

0 +

2 U [ ]

for the coef®cients 0 . The sum on the right-hand side has only ®nitely

= u 6= many nonzero terms. The initial condition for this recursion is 0 0. For

= (; ) ( ) " = 0, the equation reads then 20. Thus there is a solution only

" =( ) (; ) 6=

if 2. For generic , the coef®cient of does not vanish if 0, so

(6) gives in terms of with . We conclude that, for generic , (6) has

=( ) (; ) 2

a solution if and only if " 2, and this solution is unique.

( ) 2 A( ) U [ ] DEFINITION. We denote by E the vector space of functions 0

=( ) (; )

such that H 2.

( ) Remark 1. It looks as if the de®nition of E should depend on the choice

of simple roots, but this is not so. From the explicit expressions given below it

E ( )

follows that, for any set R of simple roots, the functions in can be analytically

A ( ) U [ ] A ( ) A( ) R continued to functions in R 0 ,where is the space de®ned

using R (see Section 6).

r D Remark 2. H is part of a commutative -dimensional algebra of Weyl-

invariant differential operators whose symbols are (Weyl-invariant) polynomials

( ())

on h . These operators have coef®cients which are polynomials in cot ,

2 A( ) D

+ . It follows that is preserved by , see (5), and that the functions

L 2 D

of the theorem are common eigenfunctions of all operators in D . Indeed if

L E ( ) H 2 E ( )

with symbol P ,then preserves since it commutes with .If

() i ()

2i 2

u 2 U [ ] L P ( i ) u

has leading term e u, 0,then has leading term 2 e .

= P ( i ) Thus L 2 .

148 GIOVANNI FELDER AND ALEXANDER VARCHENKO

U g

Remark 3. If g is a general Kac-Moody Lie algebra and is a -module with

[ ] H

®nite dimensional zero-weight space U 0 , is still a well-de®ned endomorphism

A ( ) U [ ]

of 0 : the (possibly in®nite) sum over + gives only ®nitely many

( ()) = O (X )

contributions in each ®xed degree, as sin , see (5). The above

2 A( ) U [ ] theorem holds (except for the statement that 0 ) with the same proof.

Etingof and Kirillov [3] gave a representation theoretic construction of

u 2 U [ ] M

eigenfunctions: given generic and 0 ,let be the Verma module

with highest weight . Then there exists a unique homomorphism u

v 2 M (M ;M U)

Hom , such that the generating vector is

v u +

mapped to , up to terms whose ®rst factor is of lower weight.

2 A( ) U [ ]

Then 0 is the ratio of formal power series

( i) u

tr M exp 2

()= :

u (7)

( i)

tr M exp 2

The traces are formal power series whose coef®cients are traces over the ®nite

dimensional weight spaces of the Verma modules. By de®nition, the trace of a map

! M U M

M is the canonical map (for ®nite dimensional )

(M; M U ) ' M M U ! U: C

tr M :Hom

^ ^

A( ) U [ A()

The numerator in (7) belongs to 0], and the denominator to

( ) U [ ] with leading coef®cient 1, so the ratio is a well-de®ned element of A 0 .

Combining this result with Theorem 2.1, we get

2 h u ! ( )

PROPOSITION 2.2. For any generic ,themap u de®ned by 7 is

[ ] E ( ) an isomorphism from U 0 to .

Remark. Etingof and Styrkas [6] showed that (7) viewed as a function of ,

coincides the Chalykh±Veselov -function (see [2], and [6] for the matrix case considered in this paper), which is de®ned in terms of its behavior as a function

of .

3. The ®rst formula

Our ®rst formula is an explicit calculation of the trace of the homomorphism

M ! M U

: of the previous section. The image of the generating vector

e 2 +

is a singular vector of weight (a vector of weight killed by , )

and all singular vectors of weight correspond to some homomorphism. Let

f = e i = ;::: ;r f = f ::: f I =(i :::: ;i )

I i i m

i , 1 and set for a multiindex . m

i 1 1

v u + f v u M U

I I

Let be a singular vector in of weight

. Our formula gives an eigenfunction in terms of the coef®cients I .Letus

remark that, for generic , such a singular vector is uniquely determined by its ®rst

u 2 U [ ] u

coef®cient 0 : the coef®cients I can be given rather explicitly in terms of

u and the inverse Shapovalov matrix (see [6]).

THREE FORMULAE FOR EIGENFUNCTIONS 149

2 h v u + f v u

L L L

THEOREM 3.1. Let and let be a singular vector

()

M U ()= (u + A ()u )

L L L

of weight in . Then the function e,

with

1 0

a +

j 1

X Y

(j)

A @

f ; A ()= ::: f

l l

l ;:::;l )

( (8)

( (p)

1 1)

X ::: X l

( (j)

1) j=

p 1

a m 2fj;:::;p g (m + ) <

where j is the cardinality of the set of 1such that 1

(m) E ( ) E ( )

, belongs to , and for generic all functions in are of this form.

= U U [ ] U

EXAMPLE Let g sl2.If is irreducible, 0 is one-dimensional if has

s + odd dimension and is zero otherwise. Let U bea2 1-dimensional irreducible

representation. Our formula reduces to

()

; (X = X u + u ()= ):

el 1 X

1 =

l 1 u

The components l of the singular vector are easily computed, and we get the

formula

s + l) ((; ) l) X

( !

() l

()= ( ) u:

e1 (9)

(s l) ((; )) X

l!! 1 = l 0 We will prove the more general quantum version of Theorem 3.1 in Section 8.

Note that Theorem 3.1 completes the proof of Theorem 2.1: the coef®cients are

fX = g 2 Q

meromorphic functions. They seem to have poles on 1 for general ;

however, these poles cancel, since the differential equation is regular there:

2 C

LEMMA 3.2. Let " and suppose that is a meromorphic solution of the

= " h

differential equation H on , whose poles belong to the union of the

H = f j ()=mg 2Q m2Z

hyperplanes ;m ,,.Then is regular except

H 2 m 2 Z

possibly on the hyperplanes ;m , with , .

2 Q m 2 Z

Proof.Let , , and choose a system of af®ne coordinates

z ;::: ;z h z = () m p> H

r r

1 on so that .If has a pole of order 0on ;m ,

H = z ;::: ;z )+ f (z

r then, in the vicinity of a generic point of ;m ,

r 1 1

f H z !

with nonvanishing regular . The leading term of as r 0 comes from

H p(p +

the Laplacian, as the potential term is regular on ;m and is equal to

)( ; )z p > H = "

1 f which is nonzero for 0. If , then it follows that

p +

has a pole of order 2, a contradiction. We have shown that is regular on

H V h

generic points of ;m. Therefore, on any bounded open subset of , the product

() l 2 l 2 Z of by a suitable ®nite product of factors , , is holomorphic

on the complement of a set of codimension 2, and therefore everywhere on V by

Hartogs' theorem. 2

This concludes the proof of Theorem 2.1.

4. The second formula Our second formula is an integral representation. It is obtained as the trigonometric limit of the integral representation of solutions of the Knizhnik±Zamolodchikov±

Bernard (KZB) equations on elliptic curves with one marked point.

n T ;::: ;T

One ingredient in the integrand is the rational function of 2 variables 1 n,

Y ;::: ;Y

Y ::: Y

11j 1

; T W(T; y)= = : +

n 1:1

T T Y ::: Y T

j+ j j

j 111 = j 1

Now suppose that U is an irreducible highest weight representation with non-

U Q

trivial zero-weight space. The highest weight of is then in + , i.e., of the form

= n n n = n

j j j j j

j , for some non-negative integers .Set . It is convenient to

f ;::: ;ng ! f ;::: ;rg introduce the associated `color' function c: 1 1, the unique

jc (fj g)j = n j = ;::: ;n

non-decreasing function with j , 1 . For each permutation

2 S W n + r T ;::: ;T ;X ;::: ;X

;c n r

n ,let be the rational function of variables 1 1

;::: ;T ;X ;::: ;X ): W (T; X)=W(T

) (n) c(( )) c((n)) (

11 (10)

T ;::: ;T

The other ingredient is the many-valued function of 1 n

Y Y

( ; )=

c(j )

( ; )= (; )=

c(j) c(l) c(j )

(T )= (T T ) T (T ) :

j j

l 1

;

j =

(T )W (T; X) T ::: T

;c n

We will consider integrals of ddover connected com-

; 1

ft 2 ( ; ) j t 6= t ; (i 6= j )g j

ponents of 0 1 i . These `hypergeometric integrals'

T T j

are de®ned as (meromorphic) analytic continuation in the exponents of i ,

T T j

j , 1in from a region in which the integral converges absolutely. We

; say that a hypergeometric integral exists if this analytic continuation is ®nite at the given value of the exponents.

