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DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, TODA VERSUS PFAFF LATTICE AND RELATED POLYNOMIALS M. ADLER and P. VA...

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DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1,

TODA VERSUS PFAFF LATTICE AND RELATED POLYNOMIALS M. ADLER and P. VAN MOERBEKE

Abstract The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of gl (∞) into sp (∞) and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a set of bilinear identities for the wave functions, which yield the existence of a sequence of “τ -functions”. The latter satisfy their own set of bilinear identities, which moreover characterize them. In the semi-infinite case, the τ -functions are Pfaffians, in the same way that for the Toda lattice the τ -functions are Hänkel determinants; interesting examples occur in the theory of random matrices, where one considers symmetric and symplectic matrix integrals for the Pfaff lattice and Hermitian matrix integrals for the Toda lattice. There is a striking parallel between the Pfaff lattice and the Toda lattice, and even more striking, there is a map from one to the other, mapping skew-orthogonal to orthogonal polynomials. In particular, we exhibit two maps, dual to each other, mapping Hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals. Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Splitting theorems, as applied to the Toda and Pfaff lattices . . . . . . . . 2. Wave functions and their bilinear equations for the Pfaff lattice . . . . . . 3. Existence of the Pfaff τ -function . . . . . . . . . . . . . . . . . . . . . 4. Semi-infinite matrices m ∞ , (skew-)orthogonal polynomials, and matrix integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. ∂m ∞ /∂tk = 3k m ∞ , orthogonal polynomials, and Hermitian matrix integrals (α = 0) . . . . . . . . . . . . . . . . . . . . . . . . . .

2 10 15 22 30 30

DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 9 September 1999. Revision received 22 December 2000. 2000 Mathematics Subject Classification. Primary 37K10, 70H06; Secondary 70F45, 82C23. Adler’s work supported by National Science Foundation grant number DMS-98-4-50790. Van Moerbeke’s work supported by National Science Foundation grant number DMS-98-4-50790, a North Atlantic Treaty Organization grant, a Fonds National de la Recherche Scientifique grant, and a Francqui Foundation grant. 1

2

ADLER and VAN MOERBEKE

∂m ∞ /∂tk = 3k m ∞ + m ∞ 3>k , skew-orthogonal polynomials, and symmetric and symplectic matrix integrals (α = ±1) . . . . . . . . 5. A map from the Toda to the Pfaff lattice . . . . . . . . . . . . . . . . . 6. Example 1: From Hermitian to symmetric matrix integrals . . . . . . . . 7. Example 2: From Hermitian to symplectic matrix integrals . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Free parameter in the skew-Borel decomposition . . . . . . . . . . . B. Simultaneous (skew-)symmetrization of L and N . . . . . . . . . . . C. Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.

32 37 41 46 51 51 54 55 57

0. Introduction Consider a weight on R, depending on t = (t1 , t2 , . . .) ∈ C∞ , ρt (z) dz = e

P∞ 1

ti z i

P∞

ρ(z) dz = e−V (z)+

1

ti z i

dz, with −

ρ 0 (z) g(z) = V 0 (z) = , ρ(z) f (z) (0.1)

with rational g’s and f ’s, and with ρ(z) decaying fast enough at ∞. The Toda lattice, its τ -functions and Hermitian matrix integrals (revisited) This weight leads to a t-dependent moment matrix Z m n (t) = µk+` (t) 0≤k,`≤n−1 = z k+` ρt (z) dz , R

0≤k,`≤n−1

with the semi-infinite moment matrix m ∞ , satisfying the commuting equations ∂m ∞ = 3k m ∞ = m ∞ 3k . ∂tk

(0.2)

3 is the customary shift matrix, with zeros everywhere, except for 1’s just above the diagonal; that is, (3v)n = vn+1 . Consider the Borel decomposition into a lowertriangular and an upper-triangular matrix, m ∞ = S −1 S >−1 , and the following t-dependent matrix integrals (n ≥ 0): Z P∞ i τn (t) := eTr(−V (X )+ 1 ti X ) d X = det m n and Hn

(0.3)

τ0 = 1,

(0.4)

where d X is Haar measure on the ensemble Hn = {(n × n)–Hermitian matrices}. As is well known (e.g., see E. Witten [18] or M. Adler and P. van Moerbeke [6]), integral (0.4) is a solution to the following two systems:

TODA VERSUS PFAFF LATTICE

3

(i) the KP-hierarchy∗ 1 ∂2 ˜ sk+4 (∂) − τn ◦ τn = 0 2 ∂t1 ∂tk+3

for k, n = 0, 1, 2, . . . ;

(ii) the Toda lattice, that is, the tridiagonal matrix  1/2 τ0 τ2 τ1 ∂  ∂t1 log τ0 τ12  1/2   τ0 τ2 τ2 ∂  2 ∂t1 log τ1 −1 τ  1 L(t) := S3S =  1/2  τ1 τ3  0  τ22  .. .

(0.5)



0 1/2

τ1 τ3 τ22

∂ ∂t1

τ3 τ2

log

..

.

     ,  ..  .  

(0.6)

satisfying the commuting Toda equations h1 i ∂L = (L n )sk , L , ∂tn 2 where (A)sk denotes the skew part of the matrix A for the Lie algebra splitting into skew and lower-triangular matrices. Moreover, the following t-dependent polynomials in z are defined by the S-matrix obtained from the Borel decomposition (0.3); it is also given, on the one hand, in terms of the functions τn (t) and, on the other hand, by a classic determinantal formula (for a ∈ C, define [a] := (a, a 2 /2, a 3 /3, . . .) ∈ C∞ ): pn (t; z) :=

n X i=0

Sni (t)z i = z n

τn (t − [z −1 ]) √ τn τn+1 

1 =√ τn τn+1

   det    

1 z .. .

m n (t) µn,0 (t)

...

µn,n−1 (t) z n

    .   

The pn (t; z)’s are orthonormal with respect to the (symmetric) inner product h , isy , defined by hz i , z j isy = µi j , which is a restatement of the Borel decomposition (0.3) (see [6]). The vector p(t; z) = ( pn (t; z))n≥0 is an eigenvector of the matrix L(t) in (0.6): L(t) p(t; z) = zp(t; z). P∞ i P ˜ := s` (∂/∂t1 , (1/2)(∂/∂t2 ), s` ’s are the elementary Schur polynomials e 1 ti z := i≥0 si (t)z i and s` (∂) . . .). Given a polynomial p(t1 , t2 , . . .), define the customary Hirota symbol p(∂t ) f ◦ g := p(∂/∂ y1 , ∂/∂ y2 , . . .) f (t + y)g(t − y) .

∗ The

y=0

4


The Pfaff lattice and its τ -functions For use throughout this paper, define the skew-symmetric matrix          J =       

..



. 0 1 −1 0

0 0 1 −1 0

0

0 1 −1 0 ..

.

        ,       

with J 2 = −I,

(0.7)

and the involution on the space D := gl∞ of infinite matrices, J : D −→ D : a 7 −→ J (a) := J a > J.

(0.8)

Also, consider the splitting of D = k + n into two Lie subalgebras k and n, with the corresponding projections denoted πk and πn , where k is the Lie algebra of lowertriangular matrices with some special feature (see (1.17)) and where n := {a ∈ D such that J a > J = a} = sp(∞). Given a skew-symmetric semi-infinite matrix m ∞ , consider the commuting differential equations ∂m ∞ = 3i m ∞ + m ∞ 3>i ; (0.9) ∂ti they maintain the skew-symmetry of m ∞ . The Borel decomposition of m ∞ into lower-triangular times upper-triangular matrices requires the insertion of the skewsymmetric matrix J : m ∞ (t) = Q −1 (t)J Q −1> (t). (0.10) Dressing up the shift 3 with the lower-triangular matrix Q(t) leads to the commuting equations (0.11). THEOREM 0.1 The Pfaff lattice equations

∂L = [−πk L i , L] = [πn L i , L] ∂ti

(0.11)


5

maintain the locus of semi-infinite matrices of the form (ai 6= 0):   0 1  −d1 a1 O      d1 1   .  L=  −d2 a2     ∗ d 2   .. .

(0.12)

The solutions L to (0.11) of the form (0.12) are given by L(t) = Q(t)3Q −1 (t), where Q is a lower-triangular matrix, whose entries are given by the coefficients of the polynomials, obtained by the finite Taylor expansion in z −1 of τ2n (t −[z −1 ]) below (h 2n := τ2n+2 (t)/τ2n (t)): q2n (t; z) :=

2n X

−1/2 τ2n (t

− [z −1 ]) , τ2n (t)

Q 2n, j (t)z j = z 2n h 2n

j=0

q2n+1 (t; z) :=

2n+1 X j=0

−1/2 (z

Q 2n+1, j (t)z j = z 2n h 2n

+ ∂/∂t1 )τ2n (t − [z −1 ]) , τ2n (t)

(0.13)

with τ0 , τ2 , τ4 , a sequence of functions of t1 , t2 , . . . , characterized by the following bilinear identities for all n, m ≥ 0 (τ0 = 1): I P∞ dz 0 i τ2n t − [z −1 ] τ2m+2 t 0 + [z −1 ] e 1 (ti −ti )z z 2n−2m−2 2πi z=∞ I P∞ 0 dz −i + τ2n+2 t + [z] τ2m t 0 − [z] e 1 (ti −ti )z z 2n−2m = 0. (0.14) 2πi z=0 Remark. Theorem 0.1 is robust and remains valid for the bi-infinite matrix L. In that case, the summations in the expressions q2n and q2n+1 run from j = −∞ instead of running from j = 0. The τ -functions are given by Pfaffians pf m 2n (t) and satisfy, as a consequence of the bilinear relations (0.14), the Pfaffian KP-hierarchy for k, n = 0, 1, 2, . . . , 2 ˜ τ2n+2 ◦ τ2n−2 . ˜ −1 ∂ sk+4 (∂) τ2n ◦ τ2n = sk (∂) 2 ∂t1 ∂tk+3

(0.15)

The t-dependent polynomials qn (t; z) = (Q(t) (1, z, z 2 , z 3 , . . .)> )n in z, obtained in (0.13), are “skew-orthonormal” with respect to the skew-inner product h , isk , defined

6


by hy i , z j isk = µi j (t), namely, hqi , q j isk

0≤i, j−1 N (2t)±1 = −O(±) (t)J O(±) (t).

(0.23)

Then, the lower-triangular matrices O(±) (t) map orthonormal polynomials into skew‡ ˜ orthonormal polynomials, and the tridiagonal L-matrix into an L-matrix: pn (t; z) 7−→ qn(±) (t; z) = O(±) (t) p(t; z) n , ˜ L(t) 7−→ L(t) = O(±) (t)L(2t)O(±) (t)−1 (Toda lattice)

(0.24)

(Pfaff lattice).

It also maps the weight into a new weight ρ(z) = e−V (z) 7−→ ρ˜± (z) = e−V (z) := e−(1/2)(V (z)∓log f (z)) , ˜

and the corresponding string of τ -functions into a new string of Pfaffian τ -functions P i (remember Vt (z) = V (z) − ∞ 1 ti z ): ( (+) R Z ˜ τ2n (t) := T2n eTr 2(−Vt (X )) d X (β = 4), Tr(−Vt (X )) τk (t) = e d X 7 −→ R ˜ (−) τ2n (t) := S2n eTr(−Vt (X )) d X (β = 1). Hk For the classical orthogonal polynomials pn (z), we have shown in [6] that N (0) is not only skew-symmetric but also tridiagonal; that is, 

b0  a0  L=  

a0 b1 a1

 a1 b2 .. .

.. ..

. .

  ,  



0  −c0  −N =   

c0 0 −c1

 c1 0 .. .

..

.

  .   (0.25)

∗ See

Appendix B. upper signs (resp., lower signs) correspond to each other throughout this section. ‡ We have p(t; z) := ( p (t; z), p (t; z), . . .)> . 0 1 † The

10


In Sections 6 and 7, we show that the maps O(−) and O(+) , as in (0.24), involve only three steps, in the following sense: for β = 1, r c2n (−) q2n (0; z) = p2n (0; z), a2n r a2n (−) q2n+1 (0; z) = c2n 2n c2n X × − c2n−1 p2n−1 (0, z) + bi p2n (0; z) + c2n p2n+1 (0; z) ; a2n 0

(0.26) for β = 4, r √ a2n−2 (+) (+) p2n (0; z) = −c2n−1 q2n−2 (0; z) + a2n c2n q2n (0; z), c2n−2 r a2n−2 (+) p2n+1 (0; z) = −c2n q (0; z) c2n−2 2n−2 r 2n X r c c2n (+) 2n (+) − bi q (0; z) + q (0; z). a2n 2n a2n 2n+1

(0.27)

0

The abstract map O(−) for t = 0 has already appeared in the work of E. Brézin and H. Neuberger [9]. This has been applied recently in [1] to a problem in the theory of random matrices. 1. Splitting theorems, as applied to the Toda and Pfaff lattices In this section, we show how the equations ∂m ∞ = 3i m ∞ ∂ti

and

∂m ∞ = 3i m ∞ + m ∞ 3>i ∂ti

(1.1)

both lead to commuting Hamiltonian vector fields related to a Lie algebra splitting. First recall the splitting theorem due to Adler, B. Kostant, and W. Symes in [5], and later recall the R-version due to A. Reyman and M. Semenov-Tian-Shansky [15]. The R-version allows for more general initial conditions. PROPOSITION 1.1 Let g = k + n be a (vector space) direct sum of a Lie algebra g in terms of Lie subalgebras k and n, with g paired with itself via a nondegenerate ad-invariant inner product∗ h , i; this in turn induces a decomposition g = k⊥ + n⊥ and isomorphisms ∗ hAd

g

X ; Y i = hX, Adg−1 Y i, g ∈ G, and thus h[z, x], yi = hx, −[z, y]i.


11

g ' g∗ , k⊥ ' n∗ , n⊥ ' k∗ . Let πk and πn be projections onto k and n, respectively. Let G , Gk , and Gn be the groups associated with the Lie algebras g, k, and n. Let I (g) be the Ad∗ ' Ad-invariant functions on g∗ ' g. (i) Then, given an element ε ∈ g : [ε, k] ⊂ k⊥

and [ε, n] ⊂ n⊥ ,

the functions ϕ(ε + ξ 0 )|k⊥ ,

with ϕ ∈ I (g) and ξ 0 ∈ k⊥ ,

(1.2)

respectively, Poisson commute for the respective Kostant-Kirillov symplectic structures of n ∗ ' k⊥ ; the associated Hamiltonian flows are expressed in terms of the Lax pairs∗ ξ˙ = [−πk ∇ϕ(ξ ), ξ ] = [πn ∇ϕ(ξ ), ξ ]

for ξ ≡ ε + ξ 0 , ξ 0 ∈ k⊥ .

(1.3)

(ii) The splitting also leads to a second Lie algebra g R , derived from g, such that g∗R ' g R , namely, g R : [x, y] R =

1 1 [Rx, y] + [x, Ry] = [πk x, πk y] − [πn x, πn y], 2 2

(1.4)

with R = πk − πn . The functions ϕ(ξ )|g R ,

with ϕ ∈ I (g) and ξ ∈ g R ,

respectively, Poisson commute for the respective Kostant-Kirillov symplectic structures of g∗R ' g R , with the same associated (Hamiltonian) Lax pairs ξ˙ = [−πk ∇ϕ(ξ ), ξ ] = [πn ∇ϕ(ξ ), ξ ]

for ξ ∈ g R .

(1.5)

Each of the equations (1.3) and (1.5) has the same solution expressible in two different ways:† ξ(t) = Ad K (t) ξ0 = Ad S −1 (t) ξ0 , (1.6) with‡

K (t) = πGk et∇ϕ(ξ0 )

and

S(t) = πGn et∇ϕ(ξ0 ) .

is defined as the element in g∗ such that dϕ(ξ ) = h∇ϕ, dξ i, ξ ∈ g. naively write Ad K (t) ξ0 = K (t)ξ0 K (t)−1 , Ad S −1 ξ0 = S −1 (t)ξ0 S(t). ‡ Consider the group factorization A = π Gk A πGn A. ∗ ∇ϕ

† We

12


Example 1 (The standard Toda lattice and the equations ∂m/∂ti = 3i m for the Hänkel matrix m ∞ ) Since, in particular, the matrix m ∞ is symmetric, the Borel decomposition into lowertriangular times upper-triangular matrix must be done with the same lower-triangular matrix S: m ∞ = S −1 S >−1 . (1.7) In turn, the matrix S defines a wave vector 9, and operators∗ L and M, the same as the ones defined in (0.22): P∞

9(t, z) := e(1/2) 1 ti z Sχ, L := S3S −1 , ∞ 1X M := S ∂ + iti 3i−1 S −1 , 2 i

(1.8)

1

satisfying the following well-known equations:† L9 = z9,

∂ 9, with [L , M] = 1, ∂z ∂9 1 = (L n )sk 9, ∂tn 2 ∂M 1 = (L n )sk , M . ∂tn 2

M9 =

∂S 1 = − (L n )bo S, ∂tn 2 1 ∂L = (L n )sk , L ∂tn 2

(1.9)

The wave vector 9 can then be expressed in terms of a sequence of τ -functions τn (t) = det m n (t), but it also has a simple expression in terms of orthonormal polynomials, with respect to the moment matrix m ∞ : 9(t, z) = e(1/2)

P

ti z i

= e(1/2)

P

ti z i

τn (t − [z −1 ]) zn p τn (t)τn+1 (t) n≥0 pn (t, z) n≥0 .

(1.10)

The vector fields (1.9) on L are commuting Hamiltonian vector fields, in view of the Adler-Kostant-Symes (AKS) splitting theorem (version (i)), ∂L = [−πk ∇ Hi , L] = [πn ∇ Hi , L], ∂ti

Hi =

tr L i+1 , ∇ Hi = L i , i +1

(1.11)

with L = 3> a + b + a3,

a and b diagonal matrices,

(1.12)

the formulas below, χ (z) = (z 0 , z, z 2 , . . .)> and ∂ is the matrix such that (d/dz)χ (z) = ∂χ (z). notations ( )sk and ( )bo refer to the skew part and the lower-triangular (Borel) part, respectively, that is, projection onto k and n, respectively.

∗ In

† The


13

for the splitting of the Lie algebra of semi-infinite matrices D = gl∞ = k + n := {skew-symmetric} + {lower-triangular}

= k⊥ + n⊥ := {symmetric} + {strictly upper-triangular}, (1.13) with the form (1.12) of L being preserved in time. Note that the solution (1.6) to the differential equation (1.5) in the AKS theorem can also be exressed by the factorization of m ∞ followed by the dressing up of 3. i

Example 2 (The Pfaff lattice and the equations ∂m/∂ti = 3i m + m3> ) Throughout this paper the Lie algebra D = gl∞ of semi-infinite matrices is viewed as being composed of (2 × 2)-blocks. It admits the natural decomposition into subalgebras: D = D− ⊕ D0 ⊕ D+ = D− ⊕ D0− ⊕ D0+ ⊕ D+ , (1.14) where D0 has (2×2)-blocks along the diagonal with zeros everywhere else and where D+ (resp., D− ) is the subalgebra of upper-triangular (resp., lower-triangular) matrices with (2 × 2)-zero matrices along D0 and zero below (resp., above). As pointed out in (1.14), D0 can further be decomposed into two Lie subalgebras: D0− = {all (2 × 2)-blocks ∈ D0 are proportional to Id}, D0+ = {all (2 × 2)-blocks ∈ D0 have trace zero}.

(1.15)

Remember from (0.7) and (0.8) in the introduction that the matrix J and the associated Lie algebra involution J . The splitting into two Lie subalgebras∗ D = k + n,

(1.16)

with k = D− + D0−       = algebra of      

..

.

0 Q 2n,2n 0

0 Q 2n,2n

∗

Q 2n+2,2n+2 0

0 Q 2n+2,2n+2

n = {a ∈ D , such thatJ a = a} = {b + J b, b ∈ D } = sp(∞), ∗ Note

that n is the fixed point set of J .

..

.

      ,      (1.17)

14


with corresponding Lie groups∗ Gk and Gn = Sp(∞), plays a crucial role here. Let πk and πn be the projections onto k and n. Notice that n = sp(∞) and Gn = Sp(∞) stand for the infinite-rank affine symplectic algebra and group (e.g., see [12]). Any element a ∈ D decomposes uniquely into its projections onto k and n, as follows: a = πk a + πn a 1 1 = a− − J a+ + a0 − J a0 + a+ + J a+ + a0 + J a0 . 2 2 (1.18) The following splitting, with k+ = D+ + D0−

and

n+ = n,

is also used in Section 2; the projections take on the following form: a = πk+ a + πn+ a n 1 o n 1 o = a+ − J a− + a0 − J a0 + a− + J a− + a0 + J a0 . 2 2 (1.19) Note that J intertwines πk and πk+ : J πk = πk+ J .

(1.20)

For a skew-symmetric semi-infinite matrix m ∞ , the skew-Borel decomposition m ∞ := Q −1 J Q −1> ,

with Q ∈ Gk ,

(1.21)

is unique, as was shown in [2]. Here we may assume m ∞ to be bi-infinite, as long as factorization (1.21) is unique, upon imposing a suitable normalization. Then we use Q to dress up 3: L = Q3Q −1 . i

Then letting m ∞ run according to the equations ∂m/∂ti = 3i m + m3> , we show in the next proposition and corollary that L evolves according to a system of commuting equations, which by virtue of the AKS theorem are Hamiltonian vector fields (for details, see [2]). ∗G k

is the group of invertible elements in k, that is, invertible lower-triangular matrices, with nonzero (2 × 2)-blocks proportional to Id along the diagonal.


15

PROPOSITION 1.2 For the matrices

m ∞ := Q −1 J Q −1>

and

L := Q3Q −1 ,

with Q ∈ Gk ,

the following three statements are equivalent: ∂ Q −1 (i) Q = −πk L i , ∂ti ∂ Q −1 (ii) Li + Q ∈ n, ∂ti ∂m ∞ i (iii) = 3i m ∞ + m ∞ 3> . ∂ti Whenever the vector fields on Q or m satisfy (i), (ii), or (iii), the matrix L = Q3Q −1 is a solution of the AKS-Lax pair ∂L = [−πk L i , L] = [πn L i , L]. ∂ti Proof Written out and using (1.18), Proposition 1.2 amounts to showing the equivalence of the three formulas: 1 ∂ Q −1 (I) Q + (L i )− − J (L i+ )> J + (L i )0 − J ((L i )0 )> J = 0, ∂ti 2 ∂ Q −1 ∂ Q −1 > (II) Li + Q − J Li + Q J = 0, ∂ti ∂ti ∂m ∞ (III) 3i m ∞ + m ∞ 3>i − = 0. ∂ti The point is to show that (I)+ = 0,

(I)− = (II)− = −J (II+ )> J,

(I)0 =

1 (II)0 , 2

>

Q −1 (II)J Q −1 = (III).

(1.22)

The details of this proof are found in [2]. 2. Wave functions and their bilinear equations for the Pfaff lattice Consider the commuting vector fields ∂m ∞ /∂ti = 3i m ∞ + m ∞ 3>i

(2.1)

on the skew-symmetric matrix m ∞ (t) and the skew-Borel decomposition m ∞ (t) = Q −1 (t)J Q >−1 (t),

Q(t) ∈ Gk ;

(2.2)

16


remember from (1.17) that Q(t) ∈ Gk means that Q(t) is lower-triangular, with (2 × 2)-matrices c2n I along the “diagonal,” with c2n 6 = 0. In this section, we give the properties of the wave vectors and their bilinear relations. In this and the next section, the matrices are assumed to be bi-infinite; the semi-infinite case is dealt with by specialization. Upon setting Q 1 = Q(t)

and

Q 2 = J Q >−1 (t),

(2.3)

the matrix Q(t) defines wave operators W1 (t) := Q 1 (t)e

P∞ 1

ti 3i

,

W2 (t) := Q 2 (t)e−

P∞ 1

ti 3>i

= J W1−1> (t),

(2.4)

L-matrices L := L 1 := Q 1 3Q −1 1 ,

L 2 := −J (L 1 ) = Q 2 3> Q −1 2 ,

(2.5)

and wave and dual wave vectors 91 (t, z) = W1 (t)χ(z),

91∗ (t, z) = W1−1 (t)> χ(z −1 ) = −J 92 (t, z −1 ),

92 (t, z) = W2 (t)χ(z),

92∗ (t, z) = W2−1 (t)> χ(z −1 ) = J 91 (t, z −1 ).

(2.6)

From the definition, it follows that the wave functions 91 have the following asymptotics: ( P k 91,2n (t, z) := e tk z z 2n c2n (t)ψ1,2n (t, z), ψ1,2n = 1 + O(z −1 ), P k 91,2n+1 (t, z) = e tk z z 2n+1 c2n (t)ψ1,2n+1 (t, z), ψ1,2n+1 = 1 + O(z −2 ), ( P −k −1 92,2n (t, z) = e− tk z z 2n+1 c2n (t)ψ2,2n (t, z), ψ2,2n = 1 + O(z), P −k −1 92,2n+1 (t, z) = e− tk z z 2n (−c2n (t))ψ2,2n+1 (t, z), ψ2,2n+1 = 1 + O(z 2 ), (2.7) where the ci are the elements of the diagonal part of Q. THEOREM 2.1 The following statements are equivalent: (i) m ∞ satisfies ∂m ∞ /∂ti = 3i m ∞ + m ∞ 3>i , (ii) Q 1 satisfies the hierarchy of equations (with L i defined in (2.5))

∂ Q1 = −(πk L i1 )Q 1 , ∂ti (iii)

Q 2 = J Q >−1 satisfies 1 ∂ Q2 = − J (πk L i1 ) Q 2 = (πk+ L i2 )Q 2 , ∂ti

(2.8)


(iv)

91 = e

P∞ 1

ti z i Q

1 (t)χ(z)

17

satisfies ∂91 = (πn L i1 )91 , ∂ti

(v)

92 = e−

P∞ 1

ti z −i

(2.9)

J Q >−1 (t)χ(z) satisfies 1

∂92 = −(L i2 − πk+ L i2 )92 = −(πn+ L i2 ), 92 , ∂ti 91 and 92 satisfy the bilinear identity for all n, m ∈ Z, I I dz 0 −1 dz 91,n (t, z)92,m (t , z ) + 92,n (t, z)91,m (t 0 , z −1 ) = 0. 2πi z 2πi z ∞ 0 (2.10) If any one of these six conditions is satisfied, then (vi)

∂ L1 = [−πk L i1 , L 1 ], ∂ti

∂ L2 = [πk+ L i2 , L 2 ], ∂ti

(2.11)

and L 1 91 = z91 ,

L 2 92 = z −1 92 ,

? ? L> 1 91 = z91 ,

? −1 ? L> 2 92 = z 92 .

(2.12)

For later use, we also consider the “monic” wave functions, with the factors c2n (t) removed, that is, ˆ 1 (t, z) := Q −1 91 9 0

and

ˆ 2 (t, z) := Q 0 92 9

(2.13)

and the matrix Lˆ 1 , normalized so as to have 1’s above the main diagonal, with Qˆ := Q −1 0 Q, −1 −1 −1 ˆ Qˆ −1 , Lˆ 1 = Q −1 = Q3 0 L 1 Q 0 = (Q 0 Q)3(Q 0 Q) −1 ˆ Lˆ 2 = Q 0 L 2 Q −1 0 = −Q 0 J (L 1 )Q 0 = −J ( L 1 ).

(2.14)

Then, in terms of the elements qî j of the matrix Qˆ := Q −1 0 Q, one easily computes by conjugation that Lˆ 1 has the following block structure: −1 −1 −1 Lˆ 1 = Q −1 0 L 1 Q 0 = (Q 0 Q)3(Q 0 Q)  .. .  . . . Lˆ ˆ 01 0 00 L   ˆ ˆ ˆ L 10 L 11 L 12  =  ∗ Lˆ 21 Lˆ 22   ∗ ∗ Lˆ 32 

 0 0 Lˆ 23 Lˆ 33 .. .

     ,   . . . 

18


with Lˆ ii :=

0 0 Lˆ i,i+1 := , −qˆ2i+2,2i+1 1 0 ! 2 − qˆ2i+3,2i+1 + qˆ2i+2,2i ∗ −qˆ2i+2,2i+1 . (2.15) := ∗ ∗

qˆ2i,2i−1 qˆ2i+1,2i−1 − qˆ2i+2,2i

Lˆ i+1,i

1

,

These definitions lead to a new statement that is equivalent to Theorem 2.1. 2.2 ˆ 9 ˆ 1, 9 ˆ 2 satisfy the following equations: Lˆ i , Q,

THEOREM

∂ Qˆ = − ( Lˆ n1 )− − Q −2 J (( Lˆ n1 )+ )> J Q 20 Qˆ 0 ∂tn

(2.16)

and ˆ 1 = z9 ˆ 1, Lˆ 1 9

ˆ 2 = z −1 9 ˆ 2, Lˆ 2 9

(2.17)

with ∂ ˆ 1 (t, z) = ( Lˆ n )+ + ( Lˆ n )0 + Q −2 J (( Lˆ n )+ )Q 20 9 ˆ 1 (t, z), 9 1 1 1 0 ∂tn ∂ ˆ 2 (t, z) = J ( Lˆ n )+ + ( Lˆ n )0 + Q −2 J (( Lˆ n )+ )Q 20 9 ˆ 2 (t, z) 9 1 1 1 0 ∂tn ˆ 2 (t, z). = − ( Lˆ n )− + ( Lˆ n )0 + Q 20 J (( Lˆ n )− )Q −2 9 2

2

2

0

The proof of Theorem 2.1 hinges on the following proposition. PROPOSITION 2.3 The following three statements are equivalent: (i) ∂m ∞ /∂ti = 3i m ∞ + m ∞ 3>i , (ii) the matrices W1 (t) and W2 (t) (defined in (2.4)) satisfy

W1 (t)W1 (t 0 )−1 = W2 (t)W2 (t 0 )−1 , (iii)

(2.18)

the 9i (t, z) = Wi (t)χ(z) satisfy the bilinear identity I I dz 0 −1 dz 91,n (t, z)92,m (t , z ) + 92,n (t, z)91,m (t 0 , z −1 ) = 0. 2πi z 2πi z ∞ 0

Proof The solution to (2.1) is given by m ∞ (t) = e

P∞ 1

tk 3k

m ∞ (0)e

P∞ 1

tk 3>k

.


19

Therefore, skew-Borel decomposing m ∞ (t) and m ∞ (0), we find Q −1 (0)J Q >−1 (0) = e−

P

ti 3i

Q −1 (t)J Q >−1 (t)e−

P

ti 3>i

,

(2.19)

and so, from the definition of W1 and W2 , W1−1 (0)W2 (0) = Q −1 (0)J Q >−1 (0) P∞ P i −1 i >−1 = Q(t)e 1 ti 3 J Q(t)e ti 3 = W1 (t)

−1

(using (2.19))

J W1 (t)

>−1

= W1 (t)−1 W2 (t),

(2.20)

implying the independence in t of the right-hand side of (2.20). Therefore, we have W1 (t)−1 W2 (t) = W1 (t 0 )−1 W2 (t 0 ) for all t, t 0 ∈ C∞ , and so W1 (t)W1−1 (t 0 ) = W2 (t)W2−1 (t 0 ), thus yielding (ii). Reversing the steps yields the differential equation (i). Finally, the proof of the bilinear identity (iii) proceeds as follows. Using the wellknown formula (see [3, Prop. 4.1]), I dz 0 −1 91 (t, z) ⊗ 91∗ (t 0 , z) W1 (t)W1 (t ) = , ∞ 2πi z 2 2 2 2 0 statement (ii) becomes I I dz dz 91 (t, z) ⊗ 91∗ (t 0 , z) = 92 (t, z) ⊗ 92∗ (t 0 , z) , 2πi z 2πi z ∞ 0 whose (m, n)th component is I I dz dz ∗ 0 ∗ 91,n (t, z)91,m (t , z) − 92,n (t, z)92,m (t 0 , z) = 0. 2πi z 2πi z ∞ 0 Next we use the relations 91∗ (t, z) = −J 92 (t, z −1 ) and 92∗ (t, z) = J 91 (t, z −1 ) to yield I I dz dz 91 (t, z) ⊗ J 92 (t 0 , z −1 ) + 92 (t, z) ⊗ J 91 (t 0 , z −1 ) = 0, 2πi z 2πi z ∞ 0 which again leads to (iii). That (iii) implies (ii) is obtained by reversing the arguments.

20


Proof of Theorem 2.1 The proof of statement (ii) for Q 1 , namely, ∂ Q1 = −(πk L i1 )Q 1 , ∂ti follows at once from Proposition 1.2. The proof of (iii) for Q 2 = J Q >−1 is based on the identity J πk a = πk+ J a. 1 Indeed, we compute ∂ Q> ∂ Q 2 −1 1 Q 2 = −J Q >−1 Q >−1 Q −1 1 1 2 ∂ti ∂ti i > >−1 > = −J Q >−1 Q> Q1 J 1 (πk L 1 ) Q 1 1

= −J (πk L i1 )> J = −J (πk L i1 ) = −πk+ J L i1 = −πk+ J (−J L 2 )i

(using (2.5))

= −πk+ J (−1) (J L 2 )i i

= −πk+ J (−1)i (−1)i−1 J L i2 = πk+ L i2 . Statements (iv) and (v) for 91 , 92 are straightforwardly equivalent to (ii) and (iii), respectively. According to Propositions 1.2 and 2.3 combined, the bilinear identity (2.10) in (vi) is equivalent to statement (i), (ii), or (iii). The hierarchy concerning the L i ’s follows at once from (ii) and (iii), thus ending the proof of Theorem 2.1. Proof of Theorem 2.2 To prove (2.16), remember from Theorem 2.1 that 1 ∂ Q −1 Q = −πk L n = − (L n )− − J (L n+ )> J − (L n )0 − J ((L n )0 )> J ; ∂tn 2 hence, taking the ( )0 -part of this expression yields ∂Q 1 1 ∂ log Q 0 = Q −1 = −πk (L n )0 = − (L n )0 + J (L n )> 0 J. 0 ∂tn ∂tn 2 2 ˙ Using the fact that Q 0 , Q −1 0 , Q 0 ∈ Gk ∩D0 commute among themselves and commute ˆ with J and the fact that D0 D± , D± D0 ⊂ D± , we compute for Qˆ = Q −1 0 Q, L 1


21

= Q −1 0 L 1 Q 0 (see (2.14)) ∂ Qˆ ˆ −1 −1 ˙ −1 ˙ −1 Q 0 Q = −Q −1 Q 0 + Q −1 0 Q0 Q0 Q Q 0 QQ ∂tn = −Q −1 Q˙ 0 + Q −1 Q˙ Q −1 Q 0 0

0

˙ −1 ˙ −1 = Q −1 0 (− Q 0 Q 0 + Q Q )Q 0 = Q −1 − (L n1 )− + J (L n1+ )> J Q 0 0 −1 −1 n −1 n = −(Q −1 0 L 1 Q 0 )− + Q 0 J Q 0 (Q 0 L 1 Q 0 )+ Q 0 > ˆn = −( Lˆ n1 )− + Q −2 J Q 20 . 0 J ( L 1 )+

>

J Q0

ˆ 1 (t, z) = z 9 ˆ 1 (t, z), we find Using this result and Lˆ 1 9 ˆ 1 (t, z) ∂9 ∂ P ti z i ˆ = e Qχ(z) ∂tn ∂tn P∞ i P∞ i 2 ˆ ˆ ˆn > = z n e 1 ti z Qχ(z) + e 1 ti z − ( Lˆ n1 )− + Q −2 0 J (( L 1 )+ ) J Q 0 Qχ(z) 2 ˆ ˆn > = Lˆ n1 − ( Lˆ n1 )− + Q −2 0 J ((L1 )+ ) J Q 0 91 (t, z) 2 ˆ ˆn = ( Lˆ n1 )+ + ( Lˆ n1 )0 + Q −2 (2.21) 0 (J ( L 1 )+ )Q 0 91 (t, z). ˆ 1 = Q −1 91 (t, z) and 9 ˆ 2 = Q 0 92 (t, z) satisfy, using W2 = But we also have that 9 0 −1> J W1 , ˆ 1 (t, z) ∂9 −1 −1 · · −1 ˆ = (Q −1 0 W1 ) χ(z) = (Q 0 W1 ) (Q 0 W1 ) 91 (t, z), ∂tn ˆ 2 (t, z) ∂9 = (Q 0 W2 )· χ(z) = (Q 0 W2 )· (Q 0 W2 )−1 (Q 0 92 ) ∂tn = ( Q˙ 0 W2 + Q 0 W˙ 2 )W −1 Q −1 (Q 0 92 ) 2

(2.22)

0

−1 −1 ˙ = ( Q˙ 0 Q −1 0 + Q 0 W2 W2 Q 0 )Q 0 92 −1 −1 ˙ = Q˙ 0 Q −1 0 + Q 0 J ( W1 W1 )Q 0 Q 0 92 −1 > −1 ˙ = Q˙ 0 Q −1 0 + Q 0 J ( W 1 W 1 ) J Q 0 Q 0 92 −1 ˙ −1 > = − J Q˙ 0 Q −1 0 + J (Q 0 W1 W1 Q 0 ) J Q 0 92 −1 ˙ −1 > ˙ = J (−Q −1 0 Q 0 + Q 0 W1 W1 Q 0 ) J Q 0 92 −1 · −1 = J (Q −1 (Q 0 92 ). (2.23) 0 W1 ) (Q 0 W1 )

Comparing (2.21), (2.22), and (2.23), and invoking (2.14), we have −J ( Lˆ n1 ) = Lˆ n2 ,

22


and so, in particular, −J (( Lˆ n1 )± ) = ( Lˆ n2 )∓

and

− J (( Lˆ n1 )0 ) = ( Lˆ n2 )0 ,

ˆ 2 (t, z) ∂9 ˆ 2 (t, z), = − ( Lˆ n2 )− + ( Lˆ n2 )0 + Q 20 (J (L n2 )− )Q −2 9 0 ∂tn which establishes Theorem 2.2. 3. Existence of the Pfaff τ -function The point of this section is to show that the solution of the Pfaff lattice can be expressed in terms of a sequence of functions τ , which are not τ -functions in the usual sense but which enjoy a different set of bilinear identities and partial differential equations. PROPOSITION 3.1 There exist functions τ2n (t) such that

ψ1,2n (t, z) =

τ2n (t − [z −1 ]) τ2n (t)

and

ψ2,2n (t, z) =

τ2n+2 (t + [z]) . τ2n+2 (t)

(3.1)

The proof of Proposition 3.1 is postponed until later. For future use, we define the diagonal matrix τ2n+2 . (3.2) h = diag(. . . , h −2 , h −2 , h 0 , h 0 , h 2 , h 2 , . . .) ∈ D0− , with h 2n = τ2n 3.2 91 and 92 have the following τ -function representation: THEOREM

91,2n (t, z) = e

P

−1 ti z i 2n −1/2 τ2n (t − [z ]) , z h 2n τ2n (t)

+ ∂/∂t1 )τ2n (t − [z −1 ]) , τ2n (t) P −i −1/2 τ2n+2 (t + [z]) 92,2n (t, z) = e− ti z z 2n+1 h 2n , τ2n (t) −1 − ∂/∂t )τ P −i 1 2n+2 (t + [z]) −1/2 (z , 92,2n+1 (t, z) = e− ti z z 2n+1 h 2n τ2n (t) 91,2n+1 (t, z) = e

P

ti z i 2n −1/2 (z z h 2n

(3.3)

with the τ2n (t) satisfying the following bilinear identity for all n, m ∈ Z: I P dz 0 i τ2n t − [z −1 ] τ2m+2 t 0 + [z −1 ] e (ti −ti )z z 2n−2m−2 2πi z=∞ I P 0 dz −i + τ2n+2 t + [z] τ2m t 0 − [z] e (ti −ti )z z 2n−2m = 0. (3.4) 2πi z=0


23

Conversely, this bilinear relation characterizes the τ -function for the Pfaff lattice. Remark. L then has the following representation in terms of the Pfaffian τ -functions:   .. .    . . . Lˆ ˆ 01 0 0 00 L     ˆ ˆ ˆ L L L 0   10 11 12 h 1/2 Lh −1/2 =  , ˆ ˆ ˆ   ∗ L 21 L 22 L 23    ˆ 32 Lˆ 33 . . . ∗ ∗ L   .. . with (· = ∂/∂t1 ), 

−(log τ2n )·

 Lˆ nn :=  ˜ 2n − s2 (τ∂)τ − 2n

1

˜ 2n+2 s2 (−∂)τ τ2n+2

(log τ2n+2 )·

  ,

ˆL n,n+1 := 0 0 , 1 0

∗ (log τ2n+2 )·· Lˆ n+1,n := . ∗ ∗

(3.5)

The following bilinear relations follow from (3.4) and are due to [4]. 3.3 The functions τ2n (t) satisfy the “differential Fay identity”∗ τ2n (t − [u]), τ2n (t − [v]) COROLLARY

+ (u −1 − v −1 ) τ2n (t − [u])τ2n (t − [v]) − τ2n (t)τ2n (t − [u] − [v]) = uv(u − v)τ2n−2 (t − [u] − [v])τ2n+2 (t) (3.6) and Hirota-type bilinear equations, always involving nearest neighbors, 1 ∂2 ˜ ˜ τ2n+2 ◦ τ2n−2 , k, n = 0, 1, 2, . . . . pk+4 (∂) − τ2n ◦ τ2n = pk (∂) 2 ∂t1 ∂tk+3 (3.7) LEMMA 3.4 Consider an arbitrary function ϕ(t, z) depending on t ∈ C∞ , z ∈ C, having the asymptotics ϕ(t, z) = 1 + O(1/z) for z % ∞ and satisfying the functional relation

ϕ(t − [z 1−1 ], z 2 ) ϕ(t − [z 2−1 ], z 1 ) = , ϕ(t, z 1 ) ϕ(t, z 2 ) ∗ We

define { f, g} := f 0 g − f g 0 , where 0 = ∂/∂t1 .

t ∈ C∞ , z ∈ C.

24


Then there exists a function τ (t) such that ϕ(t, z) =

τ (t − [z −1 ]) . τ (t)

Proof See Appendix C. LEMMA 3.5 The following holds for the Pfaffian wave functions 91 and 92 , as in (2.7):

ψ1,2n (t − [z 2−1 ], z 1 ) ψ1,2n (t − [z 1−1 ], z 2 ) = ψ1,2n (t, z 1 ) ψ1,2n (t, z 2 )

(3.8)

ψ2,2n−2 t − [z −1 ], z −1 ψ1,2n (t, z) = 1.

(3.9)

and

Proof Setting (2.7) in the bilinear equation (2.12), with n 7 → 2n, m 7 → 2n − 2, yields I P c2n (t) dz 0 i e (ti −ti )z ψ1,2n (t, z)ψ2,2n−2 (t 0 , z −1 ) c2n−2 (t) ∞ 2πi I P c2n−2 (t) z 2 dz 0 −i + e (ti −ti )z ψ2,2n (t, z)ψ1,2n−2 (t 0 , z −1 ) = 0. c2n (t) 0 2πi Setting t − t 0 = [z 1−1 ] + [z 2−1 ] in the above and using e

P∞ 1

x i /i

= 1/(1 − x) yields

ψ1,2n (t, z)ψ2,2n−2 (t 0 , z −1 ) dz c2n−2 ∞ (1 − z/z 1 )(1 − z/z 2 ) 2πi I 1 1 dz c2n−2 =− z2 1 − 1− ψ2,2n (t, z)ψ1,2n−2 (t 0 , z −1 ) = 0, c2n 0 zz 1 zz 2 2πi c2n

I

the latter being equal to zero, because the integrand on the right-hand side is holomorphic. The integral on the left-hand side can be viewed as an integral along a contour encompassing ∞ and the points z 1 and z 2 , thus leading to ψ1,2n (t, z 1 )ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 1−1

= ψ1,2n (t, z 2 )ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 2−1 , (3.10) with ψ1,2n (t, z) = 1 + O(z −1 ),

ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z −1 = 1 + O(z −1 ).


25

Therefore, letting z 2 % ∞, one finds ψ1,2n (t, z 1 )ψ2,2n−2 t − [z 1−1 ], z 1−1 = 1,

(3.11)

yielding (3.9), and so, upon shifting t 7 → t − [z 2−1 ], ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 1−1 =

1 ψ1,2n (t − [z 2−1 ], z 1 )

;

similarly, ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 2−1 =

1 ψ1,2n (t − [z 1−1 ], z 2 )

.

(3.12)

Setting the two expressions (3.12) in (3.10) yields ψ1,2n (t − [z 2−1 ], z 1 ) ψ1,2n (t − [z 1−1 ], z 2 ) = . ψ1,2n (t, z 1 ) ψ1,2n (t, z 2 ) Proof of Proposition 3.1 From Lemmas 3.4 and 3.5 there exists, for each 2n, a function τ2n such that the first relation of (3.1) is satisfied; that is, ψ1,2n (t, z) =

τ2n (t − [z −1 ]) , τ2n (t)

and so from (3.9) ψ2,2n−2 t − [z −1 ], z −1 =

1 τ2n (t) = , ψ1,2n (t, z) τ2n (t − [z −1 ])

thus leading to ψ2,2n−2 (t, z) =

τ2n (t + [z]) , τ2n (t)

which is the second relation of (3.1). Proof of Theorem 3.2 At first, remembering that Qˆ = Q −1 0 Q, observe that P i ˆ e ti z Qχ(z) = Q −1 0 91 (t, z) 2n 2n =e

P

ti z i 2n

z ψ1,2n (t, z)

ti z i 2n τ2n (t

− [z −1 ]) τ2n (t) ∞ P i X ˜ 2n (t) sk (−∂)τ = e ti z z 2n 1 + , τ2n (t)

=e

P

z

n=1

26


showing that a few subdiagonals of the matrix Qˆ are given by       Qˆ =      

..



. 1 0

0 1

qˆ2n,2n−2 qˆ2n+1,2n−2

qˆ2n,2n−1 qˆ2n+1,2n−1

1 0

..

with qˆ2n,2n−1 = −

∂ log τ2n , ∂t1

qˆ2n,2n−2 =

Remembering that (2.7), normalized, becomes ( P ˆ 1,2n (t, z) = e tk z k z 2n ψ1,2n (t, z), 9 P ˆ 1,2n+1 (t, z) = e tk z k z 2n+1 ψ1,2n+1 (t, z), 9 (

0 1

ˆ 2,2n (t, z) = e− tk z −k z 2n+1 ψ2,2n (t, z), 9 P ˆ 2,2n+1 (t, z) = −e− tk z −k z 2n ψ2,2n+1 (t, z), 9 P

.

˜ 2n s2 (−∂)τ . τ2n

          

(3.13)

ψ1,2n = 1 + O(z −1 ), ψ1,2n+1 = 1 + O(z −2 ),

(3.14)

ψ2,2n = 1 + O(z), ψ2,2n+1 = 1 + O(z 2 ), (3.15)

we now show (3.3). Compute, using Theorem 2.2, e

P

ti z i

∂ + z z 2n ψ1,2n (t, z) ∂t1 ∂ ˆ 1 (t, z) = 9 2n ∂t1 ˆ 1 (t, z) = ( Lˆ 1 )+ + ( Lˆ 1 )0 + Q −2 J ( Lˆ 1+ )> J Q 20 9 0

2n

(3.16)

and e−

P

ti z −i

∂ 1 2n+1 − z ψ2,2n (t, z) ∂t1 z ∂ ˆ 2 (t, z) = 9 2n ∂t1 > 2 ˆ ˆ ˆ = J ( L 1 )+ + ( Lˆ 1 )0 + Q −2 0 J ( L 1+ ) J Q 0 92 (t, z) 2n .

(3.17)


27

In expression (3.16), the matrix equals, according to (2.15), > 2 ˆ ( Lˆ 1 )+ + ( Lˆ 1 )0 + Q −2 0 J ( L 1+ ) J Q 0  .. .  . . . qˆ0,−1 1 0   q ˆ − q ˆ − q ˆ 1 1,−1 20 21   0 0 q ˆ  21 =  c02 /c22 0 qˆ31 − qˆ42   0 0 0  2  0 0 c2 /c42 

 0 0 1 −qˆ43 0 0

and, acting with J on this matrix, ˆ 1+ )> J Q 20 J ( Lˆ 1 )+ + ( Lˆ 1 )0 + Q −2 J ( L 0  .. .  . . . q ˆ 1 0 21  2 /c2  q ˆ − q ˆ − q ˆ c 1,−1 20 0,−1  0 2  0 0 qˆ43  =  qˆ31 − qˆ42 1 0   0 0 0   0 0 1 

0 0 0 1 qˆ43 qˆ53 − qˆ64

b d

−d = c

0 0 1 −qˆ21 0 0

0 0 0 c22 /c42 qˆ65 qˆ53 − qˆ64

b . −a

Therefore the 2nth rows of both matrices, respectively, have the form (0, . . . , 0, qˆ2n,2n−1 (t), 1, 0, 0, . . . ), ↑

2n (0, . . . , 0, qˆ2n+2,2n+1 (t), 1, 0, 0, . . . ), ↑

2n

       ,     . . . 



using the fact that a J c

0 0 0 0 1 −qˆ65 .. .

0 0 0 0 1 −qˆ43 .. .

       ,     . . . 

28


and thus from (3.16) and (3.17), and expansions (3.14) and (3.15), we have ∂ + z z 2n ψ1,2n (t, z) = qˆ2n,2n−1 (t)z 2n ψ1,2n + z 2n+1 ψ1,2n+1 , ∂t1 ∂ − z −1 z 2n+1 ψ2,2n (t, z) = qˆ2n+2,2n+1 (t)z 2n+1 ψ2,2n + z 2n ψ2,2n+1 . ∂t1

(3.18)

So, using the expression (3.13) for qˆ2n,2n−1 and the first expression of (3.1), z 2n+1 ψ1,2n+1 (t, z) ∂ 2n = z+ z ψ1,2n (t, z) − qˆ2n,2n−1 (t)z 2n ψ1,2n (t, z) ∂t1 ∂ ∂ 2n = z+ z ψ1,2n (t, z) + log τ2n (t) z 2n ψ1,2n (t, z) ∂t1 ∂t1 τ (t − [z −1 ]) −1 ∂ τ2n (t − [z ]) ∂ 2n = z+ z 2n + τ2n (t) z 2n 2 (t) ∂t1 τ2n (t) ∂t1 τ2n = z 2n

(z + ∂/∂t1 )τ2n (t − [z −1 ]) , τ2n (t)

(3.19)

and similarly, using the second relation (3.18), z 2n ψ2,2n+1 (t, z) = z 2n+1

(−z −1 + ∂/∂t1 )τ2n+2 (t + [z]) . τ2n+2 (t)

(3.20)

This establishes (3.3) modulo the denominators. Therefore, we also have 1 1 ∂ ∂τ2n −1 ˜ 2n z −2 + · · · ψ1,2n+1 (t, z) = z+ τ2n (t) − z + s2 (−∂)τ z τ2n (t) ∂t1 ∂t1 2 1 ∂ ˜ τ2n z −2 + O(z −3 ); =1+ − 2 + s2 (−∂) τ2n (t) ∂t1 thus, referring to the matrix Qˆ just preceding (3.13), qˆ2n+1,2n = 0,

qˆ2n+1,2n−1 =

2 ˜ 1 ˜ − ∂ τ2n = −s2 (∂)τ2n . s2 (−∂) 2 τ2n τ2n ∂t1

(3.21)

To show (3.4), setting n 7 → 2n and m 7→ 2n in bilinear relation (2.10) and substituting, using (2.7) and the expressions for ψ1,2n (t, z) and ψ2,2n (t, z) in the proof of Proposition 3.1, 91,2n (t, z) = e

P

and 92,2n (t 0 , z) = e−

P

tk z k 2n

z c2n (t)

τ2n (t − [z −1 ]) τ2n (t)

0 tk0 z −k 2n+1 −1 0 τ2n+2 (t + [z]) z c2n (t ) τ2n+2 (t 0 )


29

into I

91,2n (t, z)92,2n (t , z 0

∞

−1

dz ) + 2πi z

I

92,2n (t, z)91,2n (t 0 , z −1 ) 0

dz =0 2πi z

yields c2n (t) c2n (t 0 )

I e

P (tk −tk0 )z k τ2n (t

− [z −1 ])τ2n+2 (t 0 + [z −1 ]) dz τ2n (t)τ2n+2 (t 0 ) 2πi z 2 I P 0 0 c2n (t ) 0 −k τ2n+2 (t + [z])τ2n (t − [z]) dz + e (tk −tk )z . c2n (t) 0 τ2n+2 (t)τ2n (t 0 ) 2πi

∞

Setting t 0 = t + [α] amounts to replacing the exponential: e

P (tk −tk0 )z k

= 1 − αz,

e

P 0 (tk −tk )z −k

=

1 , 1 − α/z

so that the first integral has a simple pole at z = ∞ and the second integral has one at z = α. Evaluating the integrals yields −α

2 (t)(τ c2n 2n+2 (t)/τ2n (t)) 2 (t 0 )τ 0 0 c2n 2n+2 (t )/τ2n (t )

that is, e

P i (α /i)(∂/∂ti )

+ α = 0;

τ2n+2 (t) − 1 cn2 (t) =0 τ2n (t)

yields the following relation, which involves a constant cn , independent of time: 2 c2n (t) = cn

τ2n (t) = cn · h −1 2n (t). τ2n+2 (t)

(3.22)

Rescaling τ2n 7→ τ2n /(c1 c2 · · · cn−1 ), in effect, sets cn = 1, and then (3.22), (2.7), (3.1), (3.19), and (3.20) yield (3.3); substituting (3.3) into (2.10) yields (3.4). Finally, identity (3.22) actually says Q 0 = h −1/2 . To derive the form (3.5) of the matrix L, set (3.13) and (3.21) in the elements just below the main diagonal of matrix (2.15), to yield (· = ∂/∂t1 ) 2 −qˆ2n,2n−1 −qˆ2n+1,2n−1 + qˆ2n,2n−2 τ˙ 2 (s (−∂) ˜ − (∂ 2 /∂t 2 ))τ2n ˜ 2n s2 (−∂)τ 2 2n 1 =− − + τ2n τ2n τ2n τ¨2n τ˙2n 2 = − τ2n τ2n = (log τ2n )··

30


and ˜ − ∂ 2 /∂t 2 )τ2n ˜ 2n+2 (s2 (−∂) s2 (−∂)τ 1 − τ2n τ2n+2 ˜ 2n ˜ 2n+2 s2 (∂)τ s2 (−∂)τ =− − , τ2n τ2n+2

qˆ2n+1,2n−1 − qˆ2n+2,2n =

concluding the proof of Theorem 3.2 and the remark following it, upon substituting the relations (3.13) and (3.14) and also Q 0 = h −1/2 into (2.15). 4. Semi-infinite matrices m ∞ , (skew-)orthogonal polynomials, and matrix integrals In this section, consider the following inner product∗ for α = 0, ∓1: ZZ P i i h f, git = f (y)g(z)e ti (y +z ) 2D α δ(y − z)ρ(y) ˜ ρ(z) ˜ dy dz R2 Z  P 2ti y i  f (y)g(y)e 2ρ(y) ˜ 2 dy for α = 0,    R   Z Z  P∞ i i f (y)g(z)e 1 ti (y +z ) ε(y − z)ρ(y) ˜ ρ(z) ˜ dy dz for α = −1, =  2 R   Z  P∞  i   { f, g}(y)e 1 2ti y ρ(y) ˜ 2 dy for α = +1. R

(4.1) Each type of inner product is discussed in Sections 4.1 and 4.2. 4.1. ∂m ∞ /∂tk = 3k m ∞ , orthogonal polynomials, and Hermitian matrix integrals (α = 0) The inner product above, with α = 0, corresponds to Hermitian matrix integrals; this theory is sketched here for the sake of completeness and analogy; it mainly summarizes [6]. Consider a t-dependent weight ρt (dz) := e

P

ti z i

ρ(dz) = e−V (z)+

P

ti z i

dz

on R, as in (0.1), and the induced t-dependent measure eTr(−V (X )+

P

ti X i )

dX

(4.2)

on the ensemble Hn of Hermitian matrices, with Haar measure d X ; the latter can be decomposed into a spectral part (radial part) and an angular part: d X :=

n Y 1

d X ii

Y

(d<X i j d=X i j ) = 12 (z) dz 1 · · · dz n dU,

(4.3)

1≤i< j≤n

have ε(x) = sign x, having the property ε0 = 2δ(x). Also, consider the Wronskian { f, g} := (∂ f /∂ y)g − f (∂g/∂ y).

∗ We


31

Q where 1(z) = 1≤i< j≤n (z i − z j ) is the Vandermonde determinant. Here we form the matrix integral Z Z n P P Y ∞ i i eTr(−V (X )+ ti X ) d X = cn 12 (z) e 1 ti z k ρ(dz k ). (4.4) Hn

Rn

k=1

The weight ρt (dz) defines a (symmetric) t-dependent inner product of the type (4.1) for α = 0: Z P∞ i sy h f, git = f (z)g(z)e 1 ti z ρ(dz), with moments µi j (t) := hz

i

sy , z j it

Z

z i+ j e

=

P

tk z k

ρ(dz),

R

satisfying

Z P k ∂µi j = z i+ j+` e tk z ρ(dz) = µi+`, j (t). ∂t` R Therefore the semi-infinite moment matrix m ∞ (t) = (µi j (t))i, j≥0 satisfies ∂m ∞ i = 3i m ∞ = m ∞ 3> . ∂ti

(4.5)

The point now is that the following integral can be expressed as a determinant of moments; namely, Z Z n P Y Tr(−V (X )+ ∞ ti X i ) 1 e dX = 12 (z) ρt (dz k ) Hn

Rn

Z

k=1

X

=

Rn σ ∈S n

Z

X

=

Rn σ ∈S n

=

X

det

k−1 det(z σ`−1 (k) z σ (k) )1≤`,k≤n

ρt (dz k )

k=1

det(z σ`+k−2 (k) )1≤`,k≤n

n Y

ρt (dz σ (k) )

k=1

Z

σ ∈Sn

= n! det

n Y

R

Z

z σ`+k−2 ρ (dz ) t σ (k) (k)

z `+k−2 ρt (dz) R

1≤`,k≤n

1≤`,k≤n

= n! det(µi j )0≤i, j≤n−1 = n!τn (t) is a τ -function for the KP-equation; also, in view of (4.5) and the upper-lower Borel decomposition (0.3) of m ∞ , the integrals form a vector of τ -functions for the Toda lattice. The polynomials pn (t; z) defined by the Borel decomposition m ∞ (t) = S −1 S >−1 and p(t; z) = Sχ(z) are orthonormal with regard to the inner product sy hz i , z j it = µi j (t).

32


4.2. ∂m ∞ /∂tk = 3k m ∞ + m ∞ 3>k , skew-orthogonal polynomials, and symmetric and symplectic matrix integrals (α = ±1) Consider a skew-symmetric semi-infinite matrix m ∞ (t) = µi j (t) i, j≥0 , with m n (t) = µi j (t) 0≤i, j≤n−1 , satisfying ∂m ∞ /∂tk = 3k m ∞ + m ∞ 3>n .

(4.6)

Then we have shown in Sections 2 and 3 that, upon skew-Borel decomposing m ∞ , these equations ultimately imply the existence of functions τ (t) satisfying bilinear equations (3.4). Remember also that h(t) = diag(h 0 , h 0 , h 2 , h 2 , . . .) ∈ D0− ,

with h 2n (t) =

τ2n+2 (t) . τ2n (t)

Here we need the Pfaffian pf(A) of a skew-symmetric matrix A = (ai j )0≤i, j≤n−1 for∗ even n: pf(A) d x0 ∧ · · · ∧ d xn−1 = =

1 2n/2 (n/2)!

X

1 (n/2)!

X

ai j d xi ∧ d x j

n/2

0≤i< j≤n−1

ε(σ )ai0 ,i1 ai2 ,i3 · · · ain−2 ,in−1 d x0 ∧ · · · ∧ d xn−1 , (4.7)

σ

so that pf(A)2 = det A. We now state the following theorem due to Adler, E. Horozov, and van Moerbeke [2], in complete analogy with the discussion of the Hermitian case. THEOREM 4.1 Consider a semi-infinite skew-symmetric matrix m ∞ , evolving according to (4.6); set

τ2n (t) = pf m 2n (t)

and

h 2n =

pf(m 2n+2 (t)) . pf(m 2n (t))

(4.8)

Then, modulo the exponential, the wave vector 91 (defined by (3.3)) is a sequence of polynomials, P i 91,k (t, z) = e ti z qk (t, z), (4.9) where the qk ’s are skew-orthonormal polynomials of the forms (0.13) and (0.18), satisfying hqi , q j isk 0≤i, j−1 ,

with Q ∈ k.

(4.11)

The matrix L = Q3Q −1 , also defined by zq(t, z) = Lq(t, z), and the diagonal matrix h satisfy the equations ∂L = [−πk L i , L] ∂ti

and

h −1

∂h = 2πk (L i )0 . ∂ti

(4.12)

Sketch of proof At first, note that looking for skew-orthogonal polynomials is tantamount to the skewBorel decomposition of m ∞ , so that (4.10) and (4.11) are equivalent. The skeworthogonality of the polynomials (0.18) follows from expanding the Pfaffians explicitly in terms of z-columns, upon using the expression for the Pfaffian in terms of a column X ˆ . . . , ` − 1) = pf(0, . . . , ` − 1, i). (−1)k aki pf(0, . . . , k, 0≤k≤`−1

(For details, see [2].) On the other hand, Theorem 3.2 gives 9(t, z) and hence Q in terms of τn (t) = pf m 2n (t) of (4.8). By the uniqueness of decomposition (4.11), the two ways of arriving at Q, (0.18) and (3.3), must coincide. Important remark. The polynomials (0.18) provide an explicit algorithm to perform the skew-Borel decomposition of the skew-symmetric matrix m ∞ . Namely, the coefficients of the polynomials qi provide the entries of the matrix Q. This fact is used later in the examples. Symmetric matrix integrals (α = −1) Here we focus on integrals over the space S2n of symmetric matrices of the type Z P∞ i eTr(−V (X )+ 1 ti X ) d X, (4.13) S2n

where d X denotes Haar measure for X = U diag(z 1 , . . . , z n )U > , UU > = I , Y d X := d X i j = |1(z)| dz 1 · · · dz n dU. (4.14) 1≤i≤ j≤n

34


As appears below, integral (4.13)Pleads to a skew-inner product of the type (4.1) with P i i α = −1, with weight ρt (z) := e ti z ρ(z) = e−V (z)+ ti z , ZZ

f (x), g(y) := f (x)g(y)ε(x − y)ρt (x)ρt (y) d x dy, (4.15) R2

leading to skew-symmetric moments∗ ZZ µi j (t) = x i y j ε(x − y)ρt (x)ρt (y) d x dy 2 Z ZR = (x i y j − x j y i )ρt (x)ρt (y) d x dy x≥y Z = F j (x)G i (x) − Fi (x)G j (x) d x,

(4.16)

R

where (0 = d/d x) Z x P k Fi (x) := y i e tk y ρ(y) dy

and

−∞

G i (x) := Fi0 (x) = x i e

P

tk x k

ρ(x).

By simple inspection, the moments µk` (t) satisfy ZZ P ∂µk` n n = (x k+i y ` + x k y `+i )ε(x − y)e tn (x +y ) ρ(x)ρ(y) d x dy ∂ti R2 = µk+i,` + µk,`+i , and so m ∞ satisfies (4.6). According to M. Mehta [14], the symmetric matrix integral can now be expressed in terms of the Pfaffian, as follows, taking into account a constant c2n , coming from ∗ We

have ε(x) = 1 for x ≥ 0, and ε(x) = −1 for x < 0.


35

the integration of the orthogonal group: 1 (2n)!

Z S2n (E)

e

P Tr(−V (X )+ ti X i ) d X

Z = −∞−1 − S −1 U L >k S >−1 (using (5.7)) = −3k S −1 U S >−1 − S −1 U S >−1 3>k S > S >−1 = 3k m˜ ∞ (t) + m˜ ∞ (t)3>k

(using L k = S3k S −1 )

(by (5.4)).

For an arbitrary matrix A, we have A = A> ⇐⇒ A = (Abo )> − Asy .

(5.7)

Indeed, remembering that∗ Abo = 2A− + A0 and Asy = A+ − A− , one checks that (Abo )> − Asy − A = 2(A− )> + A0 −(A+ − A− )− A− − A+ − A0 = −2(A+ −(A− )> ), so that the left-hand side vanishes if the right-hand side does; the latter means A is symmetric. ˜ We now define L(t) by conjugation of L(2t) by O(2t): ˜ ˜ L(t) := O(2t)L(2t)O(2t)−1 = O(2t)S(2t)3S −1 (2t)O(2t)−1 = Q(t)3 Q˜ −1 (t); ˜ satisfies the Pfaff Lax equation. Therefore the sequence thus, by Proposition 1.2, L(t) of polynomials ˜ q(t, z) := O(2t) p(2t, z) = O(2t)S(2t)χ(z) = Q(t)χ(z) is skew-orthonormal,

sk qi (t, z), q j (t, z) = Ji j , with regard to the skew-inner product specified by the matrix m˜ ∞ : hz i , z j isk ˜ i j (t). t =µ sy

In the last step, we show that hϕ, ψisk t = hϕ, uψi2t . Since U (2t) = −O −1 (2t)J O >−1 (2t) = −U > (2t), ∗A

(5.8)

± means the usual strictly upper-(lower-)triangular part, and A0 means the diagonal part in the common sense.


