Zeta Values and Differential Operators on the Circle

JOURNAL OF ALGEBRA ARTICLE NO.

182, 476]500 Ž1996.

0182

Zeta Values and Differential Operators on the Circle Spencer Bloch Department of Mathematics, Uni¨ ersity of Chicago, Chicago, Illinois 60637-1404 Communicated by Walter Feit Received May 18, 1995

The Fock representation of the Virasoro Lie algebra is extended to a larger graded Lie subalgebra of the algebra of differential operators on the circle. The central cocycle is related to values of the Riemann Zeta function at odd negative integers. The corresponding generating function is related to Eisenstein series for SL 2 ŽZ.. An analogous result is obtained for values of L-series associated to even Dirichlet characters using differential operators of infinite order. Q 1996 Academic Press, Inc.

0. INTRODUCTION Let d denote the Lie algebra of derivations of the Laurent polynomial ring Qw t, ty1 x. The Virasoro algebra v is a central extension of d which is represented on the Fock space S . Write D s t ? drdt g d. Usually, one takes as a basis of v elements LŽ n. for n g Z mapping to yt n D g d together with the central generator c. The bracket structure is given by LŽ m. , LŽ n. s Ž m y n. LŽ m q n. q

1 12

Ž m3 y m . dmq n , 0 ? c, Ž 0.1.

where d a, b equals 1 or 0 depending on whether or not a s b. The endomorphism of S corresponding to LŽ0. is somewhat unnatural in that it involves ‘‘normal ordering.’’ Removing the normal ordering leads to a divergent term of the form ‘‘1 q 2 q 3 q ??? .’’ If one interprets this divergence in terms of the Riemann zeta function y

1 12

s z Ž y1 . s‘‘1 q 2 q 3 ??? ,’’ 476

0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

Ž 0.2.

DIFFERENTIAL OPERATORS ON THE CIRCLE

477

one is led to modify the basis of v, taking

¡LŽ n. , 1 ¢LŽ 0. y 24 c,

for n / 0;

L Ž n . s~

for n s 0.

Ž 0.3.

With respect to this new basis, the central term for the algebra structure simplifies to a monomial, viz. LŽ m. , LŽ n. s Ž m y n. LŽ m q n. q

1 12

m3dmq n , 0 ? c.

Ž 0.4.

All this is carefully explained in w4, 1.9x. The purpose of the present note is to explain how the values at other negative integers of z and Dirichlet L-series for even characters also arise in the central cocycle for certain projective representations on S . Recall w6, Chap. IIx, for k G 1 an integer, z Žy2 k . s 0 and z Ž1 y 2 k . s yB2 kr2 k, where B2 k is the Bernoulli number. Define D s  w : Q w t , ty1 x ª Q w t , ty1 x differential operator N w Ž 1 . s 0 4 . Ž 0.5. In words, D consists of differential operators with no degree 0 part. It is itself a Lie algebra under commutator bracket, and it contains d as a subalgebra. The Virasoro central extension is known w3, 2.4x to extend to a central extension of D. The algebra D acts on V [ L2 Ž S 1, C.rC with Hilbert space basis n t [ e 2 p i n x for n / 0. One has a non-degenerate skew form BŽ f , g . [ i

1

H0

fDg dx

Ž 0.6.

on V. There is a map u : D ª D such that u Ž w . is the skew-adjoint to w with respect to B, B Ž w f , g . s yB Ž f , u Ž w . g . .

Ž 0.7.

It follows formally that u 2 s identity and that u preserves commutator brackets. Let Dq; D be the fixed Lie subalgebra under u . We have d m Dqm D ,

Ž 0.8.

and the above representation on V gives

s : Dqª sp Ž V . .

Ž 0.9.

ŽIn fact, to make signs work out better, it is convenient to conjugate the differential operator representation of Dq by the matrix which is the identity on t n for n - 0 and y1 on t n for n ) 0..

478

SPENCER BLOCH

The metaplectic representation w8x projectively represents spŽ V . on S , and by composition we get a representation

r : V ª End Ž S .

Ž 0.10.

for some central extension V of Dq. ŽThis idea of using the metaplectic representation is due to Goncharov and Tsygan. It greatly clarifies and simplifies my original ad hoc calculations. . The algebra V has a vector space basis consisting of the central term c and elements LŽ r . Ž m. for m g Z and r g N. The elements LŽ0. Ž m. and c span v ; V, and LŽ r . Ž m. ¬ Žy1. rq1 D r t m D rq1 g D. the bracket operation is given by LŽ r . Ž m . , LŽ s. Ž n . s

a i Ž m, n . LŽ i. Ž m q n . q k Ž r q s, m . dmqn , 0 ? c.

Ý min Ž r , s .FiFrqs

Ž 0.11. Here k Ž r q s, m. is a polynomial in m of degree 2Ž r q s . q 3, and a i Ž m, n. is homogeneous in m and n of degree 2 r q 2 s q 1 y 2 i. In particular, V is graded, with LŽ r . Ž m. having degree m. As with the Virasoro representation, the representation of the LŽ r . Ž0. on S involves a non-canonical normal ordering. Removing this introduces a divergent term ‘‘12 rq1 q 2 2 rq1 q ??? .’’ If we interpret this divergence as z Žy1 y 2 r ., we are led to the change of basis LŽ r . Ž m . s

½

m/0

LŽ r . Ž m . , r

L Ž 0 . q Ž y1 . z Ž y1 y 2 r . ? c, 1 2

Žr.

m s 0.

Ž 0.12.

The interesting point is that just as in the Virasoro case, this change of basis leads to a monomial central term in the algebra law for V. Namely, writing

br s

1

H0 Ž 1 y y .

rq1

y rq1 dy,

Ž 0.13.

the bracket formula becomes LŽ r . Ž m . , LŽ s. Ž n . s

Ý min Ž r , s .FiFrqs

a i Ž m, n . LŽ i. Ž m q n . q

1 2

brqs m2 rq2 sq3dmqn , 0 ? c.

Ž 0.14.

479

DIFFERENTIAL OPERATORS ON THE CIRCLE

I thank K. Murthy and S. Vandervelde for pointing out that

bt s B Ž t q 2, t q 2 . s

Ž t q 1. ! Ž t q 1. ! , Ž 2 t q 3. !

Ž 0.15.

where B Ž s, t . is the classical beta function. In particular, bt is the reciprocal of an integer. Associated to the operators LŽ r . Ž0. together with an extra copy of the non-normalized operator LŽ0. Ž0. one can define a generating function F s Tr Ž exp Ž 2p iLŽ0. Ž 0 . t . exp Ž 2p iLŽ0. Ž 0 . t 1 . exp Ž 2p iLŽ1. Ž 0 . t 2 . ??? . s

`

3

Ł Ž 1 y q nq1nq2n q3n

5

ns1

??? .

y1

q1Ž1r2. z Žy1. q2Ž1r2. z Žy3. ??? .