THEOREM 4.1. Suppose that U is an irreducible highest weight module with

= n v n =n

j j j highest weight j and highest weight vector .Set and let

f ;::: ;ng!f ;::: ;rg c fj g

c: 1 1be the unique nondecreasing function such that

n j = ;::: ;r 2 h

has j elements, for all 1 . Fix a generic complex number and

ft 2 ( ; ) j t 6= t ; (i 6= j also generic. Then, for each connected component of 0 1 i

THREE FORMULAE FOR EIGENFUNCTIONS 151

j , the integral

()

() = (T )

;

W (T; X()) T ::: T f ::: f v ;

;c n

(( )) c((n))

d 1 dc 1(11)

( )

exists and de®nes a function in E . Moreover, for each generic and ,all

( ) functions in E can be represented in this way.

In the rest of this section we prove this theorem. The existence of the integral follows from [17], Theorem 10.7.12. Indeed, the coef®cients in (11) are linear combinations of integrals considered in [17] in the context of the Knizhnik±Zamolodchikov

equation with two points.

()

X ;::: ;X r

Clearly is of the form e times a meromorphic function of 1

X = 2 Q E ( )

with poles on the divisors 1, . To prove that belongs to it is

therefore suf®cient, thanks to Lemma 3.2, to show that is an eigenfunction of

H .

This follows from the results of [7], which we now recall. 1

= ( i ) (x)= (x)

Let us de®ne, for q exp 2 in the open unit disk, sin 1

j 0

j 2

q (x)+q ) (x)= (x)= ( ) v (x)= (1 2 cos (in the notation of [7], 11 0 )and

= x (x) v

ddln . The function is doubly periodic with periods 1 and . Then for

2 C f g

any 0, the KZB equation is a partial differential equation for a function

(; ) h f 2 C j > g U [ ]

u on Im 0 , with values in 0 :

v( ())e e u: i = 4u

4 (12)

2 2

! v (x) ! (x)

As q 0, sin , and the differential operator on the right-hand H side converges to .

PROPOSITION 4.2. Suppose that u is a meromorphic solution of the KZB equa-

) q u q jq j <

tion (12 such that is a meromorphic function of and , 1, and, as !

q 0,

a=2

u(; )=q ( ()+O(q)):

H = " " =( ) a Then obeys , with 2.

The proof consists of comparing the leading coef®cients in the expansion of both

q = sides of the KZB equation in powers of q at 0. A source of solutions of (12) with the property described in the Proposition is [7]. In that paper, we gave integral formulas for solutions.

A class of solutions with asymptotic behavior as above is constructed as follows

(see [7] for more details). Let U be an irreducible highest weight module with high-

2 Q c f ;::: ;ng! f ;::: ;rg

est weight + , and let as above : 1 1the nondecreasing

function associated to .

Solutions are labeled by an element of and a connected component of

ft 2 ( ; ) j t 6= t ; (i 6= j )g j

0 1 i .Let be a choice of branch over of the

;

many-valued function Y

i(;)=+ i(;+ t ) ( ; )=

j j

c(j) (i) c(j )

2 c

(t t ) j

e i

( ;)=

c(j )

(t )

j (13) =

j 1

w t ;::: ;t y ;::: ;y n

and let be the meromorphic function of 1 n,1

(y + + y t + t )

j j+

1j 1

w(t; y )=

(y + + y )(t t )

j j+

1j 1 =

j 1

2 S w n + r

and for each permutation n ,let be the meromorphic function of

t ;::: ;t ;x ;::: ;x r

variables 1 n 1

w (t; x)=w(t ;::: ;t ;x ;::: ;x ):

( (n) c(( )) c((n))

1) 1 (14)

U [ ] 2 h

Consider the differential n-form with values in 0 depending on ,

! (; )= w (t; ();::: ; ()) t ::: dt f ::: f v :

2S ;c r n

c(( c((n)) d )) n 1 1 1

Then the integral

(t)!(; )

;

exists (as analytic continuation in the exponents from a region of absolute conver-

! i1

gence) and is a solution of the KZB equation. As ,

(x)= (x)= + O (q ); sin

and the function (13) behaves as

;)= i ()

( 22

[ (T )+O(q)];

q e

;

T = ( it ) X = ( i ())

j j j in terms of the exponential variables j exp 2 .Let exp 2 .

The components of the differential form ! behave as

w (t; x) t ::: t = W (T; X) T ::: T + O(q):

;c n ;c n d 1 dd1d

H (C ( ); L( ))

Therefore, the solutions corresponding to cycles in n have the prop-

=(; ) erties of the Proposition, with a ,and is as stated in the claim of Theorem

4.1. We obtain in the limit an integral representation for eigenfunctions as in the

< (T ) < <

( )

Theorem but with an integration domain of the form 0 arg 1

(T ) < 2 S

(n) arg 2 for some . This integration domain may be deformed, so

that the corresponding integral can be written as linear combination of integrals

> T 0 > > T 0 >

( ) (n)

over the domains de®ned by 1 1 0. If we choose

= ia ( ; ) > j a

0,forsome 0 such that 0 j 0forall ,andlet tend to in®nity, then

the integral over is equal to the integral over (for suitable choice of branch

and orientation) plus terms that tend to zero. It follows that, for generic and all

2 S

n , the integral over can be expressed as a linear combination of integrals

over and is therefore also an eigenfunction.

This completes the proof of the ®rst part of Theorem 4.1.

( ) We must still prove that, in the generic case, all eigenfunctions in E admit

such an integral representation. In view of Theorem 2.1, it is suf®cient to show that

i ()

u 2 U [ ] e ! u X ! j

for every 0 , there exists a cycle such that as 0

= ;::: ;r U [ ]

j 1 . In other words, one must show that all vectors in 0 are of the form

0 1

n 1 Y

X 1 1

@ A

(T )

;

T T T

) (j ) (j + (n)

1 1

2S j =

n 1

f ::: f v T ::: T

(( )) c((n)) c 1 d 1 d

for some cycle . But this follows from the results in [17], Theorem 12.5.5.

5. The third formula This is a formula of the Bethe ansatz type.

THEOREM 5.1. Suppose U is an irreducible highest weight module with high-

= n v n = n

j j j est weight j and highest weight vector .Set and let

f ;::: ;ng!f ;::: ;rg c fj g

c: 1 1be the unique nondecreasing function such that

n j = ::: r T 2 C

has j elements, for all 1, ,. Then the function parametrized by

()

W (T; X())f (T; )= ::: f v ;

c( ( c((n))

e1)) (15)

( ) W E ( ) T ::: T

;c n (see 10 for the de®nition of ) belongs to if the parameters 1, ,

are a solution of the set of n algebraic equations (`Bethe ansatz equations')

0 1

c(l )

@ A

; = ; j = ;::: ;n:

(j)

c 01

T T T T

j j j

l 1

l 6=j :l

Remark. These Bethe ansatz equations are the same as the Bethe ansatz equations of a special case of the Gaudin model (cf. [14]). The solutions are the critical points

of the function

Y Y

( ; )

c(j )

( ; ) (; )

c(i) c(j ) c(j )

(T T ) T (T ) :

j j

i 1

6i6j 6n = 1 j 1

Theorem 5.1 can be understood intuitively as a consequence of Theorem 4.1: one

calculates the integrals, which up to normalization are independent of ,inthe ! limit 0, using the saddle point method. The proof of this theorem can be taken directly from [7] in the case of a

degenerate elliptic curve. =

The above result motivates the notion of Hermite±Bethe variety. Let

n n > j = ;::: ;r

j j j

j . Without loss of generality, we may take 0, for all 1 (if

f ;::: ;ng !

this condition is not ful®lled, we may pass to a subalgebra). If c: 1

f ;::: ;rg S

1is the associated nondecreasing function, we let c be the product of

S S C T

n j symmetric groups n . It acts on by permutation of the variables

1 r

(j )

with same c and is a symmetry group of the system of Bethe ansatz equations.