41

we compute

sy

sy qi (t, z), (u2t q) j (t, z) 2t = (O p)i (2t), (uO p) j (2t) 2t

sy = (O p)i (2t), (Ou p) j (2t) 2t

sy = (O p)i (2t), (O U p) j (2t) 2t

sy = O(2t) pk (2t), p` (2t) k,`≥0 (O U )> (2t) i j = O(2t)I (O U )> (2t) i j = O(2t)U > (2t)O > (2t) i j = − O(2t)U (2t)O > (2t) i j (using (5.8)).

= Ji j

(5.9)

0

Therefore, defining a new skew-inner product h , isk , 0

sy

hϕ, ψisk := hϕ, uψi2t , we have shown that 0

sk hqi , q j isk t = hqi , q j it = Ji j ,

and so by completeness of the basis qi , we have 0

sk h , isk t = h , it ,

thus ending the proof of Theorem 4.1. 6. Example 1: From Hermitian to symmetric matrix integrals Striking examples are given by using the map O(t) obtained from skew-Borel decomposing N −1 (t) and N (t) (see (5.2)). This section deals with N −1 (t), whereas Section 7 deals with N (t). PROPOSITION 6.1 The special transformation

U (t) = N

−1

(t) =

f (L)M −

−1 f0+g (L) (t) 2

maps the Toda lattice τ -functions with initial weight ρ = e−V , V 0 = −g/ f (Hermitian matrix integral) to the Pfaff lattice τ -functions (symmetric matrix integral), with initial weight ρ (z) 1/2 P∞ i 2t ρ˜t (z) := = e−(1/2)(V (z)+log f (z)−2 1 ti z ) f (z) =: e−V (z)+ ˜

P∞ 1

ti z i

= ρ(z)e ˜

P

ti z i

.

42


To be precise,

Toda lattice

 pn (t, z) orthonormal polynomialsZ in z for the inner product    P i   sy  hϕ, ψit = ϕ(z)ψ(z)e ti z ρ(z) dz,  sy

µi j (t) = hz i , z j it and  j≤n−1 ,  Z m n = (µi j )0≤i,  P∞  1 i   eTr(−V (X )+ 1 ti X ) d X  τn (t) = det m n = n! Hn       −N −1 (2t) = O −1 (2t)J O >−1 (2t),      map O(2t) such that O(2t) is lower-triangular,       O(2t)S(2t) ∈ Gk  y  qn (t, z) = O(2t) p(2t, z) n skew-orthonormal polynomials      in z for the skew-inner product (weight ρ), ˜     sy  −1 sk   hϕ, ψit := hϕ, n2t ψi2t   ZZ  P  1  ti (x i +y i )  = ϕ(x)ψ(y)ε(x − y)e    2 R2   r r ρ ρ Pfaff lattice (y) d x dy, × (x)   f f      µ˜ i j (t) = hx i , y j isk ˜ n = (µ˜ i j )0≤i, j≤n−1 ,  t and m   Z  P  1 i  Tr(−V˜ (X )+ ∞  1 ti X ) d X ,  e τ ˜ (t) = pf( m ˜ ) = 2n 2n  n (2n)!  2  S2n     1  with V˜ (z) = V (z) + log f (z) . 2

In the first integral defining τn (t), d X denotes Haar measure on Hermitian matrices (see Section 4.1), whereas the second integral τ˜2n (t) involves Haar measure on symmetric matrices (see Section 4.2). Proof At first, check that Z d −1 1 ϕ(x) = ε(x − y)ϕ(y) dy. dx 2

(6.1)


43

Indeed, d d −1 ϕ(x) = dx dx

Z

1 ∂ ε(x − y)ϕ(y) dy 2 ∂x

Z

δ(x − y)ϕ(y) dy

=

(using

∂ ε(x) = 2δ(x)) ∂x

= ϕ(x). Consider now the operator s −1 f d p −1 ut = nt = fρt , ρt dz

so that ut p = n−1 t p =N

according to (5.2). Let it act on a function ϕ(x): s d −1 ρ (x) 1 t n−1 ϕ(x) t ϕ(x) = √ f (x) f (x)ρt (x) d x s Z 1 ε(x − y) ρt (y) = ϕ(y) dy √ 2 f (y) f (x)ρt (x) R

−1

p,

(using (6.1)).

One computes sy

hϕ, ψisk t = hϕ, u2t ψi2t

sy

= hϕ, n−1 2t ψi2t s ZZ s 1 ρ2t (x) ρ2t (y) = ε(x − y) ϕ(x)ψ(y) d x dy 2 R2 f (x) f (y) ZZ P∞ 1 k k 1 tk (x +y ) ε(x − y)ϕ(x)ψ(y) d x dy. = ρ(x) ˜ ρ(y)e ˜ 2 2 R So, finally setting V˜ (x) = (1/2)(V (x) + log f (x)) yields by (4.17) Z P∞ 1 i ˜ τ˜2n (t) = pf(m˜ 2n ) = eTr(−V (X )+ 1 ti X ) d X. (2n)! S2n The map O for the classical orthogonal polynomials at t = 0 CLAIM

The matrix O, mapping orthonormal pk into skew-orthonormal polynomials qk , is

44


given by a lower-triangular three-step relation: r c2n q2n (0, z) = p2n (0, z), a2n r a2n q2n+1 (0, z) = c2n 2n c2n X bi p2n (0, z) + c2n p2n+1 (0, z) , × − c2n−1 p2n−1 (0, z) + a2n 0

(6.2) where the ai and bi are the entries in the tridiagonal matrix defining the orthonormal polynomials, and the ci ’s are the entries of the skew-symmetric matrix N . In [6], we showed that, in the classical cases below, N is tridiagonal at the same time as L (see Appendix B): 

b0  a0  L=  

a0 b1 a1

 a1 b2 .. .

  ..  , .   .. .



0  −c0  −N =   

c0 0 −c1

 c1 0 .. .

  ..  , .  

(6.3)

with the following precise entries: 2

Hermite: ρ(z) = e−z , an−1 =

p n/2, bn = 0, cn = an ; p Laguerre: ρ(z) = e−z z α I[0,∞) (z), an−1 = n(n + α), bn = 2n + α + 1, cn = an /2; Jacobi: ρ(z) = (1 − z)α (1 + z)ρ I[−1,1] (z); 1/2 4n(n + α + β)(n + α)(n + β) , (2n + α + β)2 (2n + α + β + 1)(2n + α + β − 1) α2 − β 2 bn = , (2n + α + β)(2n + α + β + 2) α + β cn = an +n+1 . 2

an−1 =

If the skew-symmetric matrix N has the tridiagonal form above, then one checks that


its inverse has the following form:  0 − c10 0  1 0 0  c0   0 0 0  c1 1  0  c0 c2 c2  0 0 0 −N −1 =   c1 c3 c 3  0 c2 c4  c0 c2 c4  0 0 0   c1 c3 c5 c3 c5  cccc 0 c2 c4 c6  0246

45

−c1 c0 c2

0 − c12 0 0 0 0 0

−c1 c3 c0 c2 c4

0 0 0 0 0

0 −c3 c2 c4

1 c4

0 c5 c4 c6

0 − c14 0 0 0

0 0 0 0 0 0 0 1 c6

−c1 c3 c5 c0 c2 c4 c6



0

                

−c3 c5 c2 c4 c6

0 −c5 c4 c6

0 − c16 0 ..

.

(6.4) In order to find the matrix O, we must perform the skew-Borel decomposition of the matrix −U : −U = −N −1 = O −1 J O >−1 . The recipe for doing so is given in Theorem 4.1 (see also the important remark following that theorem). It suffices to form the Pfaffians (0.18) by appropriately bordering the matrix −N −1 , as in (0.18), with rows and columns of powers of z, yielding skeworthonormal polynomials; we choose to call them r ’s, with Oχ(z) = r (z). They turn out to be the following simple polynomials, with 1/τ˜˜2n = c0 c2 c4 · · · c2n−2 : r2n (z) = q r2n+1 (z) = q

1 τ˜˜2n τ˜˜2n+2 1 τ˜˜2n τ˜˜2n+2

c2n z 2n 1 = √ c2n z 2n , c2n c0 c2 · · · c2n c2n z 2n+1 − c2n−1 z 2n−1 1 = √ (c2n z 2n+1 − c2n−1 z 2n−1 ). c0 c2 · · · c2n c2n

Then, also from Appendix A, in order to get O → Oˆ in the correct form, we compute ˆ the skew-orthonormal polynomials rˆk , with Oχ(z) = rˆ (z): r c2n 2n 1 z , rˆ2n (z) = √ r2n (z) = a2n a2n P2n bi √ rˆ2n+1 (z) = √0 r2n (z) + a2n r2n+1 (z) a2n r 2n a2n c2n X 2n = − c2n−1 z 2n−1 + bi z + c2n z 2n+1 . (6.5) c2n a2n 0

From the coefficients of the polynomial rˆk , one reads off the transformation matrix from orthonormal to skew-orthonormal polynomials; it is given by the matrix

46


ˆ ˆ Oˆ such that Oχ(z) = rˆ (z). Therefore q(t, z) = O(2t) p(2t, z) yields, after setting t = 0, r c2n q2n (0, z) = p2n (0, z), a2n r a2n q2n+1 (0, z) = c2n 2n c2n X × − c2n−1 p2n−1 (0, z) + bi p2n (0, z) + c2n p2n+1 (0, z) , a2n 0

(6.6) confirming (6.2).

7. Example 2: From Hermitian to symplectic matrix integrals PROPOSITION 7.1 The matrix transformation

N = f (L)M −

f0+g (L) 2

maps the Toda lattice τ -functions with t-dependent weight P∞

ρt (z) = e−V (z)+

1

ti z i

,

V 0 = g/ f

(Hermitian matrix integral), to the Pfaff lattice τ -functions (symplectic matrix integral), with t-dependent weight P∞ i 1/2 ρ˜t (z) := ρ2t (z) f (z) = e−(1/2)(V (z)−log f (z)−2 1 ti z ) =: e−V (z)+ ˜

P

ti z i

= ρ(z)e ˜

P

ti z i

.

To be precise,  pn (t, z) orthonormal polynomials in z for the inner product  P i  R  sy   hϕ, ψit = ϕ(z)ψ(z)e ti z ρ(z) dz, sy Toda lattice µi j (t) = hz i , z j it and Zm n = (µi j )0≤i, j≤n−1 ,   P  i  τ (t) = det m (t) = 1  eTr(−V (X )+ ti X ) d X n n n! Hn


47

      −N (2t) = O −1 (2t)J O >−1 (2t),      map O(2t) such that O(2t) is lower-triangular,       O(2t)S(2t) ∈ Gk  y  qn (t, z) = O(2t) p(2t, z) n skew-orthonormal polynomials      in z for the skew-inner product (weight ρ˜t ),    sy sk   hϕ, ψit := hϕ, n2t ψi2t   ZZ   P i 1    =− ϕ(z), ψ(z) e2 ti z ρ(z) f (z) dz, 2 R2 Pfaff lattice i , z j isk and m  µ ˜ (t) = hz ˜ n = det(µ˜ i j )0≤i, j≤n−1 ,  ij t   Z  P  1  2 Tr(−V˜ (X )+ ti X i )  τ ˜ (t) = pf( m ˜ (t)) = e d X,  2n 2n   (−2)n n! T2n     1   with V˜ (z) = V (z) − log f (z) . 2 Proof Representing d/d x as an integral operator Z Z Z ∂ d ϕ(x) = δ(x − y)ϕ 0 (y) dy = − δ(x − y)ϕ(y) dy = δ 0 (x − y)ϕ(y) dy, dx R R ∂y R compute s ut = nt =

f d p fρt , ρt dz

so that nt p(t, z) = N p(t, z);

remember N from (5.2). Let it act on a function ϕ(x): s f d p fρt ϕ(x) ut ϕ(x) = ρt d x s Z p f (x) 0 = δ (x − y) f (y)ρt (y)ϕ(y) dy. ρt (x) R

48


Then sy

sy

hϕ, ψisk t = hϕ, u2t ψi2t = hϕ, n2t ψi2t s ZZ p f (x) 0 δ (x − y) f (y)ρ2t (y)ψ(y) d x dy = ρ2t (x)ϕ(x) ρ2t (x) R2 ZZ p p = f (x)ρ2t (x)ϕ(x)δ 0 (x − y) f (y)ρ2t (y)ψ(y) d x dy 2 ZRZ p p ∂ =− f (x)ρ2t (x)ϕ(x) δ(x − y) f (y)ρ2t (y)ψ(y) d x dy 2 ∂x Z R p p ∂ =− f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) d x R ∂x Z p p 1 ∂ =− f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) d x 2 R ∂x Z p ∂ p 1 + f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) d x 2 R ∂x Z p p 1 =− f (x)ρ2t (x)ϕ(x), f (x)ρ2t (x)ψ(x) d x 2 R Z P∞ i 1 =− ϕ(x), ψ(x) ρ˜02 (x)e2 1 ti x d x, 2 R using the notation in the statement of this proposition. Setting ρ(x) ˜ = e−V (x) , with V˜ (x) = (1/2)(V (x) − log f ), Z P∞ i 1 hx i , x j isk = − {x i , x j }ρ˜ 2 (x)e2 1 ti x d x 2 R Z P i 1 ˜ {x i , x j }e−2(V (x)− ti x ) d x, =− 2 R ˜

and so τ2n (t) = pf m˜ 2n (t) =

1 (−2)n n!

Z

e2 Tr(−V (x)+ ˜

T2n

P

ti x i )

d x.

The map O −1 for the classical orthogonal polynomials at t = 0 CLAIM

The matrix O, mapping orthonormal pk into skew-orthonormal polynomials qk , is


49

given by a lower-triangular three-step relation: r √ a2n−2 p2n (0, z) = −c2n−1 q2n−2 (0, z) + a2n c2n q2n (0, z), c2n−2 r a2n−2 p2n+1 (0, z) = −c2n q2n−2 (0, z) c2n−2 r 2n X r c c2n 2n q2n+1 (0, z), − q2n (0, z) + bi a2n a2n 0

(7.1) where the ai and bi are the entries in the tridiagonal matrix defining the orthonormal polynomials, and the ci are the entries in the skew-symmetric matrix. In this case, we need to perform the following skew-Borel decomposition at t = 0: −U = −N = O −1 J O >−1 , where N is the matrix (6.3). Here again, in order to find O, we use the recipe given in Theorem 4.1, namely, writing down the corresponding skew-orthogonal polynomials (0.18), but where the µi j are the entries of −U = −N . Consider the Pfaffians of the bordered matrices (0.18); they have the leading term τ˜˜2n =

n−1 Y

c2 j .

0

Then one computes n i−1 n−i−1 Y X Y 1 r2n = q z 2n−2i c2 j c2n−2 j−1 , 0 0 τ˜˜2n τ˜˜2n+2 i=0 n−1 n i−1 n−i−1 Y Y X Y 1 z 2n+1 c2 j + z 2n−2i c2 j c2n−2 j−1 , r2n+1 = q 0 i=1 0 0 τ˜˜2n τ˜˜2n+2

(7.2) with

q

√ τ˜˜2n τ˜˜2n+2 = c0 c2 · · · c2n−2 c2n ,

q

√ τ˜˜0 τ˜˜2 = c0 .

Setting q q q q ˜ ˜ ˜ ˜ ˜ ˜ D := diag( τ˜0 τ˜2 , τ˜0 τ˜2 , τ˜2 τ˜4 , τ˜˜2 τ˜˜4 , . . .), the matrix O is the set of coefficients of the polynomials above; that is,

50




1  0   c  1  c  2  −1  c1 c3 O=D   c1 c4  c1 c3 c5  c1 c3 c6  =: D

−1

0 0 1 0 0 c0 0 0 0 c0 c3 0 c0 c4 0 c0 c3 c5 0 c0 c3 c6

0 0 0 c0 0 0 0 0

0 0 0 0 c0 c2 0 c0 c2 c5 c0 c2 c6

0 0 0 0 0 c0 c2 0 0

R.

0 0 0 0 0 0 c0 c2 c4 0

0 0 0 0 0 0 0 c0 c2 c4



..

.

              

(7.3)

As before, in order to get the skew-symmetric polynomials in the right form, from the orthogonal ones, one needs to multiply to the left with the matrix E, defined in (A.2): Oˆ = E O = E D −1 R,

(7.4)

Oˆ −1 = R −1 D E −1 ;

(7.5)

and so it turns out that the matrix Oˆ is complicated, but its inverse is simple. Namely, compute   1 0 0 0 0 0 0 0   0 1 0 0 0 0 0 0     c1 1   − c0 0 0 0 0 0 0 c0   c 1  − 2 0 0 0 0 0 0 c0   c0 c3   0 1 0 − c0 c2 0 0 0 0   c c 0 2 −1   1 R = 0  0 0 0 0 − c0c4c2 0 c0 c2   c5 1   0 0 0 0 − c0 c2 c4 0 0   c0 c2 c4   0 c6 1 0 0 0 − 0 0   c0 c2 c4 c0 c2 c4       .. .


and



E −1

       =      

51

α0 −β0



0 1 α0

α2 −β2

0 1 α2

α4 −β4

0 1 α4

0

 0       ,       .. .

(7.6)

with α2n and β2n as in (A.5). Carrying out the multiplication (7.5) leads to the matrix Oˆ −1 , with a few nonzero bands, yielding the map (7.1), by the recipe of Proposition 6.1 inverted.

Appendices A. Free parameter in the skew-Borel decomposition If the Borel decomposition of −H = O −1 J O >−1 is given by a matrix O ∈ Gk , with the diagonal part of O being   σ0 0  0 σ0   0      σ2 0     0 σ 2  , (A.1) (O)0 =     σ4 0     0 σ4     .. . 0 then the new matrix        Oˆ := E O :=       

1/α0 β0



0 α0 1/α2 β2

0 α2 1/α4 β4

0

0 α4

 0      ,       .. .

(A.2)

52


with free parameters α2n , β2n , is a solution of the Borel decomposition −H = Oˆ −1 J Oˆ >−1 , as well. The diagonal part of Oˆ consists of (2 × 2)-blocks σ /α 0 1/α2n 0 σ2n 0 = 2n 2n . β2n α2n 0 σ2n β2n σ2n α2n σ2n Imposing the condition that X

qi (z) =

Oˆ i j p j (z),

with pk (z) =

0≤ j≤i

k X

pki z i ,

i=0

has the required form, that is, the same leading term for q2n and q2n+1 and no z 2n -term in q2n+1 , q2n (z) = q2n,2n z 2n + · · · , q2n+1 (z) = q2n,2n z 2n+1 + q2n,2n−1 z 2n−1 + · · ·

(A.3)

implies σ2n p2n,2n = σ2n α2n p2n+1,2n+1 , α2n σ2n β2n p2n,2n + σ2n α2n p2n+1,2n = 0 yielding, upon using the explicit form of the coefficients pk` of the polynomials pk , associated with three-step relations (see Lemma A.1), 2 α2n =

p2n,2n p2n+1,2n+1

= a2n ,

p2n+1,2n β2n =− = α2n p2n,2n

P2n 0

a2n

bi

.

(A.4)

Hence α2n =

√ a2n

and

2n 1 X β2n = √ bi . a2n 0

So, if r (z) = Oχ(z),

(A.5)


53

then (A.2) yields        ˆ rˆ (z) := Oχ(z) =      

1/α0 β0



0 α0 1/α2 β2

0 α2 1/α4 β4

0 α4

0

 0      ,       .. .

r (z) = Er (z), and thus 1 rˆ2n (z) = √ r2n (z), a2n P2n bi √ rˆ2n+1 (z) = √0 r2n + a2n r2n+1 (z). a2n

(A.6) (A.7)

LEMMA A .1 Pn i A sequence of polynomials pn (z) = i=0 pni z of degree n satisfying three-step recursion relation∗

zpn = an−1 pn−1 + bn pn + an pn+1 ,

n = 0, 1, . . . ,

has the form pn+1 (z) =

n pn,n n+1 X n z − bi z + · · · . an 0

Proof Equating the z n+1 and z n coefficients of (A.8) divided by pn,n yields 1 pn+1,n+1 = pn,n an and

pn,n−1 pn+1,n = an + bn . pn,n pn,n Combining both equations leads to an ∗ We

set a−1 = 0.

pn+1,n pn,n−1 − an−1 = −bn , pn,n pn−1,n−1

(A.8)

54


yielding n

an

X pn+1,n =− bi pn,n

(using a−1 = 0).

0

B. Simultaneous (skew-)symmetrization of L and N CLAIM

For the classical polynomials, the matrices L and N can be simultaneously symmetrized and skew-symmetrized. Sketch of proof This statement was established by us in [6]. Given the monic orthogonal polynomials p˜ n with respect to the weight ρ, with ρ 0 /ρ = −g/ f , we have that the operators z and s d f0−g f d p fρ = f + n= ρ dz dz 2 acting on the polynomials p˜ n have the form 2 z p˜ n = an−1 p˜ n−1 + bn p˜ n + p˜ n+1 ,

n p˜ n = . . . − γn p˜ n+1 ,

(B.1)

in view of the fact that for the classical orthogonal polynomials,∗  d  n = dz − z,  Hermite: d Laguerre: n = z dz − 12 (z − α − 1),  d  Jacobi: − 12 ((α + β + 2)z + (α − β)). n = (1 − z 2 ) dz For the orthonormal polynomials, the matrices L and −N are symmetric and skewsymmetric, respectively. Therefore the right-hand side of these expressions must have the form 2 z p˜ n = an−1 p˜ n−1 + bn p˜ n + p˜ n+1 , 2 n p˜ n = an−1 γn−1 p˜ n−1 − γn p˜ n+1 .

Therefore, upon rescaling the p˜ n ’s, to make them orthonormal, we have zpn = (L p)n = an−1 pn−1 + bn pn + an pn+1 , n pn = (N p)n = an−1 γn−1 p˜ n−1 − an γn p˜ n+1 , ∗ They

have the respective weights ρ = e−z , ρ = e−z z α , ρ = (1 − z)α (1 + z)β . 2


from which it follows that 

0  −c0  −N =   

55

c0 0 −c1

 c1 0 .. .

  ..  , .  

with cn = an γn ,

where −γn is the leading term in expression (B.1).

C. Proof of Lemma 3.4 For future use, consider the first-order differential operators η(t, z) =

∞ −j X z ∂ j ∂t j

∞

and

B(z) = −

j=1

X ∂ ∂ + z − j−1 ∂z ∂t j

(C.1)

j=1

having the property B(z)e−η(z) f (t) = B(z) f t − [z −1 ] = 0.

(C.2)

LEMMA C .1 Consider an arbitrary function ϕ(t, z) depending on t ∈ C∞ , z ∈ C, having the asymptotics ϕ(t, z) = 1 + O(1/z) for z % ∞ and satisfying the functional relation

ϕ(t − [z 2−1 ], z 1 ) ϕ(t − [z 1−1 ], z 2 ) = , ϕ(t, z 1 ) ϕ(t, z 2 )

t ∈ C∞ , z ∈ C.

(C.3)

Then there exists a function τ (t) such that ϕ(t, z) =

τ (t − [z −1 ]) . τ (t)

(C.4)

Proof Applying B1 := B(z 1 ) to the logarithm of (C.3) and using (C.1) and (C.2) yields (e−η(z 2 ) − 1)B1 log ϕ(t; z 1 ) = −B1 log ϕ(t, z 2 ) =−

∞ X j=1

− j−1

z1

∂ log ϕ(t, z 2 ), ∂t j

which, upon setting j

f j (t) = Resz 1 =∞ z 1 B1 log ϕ(t, z 1 ),

56


yields termwise in z 1 , (e−η(z 2 ) − 1) f j (t) = −

∂ log ϕ(t, z 2 ). ∂t j

(C.5)

Acting with ∂/∂ti on the latter expression and with ∂/∂t j on the same expression with j replaced by i, and subtracting,∗ one finds ∂f ∂fj i (e−η(z 2 ) − 1) − = 0, ∂t j ∂ti yielding

∂fj ∂ fi − = 0; ∂t j ∂ti

the constant vanishes because ∂ f i /∂t j never contains constant terms. Therefore, there exists a function log τ (t1 , t2 , . . .) such that −

∂ log τ = f j (t) = Resz=∞ z j B log ϕ, ∂t j

and hence, using (C.5), ∂ ∂ log ϕ(t, z) = (e−η(z) − 1) log τ, ∂t j ∂t j or, what is the same, ∂ log ϕ − (e−η − 1) log τ = 0, ∂t j from which it follows that log ϕ − (e−η − 1) log τ = −

∞ X bi 1

i

z −i

is, at worst, a holomorphic series in z −1 with constant coefficients, which we call −bi /i. Hence τ (t − [z −1 ])e− ϕ(t, z) = τ (t) = ∗ It

is obvious that [∂/∂ti , e−η(z) ] = 0.

τ (t − [z −1 ])e τ (t)e

P∞ 1

P∞ 1

P∞ 1

(bi /i)z −i

bi (ti −z −i /i)

bi ti

;


57

that is, ϕ(t, z) =

τ˜ (t − [z −1 ]) , τ˜ (t)

where τ˜ = τ (t)e

P∞ 1

bi ti

.

Thus Lemma 3.4 is proved. References [1]

[2]

[3]

[4] [5] [6]

[7] [8]

[9] [10]

[11] [12] [13]

M. ADLER, P. J. FORRESTER, T. NAGAO, and P. VAN MOERBEKE, Classical skew

orthogonal polynomials and random matrices, J. Statist. Phys. 99 (2000), 141–170. MR 2001k:82046 10 M. ADLER, E. HOROZOV, and P. VAN MOERBEKE, The Pfaff lattice and skew-orthogonal polynomials, Internat. Math. Res. Notices 1999, 569–588. MR 2000f:37112 14, 15, 32, 33 M. ADLER, T. SHIOTA, and P. VAN MOERBEKE, A Lax representation for the vertex operator and the central extension, Comm. Math. Phys. 171 (1995), 547–588. MR 97a:58072 19 , Pfaff tau-functions, Math. Ann. 322 (2002), 423–476. 7, 8, 23 M. ADLER and P. VAN MOERBEKE, Completely integrable systems, Euclidean Lie algebras, and curves, Adv. in Math. 38 (1980), 267–317. MR 83m:58041 10 , Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials, Duke Math. J. 80 (3) (1995), 863–911. MR 96k:58101 2, 3, 8, 9, 30, 44, 54 , Group factorization, moment matrices, and Toda lattices, Internat. Math. Res. Notices 1997, 555–572. MR 98e:58089 , Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum, Ann. of Math. (2) 153 (2001), 149–189. MR 1 826 412 ´ BREZIN ´ E. and H. NEUBERGER, Multicritical points of unoriented random surfaces, Nuclear Phys. B 350 (1991), 513–553. MR 91m:81179 10 E. DATE, M. KASHIWARA, M. JIMBO, and T. MIWA, “Transformation groups for soliton equations” in Nonlinear Integrable Systems—Classical Theory and Quantum Theory (Kyoto, 1981), World Sci., Singapore, 1983, 39–119. MR 86a:58093 L. DICKEY, Soliton Equations and Hamiltonian Systems, Adv. Ser. Math. Phys. 12, World Sci., River Edge, N.J., 1991. MR 93d:58067 V. G. KAC, Infinite-dimensional Lie algebras, 3d ed., Cambridge Univ. Press, Cambridge, 1990. MR 92k:17038 14 V. G. KAC and J. VAN DE LEUR, “The geometry of spinors and the multicomponent BKP and DKP hierarchies” in The Bispectral Problem (Montreal, 1997), CRM

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Proc. Lecture Notes 14, Amer. Math. Soc., Providence, 1998, 159–202. MR 99a:17026 7 M. L. MEHTA, Random Matrices, 2d ed., Academic Press, Boston, 1991. MR 92f:82002 34, 35 A. G. REYMAN and M. A. SEMENOV-TIAN-SHANSKY, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, Invent. Math. 54 (1979), 81–100. MR 81b:58021 10 J. VAN DE LEUR, Matrix integrals and the geometry of spinors, J. Nonlinear Math. Phys. 8 (2001), 288–310. MR 1 839 189 7 P. VAN MOERBEKE, “Integrable lattices: Random matrices and random permutations” in Random Matrix Models and Their Applications, ed. P. Bleher and A. Its, Math. Sci. Res. Inst. Publ. 40, Cambridge Univ. Press, Cambridge, 2001, 321–406. MR 1 842 794 E. WITTEN, “Two-dimensional gravity and intersection theory on moduli space” in Surveys in Differential Geometry (Cambridge, Mass., 1990), Lehigh Univ., Bethlehem, Pa., 1991, 243–310. MR 93e:32028

2 Adler Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454, USA; [email protected] van Moerbeke Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454, USA; [email protected] and [email protected]


THE CRAMER-RAO INEQUALITY FOR STAR BODIES ERWIN LUTWAK, DEANE YANG, and GAOYONG ZHANG

Abstract Associated with each body K in Euclidean n-space Rn is an ellipsoid 02 K called the Legendre ellipsoid of K . It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In an earlier paper the authors showed that corresponding to each convex body K ⊂ Rn is a new ellipsoid 0−2 K that is in some sense dual to the Legendre ellipsoid. The Legendre ellipsoid is an object of the dual Brunn-Minkowski theory, while the new ellipsoid 0−2 K is the corresponding object of the Brunn-Minkowski theory. The present paper has two aims. The first is to show that the domain of 0−2 can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If K is a star-shaped set, then 0−2 K ⊂ 02 K with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory—the Cramer-Rao inequality. Associated with each body K in Euclidean n-space Rn is an ellipsoid 02 K called the Legendre ellipsoid of K . The Legendre ellipsoid is a basic concept from classical mechanics. It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In [26] the authors showed that corresponding to each convex body K ⊂ Rn is a new ellipsoid 0−2 K . The results in this paper hint at a remarkable duality between this new ellipsoid and the Legendre ellipsoid. DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 12 September 2000. Revision received 31 March 2001. 2000 Mathematics Subject Classification. Primary 52A40; Secondary 94A17. Author’s work supported in part by National Science Foundation grant numbers DMS-9803261 and DMS0104363 59

60

LUTWAK, YANG, and ZHANG

The present paper has two aims. The first is to show that the domain of 0−2 can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If K is a star-shaped set, then 0−2 K ⊂ 02 K

(1)

with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory—the Cramer-Rao inequality. The Brunn-Minkowski theory (often called the theory of mixed volumes) is the heart of analytic convex geometry. Many of the fundamental ingredients of the theory were developed by H. Minkowski a century ago. R. Schneider’s book [28] is the classical reference for the subject. Over the years the tools of the Brunn-Minkowski theory have proven to be remarkably effective in solving inverse problems for which the data involve projections of convex bodies. A quarter of a century ago, the elements of a dual Brunn-Minkowski theory were introduced in [22] (and related papers). The basic idea of the dual theory is to replace the projections of the Brunn-Minkowski theory with intersections. In [23] it was shown that there is in fact a “dictionary” between the theories. Not only do concepts like the “elementary mixed volumes” of the classical theory become the “elementary dual mixed volumes” of the dual theory, but even objects such as the “projection bodies” of the Brunn-Minkowski theory have dual counterparts, “intersection bodies,” in the dual theory. In fact, it was this dual notion of “intersection body” that played a key role in the ultimate solution of the Busemann-Petty problem (see, e.g., R. Gardner [11], [12], [13]; G. Zhang [29], [30]; A. Koldobsky [17], [18], [19], [20]; Gardner, Koldobsky, and T. Schlumprecht [15]). Gardner’s book on geometric tomography [14] is an excellent reference for the interplay between the classical and dual theories. Minkowski showed that what was to become known as the Brunn-Minkowski theory could be developed naturally by combining the notion of volume with an addition of convex bodies now known as Minkowski addition. In the early 1960s, W. Firey [9] introduced and studied an L p -generalization of Minkowski addition. In the 1990s, in [24] and [25], these Minkowski-Firey L p -sums were combined with the notion of volume to form embryonic L p -versions of the Brunn-Minkowski theory. It is easily seen that the classical Legendre ellipsoid belongs to the dual BrunnMinkowski theory. This observation led the authors to the obvious question: What is the dual analog of the Legendre ellipsoid in the Brunn-Minkowski theory? The answer was given by the new ellipsoid introduced in [26]. This new ellipsoid actually belongs to the L 2 -Brunn-Minkowski theory. The nature of the duality between the Brunn-Minkowski theory and the dual Brunn-Minkowski theory is not understood. The objects of study of the Brunn-

THE CRAMER-RAO INEQUALITY FOR STAR BODIES

61

Minkowski theory are convex bodies, while the objects of study of the dual BrunnMinkowski theory are star bodies. The basic functionals of the Brunn-Minkowski theory are often expressed as integrals involving the support and curvature functions. The basic functionals of the dual Brunn-Minkowski theory are integrals involving radial functions. A first step in understanding the nature of the duality between the Brunn-Minkowski theory and its dual is to extend some of the functionals of the Brunn-Minkowski theory so that they are defined for star bodies (rather than convex bodies) and to provide new definitions of these functionals that involve only radial functions (rather than support and curvature functions). In this article, this first step is accomplished for one object of the Brunn-Minkowski theory: the 0−2 -ellipsoid. One of the central problems in information theory is how to extract useful information from noisy signals. Let x0 ∈ Rn be the transmitted signal. A simple model for the received signal is a random vector x ∈ Rn with a probability distribution p(x − x0 ) d x on Rn , where the probability measure p(x) d x has mean zero. Suppose the same signal is transmitted repeatedly and x1 , . . . , x N are the received signals. What is the best estimate for the transmitted signal, and what is the error of this estimate? One possible estimate is the mean x=

x1 + · · · + x N . N

By the central limit theorem, as N becomes large, the distribution of the √ random variable x approaches a Gaussian with mean x0 and covariance matrix C/ N , where the matrix C is given by Z Ci j =

Rn

xi x j p(x) d x.

Another estimate is the maximum likelihood estimate x M , which is obtained by maximizing the log-likelihood function L(x) =

N X

log p(xi − x).

i=1

R. Fisher [10] and J. Doob [7] (see also [1, Appendix I]) showed that as N becomes large, the distribution of the random variable x M approaches a Gaussian with mean √ x0 and covariance matrix F −1 / N , where the matrix F is known as the Fisher information matrix and is given by Z ∂(log p) ∂(log p) p d x. Fi j = ∂xi ∂x j Rn A fundamental result in information theory is the Cramer-Rao inequality (see, e.g., [5]), which states that the covariance and Fisher information matrices satisfy the

62


inequality v · Cv ≥ v · F −1 v

(2)

for all v ∈ Rn . Moreover, equality holds for all v ∈ Rn if and only if the distribution p is Gaussian. The Cramer-Rao inequality is important because it shows that the error in the maximum likelihood estimate is smaller than the mean estimate and that the mean estimate is optimal only if the distribution is Gaussian. It has previously been observed (see [4], [5], [6]) that there exists some connection between the subject of information theory and the Brunn-Minkowski theory. The authors believe that the true connection is between information theory and the L 2 Brunn-Minkowski theory. The authors have found small bits of an embryonic “dictionary” connecting the subject of information theory and the L 2 -Brunn-Minkowski theory. In this dictionary a probability distribution corresponds to a convex body and the entropy power of the distribution to the volume of the body. Associated with the covariance matrix C of a probability distribution is the ellipsoid E C = {v ∈ Rn : v · C −1 v ≤ 1}. In our dictionary this ellipsoid corresponds to the Legendre ellipsoid 02 K of a convex body K . Associated with the Fisher information matrix F is the ellipsoid E F = {v ∈ Rn : v · Fv ≤ 1}, which corresponds to the ellipsoid 0−2 K . The Cramer-Rao inequality (2) is equivalent to the statement E F ⊂ EC

(3)

with equality holding if and only if the probability distribution is Gaussian. Using the dictionary, the main result of this paper, (1), corresponds to (3). The definition of the 0−2 -ellipsoid uses the derivative of the radial function of the star body and therefore appears to require some differentiability assumptions for the star body. We show that, remarkably, no such assumptions are necessary for either the definition or the results of this paper. 1. Notation and overview A bounded set K ⊂ Rn which is star-shaped about the origin is uniquely determined by its radial function, ρ K : Rn \{0} → R, where ρ K (x) = sup{λ ≥ 0 : λx ∈ K }. A star body is a bounded set that is star-shaped about the origin and whose radial function is positive and continuous.


63

Throughout this paper the boundary of K is denoted ∂ K , and dy denotes the density associated with the (n − 1)-dimensional Hausdorff measure on ∂ K . The unit sphere in Rn is denoted S n−1 , and the density associated with the (n − 1)-dimensional Hausdorff measure on S n−1 is denoted du. Given a convex body K ⊂ Rn containing the origin in its interior, the ellipsoid 0−2 K was defined in [26] as the body whose radial function is given by Z 1 (x · ν(y))2 −2 ρ0−2 K (x) = dy, (4) V (K ) ∂ K y · ν(y) where · denotes the standard inner product on Rn and where ν(y) ∈ S n−1 is the outer unit normal at y ∈ ∂ K . This formula can be used to define 0−2 K for any body K ⊂ Rn with sufficiently smooth boundary. In §2 we use polar coordinates to obtain the following new formula for 0−2 K . PROPOSITION 3 Given a convex body K ⊂ Rn and x ∈ Rn \{0}, Z 2 1 −2 ρ0−2 K (x) = x · ∇ρ K (u) ρ K (u)n−4 du. V (K ) S n−1

In §§3–5 we present preliminary definitions and lemmas that are needed for the remainder of the paper. We show in §6 how to define for a star body K ⊂ Rn the set 0−2 K ⊂ Rn . In §10 it is also shown that this set is an ellipsoid that is possibly degenerate. If the boundary of K is Lipschitz, then 0−2 K is a nondegenerate ellipsoid. On the other hand, if the boundary of K is sufficiently singular, then 0−2 K is just a single point, namely, the origin. The following extension of [26, Lemma 1*] is proved in §7. LEMMA 13 If K ⊂ Rn is a star body and φ ∈ GL(n),

0−2 φ K = φ0−2 K . In §8 we derive a formula for the volume of K , and in §9 we use the following formula to establish our main result. THEOREM

If K ⊂ Rn is a star body, then 0−2 K ⊂ 02 K

64


with equality holding if and only if K is an ellipsoid centered at the origin. Recall that a set E ⊂ Rn is an ellipsoid centered at the origin if there exists an (n × n)–positive definite symmetric matrix A such that E = {x : x · Ax ≤ 1}. A set E ⊂ Rn is a degenerate ellipsoid centered at the origin if there exists a proper subspace L ⊂ Rn such that E ⊂ L and E is an ellipsoid in L. 2. A new formula for the 0−2 -ellipsoid Given a convex body K ⊂ Rn containing the origin in its interior, there is a natural parameterization of the boundary ∂ K given by the map φ K : S n−1 → ∂ K , where S n−1 ⊂ Rn is the unit sphere and φ K (u) = ρ K (u)u. Moreover, the radial function ρ K is Lipschitz, and for almost every y ∈ ∂ K , there exists a unique outer unit normal ν(y) to ∂ K at y (see, e.g., [28, p. 53] for a proof of these facts). When the meaning is clear, the subscript K in ρ K and φ K may be suppressed. To change the variable of integration in (4) from y ∈ ∂ K to u ∈ S n−1 , the following two lemmas are needed. LEMMA 1 If K ⊂ Rn is a convex body containing the origin in its interior, then for almost every u ∈ S n−1 , ν(φ(u)) ∇ρ(u) =− , φ(u) · ν(φ(u)) ρ(u)2 where ∇ρ denotes the gradient of ρ in Rn .

Proof Since ρ is constant along the boundary ∂ K , its gradient is normal to the boundary. In other words, given y ∈ ∂ K , there exists λ(y) ∈ R such that ∇ρ(y) = λ(y)ν(y). On the other hand, since ρ is homogeneous of degree −1, 1 = ρ(y) = −y · ∇ρ(y) = −λ(y)y · ν(y).


65

Therefore, λ(y) = −

1 y · ν(y)

and ∇ρ(y) = −

ν(y) . y · ν(y)

Substituting in y = φ(u) and observing that ∇ρ is homogeneous of degree −2 yields the desired formula. The formula for the surface area measure of ∂ K in polar coordinates is given in the following lemma. 2 If K ⊂ Rn is a convex body containing the origin in its interior, then LEMMA

dy =

ρ K (u)n du. φ(u) · ν(φ(u))

The parameterization φ : S n−1 → ∂ K is Lipschitz and therefore differentiable almost everywhere. By [8, Theorem 3.2.3], the change of measure is given by the determinant of the Jacobian. A straightforward computation yields the formula above. PROPOSITION 3 Given a convex body K ⊂ Rn and x ∈ Rn \{0}, Z 2 1 ρ0−2 K (x)−2 = x · ∇ρ K (u) ρ K (u)n−4 du, V (K ) S n−1

(5)

where ∇ρ K is the gradient of ρ K in Rn . Proof By Lemmas 1 and 2, Z 1 (x · ν(y))2 dy V (K ) ∂ K y · ν(y) Z 1 (x · ν(φ(u)))2 = ρ K (u)n du V (K ) S n−1 (φ(u) · ν(φ(u)))2 Z 1 = (x · ∇ρ K (u))2 ρ K (u)n−4 du. V (K ) S n−1

ρ0−2 K (x)−2 =

The operator 0−2 can be extended immediately to the class of star bodies with Lipschitz radial function using either formula (4) or formula (5). To extend it to the entire

66


class of star bodies requires a little more work. The next three sections contain preliminary definitions and results that are needed to this end. 3. A projection of the sphere onto the cylinder The projection of a vector x ∈ Rn into each tangent space of S n−1 defines a vector field x(u) ˆ = x − (x · u)u. (6) We want to use the integral curves of xˆ to define coordinates on S n−1 . This can be done as follows. Given x ∈ Rn \{0}, let x ⊥ = {v ∈ Rn : v · x = 0}, and let hxi =

x . |x|

There is a natural projection of S n−1 onto the cylinder Rx + (S n−1 ∩ x ⊥ ) obtained by mapping the unit vector u to the intersection of the cylinder with the ray containing u. The inverse of this projection is given by Mx : R × (S n−1 ∩ x ⊥ ) → S n−1 \ h−xi, hxi , φ + hxi sinh |x|t (t, φ) 7 → u = . cosh |x|t A straightforward calculation shows that x · u = |x| tanh |x|t

(7)

and ∂u x − φ|x| sinh |x|t = ∂t (cosh |x|t)2 = x − (x · u)u. This implies that for each f ∈ C 1 (S n−1 ), ∂( f ◦ Mx ) ∇x(M (t, φ), ˆ x (t,φ)) f M x (t, φ) = ∂t

(8)

where ∇v f denotes the directional derivative of f in direction v. Also, du =

|x| dt dφ , (cosh |x|t)n−1

(9)

where dφ is the density corresponding to the (n − 2)-dimensional Hausdorff measure on S n−1 ∩ x ⊥ .


67

4. Differentiability along a vector field In this section we make precise the notions of continuous differentiability and L 2 differentiability along the vector field xˆ of a function on the unit sphere, without assuming any regularity in other directions. Given x ∈ Rn , let C 0 (S n−1 ∩ x ⊥ , C 1 (R)) denote the space of continuous functions f : R × (S n−1 ∩ x ⊥ ) → R such that f (·, φ) ∈ C 1 (R) for each φ ∈ S n−1 ∩ x ⊥ . Define C x1 (S n−1 ) = f ∈ C 0 (S n−1 ) : f ◦ Mx ∈ C 0 S n−1 ∩ x ⊥ , C 1 (R) . The restriction of any f ∈ C x1 (S n−1 ) to an integral curve of the vector field xˆ is C 1 . Therefore, ∇xˆ f is well defined and given explicitly by ∂ f Mx (t, φ) . ∇xˆ f Mx (t, φ) = ∂t

(10)

The following chain rule holds. LEMMA 4 Given a C 1 -function φ and f ∈ C x1 (S n−1 ), the function φ ◦ f ∈ C x1 (S n−1 ). Moreover,

∇xˆ (φ ◦ f ) = (φ 0 ◦ f )∇xˆ f. To define the notion of L 2 -differentiability, we need the following integration by parts formulas. LEMMA 5 Given x ∈ Rn and f, g ∈ C x1 (S n−1 ), Z Z f ∇xˆ g du = g −∇xˆ f + (n − 1)(x · u) f du S n−1

(11)

S n−1

and Z

Z S n−1

f ∇xˆ g du = −

Proof By (9) and (10), Z S n−1

R×(S n−1 ∩x ⊥ )

(g ◦ Mx )

Z f ∇xˆ g du =

R×(S n−1 ∩x ⊥ )

∂ h |x|( f ◦ Mx ) i dt dφ. ∂t (cosh |x|t)n−1

( f ◦ Mx )

∂(g ◦ Mx ) |x| dt dφ . ∂t (cosh |x|t)n−1

(12)

68


Equation (12) now follows by integrating by parts. By (9), (8), and (7), Z Z ∂ h |x|( f ◦ Mx ) i f ∇xˆ g du = − (g ◦ Mx ) dt dφ ∂t (cosh |x|t)n−1 S n−1 R×(S n−1 ∩x ⊥ ) Z i h ∂( f ◦ M ) x − (n − 1)( f ◦ Mx )|x| tanh |x|t =− (g ◦ Mx ) ∂t R×(S n−1 ∩x ⊥ ) |x| dt dφ × (cosh |x|t)n−1 Z = g −∇xˆ f + (n − 1)(x · u) f du. S n−1

Equation (11) can be used to define for any g ∈ C 0 (S n−1 ) the directional derivative ∇xˆ g as a distribution. This motivates the following definition. Definition Given x ∈ Rn and g ∈ C 0 (S n−1 ), we say that ∇xˆ g ∈ L 2 (S n−1 ) if there exists a function g 0 ∈ L 2 (S n−1 ) such that for any f ∈ C x1 (S n−1 ), Z Z g −∇xˆ f + (n − 1)(x · u) f du = g 0 f du. (13) S n−1

S n−1

If such a function g 0 ∈ L 2 (S n−1 ) exists, it will be denoted ∇xˆ g. Since L 2 (S n−1 ) is its own dual, we have the following lemma. LEMMA

6

If x ∈ and g ∈ C 0 (S n−1 ), then ∇xˆ g ∈ L 2 (S n−1 ) if and only if there exists c > 0 such that Z g −∇xˆ f + (n − 1)(x · u) f du ≤ ck f k2 (14) Rn

S n−1

for each f ∈ C x1 (S n−1 ). Lemma 5 now gives the following lemma. LEMMA

7

If g ∈ C x1 (S n−1 ), the pointwise definition (10) of ∇xˆ g agrees with the distributional definition (13). 5. Smoothing a function along a vector field Fix x ∈ Rn \{0}. We want to define a smoothing operator that smooths only along the integral curves of the vector field x. ˆ


69

Given τ > 0, let χτ be a smooth nonnegative function on R which is supported in the interval (−τ, τ ) and satisfies Z ∞ χτ (t) dt = 1. −∞

Using the map Mx defined in §3, we define the function gτ ∈ C 0 (S n−1 ) by Z ∞ gτ Mx (t, φ) = g Mx (t − s, φ) χτ (s) ds

(15)

for each g ∈ C 0 (S n−1 ) and τ > 0. Observe that Z ∞ g Mx (s, φ) χτ (t − s) ds gτ Mx (t, φ) =

(16)

−∞

−∞

and that gτ extends continuously across hxi and h−xi ∈ S n−1 . Before stating and proving the main proposition about gτ , we begin with two lemmas. The first is an analogue of the standard (and trivial) fact that convolution on Rn commutes with partial differentiation. 8 Given g ∈ C 0 (S n−1 ) and x ∈ Rn , if ∇xˆ g ∈ L 2 (S n−1 ), then LEMMA

∇xˆ gτ = (∇xˆ g)τ . Proof It suffices to prove that

∂(gτ ◦ Mx ) = (∇xˆ g)τ ◦ Mx . ∂t By (16) and (10), given any f ∈ C 0 (S n−1 ∩ x ⊥ ), Z ∂(gτ ◦ Mx ) f (φ) (t, φ) dφ n−1 ⊥ ∂t S ∩x Z = f (φ)g Mx (s, φ) χτ0 (t − s) ds dφ R×(S n−1 ∩x ⊥ ) Z ∂ χτ (t − s) f (φ) ds dφ =− g Mx (s, φ) ∂s R×(S n−1 ∩x ⊥ ) Z = f (φ)χτ (t − s)(∇x(u) ˆ g) M x (s, φ) ds dφ n−1 ⊥ R×(S ∩x ) Z = f (φ)(∇x(u) ˆ g)τ M x (t, φ) dφ. S n−1 ∩x ⊥

This holds for each f ∈ C 0 (S n−1 ∩ x ⊥ ) and thus yields (17).

(17)

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The next lemma is a weak form of the Young inequality. 9 There exists T > 0 such that, for each h ∈ L 2 (S n−1 ) and 0 < τ < T , LEMMA

kh τ k2 < 2khk2 . Proof Observe that

cosh |x|(t − s) ≤ e|sx| cosh |x|t

for every t ∈ R. Therefore, if 0 0 quantities δ > 0


71

and T > 0 such that g Mx (t1 , φ) − g Mx (t2 , φ) < , g Mx (t, φ) − g hxi < , 2 g Mx (−t, φ) − g h−xi < , 2 for every φ ∈ S n−1 ∩ x ⊥ , t ∈ (T −δ, ∞), and t1 , t2 ∈ [−T, T ] satisfying |t1 −t2 | < δ. It follows that, given any τ ∈ (0, δ/2), Z ∞ gτ Mx (t, φ) − g Mx (t, φ) ≤ g Mx (s, φ) − g Mx (t, φ) χτ (t − s) ds −∞

0, there exists a function f ∈ C 0 (S n−1 ) such that k f − ∇xˆ gk2 < . By (18), f τ converges uniformly to f . Therefore, there exists T > 0 such that k f τ − f k2 < for any τ ∈ (0, T ). By Lemmas 8 and 9, for any τ ∈ (0, T ), k∇xˆ gτ − ∇xˆ gk2 ≤ k∇xˆ gτ − f τ k2 + k f τ − f k2 + k f − ∇xˆ gk2

= (∇xˆ g − f )τ 2 + k f τ − f k2 + k f − ∇xˆ gk2 ≤ 2k∇xˆ g − f k2 + k f τ − f k2 + k f − ∇xˆ gk2 ≤ 4. This proves (19). We can now prove the following analogue of Lemma 4. 11 Suppose that x ∈ Rn , φ is a C 1 function, and g ∈ C 0 (S n−1 ). If ∇xˆ g ∈ L 2 (S n−1 ), then ∇xˆ (φ ◦g) ∈ L 2 (S n−1 ) and LEMMA

∇xˆ (φ ◦g) = (φ 0 ◦g)∇xˆ g. Proof By (18), as τ → 0, φ 0 ◦gτ converges uniformly to φ 0 ◦g, and φ ◦gτ to φ ◦g. Therefore,

72


(φ 0 ◦gτ )∇xˆ gτ converges in L 2 to (φ 0 ◦g)∇xˆ g, and, given any f ∈ C x1 (S n−1 ), Z (φ ◦g) −∇xˆ f + (n − 1)(x · u) f du S n−1 Z (φ ◦gτ ) −∇xˆ f + (n − 1)(x · u) f du = lim τ →0 S n−1 Z 0 = lim (φ ◦gτ )∇xˆ gτ f du τ →0 S n−1 Z 0 = (φ ◦g)∇xˆ g f du. S n−1

The lemma now follows by (13). 6. The 0−2 -ellipsoid for star bodies Given a star body K ⊂ Rn , define g K : Rn \{0} → R to be  ρ (x)(n/2)−1    K if n > 2, (n/2) − 1 g K (x) =   log ρ (x) if n = 2.

(20)

K

If ρ K : Rn \{0} → R is C 1 , then formula (5) can be rewritten as Z 2 1 ρ0−2 K (x)−2 = ∇x g K (u) du, V (K ) S n−1

(21)

where ∇x g K denotes the directional derivative of g K (viewed as a function on Rn \{0}) in direction u. In general, g K is known only to be C 0 , and its directional derivative does not necessarily exist. Nevertheless, it is still possible to define 0−2 K . Obviously, given p ∈ R, any function g ∈ C 1 (S n−1 ) has a unique extension to Rn \{0} which is homogeneous of degree p. Euler’s equation says that, for each y ∈ Rn \{0}, ∇ y g(y) = pg(y). Therefore, given x ∈ Rn , ∇x g(u) = ∇x(u) ˆ g(u) + p(x · u)g(u)

(22)

for any u ∈ S n−1 . If we assume only that g ∈ C 0 (S n−1 ) and ∇xˆ g ∈ L 2 (S n−1 ), then the function ∇x g ∈ L 2 (S n−1 ) can be defined using (22). In particular, if g K ∈ C 1 (S n−1 ), then the homogeneity of ρ K and Euler’s equation imply that, for any u ∈ S n−1 , ∇x g K (u) = ∇xˆ g K (u) − (x · u)ρ K (u)(n/2)−1 .

(23)


73

If we assume only that ∇xˆ g K ∈ L 2 (S n−1 ), then equation (23) can be used to define ∇x g K ∈ L 2 (S n−1 ). Lemma 11 ensures that (23) is consistent with the definition of ∇x ρ K , as given by (22). We can now define the set 0−2 K . Definition Given a star body K ⊂ Rn , define 0−2 K = x ∈ Rn : ∇x g K ∈ L 2 (S n−1 ) and k∇x g K k22 ≤ V (K ) .

(24)

If the body K is convex, then this definition agrees with formula (5) and therefore with the original definition given in [26]. The following integration by parts formulas are needed later. LEMMA 12 Suppose that K ⊂ Rn is a star body, and suppose that x ∈ Rn such that ∇xˆ g K ∈ L 2 (S n−1 ). Then for every f ∈ C x1 (S n−1 ),  Z  g K ∇x f du if n > 2, − Z  S n−1 f ∇x g K du = (25) Z  S n−1  − (∇x f ) g K − log | f | du if n = 2, S n−1

where f has been extended to be homogeneous of degree −(n/2) on Rn \{0}. Proof First, assume that g K ∈ C x1 (S n−1 ). The case n > 2 follows directly from equation (22) and Lemmas 4 and 5. The case n = 2 requires the additional observations that f ∇x (log | f |) ∈ L ∞ (S n−1 ) and that f (u)∇x log | f (u)| = ∇x f (u) for almost every u ∈ S n−1 . If ∇xˆ g K ∈ L 2 (S n−1 ), then equation (25) holds for (g K )τ , as defined by Proposition 10. The lemma now follows by taking the limit τ → 0.

74


7. Invariance under GL(n) 13 If K ⊂ Rn is a star body and φ ∈ GL(n), LEMMA

0−2 φ K = φ0−2 K . Proof Given a star body K ⊂ Rn , let g K be as defined by (20). Given a star body L ⊂ Rn , let n/2 f L = ρL . Recall that hxi = x/|x| for any x ∈ Rn . Let φ ∈ SL(n), and let v = hφ −1 ui. Using the fact that ρφ K (u) = ρ K (φ −1 u) and (25), we have for n > 2, Z Z f L ∇x gφ K du = − gφ K (u)∇x f L (u) du n−1 S n−1 ZS =− g K (φ −1 u)∇x f L (u) du n−1 ZS =− |φ −1 u|1−(n/2) g K hφ −1 ui ∇x f L hφvi du n−1 ZS =− g K (v)∇x f L (φv)|φ −1 u|−n du S n−1 Z =− g K (v)∇φ −1 x f φ −1 L (v) dv. S n−1

If n = 2, then Z S n−1

ρφ K ρ L ∇x log du ρL S n−1 Z ρφ K (u) = −∇x ρ L (u) log du ρ L (u) S n−1 Z ρ K (φ −1 u) = −∇x ρ L hφvi log du ρφ −1 L (φ −1 u) S n−1 Z ρ K (v) = −∇x ρ L (φv) log |φ −1 u|−2 du ρφ −1 L (v) S n−1 Z

f L ∇x gφ K du =


75

ρ K (v) dv ρφ −1 L (v) Z ρ K (v) dv = ρφ −1 L (v)∇φ −1 x log ρφ −1 L (v) S n−1 Z = f φ −1 L ∇φ −1 x g K dv. Z

=

S n−1

−∇φ −1 x ρφ −1 L (v) log

S n−1

Thus, for each dimension n ≥ 2 and star body L with a C 1 –radial function, Z Z ∇x gφ K f L du = ∇φ −1 x g K f φ −1 L du. S n−1

S n−1

It follows by linearity that, for any f ∈ C 1 (S n−1 ), Z Z ∇x gφ K f du = ∇φ −1 x g K f ◦φ du. S n−1

S n−1

Therefore, ∇x gφ K ∈ L 2 (S n−1 ) if and only if ∇φ −1 x g K ∈ L 2 (S n−1 ). Given any f ∈ C 1 (S n−1 ), extend it to Rn \{0} as a function homogeneous of degree −(n/2). There exists a star-shaped set L such that n/2

ρL

= | f |.

Observe that, for any φ ∈ SL(n), n/2

ρφ −1 L = | f ◦φ|, and therefore k f ◦φk22 = nV (φ −1 L) = nV (L) = k f k22 . It follows that nZ o k∇x gφ K k2 = sup ∇x gφ K f du : k f k2 = 1 S n−1 nZ o = sup ∇φ −1 x g K f ◦φ du : k f ◦φk2 = 1 S n−1

= k∇φ −1 x g K k2

76


and 0−2 φ K = x ∈ Rn : ∇x gφ K ∈ L 2 (S n−1 ) and k∇x gφ K k22 ≤ V (φ K ) = x ∈ Rn : ∇φ −1 x g K ∈ L 2 (S n−1 ) and k∇φ −1 x g K k22 ≤ V (K ) = φw ∈ Rn : ∇w g K ∈ L 2 (S n−1 ) and k∇w g K k22 ≤ V (K ) = φ0−2 K . For the case of dilation, let λ > 0. It is easily seen that ∇x gλK = ∇λ(n/2)−1 x g K = λn/2 ∇λ−1 x g K . Therefore, 0−2 λK = x ∈ Rn : ∇x gλK ∈ L 2 (S n−1 ) and k∇x gλK k22 ≤ V (λK ) = x ∈ Rn : ∇λ−1 x g K ∈ L 2 (S n−1 ) and k∇λ−1 x g K k22 ≤ V (K ) = λw ∈ Rn : ∇w g K ∈ L 2 (S n−1 ) and k∇w g K k22 ≤ V (K ) = λ0−2 K . 8. A formula for volume The following formula for the volume of a star body K is needed to prove that the Legendre ellipsoid contains the 0−2 -ellipsoid. LEMMA 14 Given x ∈ S n−1 and a star body K ⊂ Rn , if ∇xˆ g K ∈ L 2 (S n−1 ), then Z V (K ) = (x · u)ρ K (u)(n/2)+1 −∇x g K (u) du. S n−1

Proof For convenience we omit the subscript K , denoting ρ K by ρ and g K by g. By (23), Lemma 11, and (22), the following holds in L 2 (S n−1 ): ρ (n/2)+1 ∇x g = ρ (n/2)+1 ∇xˆ g − (x · u)ρ (n/2)−1 = ρ n−1 ∇xˆ ρ − (x · u)ρ n 1 = ∇xˆ (ρ n ) − (x · u)ρ n . n


77

Since ∇xˆ (ρ n ) ∈ L 2 (S n−1 ), it follows by (13) that Z Z i h 1 (x · u)ρ (n/2)+1 (−∇x g) du = (x · u) − ∇xˆ (ρ n ) + (x · u)ρ n du n−1 n S n−1 ZS 1 ∇xˆ (x · u) + (x · u)2 ρ n du = S n−1 n Z 1 = ρ n du n S n−1 = V (K ).

9. Inclusion and equality THEOREM

If K ⊂ Rn is a star body, then 0−2 K ⊂ 02 K with equality holding if and only if K is an ellipsoid centered at the origin. Proof By applying a linear transformation to K , we can assume that its Legendre ellipsoid 02 K is the unit ball. This means that Z 1 (x · u)2 ρ K (u)n+2 du = |x|2 (26) V (K ) S n−1 for every x ∈ Rn . Let x ∈ 0−2 K . First, it follows from the definition of 0−2 K that ∇xˆ g K ∈ L 2 (S n−1 ). By Lemma 14, the Hölder inequality, (26), and (24), Z 1 2 |x| = (x · u)ρ(u)(n/2)+1 −∇x g K (u) du V (K ) S n−1 1 Z 1/2 1 Z 2 1/2 ≤ (x · u)2 ρ K (u)n+2 du ∇x g K (u) du V (K ) S n−1 V (K ) S n−1 ≤ |x|. (27) Thus, |x| ≤ 1, and therefore, 0−2 K is contained in the unit ball. This proves that 0−2 K ⊂ 02 K .

78


Suppose that 0−2 K = 02 K . Since we are assuming that 02 K is the unit ball, so is 0−2 K . This implies that, for each x ∈ S n−1 , we have ∇x g K ∈ L 2 (S n−1 ), and equality holds in all of the inequalities in (27). From the equality conditions of the Hölder inequality, there exists for each x ∈ S n−1 a constant c(x) such that (x · u)ρ K (u)(n/2)+1 = c(x)∇x g K (u)

(28)

for almost every u ∈ S n−1 . Integrating the square of both sides with respect to u ∈ S n−1 and using (24) and (26) shows that c(x) = ±1. Since (28) holds for all x ∈ S n−1 , it follows by (13) that, for each f ∈ C 1 (Rn \{0}) homogeneous of degree −(n/2) − 1, if n > 2, Z S n−1

(n/2)+1 ρK

f du =

n Z X i=1

=±

=∓

=∓

S n−1

(u · ei )(ei · u)ρ K (u)(n/2)+1 f (u) du

n Z X i=1 S n Z X

n−1

i=1 S n Z X

n−1

i=1

Z =∓ S n−1

Z

(u · ei ) f (u)∇ei g K (u) du ∇ei (u · ei ) f g K du

S n−1

(u · ei )∇ei f + f g K du

(∇u f + n f )g K du fρ (n/2)−1 du.

=∓ S n−1

If n = 2, then for each f ∈ C 1 (Rn \{0}) homogeneous of degree −(n/2) − 1, Z (n/2)+1 ρK f du S n−1

=

n Z X i=1

=±

=±

S n−1

n Z X i=1 S n Z X i=1

=∓

(u · ei )(ei · u)ρ K (u)(n/2)+1 f (u) du

n−1

h i 1 (u · ei ) f ∇ei g K − log | f |1/2 + (u · ei )∇ei f du 2 S n−1

n Z X i=1

(u · ei ) f (u)∇ei g K (u) du

h S n−1

i 1 g − log | f |1/2 ∇ei (u · ei ) f + f ∇ei (u · ei ) du 2


79

Z f du.

=∓ S n−1

Therefore,

ρ (n/2)+1 = ±ρ (n/2)−1 .

Since ρ K is positive and continuous, it must be the constant function 1. In other words, K is the unit ball centered at the origin.

10. The set 0−2 K is an ellipsoid COROLLARY

If K ⊂ Rn is a star body, then the set 0−2 K ⊂ Rn is an ellipsoid that is possibly degenerate. Proof Suppose that K is a star body and that g K is as defined by (20). Using (13), it is easily seen that, for all x, v ∈ Rn and λ ∈ R, if ∇x g K , ∇v g K ∈ L 2 (S n−1 ), then ∇λx g K , ∇x+v g K ∈ L 2 (S n−1 ). It follows that the set L = x ∈ Rn : ∇x g K ∈ L 2 (S n−1 ) is a linear subspace of Rn . Moreover, 0−2 K ⊂ L. Let dim L = m ≤ n. Choose a basis e1 , . . . , en of Rn such that e1 , . . . , em is a basis of L. Given x ∈ L, write m X x= xi ei . i=1

Then x ∈ 0−2 K if and only if x · Ax ≤ V (K ), where A is the nonnegative symmetric matrix whose (i, j)th entry is Z Ai j = (∇ei g K )(∇e j g K ) du, S n−1

where 1 ≤ i, j ≤ m. Observe that since ei , e j ∈ L, it follows from the Hölder inequality that the integral is bounded. Now suppose that A has a zero eigenvalue. Then the set 0−2 K is unbounded. However, the Legendre ellipsoid 02 K is bounded and, by the theorem in §9, 0−2 K ⊂ 02 K . Therefore, 0−2 K is bounded. The contradiction shows that A is positive definite and that 0−2 K is a nondegenerate ellipsoid in L.