Ž 0.16.

The relation with modular forms for SL 2 ŽZ. is given by 1

d

2p i dt k

Ž log F . N t 1s t 2s? ? ? s0 s

z Ž1 y 2 k. 2

E2 k Ž t . ,

Ž 0.17.

where E2 k Žt . is the normalized Eisenstein series Ž4.5. of weight 2 k. It is possible to push the link with number theory further by introducing an even Dirichlet character x with conductor N. Working with certain differential operators of infinite order like cosŽ2p DrN ., we are led to define operators LŽ r . Ž mN .x g EndŽ S .. These satisfy commutator bracket identities of the expected sort. For example, if c is another even Dirichlet character of conductor dividing N, LŽ0. Ž mN . x , LŽ0. Ž ymN . c s 2 mNLŽ0. Ž 0 . cx q

1 2

mN

Ý Ž cx . Ž k . Ž mN y k . k ? c. Ž 0.18. ks0

Taking c s 1, it follows that the vector space span V x of c and the LŽ r . Ž mN .x for r G 0 and m g Z is a Lie module for Dq. Again the operators LŽ r . Ž0.x are defined via normal ordering, which can formally be eliminated by adding appropriate multiples of c times values of the Dirichlet L-series LŽ x , s .. For example, LŽ0. Ž 0 . x s LŽ0. Ž 0 . x q

1 2

L Ž x , y1 . ? c.

Ž 0.19.

Assuming xc is non-trivial of conductor N, Ž0.18. becomes LŽ0. Ž mN . x , LŽ0. Ž ymN . c s 2 mNLŽ0. Ž 0 . xc . This sort of argument shows

Ž 0.20.

480

SPENCER BLOCH

THEOREM Ž0.21.. For x a non-tri¨ ial e¨ en Dirichlet character of conductor N, the Dq-module V x splits, V x s Q ? c [ Ž V xrQ ? c . . One perceives vaguely a sort of dictionary between properties of Lfunctions and properties of these Fock representations

z Ž s . vanishes for s s y2 n, n g N ¦ r in Ž 0.10. is defined on Dqm D L Ž x , s . analytic at s s 1, x / 1 ¦ Q ? c ; V x splits. The author acknowledges his inexperience in this area of mathematics. Most everything he knows was learned from the course of Professor Glauberman at the University of Chicago on the book of Frenkel, Lepowsky, and Meurman w4x. He also thanks A. Goncharov, B. Tsygan, V. Kac, K. Murthy, and S. Vandervelde for helpful conversations and correspondence. The role of z Žy1. s y1r12 in the central term of the Virasoro representation is related to the curious fact that the universe has dimension 26. The current note may be viewed as an attempt to construct universes with various dimensions. The author is obliged to admit, however, that physicists he has consulted are unanimous in their opinion this project is completely daft. 1. THE ALGEBRA D Let E denote the algebra of differential operators on the ring Qw t, ty1 x. Write D s t ? drdt g E. PROPOSITION Ž1.1.. Ži. The elements t a D n for a g Z and n G 0 Ž resp. n G 1, resp. n s 1. form a ¨ ector space basis for E Ž resp. D, resp. d .. Žii. The mapping t a D n ¬ t a DŽyD y a. ny 1 gi¨ es rise to an in¨ olution u of D which fixes d and preser¨ es commutator bracket u w x, y x s w u x, u y x. Proof. Ži. We have t a D n Ž t i . s i n t iqa ,

Ž 1.2.

from which it follows that the t a D n are linearly independent and that m

t a D m t b D n s t aqb Ž D q b . D n .

Ž 1.3.

This identity implies that the t D span E. A linear combination Ý a a, n t a D n with a a, n g Q kills 1 if and only if a a, 0 s 0 for all a. a

m

DIFFERENTIAL OPERATORS ON THE CIRCLE

481

Žii. Let f and g be polynomials with Q coefficients. As an easy consequence of Ž1.3. we get the commutator bracket formula t a f Ž D . , t b g Ž D . s t aqb Ž f Ž D q b . g Ž D . y g Ž D q a . f Ž D . . . Ž 1.4. Assume f and g have no constant term, and write f Ž D . s DF Ž D . and g Ž D . s DGŽ D .. We have

u t a DF Ž D . , t b DG Ž D . s u Ž t aq b D Ž Ž D q b . F Ž D q b . G Ž D . y Ž D q a . G Ž D q a . F Ž D . . . s t aq b D Ž Ž yD y a . F Ž yD y a . G Ž yD y a y b . y Ž yD y b . G Ž yD y b . F Ž yD y a y b . . .

Ž 1.5.

On the other hand,

u Ž t a DF Ž D . . , u Ž T b DG Ž D . s t a DF Ž yD y a . , t b DG Ž yD y b . s t aq b D Ž Ž D q b . F Ž yD y a y b . G Ž yD y b . y Ž D q a . G Ž yD y a y b . F Ž yD y a . . .

Ž 1.6.

Assertion Ž1.1.Žii. follows since the right hand sides of Ž1.5. and Ž1.6. agree. Recall we have defined in Ž0.6. a Hilbert space V s L2 Ž S 1 , C.rC and a symplectic form B Ž , .. The Lie algebra D acts on V. PROPOSITION Ž1.7.. B Ž w Ž f ., g . s yB Ž f, uw Ž g ... In particular, the ufixed subalgebra Dq maps to spŽ V .. Proof. It suffices to check for w s t a DF Ž D .. We have BŽ w Ž f . , g . s i

a

H Ž t DF Ž D . . Ž f . D Ž g . dx

si

H D Ž f . Ž F Ž yD . t D . Ž g . dx a

s i D Ž f . Ž t a DF Ž yD y a . . Ž g . dx

H

s B Ž u Ž w . g , f . s yB Ž f , u Ž w . g . . Remarks Ž1.8.. Ži.

Dq is a graded sub-Lie-algebra of D.

Žii. d ; Dq. 3 5 Žiii. The degree 0 part Dq Ž 2 n. 0 is spanned by D, D , D , . . . ; and u D s yD 2 n.

482

SPENCER BLOCH

PROPOSITION Ž1.9.. d m Dqm D.