2 S

Moreover, we have, for all c ,

(T ; )= (T; ):

B (T )=

Let us write the Bethe ansatz equations as j 0, with

0 1

c(l)

@ A

B (T )= ; :

c(j )

T T T

j j

l 1

l6=j l:

Subtracting pairs of equations we may eliminate :

F =(C f ; g) [ fT j T = T g

n i j

( ; )6=

DEFINITION. Let 01i

(i) c(j )

: c 0

B (c)

Hermite±Bethe variety H is

HB (c) = fT 2 F j B (T )

n j

= B (T );j 2f ;::: ;ngfn ;n + n ;::: ;ngg=S :

+ c

j 1 1112

HB (c) ! h T 7! The remaining equations de®ne a regular map p: , .The

completeness hypothesis of Bethe states is in this case:

[ ]; [ ] [ ]) p

CONJECTURE 5.2. (cf. 18 13 and conjectures in 14 .Themap has dense

U [ ]) image, and the generic ®ber consists of (at least) dim( 0 points. The eigenfunc-

( ) E ( ) tions corresponding to points in p span .

In the case of sl2 the conjecture holds and goes back to Hermite, see [19]. We prove this conjecture in two special situations, including the scalar case

relevant for many-body systems.

g = U =

THEOREM 5.3. Conjecture 5.2 holds in the following cases: (a) slN ,

pN N

p = ; ;::: ; ( ( )) (b) g = S C ;U =

, 1 2 see 2 slN adjoint representation. N

Proof. The proof is by induction in ,viewingslN as a Lie subalgebra of

;::: ; ;::: ;

+ N N N

slN 1. We choose simple roots 1 of sl in such a way that 1 1

are simple roots of slN 1. Also, it is convenient to replace by .

N N

pN 1

U = S C =p j U [ ]

j N

In case (a), the highest weight of N is ,and 0 =

j 1

= + is one-dimensional. The ®ber over of the Hermite±Bethe variety consists

of the critical points of

Y Y

(; )

c(j)

( ; ) (; )

c(j) c(l) c(j )

(; T )= (T T ) T ( T )

N j j

l 1

T =(T ;::: ;T ) c(j )=m pm(m

(N + )=

viewed as a function of 1pN 1 2 .Wehave if

1 ) 2 1 2, 1 . We prove inductively that, for generic

, N has a nondegenerate critical point, and that the corresponding eigenfunction

E ( + ) (; T ) ( ;)="

+ N

spans . Let us consider critical points of N 1 when

( ;) j < N

tends to in®nity, and j ,are kept ®xed. Let denote the orthogonal

T j

projection of onto the Cartan subalgebra of slN . Let us replace the coordinates

j c(j )=N a T = "a

j j

indexed by such that by new coordinates j de®ned by 1

c(j )=N T =(T ;::: ;T )

pN (N + = ( ), and let 11) 2 denote the remaining coordinates.

Then

p(p+ )N pN

" (; T ) = (; T ) ( T )

+ N j

N 1 1

(j)=N

c 1 Y

+ "a ) (T j

1 l

;c(l )=N (j )=N

c 1

p(N + ) ="

1 1

"a ) a (

1 l

c(l )=N

Y 2

(a a ) : j

l (16)

This function converges as " 0 to a constant times

Y Y

) p(N+

a 1

(a a ) (; T ) a :

j N

156 GIOVANNI FELDER AND ALEXANDER VARCHENKO

a a

The factor depending on the j 's has a nondegenerate critical point in the domain

a 6= a 6= ; (j 6= k ) j

k 0 . This follows from the fact that this factor is obtained as a !1

limit M of

Y Y

p(N + ) M

1 2

( a =M ) a (a a ) :

j j

1 l

j j

The critical points of the latter function are known explicitly (see (1.3.1) and !1

(1.4.2) in [18]), and it can be easily checked that they have a limit as M in

a 6= a 6= j

the domain k 0, which is therefore a critical point of the limiting function. a

In fact, is the set of zeros of the polynomial solutions of the differential equation

00 0

+(x m)y Ny = m = p(N + )

xy 0, where 1 . By the induction hypothesis,

(; T ) T

N has a nondegenerate critical point . Moreover, in a neighborhood of

(a ; T ;" = )

0, the right-hand side of (16) is holomorphic, which implies that the = nondegenerate critical point at " 0 deforms to a nondegenerate critical point for

generic ".

It remains to show that the corresponding eigenfunction generically spans the

( + )

one-dimensional vector space E , i.e., that it does not vanish. Assuming this

" !

inductively to hold for slN , we see that as 0, the leading contribution in the sum

( (j )) = N c(j )=N

(15) is given by permutations such that c , whenever .These

f ::: f (f ) v

( ( )) c((pN (N + )= ))

permutations give terms proportional to c 1 1 2 ,for

2 S " ! T (")

pN (N + =

some 1) 2. It follows that when 0and is the above family of

i(;) i(;)

2 2

(T (");) C (a ) (T ; )

critical points, e converges to e ,

+ N

sl N 1 sl

pN N

U = S C

N N

where the representation N of sl is viewed as the sl -submodule of

U (f ) v

+ N N 1 generated by the singular vector . Using the identity

X 1 1

; =

(a a ::: a a ) (a a )a

( ( ) (N ) (N) (N)

1) 2 11

C (a )

we see that is, up to a trivial nonzero factor, the inverse of the product of the

a a components j of , and therefore nonzero. This completes the proof of part (a) of the theorem.

The case (b) is treated in a similar way. The highest weight of the adjoint

U = c =

N i N

representation N of sl is and Id. The function for sl is

2N 1

Y Y

(; )

11 1

(; T )= (T T ) T ( T ) ( T ) :

j j+ N N 11

1j 1 1

= j=

j 11

" = (; ) T = "a T = N

As before we let N go to in®nity. If we set 1 ,

;::: ;T ) = (T

1 N 1,projection of onto the Cartan subalgebra of sl ,then

1 1

" (; T ) = (; T ) ( "a) a

+ N

N 1 1

T + "a

N 11

a !

" 0 e

! (; T ) :

N a

THREE FORMULAE FOR EIGENFUNCTIONS 157

T a = Proceeding as before, we see that nondegenerate critical points , 1ofthe

limiting function deform to nondegenerate critical points for any generic ".the

corresponding eigenfunctions deform to eigenfunctions of sl N taking values in

U [ ] U [ ]

N + N N +

N 0 , viewed as a subspace of 1 0 via the inclusion of sl in sl 1.Inthis

way we get N 1 linearly independent eigenfunctions in the -dimensional space

E ( + ) +

associated to the adjoint representation of slN 1. To ®nd the remaining

T = "a j " ! "

j N + eigenfunction, we let j 1 for all .Thenas 0, 1 for suitable

m converges to

a 1 N e Y

(a a )

+ j

j 1

a a

1 N 1 =

j 1

a = j ( + =N )

The critical point of this function can be computed explicitly: j 1 1 .It

deforms to a nondegenerate critical point for " generic. The corresponding eigen- !

function converges, as " 0, to a constant function. From its explicit expression

U = U ( )f v

N N

it is clear that its value is not in slN and is therefore linearly

2 independent from the N 1 constructed before.

Remark. The proof of the preceding theorem indicates, at least in the examples

g ;U)

considered, that the construction which to a pair ( consisting of a semi- g simple Lie algebra g and a ®nite dimensional -module associates the closure

of the algebraic Bethe-Hermite variety X is functorial: to each homomorphism

0 0

g ;U) ! (g ;U )

( preserving the Lie algebra and module structures there corre-

! X sponds functorially a morphism of algebraic varieties X . This functor is compatible with the construction of eigenfunctions in a sense that should be made more precise.

6. Weyl group action, Jacobi polynomials, scattering matrices In this section we study the Weyl group action on eigenfunctions, and discuss the

relation with multivariable Jacobi polynomials [8]. The Weyl groups W acts on

[ ] N T = h

U 0 (The normalizer of the Cartan torus exp of the simply connected

U W = N=T U [ ]

group with Lie algebra g acts on ,so acts on 0 ). Therefore we

U [ ] w 2 W (w )()= have a natural action of W on 0 -valued functions: acts as

w (w ) H

. The SchrodingerÈ operator commutes with this action.