80


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[4]

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derivative information, I: Fisher information and convex duality, Math. Oper. Res. 21 (1996), 442–468. MR 97e:90114 61 H. J. BRASCAMP and E. H. LIEB, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Adv. Math. 20 (1976), 151–173. MR 54:492 , On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), 366–389. MR 56:8774 M. H. M. COSTA and T. M. COVER, On the similarity of the entropy power inequality and the Brunn-Minkowski inequality, IEEE Trans. Inform. Theory 30 (1984), 837–839. MR 86d:26029 62 T. M. COVER and J. A. THOMAS, Elements of Information Theory, Wiley Ser. Telecom., Wiley-Interscience, New York, 1991. MR 92g:94001 61, 62 A. DEMBO, T. M. COVER, and J. A. THOMAS, Information-theoretic inequalities, IEEE Trans. Inform. Theory 37 (1991), 1501–1518. MR 92h:94005 62 J. L. DOOB, Probability and statistics, Trans. Amer. Math. Soc. 36 (1934), 759–775. MR 1 501 765 61 H. FEDERER, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969. MR 41:1976 65 W. J. FIREY, p-means of convex bodies, Math. Scand. 10 (1962), 17–24. MR 25:4416 60 R. A. FISHER, Theory of statistical estimation, Philos. Trans. Roy. Soc. London Ser. A 222 (1930), 309–368. 61 R. J. GARDNER, On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies, Bull. Amer. Math. Soc. (N.S.) 30 (1994), 222–226. MR 94m:52007 60 , A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), 435–447. MR 95i:52005 60 , Geometric tomography, Notices Amer. Math. Soc. 42 (1995), 422–429. MR 97b:52003 60 , Geometric Tomography, Encyclopedia Math. Appl. 58, Cambridge Univ. Press, Cambridge, 1995. MR 96j:52006 60 R. J. GARDNER, A. KOLDOBSKY, and T. SCHLUMPRECHT, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. (2) 149 (1999), 691–703. MR 2001b:52011 60 O. G. GULERYUZ, E. LUTWAK, D. YANG, AND G. ZHANG, Information theoretic inequalities for contoured distributions, preprint, 2001, http://eeweb.poly.edu/∼ onur/online pub.html A. KOLDOBSKY, Intersection bodies and the Busemann-Petty problem, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 1181–1186. MR 98k:52007 60 , Intersection bodies in R4 , Adv. Math. 136 (1998), 1–14. MR 99j:52013 60


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, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), 827–840. MR 99i:52005 60 , Second derivative test for intersection bodies, Adv. Math. 136 (1998), 15–25. MR 99j:52014 60 E. H. LIEB and M. LOSS, Analysis, Grad. Stud. Math. 14, Amer. Math. Soc., Providence, 1997. MR 98b:00004 70 E. LUTWAK, Dual mixed volumes, Pacific J. Math. 58 (1975), 531–538. MR 52:1528 60 , Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232–261. MR 90a:52023 60 , The Brunn-Minkowski-Firey theory, I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150. MR 94g:52008 60 , The Brunn-Minkowski-Firey theory, II: Affine and geominimal surface area, Adv. Math. 118 (1996), 244–294. MR 97f:52014 60 E. LUTWAK, D. YANG, and G. ZHANG, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), 375–390. MR 2001j:52011 59, 60, 63, 73 V. D. MILMAN and A. PAJOR, “Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space” in Geometric Aspects of Functional Analysis: Israel Seminar (1987– 88), Lecture Notes in Math 1376, Springer, Berlin, 1989, 64–104. MR 90g:52003 R. SCHNEIDER, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge, 1993. MR 94d:52007 60, 64 G. ZHANG, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345 (1994), 777–801. MR 95d:52008 60 , A positive solution to the Busemann-Petty problem in R4 , Ann. of Math. (2) 149 (1999), 535–543. MR 2001b:52010 60

Lutwak Department of Mathematics, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201, USA; [email protected] Yang Department of Mathematics, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201, USA; [email protected] Zhang Department of Mathematics, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201, USA; [email protected]


THE HIGHEST LOWEST ZERO AND OTHER APPLICATIONS OF POSITIVITY STEPHEN D. MILLER

In memory of Teresa Bowen Abstract The first nontrivial zeros of the Riemann ζ -function are ≈ 1/2 ± 14.13472i. We investigate the question of whether or not any other L-function has a higher lowest zero. To do so, we try to quantify the notion that the L-function of a “small” automorphic representation (i.e., one with small level and archimedean type) does not have small zeros, and vice-versa. We prove that many types of automorphic L-functions have a lower first zero than ζ ’s (see Theorems 1.1 and 1.2). This is done using Weil’s explicit formula with carefully chosen test functions. When this method does not immediately show that L-functions of a certain type have low zeros, we then attempt to turn the tables and show that no L-functions of that type exist. Thus, the argument is a combination of proving that low zeros exist and that certain cusp forms do not. Consequently, we are able to prove vanishing theorems and improve upon existing bounds on the Laplace spectrum on L 2 (SLn (Z)\ SLn (R)/ SOn (R)). These in turn can be used to show that SL68 (Z)\ SL68 (R)/ SO68 (R) has a discrete, nonconstant, noncuspidal eigenvalue outside the range of the continuous spectrum on L 2 (SL68 (R)/ SO68 (R)), but that this never happens for SLn (Z)\ SLn (R)/ SOn (R) in lower rank. Another application is to cuspidal cohomology: we show there are no cuspidal harmonic forms on SLn (Z)\ SLn (R)/ SOn (R) for n < 27. 1. Introduction The Riemann ζ -function’s first critical zeros are surprisingly large—about 1/2 ± 14.13472i. Our main interest in this paper is the following question: Does any other automorphic L-function have a larger first zero? DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 25 September 2000. Revision received 11 April 2001. 2000 Mathematics Subject Classification. Primary 11F66; Secondary 11M26, 11M41, 11F55, 11F72, 11F75. Author’s work supported by National Science Foundation graduate and postdoctoral fellowships and by a Yale Hellman Fellowship.

83

84

STEPHEN D. MILLER

This question was raised by A. Odlyzko [O], who proved that the Dedekind ζ -function of any number field has a zero whose imaginary part is less than 14. Odlyzko also proved related conditional results for Artin L-functions. Every automorphic L-function conjecturally factors into products of standard Lfunctions of cusp forms on GLn over the rationals, and we shall be content to discuss these.∗ In fact, by twisting a cuspidal automorphic representation of GLn /Q by a power of the determinant, it is possible to shift the zeros any amount vertically, so we restrict ourselves to studying cuspidal automorphic representations π = ⊗ p≤∞ π p of GLn /Q whose central character is normalized to have finite order. In most examples coming from number theory, the archimedean type π∞ is real; that is, the gamma factors multiplying L(s, π) have real shifts. The following result answers the above question for such cusp forms. 1.1 N

THEOREM

Let π = p≤∞ π p be a cuspidal automorphic representation of GLn over Q with a real archimedean type π∞ and a unitary central character. Then L(s, π) has a low zero that either (i) is on the critical axis between 1/2 ± 14.13472i or (ii) violates the generalized Riemann hypothesis (GRH) in an effective range. When we speak of a zero violating GRH “in an effective range,” we mean that should conclusion (i) fail, then one could theoretically find an effective constant T > 0 such that the box (1/2, 1) × [−T, T ]i contains a zero. For brevity we use the following terminology. Definition. An L-function has a low zero if it either vanishes on the critical axis between 1/2 ± 14.13472i or violates GRH in an effectively bounded range (see Section 2.2). We use this definition to state unconditional results, but not much is actually gained philosophically or numerically in this problem by assuming GRH. The L-functions in Theorem 1.1 include those of Dirichlet characters, rational elliptic curves, and, conjecturally, all rational abelian varieties. Of course, they are also expected to include all Artin L-functions, for example, L-functions of Galois representations. We have been unable to squeeze our technique to answer Odlyzko’s question in full generality, but we can prove many cases. For example, we have the following theorem. ∗ Nevertheless,

our arguments work under various wider assumptions, as they are mostly sensitive to the analytic properties of the L-function.

THE HIGHEST LOWEST ZERO

85

THEOREM 1.2 Let π be a cuspidal automorphic representation of GL2 over Q with a central character of finite order. Then L(s, π) has a low zero (that is on the critical axis between 1/2 ± 14.13472i or else violates GRH in an effective range).

This includes modular form and Maass form L-functions. Other results can be proven about low zeros. For example, every L-function that is related to itself by an odd functional equation automatically vanishes at s = 1/2. For a fixed degree n, most cuspidal automorphic representations of GLn over Q have low zeros. In fact, the possible exceptions all lie in a bounded subset of the unitary dual and have bounded level. This subset tends to be devoid of cusp forms, which is why our method is successful. Thus Odlyzko’s question is related to vanishing theorems about automorphic forms. Our technique uses Weil’s explicit formula relating the coefficients and zeros of automorphic L-functions. It is a variation on the Stark-Odlyzko positivity technique, as formulated by J.-P. Serre, G. Poitou, J.-F. Mestre, and others (see [O] for a survey). In particular, one can compute an exact formula for sum of certain test functions over the critical zeros. If we use a test function that is positive only in a certain range, then finding that this sum is positive ensures a zero in that range. On the other hand, if this sum is negative, then we can often construct another test function that is positive in the critical strip yet whose sum over the zeros is negative. This contradiction shows that the L-function actually could not have existed to begin with. Our main difficulty is that it is often very difficult to construct this second test function given the failure of the first. The latter contradiction, of positive terms yielding a negative sum, can be used to prove vanishing theorems about automorphic forms since they cannot exist when their L-functions do not. Independent of our interest in low zeros, this leads to applications in group cohomology and spectral theory. Other applications One of the consequences of the Ramanujan-Selberg temperedness conjecture is that the discrete cuspidal spectrum of the laplacian 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)) is contained in the continuous spectrum of 1 on L 2 (SLn (R)/ SOn (R)). (We always normalize 1 so that this continuous spectrum is the interval [(n 3 − n)/24, ∞)). This consequence should be true more generally for congruence covers of SLn (Z)\ SLn (R)/ SOn (R), but in the following particular case slightly more was proven in [M].

86

STEPHEN D. MILLER

THEOREM 1.3 ([M]) There exists a constant c > 0 such that the Laplace eigenvalue of every cusp form φ on SLn (Z)\ SLn (R)/ SOn (R) satisfies λ(φ) > λ1 SLn (R)/ SOn (R) + cn.

The following new result is superior for small n. 1.4 Let φ be a cuspidal eigenfunction of the noneuclidean laplacian 1 on SLn (Z)\ SLn (R)/ SOn (R). Then φ’s Laplace eigenvalue satisfies THEOREM

λ(φ) >

1 n 3 − 4n + 25.92 1 + . 24 n−1

(1)

It can be applied to answer a question of A. Lubotzky: When does the eigenvalue of a noncuspidal, square-integrable eigenfunction of the laplacian on SLn (Z)\ SLn (R)/ SOn (R) lie outside [(n 3 − n)/24, ∞)? 1.5 There exists a discrete Laplace eigenfunction THEOREM

φ ∈ L 2 SL68 (Z)\ SL68 (R)/ SO68 (R) such that 1φ = λφ φ,

λφ ≈ 12916.6
0, where π˜ is the contragredient representation to π. The Jacquet-Shalika (see [JS]) bounds imply that 1 Re η j > − . (2) 2 2.1. The Weil formula The explicit formula of A. Weil equates a sum over the zeros of an L-function with a sum over its coefficients and gamma factors: X

3((1/2)+iγ ,π)=0

h(γ ) = 2 Re

m nX j=1

l(η j ) −

∞ o X cn √ g(log n) + g(0) log D, n n=1

(3)

88

STEPHEN D. MILLER

where g is an even, differentiable real function, Z g(r ˆ ) = h(r ) = g(x)eir x d x, R

0R (s) = π −s/2 0(s/2), and Z 00 1 1 h(r ) R + η + ir dr 2π R 0R 2 Z Z − log π 1 1 0 0 1 η ir dr + dr = h(r ) h(r ) + + 2π R 2 2π R 20 4 2 2 Z ∞ log π g(0) 1 g(x/2)e−(1/4+η/2)x =− − g(0) − d x. 2 2 0 1 − e−x ex x

l(η) =

Here we have made use of the fact that L(s, π) is entire; for ζ (s) and Rankin-Selberg L-functions, there is a polar term that is introduced when needed later on (see [RS] for a proof of (3)). If g is supported in the interval [− log 2, log 2], then the formula can be viewed as giving the value of the sum over the zeros from the gamma factors: X

h(γ ) = 2 Re

m X j=1

l(η j ) + g(0) log D.

(4)

The basis of the positivity technique is the observation that if h(γ ) ≥ 0 for each zero, then the sum on the right-hand side of (4) must also be positive. This immediately gives a lower bound on the conductor D, which is the original application of the positivity technique. Fortunately, the sum on the right-hand side of (4) is explicitly computable in terms of the η j ’s and D; if it is negative, then the L-function L(s, π) cannot exist, and hence neither can the original cusp form π. Upon assuming GRH, let · · · ≤ γ−2 ≤ γ−1 ≤ 0 ≤ γ1 ≤ γ2 ≤ · · · be the imaginary parts of the zeros of L(s, π). Let g and h = gˆ be chosen so that h ≥ 0 on R, and let c > 0 be a cutoff parameter. Then the function h m (r ) = h(r )(c2 − r 2 ) is positive exactly when |r | < c and it is the Fourier transform of gm = c2 g + g 00 . The support of gm is of course also contained in [− log 2, log 2], provided g is suitably P regular. If the sum 2 Re mj=1 lm (η j ) + gm (0) log D in (4) is positive, then γ1 < c or γ−1 > −c; that is, L(s, π) has a small zero. We summarize as follows.

THE HIGHEST LOWEST ZERO 0.4 0.2 0 −0.2 −0.4

0

89

5

10

15

20

Figure 1. A contour plot of the function Re h 1m (x + i y). The positive set is white and the negative one black.

Criteria 2.1 Our strategy is then, for given archimedean parameters η j and conductor D, to find a function g of support contained in [− log 2, log 2] and for which either 2 Re

m X j=1

l(η j ) + g(0) log D < 0

(which shows the L-function does not exist) or 2 Re

m X j=1

lm (η j ) + gm (0) log D > 0

(which shows that it must have a low zero or violate GRH in an effective range, as discussed below). 2.2. What low zeros mean without GRH Even if we do not assume GRH, we may still conclude from 2 Re

m X j=1

that the sum

lm (η j ) + gm (0) log D > 0 X

h m (γ ) > 0.

Thus, there are zeros ρ = (1/2) + iγ in the region where h m (γ ) > 0. We can explicitly compute the functions h m for our choices of g and examine where they are positive and negative within the critical strip. Since the density of zeros increases only logarithmically with their height (with an effective constant), and our functions h m (z) decay polynomially as z → ∞ in the critical strip, the zero must be contained in an effectively bounded region of the critical strip. As an example, Figure 1 is a contour plot of the function h 1m defined at the end of Section 3. The white regions are where Re h 1m > 0, the black where Re h 1m < 0. Figures 2 and 3 contain plots for the other functions we use.

90

STEPHEN D. MILLER

3. A library of functions The main functions we use in this paper are   π|x| πx π 1 − |x| cos + sin p p p  / cosh(x/2), g1, p (x) =  π   |x| 2π x + 3 sin 2π|x| 4π 1 − |x| p + 2π 1 − p cos p p  / cosh(x/2), g2, p (x) =  6π and  g3, p (x) =



54π 1− |x| p

cos

πx p

+6π 1− |x| p

cos

3π x p

60π

+27 sin

π|x| p

cosh(x/2)

+11 sin

3π|x| p

 

.

We have normalized g j, p (0) = 1, and we often write g j (x) = g j,log(2) (x). Ignoring the cosh(x/2)’s temporarily, the functions g j, p are rescalings of the convolutions of (cos(π x/2)) j with itself. Without the cosh(x/2) term, they would thus have a positive Fourier transform on the real line, and the cosh(x/2) term spreads the positivity into the critical strip. Were we to assume GRH, we would not need it. LEMMA 3.1 If the Fourier transform of an even function g(x) is positive on the real line, then the Fourier transform of g(x)/ cosh(x/2) is positive in the strip −(1/2) < Im r < 1/2.

Proof The Fourier transform of sech(x) is Z ∞ πr 2 eir x d x = π sech . x −x 2 −∞ e + e This has positive real part for −(1/2) < Im r < 1/2, and the Fourier transform converts multiplication into convolution, so the smeared g\ · sech remains positive in this strip. We define modified functions gm =

c2 g(x) + g 00 (x) , c2 g(0) + g 00 (0)

which also have gm (0) = 1. (Of course, we multiplicatively normalize g(0) = gm (0) = 1 to compare the explicit formulas from various test functions.) These are


91

used for showing the presence of low zeros, and since we do not assume GRH for this, we actually use g = g1, p (x) cosh(x/2), g2, p (x) cosh(x/2), or g3, p (x) cosh(x/2). Thus, g1m, p,c (x) π 2 −1+ |x| p cos

πx p

p2

=

+

π sin

π|x| p 2 p

−

c2 − π 1− |x| p cos

πx p

π

π|x| p

−sin

π2 p2

c2 −

,

g2m, p,c (x)

c2 4π

=

1− |x| p

+2π 1− xp cos 6π

2π x p

+3 sin

c2

2π|x| p

−

4π 2 3 p2

cos

−

8π 3 1− xp cos p2

2π x p

6π

+

4π 2 sin

2π|x| p p2

,

and g3m, p,c (x) =

c2 54π 1 −

|x| p

cos

πx p

+ 6π 1 −

|x| p

3π x p

+ 27 sin

π |x| p

+ 11 sin

3π |x| p

2 60π c2 − 9π 5 p2 πx 3π x + 54π 3 − 1 + |x| + 81π 2 sin π |x| − 63π 2 sin 3πp|x| 54π 3 − 1 + |x| p cos p p cos p p . + 2 60π p 2 c2 − 9π 5 p2

Since we are interested in finding zeros in the range from 1/2 ± 14.13472i, we now take c = 14.13472 and write g1m (x) = g1m,log(2),14.13472 (x),

g2m (x) = g2m,log(2),14.13472 (x), and g3m (x) = g3m,log(2),14.13472 (x). The Fourier transforms of these functions are h 1m (r ) =

−8 p 3 π 2 (c2 − x 2 )cos

px 2 2

(−c2 p 2 + π 2 )(π − px)2 (π + px)2 2 128 pπ 4 (c2 − x 2 )sin px 2 h 2m (r ) = , 2 (3c2 p 2 − 4π 2 )(−4π 2 x + p 2 x 3 )

,

92

STEPHEN D. MILLER 0.4 0.2 0 −0.2 −0.4

0

10

20

30

40

Figure 2. A contour plot of the function Re h 2m (x + i y). The positive set is white and the negative one black. 0.4 0.2 0 -0.2 -0.4 0

10

20

30

40

50

Figure 3. A contour plot of the function Re h 3m (x + i y). The positive set is white and the negative set black.

and h 3m (r ) =

px 2 2 . 2 2 2 2 4 2 2 2 (5c p − 9π )(9π − 10 p π x + p 4 x 4 )

2304 p 3 π 6 (c2 − x 2 )cos

We show the contour plots of the functions h 2m and h 3m in Figures 2 and 3. The plot of h 1m is presented in Figure 1.

4. The highest lowest zero for π∞ real THEOREM 1.1 N Let π = p≤∞ π p be a cuspidal automorphic representation of GLn over Q with a real archimedean type π∞ and a unitary central character. Then L(s, π) has a low zero that either (i) is on the critical axis between 1/2 ± 14.13472i or (ii) violates the generalized Riemann hypothesis (GRH) in an effective range.

First we note that for a fixed degree m, L-functions with large η j ’s or large conductor D must have low zeros. This is because the Stirling formula implies that Z 00 1 1 l(η) = h(r ) R + η + ir dr 2π R 0R 2 has a positive real part for η large. Thus, the lowest zero is only an issue for “small” archimedean parameters η j and small conductor—partly because l(η) is bounded from below in Re η > −1/2 (which we may assume by (2)). We present two different proofs of Theorem 1.1.


93

0.8 0.6 0.4 0.2 5

10

15

−0.2

20

25

30

η

l1 (η)

−0.4

l3m (η)

−0.6

Figure 4. The functions l1 (η) and l3m (η)

Picture proof of Theorem 1.1 Figures 4, 5, and 6 indicate that l1 (η) < l3m (η) for η ≥ −(1/2), so the theorem follows from Criteria 2.1.

Less-pictorial proof of Theorem 1.1 This proof also relies on numerical computation but demonstrates how a proof can be made even if the function l is not strictly less than the modified l m . It uses l2m instead of l3m . We noted before in Criteria 2.1 that if 2

m X j=1

l2m (η j ) + log D ≥ 0,

then there is a indeed a low zero, while if 2

m X j=1

l1 (η j ) + log D ≤ 0,

the L-function actually cannot exist to begin with. Thus, we are reduced to dismissing the situation where m X l2m (η j ) < 0 j=1

94

STEPHEN D. MILLER

20

40

60

80

100

−0.002 −0.004 −0.006 −0.008 −0.01 −0.012 −0.014 Figure 5. The difference between l 1 (η) and l3m (η)

η 60

70

80

90

−0.00002 −0.00004 −0.00006 −0.00008 −0.0001 Figure 6. The difference between l 1 (η) and l3m (η), magnified

100

η


and

95

m X l1 (η j ) − l2m (η j ) > 0 j=1

hold simultaneously. Partition the η j ∈ (−1/2, ∞) into three sets: i 1 N = η j | l2m (η j ) ≤ 0, l1 (η j ) − l2m (η j ) ≤ 0 = − , 5.4471 · · · , 2 S = η j | l2m (η j ) > 0, l1 (η j ) − l2m (η j ) ≤ 0 = (5.4471 · · · , 8.6553 · · · ], and P = η j | l2m (η j ) > 0, l1 (η j ) − l2m (η j ) > 0 = (8.6553 · · ·, ∞).

Of course, if

m X

l2m (η j ) < 0,

j=1

then also

X

l2m (η j ) < 0,

η j ∈N ∪P

and if

m X l1 (η j ) − l2m (η j ) > 0, j=1

then

X l1 (η j ) − l2m (η j ) > 0

η j ∈N ∪P

as well. Thus, we need only consider the case when S is empty. From computer investigations (see Figures 7 and 8) on the functions l 1 (η) and l2m (η), we can determine the following very precise information: η j ∈ N =⇒ −.628291 ≤ l 1 (η j ) ≤ 0,

l1 (η j ) − l2m (η j ) ≤ −.001201

and η j ∈ P =⇒ l1 (η j ) ≥ .187484,

0 ≤ l1 (η j ) − l2m (η j ) ≤ .0005801.

Thus, 0>

m X j=1

l2m (η j ) =

X

η j ∈N

l2m (η j ) +

X

η j ∈P

≥ |N |(−.628291) + |P|(.187484),

l2m (η j )

96

STEPHEN D. MILLER

η 10

20

30

40

50

−0.002 −0.004 −0.006 −0.008 −0.01 −0.012 −0.014 Figure 7. The difference between l 1 (η) and l2m (η)

0.2

η 2

4

6

8

−0.2

l1 (η) −0.4

l2m (η) −0.6

Figure 8. The graphs of l1 (η) and l2m (η)

10


97

which implies

On the other hand,

|N | .187484 < = .298403. |P| .628291

m X X X 0< l1 (η j ) − l2m (η j ) = l1 (η j ) − l2m (η j ) + l1 (η j ) − l2m (η j ) j=1

η j ∈N

η j ∈P

≤ |N |(−.001201) + |P|(.005801) forces

a contradiction.

|N | .0005801 > = .483104, |P| .001201

5. Low zeros for modular form L-functions In this section we prove that L-functions of cusp forms on GL2 over Q have low zeros. THEOREM 1.2 Let π be a cuspidal automorphic representation of GL2 over Q with a central character of finite order. Then L(s, π) has a low zero (that is on the critical axis between 1/2 ± 14.13472i or else violates (GRH) in an effective range).

Before giving the proof, we give some background on the hardest case—Maass form L-functions. In particular, we precisely describe their completions, analytic continuations, and functional equations in some important cases. 5.1. Background on Maass forms on 00 ( p)\H It is known that if D = p is a prime and π is a cuspidal automorphic representation with a trivial central character and not corresponding to a holomorphic or antiholomorphic modular form, then π instead corresponds to a Maass form φ on 00 ( p)\H. The Laplace operator 1 and the Hecke operators Tn , n ≥ 0, as well as the involutions (T−1 f )(x + i y) = f (−x + i y), −1 (W p f )(x + i y) = f , p(x + i y) all commute. Thus, after diagonalizing, we may take a basis of Maass cusp forms on 00 ( p)\H which are joint eigenfunctions of 1, Tn , T−1 , and W p . Writing 1φ = λφ,

λ=

1 − ν2, 4

98

STEPHEN D. MILLER

φ has the Fourier expansion φ(x + i y) =

X n∈Z

where K ν = K −ν =

√ cn y K ν 2π|n|y e2πinx ,

1 2

Z

∞

e−y(t+t

−1 )/2

tν

0

dt t

is the K -Bessel function of order ν. The cuspidality condition forces c0 = 0; the involution T−1 interchanges cn and c−n . There are four symmetry classes of Maass forms under the action of the involutions T−1 and W p . The standard argument of Hecke and Maass to prove that the L-functions of cusp forms are entire also describes the functional equations of Lfunctions of Maass forms having various symmetries. PROPOSITION 5.1 Suppose φ is a Maass form on 00 ( p)\H with

T−1 φ = (−1)τ φ and W p φ = (−1)ω φ,

τ, ω = 0 or 1.

Multiplicatively normalize the coefficients of φ so that a 1 = 1 and P∞ √ an y K ν (2πny) cos(2πnx), τ = 0, φ(x + i y) = Pn=1 √ ∞ τ = 1. n=1 an y K ν (2πny) sin(2πnx), Then

3(s, φ) = 0R (s + τ + ν)0R (s + τ − ν)

∞ X an n=1

ns

satisfies the functional equation 3(s, φ) = (−1)τ +ω p 1/2−s 3(1 − s, φ).

(5)


99

Proof First, consider the case τ = 0. Then Z ∞ Z ∞ ∞ X dy dy = φ(i y)y s−1/2 an K ν (2πny)y s y y 0 0 n=1 ∞ hZ ∞ X dy i = an (2πn)−s K ν (y)y s y 0 = =

n=1 ∞ X n=1

h s + ν s − ν i an (2πn)−s 2s−2 0 0 2 2

1 3(s, φ). 4

The transformation property φ(i y) = (−1)ω φ

i py

gives Z ∞ dy dy i y s−1/2 = 4(−1)ω φ(i y)y s−1/2 φ y py y 0 0 Z ∞ dy i y φ y 1/2−s = 4(−1)ω p y 0 Z ∞ dy = 4 p 1/2−s (−1)ω φ(i y)y 1/2−s y 0

3(s, φ) = 4

Z

∞

= (−1)ω p 1/2−s 3(1 − s, φ). If τ = 1, then actually φ(i y) = 0 and we instead consider the derivative d φ 0 (x + i y) := φ(x + i y) dx ∞ X √ = (2πn)an y K ν (2πny) cos(2πnx). n=1

The action under W p now reads φ 0 (i y) = (−1)ω φ 0

i −1 . py 2 py 2

We also have Z ∞ Z ∞ ∞ X dy dy φ 0 (i y)y s+1/2 (2πn)an K ν (2πny)y s+1 = y y 0 0 =

n=1 ∞ X n=1

h s + 1 + ν s + 1 − ν i an (2πn)−s 2s−1 0 0 , 2 2

100

STEPHEN D. MILLER

and the functional equation for 3(s, φ) follows as before. 5.2. Low zeros for Maass form L-functions We first prove Theorem 1.2 for Maass forms through a series of propositions. 5.2 Every Maass form L-function whose conductor satisfies PROPOSITION

D≥3

if T−1 φ = φ

or D≥2

if T−1 φ = −φ

has a low zero. Proof In these two symmetry classes, the gamma factors of 3(s, φ) are either 0R (s + ν)0R (s − ν) or 0R (s + 1 + ν)0R (s + 1 − ν), depending on whether φ is even or odd under T−1 . In each case we may assume the parameter ν is not real, and hence purely imaginary, because Theorem 1.1 already covers the case of real archimedean type. In the first case, we have Re l3m (ir ) − l1 (ir ) > 0 if − 5.1 < r < 5.1, a range in which Re l1 (ir ) < −(log 3/4) ≈ −0.274653 (see Figure 9). In the second case, Re l3m (1 + ir ) − l1 (1 + ir ) > 0 if − 5.5 < r < 5.5, where Re l1 (1 + ir ) and Re l3m (1 + ir ) are both less than −(log 2/4) (see Figure 10). Criteria 2.1 thus shows that there are low zeros in either case. The following proposition handles the case of Maass forms at full level (i.e., unramified for all primes p < ∞). PROPOSITION 5.3 If φ is a Maass form on SL2 (Z)\H, then L(s, φ) has a low zero.


101

1

2

3

4

−0.1

Re(l3m (iη) − l1 )

−0.2

Re(l1 (iη))

5

η

−0.3 −0.4 −0.5

Figure 9. The functions l3m (η) − l1 (η) and l1 (η): A plot showing that Re l1 (ir ) + (log 3/4) < 0 and Re (l3m (ir ) − l1 (ir )) > 0 for −5.1 < r < 5.1

0.02 0.015

Re(l1 (1 + iη)) +

0.01

log(2) 4

Re(l3m (1 + iη)) +

log(2) 4

0.005 5.3

5.4

5.5

5.6

−0.005

Figure 10. The functions l1 (η) and l3m (η): A plot showing that Re l1 (1 + ir ) + (log 2/4) < 0 and Re (l3m (1 + ir ) − l1 (1 + ir )) > 0 for −5.5 < r < 5.5

η

102

STEPHEN D. MILLER

0.1 2

4

6

8

η

−0.1 −0.2 −0.3

Re(l3 (iη))

−0.4

Re(l1m (iη))

−0.5

−0.6 Figure 11. The functions l3 (η) and l1m (η): A plot showing that Re l3 (ν) is negative when Re l 1m (ν) is

Proof We again break the proof up into two cases according to whether φ is even or odd under T−1 . By Theorem 1.1 we need only consider the case Re ν = 0. If φ is even, then the gamma factors of L(s, φ) are 0R (s + ν)0R (s − ν). Figure 11 shows that Re l3 (ν) is negative when Re l 1m (ν) is, which by Criteria 2.1 proves the proposition in this case. If instead φ is odd, the gamma factors are 0R (s + 1 + ν)0R (s + 1 − ν), and, similarly, Re l3 (1 + ν) is negative when Re l 1m (1 + ν) is (see Figure 12). The proposition follows by invoking Criteria 2.1. To handle the remaining case, that of even Maass forms on 00 (2)\H, we use a result about the smallest even eigenvalue of the laplacian there. Perhaps Proposition 5.5 can be proven without such explicit information. PROPOSITION 5.4 If φ is a Maass form on 00 (2)\H which is even under both T−1 and W2 , then its Laplace eigenvalue exceeds (1/4) + 6.142 .

D. Hejhal [H] has numerically computed that the first such eigenvalue is ≈ (1/4) + 8.9222 . We present the following argument to demonstrate a technique.


103

0.1 2

4

6

8

η

−0.1 −0.2

Re(l3 (1 + iη))

−0.3

Re(l1m (1 + iη))

−0.4

Figure 12. The functions l3 (η) and l1m (η): A plot showing that Re l3 (1 + ν) is negative when Re l 1m (1 + ν) is

Proof First, Figure 13 shows that log 2 1. Thus, we must have a 2 / 2 > 1 for (6) to be valid. √ At the same time, such a value of a2 / 2 is too large to achieve equality in (7). This is because it makes the second term much larger than the first and third terms could possibly be with the constraint that a3 ≤ 2 · 35/28 : 35/28 Vir (1) + 2 ·√ Vir (3) < Vir (2) , 3

6.07 ≤ r ≤ 6.14.

So (6) and (7) cannot hold simultaneously. This contradiction shows every Maass form on 00 (2)\H which is even under both T−1 and W2 has Laplace eigenvalue greater than (1/4) + 6.142 .


105

0.01 0.005 6.05

6.1

−0.005

6.15

6.2

η Re(l1 (iη)) +

−0.01

log(2) 4

Re(l1m (iη)) +

log(2) 4

Figure 13. The functions l1 (η) and l1m (η): A plot showing that Re l1m (ir ) > −(log 2/4) for r > 6.135, while Re l1 (ir ) < −(log 2/4) for r < 6.07 PROPOSITION 5.5 Maass form L-functions with conductor D = 2 (which correspond to Maass forms on 00 (2)\H) have low zeros.

Proof By Proposition 5.2 we need only consider the even Maass forms, where the gamma factors are 0R (s + ν)0R (s − ν). In fact, by Proposition 5.1 we can assume that φ is even under both W2 and T−1 ; otherwise, (5) dictates that 3

1

1 , φ = −3 , φ = 0. 2 2

The function Re l1m (ir ) > −(log 2/4) for r > 6.135 (see Figure 13), and Proposition 5.4 shows that all even eigenvalues are in that range. Proof of Theorem 1.2 Every cuspidal automorphic representation on GL2 over Q comes from either a Maass form or a holomorphic modular form. Both holomorphic modular forms and nontempered Maass forms (i.e., λ < 1/4) have real archimedean type and are thus covered under Theorem 1.1. The rest of the Maass forms (the tempered ones) are covered by Propositions 5.2, 5.3, and 5.5.

106

STEPHEN D. MILLER

6. Cuspidal eigenvalue bounds Now we move our focus completely toward automorphic representations rather than on their L-functions. In this section and the next we examine the discrete spectrum of the laplacian 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)).∗ We normalize our laplacian so that its continuous spectrum on L 2 (SLn (R)/ SOn (R)) spans the interval from n3 − n λ1 SLn (R)/ SOn (R) = 24 to ∞. Because the ring of invariant differential operators R on SLn (R)/ SOn (R) is commutative, we may take a basis of Laplace eigenfunctions that are also common eigenfunctions of the operators in R . Thus, to each discrete eigenfunction φ ∈ L 2 (SLn (Z)\ SLn (R)/ SOn (R)) we can attach Langlands parameters µ1 , . . . , µn . These describe φ’s eigenvalues under the different operators in R ; in particular, the Laplace eigenvalue satisfies 1φ = λφ,

λ=

µ2 + · · · + µ2n n3 − n − 1 . 24 2

By the Jacquet-Shalika “trivial” bound (see [JS]), | Re µ j | < Thus,

1 , 2

j = 1, . . . , n.

n

λ>

1X n 3 − 4n (Im µ j )2 + . 2 24

(10)

j=1

We use (10) to bound λ from below. Positivity functions Recall the function πx 1 π|x| |x| cos + sin g1, p = 1 − / cosh(x/2), p p π p

0 < p ≤ log 2.

Define s(r ) = max Re l1,1/2 (ir + σ ), where the maximum is taken over σ ∈ [−(1/2), 3/2]. ∗ Of course, our methods carry over to some congruence covers, but we restrict our attention to the fulllevel congruence group SLn (Z) here.


107

1.25 1 0.75 0.5 0.25 r −100

−50

50

100

−0.25 Figure 14. The graph of s(r )

Criteria 6.1 Pn If the sum j=1 s(r j ) is negative, then there is no cuspidal eigenfunction in L 2 (SLn (Z)\ SLn (R)/ SOn (R)) whose Langlands parameters µ1 , . . . , µn have Im µ j = r j . If φ is a cusp form, then the archimedean Ramanujan-Selberg conjectures assert that π∞ is tempered; that is, Re µ j = 0. A consequence is that λcusp ≥ (n 3 − n)/24. This was proven unconditionally in [M] using a similar positivity argument. Here we can derive some stronger results and different applications. 6.2 (A trivial bound) If φ is a cusp form in L 2 (SLn (Z)\ SLn (R)/ SOn (R)), then with the above notation, PROPOSITION

n X j=1

r 2j > 51.84 1 +

1 . n−1

Proof P The plot in Figure 14 shows s(r ) < 0 for |r | < 7.2. Thus, nj=1 s(r j ) ≥ 0 only if at least one |r j | ≥ 7.2. Since the r j are constrained to have r 1 + · · · + rn = 0, this means that n 7.2 2 X r 2j ≥ 7.22 + (n − 1) . n−1 j=1

108

STEPHEN D. MILLER

Theorem 1.4 follows immediately from Proposition 6.2 and (10). 6.1. Extreme values P 2 P Given d and the constraints r j = d, r j = 0, if the largest value obtained by P s(r j ) is negative, then Criteria 6.1 implies λ > (d/2) + (n 3 − 4n)/24. 6.3 If (r1 , . . . , rn ) is an extremal point of

PRINCIPLE ∗

n X

s(r j )

j=1

subject to the constraints n X j=1

r j = 0,

n X j=1

r 2j = d,

then the r j assume at most three distinct values. Proof By Lagrange multipliers, there are real constants c1 , c2 ∈ R such that s 0 (r1 ), . . . , s 0 (rn ) = c1 (r1 , . . . , rn ) + c2 (1, . . . , 1);

that is, the points (r j , s 0 (r j )) all lie on the intersection of some line and the graph of y = s 0 (x). But no line crosses this graph in more than three places. Even though Figure 15 shows only the range |x| ≤ 100, it is legal to use this principle in this paper. Another crossing would give a value of r j so large that it would not enter into our subsequent bounds. THEOREM 6.4 We have the following bounds on the Laplace eigenvalue of a cuspidal eigenfunction of 1 in L 2 (SLn (Z)\ SLn (R)/ SOn (R)):

n λ≥ ∗ Some

3 87.625

4 108

5 140.875

6 167

7 201.125

8 232

may not consider the justification to be a proof, but, as we indicate, it can be verified in the applications we use it for.


109

0.4 0.2

−100

−50

50

100

r

−0.02 −0.04

Figure 15. The graph of s 0 (r )

Proof By the last proposition, we need only consider the case where there are A r 1 ’s, B r2 ’s, and C r3 ’s, with r1 , r2 , r3 ∈ R, Ar1 + Br2 + Cr3 = 0,

Ar12 + Br22 + Cr32 = d,

and then try to find a large value of d such that As(r1 ) + Bs(r2 ) + Cs(r3 ) is always negative. Since A, B, and C are all positive integers that sum to n, this is a finite calculation. We take A, B, C > 0 by allowing some of the values of r 1 , r2 , and r3 to coincide. Then in terms of the parameter r 3 , either q ACr3 + AB −C 2r32 + A(D − C) + B(Dr 32 − Cr32 ) r1 = − A (A + B) and r2 =

− (BCr3 ) +

q

or instead r1 =

−ACr3 +

q

AB −C 2r32 + A(D − Cr32 ) + B(D − Cr32 ) B (A + B)

AB −C 2r32 + A(D − C) + B(Dr 32 − Cr32 ) A (A + B)

110

STEPHEN D. MILLER

and r2 = −

(BCr3 ) +

q

AB −C 2r32 + A(D − Cr32 ) + B(D − Cr32 )

. B (A + B) Actually, the second set of solutions and the first are interchanged upon r 3 ↔ −r3 , so they take the same values. For a given n, we need only enumerate the integer triples of A, B, C with A ≥ B ≥ C > 0 and plot q   ACr3 + AB −C 2r32 + A(D − C) + B(Dr 32 − Cr32 )  As − A (A + B) q   − (BCr3 ) + AB −C 2r32 + A(D − Cr32 ) + B(D − Cr32 )  + Cs(r 3) + Bs  B (A + B) over the range √ √ √ A+B D A+B D −√ √ ≤ r3 ≤ √ √ . C A+ B +C C A+ B +C √

One finds that the following values of d work: n d

3 174

4 212

5 273

6 318

7 376

8 424

Remark 6.5 Theorem 1.3 shows that there exists a positive constant c > 0 such that λcusp −

n3 − n > cn, 24

n = 1, 2, . . . .

The argument of Theorem 6.4 gives a much better constant.

cusp

6.2. Some open problems about λ1

(SLn (Z)\ SLn (R)/ SOn (R))

CONJECTURE 6.6 Fix a positive integer k = 1, 2, . . . , and denote the kth cuspidal eigenvalue of 1 cusp on L 2 (SLn (Z)\ SLn (R)/ SOn (R)) as λk (SLn (Z)\ SLn (R)/ SOn (R)). Then the sequence

n λcusp (SLn (Z)\ SLn (R)/ SOn (R)) − k

n

n 3 −n 24

| n = 1, 2, . . .

o

(11)


111

has a limiting distribution. Question 6.7 Is the sequence in (11) also bounded from above as well as from below? 7. Bounds on noncuspidal eigenvalues A. Lubotzky asked if the bound λ≥

n3 − n 24

might additionally hold for the entire nonzero discrete spectrum of 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)), that is, not just for cusp forms alone. Although from the point of view of automorphic forms the cusp forms are most essential, the entire discrete spectrum enters into considerations in differential geometry. In fact, there are nonconstant, noncuspidal, square-integrable residues of Eisenstein series on L 2 (SLn (Z)\ SLn (R)/ SOn (R)) which are discrete Laplace eigenfunctions, and they are never tempered (that is, they violate Re µ j = 0). The first example of one on SLn (Z)\ SLn (R)/ SOn (R) violating λ ≥ (n 3 − n)/24 occurs for n = 68. THEOREM 1.5 There exists a discrete Laplace eigenfunction

φ ∈ L 2 SL68 (Z)\ SL68 (R)/ SO68 (R) such that 1φ = λφ φ,

λφ ≈ 12916.6
1 since cusp forms obey the λ ≥ (n 3 − n)/24 bound. Thus, we can restrict to the cases r ≥ 2, a = 1, . . . , 34. Using our pre-existing bounds P for µ2j , we conclude that ra ≥ 68 (see Table 1 for details). 8. Cuspidal cohomology The positivity inequality can be applied to products of L-functions that have poles, for example, Rankin-Selberg L-functions L(s, π ⊗ π) ˜ of cuspidal automorphic forms m π on GLn . If {µ jk } j=1,k=1 are the archimedean 0R parameters, the inequality reads Z

R

m X m X g(x) e x/2 + e−x/2 d x + 2 Re l(µ jk ) + g(0) log D ≥ 0.

(13)

j=1 k=1

The new term in (13) as compared to (4) comes from the poles of L(s, π ⊗ π). ˜ Also, here we have simply dropped the coefficients entirely because (L 0 /L)(s, φ) has a Dirichlet series with nonpositive coefficients (see [RS] for a verification of this) and so there is no restriction on the support of g. N If π = p≤∞ π p comes from a constant-coefficients cohomological cusp form on GLn (AQ ), then π∞ is of either the form n π∞ = IndGL P(2,2,...,2) (D2 , D4 , . . . , Dn ),

n even,

or n π∞ = IndGL P(1,2,2,...,2) sgn(·) , D3 , D5 , . . . , Dn ,

n odd.

(sgn is the sign character, = 0 or 1, and Dk denotes the kth discrete series on GL2 , corresponding to weight k holomorphic forms). Thus, if n is written as 2m + t, t = 0 or 1, the archimedean µ jk can be computed via the recipe summarized in [RS] and are the following multisets: {µ j,k } = {t + j + k, t − 1 + j + k, |k − j|, 1 + |k − j| | 1 ≤ j, k ≤ m} ∪

{0, j, j, j + 1, j + 1 | 1 ≤ j ≤ m} . {z } | omit if t = 0

(14)

114

STEPHEN D. MILLER

Table 1. This table completes the proof of Theorem 1.5. Suppose a residue of a cusp form on GLa occurred on some GLn , n = ra < 68 with Laplace eigenvalue less than or equal to (n 3 − n)/24. The second column gives the upper bound P r ≤ [68/a]. The third column gives a lower bound for − aj=1 µ2j (from Proposition 6.2), but the fourth gives an upper bound on P − aj=1 µ2j which would be satisfied by such a residue with low eigenvalue (as derived in (12) in the proof of Theorem 1.5). The inconsistency of these two inequalities is a contradiction showing that the discrete Laplace spectrum on SLn (Z)\ SLn (R)/ SOn (R) is contained in {0} ∪ [(n 3 − n)/24, ∞) for n < 68. a

r ≤ [ 68 a ]

Lower bound for −

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

22 17 13 11 9 8 7 6 6 5 5 4 4 4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2

171.25 211 271.75 316.5 374.25 422 56.07 55.10 54.27 53.55 52.91 52.32 51.79 51.29 50.83 50.38 49.96 49.56 49.18 48.80 48.44 48.09 47.75 47.41 47.08 46.76 46.44 46.12 45.81 45.51 45.20 44.91

Pa

j=1

µ2j

Upper bound for − 120.75 96.00 70 60 46.66 42.00 36 29.16 32.08 24 26 17.5 18.75 20 21.25 12 12.66 13.33 14 14.66 5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75 8 8.25 8.5

Pa

j=1

µ2j


115

Table 2. The (numerical) proof of the cohomology theorem. The R left-hand side of (13), R g(x) e x/2 + e−x/2 d x Pm Pm +2 Re j=1 k=1 l(µ jk ) + g(0) log D, must be positive if the cusp form exists, and this table shows it is negative for n < 27. n 2 3 4 5 6 7

t 3 6 6 6 6 6

LHS(13) −2.821 −8.113 −17.02 −28.30 −38.51 −50.30

n 15 16 17 18 19 20

t 6 6 6 6 6 6

LHS(13) −111.4 −112.1 −112.4 −109.2 −105.4 −97.87

n 9 10 11 12 13 14

t 6 6 6 6 6 6

LHS(13) −71.43 −80.27 −89.68 −96.45 −103.4 −107.5

n 22 23 24 25 26

t 6 6 6 6 6

LHS(13) −77.30 −64.06 −46.70 −28.18 −5.388

THEOREM 1.6 For 1 < n < 27, the constant-coefficients cuspidal cohomology of SL n (Z) vanishes: · SLn (Z); C = 0. Hcusp

Proof Let g p (x) =

1−

πx 1 π|x| |x| cos + sin / cosh(x/2). p p π p

Then h 1, p (r ) = gˆ 1, p (r ) is positive in the critical strip | Im r | < 1/2. For our cohomological forms D = 1 at full level, and with the µ jk ’s as above, we arrive at a contradiction to the positivity inequality (see Table 2).

Remarks We proved this for n < 23 in [M] with the Rankin-Selberg L-functions but without the Weil formula (instead using the Mittag-Leffler expansion). Fermigier [F] proved a weaker result using the Weil formula but with the standard L-function. Theorem 8.1 surpasses both. Nothing is known about these cuspidal Betti numbers for n ≥ 27, let alone if they are ever nonzero. Acknowledgments. We wish to thank Don Blasius, William Duke, Benedict Gross, Alexander Lubotzky, Andrew Odlyzko, Ilya Piatetski-Shapiro, Vladimir Rokhlin, Peter Sarnak, Jean-Pierre Serre, Gunther Steil, Andrew Wiles, and Gregg Zuckerman for their discussions. We wish to thank Princeton University and the University of California at San Diego. All numerical computations were made with Mathematica v.3

116

STEPHEN D. MILLER

on an Intel Pentium II 300 MHz system running Windows NT 4.0 and Slackware Linux 2.0.30. References [BDHI] D. BUMP, W. DUKE, J. HOFFSTEIN, and H. IWANIEC, An estimate for the Hecke eigenvalues of Maass forms, Internat. Math. Res. Notices 1992, 75–81. MR 93d:11047 104 [F] S. FERMIGIER, Annulation de la cohomologie cuspidale de sous-groupes de congruence de GLn (Z), Math. Ann. 306 (1996), 247–256. MR 97h:11056 87, 115 [H] D. A. HEJHAL, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. Amer. Math. Soc. 97 (1992), no. 469. MR 93f:11043 102, 112 [J] H. JACQUET, “On the residual spectrum of GL(n)” in Lie Group Representations, II, Lecture Notes in Math. 1041, Springer, Berlin, 1984, 185–208. MR 85k:22045 111 [JS] H. JACQUET and J. A. SHALIKA, “Rankin-Selberg convolutions: Archimedean theory” in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday (Ramat Aviv, Israel, 1989), I, Israel Math. Conf. Proc. 2, Weizmann, Jerusalem, 1990, 125–207. MR 93d:22022 87, 106 [LRS] W. LUO, Z. RUDNICK, and P. SARNAK, On Selberg’s eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), 387–401. MR 96h:11045 [M] S. D. MILLER, Spectral and cohomological applications of the Rankin-Selberg method, Internat. Math. Res. Notices 1996, 15–26. MR 97a:11079 85, 86, 87, 107, 115 ´ [MW] C. MOEGLIN and J.-L. WALDSPURGER, Le spectre résiduel de GL(n), Ann. Sci. Ecole Norm. Sup. (4) 22 (1989), 605–674. MR 91b:22028 111 [O] A. M. ODLYZKO, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2 (1990), 119–141. MR 91i:11154 84, 85 ¨ [R] W. ROELCKE, Uber die Wellengleichung bei Grenzkreisgruppen erster Art, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1953/1955, 159–267. MR 18:476d [RS] Z. RUDNICK and P. SARNAK, Zeros of principal L-functions and random matrix theory, Duke Math J. 81 (1996), 269–322. MR 97f:11074 88, 113

Department of Mathematics, Yale University, Box 208283, New Haven, Connecticut 06520-8283, USA; current, Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA; [email protected]


ON LEVEL-ZERO REPRESENTATIONS OF QUANTIZED AFFINE ALGEBRAS MASAKI KASHIWARA

Abstract We study the properties of level-zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by T. Akasaka and the author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremal weight. The universal extremal weight module with level-zero fundamental weight as an extremal weight is irreducible, and it is isomorphic to the affinization of an irreducible finite-dimensional module. Contents 1. Introduction . . . . . . . . . . . . . . . . . 2. Review on crystal bases . . . . . . . . . . . 2.1. Quantized universal enveloping algebras 2.2. Crystals . . . . . . . . . . . . . . . . 2.3. Schubert decomposition of crystal bases 2.4. Global bases . . . . . . . . . . . . . . 3. Extremal weight modules . . . . . . . . . . 3.1. Extremal vectors . . . . . . . . . . . . 3.2. Dominant weights . . . . . . . . . . . 4. Affine quantum algebras . . . . . . . . . . . 4.1. Affine root systems . . . . . . . . . . 4.2. Affinization . . . . . . . . . . . . . . 4.3. Simple crystals . . . . . . . . . . . . 5. Affine extremal weight modules . . . . . . . 5.1. Extremal vectors—affine case . . . . . 5.2. Fundamental representations . . . . . . 6. Existence of global bases . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 119 119 120 121 122 123 123 124 125 126 131 132 133 133 135 142

DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 30 October 2000. Revision received 11 December 2000. 2000 Mathematics Subject Classification. Primary 20G05. This work benefits from a “Chaire Internationale de Recherche Blaise Pascal de l’Etat et de la Région d’Ile-deFrance, gérée par la Fondation de l’Ecole Normale Supérieure.” 117

118

MASAKI KASHIWARA

6.1. Regularized modified operators . 6.2. Existence theorem . . . . . . . . 7. Universal R-matrix . . . . . . . . . . . 8. Good modules . . . . . . . . . . . . . 9. Main theorem . . . . . . . . . . . . . 10. Combinatorial R-matrices . . . . . . . 11. Energy function . . . . . . . . . . . . 12. Fock space . . . . . . . . . . . . . . . 12.1. Some properties of good modules 12.2. Wedge spaces . . . . . . . . . . 12.3. Global basis of the Fock space . . 13. Conjectural structure of V (λ) . . . . . . Appendices . . . . . . . . . . . . . . . . . A. Proof of the formula (6.2) . . . . . . B. Formulas for the crystal B(U˜ q (g)) . . References . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142 143 146 148 153 155 158 160 160 164 165 169 171 171 173 174

1. Introduction In this paper, we study the level-zero representations of quantum affine algebras. This paper is divided into three parts: extremal weight modules, the conjecture in [1] on the cyclicity of the tensor products of fundamental representations, and the global basis of the Fock space. In [11], as a generalization of highest weight vectors, the notion of extremal weight vectors is introduced, and it is shown that the universal module generated by an extremal weight vector has favorable properties: this has a crystal base, a global basis, and so on. The main purpose of the first part (§2–§5) is to study such modules in the affine case and to prove the following two properties. (a) If a module is generated by an extremal vector with weight λ, then all the weights of this module are contained in the convex hull of the Weyl group orbit of λ. (b) Any module generated by an extremal vector with a level-zero fundamental weight $i is irreducible, and it is isomorphic to the affinization of an irreducible finite-dimensional module W ($i ) (see Th. 5.17 and Prop. 5.16 for an exact statement). In the second part, we prove the following theorem∗ , which is conjectured in [1] (1) (1) and proved in the case of An and Cn . ∗ M.

Varagnolo and E. Vasserot [22] prove the same conjecture in the simply laced case by a different method.

QUANTIZED AFFINE ALGEBRAS

119

THEOREM

If aν /aν+1 has no pole at q = 0 (ν = 1, . . . , m −1), then W ($i1 )a1 ⊗· · ·⊗W ($im )am is generated by the tensor product of the extremal vectors. In the course of the proof, one uses the global basis on the tensor products of the affinizations of W ($iν ), especially the fact that the transformation matrix between the global basis of the tensor products and the tensor products of global bases is triangular. Among the consequences of this theorem (see §9), we mention here the following one. Under the conditions of the theorem above, there is a unique homomorphism up to a constant multiple W ($i1 )a1 ⊗ · · · ⊗ W ($im )am −→ W ($im )am ⊗ · · · ⊗ W ($i1 )a1 , and its image is an irreducible Uq0 (g)-module. This phenomenon is analogous to the morphism from the Verma module to the dual Verma module. Conversely, combining with a result of V. Drinfeld [4], any irreducible integrable Uq0 (g)-module is isomorphic to the image for some {(i 1 , a1 ), . . . , (i m , am )}. Moreover, {(i 1 , a1 ), . . . , (i m , am )} is unique up to a permutation. In the third part (§12), we prove the existence of the global basis on the Fock space. The plan of the paper is as follows. In §2–§4, we review some of the known results of crystal bases. Then, in §5, we give a proof of (a) and (b). In §6, we prove a sufficient condition for a module to admit a global basis: very roughly speaking, it is enough to have a global basis in the extremal weight spaces. In §7, we review the universal R-matrix and the universal conjugation operator. After introducing the notion of good modules (roughly speaking, a module with a global basis), we prove in §9 the above theorem in the framework of good modules. After preparations in §10–§11 on the combinatorial R-matrix and the energy function, we prove in §12 the properties of good modules which are postulated for the existence of the wedge products and the Fock space in [13]. Finally, we show that the Fock space admits a global basis. In the case of the vector representation of (1) g = An , the global basis of the corresponding Fock space has already been constructed by B. Leclerc and J.-Y. Thibon [14] (see also [15], [21]). In the last section, we present conjectures on the structure of V (λ). 2. Review on crystal bases In this section, we review very briefly the quantized universal enveloping algebra and crystal bases. We refer the reader to [12], [8], and [11].

120

MASAKI KASHIWARA

2.1. Quantized universal enveloping algebras We define the quantized universal enveloping algebra Uq (g). Assume that we are given the following data: P : a free Z-module (called a weight lattice), I : an index set (for simple roots), αi ∈ P for i ∈ I (called a simple root), h i ∈ P ∗ = HomZ (P, Z) (called a simple coroot), ( · , · ) : P × P → Q a bilinear symmetric form. We denote by h · , · i : P ∗ × P → Z the canonical pairing. The data above are assumed to satisfy the following axioms: (αi , αi ) > 0 for any i ∈ I , 2(αi , λ) for any i ∈ I and λ ∈ P, hh i , λi = (αi , αi ) (αi , α j ) ≤ 0 for any i, j ∈ I with i 6= j.

(2.1) (2.2) (2.3)

Let us choose a positive integer d such that (αi , αi )/2 ∈ Z d −1 for any i ∈ I . Now let q be an indeterminate, and set K = Q(qs ),

where qs = q 1/d .

(2.4)

Definition 2.1 The quantized universal enveloping algebra Uq (g) is the algebra over K generated by the symbols ei , f i (i ∈ I ), and q(h) (h ∈ d −1 P ∗ ) with the following defining relations: (1) q(h) = 1 for h = 0; (2) q(h 1 )q(h 2 ) = q(h 1 + h 2 ) for h 1 , h 2 ∈ d −1 P ∗ ; (3) q(h)ei q(h)−1 = q hh,αi i ei and q(h) f i q(h)−1 = q −hh,αi i f i for any i ∈ I and h ∈ d −1 P ∗ ; (4) [ei , f j ] = δi j (ti − ti−1 )/(qi − qi−1 ) for i, j ∈ I ; here qi = q (αi ,αi )/2 and ti = q ((αi , αi )/2)h i ; (5) (Serre relation) for i 6 = j, b X

(k)

(b−k)

(−1)k ei e j ei

=

k=0

b X

(k)

(−1)k f i

(b−k)

f j fi

= 0;

k=0

here b = 1 − hh i , α j i and (k)

ei

= eik /[k]i !,

(k)

fi

[k]i = (qik − qi−k )/(qi − qi−1 ),

= f ik /[k]i !, [k]i ! = [1]i · · · [k]i .


121

Sometimes we need an algebraically closed field containing K , for example, [ b= K C (q 1/n ) ,

(2.5)

n

b. and we consider Uq (g) as an algebra over K We denote by Uq (g)Q the subalgebra of Uq (g) over Q[qs±1 ] generated by the (n)

(n)

ei ’s, the f i ’s (i ∈ I ), and q h (h ∈ d −1 P ∗ ). Let us denote by W the Weyl group, the subgroup of GL(P) generated by the simple reflections si : si (λ) = λ − hh i , λiαi . P ± Let 1 ⊂ Q = i Zαi be the set of roots. Let 1P = 1 ∩ Q ± be the set of positive and negative roots, respectively. Here Q ± = ± i Z≥0 αi . Let 1re be the set re of real roots: 1re ± = 1± ∩ 1 . 2.2. Crystals We do not review the notion of crystals, but we refer the reader to [12], [8], and [11]. We say that a crystal B over Uq (g) is a regular crystal if, for any J ⊂ I such that {αi ; i ∈ J } is of finite-dimensional type, B is, as a crystal over Uq (g J ), isomorphic to the crystal bases associated with an integrable Uq (g J )-module. Here Uq (g J ) is the subalgebra of Uq (g) generated by e j , f j ( j ∈ J ), and q h (h ∈ d −1 P ∗ ). By [11], the Weyl group W acts on any regular crystal. This action S is given by ( hh ,wt(b)i f˜i i b if hh i , wt(b)i ≥ 0, Ssi b = −hh i ,wt(b)i e˜i b if hh i , wt(b)i ≤ 0. Let us denote by Uq− (g) (resp., Uq+ (g)) the subalgebra of Uq (g) generated by the f i ’s (resp., by the ei ’s). Then Uq− (g) has a crystal base denoted by B(∞) (see [8]). The unique weight vector of B(∞) with weight zero is denoted by u ∞ . Similarly, Uq+ (g) has a crystal base denoted by B(−∞), and the unique weight vector of B(−∞) with weight zero is denoted by u −∞ . Let ψ be the ring automorphism of Uq (g) that sends qs , ei , f i , and q(h) to qs , f i , ei , and q(−h). It gives a bijection B(∞) ' B(−∞) by which u ∞ , e˜i , f˜i , εi , ϕi , wt correspond to u −∞ , f˜i , e˜i , ϕi , εi , − wt. Let us denote by U˜ q (g) the modified quantized universal enveloping algebra L ˜ ˜ λ∈P Uq (g)aλ (see [11]). Then Uq (g) has a crystal base B(Uq (g)). As a crystal, ˜ B(Uq (g)) is regular and isomorphic to G B(∞) ⊗ Tλ ⊗ B(−∞). λ∈P

Here Tλ is the crystal consisting of a single element tλ with εi (tλ ) = ϕi (tλ ) = −∞ and wt(tλ ) = λ.

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MASAKI KASHIWARA

Let ∗ be the anti-involution of Uq (g) that sends q(h) to q(−h), and qs , ei , f i to themselves. The involution ∗ of Uq (g) induces an involution ∗ on B(∞), B(−∞), B(U˜ q (g)). Then e˜i∗ = ∗ ◦ e˜i ◦ ∗, and so on, give another crystal structure on B(∞), B(−∞), B(U˜ q (g)). We call it the star crystal structure. In the case of B(U˜ q (g)), these two crystal structures are compatible, and B(U˜ q (g)) may be considered as a crystal over g ⊕ g. Hence, for example, Sw∗ , the Weyl group action on B(U˜ q (g)) with respect to the star crystal structure, is a crystal automorphism of B(U˜ q (g)) with respect to the original crystal structure. In particular, the two Weyl group actions Sw and Sw∗ 0 commute with each other. The formulas concerning B(U˜ q (g)) are given in Appendix B. Note that we always have εi (b) + ϕi∗ (b) = εi∗ (b) + ϕi (b) ≥ 0

for any b ∈ B(∞).

(2.6)

2.3. Schubert decomposition of crystal bases For w ∈ W with a reduced expression si1 · · · si` , we define the subset Bw (∞) of B(∞) by Bw (∞) = { f˜ia11 · · · f˜ia` ` u ∞ ; a1 , . . . , a` ∈ Z≥0 }. (2.7) Then Bw (∞) does not depend on the choice of a reduced expression. We refer the reader to [9] on the details of Bw (∞) and its relationship with the Demazure module. We have (see [9]) (i) Bw (∞)∗ = Bw−1 (∞); (ii) if w0 ≤ w, then Bw0 ( ∞) ⊂ Bw (∞); (iii) if si w < w, then f˜i Bw (∞) ⊂ Bw (∞); (iv) e˜i Bw (∞) ⊂ Bw (∞) t {0}; (v) if both b and f˜i b belong to Bw (∞), then all f˜ik b (k ≥ 0) belong to Bw (∞). Here ≤ is the Bruhat order. Set [ B w (∞) = Bw (∞) \ Bw0 (∞) . w0 <w

P. Littelmann [16] showed that B(∞) =

G

B w (∞).

w∈W

We have B w (∞)∗ = B w−1 (∞).

(2.8)

If si w < w, then e˜imax B w (∞) ⊂ B siw (∞),

f˜i B w (∞) ⊂ B w (∞).

(2.9)

In particular, εi (b) > 0 for any b ∈ B w (∞). Here we use the notation e˜imax b = ε (b) e˜i i b.


123

2.4. Global bases Let A ⊂ K be the subring of K consisting of rational functions in qs without pole at qs = 0. Let − be the automorphism of K sending qs to qs−1 . Set K Q := Q[qs , qs−1 ]. Let V be a vector space over K , let L 0 be an A-submodule of V , let L ∞ be an Asubmodule, and let VQ be a K Q -submodule. Set E := L 0 ∩ L ∞ ∩ VQ . Definition 2.2 ([8]) We say that (L 0 , L ∞ , VQ ) is balanced if each of L 0 , L ∞ , and VQ generates V as a K vector space, and if the following equivalent conditions are satisfied. (i) E → L 0 /qs L 0 is an isomorphism. (ii) E → L ∞ /qs−1 L ∞ is an isomorphism. (iii) (L 0 ∩ VQ ) ⊕ (qs−1 L ∞ ∩ VQ ) → VQ is an isomorphism. (iv) A ⊗Q E → L 0 , A ⊗Q E → L ∞ , K Q ⊗Q E → VQ , and K ⊗Q E → V are isomorphisms. Let − be the ring automorphism of Uq (g) sending qs , q h , ei , f i to qs−1 , q −h , ei , f i . h (n) (n) Let Uq (g)Q be the K Q -subalgebra of Uq (g) generated by ei , f i , and qn (h ∈ P ∗ ). Let M be a Uq (g)-module. Let − be an involution of M satisfying (au)− = a¯ u¯ for any a ∈ Uq (g) and u ∈ M. In this paper we call such an involution a bar involution. Let (L , B) be a crystal base of an integrable Uq (g)-module M. Let MQ be a Uq (g)Q -submodule of M such that (MQ )− = MQ

and

(u − u) ∈ (qs − 1)MQ

for every u ∈ MQ .

(2.10)

Definition 2.3 If (L , L, MQ ) is balanced, we say that M has a global basis. ∼ E := L ∩ L ∩ MQ be the inverse of E −→ ∼ L/qs L. In such a case, let G : L/qs L −→ Then {G(b); b ∈ B} forms a basis of M. We call this basis a (lower) global basis. The global basis enjoys the following properties (see [8], [10]). (i) G(b) = G(b) for any b ∈ B. (ii) For any n ∈ Z≥0 , {G(b); εi (b) ≥ n} is a basis of the K Q -submodule P (m) MQ . m≥n f i (iii) For any i ∈ I and b ∈ B, we have X i 0 f i G(b) = 1 + εi (b) i G( f˜i b) + Fb,b 0 G(b ). b0

Here the sum ranges over b0 ∈ B such that εi (b0 ) > 1 + εi (b). The coefficient 1−εi (b0 ) i Fb,b Q[qs ]. 0 belongs to qs qi

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A similar statement holds for ei G(b).