Dq is the unique graded Lie algebra such that

Proof. Here are some formulas which follow from Ž1.4.. We write f Ž D . s D d q a 1 D dy1 q a 2 D dy2 q ??? and g Ž D . s D e q ??? f Ž D . , t b D s t b D Ž f Ž D q b . y f Ž D . . s t b Ž dbD d q ??? . Ž 1.10. t a f Ž D . , tya g Ž D . s f Ž D y a . g Ž D . y f Ž D . g Ž D q a . s yaŽ d q e . D dq ey1 q lower order terms Ž 1.11. t a f Ž D . , tya D s f Ž D y a . D y f Ž D . Ž D q a . s yaŽ d q 1 . D d q lower order terms

Ž 1.12.

tya D, t a D, f Ž D . s D ? Ž 2 Df Ž D . y Ž D y a . f Ž D y a . y Ž D q a . f Ž D q a . . s ya2 Ž d q 1 . dD d q a 1 d Ž d y 1 . D dy1

ž

q 2

žž

dq1 2 dy1 a q 2 a2 4 2

/

ž

//

D dy2

/

q lower order terms.

Ž 1.13.

Now let D9 be a graded Lie algebra with d m D9 m D. LEMMA Ž1.14..

Either DX0 s D 0 or DX0 is spanned by D, D 3 , D 5, . . . .

Proof of Ž1.14.. Because d m D9 there exists t a F Ž D . g D9 with d s degŽ F . G 2. By Ž1.12. there exists H Ž D . g DX0 with degŽ H . s d G 2. By Ž1.10. for all b g Z there exists t b GŽ D . g DXb with degŽ G . s d G 2. Taking b s ya and applying Ž1.10., there exists J Ž D . g DX0 with degŽ J . G d q 1. Now applying alternately Ž1.10. and Ž1.11. it follows that there exist F Ž D . g DX0 of arbitrarily large degrees. In applying Ž1.11. we can arrange that degŽ f . s degŽ g . so we get polynomials F Ž D . g DX0 of arbitrarily large odd degree. We next claim that if DX0 contains an element F Ž D . of degree d G 3, then DX0 contains a GŽ D . of degree d y 2. To see this, since D g d ; D9 we may assume d G 4. Let DX0 2 F Ž D . s D d q a 1 D dy1 q ??? . By Ž1.13., DX0 contains

Ž y1ra2 .

tya D, t a D, F Ž D .

s y2 a 1 dD dy 1 q 2

žž

y Ž d q 1 . dF Ž D .

dq1 2 a y a 2 Ž 4 d y 2 . D dy2 4

q lower order terms.

/

/

Ž 1.15.

DIFFERENTIAL OPERATORS ON THE CIRCLE

483

For a suitable choice of a, we may arrange that the coefficient of D dy 2 is not zero. If a 1 s 0, we may take GŽ D . to be the right hand side of Ž1.15.. If a 1 / 0, divide through by y2 a 1 d to get a monic polynomial H Ž D . s D dy 1 q b 1 D dy 2 q ??? with b 1 / 0. Repeat the above with H in place of F, defining G Ž D . s DX0 2 Ž y1ra2 . tya D, t a D, H Ž D .

y Ž d y 1 . dH Ž D .

s y2 b 1 Ž d y 1 . D dy 2 q lower order terms. It is clear from the above that if F ŽyD . s yF Ž D . for all F g DX0 , then has as a basis all the odd powers of D: D, D 3 , D 5, . . . . Suppose now F Ž D . s D d q a 1 D dy1 q ??? g DX0 with F ŽyD . / yF Ž D .. I claim we can find such an F with d even. Indeed, if d is odd, there will exist j G 0 such that a 2 jq1 / 0. Let j be the minimal such. Let us say a polynomial G is odd Žresp. even. if GŽ D . is a sum of multiples of odd Žresp. even. powers of D. Replacing D by yD, one sees easily that GŽ D . ¬ w tya , w t a, GŽ D .xx preserves the parity of G. Writing our F Ž D . as a sum of an odd polynomial and an even polynomial a 2 jq1 D dy 2 jy1 q lower order terms, and using Ž1.13., we get DX0

DX0 2 Ž y1ra2 . tya D, t a D, F Ž D .

y Ž d q 1 . dF Ž D .

s  terms of odd deg.F d y 2 4 q 2 a 2 jq1 Ž 2 j q 1 . Ž j y d . D dy 2 jy1 q  terms of even deg.F d y 2 j y 3 4 . Note the coefficient of D dy 2 jy1 is non-zero. Repeating the argument, or using induction, we eventually find a polynomial of even degree d y 2 j y 1 G 2 g DX0 . As already remarked, we can find polynomials in DX0 of arbitrarily large odd degree, as well as the polynomial of even degree found above. Applying Ž1.10. and then Ž1.11., it follows that DX0 contains polynomials of arbitrarily large even degree as well. But it then follows from the above that DX0 contains polynomials of arbitrary degree, so DX0 s DQw D x, proving Ž1.14.. To finish the proof of Ž1.9., it follows easily from Ž1.10. that if DX0 s DQw D x, then D9 s D. Suppose on the other hand that DX0 consists of odd polynomials f Ž D .. Let sx : Qw D x ª Qw D x be the ring involution defined by sx Ž F Ž D .. s F ŽyD y x .. Fix an integer b and let t : DQw D x ( Qw D x; t Ž f . s f Ž D q b . y f Ž D .. ŽNote the domain of t is taken to be polynomials vanishing at 0.. One checks that s bt s yts 0 , whence t

484

SPENCER BLOCH

induces an isomorphism between the y1 eigenspace of s 0 and the q1 eigenspace of s b . It follows Žagain from Ž1.10.. that DXb contains the q1 eigenspace of u on D b . If this inclusion were strict, there would exist t a DF Ž D . g DXb such that F ŽyD y b . s yF Ž D .. By Ž1.12., this would give a non-zero element

Ž D y b . F Ž D y b . D y DF Ž D . Ž D q b . g DX0 . Substituting F Ž D y b . s yF ŽyD ., it follows that this polynomial is even, contradicting our assumption. This completes the proof of Ž1.9.. LEMMA Ž1.16.. Let f Ž m, n, k . be a homogeneous polynomial of degree N in n, m, and k. Assume f Ž m, n, k . s f Ž m, n, yn y m y k .. Then there exist a i Ž m, n. homogeneous of degree N y 2 i in m, n such that f Ž m, n, k . s

Ý ai Ž m, n . k i Ž yn y m y k . i . iG0

Proof. Use induction on N. If N s 0, f is constant and the assertion is clear. Assume N ) 0, and define a0 Ž m, n. s f Ž m, n, 0.. Since f Ž m, n, k . y a0 Ž m, n. is a polynomial in m, n, k which vanishes at k s 0 and k s yn y m, we have f Ž m, n, k . y a0 Ž m, n . s k Ž ym y n y k . g Ž m, n, k . . By induction the lemma is true for g so we are done. Fix non-negative integers r, s and consider the polynomial f Ž r , s. Ž m, n, k . [ Ž y1 .

rqsq1

Ž Ž k q n . rqsq1 Ž m q n q k . yŽ k q m.

rqsq1

r

ks s

Ž m q n q k . k r . . Ž 1.17.