S 2 h (; ) = ( ; ) 2 Q

Let be the set of such that for some + .If

2 h S E ( ) U [ ] , is isomorphic to 0 and we have the explicit expression of the isomorphism in Theorem 3.1.

i (w )

2 h S 2 E ( ) w 2 W w

LEMMA 6.1. Let .If and ,thene is a

X ;::: ;X C

rational function of 1 r which is holomorphic on the complement in of

X = 2

the root hypersurfaces 1, . = Proof. Suppose ®rst that w 1. From the explicit expression (8) it is clear that it is a rational function. By Lemma 3.2, the poles lie on the root hypersurfaces.

158 GIOVANNI FELDER AND ALEXANDER VARCHENKO

w = s

Let us assume that k is a simple re¯ection corresponding to the simple root

k . As root hypersurfaces are permuted under the action of the Weyl group, it

i (w )

w X = j

suf®ces to show that e is regular at j 0forall . This is obvious

j 6= k s X X X

j j

if since replacing by k amounts to replacing by , with k

a 6 j 6= k j = k X X k

jk 0if .If , is replaced by and we have to check that the

X !1

expression in parenthesis in (8) is regular when k , but this follows easily X

by counting powers of k in the numerator and denominator, exploiting the fact

a + 6 p j + w

that j 1 1. The case of general is reduced to this case by writing 2

w as a product of simple re¯ections.

2 h w 2 W 7! w

LEMMA 6.2. Let be generic. For all , is an isomorphism

( ) E (w ) from E onto .

i (w )

2 E ( ) w

Proof.Let . By the previous Lemma, e is holomorphic

= H w at X 0. Also, since is invariant, is an eigenfunction with eigenvalue

(; ) w 2 E (w ) 2

4 and therefore .

= U s +

EXAMPLE. Let g sl2,and be the 2 1-dimensional irreducible represen-

( ) U [ ] u 2 U [ ]

tation. The Weyl re¯ection s1 acts as 1 on 0 . Fix a nonzero 0 and

(s )() =( ) ()=S() ()

let be the eigenfunction (9). Then 1 1,

( ) X ! where S can be computed in the limit 0. This gives the `two particle

scattering matrix'

s s

X Y

s+l) ((; ) l) k +( ; )

( !

( )=( ) = :

S 1(17)

(s l) ((; )) k ( ; )

l!!

= k =

l 01

w = "(w) w 2 W

We say that a function is Weyl antiinvariant if for all .

[ ])

PROPOSITION 6.3. (cf. Proposition 3 in 7 Let be a meromorphic Weyl anti-

= "

invariant solution of the eigenvalue problem H , regular on

[ [

h f 2 h j ()=mg;

m2Z

( + p)= (p) () p P = fp 2 h j (p) 2 Zg

such that for all in the lattice

P h

and some character of ,then extends to a holomorphic function on .

2 m 2 Z

Moreover, for all and ,

l +

l 1

= O (( () m) )

e (18)

() ! m

as . _

Proof. By periodicity with respect to the coweight lattice P , we may limit our considerations to the hyperplanes through the origin. One proceeds as in Lemma 3.2 by approaching the singular hyperplane in a transversal direction. The leading

THREE FORMULAE FOR EIGENFUNCTIONS 159

= () term in the Laurent expansion in the transversal coordinate x is determined by the differential equation

d e e + e e

; ) = :

( 2 0 2 0 0

dx x

Let us decompose U into irreducible representations of the subalgebra generated

e e e + e e

by .Since is of zero weight, we may replace by

h h + e e + e e ; C =

( ; )

h = [e ;e ] l +

where .If 0 belongs to a 2 1 dimensional irreducible repre-

l +

l 1

e 6= e = C

sentation, i.e., if 0but 0, then the Casimir element acts as

+ l

l 1

; )l (l + ) x x

( 1 . Therefore either 0 or 0 . On the other hand, the

= x

Weyl re¯ection with respect to the hyperplane 0 changes the sign of and

)

multiplies the value by ( 1 .Since is antiinvariant, it follows that the ®rst

+ 2

possibility is realized and the function vanishes to order l 1. P

In particular, if belongs to the weight lattice , spanned by the fundamental

! ;::: ;! 2 E ( ) =

weights 1 r,and , then the Weyl antiinvariant function

"(w )w X ;::: ;X (C f g)

w 2W ! is a rational function of ! regular on 0.

1 r

Therefore, is a Laurent polynomial. To apply the previous lemma, should not

be in S .Thisistrueif is dominant, and we have the following result.

= 2

COROLLARY 6.4. Let where is a dominant integral weight and

E ( ) = "(w)w X ;::: ;X

2W ! ! .Then w is a Laurent polynomial in .

1 r

pN N

g = U = S C ! ;::: ;!

EXAMPLE. Let slN , , the scalar case, and let 1 1

U [ ] C

be the fundamental weights of slN . Let us ®x an identi®cation of 0 with .

Let be the eigenfunction of the previous corollary, normalized so that

i()

+ )

e (1 . By the previous corollary, it is a Laurent polynomial in the X

! . Then the vanishing property (18) of Proposition 6.3 implies that the Weyl

W 1 1

C [X ]

antiinvariant eigenfunction is divisible in the ring by where

=X ( X ):

+ +

p 1

P = = The ratio is a Weyl invariant polynomial (since is antiinvariant)

and obeys the differential equation

4P +(p+ ) ( ())@ P = "P ;

12cot (19)

2 +

@ " = ( +(p + );

( is the derivative in the direction of ), with 4 1

)) p +

( 1 . This equation de®nes multivariable Jacobi polynomials (or Jack poly-

nomials) associated to a dominant integral weight : The Jacobi polynomial

1 1

[X ;::: ;X ]

is the unique solution in C of (19), normalized in such a way

! !

P 1 1

P = X + a 2 C a X

that for some coef®cients . The leading

+ +

p 1

= X

+(p+

term of is 1) . After antisymmetrization, we obtain a leading

cX w

(+(p+ ))

term w0 1 where 0 is the longest element of the Weyl group. Thus we

P = w (p + ) w

obtain a function proportional to by choosing 0 1 ( 0 is an

involution). Therefore we have the corollary:

pN N

g = U = S C

COROLLARY 6.5. In the scalar case slN , , with ®xed identi®-

[ ] C 2 h S

cation of U 0 with ,let be a dominant integral weight, and for any ,

E ( ) : u =

let denote the eigenfunction in given in Theorem 3 1 with 1 and

= "(w)w

w2W

its antisymmetrization. Then

w (p+ )

0 1

P = c ;

p 1

c 6= w

for some constant 0.Here 0 denotes the longest element of the Weyl group,

j 7! N + j S

i.e., the permutation 1 of N . c

Aformulafor is given below.

We next study more closely the action of the Weyl group.

(j ) g e g U

Let sl be the subalgebra of generated by ,andfora -module ,

2 j

(j )

U = U U (j ) s

let s be the decomposition of viewed as sl2 -module into isotypic

(j )

U s +

components: s is isomorphic to a direct sum of 2 1-dimensional irreducible

j )

sl2 ( -modules.

j ( ) U [ ] !

THEOREM 6.6. Let be generic and denote by the isomorphism 0

()

( ) u u

E mapping to the eigenfunction with leading term e . Then there exists

S ( ) W ! (U [ ]) w 7! S ( ) = j ( )u

a family of maps : GL 0 , w such that if ,then

w = j (w )S ( )u

w . These maps have the composition property

S ( )=S (w )S ():

w w w

w1 2 122

w = s S = S

w j

If j is a simple re¯ection then has the form

Y X

k+( ;)

(j) j

(j)

S ( )=P + P ;

j (20)

k ( ;)

s> =

1k 1

(j ) (j )

P 2 (U [ ]) U \ U [ ] s where s End 0 is the projection onto 0 . Proof. The ®rst part of this theorem is just a rephrasing of the preceding lemma.