3. Extremal weight modules 3.1. Extremal vectors Let M be an integrable Uq (g)-module. A vector u ∈ M of weight λ ∈ P is called extremal (see [1], [11]) if we can find vectors {u w }w∈W satisfying the following properties: uw = u

for w = e;

if hh i , wλi ≥ 0, then ei u w = 0 and if hh i , wλi ≤ 0, then f i u w = 0 and

(3.1) (hh ,wλi) fi i u w = u si w ; (−hh ,wλi) ei i = u si w .

(3.2) (3.3)

Hence if such {u w } exists, then it is unique and u w has weight wλ. We denote u w by Sw u. Similarly, for a vector b of a regular crystal B with weight λ, we say that b is an extremal vector if it satisfies the following similar conditions: we can find vectors {bw }w∈W such that bw = b

for w = e;

(3.4)

hh ,wλi if hh i , wλi ≥ 0, then e˜i bw = 0 and f˜i i bw = bsi w ;

if hh i , wλi ≤ 0, then f˜i vw = 0 and

−hh ,wλi e˜i i bw

= bsi w .

(3.5) (3.6)

Then bw must be Sw b. For λ ∈ P, let us denote by V (λ) the Uq (g)-module generated by u λ with the defining relation that u λ is an extremal vector of weight λ. There are, in fact, infinitely many linear relations on u λ . We proved in [11]∗ that V (λ) has a global crystal base (L(λ), B(λ)). Moreover, the crystal B(λ) is isomorphic to the subcrystal of B(∞) ⊗ tλ ⊗ B(−∞) consisting of vectors b such that b∗ is an extremal vector of weight −λ. We denote by the same letter u λ the element of B(λ) corresponding to u λ ∈ V (λ). Then u λ ∈ B(λ) corresponds to u ∞ ⊗ tλ ⊗ u −∞ . Note that, for b1 ⊗ tλ ⊗ b2 ∈ B(∞) ⊗ tλ ⊗ B(−∞) belonging to B(λ), one has εi∗ (b1 ) ≤ max hh i , λi, 0 and ϕi∗ (b2 ) ≤ max − hh i , λi, 0 for any i ∈ I. (3.7) For any w ∈ W , u λ 7→ Sw−1 u wλ gives an isomorphism of Uq (g)-modules: ∼ V (wλ). V (λ) −→ [11], it is denoted by V max (λ) because I thought there would be a natural Uq (g)-module whose crystal base is the connected component of B(λ).

∗ In


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Similarly, letting Sw∗ be the Weyl group action on B(U˜ q (g)) with respect to the star ∼ crystal structure and regarding B(λ) as a subcrystal of B(U˜ q (g)), Sw∗ : B(U˜ q (g)) −→ B(U˜ q (g)) induces an isomorphism of crystals ∼ B(wλ). Sw∗ : B(λ) −→ For a dominant weight λ, V (λ) is an irreducible highest weight module of highest weight λ, and V (−λ) is an irreducible lowest weight module of lowest weight −λ.

3.2. Dominant weights Definition 3.1 For a weight λ ∈ P and w ∈ W , we say that λ is w-dominant (resp., w-regular) if −1 re hβ, λi ≥ 0 (resp., hβ, λi 6= 0) for any β ∈ 1re + ∩ w 1− . If λ is w-dominant and w-regular, we say that λ is regularly w-dominant. If w = si` · · · si1 is a reduced expression, then we have −1 re 1re + ∩ w 1− = {si 1 · · · si k−1 αi k ; 1 ≤ k ≤ `}.

Hence λ is w-dominant (resp., w-regular) if and only if hh ik , sik−1 · · · si1 λi ≥ 0

resp., hh k , sik−1 · · · si1 λi 6= 0 .

(3.8)

Conversely, one has the following lemma. 3.2 For i 1 , . . . , il ∈ I and a weight λ, assume that LEMMA

hh ik , sik−1 sik−1 · · · si1 λi > 0

for k = 1, . . . , l.

Then w = sil · · · si1 is a reduced expression. Proof By the induction on l, we may assume that sil−1 · · · si1 is a reduced expression. If l(w) < l, then there exists k with 1 ≤ k ≤ l − 1 such that sil−1 · · · sik+1 (h ik ) = −h il . Hence hh il , sil−1 · · · si1 λi = −hh ik , sik−1 sik−1 · · · si1 λi < 0, which is a contradiction. This lemma implies the following lemma.

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LEMMA 3.3 Let w1 , w2 ∈ W , and let λ be an integral weight. If λ is regularly w2 -dominant and w2 λ is regularly w1 -dominant, then `(w1 w2 ) = `(w1 ) + `(w2 ) and λ is regularly w1 w2 -dominant. Here ` : W → Z is the length function.

3.4 Let λ ∈ P, and let b1 ∈ B w1 (∞), b2 ∈ B w2 (−∞). If b := b1 ⊗ tλ ⊗ b2 belongs to B(λ), then (i) λ is regularly w1 -dominant and −λ is regularly w2 -dominant, (ii) `(w1 w2−1 ) = `(w1 ) + `(w2 ), (iii) and we have PROPOSITION

Sw∗ 2 (b1 ⊗ tλ ⊗ b2 ) ∈ Bw

−1 1 w2

(∞) ⊗ tw2 λ ⊗ u −∞ ,

Sw∗ 1 (b1 ⊗ tλ ⊗ b2 ) ∈ u ∞ ⊗ tw1 λ ⊗ Bw

−1 2 w1

(−∞).

More generally, if w1 = w0 w00 with `(w1 ) = `(w0 ) + `(w00 ), then Sw∗ 00 (b1 ⊗ tλ ⊗ b2 ) ∈ B w0 (∞) ⊗ tw00 λ ⊗ Bw2 w00 −1 (−∞). Proof Assume that w1 si < w1 . Then c := εi∗ (b1 ) > 0 by (2.9). Hence hh i , λi ≥ c > 0 by (3.7). We have e˜i∗ max b1 ∈ B¯ w1 si (∞) and b0 = Si∗ (b1 ⊗ tλ ⊗ b2 ) = (e˜i∗ max b1 ) ⊗ tsi λ ⊗ (e˜i∗ hh i ,λi−c b2 ).

(3.9)

Hence λ is regularly w1 -dominant by the induction on the length of w1 . The other statement in (i) is similarly proved, and (ii) follows from (i) and Lemma 3.3. In (3.9), e˜i∗ hh i ,λi−c b2 belongs to Bw2 si (−∞) since (ii) implies w2 si > w2 . Repeating this, we obtain (iii). 4. Affine quantum algebras In the sequel we assume that g is affine. 4.1. Affine root systems Although the materials in this subsection are more or less classical, we review the affine algebras in order to fix the notation. Let g be an affine Lie algebra, and let t be its Cartan subalgebra (assuming that they are defined over Q). Let I be the index set of simple roots, let αi ∈ t∗ be the simple roots, and let h i ∈ t be the simple coroots (i ∈ I ). We choose a Cartan subalgebra t such that {αi }i∈I and {h i }i∈I are linearly independent and dim t = rank g + 1. Let


127

us set the root lattice and coroot lattice by M Q= Zαi ⊂ t∗ and

Q∨ =

i

M

Zh i ⊂ t.

i

P P Set Q ± = ± i Z≥0 αi and Q ∨ ± = ± i Z≥0 h i . Let δ ∈ Q + be a unique element satisfying {λ ∈ Q; hh i , λi = 0 for every i} = Zδ. Similarly, we define c ∈ Q ∨ + by {h ∈ Q ∨ ; hh, αi i = 0 for every i} = Zc. We write X X δ= ai αi and c= ai∨ h i . (4.1) i

i

We take a W -invariant nondegenerate symmetric bilinear form (·, ·) on t∗ normalized by (δ, λ) = hc, λi for any λ ∈ t∗ . (4.2) Then this symmetric form has the signature (dim t − 1, 1). We sometimes identify t and t∗ by this symmetric form. By this identification, δ and c correspond to each other. We have (αi , αi ) ai∨ = ai . (4.3) 2 Note that (αi , αi )/2 takes the values 1, 2, 3, 1/2, 1/3. Hence we have for each i, (αi , αi ) ∈Z 2

or

2 ∈ Z. (αi , αi )

(4.4)

If g is untwisted, then 2/(αi , αi ) is an integer. Let us set t∗cl = t∗ /Q δ, and let cl : t∗ → t∗cl be the canonical projection. We have M t∗cl ' (Q h i )∗ . i∈I ∗0 ∗ ∗0 Set t∗0 = {λ ∈ t∗ ; hc, λi = 0} and t∗0 cl = cl(t ) ⊂ tcl . Then tcl has a positive-definite symmetric form induced by the one of t∗ .

4.1 For any a ∈ Q, LEMMA

cl : λ ∈ t∗ ; (λ, λ) = a and (λ, δ) 6 = 0 → t∗cl \ t∗0 cl is bijective. Proof Let λ ∈ t∗ such that (λ, δ) 6= 0. Setting µ = λ + xδ for x ∈ Q, we have (µ, µ) = (λ + xδ, λ + xδ) = (λ, λ) + 2x(λ, δ). Hence λ+ xδ has square length a if and only if x = (a −(λ, λ))/2(λ, δ).

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As a corollary we have the following proposition. PROPOSITION 4.2 The space t∗ endowed

with an invariant symmetric form as above, with simple roots and coroots, is unique up to a canonical isomorphism. Proof For example, take ρ ∈ t∗ such that hh i , ρi = 1 for any i and (ρ, ρ) = 0. Lemma 4.1 guarantees its existence and its uniqueness. The αi ’s and ρ form a basis of t∗ . In particular, for any Dynkin diagram isomorphism ι (i.e., a bijection ι : I → I such that hh ι(i) , αι( j) i = hh i , α j i), there exists a unique isomorphism of t∗ that sends αi to αι(i) and leaves the symmetric form invariant. Let 1 ⊂ t∗ be the root system of g, and let 1re be the set of real roots: 1re = 1 \ Z δ. For β ∈ t∗ with (β, β) 6 = 0, we set β ∨ = 2β/(β, β). Then 1∨ := {β ∨ ; β ∈ 1re } ∪ (Z c \ {0}) ⊂ t is the root system for the dual Lie algebra of g. We set 1± = 1 ∩ Q±. Let us denote by 1cl the image of 1e by cl. Then 1cl is a finite subset of t∗0 cl , and (1cl , t∗0 ) is a (not necessarily reduced) root system. We call an element of 1 a cl cl classical root. Let O(t∗ ) be the orthogonal group of t∗ with respect to the invariant symmetric form. Let O(t∗ )δ be the isotropy subgroup of δ; that is, O(t∗ )δ = {g ∈ O(t∗ ); gδ = δ}. Then there are canonical group homomorphisms cl : O(t∗ )δ → GL(t∗cl )

and

cl0 : O(t∗ )δ → O(t∗0 cl ).

The homomorphism cl : O(t∗ )δ → GL(t∗cl ) is injective. For β ∈ 1e , let sβ be the corresponding reflection λ 7→ λ − hβ ∨ , λiβ. Let W be the Weyl group, that is, the subgroup of GL(t∗ ) generated by the sβ ’s. Since W ⊂ O(t∗ )δ , there are group homomorphisms W → GL(t∗cl ) and W → O(t∗0 cl ). Let us denote by Wcl the image of W → O(t∗0 ). Then W is the Weyl group of cl cl ∗0 the root system (1cl , tcl ). For ξ ∈ t∗0 , we set T (λ) = λ + (δ, λ)ξ − (ξ, λ)δ −

(ξ, ξ ) (δ, λ)δ. 2

(4.5)

Then T belongs to O(t∗ )δ , and T depends only on cl(ξ ). For ξ0 ∈ t∗0 cl , let us define −1 ∗ t (ξ0 ) ∈ O(t )δ as the right-hand side of (4.5) with ξ ∈ cl (ξ0 ). Then ∗ ∗0 t : t∗0 (4.6) cl → Ker cl0 : O(t )δ → GL(tcl ) is a group isomorphism.


129

We have g ◦ t (ξ ) ◦ g −1 = t cl0 (g)(ξ ) for g ∈ O(t∗ )δ and ξ ∈ t∗0 cl .

(4.7)

For β ∈ t∗ such that (β, β) 6 = 0, let us denote by sβ the reflection sβ (λ) = λ − (β ∨ , λ)β . Then we have for β ∈ t∗0 such that (β, β) 6 = 0, sβ−aδ sβ = t (aβ ∨ ).

(4.8)

Wcl is generated by {si ; i 6 = i 0 }.

(4.9)

There exists i 0 such that

(2)

If g is not isomorphic to A2n , such an i 0 is unique up to a Dynkin diagram automor(2) phism and (αi0 , αi0 ) = 2, ai0 = ai∨0 = 1. In the case of A2n , there are two choices of i 0 , two extremal nodes, and (αi0 , αi0 ) = 1 or 4, and accordingly ai0 = 2 or 1, ai∨0 = 1 or 2. For α ∈ 1re or α ∈ 1cl , we set (α, α) cα = max 1, 2 and ci = cαi . Then we have, for any α ∈ 1re , {n ∈ Z; α + nδ ∈ 1} = Z cα .

(4.10)

We set ∨ Q cl = cl(Q), Q∨ cl = cl(Q ), P Here Q ∨ = α∈1re Zα ∨ . We have an exact sequence

e = Q cl ∩ Q ∨ Q cl .

cl0 t e− 1 −→ Q → W −−→ Wcl −→ 1.

(4.11)

(4.12)

e ∩ Q>0 cl(α) with the smallest length. For any α ∈ 1re , let α˜ be the element in Q We set ˜ = {α; 1 ˜ α ∈ 1re }. e is the root lattice of 1. ˜ is a reduced root system, and Q ˜ Then 1 Remark 4.3 Any affine Lie algebra is either untwisted or the dual of an untwisted affine algebra (2) or A2n .

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(i) (ii)

e = Q ∨ ⊂ Q cl , 1 ˜ = cl(1∨ re ), α˜ = α ∨ . If g is untwisted, then Q cl e = Q cl ⊂ Q ∨ , 1 ˜ = cl(1re ), If g is the dual of an untwisted algebra, then Q cl α˜ = α. (2) e = Q cl = Q ∨ , 1 ˜ = cl(1re ) = cl(1∨ re ). For any α ∈ 1re , If g = A2n , then Q cl one has ( cl(α) if (α, α) 6 = 4, α˜ = cl(α)/2 if (α, α) = 4.

(iii)

Note that (α − δ)/2 ∈ 1re if (α, α) = 4. (2) If g 6 = A2n , then α˜ = cα α ∨ . PROPOSITION

4.4

e For ξ ∈ Q, X X 1 X (β, ξ ) /cβ = (β ∨ , ξ )+ . l t (ξ ) = (β, ξ )+ /cβ = 2 β∈1cl

β∈1cl

˜ β∈1

Here a+ = max(a, 0). Proof For β ∈ 1cl , let us denote by β 0 the unique element of 1+ such that cl(β 0 ) = β and β 0 − nδ 6∈ 1+ for any n > 0. Note that (β, ξ ) ∈ cβ Z. We have t (ξ )−1 1− ∩ 1+ = γ ∈ 1+ ; γ − (γ , ξ )δ ∈ 1− , and l(t (ξ )) is the number of elements in this set. By setting γ = β 0 + ncβ δ, it is isomorphic to (β, n) ∈ 1cl × Z; n ≥ 0 and β 0 + ncβ − (β, ξ ) δ ∈ 1− = (β, n) ∈ 1cl × Z; 0 ≤ n < (β, ξ )/cβ . Since (β, ξ )/cβ is an integer, we have X l t (ξ ) = (β, ξ )/cβ + . β∈1cl

The other equalities easily follow. 4.5 e For ξ ∈ Q and w ∈ Wcl , COROLLARY

l t (wξ ) = l t (ξ ) .


131

We choose a weight lattice P ⊂ t∗ satisfying αi ∈ P and h i ∈ P ∗ for any i ∈ I ; for every i ∈ I, there exists 3i ∈ P such that hh j , 3i i = δ ji .

(4.13)

We set P 0 = λ ∈ P; hc, λi = 0 ,

Pcl = cl(P) ⊂ t∗cl ,

and

Pcl0 = cl(P 0 ). (4.14)

We have Pcl =

M (Zh i )∗ . i∈I

LEMMA 4.6 e the following two conditions are equivalent: For λ ∈ P 0 and µ ∈ Q, (i) λ and µ are in the same Weyl chamber (i.e., for any α ∈ 1re , (cl(α), µ) > 0 implies (α, λ) ≥ 0); (ii) λ is t (µ)-dominant.

Proof For α ∈ 1re , let us take α 0 ∈ (α+Zδ)∩1+ such that cl(α 0 ) = cl(α) and α 0 −nδ 6∈ 1+ for any n ∈ Z>0 . Then for α = α 0 + nδ ∈ 1+ , α ∈ 1+ ∩ t (µ)−1 1− ⇔ t (µ)α = α − (α, µ)δ = α 0 + n − (α, µ) δ ∈ 1− ⇔ 0 ≤ n < (α, µ). (i) ⇒ (ii) Now assume that α = α 0 + nδ ∈ 1+ ∩ t (µ)−1 1− . Then 0 ≤ n < (α, µ), and (i) implies (α, λ) ≥ 0 (ii) ⇒ (i) Assume that (α, µ) > 0. Then taking n = 0, α 0 ∈ 1+ ∩ t (µ)−1 1− , and hence (α, λ) = (α 0 , λ) ≥ 0. The following lemma is similarly proved. LEMMA 4.7 e the following two conditions are equivalent: For λ ∈ P 0 and µ ∈ Q, (i) for any α ∈ 1cl , (α, µ) > 0 implies (α, λ) > 0; (ii) λ is regularly t (µ)-dominant.

Let us choose i 0 ∈ I as in (4.9), and let W0 be the subgroup of W generated by e {si ; i ∈ I \ {i 0 }}. Then W is a semidirect product of W0 and Q.

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LEMMA 4.8 e and w ∈ W0 . If ξ is regularly w-dominant, then Let ξ ∈ Q l t (ξ ) = l t (ξ )w−1 + l(w).

Proof We prove the assertion by the induction on l(w). Write w = si w0 with w > w 0 and i 6 = i 0 . Then l(t (ξ )) = l(t (ξ )w0−1 ) + l(w0 ). Hence it is enough to show t (ξ )w0−1 > t (ξ )w0−1 si or, equivalently, t (ξ )w0−1 αi ∈ 1− . We have t (ξ )w0−1 αi = w0

−1

αi − (w0 ξ, αi )δ.

Since (w0 ξ, αi ) > 0, the coefficient of αi0 in t (ξ )w 0−1 αi is negative, and hence t (ξ )w0−1 αi is a negative root. 4.2. Affinization Let P and Pcl be as in (4.13). We denote by Uq (g) the quantized universal enveloping algebra with P as a weight lattice. We denote by Uq0 (g) the quantized universal enveloping algebra with Pcl as a weight lattice. Hence Uq0 (g) is a subalgebra of Uq (g) generated by the ei ’s, the f i ’s, and q h (h ∈ d −1 (Pcl )∗ ). When we talk about an integrable Uq (g)-module (resp., Uq0 (g)-module), the weight of its element belongs to P (resp., Pcl ). L Let M be a Uq0 (g)-module with the weight decomposition M = λ∈Pcl Mλ . We L define a Uq (g)-module Maff with a weight decomposition Maff = λ∈P (Maff )λ by (Maff )λ = Mcl(λ) . The actions of ei and f i are defined in an obvious way, so that the canonical homomorphism cl : Maff → M is Uq0 (g)-linear. We define the Uq0 (g)-linear automorphism ∼ Mcl(λ) = Mcl(λ+δ) −→ ∼ (Maff )λ+δ . z of Maff with weight δ by (Maff )λ −→ Let us choose 0 ∈ I satisfying Wcl is generated by {si ; i 6 = 0},

and

a0 = 1.

(4.15)

P (2) Recall that δ = i ai αi . When g = A2n , α0 is the longest simple root. Choose a section s : Pcl → P of cl : P → Pcl such that s(cl(αi )) = αi for any i ∈ I \ {0}. Then M is embedded into Maff by s as a vector space. We have an isomorphism of Uq0 (g)-modules Maff ' K [z, z −1 ] ⊗ M.

(4.16)

Here ei ∈ Uq0 (g) and f i ∈ Uq0 (g) act on the right-hand side by z δi0 ⊗ ei and z −δi0 ⊗ f i .


133

Similarly, for a crystal with weights in Pcl , we can define its affinization Baff by G Baff = Bcl(λ) . (4.17) λ∈P

Uq0 (g)-module

If an integrable M has a crystal base (L , B), then its affinization Maff has a crystal base (L aff , Baff ). For a ∈ K , we define the Uq0 (g)-module Ma by Ma = Maff /(z − a)Maff .

(4.18)

4.3. Simple crystals In [1], we defined the notion of simple crystals and studied their properties. Definition 4.9 We say that a finite regular crystal B (with weights in Pcl0 ) is a simple crystal if B satisfies the following: (1) there exists λ ∈ Pcl0 such that the weight of any extremal vector of B is contained in Wcl λ; (2) ](Bλ ) = 1. Simple crystals have the following properties (see [1]). LEMMA 4.10 A simple crystal B is connected. LEMMA 4.11 The tensor product of simple crystals is also simple. PROPOSITION 4.12 A finite-dimensional integrable Uq0 (g)-module with a simple crystal base is irreducible.

5. Affine extremal weight modules 5.1. Extremal vectors—affine case We now prove one of the main results of this paper. In the sequel we employ the notation ε (b)

e˜imax b = e˜i i

b,

ϕ (b) f˜imax b = f˜i i b,

and similarly for e˜i∗ max and f˜i∗ max .

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THEOREM 5.1 Let λ ∈ P 0 be a level zero weight. Then the weight of any extremal vector of B(λ) is contained in cl−1 cl(W λ).

Proof We regard B(λ) as a subcrystal of B(∞) ⊗ tλ ⊗ B(−∞) ⊂ B(U˜ q (g)). We show that cl(wt(b)) and − cl(wt(b∗ )) are in the same Wcl -orbit whenever b and b∗ are extremal vectors. For any b1 ⊗ tλ ⊗ b2 , we have f˜imax (b1 ⊗ tλ ⊗ b2 ) = b10 ⊗ tλ ⊗ f˜imax b2

for some b10 .

(For the action of f˜imax , etc., on B(U˜ q (g)), see Appendix B.) Hence any extremal vector b ∈ B(λ) has the form b1 ⊗ tλ ⊗ u −∞ after applying the f˜imax ’s. Hence we may further assume the following conditions on b: b has the form b1 ⊗ tλ ⊗ u −∞ ;

(5.1)

for any vector of the form b10 ⊗ tµ ⊗ u −∞ in {Sw Sw∗ 0 b; w, w0 ∈ W }, the length of wt(b10 ) is greater than or equal to the length of wt(b1 ).

(5.2)

P P Here the length of i m i αi is by definition i |m i |. Take i ∈ I . We write λi = hh i , λi and wti (b1 ) = hh i , wt(b1 )i for brevity. Note that we have εi∗ (b1 ) ≤ max(λi , 0). We show wti (b1 ) ≥ 0 for every i in several steps. (1) The case λi ≤ 0 and λi + wti (b1 ) ≤ 0. Since b1 ⊗ tλ ⊗ u −∞ is a lowest weight vector in the i-string, one has ϕi (b) = max(ϕi (b1 ) + λi , 0) = 0, and hence ϕi (b1 ) + λi ≤ 0. Similarly, εi∗ (b) = 0 because b∗ is a highest weight vector in the i-string. Therefore one has −ϕ (b )−λ Si∗ Si (b1 ⊗ tλ ⊗ u −∞ ) = f˜i∗ −λi (e˜i max b1 ⊗ tλ ⊗ e˜i i 1 i u −∞ )

= ( f˜i∗ ϕi (b1 ) e˜i max b1 ) ⊗ tsi λ ⊗ u −∞ . The last equality follows from Si∗ Si (b) = ( f˜i∗ k e˜i max b1 ) ⊗ tsi λ ⊗ u −∞ for some k. Hence the minimality of b1 gives 0 ≤ ϕi (b1 ) − εi (b1 ) = wti (b1 ). (2) The case λi > 0 and λi + wti (b1 ) ≤ 0. We show that this case cannot occur. In this case, as in (i), ϕi (b1 ) + λi ≤ 0.


135

On the other hand, ϕi∗ (b1 ⊗ tλ ⊗ u −∞ ) = max(εi∗ (b1 ) − λi , 0) = 0 implies εi∗ (b1 ) ≤ λi . Hence we obtain (the first inequality by (2.6)) 0 ≤ εi∗ (b1 ) + ϕi (b1 ) = εi∗ (b1 ) − λi + ϕi (b1 ) + λi ≤ 0, which implies εi∗ (b1 ) = λi and ϕi (b1 ) = −λi . Then we have e˜i∗ max (b1 ⊗ tλ ⊗ u −∞ ) = (e˜i∗ max b1 ) ⊗ tsi λ ⊗ u −∞ . Hence the minimality of wt(b1 ) implies εi∗ (b1 ) = 0, and this contradicts εi∗ (b1 ) = λi > 0. (3) The case λi ≥ 0 and λi + wti (b1 ) ≥ 0. In this case, one has εi (b) = ϕi∗ (b) = 0, and hence ϕi (b) = λi +wti (b1 ), which implies ϕi (b)−(λi −εi∗ (b1 )) = ϕi∗ (b1 ) ≥ 0. Hence we have λi −ε (b1 ) ϕ (b) Si Si∗ (b1 ⊗ tλ ⊗ u −∞ ) = f˜i i (e˜i∗ max b1 ⊗ tsi λ ⊗ e˜i i u −∞ ) ∗

ϕi∗ (b) ∗ max e˜i b1 ) ⊗ tsi λ

= ( f˜i

⊗ u −∞ .

Hence we have ϕi∗ (b1 ) ≥ εi∗ (b1 ) or, equivalently, wti (b1 ) ≥ 0. (4) The case λi ≤ 0 and λi + wti (b1 ) ≥ 0. We have immediately wti (b1 ) ≥ 0. In all the cases we have wti (b1 ) ≥ 0. Since wt(b1 ) is of level zero, one has P 0 = hc, wt(b1 )i = i ai∨ wti (b1 ), which implies that wti (b1 ) = 0 for every i or, equivalently, cl(wt(b1 )) = 0. COROLLARY 5.2 For any λ ∈ P, the weight of any vector in B(λ) is contained in the convex hull of W λ.

Proof In the positive level case (i.e., hc, λi > 0), λ is conjugate to a dominant weight and B(λ) is isomorphic to the crystal base of an irreducible highest weight module. In this case, the assertion is well known. We can argue similarly in the negative level case. Assume that the level of λ is zero. Note that all vectors in B(λ) can be reached at an extremal vector after applying e˜imax and f˜imax by [11]. Hence the assertion follows from Theorem 5.1. Note that cl−1 cl(W λ) is contained in the convex hull of W λ, provided that cl(λ) 6 = 0. The following theorem is an immediate consequence of Corollary 5.2.

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THEOREM 5.3 Let M be an integrable Uq0 (g)-module, and let u be a vector in M of weight λ ∈ Pcl . Then the following conditions are equivalent: (i) u is an extremal vector; (ii) the weights of Uq0 (g)u are contained in the convex hull of Wcl λ; (iii) Uq0 (g)β u = 0 for any β ∈ 1cl such that (β, λ) ≥ 0.

In particular, for any λ ∈ P, V (λ) is isomorphic to the Uq (g)-module generated by a weight vector u of weight λ with (iii) in Corollary 5.2 and the following integrability condition as defining relations: 1+hh i ,λi

fi

u=0

if hh i , λi ≥ 0

and

1−hh i ,λi

ei

u=0

if hh i , λi ≤ 0.

5.2. Fundamental representations Let us take 0∨ ∈ I such that Wcl is generated by{si ; i 6 = 0∨ }

and

a0∨∨ = 1.

(5.3)

P ∨ (2) ∨ Recall that c = i ai h i . When g = A2n , 0 is the shortest simple root. We set ∨ I0∨ = I \ {0 }. For i ∈ I0∨ , we set $i = 3i − ai∨ 30∨ ∈ P 0 . L Hence we have Pcl0 = i∈I ∨ Z cl($i ). We say that λ ∈ P is a basic weight if cl(λ) is 0 Wcl -conjugate to some cl($i ) (i ∈ I0∨ ). Note that this notion does not depend on the choice of 0∨ . 5.4 P Assume that λ = i∈J $i for some subset J of I0∨ . Then one has the following: (i) any extremal vector of B(λ) is in the W -orbit of u λ ; (ii) B(λ) is connected. PROPOSITION

Proof Part (ii) follows from (i) because any vector is connected with an extremal vector. Let us prove (i). We use arguments similar to the proof of Theorem 5.1. Let us take an extremal vector b ∈ B(λ). Among the vectors in Sw Sw∗ 0 b with the form b1 ⊗ tµ ⊗ u −∞ , we take one such that wt(b1 ) has the smallest length. Then the proof in Theorem 5.1 shows that cl(wt(b1 )) = 0. Hence one has ( ∗ f˜i εi (b1 ) e˜i∗ max (b1 ) ⊗ tsi µ ⊗ u −∞ if µi ≥ 0, Si Si∗ (b1 ⊗ tµ ⊗ u −∞ ) = f˜∗ εi (b1 ) e˜i max (b1 ) ⊗ tsi µ ⊗ u −∞ if µi ≤ 0. i


137

In both cases, the length of b1 remains unchanged after applying Si Si∗ . Therefore, applying Sw0−1 Sw∗ 0−1 , we can assume that w0 = 1 and µ = λ. For i ∈ I \ J , we have λi ≤ 0, which implies εi∗ (b1 ) = 0. If i ∈ J , then λi = 1, and hence εi∗ (b1 ) (≤ λi ) must be 0 or 1. On the other hand, we have λi −εi∗ (b1 )

Si∗ (b1 ⊗ tλ ⊗ u −∞ ) = e˜i∗ max b1 ⊗ tλ ⊗ e˜i

u −∞ .

If εi∗ (b1 ) = 1, then this contradicts the minimality of wt(b1 ). Hence εi∗ (b1 ) = 0 for every i ∈ J . Thus we have εi∗ (b1 ) = 0 for every i ∈ I , and hence b1 = u ∞ . Thus we obtain u λ = Sw b. The following theorem is a particular case of Proposition 5.4. THEOREM 5.5 If λ ∈ P is a basic weight, then any extremal vector of B(λ) is in the W -orbit of u λ .

We now study further properties of B(λ) for a basic weight λ. LEMMA 5.6 Let λ be a basic weight. Then {w ∈ W ; wλ = λ} is generated by {sβ ; β ∈ 1re + , (β, λ) = 0}.

Proof ∨ We may assume that λ = 3 j − a ∨j 3∨ 0 for some j ∈ I0 . Since a similar statement holds for (Wcl , t∗0 cl ), it is enough to show that t (ξ ) is contained in the subgroup G e generated by {sβ ; β ∈ 1re + , (β, λ) = 0}, provided that ξ ∈ Q and (ξ, λ) = 0. ∨ We have saδ−β sβ = t (aβ ) by (4.8). In particular, one has t (cβ β ∨ ) ∈ G whenever β ∈ 1re satisfies (β, λ) = 0. (2) e (ξ, λ) = 0} is (1) The case where g 6= A2n . It is enough to show that {ξ ∈ Q; ∨ e generated by {cβ β ; β ∈ 1cl , (β, λ) = 0}. In this case, Q has a basis {ci αi∨ ; i ∈ I0∨ }. e (ξ, λ) = 0} is generated by {ci α ∨ ; i ∈ I ∨ \ { j}}. Hence {ξ ∈ Q; i 0 L (2) e = Q = i∈I ∨ Zα˜ i . Hence {ξ ∈ (2) The case where g = A2n . In this case, Q 0 e (ξ, λ) = 0} has a basis {α˜ i ; i ∈ I ∨ \ { j}}. Hence the result follows from Q; 0

t (α˜ i ) =

( sδ−αi sαi s(δ−αi )/2 sαi

if (αi , αi ) = 2, if (αi , αi ) = 4.

Note that (δ − αi )/2 is a real root in the last case.

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LEMMA 5.7 For any β ∈ 1re and any λ ∈ P such that sβ λ = λ, we have Ssβ (u ∞ ⊗ tλ ⊗ u −∞ ) = u ∞ ⊗ tλ ⊗ u −∞ .

Proof Set aλ = u ∞ ⊗ tλ ⊗ u −∞ . We assume that β ∈ 1re + . We prove the assertion by the induction on the length of β. If β is a simple root, it is obvious. Otherwise, we can write β = si γ for a positive real root γ whose length is less than that of β. We have Ssβ = Si Ssγ Si . Set µ = si λ. Then sγ µ = µ, and hence we have Ssγ aµ = aµ by the induction hypothesis. Since Si Si∗ aλ = aµ or, equivalently, Si aλ = Si∗ aµ , we have Ssβ aλ = Si Ssγ Si aλ = Si Ssγ Si∗ aµ = Si Si∗ aµ = aλ . Lemmas 5.6 and 5.7 imply the following proposition. PROPOSITION 5.8 Let λ be a basic weight. (i) If w ∈ W satisfies wλ = λ, then Sw u λ = u λ and Sw∗ u λ = u λ . ∼ B(µ) does not depend on w ∈ (ii) For µ ∈ W λ, the isomorphism Sw∗ : B(λ) −→ W such that µ = wλ. Here we regard B(λ) and B(µ) as subcrystals of B(U˜ q (g)).

Remark 5.9 For a general λ ∈ P 0 , it is not true that the extremal weights of B(λ) belong to W λ. (1) For example, in λ = 2(31 − 30 ) in the A1 -case, f 0 f 1 u λ is an extremal vector with weight λ − δ. Remark 5.10 It is not true in general that wλ = λ implies Sw u λ = u λ . For example, in the case of (1) g = A2 , and λ = 31 + 32 − 230 , set w1 = t (α1 ) = s1 s0 s2 s1 and w2 = t (α2 ) = s1 s0 s1 s2 . Then w1 λ = w2 λ = λ − δ, but Sw1 u λ 6= Sw2 u λ . 5.11 For any λ ∈ P, Sw u λ = u λ if and only if w ∈ W is in the subgroup generated by {sβ ; β is a real root such that (β, λ) = 0}. CONJECTURE

Theorem 5.5 and Proposition 5.8 immediately imply the following result. 5.12 Assume that λ is a basic weight. PROPOSITION


(i) (ii)

139

We have B(λ)λ = {u λ }. B(λ) is connected.

Proof Let b ∈ B(λ)λ . Then Theorem 5.5 implies b = Sw u λ for some w ∈ W with wλ = λ, and Proposition 5.8 implies Sw u λ = u λ . In order to show the finite multiplicity theorem for B($i ), we need the following result. 5.13 Assume that λ = cl(3i1 − ai∨1 30 ) for some i 1 ∈ I0∨ and µ ∈ Wcl λ. If w ∈ W satisfies l(w) ≥ ]Wcl and if µ is regularly w-dominant, then there exist w0 , w00 ∈ W such that w = w0 w00 , l(w) = l(w0 ) + l(w00 ), and λ = w00 µ. LEMMA

Proof Let w = si1 · · · sil be a reduced expression of w. Since l ≥ ]Wcl , there exists 0 ≤ j < k ≤ l such that cl(si1 · · · si j ) = cl(si1 · · · sik ). Hence si j+1 · · · sik = t (ξ ) for some e \ {0}. Replacing µ with sik+1 · · · si` µ, we reduce the lemma to the following ξ ∈ Q sublemma. 5.14 e If ξ ∈ Q \ {0} and if µ ∈ Wcl λ is regularly t (ξ )-dominant, then there exists w1 ∈ W such that λ = w1 µ and l(t (ξ )) = l(t (ξ )w1−1 ) + l(w1 ). SUBLEMMA

Proof Let us take w ∈ W0∨ := hsi ; i ∈ I0∨ i such that µ = wλ and λ is regularly wdominant. By Lemma 4.7, for β ∈ 1cl , (β, ξ ) > 0 implies (β, µ) > 0. Hence (β, w−1 ξ ) > 0 implies (β, λ) > 0. In particular, (β, λ) = 0 (resp., (β, λ) > 0) implies (β, w−1 ξ ) = 0 (resp., (β, w−1 ξ ) ≥ 0). For i ∈ I0∨ \ {i 1 }, (αi , w−1 ξ ) = 0 because (αi , λ) = 0. Moreover, (αi1 , w−1 ξ ) ≥ 0 because (αi1 , λ) > 0. Hence we have w−1 ξ = cλ for c > 0. Hence w−1 ξ is regularly w-dominant. Corollary 4.5 and Lemma 4.8 imply that l t (ξ ) = l t (w−1 ξ ) = l t (w−1 ξ )w −1 + l(w) = l(w) + l w−1 t (ξ ) . Then the sublemma follows by setting w1 = w−1 t (ξ ). PROPOSITION 5.15 Let λ ∈ P be a basic weight. Then for every ξ ∈ P, B(λ)ξ is a finite set.

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Proof For w ∈ W and µ ∈ W λ, we define a subset Aw (µ) of B(U˜ q (g)) by Aw (µ) = b ⊗ tµ ⊗ u −∞ ∈ B(µ); b ∈ B w (∞) , and then we set Aw =

F

µ∈W λ

Aw (µ). Note that Aw (µ) is a finite set. One has

B(λ) ⊂

[ w,w1 ∈W

Sw∗ 1 Aw (w1−1 λ) .

We first show B(λ) ⊂

[ w1 ∈W, w∈W with `(w)≤N

Here N := ]Wcl . For b := b1 ⊗ tµ ⊗ u −∞ in Aw , we show [ b∈ w1 ∈W, w0 ∈W with `(w0 )≤N

Sw∗ 1 (Aw ).

(5.4)

Sw∗ 1 (Aw0 )

by the induction on `(w). Proposition 3.4 implies that µ is regularly w-dominant. We may assume that `(w) > N . By Lemma 5.13, there exists w1 = w0 w00 such that l(w) = l(w0 ) + l(w00 ), w0 6= 1, and λ0 := w00 µ satisfies cl(λ0 ) = cl(λ). By Proposition 3.4, one has Sw∗ 00 (b1 ⊗ tµ ⊗ u −∞ ) = b10 ⊗ tλ0 ⊗ b20 with b10 ∈ B w0 (∞) and b20 ∈ Bw00 −1 (−∞). Take i ∈ I such that w0 si < w 0 . Then λi0 > 0 implies i = i 1 . Hence c := εi∗ (b10 ) ≤ λi0 = 1. One has Si∗ (b10 ⊗ tλ0 ⊗ b20 ) = (e˜i∗ max b10 ) ⊗ tsi λ0 ⊗ e˜i∗ λi −c b20 . 0

(5.5)

If c = 1, then λi0 − c = 0. Take x ∈ W such that b20 ∈ B x (−∞). Then x ≤ w00 −1 since b20 ∈ Bw00 −1 (−∞). Since e˜i∗ max b10 ∈ B w0 si (∞), Proposition 3.4 implies Sx∗ (e˜i∗ max b10 ) ⊗ tsi λ ⊗ b20 ∈ Bw0 si x −1 (∞) ⊗ txsi λ ⊗ u −∞ . Since `(w0 si x −1 ) < `(w), the induction proceeds. Next assume that c = 0. Then λ j ≤ 0 for j ∈ I \ {i 1 } implies ε∗j (b10 ) = 0 for every j ∈ I . Hence b10 = u ∞ . This contradicts w0 6= 1 and b10 ∈ B w0 (∞). Thus we have proved (5.4).


141

For µ ∈ W λ, set

[

C(µ) =

Aw (µ).

w∈W with `(w)≤N

Taking w ∈ W such that µ = wλ, we set ˜ C(µ) := Sw∗ −1 C(µ) ⊂ B(λ). ˜ By Proposition 5.8, C(µ) does not depend on the choice of w. We have the following: ˜ (i) C(µ) is a finite set; ˜ (ii) there is a finite subset F of Q independent of µ such that Wt(C(µ)) ⊂ µ + F. Hence, for any ξ ∈ P, [ [ ˜ ˜ B(λ)ξ ⊂ C(µ) C(µ) ξ = ξ µ∈W λ

µ∈W λ∩(ξ −F)

is a finite set. We have thus obtained the following properties of V (λ). PROPOSITION 5.16 Let λ ∈ P 0 be a basic

weight. Wt(V (λ)) is contained in the intersection of λ + Q and the convex hull of W λ. We have dim V (λ)µ = 1 for any µ ∈ W λ. We have dim V (λ)µ < ∞ for any µ ∈ P. We have Wt(V (λ)) ∩ (λ + Z δ) ⊂ W λ. V (λ) is an irreducible Uq (g)-module. Any nonzero integrable Uq (g)-module generated by an extremal weight vector of weight λ is isomorphic to V (λ). Moreover, V (λ) has a global base. (i) (ii) (iii) (iv) (v) (vi)

For any µ ∈ W λ, let us denote by u µ the unique global basis in V (λ)µ . Since u µ is an extremal vector with weight µ, we have the Uq (g)-linear homomorphism V (µ) → V (λ) that sends u µ ∈ V (µ) to u µ ∈ V (λ). This homomorphism is in fact an isomorphism. Set λ = $i . One has {n ∈ Z; $i + nδ ∈ W $i } = Zdi ,

(5.6)

where di = ($i , α˜ i ). Note that di = max(1, (αi , αi )/2) ∈ Z except in the case di = 1 (2) when g = A2n and αi is the longest root. Hence one has M M V ($i )µ = V ($i )λ+ndi . µ∈cl−1 cl($i )

n∈Z

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MASAKI KASHIWARA

∼ V ($i ). Since there is a U 0 (g)We have a Uq (g)-linear isomorphism V ($i +di δ) −→ q ∼ linear isomorphism V ($i ) −→ V ($i + di δ) that sends u $i to u $i +di δ , we obtain a Uq0 (g)-linear automorphism z i of V ($i ) of weight di δ, which sends u $i to u $i +di δ . Let us define the Uq0 (g)-module W ($i ) by W ($i ) = V ($i )/(z i − 1)V ($i ).

(5.7)

The following result is now obvious. 5.17 W ($i ) is a finite-dimensional irreducible integrable Uq0 (g)-module. W ($i ) has a global basis with a simple crystal. For any µ ∈ Wt(V ($i )),

THEOREM

(i) (ii) (iii)

W ($i )cl(µ) ' V ($i )µ . We have dim W ($i )cl($i ) = 1. The weight of any extremal vector of W ($i ) belongs to W cl($i ). Wt(W ($i )) is the intersection of cl($i )+Q cl and the convex hull of W cl($i ). 1/d 1/d We have K [z i i ] ⊗ K [zi ] V ($i ) ' W ($i )aff . Here the action of z i i on the left-hand side corresponds to the action of z on the right-hand side defined in §4.2. (viii) V ($i ) is isomorphic to the submodule K [z di , z −di ] ⊗ W ($i ) of W ($i )aff as a Uq (g)-module. Here we identify W ($i )aff with K [z, z −1 ] ⊗ W ($i ) as in (4.16). (ix) Any irreducible finite-dimensional integrable Uq0 (g)-module with cl($i ) as an extremal weight is isomorphic to W ($i )a for some a ∈ K \ {0}. (iv) (v) (vi) (vii)

Proof The irreducibility of W ($i ) follows, for example, by Proposition 4.12, and the other assertions are now obvious. We call W ($i ) a fundamental representation (of level zero).

6. Existence of global bases 6.1. Regularized modified operators ei (n) , For n ∈ Z and i ∈ I , let us define the operator F X (n+k) (k) ei (n) = F fi ei ak (ti ). k≥0,−n

(6.1)


143

Here ak (ti ) =

k(1−n) k (−1)k qi ti

k−1 Y

(1 − qin+2ν ).

ν=0

Then it acts on any integrable Uq (g)-module M. Moreover, it acts also on any Uq (g)Q ei (n) has no pole except q = 0, ∞. Let (L , B) be a submodule MQ . In this sense, F ei (n) has no crystal base of M. Then we have the following result, which says that F n pole at q = 0 and coincides with f˜i at q = 0. PROPOSITION 6.1 ei (n) L ⊂ L, and the action of F ei (n) on L/qs L coincides with fñ . We have F i

Proof In order to prove this, it is sufficient to prove the following statement. For any weight vector u ∈ M with ei u = 0 and m ∈ Z≥0 , we have ei (n) f (m) u = c f (m+n) u F i i for some c ∈ K := Q(qs ) regular at qs = 0 and c(0) = 1. Set ti u = qil u. Then we can assume that l ≥ n + m. We have

(m)

ak (ti ) f i

(m)

u = ak (qil−2m ) f i

u.

Hence (m)

fi

u=

X k≥0 m X

(n+k) (k) (m) ei f i u

ak (qil−2m ) f i

l −m+k (m−k) = fi u k i k=0 m X l −m+k (m+n) l−2m n + m = ak (qi ) fi u. m−k i k i (n+k) ak (qil−2m ) f i

k=0

Here

n [n]i ! = m i [m]i ![n − m]i !

is the q-binomial coefficient. Hence it is enough to show that A :=

m X k=0

n+m l −m+k ∈ 1 + qi Z[qi ]. m−k i k i

ak (qil−2m )

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MASAKI KASHIWARA

This follows immediately from the following formula, whose proof due to Anne Schilling is given in Appendix A. A=

m X

k(2l−2m−n+2)

qi

k=0

n+2( j−1) m−k k Y Y 1 − q n+2 j 1 − qi i 2j

j=1

1 − qi

2j

j=1

1 − qi

.

(6.2)

6.2. Existence theorem We use the notation and terminologies in §2.4. Let M be an integrable Uq (g)-module, let − be a bar involution of M, and let (L , B) be a crystal base of M. Let MQ be a Uq (g)Q -submodule of M such that (MQ )− = MQ . Set E := L ∩ L ∩ MQ . THEOREM 6.2 Let S be a subset of P. We assume the following conditions: (i) {(ξ, ξ ); ξ ∈ Wt(M)} is bounded from above; (ii) u − u¯ ∈ (qs − 1)MQ for any u ∈ MQ ; (iii) MQ generates M as a vector space over K ; (iv) for any ξ ∈ P \ S, (L ξ , L ξ , (MQ )ξ ) is balanced; (v) any extremal weight (i.e., the weight of an extremal vector) of B is in P \ S; (vi) qs L ∩ L ∩ MQ = 0. Then we have the following: (a) (L , L, MQ ) is balanced; (b) for any n, we have M M f in M = Q(qs )G(b) and ein M = Q(qs )G(b); εi (b)≥n

(c)

MQ =

P

ξ ∈P\S

Uq (g)Q (MQ )ξ and M =

ϕi (b)≥n

P

ξ ∈P\S

Uq (g)Mξ .

Proof The rest of this section is devoted to the proof of this theorem. LEMMA 6.3 The action of − on E is the identity.

Proof For u ∈ E, we have (u − u)/(1 ¯ − qs−1 ) ∈ qs L ∩ L ∩ MQ = 0. By (vi), the homomorphism E → L/qs L is injective. Let us denote by B 0 the intersection of B and the image of this homomorphism. To see (a), it is enough to show that B = B 0 . For b ∈ B 0 , let us denote by G(b) the element E such that


145

b ≡ G(b) mod qs L. Note that G(b) = G(b) by Lemma 6.3. We prove the following statements by the descending induction on (ξ, ξ ): Bξ = Bξ0 or, equivalently, L ξ , L ξ , (MQ )ξ is balanced, (6.3) X (ε (b)) G(b) − f i i G(e˜imax b) ∈ Q[qs , qs−1 ]G(b0 ) for any b ∈ Bξ , (6.4) εi (b0 )>εi (b)

X

Q[qs , qs−1 ]G(b) =

b∈Bξ , εi (b)≥n

X

(m)

fi

(MQ )ξ +mαi

m≥n

for any n ≥ max 0, −hh i , ξ i ,

(6.5)

as well as the similar statements replacing f i with ei . If (ξ, ξ ) is big enough, those statements are trivially satisfied by (i). Now assuming (6.3) – (6.5) for ξ such that (ξ, ξ ) > a, let us prove them for ξ with (ξ, ξ ) = a. LEMMA 6.4 Let i ∈ I . Set k = max(0, −hh i , ξ i). (a) If e˜imax b ∈ B 0 , then b ∈ B 0 and (εi (b))

G(b) − f i

G(e˜imax b) ∈

X

Q[qs , qs−1 ]G(b0 ).

b0 ∈Bξ0 εi (b0 )>εi (b)

(b)

In particular, any b ∈ Bξ with εi (b) > k is contained in B 0 . P P (m) −1 We have (MQ )ξ +mαi for any n > k. b∈Bξ0 Q[qs , qs ]G(b) = m≥n f i εi (b)≥n

Similar statements hold after exchanging ei and f i . Proof Let us prove the lemma by the descending induction on n (in the case (a), n means εi (b), and hence n ≥ k). If n is big enough, they are true by hypothesis (i) in ei (n) G(b1 ) satisfies Theorem 6.2. Let us prove (a). Set b1 = e˜imax b. Then u = F b ≡ u mod qs L and X X (m) (n) (m+n) (m) Z[qs , qs−1 ] f i u − f i G(b1 ) ∈ ei G(b1 ) ⊂ f i (MQ )ξ +mαi . m>n

m>0

Induction hypothesis (b) implies that the last space is contained in X Q[qs , qs−1 ]G(b0 ). b0 ∈Bξ0 εi (b0 )>n

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MASAKI KASHIWARA

P (n) Hence we can write u − f i G(b1 ) = b0 cb0 G(b0 ), where b0 ranges over b0 ∈ B 0 with εi (b0 ) > n and cb0 ∈ Q[qs , qs−1 ]. Hence we can write cb0 − cb0 = cb0 0 − cb0 0 with P P (n) cb0 0 ∈ qs Q[qs ]. Then v := u − b0 cb0 0 G(b0 ) = f i G(b1 ) + b0 (cb0 − cb0 0 )G(b0 ) satisfies v = v, and hence it belongs to E. Moreover, one has b ≡ v mod qs L. Hence b belongs to B 0 , and G(b) = v. To complete the proof of (a), it is enough to remark that e˜imax b ∈ B 0 when εi (b) > k, because wt(e˜imax (b)), wt(e˜imax (b)) > (wt(b), wt(b)). Let us prove (b). The left-hand side is contained in the right-hand side by (a) and the induction hypothesis on n. Let us show the opposite inclusion. Set η = ξ + nαi with n > k. Then we have (η, η) > (ξ, ξ ), and (6.5) holds for η. Hence we have X X (m) (MQ )η ⊂ Q[qs , qs−1 ]G(b) + f i (MQ )η+mαi , εi (b)=0, b∈Bη0

m>0

which implies (n)

f i (MQ )η ⊂

X

(n)

Q[qs , qs−1 ] f i

G(b) +

εi (b)=0, b∈Bη0

⊂

X

(m)

fi

MQ

m>n (n)

Q[qs , qs−1 ] f i

εi (b)=0 b∈Bη0

X

G(b) +

X

Q[qs , qs−1 ]G(b).

εi (b)>n b∈Bξ0

The desired inclusion follows from (a). LEMMA 6.5 We have Bξ ⊂ B 0 .

Proof Let b ∈ Bξ . By hypothesis (v), there exists X l · · · X 1 b whose weight is outside S, where X ν is e˜imax or f˜imax . Hence, by the induction on l, we may assume that e˜imax b or f˜imax b is contained in B 0 . Then Lemma 6.4 implies b ∈ B 0 . Properties (6.3) and (6.4) are now obvious, and (6.5) easily follows from Lemmas 6.4 and 6.5. Thus the induction proceeds, and we complete the proof of (a), (b) in Theorem 6.2. Finally, let us prove (c). Set X X 0 M0 = Uq (g)Mξ and MQ = Uq (g)Q (MQ )ξ . ξ ∈P\S

ξ ∈P\S

Set L 0 = L ∩ M 0 . Then L 0 is invariant by e˜i and f˜i . By hypothesis (v), any vector in B is connected with a vector whose weight is outside S. Hence B is contained in


147

L 0 /qs L 0 ⊂ L/qs L. This shows that (L 0 , B) is a crystal base of M 0 , and L 0 /qs L 0 = 0 = L/qs L. Thus we can apply Theorem 6.2 to M 0 . Hence we obtain L 0 ∩ L 0 ∩ MQ 0 = K ⊗ (L 0 ∩ L 0 ∩ M 0 ) = M . This completes the proof of L ∩ L ∩ MQ , and MQ Q Q Q Theorem 6.2. 7. Universal R-matrix In this section, we review the universal R-matrix introduced by Drinfeld and the universal bar involution introduced by G. Lusztig. Although we mainly use the coproduct 1 in this article, 1(q h ) = q h ⊗ q h , 1(ei ) = ei ⊗ ti−1 + 1 ⊗ ei , 1( f i ) = f i ⊗ 1 + ti ⊗ f i ,

(7.1)

we introduce another coproduct 1 = (− ⊗ −) ◦ 1 ◦ −, 1(q h ) = q h ⊗ q h , 1(ei ) = ei ⊗ ti + 1 ⊗ ei , 1( f i ) = f i ⊗ 1 + ti−1 ⊗ f i .

(7.2)

Let Mν (ν = 1, 2) be Uq (g)-modules with weight decomposition. Let us denote by M1 ⊗ M2 the tensor product of M1 and M2 with the Uq (g)-module structure induced by 1, and by M1 × M2 the Uq (g)-module induced by 1. Then there is an isomorphism q −( · , · ) : M1 × M2 → M2 ⊗ M1 given by

q −( · , · ) (x × y) = q −(wt(x),wt(y)) y ⊗ x.

b Uq− (g) by Let us define the ring Uq+ (g) ⊗ M Y b Uq− (g) = Uq+ (g) ⊗ Uq+ (g)λ ⊗ Uq− (g)µ .

(7.3)

ξ ∈Q ξ =λ+µ

Uq+ (g)

Uq− (g)

b Uq− (g) → K . ModThe counits → K and → K induce ε : Uq+ (g) ⊗ ifying Drinfeld’s construction in [3] of a universal R-matrix, Lusztig has shown that b Uq− (g) satisfying the following propthere exists a unique intertwiner 4 ∈ Uq+ (g) ⊗ erties: 4 ◦ 1(a) = 1(a) ◦ 4 for any a ∈ Uq (g), normalized by ε(4) = 1. Then it satisfies 4 ◦ 4 = 4 ◦ 4 = 1.

(7.4)

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We introduce the completion of the tensor products as follows. We set Y b M2 ) = F(λ,µ) (M1 ⊗ (M1 )λ+γ ⊗ (M2 )µ−γ , γ ∈Q +

Y

b M2 ) = F>(λ,µ) (M1 ⊗

(M1 )λ+γ ⊗ (M2 )µ−γ ,

γ ∈Q + \{0}

and then b M2 = M1 ⊗

X

b M2 ) ⊂ F(λ,µ) (M1 ⊗

λ,µ∈P

Y

(M1 )λ ⊗ (M2 )µ .

λ,µ∈P

e M2 in the opposite direction, Sometimes we use another completion M1 ⊗ Y e M2 ) = F(λ,µ) (M1 ⊗ (M1 )λ−γ ⊗ (M2 )µ+γ , γ ∈Q +

and then e M2 = M1 ⊗

X

e M2 ) ⊂ F(λ,µ) (M1 ⊗

λ,µ∈P

Y

(M1 )λ ⊗ (M2 )µ .

λ,µ∈P

They have the structure of a Uq (g)-module by 1 and contain M1 ⊗ M2 as a Uq (g)-submodule. b M the same vector space M ⊗ We denote by M1 ⊗ 2 1 b M2 with the action of Uq (g) b M contains M × M as a Uq (g)-submodule. induced by 1. Then M1 ⊗ 2 1 2 We have an isomorphism b M −→ ∼ M2 ⊗ e M1 . q −( · , · ) : M1 ⊗ 2 The operator 4 induces an isomorphism bM . ∼ M1 ⊗ b M2 −→ M1 ⊗ 2 b M ), and the homomorphism inb M2 ) to F(λ,µ) (M1 ⊗ Then 4 sends F(λ,µ) (M1 ⊗ 2 duced by 4, b M2 )/F>(λ,µ) (M1 ⊗ b M2 ) M1λ ⊗ M2µ ' F(λ,µ) (M1 ⊗ b M )/F b −→ F(λ,µ) (M1 ⊗ 2 >(λ,µ) (M1 ⊗ M2 ) ' M1λ ⊗ M2µ ,

(7.5)

is equal to the identity. b M2 → M2 ⊗ e M1 , called the universal R-matrix, is The intertwiner R univ : M1 ⊗ given by q −( · , · ) 4 bM − b M2 −− e M1 . R univ : M1 ⊗ → M1 ⊗ → M2 ⊗ (7.6) 2 −−−


149

It is an isomorphism. Assume that M1 and M2 have a bar involution. Then (7.4) implies that 4 −⊗− bM − b M2 −− b M2 cuniv : M1 ⊗ → M1 ⊗ → M1 ⊗ 2 −−

b M2 as observed by Lusztig [17]. We call it the universal is a bar involution on M1 ⊗ bar involution. 8. Good modules Let us take a finite-dimensional integrable Uq0 (g)-module M. We consider the following conditions on M: M has a bar involution,

(8.1)

M has a crystal base L(M), B(M) ,

(8.2)

M has a global base,

(8.3)

B(M) is a simple crystal.

(8.4)

In this paper, we say that a Uq0 (g)-module M is a good Uq0 (g)-module if M satisfies the above conditions. The level-zero fundamental representations W (ϕi ) are good Uq0 (g)-modules. A good Uq0 (g)-module is always irreducible (see Prop. 4.12). Let M1 and M2 be good Uq0 (g)-modules. Then we have b (M2 )aff = K [z 1 /z 2 ] ⊗ K [z 1 /z 2 ] (M1 )aff ⊗ (M2 )aff , (M1 )aff ⊗ e (M1 )aff = K [z 1 /z 2 ] ⊗ K [z 1 /z 2 ] (M2 )aff ⊗ (M1 )aff . (M2 )aff ⊗ Here z ν is the Uq0 (g)-linear automorphism of weight δ on (Mν )aff introduced in §4.2. 8.1 K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] ((M1 )aff ⊗ (M2 )aff ) K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] Uq (g)[z 1±1 , z 2±1 ].

LEMMA

is

an

irreducible

module

over

Proof Since M1 ⊗ M2 has a simple crystal base by Lemma 4.11, it is irreducible by Proposition 4.12. Then the lemma follows from the fact that the specialization of (M1 )aff ⊗ (M2 )aff at the special point z 1 /z 2 = 1 is irreducible. By the result of Section 7, we have the bar involution b (M2 )aff → (M1 )aff ⊗ b (M2 )aff . cuniv : (M1 )aff ⊗ It commutes with z 1 and z 2 . Let u ν be the extremal vector with dominant weight λν b (M2 )aff of Mν (ν = 1, 2), and set u = u 1 ⊗ u 2 . Then we have (M1 )aff ⊗ = λ +λ 1

2

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MASAKI KASHIWARA

K ((z 1 /z 2 ))u. Hence, by (7.5), we have cuniv (u) = ϕ(z 1 /z 2 )u

or, equivalently,

4(u) = ϕ(z 1 /z 2 )u

(8.5)

for some ϕ(z 1 /z 2 ) ∈ K [[z 1 /z 2 ]] with ϕ(0) = 1. We define b (M2 )aff → (M1 )aff ⊗ b (M2 )aff cnorm : (M1 )aff ⊗ by cnorm = cuniv ◦ ϕ(z 1 /z 2 )−1 . Then it satisfies cnorm (u) = u. LEMMA 8.2 b (M2 )aff satisfying The operator cnorm is a unique endomorphism of (M1 )aff ⊗ cnorm (u 1 ⊗ u 2 ) = u 1 ⊗ u 2 and cnorm (av) = acnorm (v) for any a ∈ b (M2 )aff . Uq (g)((z 1 /z 2 ))[z 2±1 ], v ∈ (M1 )aff ⊗

Proof It is enough to show that a Uq (g)[z 1±1 , z 2±1 ]-linear homomorphism b (M2 )aff f : (M1 )aff ⊗ (M2 )aff → (M1 )aff ⊗ vanishes if f (u 1 ⊗ u 2 ) = 0. By Lemma 8.1, K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] (M1 )aff ⊗ (M2 )aff is an irreducible module over K (z 1 /z 2 )[z 2±1 ] ⊗ Uq (g). Hence the assertion follows. b (M2 )aff , which we call the normalHence cnorm defines a bar involution on (M1 )aff ⊗ ized bar involution. In particular, we have ϕ(z)ϕ(z) = 1. In the sequel, we use the normalized bar involution to define a global basis. The universal R-matrix b (M2 )aff → (M2 )aff ⊗ e (M1 )aff R univ : (M1 )aff ⊗ sends u 1 ⊗ u 2 to q −(λ1 ,λ2 ) ϕ(z 1 /z 2 )u 2 ⊗ u 1 with the same function ϕ given in (8.5). Hence, setting R norm = q (λ1 ,λ2 ) ϕ(z 1 /z 2 )−1 R univ , we have an intertwiner b (M2 )aff → (M2 )aff ⊗ e (M1 )aff R norm : (M1 )aff ⊗ that sends u 1 ⊗u 2 to u 2 ⊗u 1 . We call R norm the normalized R-matrix. Both R-matrices commute with z 1 and z 2 .


151

By (7.6) and (7.5), we have, for any vν ∈ (Mν )aff , R norm (v1 ⊗ v2 ) ≡ q hλ1 ,λ2 i−hwt(v1 ),wt(v2 )i v2 ⊗ v1 Y (M2 )aff wt(v )−ξ ⊗ (M1 )aff wt(v mod

1 )+ξ

2

ξ ∈Q + \{0}

.

(8.6)

We also have R norm : (M1 )aff ⊗ (M2 )aff → K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] (M2 )aff ⊗ (M1 )aff e (M1 )aff . ,→ (M2 )aff ⊗

We generalize these observations to the case of tensor products of several modules. Let Mν (ν = 1, . . . , m) be good Uq0 (g)-modules with a crystal base (L ν , Bν ). Let (Mν )aff be its affinization. Then (Mν )aff has a crystal base (L ν )aff , (Bν )aff . Let λν ∈ P be a dominant extremal weight of (Mν )aff , and let u ν be the extremal global basis with weight λν . We denote the canonical automorphism (Mν )aff of weight δ by zν . Then m O M := (Mν )aff = (M1 )aff ⊗ · · · ⊗ (Mm )aff ν=1

has a structure of a

±1 ]-module. K [z 1±1 , . . . , z m

Set

M = (M1 )aff ⊗ · · · ⊗ (Mm )aff ,

(8.7)

MQ = (M1Q )aff ⊗ · · · ⊗ (Mm Q )aff ,

(8.8)

and let (L(M), B(M)) be the tensor product of the crystal bases of the (Mν )aff ’s. We set m O b ⊗ M = K [z 1 /z 2 , . . . , z m−1 /z m ] (Mν )aff . K [z 1 /z 2 ,...,z m−1 /z m ]

ν=1

We also set

b = A [z 1 /z 2 , . . . , z m−1 /z m ] L( M) bQ = Q [z 1 /z 2 , . . . , z m−1 /z m ] M

⊗

A[z 1 /z 2 ,...,z m−1 /z m ]

⊗

Q[z 1 /z 2 ,...,z m−1 /z m ]

m O ν=1 m O

(L ν )aff , (Mν )affQ .

ν=1

Similarly to the case of the tensor product of two modules, we can define the b by universal bar involution of M Y cuniv = (− ⊗ · · · ⊗ −) ◦ 4i j , 1≤i< j≤m

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MASAKI KASHIWARA

where 4i j is the operator 4 acting on the ith and jth components of the tensor prodb It satisuct. Normalizing cuniv , we obtain the normalized bar involution cnorm on M. fies, by setting u = u 1 ⊗ · · · ⊗ u m , cnorm (u) = u. Moreover, it satisfies for vν ∈ (Mν )aff , cnorm (v1 ⊗ · · · ⊗ vm ) ≡ v1 ⊗ · · · ⊗ vm

mod

Y

(M1 )aff

ξ1 ,...,ξm

wt(v1 )+ξ1

⊗ · · · ⊗ (Mm )aff

wt(vm )+ξm

.

(8.9) P Pµ Here the product ranges over ξ1 , . . . , ξm ∈ Q with m ν=1 ξν = 0 and ν=1 ξν ∈ Q + \ {0} (µ = 1, . . . , m − 1). Since cnorm is expressed by a triangular matrix, the well-known argument of triangular matrices implies the following result. 8.3 b ∩ (cnorm L( M)) b ∩M bQ = 0. We have qs L( M) b such For any b = b1 ⊗ · · · ⊗ bm ∈ B(M), there exists a unique G(b) ∈ L( M) norm b that c (G(b)) = G(b) and b ≡ G(b) mod qs L( M). Moreover, G(b) has the form X 0 G(b) = G(b1 ) ⊗ · · · ⊗ G(bm ) + cb0 ,··· ,bm0 G(b10 ) ⊗ · · · ⊗ G(bm ).