One has f Ž r, s. Ž m, n, k . s f Ž r, s. Ž m, n, ym y n y k . so by Ž1.16. we may write rqs

f Ž r , s. Ž m, n, k . s

i

aŽi r , s. Ž m, n . k i Ž ym y n y k . . Ž 1.18.

Ý ismin Ž r , s .

PROPOSITION Ž1.19.. by the elements

A ¨ ector space basis for the Lie algebra Dq is gi¨ en r

LŽmr . [ yt m D rq1 Ž yD y m . s Ž y1 .

rq1

D r t m D rq1

Ž 1.20.

for m g Z and r g N. With respect to this basis, the commutator bracket is gi¨ en by rqs

LŽmr . , LŽns. s

Ý ismin Ž r , s .

i. aŽi r , s. Ž m, n . LŽmqn .

Ž 1.21.

485

DIFFERENTIAL OPERATORS ON THE CIRCLE

Proof. The first assertion of Ž1.19. is clear. One checks easily that LŽmr . , LŽns. Ž t k . s ykf Ž r , s. Ž m, n, k . t mqnqk . On the other hand i

LŽpi. Ž t k . s yk Ž k i Ž yk y p . . t kqp . A comparison of these two formulas yields Ž1.21..

2. THE FOCK REPRESENTATION Our reference for this section is w4, Chap. 1x. The role of the metaplectic representation was suggested by Goncharov and Tsygan. Let h be a finite dimensional Q-vector space with a non-degenerate symmetric form ² ? , ? :. We assume for simplicity that this form admits an orthonormal basis  h i 4 . Define

ˆh [ h m Q w t , ty1 x [ Q ? c. We give ˆh the structure of a Lie algebra ŽHeisenberg Lie algebra. by making the element c central and defining

w x m t m , y m t n x s ² x, y : m dmq n , 0 ? c.

Ž 2.1.

Note that h ; ˆh is central. We define

˜h [ ˆhrh s @ h t n [ Qc.

Ž 2.2.

n/0

We define the Fock space S [ Sym Q Ž h m ty1 Q w ty1 x . .

Ž 2.3.

Let b s t h w t x [ Qc viewed as an abelian Lie subalgebra of ˜h, and let U be the universal enveloping algebra of ˜h. Let F denote the one dimensional representation of b where c acts as the identity and elements in t h w t x act trivially. The Poincare]Birkoff]Witt theorem allows us to iden´ tify the induced representation with S S ( U mUŽ b . F

Ž 2.4.

so S inherits a ˜h-module structure. We define a projective representation of Dq on S by defining elements LŽ r . Ž m. g EndŽ S .. We write hŽ n. s ht n, viewed as an endomor-

486

SPENCER BLOCH

phism of S . Recall  h i 4 is an orthonormal basis of S . LŽ r . Ž m . [

1

dim h

2

is1

r Ý Ý Ž m y k . k rhi Ž m y k . hi Ž k .

if m / 0; Ž 2.5.

kgZ

LŽ r . Ž 0 . [

1 2

dim h

r

k r Ž yk . h i Ž y< k < . h i Ž < k < . .

Ý Ý is1

Ž 2.6.

kgZ

Notice that any given s g S is killed by hŽ n. for n 4 0, so these infinite sums make sense in EndŽ S .. Indeed, this is clear for LŽ r . Ž0.. For LŽ r . Ž m. it suffices to note that for m / 0, hŽ k . and hŽ m y k . commute so the one of larger degree can be placed on the right. Our next objective is to show the map LŽmr . ¬ LŽ r . Ž m. defines a projective representation. In fact, h plays no role in our construction. We will simply take it to be Q and omit it from the notation. Depending on the application he has in mind, the reader may replace V by V m h, with all operators acting as the identity on h. From Ž1.7. we have Dq; spŽ V .. Let W Žresp. W . ; V denote the span of t n s e 2 p i n x for n - 0 Žresp. n ) 0.. We give W the structure of a Hilbert space by defining ² w 1 , w 2 : s iB Ž w 1 , w 2 . s y

1

H0

w 1 Dw 2 dx.

Ž 2.7.

An orthonormal basis is given by e k s tyk r 'k , k s 1, 2, . . . . An element in the Lie algebra spŽ V . can be written as a matrix Ž ag yba t . Žcf. w8x., where a : W ª W; b , g : W ª W, and b and g are symmetric. ŽIn our calculations, b and g will be finite matrices, and a will stabilize the subspaces of finite polynomials in the t n, so we need not worry about function-theoretic conditions on these operators.. The metaplectic representation is a projective representation of the symplectic group of Ž VR , B . on the Hilbert space completion SˆŽW . of the symmetric algebra on W. The corresponding projective Lie algebra representation is given as Žcf. w8, Proposition Ž5.6.x.

ž

a

b

g

ya

t

/

¬ Da q

1 2

Mb q

1 2

MgU g End Ž SˆŽ W . . .

Ž 2.8.

Here Da is the derivation of SˆŽW . induced by the endomorphism a of W, Mb is multiplication by the element Ý e k b Ž e k . g Sˆ2 ŽW ., and MgU is the adjoint to multiplication by Ý e k g Ž e k .. We remark that SˆŽW . is a completion of Fock space S in an obvious way. The operators we work with will stabilize S .

487

DIFFERENTIAL OPERATORS ON THE CIRCLE

PROPOSITION Ž2.9.. Let i : spŽ V . ª spŽ V . denote conjugation by the matrix Ž 0I y0 I .. View D q ; sp Ž V . as abo¨ e, and define r s metaplectic( i : Dqª EndŽ S .. Then r Ž LŽmr . . s LŽ r . Ž m.. Proof. Let LŽmr . s Žy1. rq1 D r t m D rq1 correspond to the matrix Ž ag Suppose m - 0. We have

a Ž t n . s Ž y1 .

rq1

r

Ž n q m . n rq1 t nqm ,

n F y1.

b ya t

..

Ž 2.10.

To avoid confusion, write t n s hŽ n. for n - 0, and let ¨ g S be the vacuum vector. We have h Ž yn . h Ž n . ¨ s h Ž yn . , h Ž n . ¨ s yn¨ ,

n - 0.

It follows that the map Ž2.10., extended to a derivation in EndŽ S . is given by r r Ý Ž m q n . Ž yn . h Ž n q m . h Ž yn .