The composition property follows from the de®nition:

j (w w )S ( ) = w w j ( ) w

1 2 w1 2 1 2

= w j (w )S ( )

1 2 w2

= j (w w )S (w )S ( ): w 1 2 w1 2 2

THREE FORMULAE FOR EIGENFUNCTIONS 161

w = s S X l 6= j

j j

If , we may compute by letting all l , go to zero. Then the

A L = (j;j;:::;j)

coef®cients L in Theorem 3.1 tend to zero except if .andthe

i ()) formula for exp( 2 reduces to an sl2 formula, and we can apply the

previous example in each isotypic component. 2

Remark. Note the similarity between this scattering matrix and the rational

R -matrix associated with general representations of sl2 [11].

g = h C =C ( ;: :: ; )

EXAMPLE. Let slN , and identify with 1 1. The Weyl

S s C j

group is the symmetric group N and the simple re¯ections act on by

j j + S

transposition of the th and 1st coordinates. The corresponding j depends

( ;)= s s s = s s s

j j+ j j + j j + j j + only on j 1. The relations 1 1 1 translate into

`unitarity'

S ( )S ( )= ;

j + j j j j + j 1 1 Id

and the Yang±Baxter equation

S ( )S ( )S ( )

j + j + j + j j + j j j +

j 1 2 1 2 1

( )S ( )S ( ): = S

j j + j j j + j + j + j + +

j 1 1 2 1 1 2

pN N

(g = U = S C )

COROLLARY 6.7. In the scalar case slN ,

k+( ;)

S ( )= :

k ( ;)

j =

k 1

(j ) U

Proof. The only isotypic component s with non-trivial intersection with

[ ] s = p 2

U 0 has in this case. c

Let us conclude with a formula for the constant in Corollary 6.5. This constant W

can be computed from the leading term in which appears in the summand

w c = "(w )S (w ( +(p+ ))) 2 h

indexed by : w 1. If we identify with

0 0 0 0

;::: ;

the diagonal traceless matrix with diagonal entries 1 N, a simple calculation

gives

Y Y

k + (p + )(i j ) j i 1

= :

c (21)

k + +(p+ )(i j ) j

i 1

i>j =

k 0

< j This constant is clearly different from zero since for dominant we have i

if i>j.

7. Bethe ansatz in the elliptic case

E C = Z + Z

Let us ®x in the upper half-plane, and denote the torus .We®rst

quote the result of [7] on eigenfunctions of the differential operator

H = 4 + v ( ())e e ;

[ ] h v v (x)= = x (x)

on U 0 -valued functions on . The elliptic function is ddln ,

1 j i

1 2j 2

(x) = (x) ( q (x)+q q = sin 1 1 2 cos ),e. It differs from the

Weierstrass }-function by a constant.

]) U

THEOREM 7.1. ([7 Suppose is an irreducible highest weight module with

= n v n =n

j j j highest weight j and highest weight vector .Set and let

f ;::: ;ng!f ;::: ;rg c fj g

c: 1 1be the unique nondecreasing function such that

n j = ;::: ;r t 2 C

has j elements, for all 1 . Then the function parametrized by

()

(t; ) = w (t; ();::: ; ())f

;c r

( ( ))

e 1 c 1

::: f v ;

((n))

c (22)

( ( ) w ) H t

;c e

see 14 for the de®nition of is an eigenfunction of if the parameters 1,

::: t n

,n are a solution of the set of equations (`Bethe ansatz equations')

0 1

0 0

(t t ) (t )

j j

@ A

= + i; ; j = ;::: ;n:

(l ) c(j )

c 20 1 (23)

(t t ) (t )

j j

l 6=j :l

The corresponding eigenvalue " is

2 @

" = (; ) i S(t ;::: ;t ;);

4 41n

( ; ) S(t ;::: ;t ;) = (t t )

m i j

(i) c(j)

1c ln

(; ) (t ):

(i)

c ln i

Remark. Solutions of the Bethe ansatz equations are critical points of

Y Y

i(; t ) ( ; ) (; )

(i) c(i) c(j ) c(i)

(t)= (t t ) (t ) ;

i j i

e e

(t) 6=

in the domain e 0. As in the trigonometric case, we de®ne the Hermite±Bethe variety by eliminating

the spectral parameter from the Bethe ansatz equations. The resulting equations

are the n equations

0 0

(t t ) (t t )

j j + l

l 1

c(l )

(t t ) (t t )

j j + l

l 1

l 6=j;j + l : 1

THREE FORMULAE FOR EIGENFUNCTIONS 163

0 0

(t ) (t )

j +

j 1

; = ;

(j)

c 0(24)

(t ) (t )

j +

j 1

c(j )=c(j+ )

where j runs over the set of indices such that 1. )

LEMMA 7.2. The left-hand side of each of the equations (24 is a doubly periodic t

function of each of its arguments i .

This lemma follows easily from the formulas

0 0 0 0

(t) (t + ) (t) (t + )

; = i;

= 2

(t + ) (t) (t + ) (t)

=n j

taking into account the zero weight condition j .

E = E

Therefore we may view the equations as algebraic equations on

S = S S E

c n n , a product of elliptic curves. Moreover, we have an action of

1 r

t c(i)

permuting i with the same , that maps solutions to solutions, and does not

D (c) t 2 E t = t i 6= j j

change .Let be the set of such that i for some with

( ; ) 6= t = i (; ) 6=

(i) c(j) c(i) c 0, or 0forsome with 0. On this set the Bethe

ansatz equations are singular.

n > j

DEFINITION. With the notation of the Theorem, assume that j 0forall .

HB (c) (E D (c))=S

The Hermite±Bethe variety is the subvariety of c de®ned

by the equations (24).

The remaining r equations determine as a function of a solution of (24).

= n ;n + n ;::: ;n

They can be chosen to be the equations (23) with j 1 12.We

t t + n + m j

see from this formula that if j is replaced by ,then is shifted by

(j )

c . It is easy to see that these replacements do not change the eigenfunction

HB (c) ! h =Q t HB (c)

. Therefore we have a map : mapping to ,and

(!) _

( + ! )= () ! 2 P parametrizes eigenfunctions such that e, . One would like to prove that `all' eigenfunctions are obtained in this way. In the general case not much is known, however in the scalar case and the case of the

adjoint representation we have the following result:

pN N

g = U = S C

THEOREM 7.3. Let slN , or the adjoint representation. Then

2 h (U [ ]) t 2 C

for each generic there are dim 0 solutions of the Bethe ansatz

(!)

) ( + ! )= ()

equations (23 . The corresponding eigenfunctions obey e,

2 P ! , and are linearly independent.

The proof of this theorem is essentially the same as in the trigonometric case (Theorem 5.3): in the case of sl2 one has Hermite's result, and one proceeds by

induction in N . The point is that one only has to use the asymptotic behavior of the theta functions involved, and this is the same as in the trigonometric case.

8. The q -deformed case

We give here the q -deformed version of our ®rst formula. Our conventions for

quantum groups are the following. Let q be a complex number different from 0, 1

q q a 2 C

or 1. We ®x a logarithm of ,sothat is de®ned for all . We normalize the

g ( ; ) 2 Z j; l 2f ;::: ;rg j

invariant bilinear form on in such a way that l ,forall 1.

U g

The Drinfeld±Jimbo quantum universal enveloping algebra q , a Hopf algebra,

C f e k

j j

is the algebra with unit over generated by j , , commuting elements ,and

; )

1 1 ( 1

k j = ;::: ;r k e k = q e k f k =

j j

l l

their inverses , 1 with relations l ,

j j j

( ; )

1 j

l 1

q f e f f e = (k k )=(q q )

j j j

l l jl

l , , and deformed Serre relations

s (q )= a = ;::: ; m

a 0, 1 2, (see [1]). The coproduct is de®ned on generators to be

11 11

(f )=f +k f (e )=e k + e (k )=k k

j j j j j j

j 1,1and .

j j j j

M U g

We consider modules over q admitting a weight decomposition into ®nite

( ;)

M [] 2 h k q

dimensional weight spaces , ,onwhich j acts as . In particular,

M = U g =I () q

the Verma module of weight is the quotient by the left ideal

( ;)

I () k q e j = ;::: ;r j

generated by j 1, , 1 . It is generated by its highest v

weight vector , the class of 1.

U U g U g q

If is a ®nite dimensional q -module, and a homomorphism of -

M ! M U

modules : , we may de®ne, as in the classical case,

()

[]

etr M

()

()= 2 C [[X ;::: ;X ]]: P

e 1 r (25)

()

[]

e tr M 1

i ()

2 j

X = q

Recall that j e . As in the classical case, we have for generic , ,

u U [ ] U g

and for each in the zero-weight space 0 of a ®nite dimensional q -module

U U g M ! M U

, a unique homomorphism of -modules : , sending the

v v u

generating vector to [4]. So, in the generic case, the -function is

u 2 U [ ] uniquely determined by q; and a vector 0 . It is a formal power series, but

its explicit expression below shows that it is actually a meromorphic function of .

g = U pN C

If slN and is a deformation of the th symmetric power of ,it

() was shown by Etingof and Kirillov [4] that is a common eigenfunction of

a commuting family of difference operators (related by conjugation by a known

A function to the N 1-Macdonald operators).