LEMMA

(i) (ii) (iii)

1

0 Here the infinite sum ranges over b10 ⊗ · · · ⊗ bm ∈ B(M) such that Pm Pm Pµ 0 0 wt(b ) = wt(b ) and (wt(b ) − wt(b ν ν )) ∈ Q + \ {0} ν ν ν=1 ν=1 ν=1 (µ = 1, . . . , m − 1). Moreover, cb0 ,...,bm0 ∈ qs Q[qs ]. 1

Later we see that this infinite sum is in fact a finite sum. Set ±1 ]u. N = Uq (g)[z 1±1 , . . . , z m b stable by the bar involution cnorm . Set λ = Then N is a submodule of M Then we have Nλ+Zδ := =

M n∈Z m O ν=1

Nλ+nδ =

m O (Mν )aff ν=1

(Mν )aff

λν +Zδ

Pm

λ+Zδ

±1 = K [z 1±1 , . . . , z m ](u 1 ⊗ · · · ⊗ u m ).

ν=1 λν .


153

Hence one has N µ = Mµ

for any µ ∈ W λ + Zδ.

(8.10)

Define L(N ) = L(M) ∩ N , N Q = MQ ∩ N , B(N ) = B(M). Then L(N )/qs L(N ) ⊂ L(M)/qs L(M). 8.4 B(N ) is a basis of L(N )/qs L(N ), and (L(N ), B(N )) is a crystal base of N .

LEMMA

Proof Since B(N ) is a basis of L(M)/qs L(M), it is enough to show that B(N ) is contained in L(N )/qs L(N ). Since every vector in B(N ) is connected with an extremal vector with weight in λ + Zδ and since extremal vectors with such a weight are u up to the ±1 , we obtain the desired result. action of z 1±1 , . . . , z m Setting S = Wt(M) \ (W λ + Zδ), we can apply Theorem 6.2 to N . The hypotheses in the theorem are satisfied by Lemmas 8.3 and 8.4, and we obtain the following theorem. 8.5 (L(N ), cnorm L(N ), NQ ) is balanced. Hence N has a global base. We have NQ = Uq (g)Q [z 1 , . . . , z m ]u.

THEOREM

(i) (ii)

Furthermore, Lemma 8.3 implies the following proposition. 8.6 For any bν ∈ B((Mν )aff ) (ν = 1, . . . , m), we have X 0 G(b1 ⊗ · · · ⊗ bm ) = G(b1 ) ⊗ · · · ⊗ G(bm ) + cb0 ,...,bm0 G(b10 ) ⊗ · · · ⊗ G(bm ). PROPOSITION

1

Qm

Pm 0 )∈ 0 Here the sum ranges over (b10 , . . . , bm ν=1 B((Mν )aff ) such that ν=1 wt(bν ) = Pµ Pm 0 ν=1 wt(bν ) and ν=1 (wt(bν )−wt(bν )) ∈ Q + \{0} (µ = 1, . . . , m −1). Moreover, 0 ). cb0 ,...,bm0 ∈ qs Q[qs ], and cb0 ,...,bm0 vanishes except for finitely many (b10 , . . . , bm 1

1

By specializing at z ν = 1, we obtain the following proposition.

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MASAKI KASHIWARA

PROPOSITION 8.7 The tensor product of good Uq0 (g)-modules is also a good Uq0 (g)-module.

9. Main theorem The following theorem is conjectured in [1] in the special case when all the Mν are fundamental representations. Note that, as seen by the proof, the theorem holds even if we consider Uq (g) as an algebra over the algebraically closed P b := b, and replace A with the subring field K C((q 1/n )), aν as elements of K n>0 P b := b. A C[[q 1/n ]] of K n>0

9.1 Let Mν (ν = 1, . . . , m) be good Uq0 (g)-modules. Let aν ∈ K . Assume that aν /aν+1 ∈ A for ν = 1, . . . , m − 1. Then (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am is generated by u 1 ⊗ · · · ⊗ u m . Assume that (Mν )∗ (ν = 1, . . . , m) are good Uq0 (g)-modules, and aν+1 /aν ∈ A for ν = 1, . . . , m − 1. Then any nonzero submodule of (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am contains u 1 ⊗ · · · ⊗ u m .

THEOREM

(i)

(ii)

Proof Since (ii) is the dual statement of (i), it is enough to prove (i). Let us embed the crystal Bν of Mν into (Bν )aff as in (4.16). Let ψ : (M1 )aff ⊗· · ·⊗ (Mm )aff → (M1 )a1 ⊗ · · · ⊗ (Mm )am be the canonical projection. Then ψ(G(b1 ) ⊗ · · · ⊗ G(bm )) (bν ∈ Bν ) forms a basis of (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am . Since ψ(G(b1 ⊗ · · · ⊗ bm )) are in Uq0 (g)(u 1 ⊗ · · · ⊗ u m ), it is enough to show that they also generate (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am as a vector space. By Proposition 8.6, we can write G(b1 ⊗ · · · ⊗ bm ) = G(b1 ) ⊗ · · · ⊗ G(bm ) +

X

0 m cbk10 ,...,k G(z k1 b10 ) ⊗ · · · ⊗ G(z km bm ). ,...,b0 1

m

Qm 0 ) ∈ Here the summation ranges over the set of (b10 , . . . , bm ν=1 Bν and P m m (k1 , . . . , km ) ∈ Z such that ν=1 kν = 0 and k1 + · · · + kν ≥ 0 (ν = 1, . . . , m). m Moreover, we have cbk10 ,...,k 0 ∈ qs Q[qs ]. 1 ,...,bm On the other hand, we have 0 ψ G(z k1 b10 ) ⊗ · · · ⊗ G(z km bm ) 0 km = (a1k1 · · · am )ψ G(b10 ) ⊗ · · · ⊗ G(bm ) 0 = (a1 /a2 )k1 (a2 /a3 )k1 +k2 · · · ψ G(b10 ) ⊗ · · · ⊗ G(bm ) ∈ L,


155

L where L = bν ∈B(Mν ) A G(b1 ) ⊗ · · · ⊗ G(bm ) . Hence we have ψ G(b1 ⊗ · · · ⊗ bm ) ≡ ψ G(b1 ) ⊗ · · · ⊗ G(bm ) mod qs L . Then Nakayama’s lemma implies that ψ(G(b1 ⊗· · ·⊗bm )); bν ∈ B(Mν ) generates (M1 )a1 ⊗ · · · ⊗ (Mm )am . We can apply Theorem 9.1 to the fundamental representations. THEOREM 9.2 Let aν ∈ K , and let i ν ∈ I0∨ (ν = 1, . . . , m). (i) Assume that aν /aν+1 ∈ A for ν = 1, . . . , m −1. Then W ($i1 )a1 ⊗W ($i2 )a2 ⊗ · · · ⊗ W ($im )am is generated by u $i1 ⊗ · · · ⊗ u $im . (ii) Assume that aν+1 /aν ∈ A for ν = 1, . . . , m − 1. Then any nonzero submodule of W ($i1 )a1 ⊗ W ($i2 )a2 ⊗ · · · ⊗ W ($im )am contains u $i1 ⊗ · · · ⊗ u $im .

b and A, b they imply the Since these theorems hold even if we replace K and A with K following consequences, as shown in [1]. PROPOSITION 9.3 Assume that M j is a good Uq0 (g)-module with dominant extremal vector u j . The normalized R-matrix

Ri,norm j (x, y) : (Mi )x ⊗ (M j ) y → (M j ) y ⊗ (Mi )x b does not have a pole at x/y = a ∈ A. Here Ri,norm j (x, y) is the intertwiner (Mi )x ⊗ (M j ) y → (M j ) y ⊗ (Mi )x so normalized that it sends u i ⊗ u j to u j ⊗ u i . Let ψi j (x, y) be the denominator of Ri,norm j (x, y). Then one has ψi j (x, y) ∈ 1 + A[x/y]qs x/y.

(9.1)

For the sake of simplicity, we assume that M j as well as its dual M ∗j are good Uq0 (g)-modules, and we let u j be a dominant extremal vector of M j . 9.4 The extremal vector u 1 ⊗ · · · ⊗ u m generates (M1 )a1 ⊗ · · · ⊗ (Mm )am if and only if Ri,norm j (x, y) has no pole at x/y = ai /a j for any 1 ≤ j < i ≤ m. Any nonzero submodule (M1 )a1 ⊗ · · · ⊗ (Mm )am contains u 1 ⊗ · · · ⊗ u m if and only if Ri,norm j (x, y) has no pole at x/y = ai /a j for any 1 ≤ i < j ≤ m.

PROPOSITION

(i) (ii)

156

(iii)

MASAKI KASHIWARA

(M1 )a1 ⊗ · · · ⊗ (Mm )am is irreducible if and only if Ri,norm j (x, y) does not have a pole at x/y = ai /a j for any 1 ≤ i, j ≤ m (i 6 = j).

PROPOSITION 9.5 If M and M 0 are irreducible finite-dimensional integrable Uq0 (g)-modules, then M ⊗ Mz0 is an irreducible Uq0 (g)-module except for finitely many z.

10. Combinatorial R-matrices Let M1 and M2 be two good Uq0 (g)-modules. Let u ν be the extremal vector of Mν with dominant weight (ν = 1, 2). Let ψ(z 1 /z 2 ) be the denominator of the normalized R-matrix, normalized by ψ ∈ K [z 1 /z 2 ] with ψ(0) = 1. Then, by (9.1), we have ψ(z) ∈ 1 + qs z A[z].

(10.1)

We have an intertwiner ψ(z 1 /z 2 )R norm : (M1 )aff ⊗ (M2 )aff → (M2 )aff ⊗ (M1 )aff . We first prove the following proposition. PROPOSITION

10.1

We have ψ(z 1 /z 2 )R norm L(M1 )aff ⊗ L(M2 )aff ⊂ L(M2 )aff ⊗ L(M1 )aff . Proof Set M = (M1 )aff ⊗ (M2 )aff , and let L be the smallest crystal lattice of M containing A[z 1±1 , z 2±1 ](u 1 ⊗ u 2 ). Then L is contained in L(M). Since every vector in B(M) is connected with some z 1m u 1 ⊗ z 2n u 2 , L/qs L → L(M)/qs L(M) is surjective. Hence, by the following well-known lemma, there exists g such that g ∈ 1 + qs A[z 1±1 , z 2±1 ]

and

gL(M) ⊂ L .

10.2 Let R be a commutative ring, let a ∈ R, and let F be a finitely generated R-module. If F = a F, then there exists b ∈ 1 + a R such that bF = 0. LEMMA

Let us define M 0 and L 0 in a way similar to M and L by exchanging M1 and M2 . The operator T = ψ(z 1 /z 2 )R norm : M → M 0 commutes with e˜i , f˜i , z 1 , z 2 , and it satisfies T (u 1 ⊗ u 2 ) ∈ L(M 0 ) by (10.1). Hence we have T L ⊂ L(M 0 ).


157

Taking g as above, we obtain gT L(M) ⊂ T L ⊂ L(M 0 ). Since L(M 0 ) is a free A[z 1±1 , z 2±1 ]-module of finite rank, the proposition follows from the following lemma. 10.3 Let F be a free A[z 1±1 , z 2±1 ]-module, and let g be an element in 1 + qs A[z 1±1 , z 2±1 ]. If u ∈ K ⊗ F satisfies gu ∈ F, then u belongs to F. LEMMA

A

Since the proof is elementary, we do not give it. As a corollary of Proposition 10.1 and (10.1), we obtain the following. COROLLARY

10.4

We have b (M2 )aff R norm L (M1 )aff ⊗ CONJECTURE

e L(M1 )aff . ⊂ L (M2 )aff ⊗

10.5

We have ψ(z 1 /z 2 ) (M1 )aff ⊗ (M2 )aff ⊂ Uq (g)[z 1±1 , z 2±1 ](u 1 ⊗ u 2 ). Set N = Uq (g)[z 1±1 , z 2±1 ](u 1 ⊗ u 2 ) ⊂ (M1 )aff ⊗ (M2 )aff , N 0 = Uq (g)[z 1±1 , z 2±1 ](u 2 ⊗ u 1 ) ⊂ (M2 )aff ⊗ (M1 )aff . Then R norm gives an isomorphism ∼ N 0. R norm : N −→ In §8, we saw that N (resp., N 0 ) has a crystal base (L(N ), B(M1 )aff ⊗ B(M2 )aff ) resp., (L(N 0 ), B(M2 )aff ⊗ B(M1 )aff ) . Hence R norm induces an isomorphism ∼ B(M2 )aff ⊗ B(M1 )aff . R comb : B(M1 )aff ⊗ B(M2 )aff −→ We have R comb (zb1 ⊗ b2 ) = (1 ⊗ z)R comb (b1 ⊗ b2 ), R comb (b1 ⊗ zb2 ) = (z ⊗ 1)R comb (b1 ⊗ b2 ).

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MASAKI KASHIWARA

Hence we have a commutative diagram: R comb

B(M1 )aff ⊗  B(M2 )aff y

−−−→

B(M1 ) ⊗ B(M2 )

−−→

∼

B(M2 )aff ⊗  B(M1 )aff y

(10.2)

B(M2 ) ⊗ B(M1 )

Hence one obtains the following proposition. PROPOSITION 10.6 If B1 and B2 are crystal bases of good Uq0 (g)-modules, then B1 ⊗ B2 ' B2 ⊗ B1 .

By Corollary 10.4, we have R norm G(b1 ⊗ b2 ) = G R comb (b1 ⊗ b2 ) .

(10.3)

Setting R comb (b1 ⊗ b2 ) = b20 ⊗ b10 with bν , bν0 ∈ B(Mν )aff , we define S(b1 ⊗ b2 ) = wt(b10 ) − wt(b1 ) = wt(b2 ) − wt(b20 ) ∈ Q. P By (8.6), we have S(b1 ⊗ b2 ) ∈ Q + := i∈I Z≥0 αi . On the other hand, we have S(z 1 b1 ⊗ b2 ) = S(b1 ⊗ z 2 b2 ) = S(b1 ⊗ b2 ), and hence it induces a map S : B(M1 ) ⊗ B(M2 ) → Q + .

(10.4)

This map S is characterized by the following properties (note that B(M1 ) ⊗ B(M2 ) is connected): S(u 1 ⊗ u 2 ) = 0,     S(b1 ⊗ b2 ) + αi      S f˜i (b1 ⊗ b2 ) = S(b1 ⊗ b2 ) − αi         S(b ⊗ b ) 1

2

(10.5) if f˜i (b1 ⊗ b2 ) = ( f˜i b1 ) ⊗ b2 and f˜i (b0 ⊗ b0 ) = ( f˜i b0 ) ⊗ b0 , 2

1

2

1

if f˜i (b1 ⊗ b2 ) = b1 ⊗ ( f˜i b2 ) and f˜i (b0 ⊗ b0 ) = b0 ⊗ ( f˜i b0 ), 2

1

2

1

otherwise. (10.6)

11. Energy function In this section, we assume that M is good, and we investigate the properties of Maff ⊗2 . In this case, we have e Maff . R norm = ι¯ ◦ cnorm : Maff ⊗ Maff → Maff ⊗

(11.1)


159

b Maff → Maff ⊗ e Maff is given by Here ι¯ : Maff ⊗ ι¯(v ⊗ v 0 ) = q (λ,λ)−(wt(v),wt(v )) v 0 ⊗ v. 0

Indeed, R norm and ι¯ ◦ cnorm are Uq (g)-linear homomorphisms sending u ⊗ u to itself, z ⊗ 1 to 1 ⊗ z, and 1 ⊗ z to z ⊗ 1. Such a homomorphism is unique. Similarly, the identity being a unique automorphism of B(M)⊗2 , (10.2) implies that there exists a unique map H : B(M)aff ⊗2 → Z such that R comb (b1 ⊗ b2 ) = (z −H (b1 ⊗b2 ) b1 ) ⊗ (z H (b1 ⊗b2 ) b2 ). Hence one has S(b1 ⊗ b2 ) = wt(b2 ) − wt(b1 ) + H (b1 ⊗ b2 )δ. We call H the energy function. We have H ((zb1 ) ⊗ b2 ) = H (b1 ⊗ (z −1 b2 )) = H (b1 ⊗ b2 ) + 1. It is easy to see that G B(M)aff ⊗2 = H −1 (n) n∈Z

is the decomposition of B(M)⊗2 aff into the minimal regular subcrystals invariant by z ⊗ z (cf. [6]). Embedding B(M) into B(M)aff as in (4.16), the energy function restricted on B(M)⊗2 is also characterized by the following two properties: H (v ⊗ v) = 0 for any extremal vector v of B(M),   if i 6= 0 and f˜i (b1 ⊗ b2 ) 6= 0,   H (b1 ⊗ b2 )     H (b1 ⊗ b2 ) + 1 if i = 0 and  H f˜i (b1 ⊗ b2 ) = f˜i (b1 ⊗ b2 ) = ( f˜i b1 ) ⊗ b2 6= 0,    H (b ⊗ b ) − 1 if i = 0 and  1 2     f˜ (b ⊗ b ) = b ⊗ ( f˜ b ) 6= 0. i

1

2

1

i 2

Set ⊗2 N := Uq (g) (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) ⊂ Maff , ⊗2 ⊗2 N 0 := Ker R norm − 1 : Maff → K (z ⊗ z −1 ) ⊗ K [z⊗z −1 ] Maff . ⊗2 Then we have N ⊂ N 0 ⊂ Maff . Define L(N ) = L(Maff )⊗2 ∩ N , and define L(N 0 ) ⊗2 ⊗2 similarly. Then one has L(N )/qs L(N ) ⊂ L(N 0 )/qs L(N 0 ) ⊂ L(Maff )/qs L(Maff ). Set ⊗2 B0 (Maff ) := b1 ⊗ b2 ∈ B(Maff )⊗2 ; H (b1 ⊗ b2 ) = 0 .

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Then n ⊗2 ⊗2 (z ⊗ 1)b; n ∈ Z≥0 , b ∈ B0 (Maff ) t (1 ⊗ z n )b; n ∈ Z>0 , b ∈ B0 (Maff ) ⊗2 ⊗2 is a basis of L(Maff )/qs L(Maff ). ⊗2 ⊗2 Let us define the subset B 0 of L(Maff )/qs L(Maff ) by ⊗2 B 0 := z n ⊗ 1 + δ(n 6 = 0)(1 ⊗ z n ) b; n ∈ Z≥0 , b ∈ B0 (Maff ) .

Here, for a statement P, we define δ(P) by ( 1 if P is true, δ(P) = 0 if P is false. Then B 0 is linearly independent. LEMMA 11.1 We have B 0 ⊂ L(N )/qs L(N ). Moreover, (L(N ), B 0 ) and (L(N 0 ), B 0 ) are crystal bases of N and N 0 , respectively.

Proof It is enough to show that B 0 ⊂ L(N )/qs L(N ) and that B 0 is a basis of ⊗2 L(N 0 )/qs L(N 0 ). Since B0 (Maff ) is a minimal subcrystal invariant by z ⊗ z, we have ⊗2 B0 (Maff ) ⊂ L(N )/qs L(N ). Since L(N )/qs L(N ) is invariant by z n ⊗ 1 + 1 ⊗ z n , we have B 0 ⊂ L(N )/qs L(N ). It remains to prove that B 0 generates L(N 0 )/qs L(N 0 ). Since R norm = 1 on N 0 , we have R norm = 1 on L(N 0 )/qs L(N 0 ), and hence L(N 0 )/qs L(N 0 ) ⊂ F := {v ∈ (L(Maff )/qs L(Maff ))⊗2 ; R norm (v) = v}. ⊗2 Since the action of R norm on (L(Maff )/qs L(Maff ))⊗2 = Q⊕B(Maff ) is given by R comb , we can easily see that B 0 is a basis of F. PROPOSITION 11.2 N = N 0 , and it has

a global basis {G(b); b ∈ B 0 }.

Proof We apply Theorem 6.2 for N and N 0 . Set ⊗2 NQ := Uq (g)Q (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) ⊂ Maff , ⊗2 and define NQ0 similarly. Set S = Wt(Maff ) \ (W (2λ) + Zδ). For ξ = 2wλ + nδ L (w ∈ W , n ∈ Z), setting H = ν+µ=n K (z ν ⊗ z µ + z µ ⊗ z ν )u ⊗2 wλ , we have H ⊂ Nξ ⊂ Nξ0 ⊂ H . Hence Nξ = Nξ0 = H , and condition (iv) in Theorem 6.2 is satisfied for N and N 0 . Condition (v) follows from the fact that the weight of any extremal


161

vector of B(Maff )⊗2 is in W (2λ) + Zδ. Hence all the conditions in Theorem 6.2 are satisfied for N and N 0 , and both N and N 0 have a global basis. These two global bases coincide, and hence N = N 0 . COROLLARY 11.3 If b1 , b2 ∈ B(M)aff satisfy H (b1 ⊗ b2 ) = 0, then G(b1 ⊗ b2 ) ∈ Uq (g)Q (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u).

Moreover, denoting by N0 the vector subspace generated by {G(b1 ⊗ b2 ); H (b1 ⊗ b2 ) = 0}, one has Uq (g)Q (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) = Q[z ⊗ 1 + 1 ⊗ z] ⊗Q N0 , Maff ⊗2 = Q[z ±1 ⊗ 1, 1 ⊗ z ±1 ] ⊗Q[(z⊗z)±1 ] N0 = Q[z ±1 ⊗ 1] ⊗Q N0 .

12. Fock space 12.1. Some properties of good modules In [13], we defined the wedge spaces and the Fock spaces for a finite-dimensional integrable Uq0 (g)-module V . In that paper, we assumed several conditions on V . In this section, we show that all those conditions are satisfied whenever V is a good module with a perfect crystal base. In [13], we employed the reversed coproduct. Adapting the notation to ours, those conditions read as follows. We set N := Uq (g)[(z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z](u ⊗ u) ⊂ (Vaff )⊗2 with an extremal vector u of V of weight λ. (G) V is good. Let (L , B) be the crystal base of V . (P) B is a perfect crystal. (L) Let s : Q → Z be the additive function such that s(αi ) = 1, and let ` : Baff → Z be the function defined by `(b) = s(wt(b) − wt(u)). Then one has H (b1 ⊗ b2 ) ≤ 0 ⇒ `(b1 ) ≤ `(b2 ). (D) (R)

We have ψ ∈ 1 + qs z A[z]. Here ψ(x/y) is the denominator of the normalized R-matrix R norm : Vx ⊗ Vy → Vy ⊗ Vx . For every pair (b1 , b2 ) in Baff with H (b1 ⊗ b2 ) = 0, there exists Cb1 ,b2 ∈ N of the form X Cb1 ,b2 = G(b1 ) ⊗ G(b2 ) − ab0 ,b0 G(b10 ) ⊗ G(b20 ). b10 ,b20

1

2

162

MASAKI KASHIWARA 2 such that Here the sum ranges over (b10 , b20 ) ∈ Baff

H (b10 ⊗ b20 ) > 0, `(b1 ) < `(b10 ) ≤ `(b2 ), `(b1 ) ≤ `(b20 ) < `(b2 ), and the coefficients ab0 ,b0 belong to Q[qs , qs−1 ]. 1

2

12.1 ([13]) V We assume that (G), (L), (D), and (R). Then the wedge space m Vaff has a basis {G(b1 ) ∧ · · · ∧ G(bm )}, where (b1 , . . . , bm ) ranges over (Baff )m with H (b j ⊗b j+1 ) > 0 ( j = 1, . . . , m − 1). THEOREM

For the other consequences and the Fock space, see §12.2, §12.3, and [13]. In this section we prove the following theorem. THEOREM 12.2 Assume that V is a good Uq0 (g)-module. Then all the properties above except (P) are satisfied.

In fact, we prove here a little bit stronger results. In the sequel, we assume that V is a good Uq0 (g)-module. The property (D) has already been proved in (10.1). The following lemma immediately implies (L). LEMMA 12.3 If H (b1 ⊗ b2 ) ≤ 0, then wt(b2 ) − wt(b1 ) ∈ Q + .

Proof By (10.4), we have S(b1 ⊗ b2 ) = wt(b2 ) − wt(b1 ) + H (b1 ⊗ b2 )δ ∈ Q + . Hence if H (b1 ⊗ b2 ) ≤ 0, then wt(b2 ) − wt(b1 ) ∈ Q + . In order to prove the remaining property, we prove the following result on global bases. PROPOSITION 12.4 Assume that H (b1 ⊗ b2 ) = 0. Write

G(b1 ⊗ b2 ) =

X b10 , b20 ∈B(M)aff

ab0 ,b0 G(b10 ) ⊗ G(b20 ). 1

2

(12.1)


163

Then we have ab1 ,b2 = 1, ab0 ,b0 = q (λ,λ)−(wt b1 ,wt b2 ) ab0 ,b0 . 0

2

0

1

1

2

If ab0 ,b0 6 = 0, then 1

2

wt(b10 ) ∈ wt(b1 ) + Q + ∩ wt(b2 ) − Q + , wt(b20 ) ∈ wt(b1 ) + Q + ∩ wt(b2 ) − Q + . Moreover, wt(b10 ) = wt(b1 ) implies (b10 , b20 ) = (b1 , b2 ), and wt(b10 ) = wt(b2 ) implies (b10 , b20 ) = (b2 , b1 ). Proof We have seen that wt(b10 ) ∈ wt(b1 ) + Q + , wt(b20 ) ∈ wt(b2 ) − Q + , and wt(b10 ) = wt(b1 ) implies (b10 , b20 ) = (b1 , b2 ). Since R norm G(b1 ⊗ b2 ) = G(b1 ⊗ b2 ) and since cnorm G(b1 ⊗ b2 ) = G(b1 ⊗ b2 ), we have ι¯G(b1 ⊗ b2 ) = G(b1 ⊗ b2 ) by (11.1). Hence we have X 0 0 G(b1 ⊗ b2 ) = q (λ,λ)−(wt b1 ,wt b2 ) ab0 ,b0 G(b20 ) ⊗ G(b10 ), 1

b10 ,b20 ∈B(M)aff

2

which gives ab0 ,b0 = q (λ,λ)−(wt b1 ,wt b2 ) ab0 ,b0 . Hence we obtain the remaining asser1 2 2 1 tions. 0

0

CONJECTURE 12.5 Conjecturally, we have H (b10 ⊗ b20 ) ≥ 0 if ab0 ,b0 6 = 0. 1

2

Let us set I+ (b) = b0 ∈ Baff ; wt(b0 ) − wt(b) ∈ Q + \ {0} t {b}, I− (b) = b0 ∈ Baff ; wt(b) − wt(b0 ) ∈ Q + \ {0} t {b}. The following lemma immediately implies (R). 12.6 For every pair (b1 , b2 ) in Baff , there exists Cb1 ,b2 ∈ N of the form X Cb1 ,b2 = G(b1 ) ⊗ G(b2 ) − ab0 ,b0 G(b10 ) ⊗ G(b20 ). LEMMA

b10 ,b20

1

2

2 such that H (b0 ⊗ b0 ) > 0 and b0 , b0 ∈ Here the sum ranges over (b10 , b20 ) ∈ Baff 1 2 1 2 I+ (b1 ) ∩ I− (b2 ), and the coefficients ab0 ,b0 belong to Q[qs , qs−1 ]. 1

2

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MASAKI KASHIWARA

Proof We prove this by the induction on `(b2 ) − `(b1 ). Note that the assertion is trivial when H (b1 ⊗ b2 ) > 0. We may assume that H (b1 ⊗ b2 ) ≤ 0. Then (L) implies `(b2 ) − `(b1 ) ≥ 0. Set n := −H (b1 ⊗ b2 ). Then H (z n b1 ⊗ b2 ) = 0. Hence G(z n b1 ⊗ b2 ) ∈ N . By Proposition 12.4, we can write X G(z n b1 ⊗ b2 ) = z n G(b1 ) ⊗ G(b2 ) + ab0 ,b0 G(b10 ) ⊗ G(b20 ), b10 ,b20

1

2

where the sum ranges over (b10 , b20 ) with b10 , b20 ∈ I+ (z n b1 ) ∩ I− (b2 ) and b10 6 = z n b1 . In particular, one has `(z −n b10 ) > `(b1 ). Then z −n ⊗ 1 + δ(n > 0)1 ⊗ z −n G(z n b1 ⊗ b2 ) = G(b1 ) ⊗ G(b2 ) + δ(n > 0)G(z n b1 ) ⊗ G(z −n b2 ) X + ab0 ,b0 G(z −n b10 ) ⊗ G(b20 ) + δ(n > 0)G(b10 ) ⊗ G(z −n b20 ) (b10 ,b20 )∈I

1

2

⊗2 belongs to NQ = N ∩ (Maff )Q . Hence, modulo NQ , G(b1 ) ⊗ G(b2 ) is a linear combin −n nation of G(z b1 ) ⊗ G(z b2 ) (n > 0), G(z −n b10 ) ⊗ G(b20 ), and G(b10 ) ⊗ G(z −n b20 ). When n > 0, we have `(z −n b2 ) − `(z n b1 ) < `(b2 ) − `(b1 ), and the induction hypothesis implies that G(z n b1 ) ⊗ G(z −n b2 ) is, modulo NQ , a linear combination of G(b100 ) ⊗ G(b200 ) with H (b100 ⊗ b200 ) > 0 and b100 , b200 ∈ I+ (z n b1 ) ∩ I− (z −n b2 ) ⊂ I+ (b1 ) ∩ I− (b2 ). Similarly, we have `(b20 ) − `(z −n b10 ) < `(b2 ) − `(b1 ). Hence, modulo NQ , −n G(z b10 ) ⊗ G(b20 ) is a linear combination of G(b100 ) ⊗ G(b200 ) with H (b100 ⊗ b200 ) > 0 and b100 , b200 ∈ I+ (z −n b10 ) ∩ I− (b20 ) ⊂ I+ (b1 ) ∩ I− (b2 ). Finally, since `(z −n b20 )−`(b10 ) ≤ `(b20 )−`(z −n b10 ) < `(b2 )−`(b1 ), the induction hypothesis implies that G(b10 ) ⊗ G(z −n b20 ) modulo NQ is a linear combination of G(b100 ) ⊗ G(b200 ) with H (b100 ⊗ b200 ) > 0 and b100 , b200 ∈ I+ (b10 ) ∩ I− (z −n b20 ) ⊂ I+ (b1 ) ∩ I− (b2 ).

12.2. Wedge spaces Let us recall the construction of the wedge space in [13]. Let V be a good Uq0 (g)module with an extremal global basis u. Let us set ⊗2 N = Uq (g) (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) ⊂ Vaff , (12.2) Nm =

m−2 X j=0

⊗m Vaff ⊗ j ⊗ N ⊗ V ⊗(m−2− j) ⊂ Vaff .

(12.3)


The wedge space

Vm

165

Vaff is defined by m ^

⊗m /Nm . Vaff = Vaff

For v1 , . . . , vm ∈ Vaff , let v1 ∧ · · · ∧ vm denote the image of v1 ⊗ · · · ⊗ vm by the V V V ⊗m ⊗m projection Vaff → m Vaff . Let L( m Vaff ) ⊂ m Vaff be the image of L(Vaff ). ⊗m For b = b1 ⊗ · · · ⊗ bm ∈ B(Vaff ), we set G pure (b) = G(b1 ) ∧ · · · ∧ G(bm ). ⊗m ⊗m For b ∈ B(Vaff ), let G(b) be the global basis of Vaff , and let G ∧ (b) be its image Vm in Vaff . We set

B

m ^

⊗m Vaff = b1 ⊗ · · · ⊗ bm ∈ B(Vaff ); H (bν ⊗ bν+1 ) > 0

for ν = 1, . . . , m − 1 .

Then in [13], the following properties are proved: V V (i) {G pure (b); b ∈ B( m Vaff )} is a basis of L( m Vaff ); Vm Vm V (ii) identifying B( Vaff ) with a subset of L( Vaff )/qs L( m Vaff ) by G pure , V V V (L( m Vaff ), B( m Vaff )) is a crystal base of m Vaff . On the other hand, the following proposition follows from Proposition 8.6. PROPOSITION 12.7 ⊗(m +m ) ⊗m 1 ⊗m 2 For b1 ∈ B(Vaff ) and b2 ∈ B(Vaff ), one has the following equality in Vaff 1 2 :

G(b1 ⊗ b2 ) = G(b1 ) ⊗ G(b2 ) +

X b10 ,b20

cb0 ,b0 G(b10 ) ⊗ G(b20 ). 1

2

⊗m 1 ⊗m 2 Here the sum ranges over (b10 , b20 ) ∈ B(Vaff ) × B(Vaff ) such that wt(b10 ) − 0 wt(b1 ) = wt(b2 ) − wt(b2 ) ∈ Q + \ {0}, and the coefficients satisfy cb0 ,b0 ∈ qs Q[qs ]. 1

2

Set ⊗m ⊗m B0 (Vaff ) = b1 ⊗ · · · ⊗ bm ∈ B(Vaff ); H (bν ⊗ bν+1 ) = 0 or ν = 1, . . . , m − 1 , M K G(b). Nm0 = ⊗m b∈B0 (Vaff )

Arguments similar to those in Proposition 11.2 and Corollary 11.3 show the following proposition.

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MASAKI KASHIWARA

PROPOSITION

12.8

We have ±1 sym ⊗m ±1 sym Uq (g) ⊗ Q[z 1±1 , . . . , z m ] u = Q[z 1±1 , . . . , z m ]

⊗ Q[(z 1 ···z m )±1 ]

Nm0 .

⊗m Here z ν is the automorphism of Vaff induced by the action of z on the νth factor, and ±1 ±1 sym Q[z 1 , . . . , z m ] is the ring of symmetric Laurent polynomials. In particular, for any Laurent polynomial f (z 1 , . . . , z m ) symmetric in (z ν , z ν+1 ) for some ν, f (z 1 , . . . , z m )Nm0 ⊂ Nm .

Vm ±1 ] sym -module, Since Nm is a Q[z 1±1 , . . . , z m Vaff has the structure of a Vm ±1 ] sym -module. We denote by B the operator on Q[z 1±1 , . . . , z m Vaff given by n Pm n z . Then the B ’s commute with one another. n ν=1 ν LEMMA 12.9 ⊗m For any b ∈ B(Vaff ), one has either G ∧ (b) = 0 or G ∧ (b) = ±G ∧ (b0 ) for some V m 0 b ∈ B( Vaff ).

Proof Set b = z a1 b1 ⊗ · · · ⊗ z am bm with H (bν ⊗ bν+1 ) = 0. Then we have by Proposition 12.8, G(b) ≡ ±G(z aσ (1) b1 ⊗ · · · ⊗ z aσ (m) bm ) for any permutation σ . Hence we may assume that (a1 , . . . , am ) is a decreasing sequence. If there is ν such that aν = aν+1 , then G(b) ∈ Nm by Proposition 12.8. V Otherwise, b belongs to B( m Vaff ). 12.3. Global basis of the Fock space The purpose of this subsection is to define the global basis of the Fock space. Let us now assume that V is a good Uq0 (g)-module with perfect crystal base (L , B) of level `. Let us recall that a simple crystal B is called perfect of level ` if it satisfies the following conditions. P (P1) Any b ∈ B satisfies hc, ε(b)i = hc, ϕ(b)i ≥ `. Here ε(b) = i εi (b) cl(3i ) ∈ P Pcl and ϕ(b) = i ϕi (b) cl(3i ) ∈ Pcl . (`) (P2) Set Pcl = {λ ∈ Pcl ; hc, λi = ` and hh i , λi ≥ 0 for every i}, the set of dominant weights of level `, and Bmin = {b ∈ B; hc, ε(b)i = `}. Then the two maps (`) (`) ε : Bmin −→ Pcl and ϕ : Bmin −→ Pcl are bijective.


167 (1)

For example, the vector representation of An is a good Uq0 (g)-module with a perfect crystal base of level 1. Let (Baff )min be the inverse image of Bmin by the map Baff → B. Let us take a sequence {bn }n∈Z in (Baff )min such that ϕ(bn ◦ ) = ε(bn−1 ◦ )

H (bn ◦ ⊗ bn−1 ◦ ) = 1.

and

Such a sequence is called a ground state. Take a sequence {λn }n∈Z in P such that λn = λn−1 + wt(bn ◦ )

cl(λn ) = ϕ(bn ◦ ) = ε(bn−1 ◦ ).

and

In [13], the Fock spaces Fr (r ∈ Z) are constructed, and they satisfy the following properties. (F1) Fr is an integrable Uq (g)-module. (F2) We have Wt(Fr ) ⊂ λr + Q − . (F3) There exist Uq0 (g)-linear endomorphisms Bn (n ∈ Z \ {0}) of Fr with weight nδ satisfying the boson commutation relations [Bn , Bm ] = δ−n,m an for some an ∈ K \ {0}. V (F4) There exists a Uq (g)-linear map · ∧ · : Fr ⊗ m Vaff → Fr −m such that V V 0 (u ∧ v) ∧ v 0 = u ∧ (v ∧ v 0 ) for u ∈ Fr , v ∈ m Vaff , and v 0 ∈ m Vaff . (F5) Bn (u ∧ v) = (Bn u) ∧ v + u ∧ (z n v) for n ∈ Z \ {0}, u ∈ Fr , and v ∈ Vaff . (F6) There is a nonzero vector vacr ∈ Fr of weight λr , (Fr )λr = K vacr . Moreover, one has vacr +1 ∧ G(br ◦ ) = vacr . (F7) We have {u ∈ Fr ; Bn u = 0 for any n > 0 and ei u = 0 for any i} = K varr . P (F8) Let K [B−1 , B−2 , . . .] = K [Bn ; n 6= 0]/ m>0 K [Bn ; n 6= 0]Bm be the ∼ Fr as Fock space of the boson algebra. Then K [B−1 , B−2 , . . .] ⊗ V (λr ) −→ a K [Bn ; n 6= 0] ⊗ Uq (g)-module. Here 1 ⊗ u λr corresponds to vacr . (F9) Let B(Fr ) be the set of sequences {bn }n≥r satisfying H (bn+1 ⊗ bn ) > 0 bn = bn

for any n ≥ r, ◦

for n r.

For b = {bn }n≥r ∈ B(Fr ), set G pure (b) = vacn ∧ G(bn−1 ) ∧ · · · ∧ G(br ) for n r . Then {G pure (b); b ∈ B(Fr )} is a basis of Fr . S V (F10) Set L(Fr ) = n≥r vacn ∧ L( n−r Vaff ). Then (L(Fr ), B(Fr )) is a crystal base of Fr . Here B(Fr ) is identified with a subset of L(Fr )/qs L(Fr ) by G pure . (k) (F11) We have f i vacr = vacr +1 ∧ G( f˜ik br ◦ ). Now we show that the Fock space Fr has a global basis. First let us define a bar involution c on Fr such that c(vacr ) = vacr ,

(12.4)

[Bn , c] = 0

(12.5)

for any n > 0.

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MASAKI KASHIWARA

By (F8), there exists a unique bar involution on Fr satisfying the conditions above. Note that c ◦ B−n ◦ c = an an −1 B−n for n > 0, since [Bn , an −1 B−n ] = 1 implies that an −1 B−n is c-invariant. We set m−r X ^ (Fr )Q = vacm ∧ (Vaff )Q . m≥r

12.10 ⊗m Let b := b1 ⊗ · · · ⊗ bm be an element of Baff . ◦ ∧ (a) If H (br ⊗ b1 ) ≤ 0, then vacr ∧ G (b) = vacr ∧ G pure (b) = 0 hold in Fr −m . (b) We have vacr +1 ∧ G ∧ (br ◦ ⊗ b) = vacr ∧ G ∧ (b). LEMMA

Proof (a) We have G(b) =

X

cb0 ,b0 G(b10 ) ⊗ G(b0 ), 1

where the sum ranges over b10 ∈ Baff and b0 ∈ Baff ⊗(m−1) such that wt(b10 ) − wt(b1 ) ∈ Q + . Since H (br ◦ ⊗ b1 ) ≤ 0, we have `(br −1 ◦ ) < `(b1 ) ≤ `(b10 ) by [13, Lem. 4.2.2]. Since Wt(Fr −1 ) ⊂ λr −1 + Q − by (F2), one has vacr ∧ G(b10 ) = 0. Hence we obtain vacr ∧ G ∧ (b) = 0. The proof of vacr ∧ G pure (b) = 0 is similar. (b) The proof is similar to that of (a). One has X G(br ◦ ⊗ b) = G(br ◦ ) ⊗ G(b) + cb0 ,b0 G(b00 ) ⊗ G(b0 ), 0

where the sum ranges over b00 ∈ Baff and b0 ∈ Baff ⊗m such that wt(b00 ) − wt(br ◦ ) ∈ Q + \ {0}. Then by the same reasoning on the weight of Wt(Fr ), we have vacr +1 ∧ G(b00 ) = 0. By Lemma 12.10, for b = {bn }n≥r ∈ B(Fr ), G(b) := vacm ∧ G ∧ (bm−1 ⊗ · · · ⊗ br ) does not depend on m such that b j = b j ◦ for j ≥ m. LEMMA 12.11 {G(b); b ∈ B(Fr )} is a basis of the A-module L(Fr ).

Proof Since b ≡ G(b) mod qs L(Fr ), {G(b); b ∈ B(Fr )} is linearly independent. Hence it is enough to show that it generates L(Fr ).


169

V Let b = (b1 , . . . , bm ) ∈ B( m Vaff ). For any integer N , we can write X X G pure (b) = ab0 G ∧ (b0 ) + cb00 G pure (b00 ). b0

b00

⊗m 00 ranges over B(V ⊗m ) with Here b0 ranges over B(Vaff ) and b00 = b100 ⊗ · · · ⊗ bm aff 00 ◦ `(b1 ) > N . Taking `(bm+r −1 ) as N , one has vacm+r ∧ G pure (b00 ) = 0. Hence one has X ab0 vacm+r ∧ G ∧ (b0 ). vacm+r ∧ G pure (b) = ⊗m ) b0 ∈B(Vaff

Now it is enough to apply Lemmas 12.9 and 12.10. THEOREM 12.12 {G(b); b ∈ B(Fr )} is a global basis of Fr .

Proof It remains to prove that the G(b)’s are invariant by the bar involution c. Let E be the vector space over Q generated by {G(b); b ∈ B(Fr )}. Then vacr +m ∧ G ∧ (b) ⊗m is contained in E for any b ∈ B(Vaff ) by Lemmas 12.9 and 12.10. We define the 0 involution c of Fr by c 0 (v) = v

for any v ∈ E

and c 0 (av) = a c 0 (v)

for any v ∈ Fr and a ∈ K .

We show that c 0 = c. In order to see this, it is enough to show the following properties: c 0 (vacr ) = vacr , 0

c commutes with Bn

(12.6) if n > 0,

c (av) = ac (v) for any v ∈ Fr and a ∈ Uq (g). 0

0

(12.7) (12.8)

The property (12.6) is obvious. Let us first show that c 0 commutes with Bn (n > 0). This follows from the fact V that Bn (vacr +m ∧ G ∧ (b)) = vacr +m ∧ Bn G ∧ (b) holds for b ∈ B( m Vaff ) and from the fact that Bn G ∧ (b) belongs to E. Let us show (12.8). We evidently have q h ◦ c 0 = c 0 ◦ q −h for every h ∈ P ∗ . The conjugation c 0 commutes with ei because, for b ∈ B(Fr ), ei G(b) belongs to Q[qs + qs−1 ] ⊗ E.

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MASAKI KASHIWARA

Finally, let us show that c 0 commutes with f i . To see this, we prove f i c 0 (v) = i v) for any weight vector v ∈ Fr by the induction on wt(v). For any j ∈ I , one has, by using the commutativity of c 0 and ei ,

c0( f

ti − ti−1 0 e j f i c 0 (v) − c 0 ( f i v) = f i e j + δi j c (v) qi − qi−1 ti − ti−1 v − c 0 f i e j + δi j qi − qi−1 = f i c 0 (e j v) − c 0 ( f i e j v). Since this vanishes by the induction hypothesis, f i c 0 (v) − c 0 ( f i v) is a highest weight vector. Similarly, it is annihilated by all the Bn ’s (n > 0). Since the weight of f i c 0 (v)− c 0 ( f i v) is not λr , it must vanish by (F7). Thus we obtain (12.8). Remark 12.13 (1) In the case when g = An and V is the vector representation, the global basis of the Fock space was introduced by Leclerc and Thibon [14], [15]. D. Uglov [20] general(1) (1) ized this to the case when g = An ⊕ Am and when V is the tensor product of the vector representations. The connection of global bases of Fock space and KazhdanLusztig polynomials has also been studied by Varagnolo and Vasserot [21] and O. Schiffmann [19]. 13. Conjectural structure of V (λ) In this section, we present conjectures that clarify the structure of V (λ) and its crystal base B(λ) for λ ∈ P 0 . The paper by J. Beck, V. Chari, and A. Pressley [2] should help to solve them. These conjectures are closely related to those of Lusztig [18]. P Let λ be a dominant integral weight of level zero. We write λ = i∈I ∨ m i $i . 0 N N Then the module i∈I ∨ V (m i $i ) contains the extremal vector i∈I ∨ u m i $i whose 0 0 weight is λ. Here we can take any ordering of I0∨ to define the tensor product. Hence we have a Uq (g)-linear morphism O 8λ : V (λ) → V (m i $i ) i∈I0∨

sending u λ to

N

i∈I0∨

u m i $i .

13.1 8λ is a monomorphism. We have 8−1 λ (⊗i∈I ∨ L(m i $i )) = L(λ).

CONJECTURE

(i) (ii)

0


171

By 8λ , we have an isomorphism of crystals O ∼ B(λ) −→ B(m i $i ).

(iii)

i∈I0∨

Next we consider the case when λ is a multiple of a fundamental weight. There is a morphism of Uq (g)-modules 9m,i : V (m$i ) → V ($i )⊗m 0 sending u m$i to u ⊗m $i . Let z i be the Uq (g)-linear automorphism of V ($i ) of weight di δ introduced in §5.2, and let z ν (ν = 1, . . . , m) be the operator of V ($i )⊗m obtained by the action of z i on the νth factor. It is again a Uq0 (g)-linear automorphism of V ($i )⊗m of weight di δ. Let B0 (m$i ) be the connected component of B(m$i ) containing u m$i , and let B0 (V ($i )⊗m ) be the connected component of B($i )⊗m containing u ⊗m $i .

13.2 9m,i is a monomorphism. −1 We have 9m,i L(V ($i )⊗m ) = L(m$i ). ∼ B0 (V ($i )⊗m ) by 9m,i . Moreover, the global basis G(b) with B0 (m$i ) −→ b ∈ B0 (m$i ) is sent to the corresponding global basis of Uq (g)u ⊗m $i ⊂ W ($i )⊗m constructed in Theorem 8.5. aff Let S be the set of Schur Laurent polynomials in z 1 , . . . , z m , that is, the set of characters of GL(m) ((z 1 , . . . , z m ) that are the components of the diagonal matrices). Then {G(b); b ∈ B(m$i )} is sent by 9m,i to {aG(b); b ∈ B0 (V ($i )⊗m ), a ∈ S}.

CONJECTURE

(i) (ii) (iii)

(iv)

Note that, for a, a 0 ∈ S and b, b0 ∈ B0 (V ($i )⊗m ), a G(b) = a 0 G(b0 ) holds if and only if a 0 = (z 1 · · · z m )r a and b = (z 1 · · · z m )r b0 for some r ∈ Z. These conjectures imply the following conjecture on U˜ q (g) analogous to the Peter-Weyl theorem. For λ ∈ P, let B0 (λ) be the connected component of B(λ) conF taining u λ . Note that if hc, λi 6 = 0, then B0 (λ) = B(λ). We consider λ∈P B0 (λ) × F B(−λ) as a crystal over g ⊕ g. The Weyl group W acts on λ∈P B0 (λ) × B(−λ) by Sw∗ × Sw∗ : B0 (λ) × B(−λ) → B0 (wλ) × B(−wλ). 13.3 ∼ B(U˜ q (g)) as a crystal over g × g. B0 (λ) × B(−λ) /W −→

CONJECTURE

F

λ∈P

Here the usual crystal structure on B(U˜ q (g)) corresponds to that of B0 (λ), and the star crystal structure on B(U˜ q (g)) corresponds to that of B(−λ). The isomorphism

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MASAKI KASHIWARA

sends u λ ⊗ b ∈ B0 (λ) × B(−λ) to b∗ ∈ B(U˜ q (g)).

Appendices A. Proof of the formula (6.2) In this appendix, we give a proof of (6.2) due to Anne Schilling. Let us define (a)n = (a; q)n =

n−1 Y

(1 − aq i ).

i=0

Then in terms of (a; q)n , the q-binomial in this paper is given as m (q 2 ; q 2 )m = q n(n−m) 2 2 . n i (q ; q )n (q 2 ; q 2 )m−n Hence, replacing n → 2n and q → q 1/2 in (6.2), it reads as follows. LEMMA

A.1

We have m X

(−1)k q (1/2)k(k+1−2m)−nm

k=0

(q n )k (q)2n+m (q)`−m+k (q)m−k (q)2n+k (q)k (q)`−m =

m X

q k(`−m−n+1)

k=0

(q n )k (q n+1 )m−k . (A.1) (q)k (q)m−k

Proof Using (see [5, (I.10) in Appendix]) (a)m−k =

(a)m q k (k )−mk q 2 , − a (q 1−m /a)k

equation (A.1) may be rewritten in hypergeometric notation as q

−nm (q

2n+1 ) m

(q)m

3 82

q −m , q n , q `−m+1 ;q q 2n+1 , 0

−m n (q n+1 )m q ,q `−m−2n+1 = ;q . 2 81 (q)m q −m−n However, this formula readily follows from [5, (III.7) in Appendix] with the replacements n → m,

b → qn,

c → q −n−m ,

z → q `−m−2n+1 .


173

B. Formulas for the crystal B(U˜ q (g)) In this table, b1 ∈ B(∞), b2 ∈ B(−∞), λ ∈ P, b = b1 ⊗ tλ ⊗ b2 , λi = hh i , λi, and wti (b1 ) = hh i , wt(b1 )i: b∗ = b1∗ ⊗ t−λ−wt(b1 )−wt(b2 ) ⊗ b2∗ ; εi (b) = max εi (b1 ), εi (b2 ) − λi − wti (b1 ) ; ϕi (b) = max ϕi (b1 ) + λi + wti (b2 ), ϕi (b2 ) ; wt∗ (b) = wt(b∗ ) = −λi ; εi∗ (b) = max εi∗ (b1 ), ϕi∗ (b2 ) + λi ; ϕi∗ (b) = max εi∗ (b1 ) − λi , ϕi∗ (b2 ) ; e˜i b1 ⊗ tλ ⊗ b2 if ϕi (b1 ) ≥ εi (b2 ) − λi , e˜i b = b1 ⊗ tλ ⊗ e˜i b2 if ϕi (b1 ) < εi (b2 ) − λi ; ( f˜i b1 ⊗ tλ ⊗ b2 if ϕi (b1 ) > εi (b2 ) − λi , f˜i b = b1 ⊗ tλ ⊗ f˜i b2 if ϕi (b1 ) ≤ εi (b2 ) − λi ; ( e˜i∗ b1 ⊗ tλ−αi ⊗ b2 if εi∗ (b1 ) ≥ ϕi∗ (b2 ) + λi , ∗ e˜i b = b1 ⊗ tλ−αi ⊗ e˜i∗ b2 if εi∗ (b1 ) < ϕi∗ (b2 ) + λi ; ( f˜i∗ b1 ⊗ tλ+αi ⊗ b2 if εi∗ (b1 ) > ϕi∗ (b2 ) + λi , ∗ f˜i b = b1 ⊗ tλ+αi ⊗ f˜i∗ b2 if εi∗ (b1 ) ≤ ϕi∗ (b2 ) + λi ; e˜imax b = e˜imax b1 ⊗ tλ ⊗ e˜i c b2 , where c = max εi (b2 ) − ϕi (b1 ) − λi , 0 ; f˜imax b = f˜i c b1 ⊗ tλ ⊗ f˜imax b2 , where c = max ϕi (b1 ) − εi (b2 ) + λi , 0 ;  ∗ max ∗ ∗ e˜i b1 ⊗ tλ−(ϕi∗ (b2 )+λi )αi ⊗ e˜i∗ ϕi (b2 )−εi (b1 )+λi b2      if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≤ 0, ∗ max e˜i b=  e˜i∗ max b1 ⊗ tλ−εi∗ (b1 )αi ⊗ b2     if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≥ 0;  ∗ ε∗ (b )−ϕ ∗ (b )−λ f˜i i 1 i 2 i b1 ⊗ tλ+(εi∗ (b1 )−λi )αi ⊗ f˜i∗ max b2      if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≥ 0 , f˜i∗ max b =  b1 ⊗ tλ+ϕi∗ (b2 )αi ⊗ f˜i∗ max b2     if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≤ 0 .

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Now assume that b = b1 ⊗ tλ ⊗ u −∞ . If b is extremal, wt (b )+λ if εi (b) = 0, f˜i i 1 i b1 ⊗ tλ ⊗ u −∞ Si b = max −ϕ (b )−λ i i 1 e˜i b1 ⊗ tλ ⊗ e˜i u −∞ if ϕi (b) = 0. If b∗ is extremal, ( Si∗ b

=

if εi∗ (b) = 0,

f˜i∗ −λi b1 ⊗ tsi λ ⊗ u −∞ ∗ λi −εi∗ (b1 )

e˜i∗ max b1 ⊗ tsi λ ⊗ e˜i

u −∞

if ϕi∗ (b) = 0.

Acknowledgment. The author would like to thank Anne Schilling who kindly provided a proof of the formula (6.2). References [1]

[2] [3]

[4] [5]

[6]

[7] [8] [9] [10] [11] [12]

T. AKASAKA and M. KASHIWARA, Finite-dimensional representations of quantum

affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), 839–867. MR 99d:17017 117, 118, 123, 132, 153, 154 J. BECK, V. CHARI, and A. PRESSLEY, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), 455–487. MR 2000g:17013 169 V. G. DRINFELD, “Quantum groups” in Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), Vol. 1, Amer. Math. Soc., Providence, 1987, 798–820. MR 89f:17017 147 , A new realization of Yangians and the quantum affine algebras, Soviet Math. Dokl. 36, no. 2 (1988), 212–216. MR 88j:17020 118 DZH. GASPER and M. RAKHMAN, Basic Hypergeometric Series (in Russian), “Mir,” Moscow, 1993, MR 95d:33017; English translation in Encyclopedia Math. Appl. 35, Cambridge Univ. Press, Cambridge, 1990. MR 91d:33034 171, 172 S.-J. KANG, M. KASHIWARA, K. MISRA, T. MIWA, T. NAKASHIMA, and A. NAKAYASHIKI, “Affine crystals and Vertex models” in Infinite Analysis (Kyoto, 1991), Part A, Adv. Ser. Math. Phys. 16, World Sci., River Edge, N.J., 1992, 449–484. MR 94a:17008 158 , Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), 499–607. MR 94j:17013 M. KASHIWARA, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516. MR 93b:17045 119, 120, 122 , The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), 839–858. MR 95b:17019 121 , Global crystal bases of quantum groups, Duke Math. J. 69 (1993), 455–485. MR 94b:17024 122 , Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), 383–413. MR 95c:17024 117, 119, 120, 123, 135 , “On crystal bases” in Representations of Groups (Banff, Alberta, 1994), CMS Conf. Proc. 16, Amer. Math. Soc., Providence, 1995, 155–197. MR 97a:17016 119, 120


[13]

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M. KASHIWARA, T. MIWA, J.-U. H. PETERSEN, and C. M. YUNG, Perfect crystals and

q-deformed Fock spaces, Selecta Math. (N.S.) 2 (1996), 415–499. MR 98f:17012 118, 160, 161, 164, 166, 167 B. LECLERC and J.-Y. THIBON, Canonical bases of q-deformed Fock spaces, Internat. Math. Res. Notices 1996, 447–456. MR 97h:17023 118, 169 , “Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials” in Combinatorial Methods in Representation Theory (Kyoto, 1998), Adv. Stud. Pure Math. 28, Kinokuniya, Tokyo, 2000, 155–200. MR 1 864 481 118, 169 P. LITTELMANN, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), 499–525. MR 96m:17011 121 G. LUSZTIG, Introduction to Quantum Groups, Progr. Math. 110 Birkhäuser, Boston, 1993. MR 94m:17016 148 , “Quantum groups at v = ∞” in Functional analysis on the Eve of the 21st century (New Brunswick, N.J., 1993), Vol. 1, Progr. Math. 131 Birkhäuser, Boston, 1995, 199–221. MR 97g:17014 169 O. SCHIFFMANN, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 2000, 413–440. MR 2001g:16032 169 D. UGLOV, “Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials” in Physical Combinatorics (Kyoto, 1999), ed. M. Kashiwara and T. Miwa, Progr. Math. 191, Birkhäuser, Boston, 2000, 249–299. MR 2001k:17030 169 M. VARAGNOLO and E. VASSEROT, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267–297. MR 2001c:17029 118, 169 , Standard modules of quantum affine algebras, Duke Math. J. 111 (2002), 509–533. 118

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606–8502, Japan.


CUSPIDALITY OF SYMMETRIC POWERS WITH APPLICATIONS HENRY H. KIM and FREYDOON SHAHIDI

Abstract The purpose of this paper is to prove that the symmetric fourth power of a cusp form on GL(2), whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the Ramanujan1/9 Petersson and Sato-Tate conjectures. In particular, we establish the bound qv for unramified Hecke eigenvalues of cusp forms on GL(2). Over an arbitrary number field, this is the best bound available at present. 1. Introduction In this paper we prove a criterion for cuspidality of the fourth symmetric powers of cusp forms on GL(2), whose existence was established earlier by the first author. As a consequence, we show that a cuspidal representation has a noncuspidal symmetric fourth power if and only if it is of either dihedral, tetrahedral, or octahedral type. We then prove a number of corollaries toward both the Ramanujan-Petersson and Sato-Tate conjectures for cusp forms on GL(2) by establishing analytic properties of several new symmetric power L-functions attached to them. N More precisely, let A be the ring of adeles of a number field F. Let π = v πv be a cuspidal automorphic representation of GL2 (A) with central character ωπ . Fix a positive integer m, and let Symm : GL2 (C) −→ GLm+1 (C) be the mth symmetric power representation of GL2 (C) on symmetric tensors of rank m (cf. [28], [30]). By the local Langlands correspondence (see [4], [5], [17]), Symm (πv ) is well defined for every v. Then Langlands functoriality in this case is equivalent to the fact N m that Symm (π) = v Sym (πv ) is an automorphic representation of GLm+1 (A). It is convenient to introduce Am (π) = Symm (π ) ⊗ ωπ−1 (denoted by Adm (π) in [28]). DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 30 March 2001 2000 Mathematics Subject Classification. Primary 11F30, 11F70, 11R42 Kim’s work partially supported by National Science Foundation grant numbers DMS9988672 and DMS9729992 (at the Institute for Advanced Study) and by the Clay Mathematics Institute, as a Prize Fellow. Shahidi’s work partially supported by National Science Foundation grant number DMS9970156 and by the Clay Mathematics Institute, as a Prize Fellow. 177

178

KIM and SHAHIDI

If m = 2, A2 (π) = Ad(π) and it is the well-known Gelbart-Jacquet lift in [2]. If m = 3, we proved in [13] and [12] that Sym3 (π) is an automorphic representation of GL4 (A) and gave a criterion for when it is cuspidal. In [10], the first author proved that Sym4 (π) is an automorphic representation of GL5 (A). If Sym3 (π) is cuspidal, Sym4 (π)is either cuspidal or unitarily induced from cuspidal representations of GL2 (A) and GL3 (A). In this paper, we give a criterion for when Sym4 (π ) is cuspidal. More precisely, we have the following. 3.3.7 Sym4 (π) ⊗ ωπ−1 is a cuspidal representation of GL5 (A), except in the following three cases: (1) π is monomial; (2) π is not monomial and A3 (π) is not cuspidal; this is the case when there exists a nontrivial grössencharacter µ such that Ad(π) ' Ad(π) ⊗ µ; (3) A3 (π) is cuspidal, but there exists a nontrivial quadratic character η such that A3 (π) ' A3 (π ) ⊗ η, or, equivalently, there exists a nontrivial grössencharacter χ of E such that Ad(π E ) ' Ad(π E ) ⊗ χ , where E/F is the quadratic extension determined by η and π E is the base change of π. In this case, A4 (π) = σ1 σ2 , where σ1 = π(χ −1 ) ⊗ ωπ and σ2 = Ad(π) ⊗ (ωπ η). Cases (1), (2), and (3) are equivalent to π being of dihedral, tetrahedral, and octahedral type, respectively. THEOREM A4 (π ) =

We give several applications of the cuspidality of third and fourth symmetric powers. First, following Ramakrishnan [23], we prove that given a cuspidal representation of GL2 (A), the set of tempered places has lower Dirichlet density of at least 34/35. Next, we prove the meromorphic continuation and a functional equation for each of the sixth, seventh, eighth, and ninth symmetric power L-functions for cuspidal representations of GL2 (A). An immediate corollary (cf. [27, Lemma 5.8]) is that if N πv is an unramified local component of a cuspidal representation π = v πv , then −1/9 1/9 qv < |αv |, |βv | < qv , where diag(αv , βv ) is the Satake parameter for πv . The archimedean analogue of 1/9, using this approach, is proved in [11]. When F = Q, using the ideas in [19], the bound 1/9 can be improved to 7/64 + ε, ∀ε > 0. This is the subject of an appendix in [10] by Kim and Sarnak. For an arbitrary number field, 1/9 remains the best bound available at present. Finally, we prove that partial fifth, sixth, seventh, and eighth symmetric power L-functions attached to a cuspidal representation π of GL2 (A) with trivial central character such that Sym4 (π) is cuspidal are all invertible at s = 1, and we apply this fact to the Sato-Tate conjecture (see [25]), following Serre’s method (see [30, Appendix]). Namely, we show that for every > 0 there are sets T + and T − of

SYMMETRIC POWERS

179

positive lower (Dirichlet) densities such that av > 2 cos(2π/11) − for all v ∈ T + , and av < −2 cos(2π/11) + for all v ∈ T − , where av = αv + βv . Note that 2 cos(2π/11) = 1.68 . . . . 2. Cuspidality of the symmetric cube Suppose π is a cuspidal representation of GL2 (A). We review the properties of the symmetric cube Sym3 (π) (see [13]). Recall that A3 (π) = Sym3 (π) ⊗ ωπ−1 . 2.1. π a monomial cuspidal representation That is, π ⊗ η ' π for a nontrivial grössencharacter η. Then η2 = 1 and η determines a quadratic extension E/F. According to [14], there is a grössencharacter χ of E such that π = π(χ), where π(χ) is the automorphic representation whose local factor at v is the one attached to the representation of the local Weil group induced from χv . Let χ 0 be the conjugate of χ by the action of the nontrivial element of the Galois group. Then the Gelbart-Jacquet lift (adjoint) of π is given by Ad(π ) = π(χχ 0

−1

) η.

There are two cases. Case 1: χ χ 0 −1 factors through the norm. That is, χχ 0 −1 = µ ◦ N E/F for a grössencharacter µ of F. Then π(χχ 0 −1 ) is not cuspidal. In fact, π(χχ 0 −1 ) = µ µη. In this case, −1 A3 (π) = π(χχ 0 ) π = (µ ⊗ π ) (µη ⊗ π ). Case 2: χ χ 0 −1 does not factor through the norm. In this case, π(χχ 0 −1 ) is a cuspidal representation. Then A3 (π) = π(χχ 0

−1

) π = π(χ 2 χ 0

−1

) π.

Here we use the fact that π(χ) E = χ χ 0 (see [24, Proposition 2.3.1]) and that π 0 π = I FE (π E0 ⊗ χ ) if π = π(χ) (see [24, §3.1]). 2.2. π not monomial In this case, Ad(π) is a cuspidal representation of GL3 (A). We recall from [13] the following. THEOREM 2.2.1 Let σ be a cuspidal representation of GL2 (A). Then the triple L-function L S (s, Ad(π) × π × σ ) has a pole at s = 1 if and only if σ ' π ⊗ χ and Ad(π ) ' Ad(π ) ⊗ (ωπ χ) for some grössencharacter χ .

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By Theorem 2.2.1, we have the following. 2.2.2 Let π be a nonmonomial cuspidal representation of GL2 (A). Then A3 (π ) is not cuspidal if and only if there exists a nontrivial grössencharacter µ such that Ad(π ) ' Ad(π ) ⊗ µ. In that case, THEOREM

A3 (π) = (π ⊗ µ) (π ⊗ µ2 ). 3. Cuspidality of the symmetric fourth Suppose π is a cuspidal representation of GL2 (A F ). Let A4 (π ) = Sym4 (π ) ⊗ ωπ−1 . We review the properties of Sym4 (π) (see [10]). 3.1. π monomial Suppose π is a monomial cuspidal representation given by π = π(χ). Then the Gelbart-Jacquet lift of π is given by Ad(π ) = π(χχ 0 −1 ) η. Case 1: χ χ 0 −1 factors through the norm. Then (see Section 2.1) since ∧2 (A3 (π)) = A4 (π) ⊕ ωπ , A4 (π) = (π π ) ωπ = ωπ ωπ µωπ ηωπ µηωπ . We used the fact that η and µ are quadratic grössencharacters. Case 2: χ χ 0 −1 does not factor through the norm. Then (see Section 2.1) A4 (π) = π(χ 2 χ 0

−1

−1 ) π ωπ = π(χ 3 χ 0 ) π(χ 2 ) ωπ .