Da s

nFy1

s

r Ý Ž m y k . k rhŽ m y k . hŽ k .

kG1

s

1 2

žÝq kG1

Ý kFmy1

r

/ Ž m y k . k hŽ m y k . hŽ k . . r

Ž 2.11.

Note LŽmr . stabilizes W, so g s 0. On the other hand, b Ž t n . is given by the same formula Ž2.10. for 1 F n F ym y 1, and zero otherwise. We see that Mb is multiplication by tyn

ymy1

r Ý Ž m q n . Ž yn . rq1 'n

ns1

m

t mqn

'n

.

In terms of the hŽ k ., this looks like ksmq1

Mb s y

Ý

r

k r Ž m y k . hŽ k . hŽ m y k . .

Ž 2.12.

ksy1

The conjugation map i replaces b and g by their negatives, so the desired formula Ž2.5. follows from Ž2.8., Ž2.11., and Ž2.12.. The case m s 0 is easier. We have a Ž t n . s Žy1. rq1 n2 rq1 t n, which extends to the derivation on S given by L Ž r . Ž 0. s

1 2

Ý kgZ

r

k r Ž yk . h Ž y< k < . h Ž < k < . ,

488

SPENCER BLOCH

the desired formula. Finally, the case m ) 0 is analogous and is left for the reader. An explicit calculation using Ž2.5. and Ž2.6., yields the following proposition Žproof omitted., which should be compared with Ž1.21.. PROPOSITION Ž2.13..

For m / yn and r, s G 0 rqs

LŽ r . Ž m . , LŽ s. Ž n . s

aŽi r , s. Ž m, n . LŽ i. Ž m q n . ,

Ý ismin Ž r , s .

with aŽi r, s. Ž m, n. as in Ž1.18.. It remains to compute the commutator of LŽ r . Ž m. and LŽ r . Žym.. In this calculation, dimŽ h . plays a role, so we return to the notation at the beginning of Section 2. For t G 0 define g Ž t . Ž m, k . [ Ž m y k .

tq1

k t q Ž k q m.

tq1

t Ž yk . .

Ž 2.14.

We have t

g Ž t . Ž m, k . s g Ž t . Ž m, yk . s yg Ž t . Ž ym, k . s

Ý

js n

aŽj t . m2 tq1y2 j k 2 j Ž 2.15.

with n s wŽ t q 1.r2x and aŽj t . g Q. Note that f Ž r , s. Ž m, ym, k . s g Ž rqs. Ž m, k . .

Ž 2.16.

Define

k Ž t , m. [

1 2

m

dim Ž h .

Ý Ž m y k . tq1 k tq1 .

Ž 2.17.

ks0

PROPOSITION Ž2.18.. Let r, s be non-negati¨ e integers, t s r q s, and let m be a non-zero integer. Then, with notation as abo¨ e, t

LŽ r . Ž m . , LŽ s. Ž ym . s

Ý Ž y1. j aŽj t . m2 tq1y2 j LŽ j. Ž 0. q k Ž t , m . ,

js n

where the term k is the central term Ž multiple of the identity in EndŽ S ... Proof Žcompare w4, 1.9x.. If t s 0 we get a00 s 2, k Ž0, m. s Ž1r12.dimŽ h .Ž m3 y m. and the desired formula is just Ž0.1.. Assume now

DIFFERENTIAL OPERATORS ON THE CIRCLE

489

t / 0. We write LŽ r . Ž m . , LŽ s. Ž ym . 1

s LŽ r . Ž m . ,

2

s s Ý Ý Ž ym q k . Ž yk . h i Ž ym q k . h i Ž yk .

i

1

q LŽ r . Ž m . , s

s s Ý Ý Ž ym q k . Ž yk . h i Ž ym q k . h i Ž yk .

2

i

k)m

1

rqsq1 rqs k h i Ž k . h i Ž yk . Ž m y k. Ý Ý 2

i

q q q

kFm

kF0

1 2 1 2

Ý Ž m y k . rqsq1 k rqs h i Ž k . h i Ž yk .

Ý i

0-kFm

rqs Ý Ý Ž m q k . rqsq1 Ž yk . h i Ž k . h i Ž yk .

i

kF0

1

rqsq1 rqs k h i Ž k . h i Ž yk . Ž m y k. Ý Ý 2

i

kF0

q y

1 2

rqs Ý Ý Ž m q k . rqsq1 Ž yk . h i Ž k . h i Ž yk .

i

kF0

1 2

Ý

Ž m q k.

Ý

rqsq1

Ž yk .

rqs

ymFk-0

i

h i Ž k . h i Ž yk . .

The infinite sums in the above expression yield

Ý Ý i

g Ž t . Ž m, k . h i Ž k . h i Ž yk . .

kF0

The two finite sums Žnote the minus sign before the last. give 1 2

Ý Ž m y k . rqsq1 k rqs

Ý i

h i Ž k . , h i Ž yk .

0-kFm

s s

1 2 1 2

Ý i

Ý Ž m y k . rqsq1 k rqsq1 0-kFm

dim Ž h .

Ý Ž m y k . rqsq1 k rqsq1 . 0-kFm

This completes the proof of Ž2.18..

490

SPENCER BLOCH

DEFINITION Ž2.19.. V is the central extension of Dq corresponding to the projective representation of Dq on S . We view V as a Lie algebra with vector space basis LŽ r . Ž m., r G 0, m g Z together with central basis element c. V is represented on S with c acting by multiplication by dimŽ h .. We have for t s r q s t

LŽ r . Ž m . , LŽ s. Ž ym . s

Ý

js wŽ tq1 .r2 x m

1

q

2

j Ž y1. aŽj t . m2 tq1y2 j LŽ j. Ž 0 .

Ý Ž m y k . tq1 k tq1 ? c. ks0

It is frequently convenient to view V as a subalgebra of EndŽ S ..

3. ZETA NORMALIZATION The zeta normalized basis of V arises by writing formally LŽ t . Ž 0 . s

1 2

Ž y1.

t

Ý Ý i

1 t s‘‘ Ž y1 . Ý 2 i

kgZ

k 2 t h i Ž yk . h i Ž k . ’’

Ý kgZ

1 t q ‘‘ Ž y1 . Ý 2 i 1 t s‘‘ Ž y1 . Ý 2 i

k 2 t h i Ž y< k < . h i Ž < k < .

Ý

k 2 t h i Ž k . , h i Ž yk . ’’

k-0

k 2 t h i Ž yk . h i Ž k . ’’

Ý kgZ

1 t y ‘‘ Ž y1 . Ý 2 i

Ý

k 2 tq1 .’’