As in the classical case, the image of the generating vector is a singular vector

e i = ;::: ;r

(a vector killed by i , 1 ) of weight and all singular vectors

f = f ::: f

L l of weight correspond to some homomorphism. Set l for a

1 m

L = (l :::: ;l ) u 2 U [ ]

multiindex 1 m . Our formula gives, for each 0 , in terms u

of the coef®cients L in the formula of the the unique singular vector of the form

v u + f v u u

L L L

These coef®cients are given explicitly in terms of and

the inverse Shapovalov matrix, see [5].

2 h v u + f v u

L L L

THEOREM 8.1. Let be generic and let be a

M U 2 (M ;M U)

U g

singular vector of weight in .Let Hom q

THREE FORMULAE FOR EIGENFUNCTIONS 165

( ) () =

be the corresponding homomorphism, and the -function 25 .Then

()

(u + A ()u )

L L

e L , with

0 1

a ( )+

j 1

X Y

B (j ) C

a(L; )

A () = q

@ A

(l ;:::;l )

1 p 2

j ++ j

l l

( (j )

q X ::: X

2S j =

l l

p 1 1

( (j)

; f ::: f

l l

( (p)

a ( ) m 2fj;:::;p g (m + ) <

where j is the cardinality of the set of 1such that 1

(m) X = ( i ()) j

, j exp 2 , and

X X

) ; ( a(L; ) = ( ; )+

l l l

(k) (j )

j =

(k )> (j )

1 k

0 1

m m

X X X

@ A

+ ; ;

l l

(j) (j)

m2S j= =

1j 1

S = fm 2f gj(m)>(m+ )g[fpg: 1;::: ;p 11

We conclude this section by giving a conjectural formula for Macdonald poly-

q A

nomials, which is a -deformation of Corollary 6.5. The N 1 Macdonald poly-

P (x; q ; t)

nomials [12] are symmetric polynomials (we use the notation of [4])

N x ::: x q t

in variables 1, N, depending on parameters and . They are labeled by

> > N

dominant integral weights of glN , i.e., decreasing sequence 1 of

= q

non-negative integers. We consider the case where t for some positive integer

P (x; q ; q )

k as in [4]. The symmetric polynomials are uniquely characterized by

1 N

x ::: x x =x ! 1 i = ;::: ;N

i+ having leading term (as i , 1 1) with unit

1 N 1

f; g) =

coef®cient and by being orthogonal with respect to the inner product (

f (x ;::: ;x )g(x ;::: x )(x)

constant term of N ,where

1 1N

Y Y

(x)= ( q x =x ): j

6=j

i 0

P = x ::: x P

+( ;:::; )

Obviously 1 1 1 so it is suf®cient to evaluate Macdonald

x ::: x =

polynomials on the hypersurface 1 N 1. Then may be considered modulo

Z( ;:::; ) q

1 1, i.e., as slN dominant weight. The -analogue of Corollary 6.5 is the

following conjecture, which can be proved with the same method as in the classical =

case if N 2 or 3, but is open in the general case.

g = h = f 2 C j

CONJECTURE 8.2. Consider the scalar case slN , with

pN N i

2 j

= g U = S C U [ ] C x = j i 0 , , and ®x an identi®cation of 0 with .Set e .

166 GIOVANNI FELDER AND ALEXANDER VARCHENKO

2 h 2 h

Let be a dominant integral weight, and for ,let denote the =

function given in Theorem 8.1 with u 1. Then

(w)

w (p+ ) +

p 1 01

; P (x; q ; q )=c

(w)

w2S N

where

Y Y

i( ) m i( )

j j l

l 2

()= ( q );

e e

c 6= w

for some constant 0. Here 0 denotes the longest element of the Weyl group,

j 7! N + j S

i.e., the permutation 1 of N .

c q

As before, the constant may be computed in terms of the -analogue of the

scattering matrix (21)

x x

Y Y

q [m + (p + )(i j )] q

j q i 1

c = ; [x] = :

[m + +(p+ )(i j )] q q

j q

i 1

i>j = m 0

9. Proof of Theorems 3.1 and 8.1 We give the proof of Theorem 8.1. The proof of Theorem 3.1 is essentially a special case. The proof is in two parts. We ®rst prove that the calculation can be done in the algebra without Serre relation (Proposition 9.2), and then do this calculation, reducing it to a combinatorial problem.

U b U g f ;k j = ;::: ;r

q j

We denote by q the subalgebra of generated by ,1

U n f j = ;::: ;r (U n)

j q

and by q the subalgebra generated by , 1 .Wehave

U b U n Q (f )=

q j j

q . Let us introduce a -grading of these algebras by setting deg ,

(k ) =

deg j 0. What we need to know about the deformed Serre relations is that

U n f ;::: ;f J

r q

(i) q is the quotient of the free algebra generated by 1 by the ideal

s (q ) ::: s (q) s (q );::: ;s (q) m generated by the deformed Serre relations 1 , ,m , (ii) 1

f Z[q; q ]

are homogeneous polynomials in the j with coef®cients in reducing to

q = (s (q )) = s (q ) + K s (q )

a a

the (classical) Serre relations at 1, and (iii) a 1

= k

for some K . j

We will also need the fact that the dimensions of the spaces of ®xed degree

U n q

in q are independent of and coincide with the dimensions of the classical

U n n g f = e

enveloping algebra of the Lie subalgebra generated by j ,

= ;::: ;r j 1 (see, e.g., [1]).

The ®rst observation is that the computation of the trace, given the components

U n

of the singular vector is a computation in q :

THREE FORMULAE FOR EIGENFUNCTIONS 167

w 2 M (w ) M ! M U n

LEMMA 9.1. Let for , : be the unique

U n v w

homomorphism of -modules mapping to 1. Then, in the notation of

Theorem 8.1,

() n

(f v )

[]

etr M

A ()= :

()

[]

e tr M 1

This Lemma follows immediately from the fact that Verma modules are free over

U n

q .

Let us introduce algebras without Serre relations. 1

e 1

b U C f ;::: ;f ;k ;::: ;k r DEFINITION. Let q be the algebra over generated by

1 1r

( ; )

1 1

k f k = q f k k = k k k k =

j j j j

k l l

with relations k , , 1 and with coproduct

j j

b U g U n U f ;::: ;f

q q r

as in q .Let be the free algebra on generators 1 .TheVerma

e e

b b=I () M U 2 h U

q q

module over of highest weight is the left module where

(; )

I () k q j = ;::: ;r

is the left ideal generated by j 1, 1 . The image of 1 in

M v

is denoted by .

f M

The Verma module has a weight decomposition with ®nite dimensional

v U n q

weight spaces, and is freely generated by as a -module. In particular,

e e

M = M =J v (U n) U b U n

q q q q

. Moreover, . Therefore the construction of

f f

(w ) M ! M U n

the homomorphisms : worksalsointhiscaseand

traces make sense (as a formal power series).

w 2 M (w ) M ! M U n

PROPOSITION 9.2. Let for , : be the

U n v w w 2 M

unique homomorphism of -modules mapping to 1, and for

(w )

let be the same object for the algebra without Serre relations. Then for any

w w M ! M

projecting to under the canonical projection ,

() n

i() n

(w )

2 e tr (w )

[] M []

e tr M

= : P

P (26)

() i()

2i 2

[]

e tr M 1 e tr 1

M []

The proof of this proposition requires some preparation, and will be completed

after Lemma 9.7 below.

x 2 U n = m 2 Q

j j j +

LEMMA 9.3. If q is homogeneous of degree ,then

0 00 00

x+x x x) (x)= k x

( can be written as ,where are homoge-

j j j j

neous of degree < .

This lemma is an easy consequence of the form of the coproduct of the generators f

j .