3.2. π a nonmonomial representation such that Sym3 (π) is not cuspidal This is the case when there exists a nontrivial grössencharacter µ such that Ad(π ) ' Ad(π) ⊗ µ. Note that µ3 = 1. Then A3 (π) = (π ⊗ µ) (π ⊗ µ2 ). Hence ∧2 (A3 π) = Sym2 (π ) ωπ ωπ µ ωπ µ2 . So A4 (π) = Sym2 (π ) ωπ µ ωπ µ2 . PROPOSITION 3.2.1 Suppose π is a nonmonomial representation such that A3 (π) is not cuspidal; that is, Ad(π) ' Ad(π) ⊗ µ. Then L(s, π, Sym4 ⊗ ωπ−1 ) has a pole at s = 1 if and only if ωπ = µ or µ2 . In particular, if ωπ = 1, L(s, π, Sym4 ) is holomorphic at s = 1.

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3.3. Both Ad(π) and Sym3 (π) cuspidal Throughout this paper S is always a set of places of F such that for v 6∈ S every representation is unramified. The first author showed in [10] that ∧2 (A3 (π )) is an automorphic representation of GL6 (A) and that ∧2 (A3 (π )) = A4 (π) ωπ . Hence A4 (π) is an automorphic representation of GL5 (A), either cuspidal or induced from cuspidal representations of GL2 (A) and GL3 (A). We want to give a criterion for when A4 (π) is cuspidal. First we note that (1) L S s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 = L S s, σ × A4 (π) L S (s, σ ⊗ ωπ ). LEMMA 3.3.1 A4 (π ) is not cuspidal if and only if

L S (s, σ ⊗ A3 (π), ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1 for some cuspidal representation σ of GL2 (A). Proof Since A4 (π ) is either cuspidal or induced from cuspidal representations of GL2 (A) and GL3 (A), A4 (π) is not cuspidal if and only if L S (s, σ × A4 (π )) has a pole at s = 1 for some cuspidal representation σ of GL2 (A). Our assertion follows from (1) since L S (s, σ ⊗ ωπ ) is invertible at s = 1. In order to find a criterion for the pole of L(s, σ ⊗ A3 (π), ρ2 ⊗ ∧2 ρ4 ), we need the following unpublished result of H. Jacquet, I. Piatetski-Shapiro, and J. Shalika (cf. [22]). THEOREM 3.3.2 (Jacquet, Piatetski-Shapiro, and Shalika) Let π be a cuspidal automorphic representation of GL4 (A) such that there exist a grössencharacter χ and a finite set S of places as above for which L S (s, π, ∧2 ⊗ χ −1 ) has a pole at s = 1. Then there exists a globally generic cuspidal automorphic representation τ of GSp4 (A) with central character χ such that π is the Langlands functorial lift of τ under the natural embedding L GSp4 = GSp4 (C) ,→ GL4 (C).

The following paragraph is a brief sketch of the steps of the proof of the theorem, which we are including at a referee’s suggestion. We thank Dinakar Ramakrishnan for helping us with its preparation. Theorem 3.3.2 is proved using the dual reductive pair (GO6 (A), GSp4 (A)). More precisely, one considers the low-rank isogeny of SL4 and SO6 to lift a cuspidal repreN sentation π = v πv of GL4 (A) to one, still denoted by π, on GO6 (A), provided that the central character of πv is trivial on ±I for each v. One can then compute the theta lift of π to an automorphic representation of GSp4 (A) by integrating functions in the

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space of π in the usual way against the θ -function, a function on the two-fold cover of GSp12 (A). It is this integral that, in view of [6], is in fact equal to the residue of L S (s, π, ∧2 ⊗ χ −1 ) at s = 1, where χ is the central character of the theta lift. Hence the nonvanishing of the theta lift of π to GSp4 (A) is equivalent to the existence of a pole for L S (s, π, ∧2 ⊗ χ −1 ) at s = 1. Here S is a finite set of places for which v ∈ /S implies that πv is unramified. The lift is irreducible and globally generic. We now look at how the Satake parameter behaves under the map L GSp4 = GSp GSp4 (C) ,→ GL4 (C). Suppose τv is an unramified representation given by Ind B 4 µ ⊗ ν ⊗ λ, where µ, ν, λ are unramified quasicharacters of Fv× and µ ⊗ ν ⊗ λ is the character of the torus which assigns to diag(x, y, t y −1 , t x −1 ) the value µ(x)ν(y)λ(t). Note that the central character is χv = µνλ2 . Then the Satake parameter corresponding to τv is (see, e.g., [31, p. 95]) diag(µνλ, µλ, λ, νλ). Here we identify µ with µ($ ); we do the same with ν and λ. The Satake parameter for ∧2 (πv ) is diag(µ2 νλ2 , µνλ2 , µν 2 λ, µλ2 , νλ2 , µνλ2 ) = diag(χv µ, χv ν, χv , χv µ−1 , χv ν −1 , χv ). Hence, if σv is an unramified representation of GL2 (Fv ) given by π(η1 , η2 ), then L(s,σv ⊗ πv , ρ2 ⊗ ∧2 ρ4 )−1 =

2 Y (1 − χv ηi µ±1 qv−s )(1 − χv ηi ν ±1 qv−s )(1 − χv ηi qv−s ) i=1 2 Y × (1 − χv ηi qv−s ) i=1

= L s, (σv ⊗ χv ) × τv

−1

L(s, σv ⊗ χv )−1 .

(2)

Here L(s, (σv ⊗χv )×τv ) is the degree 10 Rankin-Selberg L-function for GL2 × GSp4 . Note that if τ 0 is any irreducible constituent of τ |Sp4 (A) , then L s, (σv ⊗ χv ) × τv = L s, (σv ⊗ χv ) × τv0 . We now apply the above observation to A3 (π ), where π is a cuspidal representation of GL2 (A). Since ∧2 (A3 (π)) = A4 (π) ωπ , L S (s, A3 (π ), ∧2 ⊗ ωπ−1 ) has a pole at s = 1. So there exists a generic cuspidal representation τ of GSp4 (A) with central character ωπ . Let τ 0 be any irreducible constituent of τ |Sp4 (A) . Then by (1) and (2), we have L S s, σ × A4 (π) = L S s, (σ ⊗ ωπ ) × τ 0 .

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Recall that if we consider the maximal Levi subgroup GL2 × Sp4 ⊂ Sp8 with the cuspidal representation (σ ⊗ ωπ ) ⊗ τ 0 (see [27]), we obtain the following L-function as a normalizing factor in the constant term of the Eisenstein series: L S s, (σ ⊗ ωπ ) × τ 0 L S (2s, ωσ ωπ2 ). Note that if (ωσ ωπ2 )2 6= 1, (σ ⊗ ωπ ) ⊗ τ 0 is not self-contragredient. Hence we have the following. PROPOSITION 3.3.3 If (ωσ ωπ2 )2 6= 1, then L S (s, (σ ⊗ ωπ ) × τ 0 ) is holomorphic at s = 1.

Proof By following the proof of [8, Theorem 3.2], we can show that the completed Lfunction L(s, (σ ⊗ ωπ ) × τ 0 ) is entire. Since (σ ⊗ ωπ ) ⊗ τ 0 is not self-contragredient, by [7, Proposition 2.1] the global intertwining operator M(s, (σ ⊗ ωπ ) ⊗ τ 0 , w0 ) is holomorphic for Re s > 0. Using [8, Proposition 3.4], the local normalized intertwining operators N (s, (σv ⊗ωπv )⊗τv0 , w0 ) are holomorphic and nonzero for Re s ≥ 1/2. Hence we see that the L-function L(s, (σ ⊗ωπ )×τ 0 )L(2s, ωσ ωπ2 ) is holomorphic for Re s ≥ 1/2. Since L(2s, ωσ ωπ2 ) has no zeros for Re s ≥ 1/2, L(s, (σ ⊗ ωπ ) × τ 0 ) is holomorphic for Re s ≥ 1/2. Our assertion follows from the functional equation. Using the integral representation, we have the following. THEOREM 3.3.4 (Ginzburg-Rallis-Soudry [3]) If ωσ ωπ2 = 1, then L S (s, (σ ⊗ ωπ ) × τ 0 ) is holomorphic at s = 1.

Proof This follows immediately from the integral representation for L S (s, (σ ⊗ ωπ ) × τ 0 ). The possible poles come from the poles of the Eisenstein series attached to (GL2 , σ ⊗ ωπ ) for the split group SO5 , which in turn come from the poles of symmetric square L-function L S (s, σ ⊗ ωπ , Sym2 ). The last L-function is entire if ωσ ωπ2 = 1. COROLLARY 3.3.5 If ωσ ωπ2 = 1, or if (ωσ ωπ2 )2 6= 1, then L S (s, σ × A4 (π)) is holomorphic at s = 1. PROPOSITION 3.3.6 Let π be a cuspidal representation of GL2 (A) such that both 5 = Ad(π ) and A3 (π ) are cuspidal. Then the following are equivalent. (1) A4 (π) is not cuspidal.

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(2)

There exists a quadratic extension E/F such that A3 (π) ' A3 (π ) ⊗ η, where η is the quadratic grössencharacter attached to E/F via class field theory. This means that the base change (A3 (π)) E is not cuspidal. (Note that A3 (π E ) ' (A3 (π )) E .) There exists a quadratic extension E/F such that 5 E ' 5 E ⊗ χ for a nontrivial grössencharacter χ of E, where 5 E is the base change of 5 to E.

(3)

Proof Consider L S s, (σ ⊗ ωπ ) × 5 × 5 = L S s, (σ ⊗ ωπ ) × 5 L S s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 . Since L S (s, (σ ⊗ ωπ ) × 5) does not have a pole or a zero at s = 1, L S (s, (σ ⊗ ωπ ) × 5 × 5) has a pole at s = 1 if and only if L S (s, σ ⊗ A3 (π), ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1. Statement (1) implies statement (2). Since A4 (π) is not cuspidal, L S (s, σ ⊗ 3 A (π ), ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1 for some σ by Lemma 3.3.1. Hence, by Corollary 3.3.5, ω2 = 1, ω 6= 1, where ω = ωσ ωπ2 . Let E/F be the quadratic extension attached to ω via class field theory. Let (A3 (π)) E be the base change of A3 (π ). Consider the equality L S s,σ E ⊗ A3 (π) E , ρ2 ⊗ ∧2 ρ4 = L S s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 L S s, (σ ⊗ ω) ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 . Note that L S s, σ E ⊗ (A3 (π)) E , ρ2 ⊗ ∧2 ρ4 has a pole at s = 1 if and only if L S (s, (σ E ⊗ ωπ E ) × 5 E × 5 E ) has a pole at s = 1. Suppose that (A3 (π)) E is cuspidal. Then 5 E 6 ' 5 E ⊗ χ for any nontrivial character by Theorem 2.2.2. If σ is not monomial, then σ E is cuspidal, and hence L S (s, σ E ⊗(A3 (π)) E , ρ2 ⊗∧2 ρ4 ) is holomorphic at s = 1. If σ is monomial, then σ E is an automorphic representation induced from two grössencharacters. Hence again L S (s, (σ E ⊗ ωπ E ) × 5 E × 5 E ) is holomorphic at s = 1. Therefore L S (s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 ) is holomorphic at s = 1 for any σ . This is a contradiction. Statement (2) is equivalent to statement (3). Suppose (A3 (π )) E is not cuspidal. Since (A3 (π)) E is equivalent to A3 (π E ), A3 (π E ) is not cuspidal. So Ad(π E ) ' Ad(π E )⊗χ for a nontrivial grössencharacter of E. Since 5 E is equivalent to Ad(π E ), we have 5 E ' 5 E ⊗ χ for a nontrivial grössencharacter χ of E. Statement (3) implies statement (1). Let σ 0 = σ ⊗ ωπ = π(χ ) be the monomial representation of GL2 (A F ) attached to χ . Let η be the quadratic character attached to E/F. Then σ E0 = χ χ 0 . Consider the equality L S (s, σ E0 × 5 E × 5 E ) = L S (s, σ 0 × 5 × 5)L S s, (σ 0 ⊗ η) × 5 × 5 .

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Then L S (s, σ E0 × 5 E × 5 E ) = L S (s, (χ ⊗ 5 E ) × 5 E )L S (s, (χ 0 ⊗ 5 E ) × 5 E ). Since χ ⊗ 5 E ' 5 E , L S (s, (χ ⊗ 5 E ) × 5 E ) has a pole at s = 1. Hence either L(s, σ 0 × 5 × 5) or L S (s, (σ 0 ⊗ η) × 5 × 5) has a pole at s = 1. This implies that either L S (s, σ ⊗ τ, ρ2 ⊗ ∧2 ρ4 ) or L S (s, (σ ⊗ η) ⊗ τ, ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1. Hence A4 (π) is not cuspidal. By the above proposition, we see the following. THEOREM 3.3.7 A4 (π ) = Sym4 (π)

⊗ ωπ−1 is a cuspidal representation of GL5 (A) except in the fol-

lowing three cases: (1) π is monomial; (2) π is not monomial and A3 (π) is not cuspidal; this is the case when there exists a nontrivial grössencharacter µ such that Ad(π) ' Ad(π) ⊗ µ; (3) A3 (π) is cuspidal and there exists a nontrivial quadratic character η such that A3 (π) ' A3 (π ) ⊗ η, or, equivalently, there exists a nontrivial grössencharacter χ of E such that Ad(π E ) ' Ad(π E ) ⊗ χ , where E/F is the quadratic extension determined by η. In this case, A4 (π) = σ1 σ2 , where σ1 = π(χ −1 ) ⊗ ωπ and σ2 = Ad(π ) ⊗ (ωπ η). Proof We only need to prove the last assertion. By the proof of Proposition 3.3.6, L S (s, σ˜ 1 × ^ 3 (π) × (A3 (π) ⊗ η)). It has a pole at A4 (π )) has a pole at s = 1. Consider L(s, A s = 1 since A3 (π) ' A3 (π) ⊗ η. By formal calculation, ^ 3 (π) × A3 (π) ⊗ η L S s, A = L S s, π, Sym6 ⊗ (ωπ−3 η) L S s, A4 (π) ⊗ (ωπ−1 η) × L S s, Ad(π) ⊗ η L S (s, η) = L S s, Ad(π) ⊗ (ωπ−1 η) × A4 (π) L S (s, η). The L-function L(s, η) has no zeros at s = 1, and therefore L S s, Ad(π) ⊗ (ωπ−1 η) × A4 (π) has a pole at s = 1. Hence σ2 = Ad(π) ⊗ (ωπ η). We note that since χ is a nontrivial cubic character, χ|A∗F = 1, and the central character of π(χ) is η · χ|A∗F = η. (of the proof ) Let π be a cuspidal representation of GL2 (A F ) such that A3 (π) is cuspidal. If A4 (π ) COROLLARY

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is not cuspidal, then L(s, π, Sym6 ⊗ (ωπ−3 η)) has a pole at s = 1. Here η is as in Theorem 3.3.7(3). Finally, based on Langlands’s calculations in [18], it is reasonable to claim the following. CONJECTURE

Let π be a nonmonomial cuspidal representation of GL2 (A). Then (1) Sym3 (π) is not cuspidal if and only if π is of tetrahedral type; (2) Sym3 (π) is cuspidal, but Sym4 (π) is not, if and only if π is of octahedral type; (3) Sym4 (π) and Sym5 (π ) are cuspidal, but Sym6 (π) is not, if and only if π is of icosahedral type. Remark. The final form of part (3) of the conjecture is an outcome of a number of communications with J.-P. Serre as well as calculations done in [9]. The purpose of our next proposition is to demonstrate the first two parts of the conjecture. PROPOSITION 3.3.8 Let π be a nonmonomial cuspidal representation of GL2 (A). Then (1) Sym3 (π) is not cuspidal if and only if π is of tetrahedral type; (2) Sym3 (π) is cuspidal, but Sym4 (π) is not, if and only if π is of octahedral type.

Proof Part (1) is [13, Lemma 6.5]. Part (2) is proved the same way. In fact, observe first that by Proposition 3.3.6(3) there exists a quadratic extension E/F such that 5 E ∼ = 5E ⊗ χ for a nontrivial grössencharacter χ . Notice that 5 E = Ad(π E ) and that therefore Ad(π E ) ∼ = Ad(π E ) ⊗ χ . By Theorem 2.2.2, Sym3 (π E ) is not cuspidal, and therefore, by part (1), π E is of tetrahedral type. Consequently, there exists a two-dimensional tetrahedral representation σ E of W E such that π E = π(σ E ). Since σ E is invariant under Gal(E/F), it can be extended to a two-dimensional continuous representation σ of W F , which is now octahedral. Let π 0 = π(σ ), which is of octahedral type. Clearly, π E0 ∼ = π E , and therefore π 0 ∼ = π ⊗ ηa for some a = 0, 1, where η is the grössencharacter attached to E/F. But σ is unique only up to twisting by a power of η, and therefore by changing the choice of η if necessary, we have π ∼ = π 0 = π(σ ). We are done.

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4. Applications We give several applications of cuspidality of third and fourth symmetric powers. In N this section we let π = v πv be a cuspidal representation of GL2 (A) unless otherwise specified. Let πv be an unramified local component with the Satake parameter diag(αv , βv ). Set av (π) = αv + βv . THEOREM 4.1 Let S(π ) be the set of places where πv is tempered. Then

34 δ S(π ) ≥ . 35 Proof We follow [23]. Let ω be the central character of π . Let av = av (π ). Then by direct computation we see that av (Sym2 (π)) = av2 −ωv and av (A4 (π)) = ωv−1 av4 −3av2 +ωv . For m, k, l nonnegative integers to be chosen below, let η = m[ω] k Sym2 (π) l A4 (π). Then av (η) = mωv + k(av2 − ωv ) + l(ωv−1 av4 − 3av2 + ωv ) = (m − k + l)ωv + (k − 3l)av2 + lωv−1 av4 . Let T (π, 2) = {v| |av | ≥ 2}. Then note that for v ∈ T (π, 2), αv = αq r , βv = for |α| = 1 and r ≥ 0 (see [23, Claim 4.6]). So except for finitely many places, S(π) = {v| |av | ≤ 2}. If v ∈ T (π, 2), av = α(q r + q −r ). So av (η) ≥ m + 3k + 5l.

αq −r ,

Hence v ∈ T (η, m + 3k + 5l). Thus, by [23, (4.4)], m2 + k2 + l2 δ T (η, m + 3k + 5l) ≤ . (m + 3k + 5l)2 This holds for every choice of (nonnegative) triples (m, k, l). It can be verified that the minimum of the right-hand side occurs when k = 3m, l = 5m, yielding 1 δ T (π, 2) ≤ . 35 Higher symmetric power L-functions and the Sato-Tate conjecture N In the following, let π = v πv be a cuspidal representation of GL2 (A), and let S be a finite set of places, including the archimedean ones, such that πv is unramified for

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v ∈ / S. Let diag(αv , βv ) be the Satake parameter for πv for v ∈ / S. Then the partial mth symmetric power L-function is defined to be L(s, πv , Sym )

m −1

m Y = (1 − αvm−i βvi qv−s ), i=0

L S (s, π, Sym ) = m

Y

L(s, πv , Symm ).

v ∈S /

PROPOSITION 4.2 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then L S (s, π, Sym5 ) is invertible for Re s ≥ 1; that is, it is holomorphic and nonzero for Re s ≥ 1.

Proof Consider L S s, Sym2 (π ) × Sym3 (π) = L S (s, π, Sym5 )L S s, Sym3 (π) ⊗ ωπ L S (s, π ⊗ ωπ2 ). Note that L S (s, Sym3 (π) ⊗ ωπ )L S (s, π ⊗ ωπ2 ) is invertible for Re s ≥ 1. Note also that the left-hand side is invertible for Re s ≥ 1. Hence our result follows. PROPOSITION 4.3 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then every partial sixth symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. Moreover, if ωπ3 = 1, they are all invertible for Re s ≥ 1.

Proof By standard calculations, L S s, Sym3 (π) × Sym3 (π) = L S (s, π, Sym6 )L S s, Sym4 (π) ⊗ ωπ L S s, Sym2 (π) ⊗ ωπ2 L S (s, ωπ3 ). Meromorphic continuation and a functional equation follow. If ωπ3 = 1, then Sym3 (π ) is self-contragredient. This implies that the lefthand side has a pole at s = 1, while in the right-hand side L(s, ωπ3 ) has a pole at s = 1. By assumption, Sym4 (π) is either cuspidal or of the form σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. Hence L S (s, Sym4 (π) ⊗ ωπ )L S (s, Sym2 (π) ⊗ ωπ2 ) is invertible for Re s ≥ 1. This implies our last claim.

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PROPOSITION 4.4 If Sym4 (π ) is cuspidal, then partial sixth symmetric power L-functions are all invertible for Re s ≥ 1.

Proof Consider the equality L S s, Sym2 (π) × Sym4 (π) = L S (s, π, Sym6 )L S s, Sym4 (π) ⊗ ωπ L S s, Sym2 (π ) ⊗ ωπ2 . Since the left-hand side and L S (s, Sym4 (π) ⊗ ωπ )L S (s, Sym2 (π ) ⊗ ωπ2 ) are invertible for Re s ≥ 1, the same holds for L S (s, π, Sym6 ). COROLLARY

Let π be a cuspidal representation of GL2 such that Sym3 (π) is cuspidal. If ωπ3 6= 1 and Sym3 (π) is self-contragredient, then L(s, π, Sym6 ) has a pole at s = 1. PROPOSITION 4.5 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then every partial seventh symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. Moreover, they are all invertible for Re s ≥ 1.

Proof By standard calculations, L S s, Sym3 (π) × Sym4 (π ) = L S (s, π, Sym7 )L S (s, π, Sym5 ⊗ ωπ )L S s, Sym3 (π) ⊗ ωπ2 L S (s, π ⊗ ωπ3 ). Meromorphic continuation and functional equations then follow. By assumption, Sym4 (π ) is either cuspidal or of the form σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. Hence the left-hand side is invertible for Re s ≥ 1. In the proof of Proposition 4.2, using L S (s, (Sym2 (π) ⊗ ωπ ) × Sym3 (π )), one can see that L S (s, π, Sym5 ⊗ ωπ ) is invertible for Re s ≥ 1. From this, we obtain our assertion. 4.6 Let π be a cuspidal representation of GL2 such that Sym3 (π) is cuspidal. Then every partial eighth symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. If Sym4 (π) is cuspidal and ωπ4 = 1, then they are all invertible for Re s ≥ 1. PROPOSITION

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Proof By standard calculations, L S s, Sym4 (π) × Sym4 (π) = L S (s, π, Sym8 )L S (s, π, Sym6 ⊗ ωπ )L S s, Sym4 (π ) ⊗ ωπ2 × L S s, Sym2 (π) ⊗ ωπ3 L S (s, ωπ4 ).

This proves the meromorphic continuation and functional equation. If Sym4 (π ) is cuspidal and ωπ4 = 1, then Sym4 (π) is self-contragredient. Hence the left-hand side has a simple pole at s = 1, while in the right-hand side L(s, ωπ4 ) has a simple pole at s = 1. By Proposition 4.4, and by considering L S (s, Sym2 (π) × (Sym4 (π ) ⊗ ωπ )), we see that L S (s, π, Sym6 ⊗ ωπ )L S s, Sym4 (π) ⊗ ωπ2 L S s, Sym2 (π ) ⊗ ωπ3 is invertible for Re s ≥ 1. This completes our claim. 4.7 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then every partial ninth symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. If Sym4 (π) is cuspidal, then L S (s, π, Sym9 ) has at most a simple pole or a simple zero at s = 1. If Sym4 (π ) is not cuspidal, then L S (s, π, Sym9 ) is invertible for Re s ≥ 1. PROPOSITION

Proof Suppose first that Sym4 (π) is cuspidal. Consider the case E 8 − 2 of [27]. Let M be a maximal Levi subgroup, and denote by A the connected component of its center. Since E 8 is simply connected, the derived group M D of M is simply connected as well, and hence M D = SL4 × SL5 . Thus M = (GL1 × SL4 × SL5 )/(A ∩ M D ), where A ∩ M D ' Z/20Z. Let πi , i = 1, 2, be cuspidal representations of GL4 (A) and GL5 (A) with central characters ωi , i = 1, 2, respectively. Let πi0 , i = 1, 2, be irreducible constituents of π1 |SL4 (A) and π2 |SL5 (A) , respectively. Then 6 = ω15 ω28 ⊗ π10 ⊗ π20 can be considered a cuspidal representation of M(A). We then get the L-function L(s, π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ) as our first L-function. In fact, there are five

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L-functions in the constant term of the Eisenstein series, namely, L S (s, 6, r1 ) = L S (s, π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ), L S (s, 6, r2 ) = L S s, π1 ⊗ (π˜ 2 ⊗ ω2 ), ∧2 ρ4 ⊗ ρ5 , L S (s, 6, r3 ) = L S s, π˜ 1 × (π2 ⊗ ω1 ω2 ) , L S (s, 6, r4 ) = L S (s, π˜ 2 , ∧2 ρ5 ⊗ ω1 ω22 ), L S (s, 6, r5 ) = L S (s, π1 ⊗ ω1 ω22 ). (See [27] for the trivial central character case and [11] for the general case for detailed calculations.) Each of the L-functions, especially L S (s, 6, r1 ), has a meromorphic continuation and satisfies a standard functional equation (see [27]). We apply the above to π1 = A3 (π ) and π2 = Sym4 (π ). By standard calculations, we have L S (s,π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ) = L S (s, π, Sym9 )L S (s, π, Sym7 ⊗ ωπ )L S (s, π, Sym5 ⊗ ωπ2 )2 2 × L S s, Sym3 (π) ⊗ ωπ3 L S (s, π ⊗ ωπ4 ). The meromorphic continuation and functional equation of L S (s, π, Sym9 ) now follow from those of L S (s, π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ). Moreover, 2 L S (s, π, Sym7 ⊗ ωπ )L S (s, π, Sym5 ⊗ ωπ2 )2 L S s, Sym3 (π) ⊗ ωπ3 L S (s, π ⊗ ωπ4 ) is invertible at s = 1 by Propositions 4.2 and 4.5. So it is enough to prove that L S (s, 6, r1 ) has at most a simple pole or simple zero at s = 1. Q5 By [26], the product i=1 L S (1 + is, 6, ri ) does not have a zero at s = 0. But none of the L-functions L S (s, 6, ri ), i = 3, 4, 5, has a pole at s = 1. In fact, they are all entire and have no zeros for Re s ≥ 1. (See [7] for the case r4 and ω1 ω22 = 1. The general case can be seen to be the same by observing that the twisted exterior square L-function appears as the normalizing factor of a certain Eisenstein series if we consider Spin2n (cf. [11]). Alternatively, by direct calculation, we have L S (s, 6, r4 ) = L S (s, π, Sym6 ⊗ ωπ15 )L S (s, Sym2 (π) ⊗ ωπ17 ). The necessary properties of L S (s, π, Sym6 ⊗ ωπ15 ), and in particular its invertibility at s = 1, are now obtained by considering L S (s, Sym2 (π ) × (Sym4 (π ) ⊗ ωπ15 )) in the proof of Proposition 4.4.) The second L-function, L S (s, 6, r2 ), appears as the first L-function in the case D8 − 3. Hence it has at most a simple pole at s = 1. Alternatively, by direct calculation, we see that, since ∧2 (A3 (π)) = A4 (π) ⊕ ωπ , L S s,π1 ⊗ (π˜ 2 ⊗ ω2 ), ∧2 ρ4 ⊗ ρ5 = L S s, Sym4 (π ) × Sym4 (π) ⊗ ωπ5 L S s, Sym4 (π) ⊗ ωπ7 .

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Hence L S (s, π1 ⊗ (π˜ 2 ⊗ ω2 ), ∧2 ρ4 ⊗ ρ5 ) has at most a simple pole at s = 1 and is invertible for Re s > 1. Therefore L S (s, 6, r1 ) has at most a simple zero at s = 1. Q5 By [27], the product i=1 L S (is, 6, ri ) has at most a simple pole at s = 1. Q5 However, i=2 L S (is, 6, ri ) has no zeros at s = 1. Therefore L S (s, 6, r1 ) has at most a simple pole at s = 1. Next, suppose Sym4 (π) is not cuspidal. Then Sym4 (π) = σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. In this case, by standard calculations, L S (s,π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ) = L S (s, π1 × σ1 × σ2 )L S (s, π1 ⊗ ωσ1 )L S s, π1 × (σ˜ 2 ⊗ ωσ2 ) . Here L S (s, π1 ⊗ ωσ1 )L S (s, π1 × (σ˜ 2 ⊗ ωσ2 )) is invertible for Re s ≥ 1. By [13], L S (s, π1 × σ1 × σ2 ) = L S s, π1 × (σ1 σ2 ) , where σ1 σ2 is the functorial product that is an automorphic representation of GL6 (A). Since σ1 is monomial (see Theorem 3.3.7), by the main theorem of [13], σ1 σ2 is either cuspidal or unitarily induced from two cuspidal representations of GL3 (A). Hence L S (s, π1 × σ1 × σ2 ) is invertible for Re s ≥ 1, and therefore the same conclusion holds for L S (s, π, Sym9 ). PROPOSITION 4.8 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Let diag(αv , βv ) be the Satake parameter for an unramified local component. Then 1/9 |αv |, |βv | < qv . If Sym4 (π) is not cuspidal, then the full Ramanujan conjecture is valid.

Proof If Sym4 (π ) is cuspidal, use Proposition 4.7 and [27, Lemma 5.8]. If Sym4 (π) is not cuspidal, then by Proposition 3.3.8, π is of Galois type for which |αv | = |βv | = 1. The following result coincides with Langlands’s calculations in [18]. PROPOSITION 4.9 Let π be a nonmonomial cuspidal representation of GL2 (A) with a trivial central character. Suppose m ≤ 9. (1) Suppose Sym3 (π) is not cuspidal. Then L S (s, π, Symm ) is invertible at s = 1, except for m = 6, 8; the L-functions L S (s, π, Sym6 ) and L S (s, π, Sym8 ) each have a simple pole at s = 1.

SYMMETRIC POWERS

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193

Suppose Sym3 (π) is cuspidal, but Sym4 (π) is not. Then L S (s, π, Symm ) is invertible at s = 1 for m = 1, . . . , 7 and m = 9; the L-function L S (s, π, Sym8 ) has a simple pole at s = 1.

Proof (1) By Theorem 2.2.2 and Section 3.2, Sym3 (π) = (π ⊗ µ) (π ⊗ µ2 ),

Sym4 (π) = Sym2 (π ) µ µ2 ,

where µ is a nontrivial grössencharacter such that Ad(π) ' Ad(π )⊗µ. We explicitly calculate L S (s, π, Symm ). Let 5 = Sym2 (π). Then L S (s, π, Sym3 ) = L S (s, π ⊗ µ)L S (s, π ⊗ µ2 ), L S (s, π, Sym4 ) = L S (s, 5)L S (s, µ)L S (s, µ2 ). They are both invertible for Re s ≥ 1. From the equality in Proposition 4.2, we have L S (s, π, Sym5 ) = L S (s, π)L S (s, π ⊗ µ)L S (s, π ⊗ µ2 ), which is clearly invertible for Re s ≥ 1. Using the equality in Proposition 4.3, we have L S (s, π, Sym6 ) = L S (s, 5)2 ·L S (s, 1). Since L(s, 5) is invertible at s = 1, L S (s, π, Sym6 ) has a simple pole at s = 1. From the equality in Proposition 4.5, we have L S (s, π, Sym7 ) = L S (s, π)2 L S (s, π ⊗ µ)L S (s, π ⊗ µ2 ). Hence it is invertible for Re s ≥ 1. For L S (s, π, Sym8 ), consider the equality in Proposition 4.4 with ωπ = 1; we have L S (s, 5 × 5) = L S (s, π, Sym6 )L S (s, µ)L S (s, µ2 ). Hence L S (s, 5 × 5) = L S (s, 5)2 L S (s, 1)L S (s, µ)L S (s, µ2 ). Then from the equality in Proposition 4.6, L S (s, π, Sym8 ) = L S (s, 5)2 L S (s, µ)L S (s, µ2 )L S (s, 1), and therefore L S (s, π, Sym8 ) has a simple pole at s = 1. For L S (s, π, Sym9 ), consider the equality in Proposition 4.7 with ωπ = 1; by standard calculations, we see that L S (s, π, Sym9 ) = L S (s, π ⊗ η)2 L S (s, π ⊗ η2 )2 L S (s, π ). We therefore conclude that L S (s, π, Sym9 ) is invertible for Re s ≥ 1. (2) In this case, Sym4 (π ) = σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. If ωπ = 1, then σ1 and σ2 are self-contragredient. We only have to discuss the case m = 8. Consider the equality in Proposition 4.6 with ωπ = 1 in which the left-hand side has a double pole at s = 1. But L S (s, π, Sym6 )L S (s, π, Sym4 )L S (s, π, Sym2 ) is invertible at s = 1. Thus L S (s, π, Sym8 ) has a simple pole at s = 1.

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Remark. The above proposition is no longer true if the central character is not trivial (see Proposition 3.2.1, Corollaries to Theorem 3.3.7 and Proposition 4.4). We now give an application of these properties of symmetric power L-functions to the Sato-Tate conjecture (see [25], [30]). This we do by following Serre’s method N (see [30, Appendix]). In what follows, let π = v πv be a cuspidal representation of GL2 (A) with a trivial central character such that Sym4 (π) is cuspidal. We also assume that π satisfies the Ramanujan-Petersson conjecture. Recall that we let av = αv + βv , where πv is an unramified local component with the Satake parameter diag(αv , βv ). We use exactly the same notation as in [30, Appendix]. First let us calculate Tn (x), the polynomial that gives the trace of the nth symmetric power of an element of SL2 (C) whose trace is x: T0 = 1,

T2 = x 2 − 1,

T1 = x,

T4 = x 4 − 3x 2 + 1,

T3 = x 3 − 2x,

T5 = x 5 − 4x 3 + 3x,

T7 = x 7 − 6x 5 + 10x 3 − 4x,

T6 = x 6 − 5x 4 + 6x 2 − 1,

T8 = x 8 − 7x 6 + 15x 4 − 10x 2 + 1,

T9 = x 9 − 8x 7 + 21x 5 − 20x 3 + 5x. Next recall the quantity P I (Tn ) = lim

N →∞

qv ≤N

Tn (av )

π(N )

= kn ,

where the nth symmetric power L-function has an order −kn at s = 1 and π(N ) is the number of places such that qv ≤ N . Hence I (T0 ) = 1 and I (Tn ) = 0 for n = 1, . . . , 8 by Propositions 4.2–4.6. Let I (T9 ) = k. By Proposition 4.7, we know that k ∈ {−1, 0, 1}. We now calculate I (x n ): I (x) = 0, I (x 5 ) = 0,

I (x 2 ) = 1,

I (x 6 ) = 5,

I (x 3 ) = 0,

I (x 7 ) = 0,

I (x 4 ) = 2,

I (x 8 ) = 14,

I (x 9 ) = k.

With notation as in [30, Appendix], d = 9 and m = 4. Using this, we can calculate the orthogonal polynomials, P0 , P1 , . . . , P4 to get P0 = 1,

P1 = T1 = x,

P2 = T2 = x 2 − 1,

P4 = T4 = x 4 − 3x 2 + 1, (1) (2)

P3 = T3 = x 3 − 2x,

P5 = T5 − k(x 4 − 3x 2 + 1).

There are three possibilities: √ √ k = 0: P5 = T5 = x 5 − 4x 3 + 3x; the roots are − 3, −1, 0, 1, 3; k = 1: P5 = T5 − T4 = x 5 − x 4 − 4x 3 + 3x 2 + 3x − 1; the roots are −2 cos(2kπ/11), k = 1, 2, 3, 4, 5;

SYMMETRIC POWERS

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195

k = −1: P5 = T5 + T4 = x 5 + x 4 − 4x 3 − 3x 2 + 3x + 1; the roots are 2 cos(2kπ/11), k = 1, 2, 3, 4, 5. Note that 2 cos(2π/11) ∼ = 1.68 . . . .

THEOREM 4.10 For every > 0, there are sets T + and T − of positive lower (Dirichlet) density such that av > 1.68 . . . − for all v ∈ T + and av < −1.68 . . . + for all v ∈ T − .

Remark. In [30, Appendix] and [23], where the √ third, fourth, and fifth symmetric power L-functions were used, the result is av > 2 − . In the next proposition, we do not assume the Ramanujan-Petersson conjecture for π. PROPOSITION 4.11 Let π be a cuspidal representation of GL2 (A) with a trivial central character. Then for every > 0, there exists a set T of positive lower (Dirichlet) density such that |av | > 1.68 . . . − for all v ∈ T .

Remark. Let σ be a two-dimensional continuous representation of W F , the Weil group of F/F. Assume that there exists an automorphic cuspidal representation π(σ ) of GL2 (A) preserving root numbers and L-functions for pairs (cf. [16], [13]). This is possible except perhaps when σ is icosahedral of special kind (cf. [13, §10]). When σ is of icosahedral type and π(σ ) exists, L(s, Sym4 σ ) is a five-dimensional irreducible, and therefore entire, Artin L-function. But it is not primitive since Sym4 σ is monomial (cf. [9]). We refer to [9] for an interesting application of this to automorphic induction. Acknowledgments. We would like to thank Joseph Shalika for his suggestion that we use an unpublished result of his with Hervé Jacquet and Ilya Piatetski-Shapiro in the proof of Corollary 3.3.5. Thanks are also due to Dinakar Ramakrishnan and JeanPierre Serre for many useful discussions and communications. Finally, the authors would like to thank the Institute for Advanced Study and particularly the organizers of the Special Year in the Theory of Automorphic Forms and L-functions, E. Bombieri, H. Iwaniec, R. P. Langlands, and P. Sarnak, for inviting them to participate. References [1]

J. ARTHUR and L. CLOZEL, Simple Algebras, Base Change, and the Advanced Theory

of the Trace Formula, Ann. of Math. Stud. 120, Princeton Univ. Press, Princeton, 1989. MR 90m:22041

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[8] [9] [10] [11] [12]

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[18]

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S. GELBART and H. JACQUET, A relation between automorphic representations of

´ GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), 471–552. MR 81e:10025 178 D. GINZBURG, S. RALLIS, and D. SOUDRY, L-functions for symplectic groups, Bull. Soc. Math. France 126 (1998), 181–244. MR 2000b:22017 183 M. HARRIS and R. TAYLOR, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Stud. 151, Princeton Univ. Press, Princeton, 2001. 177 G. HENNIART, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), 439–455. MR 2001e:11052 177 H. JACQUET and J. A. SHALIKA, “Exterior square L-functions” in Automorphic Forms, Shimura Varieties, and L-functions, II (Ann Arbor, Mich., 1988), ed. L. Clozel and J. Milne, Academic Press, Boston, 1990, 143–226. MR 90j:11003 182 H. KIM, Langlands-Shahidi method and poles of automorphic L-functions: Application to exterior square L-functions, Canad. J. Math. 51 (1999), 835–849. MR 2000f:11058 183, 191 , Langlands-Shahidi method and poles of automorphic L-functions, II, Israel J. Math. 117 (2000), 261–284. MR 2001i:11059a 183 , An example of a non-normal quintic automorphic induction, preprint, 2000. 186, 195 , Functoriality for the exterior square of GL4 and symmetric fourth of GL2 , preprint, 2000. 178, 180, 181 , On local L-functions and normalized intertwining operators, preprint, 2000. 178, 191 H. KIM AND F. SHAHIDI, Functorial products for GL2 × GL3 and functorial symmetric cube for GL2 , C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 599–604. MR 1 799 096 178 , Functorial products for GL2 × GL3 and the symmetric cube for GL2 , to appear in Ann. of Math. (2). 178, 179, 186, 192, 195 J.-P. LABESSE AND R. P. LANGLANDS, L-indistinguishability for SL(2), Canad. J. Math. 31 (1979), 726–785. MR 81b:22017 179 R. P. LANGLANDS, “Problems in the theory of automorphic forms” in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math. 170, Springer, Berlin, 1970, 18–61. MR 46:1758 , Base change for GL(2), Ann. of Math. Stud. 96, Princeton Univ. Press, Princeton, 1980. MR 82a:10032 195 , “On the classification of irreducible representations of real algebraic groups” in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, ed. P. J. Sally Jr. and D. A. Vogan, Math. Surveys Monogr. 31, Amer. Math. Soc., Providence, 1989, 101–170. MR 91e:22017 177 , “Beyond endoscopy” to appear in Volume in Honor of Joseph Shalika’s Sixtieth Birthday, ed. D. Ramakrishnan and F. Shahidi, Johns Hopkins Univ. Press, Baltimore. 186, 192 W. LUO, Z. RUDNICK, and P. SARNAK, “On the generalized Ramanujan conjecture for GL(n)” in Automorphic Forms, Automorphic Representations, and Arithmetic

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(Fort Worth, Tex., 1996), Proc. Sympos. Pure Math. 66, Part 2, Amer. Math. Soc., Providence, 1999, 301–310. MR 2000e:11072 178 ´ C. MOEGLIN and J.-L. WALDSPURGER, Le spectre résiduel de GL(n), Ann. Sci. Ecole Norm. Sup. (4) 22 (1989), 605–674. MR 91b:22028 , Spectral decomposition and Eisenstein series: Une paraphrase de l’Ecriture, Cambridge Tracts in Math. 113, Cambridge Univ. Press, Cambridge, 1995. MR 97d:11083 D. PRASAD and D. RAMAKRISHNAN, “On the global root numbers of GL(n) × GL(m)” in Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, Tex., 1996), Proc. Sympos. Pure Math. 66, Part 2, Amer. Math. Soc., Providence, 1999, 311–330. MR 2000f:11060 181 D. RAMAKRISHNAN, On the coefficients of cusp forms, Math. Res. Lett. 4 (1997), 295–307. MR 98e:11064 178, 187, 195 , Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. (2) 152 (2000), 45–111. MR 2001g:11077 179 J.-P. SERRE, Abelian `-adic Representations and Elliptic Curves, W. A. Benjamin, New York, 1968. MR 41:8422 178, 194 F. SHAHIDI, On certain L-functions, Amer. J. Math. 103 (1981), 297–355. MR 82i:10030 191 , On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), 547–584. MR 89h:11021 178, 183, 190, 191, 192 , Third symmetric power L-functions for GL(2), Compositio Math. 70 (1989), 245–273. MR 90m:11081 177 , A proof of Langlands’ conjecture on Plancherel measures: Complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), 273–330. MR 91m:11095 , “Symmetric power L-functions for GL(2)” in Elliptic Curves and Related Topics, ed. H. Kisilevsky and M. R. Murty, CRM Proc. Lecture Notes 4, Amer. Math. Soc., Providence, 1994, 159–182. MR 95c:11066 177, 178, 194, 195 D. SOUDRY, The CAP representations of GSp(4, A), J. Reine Angew. Math. 383 (1988), 87–108. MR 89e:11032 182

Kim Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada; [email protected] Shahidi Department of Mathematics, Purdue University, West Lafayette, Indiana 47906, USA; [email protected]


GLOBAL ASYMPTOTICS FOR MULTIPLE INTEGRALS WITH BOUNDARIES E. DELABAERE and C. J. HOWLS

Dedicated to Frédéric Pham on the occasion of his 65th birthday. Abstract Under convenient geometric assumptions, the saddle-point method for multidimensional Laplace integrals is extended to the case where the contours of integration have boundaries. The asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points. The incidence of the Stokes phenomenon is related to the monodromy of the homology via generalized Picard-Lefschetz formulae and is quantified in terms of geometric indices of intersection. Exact remainder terms and the hyperasymptotics are then derived. A direct consequence is a numerical algorithm to determine the Stokes constants and indices of intersections. Examples are provided. 1. Introduction The asymptotic behaviour as k → ∞ in the complex plane C of complex oscillatory integrals Z I0 (k) =

0

e−k f (z) g(z) dz (1) ∧ · · · ∧ dz (n) ,

(1)

with f, g : Cn → C analytic functions of the variable z = (z (1) , . . . , z (n) ) and 0 a chain of real dimension n, has been the study of much work, both theoretical and practical. A discussion of the history of the problem can be found in V. Arnold, A. Varchenko, and S. Gussein-Zadè [3] and D. Kaminski and R. Paris [40]. Applications of these integrals in optics are detailed in [53] and the references therein. Much work has focused on obtaining the asymptotic expansions themselves. Here we focus on deriving “global asymptotics” in all sectors of the complex k-plane for these integrals when the contours of integration are finitely bounded. Interest in this area has been renewed recently following the ideas of R. Balian and C. Bloch [6] and F. Pham [63], DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 21 March 2000. Revision received 31 January 2001. 2000 Mathematics Subject Classification. Primary 30E15, 30E20, 34M40, 34M60, 41A80; Secondary 32C30, 32S05, 34M35 44A10. ´ Authors’ work supported by the French Ministère des Affaires Etrang` eres and the British Council under Alliance Partnership Programme grant number 98018. 199

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[65], the development of the resurgence (see [23]) and hyperasymptotic theories (see [7]), and the work of Kaminski and Paris [40], [41], and C. Howls [36]. The main approach used in deriving the (global) asymptotics of such Laplace integrals is a generalisation of the Riemann-Debye saddle-point method, which can be reduced to the following algorithm (see [25]): (1) the identification of all possible critical points; (2) the topological operation of pushing the integration contour in Cn toward the directions of steepest descent, forming a chain of integration hypersurfaces; (3) the local study near the critical points of the phase function and near the boundary of the hypersurfaces of integration 0, and the computation of the relevant asymptotic expansions; (4) the derivation of the exact remainder terms, their reexpansions in terms of distant critical points, and the calculation of the associated Stokes constants, thereby explicitly linking the contributions from all relevant critical points. The first and third parts have been extensively discussed at leading order (see, e.g., Arnold, Varchenko, and Gussein-Zadè [3], V. Vassiliev [71], and B. Gaveau [26]) and for real variables (see R. Wong [76]). The second topic has been studied practically for real variables in terms of flows by Kaminski [39]. The third topic has been studied in great detail by Varchenko from the theoretical viewpoint, relating the characteristic exponents of the asymptotic expansions to mixed Hodge structure in vanishing cohomologies (see [69], [70]). Substantial practical progress in the derivation of asymptotic expansions with exact remainder terms for polynomial exponents using Mellin integral representations has been made by Kaminski and Paris [40], [41] and by G. Liakhovetski and Paris [43] using a Newton polygon to identify the appropriate contributions. The fourth point was studied for unbounded integration contours by Howls [36]. Our goal here is to combine the four points above so as to produce for the first time exact remainder terms and a self-consistent numerical algorithm to determine the Stokes constants when 0 is a bounded domain. The asymptotic expansions are well known when f is a polynomial function (and g is “well behaved at infinity”) and the contour of integration 0 is an unbounded nchain of hypersurfaces of integration, satisfying a convergence criterion at infinity −1, belong to the finite set of distinct monodromy exponents of the classical monodromy operator in homology∗ (cf. [45]); to each r is associated a set of s ∈ N with s ≤ inf{µα − 1, n − 1} (see [3] for more details); of course, s = 0 for nondegenerate critical points; P ar,s (t) = j≥0 ar,s, j t j are convergent Taylor series. It remains to use the standard integral Z +∞ d l 0(λ + 1) (31) e−kt t λ (ln t)l dt = dλ k (λ+1) 0 t

to conclude from Watson’s lemma that Iα0 (k); hence Iα (k) has Jα (k) =

X r,s

Tr,s (k)

(ln k)s k r +1

(32)

for its asymptotics when |k| → ∞ in all closed subsectors of 6θ , where Tr,s (k) are formal Gevrey-1 series expansions.† ∗ Define

a base point t0 near (but different from) the critical value, and consider (for t0 ) a basis of vanishing cycles generating the vanishing homology. Consider the deformation of these cycles when t goes around the critical value, starting at and coming back to t0 . The resulting cycles still define a basis of the homology: comparing the two bases, one gets an invertible matrix with integer coefficients describing the monodromy of the vanishing homology. In the case of a nondegenerate critical point (only one vanishing cycle), the possible monodromy arises from the possible self-intersection of the vanishing cycle. In the case of a self-intersection, we get a square root singularity at the origin for Jbα (t); otherwise, Jbα (t) is analytic near t0 . In the case of a degenerate critical point, each of the deformed cycles is described in general as a linear combination (with integer coefficients) in terms of the preliminary basis. Write the eigenvalues of this matrix under the form exp(−2iπλ). Then these λ are rationals (see [45]), and, moreover, they are (up to an addition of an integer) the exponents r in the series expansion (30). It may happen that the monodromy matrix has multiple eigenvalues, resulting in possible logarithmic terms in (30). In Kaminski and Paris’s scheme (see [40], [41]), this corresponds to multiple poles for the integrand for the associated Mellin-Barnes integral representation. P j † That is, T (k) = j r,s j≥0 Tr,s; j /k , and there exist Cr,s > 0 and Ar,s > 0 such that |Tr,l; j | ≤ Cr,s Ar,s 0( j) (see Malgrange [47], J. P. Ramis [66], or M. Loday-Richaud [44]).

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Now we can use our freedom to change θ slightly so that L α still does not meet other critical values. The resulting integral is just the analytic continuation of the previous one, the asymptotics at infinity being preserved. This proves that the asymptotics (32) are valid inside a wider sector of aperture greater than π. Therefore formal expansion (32) is Borel resummable with Iα (k) as its Borel sum (Watson’s theorem;∗ see [47]), and we thus obtain the following equivalent, one-dimensional, integral representation: Z ∞eiθ

Iα (k) =

dte−kt Jbα (t).

0

(33)

In summary, for general value f α , we have the following theorem. THEOREM 3.1 Let f α be a first-type critical value. The integral Iα (k) admits e−k fα Jα (k) as its asymptotic series expansion for k → ∞ in 6θ = {|k| > 0, | arg(k) + θ| < π/2}; in other words, Iα (k) ∼ e−k fα Jα (k) in 6θ , with

Jα (k) =

X r,s

Tr,s (k)

(ln k)s , k r +1

(34)

where Tr,s (k) =

X

Tr,s; j /k j

(35)

j≥0

belongs to C[[k −1 ]]1 , the differential algebra of Gevrey-1 series expansions (see [66]). The r ∈ Q run over a finite spectral set, and to each r is associated a set of s ∈ N satisfying s ≤ inf{µα − 1, n − 1}. Conversely, Iα (k) (resp., Jbα (t)) can be considered as the Borel sum (resp., the mi-

nor in the resurgence theory of J. Ecalle† ) of e−k fα Jα (k) in the direction of argument θ.

Remark. Note that we are here treating the Tr,s and Jα as formal series expansions, although they can be interpreted also as (Poincaré) asymptotic expansions. The notational approach used here may be unfamiliar to some readers, but it is used to maintain consistency with the Borel and resurgence viewpoint (see [47], [23], [12]). Note also that in Theorem 3.1 the direction of summation should be considered ^ the universal covering of C\{0}. as a direction in C\{0}, ∗ This † See

is not to be confused with his lemma for standard integral expansions. Ecalle [23], or [12], [18], or [16] for a short introduction to resurgence theory.

GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS

217

COROLLARY 3.1 If z α is a nondegenerate quadratic critical point of f , then Iα (k) ∼ e−k fα Jα (k) in 6θ , with

Jα (k) =

1

k

T (k) n/2 α

and

X g(0)(2π)n/2 Tα (k) = √ + T α /k j ∈ C[[k −1 ]], Hess( f )(0) j≥1 j

(36) where Hess( f ) is the hessian determinant. The choice of the root depends on the orientation of the vanishing cycle. This corollary is well known (cf. [25], [76]). One method for the practical computation of the T jα is described in [17]. 3.2. Relative steepest-descent contours (bounded case) This case corresponds to a critical value of the second type and is the main focus of this paper. Let f α ∈ 3(1,..., p) be a critical value of the restricted function f (1,..., p) with p > 0 the depth. We apply the reduction process developed in Section 2.3.2 (resp., Section A.4.2) to analyse the asymptotics of our bounded integral. Again without loss of generality, we assume that f α = 0 and θ = 0. To compute the asymptotics, it is enough to consider the truncated integral. Since f = 0 is not a singular level, by following formula (17) and its comments, we can apply Fubini’s theorem to write Z t0 Z ω t0 −kt Iα (k) = dte , (37) 0 ∂[0 t ] d f t where ω/d f t is the (holomorphic) restriction of the differential quotient (n − 1)form ω/d f to the level f = t, and ∂[0 t ] is that part of the boundary of 0 t lying on the fibre f −1 (t).∗ The same reduction can be repeated step by step. We use the local coordinates (s (1) , . . . , s ( p) , s ( p+1) , . . . , s (n) ) so that f is given by formula (8), and we define the set of functions f 1,2,...,i : (s (1) , . . . , s (n) ) 7 → f (0, . . . , 0, s (i+1) , . . . , s (n) ). At the second step (if p ≥ 2), from Fubini’s theorem we have Z Z t Z ω ω b Jα (t) = = dt1 | f =t, f1 =t1 , (38) t ∂[0 t ] d f 0 ∂1 ◦∂[0 1 ] d f ∧ d f 1 where ω/d f ∧ d f 1 | f =t, f1 =t1 denotes the (holomorphic) restriction of the differential quotient (n − 2)-form ω/d f ∧ d f 1 along the nonsingular level f = t, f 1 = t1 : here again, (38) is a simple translation of formula (18) with its comments. At the pth step the class of homology of [0 t ] belongs to Hn (X α , X αt ∪ Y1 ∪ · · · ∪ Y p ), and we select that part of its boundary lying on X αt . ∗ Precisely,

218

DELABAERE and HOWLS

we obtain Iαt0 (k)

t0

Z

dte

=

−kt

0

t

Z

t p−2

Z dt1 · · ·

0

Z dt p−1

0

∂( p−1,...,1,0) [0

t p−1

]

ω( p−1,...,1,0) ,

(39)

where ∂( p−1,...,1,0) = ∂ p−1 ◦ · · · ◦ ∂1 ◦ ∂,

(40)

while ω( p−1,...,1,0) =

ω | f =t, f1 =t1 ,..., f1,...,( p−1) =t p−1 d f ∧ d f 1 ∧ · · · ∧ d f 1,...,( p−1)

(41)

is the corresponding Leray quotient (n − p)–differential form. If p = n (corner critical point), then ω( p−1,...,1,0) is just a holomorphic function of (t, t1 , . . . , t p−1 ) and the reduction process stops here. If p ≤ n − 1, it then remains to use the same argument as in Section 3.1 to obtain Z t0 Z t Z t p−1 Z t0 −kt Iα (k) = dte dt1 · · · dt p ω( p,...,1,0) , (42) 0

0

0

∂( p,...,1,0) [0 t p ]

where ∂ p ◦ · · · ∂1 ◦ ∂[0 t p ] is the vanishing cycle of the critical point. In the case p = n, the integral Z t Z t p−2 Z Jbα (t) = dt1 · · · dt p−1 ω( p−1,...,1,0) 0

0

∂( p−1,...,1,0) [0

t p−1

(43)

]

P j+ p−1 as its convergent Taylor defines an analytic function on Dα , with j≥0 a j t expansion. In the case p ≤ n − 1, by using the results from Section 3.2, the function Z h α (t, t1 , . . . , t p ) = ω( p,...,1,0) , (44) ∂( p,...,1,0) [0 t p ]

considered as a function of t p with (t, t1 , . . . , t p−1 ) as a parameter, is defined as an analytic function on the universal covering D^ α \{ f α } and admits in each sector a < arg(t) < b a convergent series expansion of the form X ar,s (t, t1 , . . . , t p−1 , t p )t rp (ln t p )s ,

(45)

r,s

where •

•

the r belong to a finite set of distinct monodromy exponents of the classical monodromy operator in homology, and the s ∈ N satisfy s ≤ inf{µα − 1, n − p − 1}; P j ar,s (t, t1 , . . . , t p−1 , t p ) = j≥0 ar,s, j (t, t1 , . . . , t p−1 )t p are convergent Taylor series with analytic dependence with respect to (t, t1 , . . . , t p−1 ).


It thus follows that the function Z t Z Jbα (t) = dt1 · · · 0

t p−1

Z dt p

0

∂( p,...,1,0) [0 t p ]

219

ω( p,...,1,0)

(46)

has the same properties, with Jbα (t) =

X

br,s (t)t r + p (ln t)s

(47)

r,s

P j for its convergent series expansion, where br,s (t) = j≥0 br,s, j t p are convergent Taylor series. As in Section 3.1, we finally recast Iα (k) as a one-dimensional integral representation Z ∞eiθ

Iα (k) =

0

dte−kt Jbα (t),

(48)

and we obtain the following theorem. THEOREM 3.2 If f α is a second-type critical value of depth p, the integral Iα (k) admits e−k fα Jα (k) as its asymptotic series expansion for k → ∞ in 6θ (i.e., Iα (k) ∼ e−k fα Jα (k) in 6θ ), where 1 X T j /k j ∈ C[[k −1 ]]1 , (49) for p = n, Jα (k) = p k j≥0

for p ≤ n − 1,

Jα (k) =

X r,s

Tr,s (k)

(ln k)s , k r + p+1

(50)

P j −1 where Tr,s (k) = j≥0 Tr,s; j /k belongs to C[[k ]]1 . The rational r ’s run over a finite spectral set associated with the critical point, and to each r is associated a set of s ∈ N satisfying s ≤ inf{µα − 1, n − p − 1}. Conversely, Iα (k) (resp., Jbα (t)) is the Borel sum (resp., the minor) of e−k fα Jα (k) in the direction of argument θ. The asymptotics are simpler in the case of a nondegenerate critical point (see, for instance, [17], [76], [3]), arising from p boundaries. COROLLARY 3.2 When the singular point is quadratic, Iα (k) ∼ e−k fα Jα (k) in 6θ , with

Jα (k) =

1

k

T (k) (n+ p)/2 α

for the asymptotic series expansion.

and

Tα (k) =

∞ Tα X j j=0

kj

(51)

220

DELABAERE and HOWLS

4. Stokes phenomenon We described in the previous section the asymptotics of the multiple Laplace integrals of type (1), as well as their representation as Borel sums for generic summation directions (θ ). To obtain the global asymptotics, we must analyse the Stokes phenomena. This amounts to analysing the singularities in the Borel plane of the analytic continuations of (the minors) Jbα (t). How this can be obtained from generalised PicardLefschetz formulae (detailed in Section B.1) is discussed in Section 4.1. Against this Borel viewpoint, we demonstrate in Section 4.2 how the Stokes phenomenon can be understood directly in terms of steepest-descent contours, thus keeping the geometric ideas of the saddle-point method as in the one-dimensional case. In what follows, it is convenient to assume the following hypothesis. HYPOTHESIS H6 All singular points are nondegenerate.

4.1. Ramifications We return to the Jbα (t) of Theorems 3.1 and 3.2, together with their corollaries. From (36) and (51), if f α has depth pα , then locally near the origin, Jbα (t) = t (n+ pα −2)/2 Hα (t),

(52)

where Hα is holomorphic near zero. Thus when n + pα is even, Jbα (t) is an analytic function at the origin, but when n + pα is odd, Jbα (t) has a square-root singularity there.∗ Note that the Jbα (t) have been defined as (germs of ramified) analytic functions at the origin. If f α 6= 0, it is necessary to take into account translations by the complex numbers f αβ = f β − f α . (53) The generalised Picard-Lefschetz formulae described in Section B.1 allow us to identify the type of singularity in the analytic continuations of each Jbα (t). Denoting by Var = ρ − I the “variation” operator, where I is the identity operator and ρ is the analytic continuation around f αβ anticlockwise, it follows from formula (B.3) that for t near f αβ , Var Jbα (t) = καβ Jbβ (t − f αβ ), (54) where καβ is a (positive or negative) integer. This is consistent with the geometric argument (B.5). The vanishing homology Hn− pα −1 (S p ∩ · · · ∩ S1 ∩ X αt ) is a trivial covering on the circle of directions S when n + pα is even, but it is a two-fold covering when n + pα is odd.

∗


221

PROPOSITION 4.1 If pβ is the depth of the critical value f β , then • when n − pβ is even (so that Jbβ (t) is not ramified around t = 0),

Jbα (t) = καβ Jbβ (t − f αβ )

•

ln(t − f αβ ) + Hol(t − f αβ ) 2iπ

(55)

with Hol a holomorphic function near zero,∗ and when n − pβ is odd, Jbα (t) = −

καβ Jbβ (t − f αβ ) + Hol(t − f αβ ). 2

(56)

Arguments developed in Malgrange [45] show that Jbα (t) remains bounded (when n ≥ 2, the analysis being obvious when n = 1) near f αβ . Proposition 4.1 now results from the variation formulae (54) and Riemann’s removable singularity theorem (see [28]). We show in Section 4.2 how the above properties can be derived independently. 4.2. Stokes phenomenon We consider a singular direction (θ ) and assume that the closed half-line L α = L α (θ) meets a singular value f β ; this (possibly) gives rise to a Stokes phenomenon, which can be described as follows. Assume for simplicity that L α meets no singular point − − + + other than f β . We define the half-lines L − α = L α (θ ) and L β (resp., L α = L α (θ ) + and L β ) by slightly rotating L α and L β clockwise (resp., anticlockwise; see Figure 4). We perform the same rotations for the other half-lines L λ . For fixed θ − (resp., θ + ), define a basis of steepest-descent n-folds (0α − , 0β − , . . . , 0λ− , . . .) (resp., (0α + , 0β + , . . . , 0λ+ , . . .)). From our hypothesis, we can assume that (0β + , . . . , 0λ+ , . . .) is deduced from (0β − , . . . , 0λ− , . . .) by an isotopy (continuous deformation) when the argument runs from θ − to θ + , so that we can remove the upper scripts ± in the notation. Concerning the 0α ± ’s, we can assume only that the one is deduced from the other by a local deformation near the critical point z α . This may not work globally due to the critical value f β . We thus get the decomposition 0α − = 0α + + 0α ± ,

(57)

0α ± = 0α − − 0α + = καβ 0β .

(58)

where necessarily

∗ In other words, J bα (t) and καβ Jbβ (t − f αβ ) ln(t − f αβ )/2iπ are equal when one considers them as microfunctions.

222

DELABAERE and HOWLS

L+ α L+ β fα

fβ fβ

fα

L− β L− α

Figure 4. Generic Stokes phenomenon: Below, before the Stokes phenomenon; above, after the Stokes phenomenon

In this formula, καβ can be understood geometrically as an index of intersection,∗ and therefore it is a positive or negative integer, the sign depending on the orientations of 0α and 0β . We now apply (58) to our integral representation. We start with the integral Iα (k) =

Z 0α−

e

−k f (z)

ω=e

−k f α

∞eiθ

Z

−

e−kt Jbα (t) dt,

0

(59)

which defines an analytic function within the sector 6θ − = {|k| > 0, | arg(k)+θ − | < π/2}. We know that Iα (k) extends analytically over C\{0} as a multivalued function, and from (58) we see that Z Z Iα (k) = e−k f (z) ω + καβ e−k f (z) ω (60) 0α+

0β+

for k ∈ 6θ + , that is, that Iα (k) = e

−k f α

∞eiθ

Z

+

e 0

−kt

Jbα (t) dt + καβ e−k fβ

∞eiθ

Z 0

+

e−kt Jbβ (t) dt.

(61)

Formula (61) provides a complete description of the Stokes phenomenon. We can recast these results in the framework of resurgence theory, using the notation of Section 3. From (59), Iα (k) is the left (lateral) Borel sum in the direction of argument θ of the ∗ Compare

h0β? , 0α ± i.

with Section B.2. With the convention of formula (B.8), καβ is given by the equality καβ =


223

resurgent symbol e−k fα Jα (k), Iα (k) = e−k fα S(θ − ) Jα (k),

(62)

and from (61) the action on Jα (k) of the Stokes automorphism S(θ) is given by S(θ) Jα (k) = Jα (k) + καβ e−k fαβ Jβ (k),

(63)

1 fαβ Jα (k) = καβ Jβ (k),

(64)

which finally yields where 1 fαβ is the “alien derivative operator” (see [23], [12], [18], [16]) at f αβ . It is important to notice that the alien derivatives (as well as the directions of summations) have to be indexed over the two-fold covering of C\{0} (i.e., the Riemann surface of the square root) when n + pα is odd. The sign of καβ depends on which sheet of this covering is under consideration. 4.3. Conclusion In (64), the index of intersection καβ now appears as a so-called Stokes multiplier. Note that (64) can be understood in terms of singularity in the Borel plane. The analytic continuation of the minor Jbα (t) of Jα (k) along a straight half-line (0, f αβ ∞) encounters a singularity at t = f αβ if καβ 6= 0, and its variation there (as defined in Section 4.1) is καβ Jbβ (t − f αβ ). This is nothing but (54). To compute the constants καβ and thus to obtain the complete resurgence structure, it is helpful to keep in mind their two interpretations. The geometric description in terms of indices of intersection has already demonstrated their quantised nature. This helps again to show that some of them necessarily vanish. This stems from the S fact that the set of critical values 3 = 3(i1 ,...,i p ) has a natural hierarchy that follows directly from the stratification of S. This means that a relative steepest-descent n-fold 0α having S1 , . . . , S p (say) as its boundary can be eventually affected only by critical values f β belonging to the subset 3α ⊂ 3 of critical values of f or of the restricted f | to one of the Si , i = 1, . . . , p, or their different intersections. Otherwise, καβ = 0 necessarily.∗ This is almost all that we can learn from the geometry. To get quantitative information about the remaining indices of intersections καβ , we have to turn to the hyperasymptotic analysis, interpreting this time the καβ as Stokes multipliers. This is the aim of the next section. It is convenient for this purpose to represent the results of (64) in terms of the series expansions Tα introduced in formulae (36) – (51). Hereafter, formula (65) is simply derived from formula (64) by the Leibniz rule (see [23], [16]). ∗ See

also the second remark of Section B.1.