Ž 3.1.

kG1

Of course, the expressions in quotes don’t make sense. Still, at least formally, we can remove the unnatural normal ordering by defining LŽ t . Ž m . [

½

for m / 0;

LŽ t . Ž m . , t

L Ž 0 . q Ž y1 . z Ž y1 y 2 t . ? c. Žt.

1 2

Here z Ž s . s Ý nG 1 nys is the Riemann Zeta function.

for m s 0.

Ž 3.2.

491

DIFFERENTIAL OPERATORS ON THE CIRCLE

Let B Ž x, y . denote the beta function w1, 6.2x, and define for t g N

bt s B Ž t q 2, t q 2 . s

1

H0 Ž 1 y y .

tq1

y tq1 dy s

Ž t q 1. ! Ž t q 1. ! . Ž 3.3. Ž 2 t q 3. !

Let r, s g N and write t s r q s. Let n s wŽ t q

PROPOSITION Ž3.4.. . 1 r2x. Then

jst

LŽ r . Ž m . , LŽ s. Ž ym . s

1

Ý Ž y1. j aŽj t . m2 ty2 jq1 LŽ j. Ž 0. q 2 bt m2 tq3 ? c.

js n

Proof. Write f Ž m, y . s Ž m y y . tq1 y tq1. We use the Euler]Maclaurin sum formula w1, 25.4.7; 5, p. 200x to write my1

`

m

Ý f Ž m, k . s H Ž m y y . tq1 y tq1 dy q Ý Ž B2 ir Ž 2 i . ! . 0

ks1

is1

=Ž d

2 iy1

frdy

2 iy1

Ž m, m . y d 2 iy1 frdy 2 iy1 Ž m, 0 . . . Ž 3.5.

Here the B2 i are Bernoulli numbers w6, IIx. The change of variables z s yrm yields m

H0

Ž m y y.

tq1

y tq1 dy s bt m2 tq3

Ž 3.6.

with bt as above. The function f Ž m, y . has degree 2 t q 2 in y and vanishes to order t q 1 at y s 0 and y s m. It follows that the above infinite sum Ž3.5. need only be taken over the finite range tr2 q 1 F i F t q 1. Using the Leibnitz rule p n pyn Ž drdy . Ž ab . s Ý np Ž drdy . Ž a. Ž drdy . Ž b . ,

n

ž /

we find for p G t q 1 p

Ž drdy . f Ž m, 0 . s Ž y1 .

py tq1

Ž t q 1. !

p tq1

ž /ž

p

tq1 p y t y 1 . !m2 tq2yp pyty1 Ž Ž 3.7.

/

Ž drdy . f Ž m, m . s Ž y1 .

tq1

Ž t q 1. !

p tq1

ž /ž

tq1 p y t y 1 . !m2 tq2yp . pyty1 Ž

/

492

SPENCER BLOCH

Substituting Ž3.6. and Ž3.7. into Ž3.5. and writing j s i y 1, p s 2 j q 1, n s wŽ t q 1.r2x my1

Ý f Ž m, k . ks1

s bt m2 tq3 q 2 Ž y1 . t

=

B2 jq2

ž

Ý js n Ž 2 j q 2 . !

s bt m2 tq3 q Ž y1 .

tq1

Ž t q 1. !

tq1 2jq1 2 j y t. ! m2 tq1y2 j 2jyt Ž tq1

tq1

/

t

Ý

js n

ž

B2 jq2

/

tq1 m2 tq1y2 j . jq1 2jyt

ž

/

Ž 3.8.

We have used here the ugly identity 1 jq1

s 2 Ž t q 1. ! Ž 2 j y t . !

ž

2jq1 r Ž 2 j q 2 . !. tq1

/

On the other hand, a well-known formula for the Riemann Zeta function w6, IIx gives for k G 1 an integer

z Ž 1 y 2 k . s yB2 kr2 k.

Ž 3.9.

Substituting in Ž3.2. LŽ j. Ž 0 . s LŽ j. Ž 0 . q

1 2

Ž y1.

B2 jq2

jq1

2jq2

? c.

Ž 3.10.

Note that aŽj t . in Ž3.4. is the coefficient of m2 tq1y2 j in

Ž m y k.

tq1

kt q Ž m q k.

tq1

t Ž yk . ,

from which one deduces aŽj t . s 2 Ž y1 .

t

ž

tq1 . 2jyt

/

Ž 3.11.

Comparing the desired formula Ž3.4. with Ž2.19., Ž3.5., and Ž3.10., we find we must show j jq1 Ž y1. aŽj t . Ž y1.

s

1 2

½

1 B2 jq2 2 2jq2 my1

coefficient of m2 tq1y2 j in

Ý Ž m y k . tq1 k tq1

ks1

5

. Ž 3.12.

DIFFERENTIAL OPERATORS ON THE CIRCLE

493

Plugging in Ž3.11. and Ž3.8., we find that both sides in Ž3.12. equal

Ž y1.

tq1

ž

t q 1 B2 jq2 . 2jyt 2jq2

/

This completes the proof of Ž3.4..

4. THE GENERATING FUNCTION Let A1 , A 2 , . . . be endomorphisms of a real vector space S . Assume the A i commute, are simultaneously diagonalizable, and have finite dimensional eigenspaces. The generating function is the formal expression F Ž A1 , A 2 , . . . . [ Tr Ž exp Ž 2p it 1 A1 . exp Ž 2p it 2 A 2 . ??? . .

Ž 4.1.

Note replacing A k with A k q e for a constant e replaces F Ž A1 , A 2 , . . . . with exp Ž 2p i et k . F Ž A1 , A 2 , . . . . . We want to apply this to S s Fock Space and A rq1 s Žy1. r LŽ r . Ž0.. Here the LŽ r . Ž0. are the Zeta-normalized operators from Section 3 and the sign Žy1. r is inserted to make the eigenvalues of LŽ r . Ž0. equal k 2 rq1 for k s 0, 1, 2, 3 ??? . Actually, the picture is nicer if we throw in a copy of the non-normalized operator LŽ0. Ž0., so our generating function looks like F Ž LŽ0. Ž 0 . , LŽ0. Ž 0 . , LŽ1. Ž 0 . , . . . . s Tr Ž exp Ž 2p iLŽ0. Ž 0 . t . exp Ž 2p iLŽ0. Ž 0 . t 1 . exp Ž 2p iLŽ1. Ž 0 . t 2 . ??? . .