168 GIOVANNI FELDER AND ALEXANDER VARCHENKO

n r f ;::: ;f

DEFINITION. Let be the free Lie algebra on generators 1 r (see [10]). f

An iterated bracket of length one is an element j of this set. An iterated bracket of

[a; b] a

length l>1 is de®ned recursively as an expression of the form ,where ,and

l l l l 2f ;::: ;l g

b are iterated brackets of length 1, 1 respectively, for some 1 11.

l [f ; [f ; [::: ;f ]]]

i i An iterated bracket of length is called simple if it is of the form i .

1 2 l

2 n a [a; b] LEMMA 9.4. Let a; b , and be an iterated bracket. Then is a linear

combination of elements of the form

[f ; [f ; [::: ;[f ;b]]]]:

i i i (27)

1 2 l a

Proof. We use induction in the length of a. If the length of is one there

a = [a ;a ] l a l < l i

is nothing to prove. If 1 2is of length , with i of length ,the

a; b]=[a ;[a ;b]] [a ; [a ;b]]

Jacobi identity gives [ 122 1 . By the induction hypothesis,

0 00

= [a ;b] b = [a ;b]

b 1 and 2 are a linear combination of elements of the desired

0 b

form (27). Then we use the induction hypothesis once more with b replaced by

00 2

and b .

We apply this lemma to the following situation. The Lie algebra n is the

r s

quotient of the free Lie algebra n on generators by the Lie ideal generated by

s ::: s n

Serre relations 1, ,m, which are simple iterated brackets in .

n r f ;::: ;f

LEMMA 9.5. Denote by the free Lie algebra on generators 1 r, and

s ;::: ;s s n

let 1 m be simple iterated brackets in .Let be the Lie ideal generated

s ;::: ;s d = (n=s) < 1

by 1 m, and assume that dim . Then there exists a basis

b ;b ;::: n b ;b ;:::

+ d+

1 2of consisting of simple iterated brackets such that d 1 2is a

s j>d b

basis of , and such that, for ,jis of the form

[f ; [f ; [::: ;[f ;s ]]]]:

i i i k

1 2 l e Proof. By de®nition every element in n is a linear combination of iterated

brackets. The ideal s is spanned by iterated brackets containing at least one of the

s s

j . Using the skew-symmetry of the bracket, we see that every element of is a

[a ; [a ; [::: ;s ]]]

linear combination of elements of the form 1 2 k for some iterated

a 2

brackets j . The claim then follows from the preceding lemma.

e n

By the Poincare±Birkhoff±WittÂ theorem, the universal enveloping algebra U has

j j j J

k 21

b = b b J J = (j ;j ;:::) abasisb labeled by the set of sequences of

k 21 1 2

j = l

non-negative integers with l 0 for all suf®ciently large . As the Verma module

e e

U n (b v ) M v

J 2J

is freely generated by the highest weight vector as a -module,

f M

is a basis of .

q U n

Let us extend this to the -deformed case. Note that q as an algebra is the

same for all q .

THREE FORMULAE FOR EIGENFUNCTIONS 169

(q );b (q);::: LEMMA 9.6. There exists a sequence b1 2of homogeneous elements

n C [ q; q ] in U such that

b (q )=b (q )C [ q; q ] j

(i) j mod 1

j j

J 21

b (q )= b (q) b (q)

(ii) If q is a transcendental number, the elements 21form

U n

a basis of q .

b J j > d + (b (q )) = b (q ) U

q j q

(iii) For 1, j 1mod .

b (q ) = b j 6 d a 2 U n

j q

Proof.Wemaytake j if .If is a homogeneous

( ; )

2 Q (f )a = f a q af j > d b =

j j j j

element of degree , set adq .If and

(f ) ::: (f )s

j j ad ad k ,set

1 l

b (q )= ) ::: (f (f )s (q ):

j q j q j ad ad k 1 l

It is clear that (i) holds. (ii) holds too, since the determinant of the matrix expressing,

J L

n C [q; q ] b (q) b in each homogeneous component of U ,in terms of has a

[q; q ] q = q

determinant in Q with the value 1 at 1, and is therefore invertible if

s (q )

is transcendental. As for (iii), we know that k has the required property since

(s (q )) = s (q ) + K s (q ) K a 2 U n

k k

k 1 for some . Moreover if of degree

(a)=a U b J q

obeys 1mod q ,then

( ; )

1 j

(f )a) = f a ( + k a f q

q j j

ad 1 j j

+ ak f ) U b J (af

j q q

j 1 mod

= (f )a U b J ;

q j q

ad 1mod q

( ; )

1 1 e

k a = q ak b (q )=b (q) U b

j q

since . Therefore, by induction, j 1mod

j j

J 2

q .

q b (q )

Suppose that is transcendental, and ®x a sequence j ,asinLemma9.6and

b (q ) U n J 2J

thus a basis ( )of q . To simplify the notation, we will write, for any ,

J 00

(J )= (b (q )) J J =(j ;j ;:::) 2J j =

deg deg .Let ,bethesetof 12such that l 0

0 0 00

l > d J J j = l 6 d J = J J

if ,and be the set of such that l 0if .Then

0 00

J 0 00 0 00 J

b (J ;J ) 2J J (q )b q )

canonically, and we may write the basis as ( , .Then

0 00

J 0 J

b J (q )b (q ) 2 J

q if is nontrivial. Since the dimensions of the homogeneous

U n J

components of q are the same as in the classical case, the basis elements with

J b (q ) U n = U n=J

q q q

nontrivial form a basis of q and the classes of form a basis of .

2 C b (q )

LEMMA 9.7. Suppose that q is transcendental. Then the elements ,

J 2J U n x

form a basis of the subalgebra of q consisting of elements such that

(x)=x U b J q 1mod q

Proof.CallB this subalgebra. By Lemma 9.6(iii), the linearly independent ele-

J 0

(q ) J 2J B x 2 B

ments b , belong to . What is left to prove is that any can be writ-

0 00

J J

0 00 0 00

x = a b (q )b (q )

J J ;J ten as a linear combination of these elements. Write J ,

170 GIOVANNI FELDER AND ALEXANDER VARCHENKO

0 00

a Z Q

+ ;J

with complex coef®cients J .Let be the set of such that there exist

0 00 00

0 00

J ;J (J ) = a 6= 2 Z ;J

with deg and J 0. An element is called maximal

0 0

> 62 Z Z 6 if implies . Obviously, every element in is some maximal element. Therefore, if we show that the only maximal element is 0, we have proved the Lemma.

By Lemma 9.6,

0 00

J J

0 00

b J : (x)= a (b (q ) )(b (q )) U

q q ;J

J 1 mod

Let be maximal. The terms whose second factor has degree are, by Lemma 9.3,

0 00

J J

0 00

a b b J : (x)=+ + (q ) b U (q )+

q q ;J

J mod

J )=

deg(

6= x 2 B b

These terms must vanish if 0since . But the elements are linearly

0 00

J a 6= J ;J

q ;J

independent, even mod . And, by construction, J 0forsome with

J )= = 2

deg( . Therefore 0.

b (q )v

The proof of Proposition 9.2 is a calculation of the traces using the basis ,

J 00

J 2J M b (q )v J 2J M

of and , of . It is suf®cient to prove Proposi-

tion free in the case where q is transcendental, since all traces over weight spaces

w w J 2J

are clearly rational functions of q .Let and be as in Proposition 9.2. If ,

J L

e e

b (q )w = b (q ) A A 2 U n

2J JL JL q L ,forsome .

The numerator of the left-hand side of (26) is then

X X

() n

(w )= A X J :

JJ q

[] + (J )

e tr M mod

deg

J2J On the other hand, the calculation of the numerator on the right-hand side of (26)

involves (see Lemma 9.6(iii))

0 00 0

J J J L

(b (q )b (q ))w = b (q )b (q ) A U b J :

q q L

1 J mod

L2J

Now we claim that the only terms in this summation that may give a nontrivial

0 00

00 L L L

2J b (q )=b (q)b (q )

contribution to the trace are those with L . Let indeed ,

0 0

0 0 00 00 J L

2J L 2J b (q ) b (q ) with L and . Since both and are in the algebra de®ned

in Lemma 9.7, their product is also in this algebra and is thus a linear combination

M 0 0 0 0 0 0

b M 2 J (M ) = (J )+ (L ) L 6= q )

of ( , ,deg deg deg .If 0, this degree is 0

strictly larger than deg(J ) and therefore does not contribute to the trace. The trace

is therefore

n ()

(w )

e tr

[]

X X

00 00

00 0

= A X X J

J J

+ J ) (J )

deg ( deg mod

0 0 00 00

J 2J J 2J

THREE FORMULAE FOR EIGENFUNCTIONS 171

0 1

X X

() n

@ A

= X (w ) J :

J ) M []

( e tr mod

deg

0 0

J 2J

On the other hand,

X X X

()

= 00 0 X

(J )+ (J ) + e tr 1

e deg deg

M []

0 0 00 00

J 2J J 2J

0 1

X X

()

@ A

= X 0 :

J ) M []

( e tr 1

deg

0 0

J 2J

Taking the ratio of the last two expression completes the proof of Proposition 9.2.