224

DELABAERE and HOWLS

THEOREM 4.1 The series expansions Tα (k) ∈ C[[k −1 ]]1 are Gevrey-1 resurgent resummable. If f α is a critical value of depth pα , the adjacent singularities∗ of the minor Tbα (t) of Tα (k) are (at most) the f αβ = f β − f α , with f β ∈ 3α . Moreover, in the (generic) case where these relevant f αβ have distinct phases,

1 fαβ Tα (k) = καβ k ( pα − pβ )/2 Tβ (k),

(65)

where pβ (≤ pα ) is the depth of f β , whereas the καβ are integers. Remark. In (65), the alien derivatives can now be indexed over C\{0}. This does not make (65) ambiguous. One has to keep in mind that the sign of the index of intersection καβ depends on the orientations of the steepest-descent n-folds 0α and 0β , on which the determination of the square root k ( pα − pβ )/2 also depends. 5. Hyperasymptotic analysis: Calculation of Stokes constants We assume here for simplicity that the values f αβ have distinct phases. Theorem 4.1 thus applies, allowing us to identify the singularity types. The results from the previous section can now be used to deduce the exact remainder term for a truncated asymptotic expansion about any of the singularities. As we have converted integral (1) into a one-dimensional Laplace integral (Borel sum), the procedure follows closely that of Howls [36] and Olde Daalhuis [55], allowing us to be brief. It is important to stress that at no point in the hyperasymptotic procedure detailed in this section are the full infinite asymptotic/formal power series used. All the expansions are finite and exactly terminated by the appropriate (hyperasymptotic) remainder term. We now describe the hyperasymptotic analysis for the (slowly varying) related analytic function Tα (k) = k (n+ pα )/2 ek f α Iα (k) = S(θ ) Tα (k) = T0α +

∞eiθ

Z 0

dt e−kt Tbα (t).

(66)

Here S(θ ) Tα (k) is the Borel sum of the formal series expansions Tα (k) in the nonsingular direction of argument θ , and therefore Tα (k) ∼ Tα (k) for k → ∞ in 6θ . We represent the local behaviour of the function Tbα (t) in terms of a Cauchy integral representation I Tbα (u) 1 b du . (67) Tα (t) = 2iπ u=t u−t ∗

The adjacent singularities of a germ of analytic functions at the origin are the singularities of the analytic continutions of this germ along half-lines emanating from the origin.


225

γαβ1 f αβ1 γα

f αβ2

γαβ2

Figure 5. The path γα on the left. Its deformation is on the right with the contributions γαβ from the adjacent singularities.

After a binomial expansion to the order of truncation N required, we have exactly Tα (k) =

N −1 X r =0

Trα + Rα (k, N ), kr

(68)

where Rα (k, N ) =

1 2iπk N

Z 0

+∞

dwe−w w N −1

Z γα

du

Tbα (u) . (1 − w/ku)u N

(69)

The contour γα encircles the positive real axis as in Figure 5. We then deform γα (see Figure 5) to encounter the other singularities f αβ that are adjacent (see [36]). By a suitable restriction of the type of integrand functions we consider, or by going to sufficiently high truncation N , the arcs at infinity (see Figure 5) make no contribution (see [36]).∗ We thus have Z X Z +∞ 1 Tbα (u) −w N −1 Rα (k, N ) = dwe w du . (70) N 2iπk (1 − w/ku)u N 0 γαβ adjacent f αβ

∗ The general case can be treated as in Olde Daalhuis [55]: instead of working with full Borel sums, one uses truncated Laplace integrals (essentially changing summation to presummation in resurgence-speak; cf. [12], [18]). Pushing the circular arcs far enough away, their contributions can be bounded away to an exponential level smaller than the one to which the hyperasymptotics are eventually taken, that is, less than exp(−M|k|) for any chosen M > 0.

226

DELABAERE and HOWLS

At each of the singularities β, we make a change of variables and collapse the contour γαβ onto the associated cut. The results of the Picard-Lefschetz analysis, embodied in (65), then guarantee that the discontinuities generate self-similar integrands, with contours now over the critical point β. The final result is Tα (k) =

N −1 X r =0

Trα + Rα (k, N ) kr

(71)

with Rα (k, N ) = −

Z [−θαβ ] N −1+( pα − pβ )/2 1 X καβ −t0 f αβ t0 dt e Tβ (t0 ), 0 2iπ k − t0 k N −1 0

(72)

καβ 6 =0

R [η] R ∞eiη (see [55]), with θαβ denoting the phase where we have used the notation 0 = 0 ∗ of f αβ . The indices pα and pβ are the depths of (number of boundaries associated with) α and β, respectively. The minus sign comes from the fact that the orientation of the contours γαβ is in the opposite sense to the convention used to define the index of intersection καβ (compare the γαβ in Figure 5 with the path L + β in Figure 4, for instance). The (quantised and, as yet, unknown) Stokes constants associated with α and β are nonzero integers if the singularity is adjacent, defined up to a sign depending ( p − p )/2 on the branch of t0 α β (see the remark following Theorem 4.1). We determine the Stokes constants by resorting to a resurgence formula for the coefficients in the expansions themselves. Using the fact that TNα = k N Rα (k, N ) − Rα (k, N + 1) Z +∞ v καβ 1 X −v N −1+( pα − pβ )/2 =− dv e v T , (73) β N +( pα − pβ )/2 2iπ f αβ 0 f καβ 6=0

αβ

we substitute the corresponding exactly terminated asymptotic expansions of all the adjacent Tβ of type (71) with v/ f αβ playing the role of k. If the next-nearest singularity is some distance further away from α than the nearest, we obtain the usual (Dingle type; see [21]) leading-order approximation to the late terms: N −N καβ 0(N + ( pα − pβmin )/2) βmin TNα = − min T 1 + O 0 N +( pα − pβmin )/2) 2iπ |k f αβmin | f αβmin as N → ∞.

(74)

analytic function Tβ can be thought of as the Borel sum of the series expansion Tβ in the direction of arg( f αβ ). ∗ The


227

Here f αβmin is the distance in the Borel plane to the nearest potential singularity (which may or may not be adjacent). As the Stokes constants are quantised integers, we only need to determine καβ to an accuracy of within 1/2 to infer its value. The large parameter in (74) is now N . An appropriately high value of N will give that accuracy if we know the N th term, f αβmin , and the number of boundaries that α, β sit on ( pα , pβ ). We can then move to determine whether the next-nearest neighbour is adjacent using the same level of approximation. The calculations may be checked by including further terms in the series expansion taking larger values of N and/or higher |k|. Note that this procedure is a simplification of the method in [36] and [55] and reduces the work required to determine the full set of Stokes constants. However, in this form it only works for integrals due to the quantised nature of the Stokes constants. Even this new method cannot determine all the καβ at this stage. Stokes constants from next-nearest neighbours β1 can be determined, provided that | f βmin β1 | < | f αβmin |. By summing the nearest neighbour contributions to the least term and including a single term of the next-nearest contribution, we may again shorten the method of [36] and [55] and use the following to determine καβ1 : N1 −1 καβmin X 0(N0 + ( pα − pβmin )/2 − r1 ) βmin Tr1 N0 +( pα − pβmin )/2−r1 2πi f αβmin r1 =0 N −N0 καβ1 0(N0 + ( pα − pβ1 )/2) β1 0 =− , T0 1 + O N +( p − p )/2 α β 0 2πi |k f αβ1 | 1 f αβ1 N0 = |k| | f αβmin | + | f βmin β1 | , and N1 = N0 − |k f αβ1 |.

TNα0 +

(75)

This can be demonstrated by an intricate use of Stirling’s approximation. If this is not sufficient to determine all the καβ , one must resort to hyperasymptotic approximations and successively reexpand the remainders for each β. The result of this incestuous iterative reexpansion and substitution of the exact remainder term into itself is the treelike hyperasymptotic expansion (see Howls [36]) TNαα 0

X καβ 1 = 2iπ καβ1 6 =0

−

X

αβ1

N1

X−1

Trβ1 K (1) (0; α, β1 , N0α + 1, r )

r =0

X καβ κβ β 1 1 2 (2iπ)2

καβ1 6 =0 κβ1 β2 6 =0

αβ1 β2

N2

X−1 r =0

αβ

Trβ2 K (2) (0; α, β1 , β2 , N0α + 1, N1 1 , r )

228

DELABAERE and HOWLS

+ · · · + (−1)m−1

X καβ1 6 =0

···

X κβm−1 βm 6 =0

καβ1 · · · κβm−1 βm (2iπ )m

αβ1 ···βm −1

Nm

X

Trβm

r =0

α···β

αβ

× K (m) (0; α, β1 , . . . , βm , N0α + 1, N1 1 , . . . , Nm−1 m−1 , r ) X X καβ1 · · · κβm−1 βm αβ rαβ1 ···βm (N0α , N1 1 , . . . , Nmαβ1 ···βm ), + ··· (2iπ )m καβ1 6 =0

κβm−1 βm 6 =0

(76) where K (1) k; α, β1 , N0α , r N0α + ( pα − pβ1 )/2 − r (1) =F k; , f αβ1 αβ

K (2) (k; α, β1 , β2 , N0α , N1 1 , r ) =F

(2)

αβ1

k;

N0α − N1

+ 1 + ( pα − pβ1 )/2, f αβ1 ,

αβ1

N1

+ ( pβ1 − pβ2 )/2 − r f β1 β2

! , (77)

and, more generally, α...β αβ K (m) k; α, β1 , . . . , βm , N0α , N1 1 , . . . , Nm−1 m−1 , r αβ N α − N1 1 + 1 + ( pα − pβ1 )/2, . . . , (m) =F k; 0 f αβ1 , α···βm−1 α···βm−2 + ( pβm−2 − pβm−1 )/2, − Nm−1 Nm−2 f βm−2 βm−1 ,

...,

α···β

Nm−1 m−1 + ( pβm−1 − pβm )/2 − r f βm−1 βm

.

(78) αβ ···β

The Fr 1 m are the canonical hyperterminants (see [7], [35], [36], [55])  (0) F (z) = 1,      M −1  R [−θ ] t 0  (α) z; M0  F = 0 0 e−t0 σ0 0z−t0 dt0 ,    σ0    (l+1) z; M0 , . . . , Ml  F   σ0 , . . . , σl    M −1 M −1  R [−θ0 ] R [−θ ] t0 0 ···tl l  = 0 · · · 0 l e−(t0 σ0 +···+tl σl ) (z−t0 )(t dt0 · · · dtl , 0 −t1 )···(tl−1 −tl )

(79)

where θi is the phase of σi ∈ C\{0}, <Mi > 1. When ph σ j = ph σ j+1 (mod 2π), the t j -path of integration is deformed to the left or the right as in [55], yielding


229

lateral canonical hyperterminants, intrinsically linked to the lateral summations of the resurgence theory (see [12], [16], [18]). These multiple integrals converge when −θ0 < ph z < 2π − θ0 and can be evaluated easily by the methods of Olde Daalhuis [54], [56] with the convention that µαβ |Olde

Daalhuis

=

( pα − pβ ) . 2

(80)

Thus we can employ his truncations and error estimates directly to minimise the overall remainder terms. After m stages of hyperasymptotics, the optimal truncations that globally minimise the remainder term are X N0α = |k| × min | f βl βl+1 | , | f αβ1 | + καβ1 6=0 κβl βl+1 6 =0

αβ1

N1

αβ1 β2

N2

l=1···m

= max 0, N0α − |k f αβ1 | , αβ1

= max 0, N1 .. .

− |k f β1 β2 | ,

αβ ···βm−1

1 Nmαβ1 ···βm = max 0, Nm−1

− |k f βm−1 βm | .

(81)

At each level of hyperasymptotics, if a καβ is not yet known, then it is initially included, regardless of whether it subsequently turns out to be zero (at which point all branches containing this constant may be immediately pruned). Knowing only the relevant f αβ , pα , pβ along each of the branches and using the truncations (81) allows each καβ to be determined from an algebraic system of equations as outlined in [36] and [55]. It might appear that hyperasymptotic expansions are an unnecessary technical numerical detail. However, we know of no other general and systematic analytical or numerical method that is a practical tool for calculating the Stokes constants of these types of integrals. Following Theorem 4.1, there is one qualitative difference between the unbounded and bounded integral case. In the unbounded integrals, every distant quadratic critical point β that could be seen by the initial one α could, in turn, see α itself. In the bounded case, some of the α (and even β) arise only because of the presence of the boundaries. If α lies on a boundary S1 ∩ S2 , then α can see β only if β lies on one of the strata S1 ∩ S2 , S1 , or S2 , or if it arises from the phase function f itself. If β is a quadratic critical point arising from the phase function f , since this is a fundamental property of the integrand, β is a likely candidate for adjacency to all the boundary α. There is thus a hierarchy that can be inferred and can be used to simplify the hyperasymptotic analysis and deduction of Stokes constants. If pβ > pα , we may

230

DELABAERE and HOWLS

deduce that καβ = 0 immediately since β exists only because of the presence of an extra boundary that α knows nothing about. Note that the opposite inference cannot be made; unlike the unbounded case, κβα may differ from καβ , and so care must be taken in calculations. Once the Stokes constants have been determined, it is possible to obtain a (hyper)exponential accurate approximation to (1) via the full expansion αβ1

N0α −1

N1 −1 X X X Tα καβ1 r − Trβ1 K (1) (k; α, β1 , N0α , r ) Tα (k) = α N0 −1 kr 2iπk κ 6 =0 r =0 r =0 αβ1

+

X

X

καβ1 6=0 κβ1 β2

αβ1 β2

N2

καβ1 κβ1 β2 α

(2iπ )2 k N0 −1 6=0

+ · · · + (−1)

m

X καβ1 6=0

···

X−1

αβ

Trβ2 K (2) (k; α, β1 , β2 , N0α , N1 1 , r )

r =0

καβ1 · · · κβm−1 βm

X κβm−1 βm 6 =0 αβ

αβ1 ···βm −1

Nm

α (2iπ )m k N0 −1

X

Trβm

r =0

α···β

× K (m) (k; α, β1 , . . . , βm , N0α , N1 1 , . . . , Nm−1 m−1 , r ) X X καβ1 · · · κβm−1 βm + ··· Rαβ1 ···βm α m k N0 −1 (2iπ ) κ 6 =0 κ 6 =0 αβ1

× (k,

βm−1 βm

N0α ,

αβ N1 1 , . . . ,

Nmαβ1 ···βm ).

(82)

Using truncations (81), this yields an accuracy of O(e−M|k| ) at the Mth iterαβ ···β ation (see [55]), leaving an unevaluated remainder Rαβ1 ···βm (k, N0α , . . . , Nm 1 m ). This expression also widens the domain of validity of the original asymptotics and automatically and exactly incorporates any Stokes phenomenon through the hyperterminants, as explained in [36] and [55]. 6. Example We illustrate the theory with the following example, for which explicit Borel transforms can be deduced as benchmarks against which to test the hyperasymptotic analysis. We take 2

f (z) = z (1) + 2z (2) + 3z (3) + z (1) z (2) + z (2) z (4) + z (3) z (4) + z (2) + z (4)

2

(83)

and g(z) = 1. The boundary is S = S1 ∪ S2 ∪ S3 , where S1 , S2 , S3 are hypersurfaces defined by the equations S1 : z (1) = 2i, S2 : z (2) = 1 + 2i, and S3 : z (3) = 2 − 3i. The set of isolated critical values on the different strata is thus • 3() = { f 0 = 11} and z 0 = (3, −1, 7, −3),


231

3(1) = { f (1) = 39/4 + 3i} and z (1) = (2i, 1/2 − i, 11/2 + i, −3), 3(3) = { f (3) = 7 − 15i/2} and z (3) = (1/2 − 3i/2, −1, 2 − 3i, −1/2 + 3i/2), • 3(1,2) = { f (1,2) = 1 + 6i} and z (12) = (2i, 1 + 2i, 5 − 2i, −3), • 3(1,3) = { f (1,3) = 11 − 19i/3} and z (13) = (2i, −2/3 − 7i/3, 2 − 3i, −2/3 + 8i/3), • 3(1,2,3) = { f (1,2,3) = −1 + 9i/2} and z (123) = (2i, 1 + 2i, 2 − 3i, −3/2 + i/2). We choose θ = 0 for the (generic) direction of summation. To the family of closed half-lines drawn on Figure 6 corresponds a basis of six steepest-descent contours ([00 ], . . . , [0(1,2,3) ]) generating the space H49 (C4 , S) of allowed cycles of integration (cf. Theorem 2.1). • •

[1, 2] 6

[1, 2, 3] 4

[1] 2 2

−2

4

6

8

10

[]

12

0

−2

−4

−6

[1, 3] [3]

−8

Figure 6. The set of singularities and the family of half-lines L α in the complex Borel t-plane

6.1. Asymptotics We now describe the representation of the integration contours space H49 (C4 , S) via Laplace integrals. From Theorem 2.1, it is enough to consider the Laplace integrals over each of the steepest-descent contours [00 ], . . . , [0(1,2,3) ]. We recall that the Lefschetz thimble, in principle, determines completely the homology of the steepest-descent contour, and thus we do not provide a detailed global description of [00 ], . . . , [0(1,2,3) ]. We have just to prescribe the orientations, which we take to be those of the standard Lefschetz thimble (cf. Sections 2.3.1 and 2.3.2).

232

DELABAERE and HOWLS

√ In what follows, the square root k refers to the usual determination (real positive for positive real k). • Straightforward computations give I00 (k) = •

Z 00

e−k f (z) dz (1) ∧ · · · ∧ dz (4) =

4π 2 −11k e . k2

Convenient reductions show that the integral Z I0(1) (k) = e−k f (z) dz (1) ∧ · · · ∧ dz (4) 0(1)

(84)

(85)

can be written as 4iπ 3/2 I0(1) (k) = 3/2 e−(39/4+3i)k k

+∞

Z 0

dt e−kt √ . 5 − 12i − 4t

(86)

It follows that I0(1) (k) is the Borel sum of T(1) (k)k −(4+1)/2 e−(39/4+3i)k , where T(1) (k) = 4iπ

3/2 3 + 2i

13

∞ X 20 + 48i j 0( j + 1/2) − j k ∈ C[[k −1 ]]1 . 169 0(1/2)

(87)

j=0

I0(1) (k) can be analytically continued by rotating the direction of summation, but a Stokes phenomenon occurs due to the adjacent singularity at t = 5/4 − 3i = f 0 − f (1) = f (1),0 . R −k f (z) dz (1) ∧ · · · ∧ dz (4) yields • Similarly, I0 (k) = (3) 0(3) e I0(3) (k) =

4iπ 3/2 −(7−15i/2)k e k 3/2

+∞

Z 0

dt e−kt √ . 16 + 30i − 4t

(88)

This is the Borel sum of T(3) (k)k −5/2 e−(7−15i/2)k , where ∞

T(3) (k) = 2iπ 3/2

5 − 3i X 16 − 30i j 0( j + 1/2) − j k ∈ C[[k −1 ]]1 . 17 289 0(1/2)

(89)

j=0

A Stokes phenomenon occurs due to the adjacent singularity at t = 4 + 15i/2 = f 0 − f (3) = f (3),0 . R −k f (z) dz (1) ∧ · · · ∧ • In the same way as above, the integral I0 (12) (k) = 0(12) e dz (4) is conveniently reduced to Z +∞ dt π I0(12) (k) = 2 e−(1+6i)k e−kt √ . √ k 35/4 − 3i − t(3/2 − i + i 35/4 − 3i − t) 0 (90)


233

This is the Borel sum of T(12) (k)k −(4+2)/2 e−(1+6i)k , where j ∞ X 5 + 7i j+1 X 14 − 10i l 0( j + l + 1) − j T(12) (k) = −2iπ k 148 37 0(l + 1) j=0

∈ C[[k

−1

l=0

]]1 .

(91)

Stokes phenomena occur due to adjacent singularities at t = 35/4 − 3i = f (1) − f (12) = f (12),(1) , and at t = 10 − 6i = f 0 − f (12) = f (12),0 for the Borel transform Td (12) (t). R −k f (z) dz (1) ∧ · · · ∧ dz (4) , which can be • Continuing, we have I0 (13) (k) = 0(13) e written as √ Z Z t 3 3π −(11−19i/3)k +∞ −kt I0(13) (k) = e dte dt1 2k 0 0 1 p ×√ . √ 15/4 − 28i + 3t1 3(t − t1 ) + (3/2 + 9i/4 + 15/4 − 28i + 3t1 /2)2 (92) This gives I0(13) (k) as the Borel sum of T(13) (k)k −3 e−(11−19i/3)k , with ∞

2π X (−1) j 567 + 369i j+1 T(13) (k) = √ 5650 3 j=0 3 j ×

X j X j−l1 l1 =0 l2 =0

49 + 57i l1 −l2 0( j − l1 + l2 + 1)0( j + l1 − l2 + 1) − j k 113 0(l1 + 1)0(l2 + 1)0( j − l1 − l2 + 1)

∈ C[[k −1 ]]1 .

(93)

From (92), a Stokes phenomenon occurs due to an adjacent singularity at t = −5/4 + 28i/3 = f (1) − f (13) = f (13),(1) (to see this, set t1 = t in the integrand) and also at t = −4 − 7i/6 = f (3) − f (13) = f (13),(3) (t1 = 0). Referring to Section 4.1, a singularity at t = 19i/3 = f 0 − f (13) = f (13),0 is also expected when one continues onto another sheet. R −k f (z) dz (1) ∧ · · · ∧ dz (4) may be written • The last integral I0 (123) (k) = 0(123) e in the form I0(123) (k) =

√ Z Z t 12i π +∞ −kt dte dt1 k 3/2 0 0

1 p ×√ √ 8 + 6i − 4t1 ( 16t1 − 12t + 132 − 54i − (4 + 24i) 8 + 6i − 4t1 ) 1 p × ; √ √ (4 − 6i − 8 + 6i − 4t1 + i 16t1 − 12t + 132 − 54i − (4 + 24i) 8 + 6i − 4t1 )

(94)

234

DELABAERE and HOWLS

I0(123) (k) is then the Borel sum of T(123) (k)k −(4+3)/2 with T(123) (k) ∈ C[[k −1 ]]1 , T(123) (k) √ X j j−l j−l1 −l2 ∞ 5 − 13i l1 +l2 +2l3 +1 π (−1) j X X1 X = (2i)l2 +1 3l3 (1 − i)l1 +1 2 j 8 97 2 j=0 l1 =0 l2 =0 l3 =0 1 + 3i 2 j−2l1 −l2 −2l3 +1 0(l + l + 2l + 1)0(2 j − 2l − l − 2l + 1) 1 2 3 1 2 3 × k− j . 5 0(l2 + 1)0(l3 + 1)0( j − l1 − l2 − l3 + 1) (95) Moreover, one can deduce from the integral representation above that a Stokes phenomenon occurs due to an adjacent singularity at t = 2 + 3i/2 = f (12) − f (123) = f (123),(12) (t1 = t) and also at t = 12 − 65i/6 = f (13) − f (123) = f (123),(13) (t1 = 0). For reasons explained in Section 4.1, singularities can be expected at f (123),(1) , f (123),(3) , and f (123),() for some analytic continuations. 6.2. Resurgence formulae The sheet structure in this example can be derived from knowledge of explicit formulae for the Borel transforms of the multiple integrals. In general, the situation is not so simple. Such a study can nevertheless be done numerically with only the asymptotic expansions T(··· ) as inputs by appealing to hyperasymptotics. Since the values f αβ have distinct phases, formulae (74), (75), (76), (81), (82) can thus be applied. Note that from the “one way” adjacency property and the fact that there is only one critical point on each stratum, the hyperasymptotic expansion (82) (or (76)) terminates at most (when T(123) is concerned) at the m = 3 level. By comparison with Figure 6, part of the first sheet structure can be recovered from the leading order formula (74), which yields information about the indices of intersections corresponding to the nearest adjacent singularities for each critical point in turn. One obtains κ(1,2,3),(1,2) = +1,

κ(1,3),(3) = −1,

κ(3),() = +1,

κ(1,2),(1) = −1,

κ(1),() = +1.

(96)

Now one turns to hyperasymptotics to gain more information. We know that α = (1, 2) has βmin (1) for its nearest adjacent singularity, while its next-nearest singularity is β1 = (). Figure 6 shows that expansion (75) may be applied. With k = 1, (75) yields N0 = 13, N1 = 4, and κ(1,2),() ' −0.924 + 0.04i. From the arguments of Section 5, this is quite enough to conclude that κ(1,2),() = −1. Formula (76) at the first level (m = 1) yields κ(1,2,3),(1) = 0,

κ(1,3),() = 0,

κ(1,3),(1) = −1,

(97)


235

Table 1. The Stokes multipliers

καβ α= () (1) (3) (1,2) (1,3) (1,2,3)

β=() 0 +1 +1 −1 0 0

(1) 0 0 0 −1 −1 0

(3) 0 0 0 0 −1 −1

(1,2) 0 0 0 0 0 +1

(1,3) 0 0 0 0 0 +1

(1,2,3) 0 0 0 0 0 0

but it may also be used to recover our previous results (cf. [36], [55]). For instance, (1,3) (1,3),(1) (1,3),(3) replacing k = 5 in (81) gives N0 = 63, N1 = 16, N1 = 42, and (1,3),() (1,3) (1,3) N1 = 31 as optimal truncations. On replacing N0α by N0 , N0 − 1, and (1,3) (1,3) N0 − 2 in formula (76), ignoring the error term, and explicitly substituting T (1,3) , T

(1,3) , and (1,3) N0 −1

T

N0 (1,3) , we get a system of three linear equations. Solving these equa(1,3) N0 −2

tions yields κ(1,3),(1) , κ(1,3),(3) , and κ(1,3),() . Using some higher hyperasymptotic levels in the same spirit, one can completely determine the sheet structure. The level-two expansion yields κ(1,2,3),(3) = −1, κ(1,2,3),() = 0, (98) while the level-three expansion gives the last unknown index of intersection, κ(1,2,3),(1,3) = +1 (see Table 1). The results of the adjacency calculations are displayed in Table 2. The sheet structure is now in its complete form.

6.3. Hyperasymptotic computations The Stokes multipliers being known, (82) gives a (hyper)exponential accurate approximation for the integrals, yielding the results in Table 2. Based on the adjacency calculations, we have calculated hyperasymptotic approximations to the integrals T(1) , T(3) , T(12) , T(13) for various values of large parameter k. “Exact” values of the integrals were obtained from numerical integration schemes. The accuracy obtainable for T(123) by this approach was sufficient only for comparison with the first hyper-level, and so we have not included this. Nevertheless, the agreement was as expected and consistent with the adjacency of (3), (12), and (13) to (123).

236

DELABAERE and HOWLS

Table 2. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(1) and k = 4

Level Lowest

Truncations N (1) = 1

Level 0 (Super) Level 1

N (1) = 13 N (1) = 26, N (1, 0) = 1 k=4

Exact

Approximation −3.42666338266567+ 5.13999507399850i −3.66008976352172+ 5.06575794497829i −3.66007435340317+ 5.06575062056987i −3.66007435340317+ 5.06575062056987i

|1 − approx./exact| 3.9 × 10−2 2.7 × 10−6 0 0

Table 3. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(3) and k = 1

Level Lowest



N (3) = 9

Exact

N (3) = 17, N (3, 0) = 1 k=1

Approximation 1.96529223417590+ 3.27548705695983i 2.19659657698560+ 3.22186580822527i 2.19566048409404+ 3.22216423713061i 2.19566048409404+ 3.22216423713061i

|1 − approx./exact| 6.1 × 10−2 2.5 × 10−4 0 0

Table 4. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(12) and k = 1 Level Lowest



N (12) = 9

Level 2 Exact

N (12) = 13, N (12, 1) = 3, N (12, 0) = 1 N (12) = 15, N (12, 1) = 5, N (12, 0) = 1, N (12, 1, 0) = 1 k=1

Approximation 0.29717768344768 −0.21226977389120i 0.34290482693409 −0.20268045850289i 0.34300280953350 −0.20290401468035i 0.34299544259126 −0.20290986661454i 0.34299544259128 −0.20290986661454i

|1 − approx./exact| 1.2 × 10−1 6.2 × 10−4 2.4 × 10−6 < 5.1 × 10−14 0


237

log10 |Hyperterms|

0 −1 −2

(1)

−3

(1, 0)

−4 0

5

10

15

20

25

Figure 7. The size of the terms in the hyperasymptotic expansion of T(1) , from the critical points (1) (ordinary asymptotics) and (0) (hyperasymptotics along path (1,0)), with k = 4. From the truncations (see Table 2 and (81)) and adjacency of the singularities, only the (0) critical point contributes at the first hyperlevel. As this contributes an exact exponential, only one term is generated, and the hyperasymptotics terminates, generating the exact result.

7. Discussion We first discuss our hypotheses, suggesting how the above results may be generalised. 7.1. Critical points at infinity (Hypothesis H2) Throughout this article, we have assumed that no critical point at infinity occurred (see Hypothesis H2). This assumption was a consequence partly of our own ignorance and partly of a lack of general results about critical points at infinity, although this subject is a matter of an intensive current research (see [59], [31], [33], [67], [13]). However, the following simple example may suggest that Hypothesis H2 can be removed in some instances. We go back to the family of polynomials (7), and we study the behaviour near k = 0 of the integral Z (1) (2) I (k) = e− f (z ,z ;k) dz (1) ∧ dz (2) . (99) 0

For nonzero k, this integral (99) fits into our frame, and we may take for 0 an unbounded chain of integration. By a simple argument of quasi-homogeneity, we first

238

DELABAERE and HOWLS

log10 |Hyperterms|

0

−1 (3) (3, 0)

−2

−3 0

5

10

15

Figure 8. As for Figure 7, but for T(3) , with k = 1. Again only critical point (0) is adjacent, so the hyperasymptotics terminates at the first reexpansion, generating the exact result.

notice that I (k) =

Z 0

e−k f (z

(1) ,z (2) ;1)

dz (1) ∧ dz (2) .

(100)

(We keep the same notation for the chain 0.) Instead of analysing the behaviour near k = 0, we first compute the asymptotics when k → ∞. We consider the critical value +2i (resp., −2i), choose the nonsingular direction (θ = 0), define the associated Lefschetz thimble 02i (resp., 0−2i ), and consider the asymptotics inside the sector | arg(k)| < π/2. With a convenient orientation of 02i and 0−2i , computations (thanks to the algorithm of [17]) suggest that I2i (k) = (π/k + o(k −N ))e−2ik and I−2i (k) = (π/k + o(k −N ))e2ik for all integers N > 0; hence I2i (k) = πe−2ik /k and I−2i (k) = πe+2ik /k exactly by a Borel-summability argument (Watson theorem; see [47]).∗ This would mean that ? I (k) = h02?i , 0iI2i (k) + h0−2 i , 0iI−2i (k)

(101)

∗ More generally, we suspect (by direct computations) that for every polynomial function g(z (1) , z (2) ), we have the equality

Z 0

e−k f (z

(1) ,z (2) ;1)

g(z (1) , z (2) ) dz (1) ∧ dz (2) =

where the a j , b j are complex coefficients depending on g.

6 6 X a j −2ik X b j 2ik e + e , j k kj j=1

j=1


239

0

log10 |Hyperterms|

−1 −2 (12)

−3

(1, 0)

−4 (12, 1)

(12, 1, 0)

−5 0

5

10

15

20

Figure 9. As for Figure 7, but for T(12) , with k = 1 showing the various sequence of critical points that contribute to the hyperasymptotics. Here critical points (1) and (0) are adjacent to (12) and so generate first-level hyperasymptotic corrections. A second level is generated by (0) as this is adjacent to the (first-level) contribution (1).

satisfies the differential equation L (k, ∂k )I = 0

with L (k, ∂k ) =

d2 2 d + +4 2 k dk dk

(102)

with a regular singular point at the origin. This property is satisfied if the differential 2(1) (2) 2 form ω0 = (4k 2 z (2) −6z (2) +4)e− f (z ,z ;k) dz (1) ∧dz (2) is exact (Stokes theorem), and actually, 2

ω0 = d 4z (2) e− f (z

(1) ,z (2) ;k)

dz (1) + (−2 + 2z (1) z (2) )e− f (z

(1) ,z (2) ;k)

dz (2) .

Thus I (k) defined by (100) has a simple pole at k = 0, except when h02?i , 0i = ? , 0i where I (k) can be extended analytically at the origin. This suggests the −h0−2 i existence of convenient valleys at infinity such that the integral (99) still converges for k = 0.∗ ∗ We

formulate a conjecture. The integral Z (1) (2) 8(k) = e−k f (z ,z ;0) g(z (1) , z (2) ) dz (1) ∧ dz (2) 0

might be defined along a (steepest-descent) chain around the connected component z (1) = 0 of the special fiber f (·; 0) = 0, whose image in f would be a path λ starting at infinity along the half-line L 0 , running R ∨ around the origin, and returning to infinity. This integral could be recast as 8(k) = λ e−kt ϕ (t) dt, where

240

DELABAERE and HOWLS

0

log10 |Hyperterms|

−1 −2 −3 −4 −5 0

5

10

15

20

Figure 10. As for Figure 9, but with the terms ordered according to absolute size

Taking for granted that our oscillating integrals can be defined without Hypothesis H2, can we expect an analogy of Theorem 4.1, at least when the critical points at infinity are isolated (see [59], [67])? Consider the integral with boundaries Z (1) (2) I (k) = e−k f (z ,z ;0) dz (1) ∧ dz (2) , (103) 0

where, for instance, 0 = {(z (1) , z (2) ) ∈ [0, +∞[×[1, +∞[}. For 0, integral (103) defines an analytic function in k and can be recast as Z ∞ 2 −2k I (k) = e dt e−kt 1 − . (104) t +2 0 Integral (104) has obvious resurgence properties: t = −2 is the sole singularity in the Borel plane, and, taking into account the translation in the Borel plane induced by the exponential factor e−2k = e−k f (1,0;0) , we see that the location of the singularity arises from the sole bifurcation value corresponding to the critical point at infinity of f (z (1) , z (2) ; 0). Nevertheless, comparing this result with Theorem 4.1, we note a major novelty: the singularity is nonintegrable. Removing Hypothesis H2 thus generates a challenging problem for a geometer—to relate these resurgence properties ∨

O

ϕ (t) is an endlessly continuable major (see [18]), whose singularity ϕ (corresponding microfunction) at the origin could be represented by Z O g(z (1) , z (2) ) dz (1) ∧ dz (2) ϕ (t) ≡ , df γ (t) where γ (t) is a semicycle vanishing at infinity (see [29]).


241

Table 5. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(13) and k = 1 Level Lowest



N (13) = 4

Level 2

Exact

N (13) = 12, N (13, 1) = 2, N (13, 3) = 7 N (13) = 15, N (13, 1) = 5, N (13, 3) = 10, N (13, 1, 0) = 1, N (13, 3, 0) = 1 k=1

Approximation 0.36404397859143+ 0.23691750987697i 0.32105083513787+ 0.19423741497334i 0.32676405233855+ 0.19354284669953i 0.32676199784979+ 0.19354375274768i 0.32676199784980+ 0.19354375274769i

|1 − approx./exact| 1.5 × 10−1 1.5 × 10−2 5.9 × 10−6 < 10−14

0

to the geometry of the phase function f (z (1) , z (2) ; 0) through, for instance, extended Picard-Lefschetz formulae (see [31]). 7.2. Nonisolated critical points (Hypothesis H3) Although the asymptotics of oscillating integrals have been calculated for a class of nonisolated critical points in [76], Hypothesis H3 plays an essential role in all parts of our study, from the proof of Theorem A.2 and the description of the asymptotics (see Theorems 3.1 and 3.2) to the Picard-Lefschetz formulae and the resurgence properties (see Theorem 4.1) and hence the hyperasymptotics. However, we believe it is still possible to generate exact remainder terms by other means. 7.3. “Multiple” critical values (Hypothesis H4) We discuss condition H4 using the following example.∗ For q ∈ C{0}, Z (1) (2) I (q, k, s) = e−k f (z ,z ;q) g(z (1) , z (2) ) dz (1) ∧ dz (2) 0

(105)

with 3

3

f (z (1) , z (2) ; q) = qz (1) z (2) − 4z (1) /3 − z (2) /24 − q 3 /3 and 0 an unbounded chain of integration of real dimension 2. The phase function f has four (isolated) nondegenerate critical points, and a little thought shows that the s

When g = z (1) with s ∈ C, (105) is the solution of the differential equation (∂q2 − k 2 q 4 + 2ksq)I = 0. For s = 0, (105) can be reduced to a Hardy integral (see [34]). Note that, for noninteger s, the ramification of the amplitude function around the complex curve z (1) = 0 enlarges the space of independent contours (see [64] for a similar one-dimensional feature).

∗

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DELABAERE and HOWLS

(13)

log10 |Hyperterms|

0 (13, 3)

−2

−4

(13, 1) (13, 1, 0)

−6 0

5

10

15

20

25

30

(13, 3, 0)

Figure 11. As for Figure 9, but for T(13) , with k = 1

space of allowed contours of integration is generated by four independent cycles.∗ Nevertheless, Hypothesis H4 is here violated. While one of the critical points (the origin) corresponds to the critical value −q 3 /3, the three others have the same critical value +q 3 /3 for their image by f . However, we can still define four Lefschetz 1 , 0 2 , 0 3 , and compute the asymptotics of the corthimbles properly, say, 0− and 0+ + + 3 1 2 responding I− and I+ , I+ , I+ . This suggests that Hypothesis H4 is essentially a technicality. It remains to extend the generalised Picard-Lefschetz formulae to understand the Stokes phenomenon. The 1 , 0 2 , and 0 3 simultaneously when a contour 0− may intersect (the duals of) 0+ + + Stokes phenomenon occurs. To compute the indices of intersection, one can introduce a suitable deformation of f so that the “multiple critical value” splits into distinct critical values.† For near but different from zero, the family f = f + z (1) 1 , 0 2 , and 0 3 are deformed into 0 , 0 1 , 0 2 , and 0 3 , and can be used: 0− , 0+ − + + + + + ? l l ? , 0 i is also the indices h0+ , 0− i (l = 1, 2, 3) are now computable; hence h0+ − i ? , 0 j i, computable by continuity. Note in this case that the indices of intersection h0+ + i 6= j, are necessarily zero for topological reasons. (The dependence in is regular in the sense of [18].) 3

3

phase function is governed by the monomials z (1) and z (3) near infinity; therefore each variable has three possible asymptotic valleys. This gives a basis of ((3 − 1) × (3 − 1))-cycles. † Note here that Hypothesis H2 remains true under the deformation. In general, under our hypotheses, this is not guaranteed. For instance, although the set of tame polynomials is a dense (constructible) set in the set of polynomials of a given degree (see [11]), it is not open in dimension 3 (see [13]). ∗ The


243

log10 |Hyperterms|

0

−2

−4

−6 0

5

10

15

20

25

30

Figure 12. As for Figure 11, but with the terms ordered according to absolute size. Note that here the final exact result is 0.32676 . . . + i0.19354 . . . , although there are several terms larger than this. Cancellations occur between these larger numbers.

7.4. Confluent cases (Hypothesis H5) We give here a flavour of the difficulty arising from third-type singularities. The main difference between third-type and first- or second-type critical values is primarily topological in nature. When a first- or a second-type critical value is considered, we have seen (see Section 2) that the corresponding Milnor fiber has the homotopy of a bouquet of µ spheres, defining a single group of vanishing homology. The situation is quite different for a third-type critical point, where different groups of vanishing homologies can be defined, depending on which strata are considered, with each of these groups playing a role. Figure 13 exemplifies this situation for n = 3. Here z α is a nondegenerate critical point for f ( f α ∈ 3() ), but the same z α is also a nondegenerate critical point for f |S1 and f |S2 . To such a geometry correspond four vanishing homologies: the vanishing cycle γ (the sphere) is a generator of the vanishing homology group H2 (X αt ), the vanishing cycle γ1 (resp., γ2 ) generates the vanishing homology H1 (X αt ∩ S1 ) (resp., H1 (X αt ∩ S2 )), and γ12 is a basis for the remaining vanishing (reduced) homology group H0 (X αt ∩ S1 ∩ S2 ). These four groups can be understood in terms of our “allowed” cycles of integration. In Figure 13, consider the ball B1 , bounded by the vanishing sphere, as a representation of the Lefschetz thimble. Consider similarly one of the two half-balls 1 (resp., B 2 ), bounded by the vanishing sphere and S (resp., S ), and finally one B1/2 1 2 1/2 of the four quarter-balls B1/4 , bounded by the sphere, S1 , and S2 . Then, obviously, all possible allowed (localized) chains of integration can be described as combinations

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DELABAERE and HOWLS

1 , B 2 , and B (with integer coefficients) of B1 , B1/2 1/4 . Hence we have the following 1/2 lemma.

LEMMA 7.1 The local homology is a free Z-module of finite rank, isomorphic to the direct sum of H2 (X αt ), H1 (X αt ∩ S1 ), H1 (X αt ∩ S2 ), and H0 (X αt ∩ S1 ∩ S2 ).

With this kind of decomposition in hand, one can then concentrate on the integral representation. With the notation of Section 3.2 and in convenient local coordinates, the phase function f reads f = s (1) + · · · + s (minα −1) + F(s (minα ) , . . . , s (n) ),

(106)

where F has an (isolated) critical point at the origin, while the (considered) boundary is given by the set of local equations s (1) = 0, . . . , s ( p) = 0, with minα ≤ p. Section 3.2 suggests a way to derive the asymptotics by reducing step by step the multiple integral into a one-dimensional Laplace integral. But a new difficulty arises in the confluent case: the reduction process means considering integrals of differential quotient forms where the quotients are differential forms d f (1,...,q) that vanish at the critical point as soon as q ≥ minα . This makes the analysis of the analytic behaviour of the Borel transform Jbα (t) (analogous to (46)) more complicated. It seems that a result similar to Theorem 3.2 works also in this case. (1) The asymptotics are essentially described by Gevrey-1 resurgent (see [23], [18]) asymptotic expansions. Generically, its leading term is Const/k minα +1/d , where d is the distance to the Newton diagram of F (cf. Figure 14; see [71]). (2) The asymptotics are essentially governed by the geometry of the singularity (monodromy of the vanishing homologies). The first assertion boils down to proving that the integral may be written as a Laplace transform of a solution of a Picard-Fuchs differential equation with (at most) a regular singular point at the origin; similar results have been proved in [52]. To us, to provide a precise statement of the second assertion seems to be much harder. It is worthwhile noting here that the class of integrals considered by Kaminski and Paris [40], [41] (see also [49]) enters into the framework of our third-type critical values (when the phase function is a polynomial function). This class of integrals can thus be used to experiment. Consider, for instance, the function of the example in Section 2.4, 4 2 3 f (z (1) , z (2) ) = z (1) + z (1) z (2) + z (2) . (107) We have seen that the origin in C2 is an isolated critical point with µ = 5 for its Milnor number. We concentrate on Z (1) (2) I (k) = e−k f (z ,z ) dz (1) dz (2) , (108) 0


245

S2

γ2

γ

γ1

S1

γ12 Figure 13. Confluent case

with 0 = [0, +∞] × [0, +∞], so that the origin is a critical point of type three. As shown in [40], the asymptotics of I (k) when k → +∞ depend heavily on the Newton diagram of the singularity f . In the terminology of Kaminski and Paris, we are here in the one-internal-point case, and the point (1, 2) lies behind the back face of the Newton diagram. Following the key idea of Kaminski and Paris, we transform our integral representation into a new Mellin-Barnes-type integral representation I (k) =

k −7/12 24iπ

Z

i∞

−i∞

0(t)0

1 − t 1 − 2t 0 k −t/12 dt, 4 3

(109)

where the path of integration avoids the origin to the right. The asymptotics are now simply obtained by taking into account the right-hand poles of the integrand.∗ They occur at the points t = 4 j + 1 and t = (3 j + 1)/2, with a sequence of double poles at t = 12 j + 5, j ∈ N. This yields J = I1 + I2 + I3 for the asymptotic expansion, Mellin-Barnes representation can also be used to compute the behaviour of I (k) near the origin: I (k) is analytic on the twelve-fold covering of C\{0}, that is, the Riemann surface of k 1/12 .

∗ The

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DELABAERE and HOWLS

where I1 :=

k −2/3 3

k −5/8 I2 := 8 I3 :=

k −1 24

X j∈N, j−1∈3N /

X j∈N, j−3∈8N /

X j∈N

1 + 8j (−1) j 0(1 + 4 j)0 − k − j/3 , 0( j + 1) 3 (−1) j 1 + 3 j 1 − 3 j − j/8 0( )0 k , 0( j + 1) 2 8

0(5 + 12 j) 0(2 + 3 j)0(4 + 8 j)

× ln(k) − 129(5 + 12 j) + 89(4 + 8 j) + 39(2 + 3 j) k − j

(110)

(9 denotes the digamma function (see [57])). It is easy to check that each of the series expansions in J is Gevrey-1, as expected (see assertion (1) above), but more can be said. While the asymptotics are governed by the local properties of f near the critical point, the reduction to a Mellin-Barnes integral representation proceeds from global information, and, as shown in [40], the asymptotics remain valid in a whole open sectorial neighbourhood at infinity with opening | arg k| < 23π/2. We thus deduce (Watson theorem; see [47]) that J is actually Borel resummable for an argument of summation running over ] − 11π, 11π[. One may guess here that when | arg k| = 11π, then the deformed (steepest-descent) chain of integration 0 encounters the other critical point, so that the minor of J has the corresponding negative critical value −1/19683 as singularity for its analytic continuations, thus inducing a Stokes phenomenon. To illustrate assertion (2) above, it is interesting to compare asymptotics (110) to what we would get in the absence of a boundary. The so-called spectral set of r (see [3]) of Theorem 3.1 can be computed in various ways,∗ but the best references here are certainly the works of A. Varchenko [69], [70]. Taking into account that only 4 2 the quasi-homogeneous part z (1) + z (1) z (2) (with type (1/4, 3/8) and weight 1) of f plays a role, one can directly apply known results from [3, Section 13, Theorem 5], which gives {−3/8, −1/8, 0, 1/8, 3/8} for the spectral set. Moreover, all the s of Theorem 3.1 are less than or equal to n − 1 = 1. We thus get I (k) ∼ J (k) when |k| → +∞ with J (k) =

1 X

X

s=0 r ∈{−3/8,−1/8,0,1/8,3/8}

Tr,s (k)

(ln k)s k r +1

(111)

is known, for instance (cf. [46]), that the eigenvalues of the monodromy are exp(−2iπλ), where the λ are the zeros of the polynomial function e b(λ) related to the Bernstein polynomial b(t) by b(t) = (t +1)e b(t). This polynomial can be computed directly from the Newton diagram (cf. [8]), giving the exponents up to translations. ∗ It


247

d Figure 14. Newton diagram of the singularity. The dots correspond to the cosets of the monomials 2

3

1, z (2) , z (1) , z (1) , z (1) after multiplication by z (1) z (2) , which may be chosen as a C-basis for the Milnor algebra O/(∂ f ). The distance to the Newton diagram is d = 8/5.

and Tr,s (k) ∈ C[[k −1 ]]. Comparing now (110) with (111), one can remark that the presence of a boundary has enriched the spectral set; while only the quasihomogeneous part of the phase function plays a role in the unbounded case, in the bounded case the two faces of the Newton diagram must be taken into account. 7.5. Integrals and differential equations When unbounded integration contours are considered, integrals (1) with polynomials f and g belong to a class of functions known as “Bernstein functions”; that is, they satisfy a system of holonomic differential equations in C[k]h∂k i (cf. [63]). It is beyond our scope to study in detail how this property extends to our integrals with boundaries. We discuss our example here (see Section 6) only from the viewpoint of differential equations. This suggests new links between our results and those developed in [55] or [44] and gives a new insight into the hierarchy property. The basic integral (84) obviously satisfies L0 (k, ∂k ) I00 (k) = 0,

with L0 (k, ∂k ) =

d 2 + 11 + , dk k

(112)

while (86) is not only a solution of the second-order differential equation L1 (k, ∂k )L0 (k, ∂k ) I0(1) (k) = 0,

with L1 (k, ∂k ) =

d 5 + 39/4+3i+ (113) dk 2k

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DELABAERE and HOWLS

but also, more precisely, a solution of the inhomogeneous equation √ iπ 3/2 5 − 12i −(39/4+3i)k L0 (k, ∂k ) I0(1) (k) = e . k 5/2

(114)

It is important to note that in (114) the homogeneous part is a property of the phase function, while the inhomogeneous part is a consequence of the boundary. This inhomogeneity provides a new insight into the hierarchy property as stated in Theorem 4.1. The characteristic equation associated with the irregular singular point at infinity of (113) shows that 5/4 − 3i is the sole (possible) singularity for the Borel transform of I0(1) . Appealing now to resurgence and using the fact that the dotted alien derivative ˙ ω commutes with the usual differentiation (cf. [23], [12], [16]), (114) yields∗ 1 iπ 3/2 √5 − 12i −(39/4+3i)k ˙ ˙ e = 0. (115) L0 (k, ∂k )(1(5/4−3i) I0(1) ) = 1(5/4−3i) k 5/2 The inhomogeneous part thus disappears, and one obtains ˙ 1 (5/4−3i) I0(1) = κ(1),() I00 ,

(116)

where κ(1),() ∈ C is the Stokes multiplier. More interestingly, (90) satisfies Z +∞ π i 3 1 L0 (k, ∂k ) I0(12) (k) = 2 e−(1+6i)k − + −i e−kt dt √ , k 2 k 35/4 − 3i − t 0 (117) 1 − 25i/2 i L1 (k, ∂k )L0 (k, ∂k ) I0(12) (k) = π e−(1+6i)k + 2 , (118) 3 k 2k L2 (k, ∂k )L1 (k, ∂k )L0 (k, ∂k ) I0(12) (k) = 0, (119) d 3 1 with L2 (k, ∂k ) = + 1 + 6i + + . (120) dk k (1 − (25 + 2i)k)k Starting with (119), a third-order linear differential equation with a singularity of rank one at infinity, we see that (35/4 − 3i) and (10 − 6i) are the two possible adjacent singularities. The one-way adjacency (hierarchy property), which is hidden in (119), clearly appears in the lower-order differential equations (117) and (118). One can analyse each Stokes phenomenon as before. We see, for instance, that the right-hand part of (117) encounters a Stokes phenomenon due to the singularity at t = 35/4 − 3i; replacing the functions by their asymptotic series expansion and applying the alien ˙ derivatives 1 (35/4−3i) to (117), we obtain an equation similar to (114).

∗ With

an abuse of notation, in (115) and (116), I0(1) and I00 stand for their asymptotics.


249

Appendices A. The space of homology classes We detail here the proofs for Section 2. We freely appeal to homology and geometric integration theories. We refer readers unfamiliar with these topics to, for instance, [75], [68], [60], and [14] (see also [37] for a physicist’s approach). We recall that S1 , S2 , . . . , Sm are m ≤ n smooth irreducible affine hypersurfaces. A.1. Chains of integration and their homology classes First we describe the space of allowed endless contours of integration running between valleys at infinity where 0, Sc+ = Sc+ (θ ) = t ∈ C, 0 such that the set {z ∈ Cn , k grad f (z) k≤ δ} is compact, or even M -tame (see [51]). Of course, the sole tameness condition on f does not prevent critical points at infinity for the restricted function 2 f | on the strata (just consider f (z (1) , z (2) , z (3) ) = z (1) z (2) − z (1) z (3) + z (2) (z (3) − 1) with the boundary z (3) = 1), and this has to be checked otherwise.† Under Hypothesis H2, f realises a topological trivial stratified fibration outside an open ball B ⊂ Cn of large enough radius (topological triviality at infinity). The following theorem is then a direct consequence of the Thom-Mather first isotopy lemma (see, e.g., [27], [60], [20]). A.1 f| We consider S with its natural stratification; then the mapping Cn \ f −1 (3) −→ C\3 is a topological locally trivial stratified fibration. THEOREM

Thus, for all t0 ∈ C\3, there exists an open neighbourhood U ⊂ C\3 of t0 , a stratified set V , and a homeomorphism ϕ : f −1 (U ) → V × U which maps any stratum of f −1 (U ) onto the product of a stratum of V by U . This result is central to our analysis of the homology through its main consequence, the lifting property. As a first consequence, if c > c0 > 0 are large enough that f −1 (Sc+0 ) does not contain any element of 3, then the natural mapping Cn , f −1 (Sc+ ) ,→ Cn , f −1 (Sc+0 ) (A.3) as well as S, S ∩ f −1 (Sc+ ) ,→ S, S ∩ f −1 (Sc+0 ) can be considered as equivalences of homotopy. Thus the complex of chains (resp., C?9 (S)) can be identified with the projective limit C?9 (Cn ) = lim proj C? Cn , f −1 (Sc+ ) ,

(A.4) C?9 (Cn )

c→+∞

respectively, C?9 (S) = lim proj C? S, S ∩ f −1 (Sc+ ) . c→+∞

S S strata are Cn \S, Si \ j6=i Si ∩ S j , Si ∩ S j \ k6=i, j Si ∩ S j ∩ Sk , and so on (see [20]). We have no general practical way of doing this, the gradient tool not being available, as a rule.

∗ The †

(A.5)


We therefore get the following projective limit of isomorphisms: Hn9 (Cn , S) = Hn Cn ; S, f −1 (Sc+ ) = Hn Cn , S ∪ f −1 (Sc+ )

251

(A.6)

for all c large enough. Now Cn being contractible (and working with a reduced homology), we finally get the isomorphism ∂ Hn9 Cn , S) → Hn−1 (S ∪ f −1 (t)

(A.7)

for t ∈ Sc+ . In the absence of boundaries and if f has only isolated critical points, it is well known that the (reduced) homology Hn−1 ( f −1 (t)) (hence Hn9 (Cn )) of the generic fiber f −1 (t) is a free Z-module of finite rank, its rank µ f ∗ being the sum of the Milnor numbers at the critical points of f in Cn . In the language of the saddle-point method, this translates to being able to decompose the unbounded chain of integration into a chain of steepest-descent n-folds. Following (A.7) and from an algebraic viewpoint, analysing Hn9 (Cn , S) reduces to computing the total Milnor number of the (generic) fiber P = 0, where P is the Qn polynomial ( f − t) i=1 Pi (each Si being defined by the polynomial Pi ). We, however, follow another way for two reasons: (1) our hypotheses (especially Hypothesis H2) do not translate easily in terms of fibration properties of P; (2) having the saddle-point method in mind, it is useful to understand Hn9 (Cn , S) directly in terms of steepest-descent cycles. We do this in Section A.3. A.3. Localization at the target Following ideas developed in [63], in the complex plane we draw the family (L α ) of closed half-lines L α = f α + eiθ R+ for all f α belonging to 3. Assume also that θ has been chosen generically so that no Stokes phenomenon is currently occurring; that is, all these half-lines are two-by-two disjoint. For all f α ∈ 3, let Tα be a closed neighbourhood of L α , retractable by deformation onto L α . It is assumed that all these Tα are disjoint from one another, as shown in Figure A.1. The reduction process hereafter draws heavily on the work of Pham [63], and so the discussion is brief. We start with (A.6), construct a deformation-retraction of C onto Sc+ ∪ fα Tα , and lift it by f (by virtue of Theorem A.1). This gives [ [ Hn Cn , S ∪ f −1 (Sc+ ) = Hn f −1 (Tα ∪ Sc+ ), ( f |S)−1 (Tα ) ∪ f −1 (Sc+ ) . fα

fα

(A.8) ∗ This

is the total Milnor number, usually defined as µ f = dimC C[z]/(∂ f ).

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DELABAERE and HOWLS

Tα fα

Lα

Sc+

Figure A.1. The family of half-lines L α and their closed neighbourhoods Tα for θ = 0

By excision and deformation-retraction (using Theorem A.1), we have [ [ ( f |S)−1 (Tα ) ∪ f −1 (Tα ∩ Sc+ ) Hn Cn , S ∪ f −1 (Sc+ ) = Hn f −1 (Tα ), fα

=

M

Hn f

fα −1

(Tα ), ( f |S)−1 (Tα ) ∪ f −1 (Tα ∩ Sc+ ) .

f α ∈3

(A.9) For each f α ∈ 3, let Dα be an open disc centered at f α with a very small radius r and Dα+ (θ) = Dα ∩ { 0). This means that the restricted function f | f −1 (Dα ) ∩ Bα from f −1 (Dα ) ∩ Bα to Dα is a trivial fibration, thus allowing us to analyse the homology group by localisation at the source. We use the notation X α = Bα ∩ f −1 (Dα ), X α+ = Bα ∩ f −1 (Dα+ (θ)). It follows from our hypotheses that Hn f −1 (Dα ), ( f |S)−1 (Dα ) ∪ f −1 (Dα+ (θ)) = Hn f −1 (Dα ), f −1 (Dα+ (θ )) (A.11) by an easy argument of deformation-retraction. Moreover, from [63] or [3, Section 11], one has the following isomorphisms: ∂ Hn f −1 (Dα ), f −1 (Dα+ (θ)) = Hn (X α , X α+ ) → Hn−1 (X αt )

(A.12)

for all t ∈ Dα+ (θ). From Milnor [50] again, we obtain that the fibre X αt has the homotopy type of a bouquet (wedge) of µ spheres, where µ = µα is the Milnor number of the critical point z α ; hence Hn−1 (X αt ) = Zµ , the so-called vanishing homology of the critical

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DELABAERE and HOWLS

point. When µ = 1, the relative homology Hn (X α , X α+ ) is generated by the Lefschetz thimble 0αt , as shown in Figure 1. A.4.2. Critical values from the boundary: Second-type case We concentrate now on the case where f α is of the second type, with depth p > 0. Up to a reordering, we can assume that f α belongs to 3(1,..., p) . We introduce as above the sets X α = Bα ∩ f −1 (Dα ), X α+ = Bα ∩ f −1 (Dα+ (θ )), and X αt = Bα ∩ f −1 (t) for t ∈ Dα (θ). We write also Yi = Bα ∩ ( f |Si )−1 (Dα ) = X α ∩ Si . Note that Yi = ∅ for all i ∈ / {1, . . . , p} as a consequence of our hypothesis. −1 The restricted function f | f (Dα ) ∩ Bα now realises a trivial fibration of the stratified set. Analysing the homology thus reduces to a local analysis near the critical point z α , and copying arguments in [63, Section I.3], we obtain Hn f −1 (Dα ), ( f |S)−1 (Dα ) ∪ f −1 (Dα+ (θ)) = Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) (A.13) as a preliminary result. Following ideas developed in [60, Section 5.2], we now reduce the homology step by step so as to reach the vanishing homology. We first use the fact that X α and Y1 ∪ · · · ∪Y p are contractible;∗ hence the exact sequence of a triple and a deformationcontraction argument yield the isomorphism ∂ Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) → Hn−1 X αt , X αt ∩ (Y1 ∪ · · · ∪ Y p ) , [0] 7→ ∂[0]

(A.14)

for t ∈ Dα+ (θ), where ∂ is the boundary operator that selects the part of the boundary lying on X αt . Second, we observe that both X αt and X αt ∩ (Y2 ∪ · · · ∪ Y p ) are contractible, so using the exact sequence of a triple again gives the isomorphism ∂1 Hn−1 X αt , X αt ∩(Y1 ∪ · · · ∪Y p ) → Hn−2 Y1 ∩X αt , Y1 ∩X αt ∩(Y2 ∪ · · · ∪Y p ) , (A.15) where ∂1 is the boundary operator that takes the part of the boundary lying on Y1 . The same argument can be used p times, yielding the following sequence of isomorphisms: Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) ∂

→ Hn−1

(A.16)

X αt , X αt ∩ (Y1 ∪ · · · ∪ Y p )

∂1 → Hn−2 Y1 ∩ X αt , Y1 ∩ X αt ∩ (Y2 ∪ · · · ∪ Y p ) ∂i ∂2 → · · · → Hn−i−1 Yi ∩ · · · ∩ Y1 ∩ X αt , Yi ∩ · · · Y1 ∩ X αt ∩ (Yi+1 ∪ · · · ∪ Y p ) ∂i+1

∂ p−1

→ · · · → Hn− p (Y p−1 ∩ · · · ∩ Y1 ∩ X αt , Y p−1 ∩ · · · Y1 ∩ X αt ∩ Y p ),

∗

Just use the local representation (8).

(A.17)


255

where ∂i is the boundary operator that takes the part of the boundary lying on Yi . Now assuming that p ≤ n − 1 and taking into account that Y p−1 ∩ · · · ∩ Y1 ∩ X αt is still contractible by virtue of Hypothesis H3, it follows from the exact sequence of a pair that ∂p

Hn− p (Y p−1 ∩ · · · ∩ Y1 ∩ X αt , Y p−1 ∩ · · · Y1 ∩ X αt ∩ Y p ) → Hn− p−1 (Y p ∩ · · · ∩ Y1 ∩ X αt ) (A.18) is again an isomorphism. We finally get the isomorphism Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p )

∂ p ◦···◦∂1 ◦∂

−→

Hn− p−1 (Y p ∩ · · · ∩ Y1 ∩ X αt ).

(A.19)

Of course, this isomorphism depends on the chosen ordering of Y1 , . . . , Y p (cf. [5, Chapter 4, Section 1.14]). The homology group Hn− p−1 (Y p ∩· · ·∩Y1 ∩ X αt ) = Hn− p−1 (S p ∩· · ·∩ S1 ∩ X αt ) is the vanishing homology of the critical point; it is isomorphic to the free Z-module Zµα , where µα is the Milnor number of the critical point z α of f (1,..., p) . When µα = 1, the relative homology Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) is generated by the relative Lefschetz thimble 0αt , as described in Figure 2. The case of a corner, where p = n, corresponds to the so-called linear pinching case (see [60]) where ∂ p ◦ · · · ◦ ∂ˇi ◦ · · · ∂1 ◦ ∂[0] is just reduced to a point: no vanishing cycle exists, and the local homology Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) is generated by a single relative Lefschetz thimble (see Figure 3). A.4.3. Concluding theorem Putting the pieces together, we have shown how the homology group Hn9 (Cn , S) of the allowed contours of integration in the presence of boundaries can be decomposed into a direct sum (see Section A.3) of free Z-modules of finite rank (see Section A.4), at least under Hypothesis H5 (no critical value of the third type). We have also demonstrated a natural way to define the rank by reduction to the vanishing homology (if defined) through localisation on each stratum. Moreover, when the closed halflines L α = f α + eiθ R+ ( f α ∈ 3) are two-by-two disjoint (θ is “generic in the Stokes sense”), then a basis is given by the set of (relative) steepest-descent n-folds (0α )α∈3 , where each of the 0α projects by f onto L α ∈ C. We have thus obtained the following theorem. THEOREM A.2 The space Hn9 (Cn , S) of relative homology classes is a free Z-module of finite rank. Moreover, if all the half-lines L α are two-by-two disjoint, then every cycle can be decomposed into a chain of (relative) steepest-descent n-folds (0α )α∈3 of Hn9 (Cn , S).

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B. Duality and Stokes phenomenon The aim of this appendix is to provide the geometrical tools to understand the Stokes phenomenon, as discussed in Section 4. We discuss the now more or less classical Picard-Lefschetz formulae, which help in understanding the Stokes phenomena as singularities in the Borel plane. Then we see how a suitable duality provides a direct insight into the Stokes phenomenon, in the spirit of the usual saddle-point method. We here make use of Hypothesis H6, namely, that all singular points are nondegenerate, with the following comment. It is possible to extend the results hereafter by allowing degenerate critical points. The generalised Picard-Lefschetz formulae, as well as the nondegeneracy properties of the Kronecker index (see B.6), follow from a local analysis near the critical points, and this can be studied in the degenerate case by local generic deformations.∗ However, the information we would obtain does not translate easily into the language of resurgence of asymptotic expansions. B.1. Generalised Picard-Lefschetz We assume that θ has been chosen generically so that no Stokes phenomenon is occurring. We denote by (0α ) fα a basis of relative steepest-descent n-cycles (or relative 9(θ ) Lefschetz thimbles) of Hn (Cn , S) considered in Section 2 (with a given orientation). We introduce γα (t) = ∂ 0α (t) ∈ Hn−1 X αt , X αt ∩ (S1 ∪ · · · ∪ Sm ) (B.1) as well as its corresponding vanishing cycle. If f α belongs to 3(i1 ,...,iq ) , then eα (t) = ∂iq ◦ · · · ◦ ∂i1 γα (t) ∈ Hn−q−1 (Siq ∩ · · · ∩ Si1 ∩ X αt ),

(B.2)

which vanishes when t → f α along L α (θ). Let t ? be a fixed regular point in the half-complex plane Sc+ (θ) for c > 0 large enough. We denote by (lα ) a system of paths, where lα starts from t ? and travels to f α along a straight line. This system (lα ) is a so-called distinguished system of paths (see [3]), where the cycle eα (t) vanishes when t → f α along lα . For each lα we associate a closed path `α starting from t ? , following lα , running around f α in the positive sense, and returning to t ? along lα (see Figure B.1). This defines a basis (`α ) fα of the free fundamental group π1 (C\3, t ? ). Hence the variation of the homology when t runs along a loop in C\3 with t ? as base point reduces to a description of what happens for each of the `α . We now apply the generalised Picard-Lefschetz formula described in [60] (see also [72], [5]). Starting with the cycle γα (t), we follow its deformation when t runs ∗ Such

local Morsification does not bring into play any global deformation of f , which may destroy Hypothesis H2 (cf. [13]).