Ž 4.2. In some sense, this is like looking at a Cartan subalgebra of Dq= Q ? ŽyD . Žsemidirect product.. Let qk s expŽ2p it k ., and q s expŽ2p it .. Our generating function becomes F Ž LŽ0. Ž 0 . , LŽ0. Ž 0 . , LŽ1. Ž 0 . , . . . . s

`

3

Ł Ž 1 y q nq1nq2n q3n

ns1

5

??? .

y1

q1Ž1r2. z Žy1. q2Ž1r2. z Žy3. ??? .

Ž 4.3.

Consider the operator d k which associates to F the function of q

dk F [

1

d

2p i dt k

Ž log F . N t 1s t 2s? ? ? s0 .

Ž 4.4.

494

SPENCER BLOCH

Recall the familiar function in number theory sr Ž n. [ Ý d N n d r. One has defined w10, p. 111x the normalized Eisenstein series of weight 2 k E2 k [ 1 q

`

2

Ý s Ž n. q n . z Ž 1 y 2 k . 1 2 ky1

Ž 4.5.

E2 k is a modular form of weight 2 k for k G 2 and E2 satisfies the ‘‘modular-like’’ equation y1

ž /

E2

t

ty2 s E2 Ž t . q

12 2p it

.

Ž 4.6.

d k F s Ž z Ž1 y 2 k .r2. E2 k .

PROPOSITION Ž4.7..

Proof. A direct calculation gives

dk F s

z Ž1 y 2 k. 2

ž

2

1q

z Ž1 y 2 k.

Ý s 2 ky1Ž n . q n n

/

.

5. V AND VERTEX OPERATOR ALGEBRAS Here is another characterization of the elements of V ; EndŽ S .. PROPOSITION Ž5.1.. the form Xs

Let f Ž k . be a polynomial in k. Then any element of dim Ž h .

Ý is1

Ý f Ž k . hi Ž k . hi Ž n y k . kgZ

for n / 0, and any element of the form Ys

dim Ž h .

Ý f Ž k . h i Ž y< k <. h i Ž < k <.

Ý is1

kgZ

lie in V. Proof. If n / 0, h i Ž k ., and h i Ž n y k . commute, so replacing f Ž k . by Ž f Ž k . q f Ž n y k .., we may assume f Ž k . s f Ž n y k .. An obvious variant of Ž1.16. permits us to write f Ž k . s Ý i a i k i Ž n y k . i so X s Ý i a i LŽ i. Ž n.. Similarly, for Y we may assume f Ž k . s f Žyk ., so f Ž k . s Ý i a i k 2 i and Y s Ý i a i LŽ i. Ž0., proving Ž5.1.. 1 2

Recall, for h g h we write hŽ n. s h m t n, viewed as an endomorphism of S . Define, for r s 0, 1, 2 ???

vŽr. [

1 2

Ý h i Ž y1 y r . 2 . i

Ž 5.2.

DIFFERENTIAL OPERATORS ON THE CIRCLE

495

ŽHere, as usual, the h i are an orthonormal basis of h.. v Ž0. is denoted v in w4, 8.7.2x. The vertex operator algebra is given by VL ( S m Q L4 , where Q L4 is a ‘‘twisted’’ group ring of a lattice L ; h. ŽFor more details, cf. w4, Chap. 8x.. The vertex algebra structure is a map Y Ž ?, z . : VL ª End Ž VL . w z, zy1 x .

Ž 5.3.

ŽIn fact, the map Y Ž?, z . on S m 1 ; VL factors through EndŽ S . ; EndŽ VL .. It is this part that we will use, so the reader can ignore the group ring side.. We have by definition Y Ž h Ž yn . , z . [

1

Ž n y 1. !

Ž drdz .

ny 1

Ž hŽ z . . ,

Ž 5.4.

where hŽ z . [

Ý h Ž r . zyry1 .

Ž 5.5.

rgZ

Also Y Ž h1 Ž yn1 . ? h 2 Ž yn 2 . ? ??? ? h r Ž yn r . , z . [ Norm.Ord.Prod. Ž Y Ž h1 Ž yn1 . , z . , . . . , Y Ž h r Ž yn r . , z . . , Ž 5.6. where the notation refers to the normal ordered product on EndŽ VL . ww z, zy1 xx. Finally, for ¨ g VL one writes YŽ¨ , z. s

Ý ¨ n zyny1 ,

Ž 5.7.

n

so ¨ n g EndŽ VL .. It is known that the vector space spanned by the ¨ n as ¨ runs through VL is closed under commutator bracket and forms a Lie subalgebra of EndŽ VL .. Define operators l Ž r . Ž n. g EndŽ VL . by Y Ž vŽr., z . s

Ý l Ž r . Ž n . zyny2y2 r .

Ž 5.8.

n

One has l Ž0. Ž n. s LŽ0. Ž n. by w4, 8.7.6x. PROPOSITION Ž5.9.. The elements l Ž r . Ž n. for n g Z and r g N lie in V and together with the central element c form a ¨ ector space basis.

496

SPENCER BLOCH

Proof. By definition w4, 8.5x

Ý l Ž r . Ž n . zyny2y2 r n

s Y Ž vŽr., z . s

1 2

ž s

Ý Norm.ord.prod. i

1

Ž drdz .

r!

r

1

Ý h i Ž k . zyky1 ,

r!

k

1 2Ž r !.

2

Ž drdz .

r

Ý h i Ž m . zymy1 m

/

Norm.ord.prod.

= Ž yk y 1 . ??? Ž yk y r . Ý h i Ž k . zyky1yr ,

ž

k

Ž ym y 1 . ??? Ž ym y r . Ý h i Ž m . zym y1yr .

/

m

It follows that l Ž r . Ž 0. s l

Žr.

Ž n. s

1 2

Ý Ý i

kgZ

1 2

Ý Ý i

kgZ

ž r qr k / ž r yr k / h Ž y< k <. h Ž < k <. i

i

Ž 5.10.

ž

rqk r

/ž

r q n y k h Ž k. h Ž n y k. . i i r

/

Arguing as in Ž1.16. we can write

ž r qr k / ž r q nr y k / s Ý a k Ž n y k . j

j

j

jFr

with a j independent of k and a r / 0. Recalling that a basis of V is given by c and the elements LŽ r . Ž 0 . s LŽ r . Ž n . s

1 2

r Ý Ý Ž y1. k 2 r h i Ž y< k <. h i Ž < k <.

i

kgZ

1

Ý Ý 2 i

r

k r Ž n y k . hi Ž k . hi Ž n y k . ,

kgZ

the assertion of Ž5.9. follows easily.