We now turn to the actual calculation of the ratio of traces on the right-hand side

(f = f ::: f ) U n

j j q of (26). We use the basis J of , labeled by ®nite sequences

1 m

()

f ;::: ;rg m > m = f =

J in 1 of arbitrary length 0(if 0, we set 1), and we

e e

w = f v L =(l ;::: ;l )

L p

may assume that , with 1. By de®nition, the trace in the

J =(j ;::: ;j ) B JJ numerator of (26) is the sum over 1m of the diagonal elements

e e

f v B (f )f v =

M MJ J L

1 (28)

X X

weighted by a factor . Expanding the coproduct on the left-hand

k = 1 k

side of (28) gives 2m terms, each of them of the form

q f ::: f f v f ::: f

j j L j j

b b d d

1 1m

B = fb < < b g f ;::: ;mg D =

labeled by subsets 1 s of 1 with complement

fd <

1 m . The exponent of is

0 1

X X X

@ A

a = + ; + ( ; ):

j j j

l (29)

d b d

d2D B 3b>d2D

This term gives a contribution to the trace if and only if

(j ;::: ;j ;l ;::: ;l )=(j ;::: ;j ):

p m b b (30) 1 s 11

If it does, the contribution is

q X X ::: X ::: f f :

j j j j

m d

1 d

j l m s = i

Note that the d are necessarily equal to the up to ordering. In particular i

p. The terms contributing to the trace in the numerator are therefore in one-to-one

(j ;::: ;j )

correspondence with pairs consisting of a ®nite sequence 1 m and a subset

172 GIOVANNI FELDER AND ALEXANDER VARCHENKO

B f ;::: ;mg B = f 6 b

1obeying (30). Let 1 1 2s be given.

b = j j t = s + t b >t t

If j for all ,set 1. Otherwise let be the smallest integer so that .

J j ;j ;::: ;j L

t+ m

Then the condition (30) on determines uniquely t 1in terms of ,

j ;::: ;j

and gives no constraint on 1 t 1. Therefore we can restrict our attention to

0 0

t = (J;B) J =(j ;::: ;j ;j ;::: ;j ) the case 1: the general solution of (30) is t ,

111m

0 0 0 0

= f ;::: ;t ;b + t ;::: ;b + t g (J ;B ) B 1111where is a solution with

b > j ;::: ;j

1 1, and 1 t 1are arbitrary. The contribution to the trace of such a

0 0

(J;B) X ::: X (J ;B ) j

solution is j times the contribution of . The sum over

1 t 1

0 0

J ;B )

all such solutions with ®xed ( gives a factor that cancels with the trace in X

the denominator (except for ).

X X X

()

2i

= X X ::: X :

j j

e tr 1 m

e 1

[]

m= j ;:::;j 0 1 m

This implies

L =(l ;::: ;l )

LEMMA 9.8. Let 1p .

() n

2i

 (f v )

etr 

M []

 

()

e2i tr 1

M []

() a

2i

= q X ::: X f ::: f ;

j j j

e j (31)

d d

1 1p

(J;B )

J;B) ( ) b >

where ( runs over all solutions of 30 such that 1 1. The exponent

a = a(J;B) ( ) fd <

is 29 , and 1 p is the complement of .

Our problem is therefore reduced to the combinatorial problem of ®nding all

J;B) b > solutions ( of (30) with 1 1 and computing their contribution to the trace.

The problem can be reformulated as follows: consider a circle with p distinct

marked points numbered counterclockwise from 1 to p starting from a base point j

distinct from the marked points. The marked points are assigned labels: the th

l (J;B)

point is assigned the label j . Solutions are in one-to-one correspondence with games that a player can play erasing marked point on this circle according

to the following rules. The player walks around the circle in clockwise direction starting from a ®nite number of times to return to . Each move consists of proceeding to the next (not yet erased) marked point. When the player meets a

marked point, he or she has the option of erasing it. If he or she does not erase it,

X l

the score is l where is the label of the point, and we say that the player `visited'

( ; )

X q = m

j l

the point. The score for erasing the point is l ,where and

m j

j is the number of times the th point has been visited previously. The game continues until all marked points have been erased. The score of the game is the

THREE FORMULAE FOR EIGENFUNCTIONS 173

= ( ; ; ) product of scores collected during the game. For instance, if L 1 1 2 , one game consists of the sequence VEVEE of visits (V) and erasures (E). Its score is

; ) ( ; + ) ( ; + ) ( ; )+j + j

( 1 2 2 1 2 1 1 2 3 2 1 2 1 2

X q X X q X q = X X q :

X2 1 1 2 1 1 2

j ;::: ;j ;j

The relation with the previous description is that m 21is the list of labels

of marked points met (visited or erased) at each move of the game, and B indicates

J;B) at which moves points are visited. The contribution of a solution ( to (31) is

the overall factor

( ; )

i ()

j j l

l 2

j j ;

q e (32)

f ::: f i ;::: ;i

i p times the score of the corresponding game times i where are the p 11 labels of the erased points, in order of erasure. The next step is a reduction to the classi®cation of `minimal' games. An empty round in a game is a sequence of of subsequent visits of all marked points exactly once without any erasures. A game without empty rounds is called minimal.Out

of a game we can construct a new game by inserting an empty round just before

X q

an erasure. The score of the new game is l times the score of of the old

s2E s

game, where the product is taken over the set E of marked points yet to be erased,

b = j j

s2E

and l . Any game can be obtained from a unique minimal game by s doing this construction suf®ciently many times. The games obtained this way have all the same sequence of erased points, and give thus proportional contributions to

the trace.

 2 S

Minimal games are in one-to-one correspondence with permutation p :we

(p) (p ) erase ®rst ,then 1, and so on, always at the earliest opportunity. The

example above gives the minimal game corresponding to the permutation (231).

 X ::: X l If is the identity, the score of the corresponding minimal game is l .

1 p

 b (k )

For general , we must calculate the numbers jk of times the point has been

(j ) visited before is erased. The score of the minimal game corresponding to is

then

p p

Y X

b + jj

c 1

q X ; c = b ( ; ):

l l

jk (33)

(j) (k)

(j)

j = = 1 j;k 1 The sum of the scores of all games obtained from this game by inserting empty

rounds is then

b + p

jj 1

q X

j =

1 l

(j)

: (34)

Q 2

j ++ j

l l

) ( (j )

 1

X ::: X q l

j =

) ( (j)

1 1

b b jk

We proceed to compute the numbers jk . Roughly speaking, is the number of

(j ) (k ) times one goes around the circle before erasing either or . More precisely,

let

= fm 2f ;::: ;p gj(m) >(m+ )g: S 111

174 GIOVANNI FELDER AND ALEXANDER VARCHENKO

k < j (k ) > (j ) (j ) b

If and , is erased ®rst, and jk is the number of times

the player passes through the base point , including at the beginning of the game,

 (j ) b = jfm 2 S jm > j gj+ k

before erasing . This number is jk 1. If ,and

 (k ) <(j) b (j )

,jk is the number of passages through before erasing , minus

b = jfm 2 S j m > j gj k > j

1: jk .If , we get in all cases the number of passages

 (k ) b = jfm 2 S j m > k gj

through before erasing , minus 1: jk . In particular,

b = jfm 2fj;:::;p gj (m) >(m+ )g c q

jj 11. The exponent of in (33) is

then

0 1

m m

X X X X

@ A

)+ ; ( ; : c =

l l l l

(k) (j ) (j) (j)

j= j=

k (j )

2S m 11

The formula in the Theorem is then obtained by combining this formula with (34), (32).

Acknowledgement We are grateful to Pavel Etingof for various useful explanations and suggestions.

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