497

DIFFERENTIAL OPERATORS ON THE CIRCLE

6. DIRICHLET L-SERIES In this section we show how extending our Fock representation to suitable differential operators of infinite order leads to a central cocycle involving values at negative integers of Dirichlet L-series. Recall we have t s e 2 p i x and D s tdrdt s Ž1r2p i . drdx. Our projective representation of Dq is given by LŽmr . [ Ž y1 . s

1 2

½

1 2

rq1

D r t m D rq1 ¬ LŽ r . Ž m . r

Ýi Ýk g Z Ž m y k . k r h i Ž m y k . h i Ž k . , r

Ý i Ý k g Z k r Ž yk . h i Ž y< k < . h i Ž < k < . ,

if m / 0; if m s 0.

Let N G 2 be an integer and let a g Z. Let x be a Dirichlet character. We assume

x / 1;

x Ž x q N . s x Ž x. ;

x Ž x . s 0 if Ž x, N . / 1;

x Ž y1 . s 1.

Then

Ž y1.

rq1

¬

D 2 rq1 cos Ž 2p aDrN . 1 2

Ý Ý i

r

cos Ž 2p akrN . k r Ž yk . h i Ž y< k < . h i Ž < k < . .

Ž 6.1.

kgZ

Writing the Gauss sum g Ž x . s Ý a modŽ N . x Ž a.expŽ2p iarN ., we have

x Ž k. s gŽ x .

y1

Ý x Ž a. exp Ž 2p iakrN . a

s gŽ x .

y1

Ý x Ž a. cos Ž 2p akrN . .

Ž 6.2.

a

It follows that

Ž y1.

rq1

D 2 rq1 g Ž x .

y1

Ý x Ž a. cos Ž 2p aDrN . a

Žr.

¬ L Ž 0. x [

1 2

r Ý Ý x Ž k . k r Ž yk . h i Ž y< k <. h i Ž < k <. . Ž 6.3.

i

kgZ

For any integer d / 0. The function cosŽ2p x . is invariant under x ¬ d y x, so we can write cosŽ2p x . s Ý n a nŽ d . x n Ž d y x . n. If we fix r G 0 and

498

SPENCER BLOCH

an integer m / 0, we can write gŽ x .

y1

`

x Ž a.

Ý

ns0

a modŽ N .

[

1 2

. Žr. a n Ž am . Ny2 na2 n LŽmnqr N ¬ L Ž mN . x

Ý

r Ý Ý x Ž k . Ž mN y k . k r h i Ž mN y k . h i Ž k . .

i

Ž 6.4.

kgZ

A calculation analogous to Ž2.18. yields PROPOSITION Ž6.5.. Let x and c be e¨ en Dirichlet characters mod N as abo¨ e, and let r, s G 0. Write t s r q s. Then with aŽj t . as in Ž2.15. LŽ r . Ž mN . x , LŽ s. Ž ymN . c t

s

Ý

js wŽ tq1 .r2 x

q

1 2

j 2 tq1y2 j Ž j. L Ž 0 . cx Ž y1. aŽj t . Ž mN .

mN

dim Ž h .

Ý Ž cx . Ž k . Ž mN y k . tq1 k tq1 .

ks0

Warning: these characters are not assumed primitive mod N, but they are still taken to vanish at any k with Ž N, k . / 1. As in Ž3.1. and Ž3.2. we define LŽ t . Ž m . x [

½

for m / 0;

LŽ t . Ž m . x , t

L Ž 0 . q Ž y1 . L Ž x , y1 y 2 t . ? c 1 2

Žt.

for m s 0,

Ž 6.6. where LŽ x , s . s Ý x Ž n. nys is the Dirichlet L-series, and c acts on S by the scalar dimŽ h .. We now consider the generalized Bernoulli numbers Bn, x associated to x . By definition, if x is primiti¨ e mod N, i.e., if N s conductorŽ x . then N

Ý as1

x Ž a . te at e Nt y 1

s

`

Ý

Bn , x t nrn!.

Ž 6.7.

ns0

By w9, Theorem 1, p. 11x we have for any integer n G 1 and x primitive mod N LŽ x , 1 y n. s y

Bn , x n

.

Ž 6.8.

499

DIFFERENTIAL OPERATORS ON THE CIRCLE

PROPOSITION Ž6.9.. Let x and c be Dirichlet characters mod N, and assume xc is primiti¨ e mod N. Then for t s r q s LŽ r . Ž mN . x , LŽ s. Ž ymN . c t

s

j 2 tq1y2 j Ž j. L Ž 0 . cx , Ž y1. aŽj t . Ž mN .

Ý

js wŽ tq1 .r2 x

i.e., the central term ¨ anishes. Proof. By Ž6.6. and Ž6.8., we must show

Ž y1.

t

t

Ý

js wŽ tq1 .r2 x

j 2 tq1y2 j y Ž y1. aŽj t . Ž mN .

ž

B2 tq2, xc 2t q 2

/

mN

s

Ý Ž cx . Ž k . Ž mN y k . tq1 k tq1 .

Ž 6.10.

ks0

LEMMA Ž6.11.. Let s be a non-tri¨ ial Dirichlet character modŽ N .. Let m G 1 be an integer and let f Ž x . be a polynomial. Then Nm

Ý

s Ž a. f Ž a. s

`

Bn , s

Ý

n s1

as1

n!

Ž f Ž ny1. Ž Nm . y f Ž ny1. Ž 0. . .

Proof of Lemma. Let d s drdx. Write J Ž f . s HxxqN f dx. We have formally N

Ý

N

s Ž a. f Ž x q a. s

as1

Ý as1

s Ž a . e ad ? d e Nd y 1

JŽ f . .

On the other hand, using that B0, s s 0 since s / 1, we get N

Ý as1

s Ž a . e ad ? d e

Nd

y1

JŽ f . s

`

Ý

n s1

Bn , s

n!

Ž f Ž ny1. Ž x q N . y f Ž ny1. Ž x . . .

Replacing f Ž x . by f Ž x q nN . for 0 F n F m y 1 and summing in the above formulas, we get the lemma. Returning to the proof of the proposition, we apply the lemma with s s xc and f Ž x . s x tq1 Ž mN y x . tq1. The computation now mimics Ž3.7. ] Ž3.12. in the proof of Ž3.4.. Simply replace m by mN and B2 jq2 by B2 jq2, xc in that calculation. Note, however, that because xc is non-trivial, the present calculation has no integral term like Ž3.6..

500

SPENCER BLOCH

One way to formulate this result is: COROLLARY Ž6.12.. Let x be an e¨ en Dirichlet character which is primiti¨ e mod N. Let V x ; EndŽ S . be the ¨ ector space spanned by the identity operator together with LŽ r . Ž m.x for r G 0 and m g Z. Then V x is a module for Dq under commutator bracket and the inclusion Q ? identity ; V x splits, i.e., V x ( Ž V xrQ ? id . [ Q ? id as Dq-modules.

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