257

Lα

fα t∗

fβ

`β

Figure B.1. The basis (`α ) f α

along the path lβ . We consider the trace γα? of γα (t) in the Milnor ball Bβ as beF (X t \X t ∩ (S ∪ · · · ∪ S )), the closed homology group dual from longing to Hn−1 m 1 β β Hn−1 (X βt , X βt ∩ (S1 ∪ · · · ∪ Sm )) by the Poincaré duality (covanishing homology). We can now define the variation operator Var when t goes around f β along `β . Assuming that f β belongs to 3(1,..., p) , [60] now yields (

Var γα? = καβ γβ , καβ = (−1)(n− p)(n− p+1)/2 heβ , α i,

(B.3)

where h , i is the (Kronecker) index of intersection. The cycle α ∈ Hn− p−1 (S1 ∩ · · · ∩ S p ∩ X βt ) is deduced from γα? by γα? = δ1 ◦ · · · ◦ δ p α ,

(B.4)

where δi denotes the Leray coboundary operator (cf. [42]) with respect to Si . Moreover, the index of self-intersection is given by ( 2(−1)(n− p)(n− p−1)/2 if n − p is odd, heβ , eβ i = (B.5) 0 if n − p is even.

Remarks. When m = n (corner critical point) and f β belongs to 3(1,...,n) , then the vanishing cycle eβ does not exist. We can give a meaning to the previous equations with the convention eβ = 0; hence heβ , α i− = 0. One can remark also that if f α belongs to 3(i1 ,...,iq ) , equalities (B.3) and (B.4) show that when {1, . . . , p} is not a subset of the set {i 1 , . . . , i q }, then γα? has a trivial variation around f β (i.e., καβ = 0).

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B.2. Duality from the viewpoint of Laplace integrals The generalised Picard-Lefschetz formulae recalled in Section B.1 follow from the duality between vanishing homology and covanishing homology. Here we realise this duality directly from the viewpoint of Laplace integrals by describing the dual space of the homology group Hn9 (Cn , S) with which we started. 9(θ ) We consider a relative n-cycle 0 ∈ Hn (Cn , S) with support in 9(θ) and 9(θ +π ) an n-cycle 0 0 ∈ Hn (Cn \S) with support in 9(θ + π ). Up to a deformation by isotopy, these two cycles can be assumed to be in a general position. It follows now by definition that the intersection 9(θ) ∩ 9(θ + π ) of the families of supports 9(θ) and 9(θ + π) is the family of compact subsets of Cn . This implies that the intersection in Cn of 0 with 0 0 defines a 0-cycle with compact support and hence an integer because H0 (Cn ) = Z. This allows us to define the index of intersection h0 0 , 0i by the bilinear map (B.6) h , i : Hn9(θ+π ) (Cn \S) ⊗ Hn9(θ) (Cn , S) → Z. This is a direct generalisation of the Kronecker index introduced in [63]. Moreover, arguments based on [63] show the following lemma. LEMMA B.1 The bilinear map (B.6) is nondegenerate. 9(θ)

9(θ+π )

This induces a duality between Hn (Cn , S) and Hn (Cn \S). We recall here that θ has been chosen generically so that the lines L α (θ) ∪ L α (θ + π) are disjoint. We first remark that our bilinear map is diagonal for the decomposition of the homology shown in Appendix A. This allows us to localise the study. Assuming that f α belongs to 3(1,...,q) , we focus on Hn (X α , Sq ∪ · · · ∪ S1 ∪ X α+ ). Following Section A.4.2, this group is isomorphic to Hn−q (Sq ∩· · ·∩ S1 ∩ X α , Sq ∩· · ·∩ S1 ∩ X α+ ) via the isomorphism ∂q ◦ · · · ◦ ∂1 . Returning to the notation of Section A.4.2, we now define Dα− = Dα+ (θ + π) = Dα ∩ < (t − f α )e−i(θ +π) ≥ r/2 (B.7) and, respectively, X α− . From [63, Section I.5] (see also the remark hereafter), we know that the spaces of homology Hn−q (Sq ∩ · · · ∩ S1 ∩ X α , Sq ∩ · · · ∩ S1 ∩ X α+ ) and Hn−q (Sq ∩ · · · ∩ S1 ∩ X α , Sq ∩ · · · ∩ S1 ∩ X α− ) are dual. 9(θ ) As a consequence, each n-cycle 0 ∈ Hn (Cn , S) can be decomposed as X 0= h0α? , 0i0α (B.8) fα

with respect to the basis (0α ) fα of (oriented) relative Lefschetz thimbles, where (0α? ) fα is the dual basis (hence h0α? , 0α i = +1). Figure B.2 describes this duality for the Airy pattern.


259

Remark. A “concrete dual basis” can be built as follows. We start with a basis of 9(θ ) relative Lefschetz thimbles (0α ) fα of Hn (Cn , S) with the standard orientation. Assuming that f α belongs to 3(1,..., p) , we introduce χα = ∂ p ◦ · · · ◦ ∂1 0α ∈ Hn− p (S p ∩ · · · ∩ S1 ∩ X α , S p ∩ · · · ∩ S1 ∩ X α+ ).

(B.9)

We define χα? ∈ Hn− p (S p ∩ · · · ∩ S1 ∩ X α , S p ∩ · · · ∩ S1 ∩ X α− ) as this cycle deduced from χα by rotating the direction from eiθ to ei(θ+π) on S in the positive sense; then χα (resp., χα? ) can be identified with the ascent (resp., descent) 2 2 2 gradient surfaces of a Morse function F = x ( p+1) + · · · + x (n) − y ( p+1) − 2 · · · − y (n) of index n − p in R2(n− p) = C(n− p) . We compare the orientation (∂/∂ x ( p+1) , . . . , ∂/∂ x (n) , ∂/∂ y ( p+1) , . . . , ∂/∂ y (n) ) with the canonical orientation (∂/∂ x ( p+1) , ∂/∂ y ( p+1) , . . . , ∂/∂ x (n) , ∂/∂ y (n) ) of C(n− p) . This yields hχα? , χα i = (−1)(n− p)(n− p−1)/2 .

(B.10)

Using now the Leray coboundary isomorphisms, we can define 0α? = δ1 ◦ · · · ◦ δ p χα? ,

(B.11)

9(θ+π )

which extends as an element of Hn (Cn \S). Using (B.10), we thus obtain a basis 9(θ+π ) (0α? ) fα of n-cycles of Hn (Cn \S) dual to (0α ) fα . Hill

Valley

Valley

01∗

02

01

0a

a

02∗

Hill

0a∗ Hill

Valley

Figure B.2. The 1-cycles (01 , 0−1 , 0a ) as a basis of H19 (C, a) ? , 0 ? ) for the Airy pattern and its dual basis (01? , 0−1 a

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Delabaere Département de Mathématiques, Unité Mixte de Recherche Centre National de la Recherche Scientifique 6093, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers CEDEX 01, France; [email protected] Howls Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom; [email protected]


THE TOPOLOGICAL IHX RELATION, PURE BRAIDS, AND THE TORELLI GROUP SYLVAIN GERVAIS and NATHAN HABEGGER

Abstract We prove that the filtration on the pure braid group on g strands, induced by the lower central series of the Torelli group of a genus g surface with one boundary component, coincides with its lower central series, shifted by one. In particular, the cubic Jacobi relations in the pure braid group are quadratic relations in the Torelli group. 1. Introduction and statement of results The study of the Torelli subgroup of the mapping class group has progressed in recent years. Beginning with the work of D. Johnson [J1], [J2], [J3] in the early 1980s, which computed its abelianization and gave a finite set of generators, and continuing with the work of S. Morita [Mo1] on the Casson invariant and higher Johnson homomorphisms, the work of S. Garoufalidis and J. Levine [GL] has tied the Torelli group and Johnson homomorphisms to the rich theory of perturbative (or finite-type) 3-manifold invariants (see [O], [Le], [LMO], [G], [Ha]). From another direction, a major leap forward in our understanding is the work of R. Hain [Hn], who gave a presentation of the Malcev Lie algebra of the Torelli group. At the group level, a fundamental open problem is to determine whether the Torelli group is finitely presentable or not. Indeed, to our knowledge, there does not currently exist any presentation of the Torelli group with explicit sets of generators and relators in genus greater than or equal to 3.∗ We still lack an understanding of the topological nature of relations within the Torelli group.† In this paper we exhibit a relation, known as the IHX relation in perturbative theory, that exists in the Torelli group of a surface of genus g ≥ 3. This relation seems to have appeared at least implicitly in calculations of Morita [Mo1]. To see how this relation comes about, recall from [H] that the topological IHX DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 21 November 2000. Revision received 9 May 2001. 2000 Mathematics Subject Classification. Primary 57M27, 57M99, 57R50, 20F36, 20F38, 20F05. ∗ In

genus 2, the Torelli group is free (see [M]). is the obvious commutativity of diffeomorphisms with disjoint support.

† There

265

266

GERVAIS and HABEGGER

relation in the theory of homology spheres can be considered as arising from the Jacobi relation in the pure braid group on 4 strands. We denote by 6g,r a surface of genus g with r boundary components, and we denote by Mg,r its mapping class group. Note that M0,g+1 is the framed pure braid group on g strands. There is an embedding of 60,g+1 in 6g,1 (given by considering 6g,0 to be the double of 60,g+1 and removing a disk). Then it turns out that the cubic Jacobi relations in M0,g+1 induce quadratic relations in the Torelli group Tg,1 . Before stating our main result, we recall that Mg,1 comes equipped with a filtration · · · ⊂ Mg,1 [2] ⊂ Mg,1 [1] = Tg,1 ⊂ Mg,1 [0] = Mg,1 , where Mg,1 [n] is the subgroup of Mg,1 acting trivially on F/Fn+1 and where F = π1 (6g,1 ); and for a group G, G n denotes its lower central series.∗ We denote by P(g) the pure braid group on g strands. THEOREM 1.1 Let P(g) → Mg,1 denote the standard inclusion (defined above; see also Section 2). Then for all n ≥ 1,

P(g)n+1 = P(g) ∩ (Tg,1 )n = P(g) ∩ Mg,1 [n]. L Let H denote F/F2 , and let L(H ) = ∞ n=1 L n (H ) denote the free Lie algebra on the Z-module H . Define Dn (H ) to be the kernel of the Lie bracket map [ , ] : H ⊗ L n (H ) → L n+1 (H ). The degree n Johnson homomorphism (see, e.g., [GL]) is a map Jn : Mg,1 [n] → Dn+1 (H ) whose kernel is Mg,1 [n + 1]. One has (Tg,1 )n ⊂ Mg,1 [n], and it is an open problem to study the map jn0 : (Tg,1 )n /(Tg,1 )n+1 → Dn+1 (H ). 1.2 The homomorphism jn0 is injective on the image of P(g)n+1 = P(g) ∩ (Tg,1 )n in (Tg,1 )n /(Tg,1 )n+1 . COROLLARY

Recall that in any group, one has the Jacobi identity c a , [b, c] cb , [a, b] ba , [c, a] = 1, ∗ The

lower central series of a group G is defined by G 1 = G, and inductively, G k+1 = [G, G k ] is the subgroup of G k generated by the commutators of G and G k .

PURE BRAIDS AND THE TORELLI GROUP

267

where x y denotes yx y −1 and [x, y] denotes x yx −1 y −1 . 1.3 Every Jacobi relation in the pure braid group is a quadratic relation in the Torelli group. COROLLARY

Remark. Theorem 1.1 (and hence Corollary 1.3) holds also for a surface 6 = 6g,0 without boundary, where Mg [n] is defined to be the image of Mg,1 [n] under the natural map Mg,1 → Mg = Mg,0 . Note first that P(g) ⊂ M0,g+1 → Mg is an inclusion (see, e.g., [PR]). Thus, we have P(g)n+1 = P(g) ∩ (Tg )n = P(g) ∩ Mg [n]. To get an equivalent statement of Corollary 1.2 in the closed case, we just have to replace Dn (H ) by a suitable quotient of Dn0 (H ), the kernel of the Lie bracket map [ , ] : H ⊗ Ln (g) → Ln+1 (g), where Ln (g) = π1 (6g )n /π1 (6g )n+1 (see [Mo3]). 2. The standard embedding of 60,g+1 in 6g,1 There are a number of models available for a surface 6g,1 , but one which we find convenient for drawing pictures is the description as a 2-dimensional handlebody with a single index 0-handle and 2g orientation-preserving index 1-handles grouped together in interlaced pairs. Let B denote a 2-disk (considered as the 0-handle) such that its boundary ∂ B is divided into two segments ∂+ B, ∂− B. We choose a point of ∂− B as the base point. Consider a collection p1 , . . . , p4g of points in ∂+ B ordered using an orientation β of ∂+ B. We attach orientation preserving 1-handles, Hiα (resp., Hi ) along neighborhoods of the 0-spheres { p4i−3 , p4i−1 } (resp., { p4i−2 , p4i }) in ∂+ B. The handlebody Sg β B ∪ ( i=1 (Hiα ∪ Hi )) is a surface 6g,1 of genus g with 1 boundary component. β β The core of the handle Hiα (resp., Hi ) is an arc Aiα (resp., Ai ) whose boundary is the 0-sphere { p4i−3 , p4i−1 } (resp., { p4i−2 , p4i }) which we orient so as to go from p4i−3 to p4i−1 (resp., from p4i to p4i−2 ). β We join the endpoints of the arc Aiα (resp., Ai ) by an arc whose interior lies in β the interior of B. This produces a simple closed curve Ciα (resp., Ci ) which we orient β so as to induce the chosen orientation on Aiα (resp., Ai ). These curves can be chosen β to be mutually disjoint, except that Ciα and Ci meet transversely in a single point (see Figure 1). β We choose paths lying in B to connect the curves Ciα (resp., Ci ) to the base point. This yields well-defined elements of π1 (6g,1 ), which are denoted by αi (resp.,

268


C1α

C gα

β

C1

β

Cg

β

Figure 1. The curves Ciα and Ci

b

α1

β1

b

b

αg

βg

Figure 2. The curves αi and βi

βi ; see Figure 2). Then, the set {αi , βi |i = 1, . . . , g} is a basis of the free group F = F(2g) = π1 (6g,1 ). With these conventions, ∂(6g,1 ) is represented by the element [α1 , β1 ] · · · [αg , βg ]. Sg The subsurface of 6g,1 consisting of B ∪( i=1 Hiα ) is a surface 60,g+1 of genus zero with g + 1 boundary components. If we remove a small collar neighborhood of ∂60,g+1 \ ∂− B, the resulting surface is still denoted by 60,g+1 . The closure of its complement in 6g,1 is now a connected surface, which can be made planar. It has g + 1 boundary components and hence is homeomorphic to 60,g+1 . 6g,1 is just the double of 60,g+1 , slit open along ∂− B to obtain a boundary component. We call this embedding of 60,g+1 in 6g,1 the standard embedding. It was studied by T. Oda [Od] and Levine [L] (see also A. Hatcher and W. Thurston [HT] for an embedding of 60,2g in 6g,0 ). The fundamental group π1 (60,g+1 ) is the free group F(g) on the set {αi |i = 1, . . . , g}, π1 (60,g+1 ) = F(g) ⊂ F(2g) = π1 (6g,1 ). We define simple closed curves Ci, j in 60,g+1 by taking the connected sum of Ciα and C αj along an arc lying in B (see Figure 3). Let τi denote the Dehn twist along


Hiα

β Hi

269

H jα

β

Hj

Ci, j

Figure 3. The curve Ci, j

Ciα , and let τi, j be the Dehn twist along Ci, j . The elements σi, j = τi, j τi−1 τ −1 j are the generators of P(g), the pure braid group on g strands, considered as a subgroup of M0,g+1 = P(g) × Zg . As the τi are central (generating the Zg summand of M0,g+1 ), when writing commutators in the σi, j , we may dispense with the framing corrections τi−1 , τ −1 j and simply write τi, j . We have that the curves Ci, j (and thus the twists τi, j ) are all conjugate in the following sense: there is a diffeomorphism F˜ of 6g,1 which is the identity on ∂6g,1 and which restricts a diffeomorphism F of 60,g+1 to itself and sends Ci, j to C1,2 .∗ Such an F exists since cutting 60,g+1 open along Ci, j results in 2 surfaces, one of which is 60,3 (and hence the other is 60,g ). F can be extended to 6g,1 simply by taking the double of F, slit open along ∂− B. 3. The main construction Consider a simple closed curve D1,2 lying in 62,1 whose homotopy class is the element α1−1 β1−1 α2 β2 . The curve D1,2 may be taken to intersect C1α and C2α transversely in one point. These two intersection points divide D1,2 into two subarcs, one of which may be taken to lie entirely in B and is called A. The other, A0 , can be taken to miss a third arc A0 connecting A to the base point in B. Note that a small neighborhood of A0 ∪ A ∪ C1α ∪ C2α is a surface 60,3 ⊂ 62,1 , isotopic to the standard embedding. 0 the The connected sum of C1α and C2α along A is the curve C1,2 . Denote by C1,2 α α α 0 0 connected sum of C1 and C2 along A (see Figure 4). By pushing A ∪ C1 ∪ C2α into the complement of A0 ∪ A ∪ C1α ∪ C2α , that is, into 62,1 \ 60,3 ⊂ 6g,1 \ 60,g+1 , we 0 along C 0 commutes with elements of M see that the Dehn twist τ1,2 0,g+1 . 1,2 0 0 −1 Since C1,2 and C1,2 are homologous, τ1,2 (τ1,2 ) lies in Tg,1 . necessarily permutes the components of ∂60,g+1 . It thus corresponds to an element of the braid group and not to the pure braid group. ∗F

270


C1,2

D1,2

0 Figure 4. The curves D1,2 and C1,2

4. Proof of the inclusion P(g)n+1 ⊂ (Tg,1 )n Recall that we have P(g) ⊂ M0,g+1 ⊂ Mg,1 . Set P = P(g), Pn = P(g)n , T = Tg,1 , Tn = (Tg,1 )n . Since P ∩ Tn is a normal subgroup of P, it suffices to see that a normal set of generators of Pn+1 lies in Tn . For n ≥ 1, Pn+1 is normally generated by [σi, j , f ] = [τi, j , f ], f ∈ Pn . Recall that f τi, j is the Dehn twist along the image of ci, j by the diffeomorphism f (or rather a representative of its class). f Since f (ci, j ) is homologous to ci, j , [τi, j , f ] = τi, j (τi, j )−1 lies in T . This proves that P2 ⊂ T . Suppose inductively that Pn lies in Tn−1 . 0 commutes with M We first suppose that {i, j} = {1, 2}. Since τ1,2 0,g+1 , we have 0 0 −1 −1 [τ1,2 , f ] = [τ1,2 (τ1,2 ) , f ]. But as τ1,2 (τ1,2 ) ∈ T and f ∈ Pn ⊂ Tn−1 , we have [τ1,2 , f ] ∈ Tn . ˜ F) be a diffeomorphism (cf. Section 2) of the Now let {i, j} 6= {1, 2}. Let ( F, pair (6g,1 , 60,g+1 ) sending ci, j to c1,2 . Then [τi, j , f ] F = [τ1,2 , f F ] ∈ Tn since f F lies in Pn , because conjugation by F sends Pn to itself.∗ It follows that [τi, j , f ] ∈ Tn since Tn is normal in Mg,1 . 5. Proof of Theorem 1.1 We have seen that for all n ≥ 1, P(g)n+1 ⊂ (Tg,1 )n . Since (Tg,1 )n ⊂ Mg,1 [n], we have P(g)n+1 ⊂ (Tg,1 )n ∩ P(g) ⊂ Mg,1 [n] ∩ P(g). ∗ Note

well that f 7→ f F is not an inner automorphism of P(g).


271

To prove equality it is therefore enough to show that M[n] ∩ P ⊂ Pn+1 , where P = P(g) and M[n] = Mg,1 [n]. This was shown by Oda [Od] (see also Levine [L]). For the convenience of the reader, we give the details of the argument, as in [L]. For n = 1, the map P/P2 → M[0]/M[1] = Sp(2g) sends the elements σi, j to j,i the matrix OI Ei, j +E in the basis α1 , . . . , αg , β1 , . . . , βg , (or rather their homology I classes). Since the σi, j are a basis of P/P2 and the E i, j +E j,i are linearly independent, it follows that P/P2 injects into Sp(2g). Thus, we get M[1] ∩ P ⊂ P2 . Let Dn (H ) = ker([ , ] : H ⊗ L n (H ) → L n+1 (H )), where H = H1 (6g,1 ) and L where L(H ) = ∞ n=1 L n (H ) denotes the free Lie algebra on the Z-module H . The degree n Johnson homomorphism (see, e.g., [GL]) is a map Jn : M[n] → Dn+1 (H ) which has kernel M[n +1]. On the other hand, if we set H 0 = H1 (60,g+1 ), the degree n Milnor invariants give a map µn : Pn → Dn (H 0 ) whose kernel is Pn+1 (see, e.g., [HL], [HM]). Thus, the inclusion M[n] ∩ P ⊂ Pn+1 is a result of the following claim by induction. CLAIM

Denote by pn : Dn (H ) → Dn (H 0 ) the morphism induced by the map H → H 0 obtained by forgetting the βi ’s. Then the diagram Pn

⊂

- M[n − 1]

µn ? pn Dn (H 0 )

Jn−1 ? Dn (H )

commutes up to sign. We recall briefly the definitions of the maps µn , Jn . Recall that if F is a free group and H = F/F2 , then Fn /Fn+1 ' L n (H ). Recall that a string link σ induces an automorphism Aσ of F(g)/F(g)n+2 . If we let ì denote the longitude in π = π1 (D 2 × I \ σ ), then Aσ (xi ) = xiλi , where the xi ’s are the free generators of π and λi denotes the image of ì in π/πn+1 ≈ F(g)/F(g)n+1 . Thus, if the λi lie in F(g)n /F(g)n+1 = P 0 0 L n (H 0 ), we define µn (σ ) = i x i ⊗ λi ∈ H ⊗ L n (H ). Actually, µn (σ ) lies in 0 Dn (H ). Similarly, if f is an element of Mg,1 [n], then the induced automorphism f ∗ of β β F(2g)/F(2g)n+2 takes αi to αi ηiα and βi to βi ηi , where ηiα , ηi are elements of P β α F(2g)n+1 /F(2g)n+2 = L n+1 (H ), and we set Jn ( f ) = i αi ⊗ ηi − βi ⊗ ηi . Actually, Jn ( f ) lies in Dn+1 (H ).

272


i + (α1 )

i − (αi )

l1

l1

l1

γ2

γi

li

li

li

γi+1

Figure 5. The mapping cylinder of f σ and the curves li , li0 , li00 , γi

For σ ∈ P, let us denote by f σ the corresponding element in Mg,1 . If C( f σ ) is the mapping cylinder of f σ , then C( f σ ) is a handlebody of genus 2g (see Figure 5). + − The boundary of C( f σ ) can be decomposed in two pieces: ∂(C( f σ )) = 6g,1 ∪ 6g,1 . ± + ± We denote by i the inclusions of 6g,1 in C( f σ ), and we identify 6g,1 with 6g,1 , and F(2g) with π1 (C( f σ )) via i ∗+ . Then, by definition, we have ηiα = i ∗+ (αi )−1 i ∗− (αi )

and

β

ηi = i ∗+ (βi )−1 i ∗− (βi ).

Now, if li , li0 , li00 and γi are the curves described in Figure 5, then one has (after connecting these curves to the basepoint) i − (αi ) = li−1 i + (αi )li0

and

i − (βi ) = (li00 )−1 i + (βi )li0 ;

thus, one gets ηiα = [αi−1 , li−1 ]li−1li0

and

β ηi = βi−1 , (li00 )−1 (li00 )−1li0 .

But one has (li00 )−1li0 = i − (αi )γi−1li i − (αi )−1 ; thus, one has β ηi = βi−1 , (li00 )−1 i − (αi ), (γi )−1li (γi )−1li . β

On the other hand, since f σ ∈ M0,g+1 ⊂ Mg,1 , the handles Hi remain fixed via f σ and it follows that we have li = li0 , and li00 = γi+1 . Thus, we get β ηiα = [αi−1 , li−1 ] and ηi = βi−1 , (γi+1 )−1 i − (αi ), (γi )−1li (γi )−1li . β

Now, suppose that σ ∈ Pn . Then f σ ∈ M[n−1] and ηiα , ηi ∈ F(2g)n /F(2g)n+1 . Thus, via the projection H → H 0 , we get ηeiα = xi−1 , (l˜i )−1 ∈ L n (H 0 ) and eβ − ηi = i^ (αi ), (γ˜i )−1l˜i (γ˜i )−1l˜i ∈ L n (H 0 ),


273

where z˜ denotes the projection of z. On the other hand, one has Aσ (xi ) = (l˜i )−1 xi l˜i , and we have l˜i ∈ F(g)n since σ ∈ Pn . Furthermore, if δi = li00 (li0 )−1li , we have the following inductive relations: δi = [αi−1 , γi li−1 ]γi and γi+1 = βi li0 (li00 )−1 , δi (li00 )−1 δi , which implies, since li = li0 and li00 = γi+1 , −1 −1 δ˜i = γg γ˜i . i+1 = x i , γ˜i (l˜i ) So, we have γ˜i ∈ F(g)n+1 by induction and we get ηeiα = 0

eβ ηi = l˜i

and

in L n (H 0 ).

This proves that pn Jn−1 ( f σ ) = pn

X

X β αi ⊗ ηi − βi ⊗ ηiα = αi ⊗ l˜i = −µn (σ ).

This concludes the proof of Theorem 1.1. 6. A relation in (Tg,1 )2 We want to give explicitly the relation in (Tg,1 )2 obtained from the Jacobi relation via the map P(g)3 → (Tg,1 )2 . We use Johnson’s notation (see [J1]). Let (a1 , . . . , a2k+1 ) be an odd chain, that is, an ordered collection of oriented simple closed curves in a surface 6 whose intersection graph is the one below (defined by using a vertex for each curve, an edge between two vertices if the corresponding curves intersect transversely in a single point, and no edge if they are disjoint), and such that the algebraic intersection ai · ai+1 is +1. The chain

a1

a2

a3

a2k

a2k+1

defines an element of Tg,1 as follows. A neighborhood of a1 ∪ · · · ∪ a2k+1 in 6 is homeomorphic to 6k,2 . Denote by γ (resp., γ 0 ) the boundary component that is on the left (resp., on the right) of a1 , a3 , . . . , a2k+1 . Then γ and γ 0 are homologous and τγ τγ−1 0 is in the Torelli group of 6. This element is written [a1 ; . . . ; a2k+1 ] and is called the k-chain map induced by (a1 , . . . , a2k+1 ). Now, consider the curves C1α , D1,2 , C2α , C3α introduced in Section 2 and 3, and denote by D2,3 a curve in 6g,1 whose homotopy class is α2−1 β2−1 α3 β3 . Then (C1α , D1,2 , C2α , D2,3 , C3α ) forms a 5-chain.

274

GERVAIS and HABEGGER C2α

C1,2

C3α

e1,3

C1,2,3

C1,3

C2,3

C1α D1,2

e1,3

C2,3

D2,3

C1,2

C1,2,3

Figure 6

LEMMA

6.1

One has [τ1,2 , τ2,3 ] = [C1α ; D1,2 + C2α + D2,3 ; C3α ]−1 [C1α ; D1,2 ; C2α ]−1 × [C2α ; D2,3 ; C3α ]−1 [C1α ; D1,2 ; C2α ; D2,3 ; C3α ], where D1,2 + C2α + D2,3 denotes the curve τ D−12,3 τ2−1 (D1,2 ). Remark. It is more convenient here to use the model of 6g,1 given in Figure 6. Proof 0 Let us denote by C1,2,3 and C1,2,3 the two boundary curves of a neighborhood in 6g,1 of C1α ∪ D1,2 ∪ C2α ∪ D2,3 ∪ C3α (see Figure 6). Then one has 0 [C1α ; D1,2 ; C2α ; D2,3 ; C3α ] = τ1,2,3 (τ1,2,3 )−1 , 0 0 ). where τ1,2,3 (resp., τ1,2,3 ) is the Dehn twist along C1,2,3 (resp., C1,2,3 α α α The four curves C1 , C2 , C3 , and C1,2,3 bound a subsurface of 60,g+1 which is homeomorphic to 60,4 . Thus, they yield the following lantern relation (see [J1]):

τ1 τ2 τ3 τ1,2,3 = τ1,3 τ2,3 τ1,2 = τ1,2 τ1,3 (τ1,2 )−1 τ1,2 τ2,3 .

(L1 )

From this we get [τ1,2 , τ2,3 ] = τ1,2 (τ1,3 )−1 (τ1,2 )−1 τ1,3 .

(?)

0 On the other hand, C1α , C2α , C3α , and C1,2,3 also bound a subsurface homeomorphic to 60,4 and so yield the lantern relation 0 0 0 τ1 τ2 τ3 τ1,2,3 = τ1,2 τ2,3 τe 0 . 1,3

(L2 )


275

0 are precisely the twists considRemark. The twists τ1 , τ2 , τ3 , τ1,2 , τ1,3 , τ2,3 , and τ1,2 0 are the twists ered in the preceding sections, whereas τ1,2 τ1,3 (τ1,2 )−1 , τe0 , and τ2,3 1,3 0 , and C 0 described in Figure 6. along the curves e1,3 , e1,3 2,3

Multiplying (L1 ) by the inverse of (L2 ), we get 0 0 −1 0 −1 τ1,2,3 (τ1,2,3 )−1 = τ1,3 (τe0 )−1 τ2,3 (τ2,3 ) τ1,2 (τ1,2 ) , 1,3

which yields by (?), [τ1,2 , τ2,3 ] = (τe1,3 )−1 τe0 (τe0 )−1 τ1,3 1,3

1,3

0 0 0 = (τe1,3 )−1 τe0 (τ1,2 )−1 τ1,2 (τ2,3 )−1 τ2,3 τ1,2,3 (τ1,2,3 )−1 . 1,3

Now, noticing that 0 −1 τ1,2 (τ1,2 ) = [C1α ; D1,2 ; C2α ],

0 −1 τ2,3 (τ2,3 ) = [C2α ; D2,3 ; C3α ],

and τe1,3 (τe0 )−1 = [C1α ; D1,2 + C2α + D2,3 ; C3α ], 1,3

we get the required relation. Now, suppose that we are given a collection of seven curves λ, λ1 , λ2 , λ01 , λ02 , λ001 , λ002 whose intersection graph is λ2 λ1 λ2

λ1

λ λ1 λ2

and consider the following elements of Tg,1 : Y1 = [λ2 ; λ1 ; λ; λ01 ; λ02 ], Y2 = [λ02 ; λ01 ; λ; λ001 ; λ002 ], A = [λ; λ1 ; λ2 ],

Y3 = [λ002 ; λ001 ; λ; λ1 ; λ2 ],

A0 = [λ; λ01 ; λ02 ],

B1 = [λ2 ; λ1 + λ + λ01 ; λ02 ],

A00 = [λ; λ001 ; λ002 ],

B2 = [λ02 ; λ01 + λ + λ001 ; λ002 ],

B3 = [λ002 ; λ001 + λ + λ1 ; λ2 ].

276


THEOREM 6.2 One has the following relation in (Tg,1 )2 ⊂ Tg,1 :

Y3 B3−1 A00 , Y2 B2−1 A0 (A00 )−1 Y2 B2−1 A0 , Y1 B1−1 A(A0 )−1 × Y1 B1−1 A, Y3 B3−1 A00 (A)−1 = 1.

Proof With the notation of Section 2, consider curves D1,3 and D1,4 in 64,1 whose homotopy classes are, respectively, β1−1 α2 α3 β3 and β1−1 α2 α3 β3 α3−1 β3−1 α4 β4 . Then the intersection graph of C1α , D1,2 , C2α , D1,3 , C3α , D1,4 , C4α is C3α D1,3 C2α

D1,2

C1α D1,4 C4α

and there exists a homeomorphism of 6g,1 which sends, respectively, λ, λ1 , λ2 , λ01 , λ02 , λ001 , and λ002 to C1α , D1,2 , C2α , D1,3 , C3α , D1,4 , and C4α . Thus, it suffices to prove the relation in this case. Consider first the 5-chain (C2α , D1,2 , C1α , D1,3 , C3α ). We claim that B1−1 A(A0 )−1 Y1 = [τ1,2 , τ1,3 ]. Indeed, there is a diffeomorphism h of 6g,1 (e.g., h = (τ1 τ2 τ D1,2 )2 ) which sends, respectively, C2α , D1,2 , C1α , D1,3 , C3α , C1,2 , and C1,3 to C1α , D1,2 , C2α , D2,3 , C3α , C1,2 , and C2,3 . Thus, up to a conjugation, the required relation is the one of Lemma 6.1. In the same way, we have the following two relations: B2−1 A0 (A00 )−1 Y2 = [τ1,3 , τ1,4 ]

and

B3−1 A00 A−1 Y3 = [τ1,4 , τ1,2 ].

From these three relations, we deduce 0 −1 (τ1,2 )τ1,4 (τ1,2 ) = [τ1,4 , τ1,2 ]A = Y3 B3−1 A00 , 0 −1 (τ1,4 )τ1,3 (τ1,4 ) = [τ1,3 , τ1,4 ] A00 = Y2 B2−1 A0 , 0 −1 (τ1,3 )τ1,2 (τ1,3 ) = [τ1,2 , τ1,3 ] A0 = Y1 B1−1 A, 0 )−1 , A0 = τ (τ 0 )−1 , and A00 = τ (τ 0 )−1 . where A = τ1,2 (τ1,2 1,3 1,3 1,4 1,4


277

Now, the Jacobi identity in P3 gives (τ1,2 )τ1,4 , [τ1,3 , τ1,4 ] (τ1,4 )τ1,3 , [τ1,2 , τ1,3 ] (τ1,3 )τ1,2 , [τ1,4 , τ1,2 ] = 1. 0 , τ 0 , and τ 0 commute with τ , τ , and τ , we get in (T ) the folSince τ1,2 1,2 1,3 1,4 g,1 2 1,3 1,4 lowing relation: Y3 B3−1 A00 , Y2 B2−1 A0 (A00 )−1 Y2 B2−1 A0 ,Y1 B1−1 A(A0 )−1 × Y1 B1−1 A, Y3 B3−1 A00 (A)−1 = 1.

7. The topological IHX relation L Let G be a group. We set L (G) = n≥0 Ln (G), where Ln (G) = G n /G n+1 . ¯ = [a, b], where a ∈ G n , It is a Lie algebra with bracket defined by [a, ¯ b] b ∈ G m represent a¯ ∈ G n /G n+1 , b¯ ∈ G m /G m+1 . The Jacobi group relation [a c , [b, c]][cb , [a, b]][ba , [c, a]] = 1 implies the Jacobi relation in L (G). We will write LnQ (G) for L (G) ⊗ Q. If G ab = G/G 2 , then one has a surjective mapping L(G ab ) → L (G) given by the isomorphism L 1 (G ab ) = G ab = G/G 2 = L1 (G), where L(G ab ) is the free Lie algebra on G ab . We write L(G ab Q ) for the free Q Lie ab ⊗ Q. Then L(G ab ) → L Q (G) is also surjective. The kernel algebra on G ab = G Q Q R(G) of this map is the ideal of relations. If T = Tg,1 , g ≥ 4, we deduce a relation in R2 (T ), which we call the IHX relation, as it arises from the Jacobi relation in P = P(g). First, recall that the degree 1 Johnson homomorphism is a map J1 : T → ∧3 (H ), H = H1 (6g,1 ). Rationally, this gives an isomorphism L1Q (T ) = TQab → ∧3 (H Q ). Thus, R2 (T ) lies in L 2 (∧3 (H Q )). There is an obvious relation in R2 (T ) (see [Hn]) coming from the commutativity of diffeomorphisms having disjoint support, namely, [a1 ∧ a2 ∧ b2 , b3 ∧ a3 ∧ a4 ], where (ai , bi )1≤i≤g is a symplectic basis of H . Substituting a1 + b3 for b3 also gives a relation. The difference gives the relation [a1 ∧ a2 ∧ b2 , a1 ∧ a3 ∧ a4 ]. The Jacobi relation in P3 /P4 determines a relation in T2 /T3 and hence in Q L2 (T ). We can compute examples as follows. For a = σ1,2 , we have [a, [b, c]] = 0 )−1 , [b, c]]. We claim that J (τ (τ 0 )−1 ) = α ∧ α ∧ β − α ∧ α ∧ β . [τ1,2 (τ1,2 1 1,2 1,2 1 2 2 1 2 1 To see this, note that the curves C1α , D1,2 , C2α of Section 3 define a 3-chain and that J1 ([x, y, z]) = −x ∧ y ∧ z for a 3-chain [x, y, z] (see [J2]). Thus, 0 −1 J1 τ1,2 (τ1,2 ) = −α1 ∧ (−β1 + α2 + β2 ) ∧ α2 = α1 ∧ α2 ∧ β2 − α1 ∧ α2 ∧ β1 . If we take b = σ1,3 , c = σ1,4 , then J1 ([b, c]) = α1 ∧ α3 ∧ α4 . Thus, [σ1,2 , [σ1,3 , σ1,4 ]] defines [α1 ∧ α2 ∧ β2 − α1 ∧ α2 ∧ β1 , α1 ∧ α3 ∧ α4 ] in L2Q (T ). Now note that there is a diffeomorphism f taking M4,1 to itself which on homology takes α1 , β1 to themselves and cyclically permutes αi , βi for i = 2, 3, 4. Thus, f cyclically

278


permutes the curves C1,2 , C1,3 , C1,4 . This proves that the Jacobi relation yields the element [α1 ∧α2 ∧β2 −α1 ∧α2 ∧β1 , α1 ∧α3 ∧α4 ]+[α1 ∧α3 ∧β3 −α1 ∧α3 ∧β1 , α1 ∧α4 ∧α2 ] + [α1 ∧ α4 ∧ β4 − α1 ∧ α4 ∧ β1 , α1 ∧ α2 ∧ α3 ] in R2 (T ). Since terms like [α1 ∧ α2 ∧ β2 , α1 ∧ α3 ∧ α4 ] are in R2 (T ), this reduces to [α1 ∧α2 ∧β1 , α1 ∧α3 ∧α4 ]+[α1 ∧α3 ∧β1 , α1 ∧α4 ∧α2 ]+[α1 ∧α4 ∧β1 , α1 ∧α2 ∧α3 ]. In genus g ≥ 5, we may substitute α1 + α5 for α1 , and the difference yields [α5 ∧ α2 ∧ β1 , α1 ∧ α3 ∧ α4 ] + [α5 ∧ α3 ∧ β1 , α1 ∧ α4 ∧ α2 ] + [α5 ∧ α4 ∧ β1 , α1 ∧ α2 ∧ α3 ] plus terms like [α1 ∧ α2 ∧ β1 , α5 ∧ α3 ∧ α4 ] and [α5 ∧ α2 ∧ β1 , α5 ∧ α3 ∧ α4 ] which lie in R2 (T ). Permuting the indices, we show the following theorem. THEOREM 7.1 If g ≥ 5, the relation

[α1 ∧α4 ∧β5 , α2 ∧α3 ∧α5 ]+[α2 ∧α4 ∧β5 , α3 ∧α1 ∧α5 ]+[α3 ∧α4 ∧β5 , α1 ∧α2 ∧α5 ] = 0 holds in L2Q (Tg,1 ). Remark. This relation appears at least implicitly in work of Morita [Mo1], [Mo2], when he computed the image of the second Johnson homorphism. Remark. In [HS] (following Hain [Hn]) a presentation of the Malcev Lie algebra of Tg,1 is given. Theorem 7.1 is related to another quadratic relation R2 (see [HS, §3]) determined by the boundary of 6g,1 . Acknowledgment. The authors would like to thank Christoph Sorger for an introduction to the program Lie which served as a tool in understanding the IHX relation. References [GL]

[G]

S. GAROUFALIDIS and J. LEVINE, Tree-level invariants of three-manifolds, Massey

products and the Johnson homomorphism, preprint 1999, http://www.math.gatech.edu/˜stavros/publications.html 265, 266, 271 M. GOUSSAROV, Finite type invariants and n-equivalence of 3-manifolds, C. R. Acad. Sci. Paris Sèr. I Math. 329 (1999), 517 – 522. MR 2000g:57019 265


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[HL] [HM] [HS]

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[Hn] [HT] [J1] [J2]

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N. HABEGGER, “The Topological IHX Relation” in Knots in Hellas ’98 (Delphi,

Greece), Vol. II, J. Knot Theory Ramifications 10 (2001), World Sci., Singapore, 2001, 309 – 329. CMP 1 822 495 265 N. HABEGGER and X.-S. LIN, On link concordance and Milnor’s µ invariants, Bull. London Math. Soc. 30 (1998), 419 – 428. MR 2000e:57023 271 N. HABEGGER and G. MASBAUM, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), 1253 – 1289. MR 2002b:57011 271 N. HABEGGER and C. SORGER, An infinitesimal presentation of the Torelli group of a surface with boundary, preprint, 2000, http://www.math.sciences.univ-nantes.fr/˜habegger 278 K. HABIRO, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1 – 83, http://www.maths.warwick.ac.uk/gt/GTVol4/paper1.abs.html MR 2001g:57020 265 R. HAIN, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), 597 – 651. MR 97k:14024 265, 277, 278 A. HATCHER and W. THURSTON, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), 221 – 237. MR 81k:57008 268 D. JOHNSON, The structure of the Torelli group, I: A finite set of generators for I , Ann. of Math. (2) 118 (1983), 423 – 442. MR 85a:57005 265, 273, 274 , The structure of the Torelli group, II: A characterization of the group generated by twists on bounding curves, Topology 24 (1985), 113 – 126. MR 86i:57011 265, 277 , The structure of the Torelli group, III: The abelianization of T , Topology 24 (1985), 127 – 144. MR 87a:57016 265 T. T. Q. LE, “An invariant of integral homology 3-spheres which is universal for all finite type invariants” in Soliton Geometry and Topology: On the Crossroad, Amer. Math. Soc. Transl. Ser. 2 179, Amer. Math. Soc., Providence, 1997, 75 – 100. MR 99m:57008 265 T. T. Q. LE, J. MURAKAMI, and T. OHTSUKI, On a universal perturbative invariant of 3-manifolds, Topology 37 (1998), 539 – 574. MR 99d:57004 265 J. LEVINE, “Pure braids, a new subgroup of the mapping class group and finite-type invariants of 3-manifolds” in Tel Aviv Topology Conference: Rothenberg Festschrift (Tel Aviv, 1998), Contemp. Math. 231, Amer. Math. Soc., Providence, 1999, 137 – 157. MR 2000i:57023 268, 271 G. MESS, The Torelli groups for genus 2 and 3 surfaces, Topology 31 (1992), 775 – 790. MR 93k:57003 265 S. MORITA, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles, I, Topology 28 (1989), 305 – 323. MR 90h:57020 265, 278 , On the structure of the Torelli group and the Casson invariant, Topology 30 (1991), 603 – 621. MR 92i:57016 278 , “A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles” in Topology and Teichmüller Spaces (Katinkulta, Finland, 1995), World Sci., River Edge, N.J., 1996, 159 – 186. MR 2000b:57024 267

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filtration on the Teichmüller group, preprint. 268, 271 Ramifications 5 (1996), 101 – 115. MR 97i:57019 265 Angew. Math. 521 (2000), 47 – 83. MR 2001b:57035 267

Gervais Unité Mixte de Recherche 6629 du Centre National de la Recherche Scientifique, Université de Nantes, Département de Mathématiques, 2 rue de la Houssinière, 44072 Nantes CEDEX 03, France; [email protected]; http://www.math.sciences.univ-nantes.fr/˜gervais Habegger Unité Mixte de Recherche 6629 du Centre National de la Recherche Scientifique, Université de Nantes, Département de Mathématiques, 2 rue de la Houssinière, 44072 Nantes CEDEX 03, France; [email protected]; http://www.math.sciences.univ-nantes.fr/˜habegger


HEATING OF THE AHLFORS-BEURLING OPERATOR: WEAKLY QUASIREGULAR MAPS ON THE PLANE ARE QUASIREGULAR STEFANIE PETERMICHL and ALEXANDER VOLBERG

Abstract We establish borderline regularity for solutions of the Beltrami equation f z −µf z¯ = 0 on the plane, where µ is a bounded measurable function, kµk∞ = k < 1. What is the 1,q minimal requirement of the type f ∈ Wloc which guarantees that any solution of the Beltrami equation with any kµk∞ = k < 1 is a continuous function? A deep result of 1,1+k+ε K. Astala says that f ∈ Wloc suffices if ε > 0. On the other hand, O. Lehto and T. Iwaniec showed that q < 1 + k is not sufficient. In [2], the following question was asked: What happens for the borderline case q = 1 + k? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia’s extrapolation technique and two-weight estimates for the martingale transform from [26]. Contents Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 0. Introduction: Main objects and results . . . . . . . . . . . . . . . . . . 282 1. The sharp weighted estimate for the Ahlfors-Beurling operator in L 2 (w d A) 287 2. The sharp weighted estimate for the Ahlfors-Beurling operator in L p (w d A) for p > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 3. The comparison of classical and heat A p -characteristics . . . . . . . . . 299 4. Injectivity at the critical exponent and regularity of solutions of the Beltrami equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 8 December 2000. Revision received 2 May 2001. 2000 Mathematics Subject Classification. Primary 42B20, 42C15, 42A50, 47B35, 47B38. Volberg’s work partially supported by National Science Foundation grant number DMS-9900375 and by United States–Israel Binational Science Foundation grant number 00030. 281

282

References

PETERMICHL and VOLBERG

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

302

Notation We have the following notation: x = (x1 , x2 ), |x| = (x12 + x22 )1/2 ; B(x0 , r ) is the open ball, where B(x0 , r ) := {x ∈ R2 : |x − x0 | < r }; h f i B is the average of the function f over the ball B; x = (x1 , x2 ), k(x, t) := (1/πt) exp(−(|x|2 /t)) is the heat kernel on the plane; RR RR −1 Q heat w(x − y)k(y, t) dy1 y2 · w (x − y)k(y, t) dy1 y2 is the w,2 = supx∈R2 ,t>0 heat A2 characteristic of weight w (see Section 0); RR w(x − y)k(y, t) dy1 y2 Q heat = supx∈R2 ,t>0 w, p RR −(1/( p−1)) ·( w (x − y)k(y, t) dy1 y2 ) p−1 is the heat A p -characteristic of weight w (see Section 0); −1 Q class w,2 = sup B(x,r ) hwi B(x,r ) hw i B(x,r ) is the standard A2 -characteristic of weight w (see Section 0); −(1/( p−1)) i p−1 is the standard A -characteristic Q class p B(x,r ) ) w, p = sup B(x,r ) hwi B(x,r ) (hw of weight w (see Section 0); dyadic

Q w,2 = sup I,I ∈D hwi I hw−1 i I is the dyadic A2 -characteristic of weight w (see Section 1); dyadic

Q w, p = sup I,I ∈D hwi I (hw−(1/( p−1)) i I ) p−1 is the dyadic A p -characteristic of weight w (see Section 1); D is a collection of dyadic intervals (see Section 1).

0. Introduction: Main objects and results Our goal is to present a sharp estimate of the weighted Ahlfors-Beurling operator. This estimate is sufficient to prove that any weakly quasiregular map is quasiregular. Thus, we are interested in the following Ahlfors-Beurling operator (d A denotes area Lebesgue measure on C): Z Z 1 ϕ(ζ ) d A(ζ ) T ϕ(z) := π (ζ − z)2 understood as a Calderón-Zygmund operator. Our starting point is the following theorem of Astala, Iwaniec, and E. Saksman [2]. THEOREM 0.1 Let µ ∈ L ∞ , kµk∞ = k < 1. Then the operators I − µT and I − T µ are invertible

HEATING OF THE AHLFORS-BEURLING OPERATOR

283

in L p (C, d A) for p ∈ (1 + k, 1 + (1/k)). Remark. In [2], the authors suggest the following problem: Prove that these operators have dense range for p = 1+(1/k) and are injective for p = 1+k. Actually, this problem has a long history and has reappeared in many papers on regularity of quasiconformal homeomorphisms and quasiregular maps. The L p -theory of quasiregular mappings was essentially formulated by B. Bojarski [6] and [7]. Later this subject came under intensive investigation. In particular, the best integrability of K -quasiconformal mappings and the (in a sense dual; see [25]) problem of minimal regularity of the quasiregular mappings are discussed in [16], [13] – [15], [17], [20], [21], [22], [24], and [25]. The best integrability result was finally established in [1]. Notice that the p − 1 problem for the operator T is still open. We recall that this problem consists of proving that kT k L p →L p = p − 1, p > 2. A discussion of the fantastically beautiful connections of this problem with the calculus of variations and C. Morrey’s problem can be found in [3]. Until recently the best result for all p appeared in [5], where probabilistic methods were used to obtain the estimate kT k L p →L p ≤ 4( p − 1), p > 2. Discussion of related results can be found in [4] and [30]. Recently, in [27] this estimate was slightly improved to kT k L p →L p ≤ 2( p − 1), p > 2. The main result of this paper may serve as yet another indication that this norm is p − 1. We establish the dense range/injectivity property in the present paper. Our principal tool is a sharp weighted estimate of the Ahlfors-Beurling operator. The main consequence (see [2]) is the above-mentioned geometric fact: Every weakly quasiregular map is quasiregular. We first quote several results and notions that we use. The symbol f = f µ represents the homeomorphic solution of the Beltrami equation f z¯ − µf z = 0,

f (z) = z + o(1),

z → ∞.

Here we consider compactly supported µ, µ ∈ L ∞ (C), kµk∞ = k < 1. The function µ is called the Beltrami coefficient of the Beltrami equation written above. The constant K = (1 + k)/(1 − k) has the geometric meaning of the bound on the distortion of infinitely small discs by the map f . (The resulting infinitely small ellipses have the ratio of their axes at most K .) The map f is called K -quasiconformal. The image of a disc by a K -quasiconformal homeomorphism is called a K -quasidisc. We need the following results from Astala, Iwaniec, and Saksman [2]. 0.2 Let f be a K -quasiconformal homeomorphism built by a certain µ as it is described above. Consider ω = | f z ◦ f −1 | p−2 for p ∈ (1 + k, 1 + (1/k)). THEOREM

284


Then k(I − µT )−1 k L p →L p , k(I − T µ)−1 k L p →L p are bounded by C(k)λω, p , where λω, p := kT k L p (ω d A)→L p (ω d A) . THEOREM 0.3 Let f be a K -quasiconformal homeomorphism. Let k = (K + 1)/(K − 1), and let p ∈ (0, 1 + (1/k)). Then for any disc or K -quasidisc B, one has Z | f (B)| p/2 p 1 C(k) | f z | + | f z¯ | d A ≤ . |B| B 1 + (1/k) − p |B|

R Remark. Obviously, we have the same type of estimate for (1/|B|) B | f z | p d A and R p/2 for (1/|B|) B J f d A, where J f := | f z |2 − | f z¯ |2 is the Jacobian of the mapping f . Next, a very elegant result from [2] reduces the critical exponent cases p = 1 + (1/k), p = 1 + k to a weighted estimate of the Ahlfors-Beurling operator. This result is quoted for the convenience of the reader. THEOREM 0.4 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). If

kT k L p (ω d A)→L p (ω d A) ≤

C , 1 + a(1/k) − p

(0.1)

then I − µT, I − T µ have dense ranges in L 1+1/k (C) and are injective in L 1+k (C). Our main result implies the weighted estimate (0.1). Theorem 0.4 served as motivation for our main result. In its turn, injectivity at the critical exponent 1 + k proves that weakly quasiregular maps are quasiregular (see [2]). Proof The operator T is invertible in all L q , q ∈ (1, ∞). We notice that I − T µ = T (I − µT )T −1 . Thus, it is enough to prove that I − µT has closed range in L 1+(1/k) (C). Let p0 := 1 + (1/k). Take any ϕ ∈ C0∞ (C). By Theorem 0.1, ϕε := I − (1 − ε)µT

−1

ϕ ∈ L p0 (C).

By Theorem 0.2, kϕε k L p0 ≤ C(k)kT k L p0 (ωε d A)→L p0 (ωε d A) , where ωε = | f z ◦ f −1 | p−2 , and f denotes (as always) a homeomorphic solution of the Beltrami equation, but for Beltrami coefficient (1 − ε)µ instead of µ.


285

Now assuming estimate (0.1), we conclude that kϕε k L p0 ≤

C C ≤ . 1 + (1/(1 − ε)k) − p0 ε

Thus, kεϕε k L p0 ≤ C1 < ∞,

ε → 0.

(0.2)

On the other hand, the definition of ϕε implies that (I − µT )ϕε = ϕ − εµϕε

(0.3)

and kϕε k L 2 ≤ C(k) < ∞,

∀ε ∈ [0, 1].

(0.4)

Now (0.2) and (0.4) imply that εϕε converge weakly to zero in L p0 . Keeping this in mind, we see that (0.3) implies that the range of I − µT is weakly dense in L p0 . So the range of I − µT is automatically dense in L p0 in the norm topology. We are now ready to formulate one of our main results. 0.5 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). Then THEOREM

kT k L p (ω d A)→L p (ω d A) ≤

C . 1 + (1/k) − p

This result is obtained as a corollary of the following theorems, which may be of interest in their own right. Let ω be any weight on R2 , and denote its heat extension into R3+ by ω(x, t) = ω(x1 , x2 , t): Z Z kx − yk2 1 ω(x, t) = ω(y) exp − dy1 dy2 . πt t R2 We define Q heat ω, p :=

sup

p−1 ω(x, t) ω−(1/( p−1)) (x, t) .

(x,t)∈R3+

The weights with finite Q heat ω, p are called A p -weights. There is an extensive theory of A p -weights (see, e.g., [29], [12]). The usual definition differs from the one above, but it describes the same class of weights. Actually, we say more about the relationship between the classical definition and ours. But first we state two more theorems, whose combined use gives Theorem 0.5.

286


THEOREM 0.6 For any A p -weight w and any p ≥ 2, we have

kT k L p (w d A)→L p (w d A) ≤ C( p)Q heat ω, p . THEOREM 0.7 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). Then the weight ω belongs to A p and

Q heat ω, p ≤

C . 1 + (1/k) − p

Clearly, these last two results together imply Theorem 0.5. And Theorem 0.5 and Theorem 0.4 give positive answers to the Beltrami equation critical exponent questions discussed above and raised in [2]. At the end of our introduction, we discuss the connection between Q heat w, p and class Q w, p . Here Q class denotes the following supremum over all discs in the plane: w, p Z Z p−1 1 1 ω d A · ω−(1/( p−1)) d A . Q class := sup w, p |B(x, R)| B(x,R) B(x,R) |B(x, R)| B(x,R) Obviously, there exists a positive absolute constant a such that for any function w, heat a Q class w, p ≤ Q w, p .

Remark. The opposite inequality is easy to prove, too. We are grateful to F. Nazarov for indicating this to us. Originally, we did not have the following two theorems. This was not an obstacle to Theorem 0.5 because we still could estimate (see Theorem 0.7) Q heat ω, p for special weights ω. 0.8 There exists a finite absolute constant b such that THEOREM

class Q heat w, p ≤ bQ w, p .

In conjunction with our main Theorem 0.5, this gives us the following weighted estimate of the Ahlfors-Beurling operator, which answers another question of [2] positively. THEOREM 0.9 For any A p -weight w and any p ≥ 2, we have

kT k L p (w d A)→L p (w d A) ≤ C( p)Q class ω, p .


287

In [2], it is proved that if k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , and p ∈ [2, 1 + (1/k)), then the weight ω belongs to A p and Q class ω, p ≤

C . 1 + (1/k) − p

(0.5)

So instead of using Theorem 0.7, one can use Theorem 0.9 in conjunction with (0.5). As a result, for the special weight function ω = | f z ◦ f −1 | p−2 , where p ∈ [2, 1 + (1/k)), f := f µ , kµk∞ = k, and k ∈ [0, 1), we always get Q heat ω, p ≤

C . 1 + (1/k) − p

(0.6)

Theorem 0.6 then proves our main Theorem 0.5. Plan of the paper In Section 1, we prove Theorem 0.6 for p = 2. In Section 2, we prove Theorem 0.6 for p > 2. In Section 3, we give an easy proof of Theorem 0.8. In Section 4, we repeat (for the convenience of the reader) the reasoning of [2] that deduces the quasiregularity of weakly quasiregular maps from the injectivity of I −µT at the critical exponent. 1. The sharp weighted estimate for the Ahlfors-Beurling operator in L 2 (w d A) In this section, w is an arbitrary positive function on the plane. We want the estimate kT k L 2 (w d A)→L 2 (w d A) ≤ C Q heat w,2 .

(1.1)

As we see in Section 3, estimate (1.1) readily implies the estimate kT k L 2 (w d A)→L 2 (w d A) ≤ C Q class w,2 .

(1.2)

However, at this moment we do not know how to get the latter estimate without using the first. So now our immediate goal is to obtain (1.1). The operator T is given in the Fourier domain (ξ1 , ξ2 ) by the multiplier ξ12 ξ22 ζ¯ ξ1 ξ2 ζ¯ 2 (ξ1 − iξ2 )2 = − − 2i 2 . = = 2 2 2 2 2 2 2 ζ |ζ | ξ1 + ξ2 ξ1 + ξ2 ξ1 + ξ2 ξ1 + ξ22 Thus, T can be written as T = R12 − R22 −2i R1 R2 , where R1 , R2 are Riesz transforms on the plane (see [29] for their definition and properties). Another way of writing T is T = m 1 − im 2 , where m 1 , m 2 are Fourier multiplier operators. Notice that the multipliers themselves (as functions, not as multiplier operators) are connected by m 2 = m 1 ◦ ρ,

288


where ρ is a π/4 rotation of the plane. So the multiplier operators are related by m 2 = Uρ m 1 Uρ−1 , where Uρ is an operator of ρ-rotation in the (x1 , x2 )-plane. But for any operator K , we have kUρ K Uρ−1 k L 2 (w d A)→L 2 (w d A) = kK k L 2 (w◦ρ −1 d A)→L 2 (w◦ρ −1 d A) . heat Combining this with the fact that Q heat w,2 = Q w◦ρ −1 ,2 for any rotation, we conclude

that we need the desired estimate (1.1) only for m 1 = R12 − R22 . Actually, we show that kRi2 k L 2 (w d A)→L 2 (w d A) ≤ C Q heat i = 1, 2. (1.3) w,2 , To prove (1.3) we fix, say, R12 and two test functions ϕ, ψ ∈ C0∞ . We use heat extensions. For f on the plane, its heat extension is given by the formula ZZ |x − y|2 1 f (y, t) := f (x) exp − d x1 d x2 , (y, t) ∈ R3+ . 2 πt t R We usually use the same letter to denote a function and its heat extension. LEMMA 1.1 RRR ∂ϕ ∂ψ Let ϕ, ψ ∈ C0∞ . Then the integral ∂ x1 · ∂ x1 d x 1 d x 2 dt converges absolutely and ZZ ZZZ ∂ϕ ∂ψ R12 ϕ · ψ d x1 d x2 = −2 · d x1 d x2 dt. (1.4) ∂ x1 ∂ x1

Proof The proof of this lemma is actually trivial. It is based on the well-known fact that a function is an integral of its derivative, and it also involves Parseval’s formula. Consider ϕ, ψ ∈ C0∞ and now ZZ ψ R12 ϕ d x1 d x2 ZZ

ξ12

ˆ ϕ(ξ ˆ 1 , ξ2 )ψ(−ξ 1 , −ξ2 ) dξ1 dξ2 ξ12 + ξ22 ZZ Z ∞ 2 2 ˆ 1 , ξ2 ) dξ1 dξ2 dt =2 e−2t (ξ1 +ξ2 ) ξ12 ϕ(ξ ˆ 1 , ξ2 )ψ(ξ 0 Z ∞ ZZ 2 2 −t (ξ12 +ξ22 ) ˆ = −2 iξ1 ϕ(ξ ˆ 1 , ξ2 )e−t (ξ1 +ξ2 ) iξ1 ψ(−ξ dξ1 dξ2 dt 1 , −ξ2 )e Z0 ∞ Z Z ∂ϕ ∂ψ = −2 (x1 , x2 , t) (x1 , x2 , t) d x1 d x2 dt ∂ x1 ∂ x1 Z0Z Z ∂ϕ ∂ψ = −2 (x1 , x2 , t) (x1 , x2 , t) d x1 d x2 dt. 3 ∂ x ∂ x1 1 R+ =


289

We used Parseval’s formula twice above, and we also used the absolute convergence of the integrals ZZZ 2 2 ˆ 1 , ξ2 ) dξ1 dξ2 dt, e−2t (ξ1 +ξ2 ) ξ12 ϕ(ξ ˆ 1 , ξ2 )ψ(ξ R3+

ZZZ R3+

∂ψ ∂ϕ (x1 , x2 , t) (x1 , x2 , t) d x1 d x2 dt. ∂ x1 ∂ x1

For the first integral this is obvious. The absolute convergence of the second integral can be easily proved. We leave this as an exercise for the reader. Our next goal is to estimate the right-hand side of (1.4) from above. 1.2 For any ϕ, ψ ∈ C0∞ , and any positive function w on the plane, we have THEOREM

ZZZ

∂ϕ ∂ψ d x1 d x2 dt 3 ∂ x ∂ x 1 1 R+ ≤ AQ heat w,2

ZZ

|ϕ|2 w d x1 d x2 +

ZZ

|ψ|2

1 d x1 d x2 , w

where A is an absolute constant. In the proof we use the following key result. (In what follows, d 2 f denotes the Hessian form that is the second differential form of f .) THEOREM 1.3 For any Q > 1, define the domain D Q := {0 < (X, Y, x, y, r, s) : x 2 < X s, y 2 < Y r, 1 < r s < Q}. Let K be any compact subset of D Q . Then there exists a function B = B Q,K (X, Y, x, y, r, s) infinitely differentiable in a small neighborhood of K such that (1) 0 ≤ B ≤ 5Q(X + Y ), (2) −d 2 B ≥ |d x||dy|.

We prove Theorem 1.3 later. Now we use it to obtain the proof of Theorem 1.2. Proof of Theorem 1.2 Given a nonconstant smooth w that is constant outside some large ball, we consider Q = Q heat w,2 . We treat only the case w ∈ A2 , that is, Q < ∞, for otherwise there is nothing to prove. Consider two nonnegative functions ϕ, ψ ∈ C0∞ . Now take B = B Q,K , where a compact K remains to be chosen.

290


We are interested in b(x, t) := B (ϕ 2 w)(x, t), (ψ 2 w−1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w−1 (x, t) . This is a well-defined function because the choice of Q ensures that the 6-vector v, defined by v := (ϕ 2 w)(x, t), (ψ 2 w−1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w−1 (x, t) , lies in D Q for any (x, t) ∈ R3+ . Also, we can fix any compact subset M of the open set R3+ and guarantee that for (x, t) ∈ M, the vector v lies in a compact K . In fact, notice that for our w and for compactly supported ϕ, ψ, the mapping (x, t) → v(x, t) maps compacts in R3+ to compacts in D Q . Now just take K large enough. We want to apply the Green formula to b(x, t). To do this, we introduce a Green function G(x, t), as in [11]:  ∂   + 1 G = −δ0,1/2 in C(1, 1) = B(0, 1) × (0, 1),   ∂t G=0 on ∂ 0 C(1, 1) = ∂ B(0, 1) × (0, 1),     G=0 when t = 1. Here δ0,1/2 is a δ-function at the point (0, 1/2). It is important to keep in mind that G(x, 0) ≥ a 1 − kxk ,

(1.5)

where a is a positive absolute constant. We also need a Green function in the cylinder C(R, R 2 ) = B(0, R) × (0, R 2 ):  ∂   + 1 G R = −δ0,R 2 /2 in C(R, R 2 ) = B(0, R) × (0, R 2 ),   ∂t GR = 0 on ∂ 0 C(R, R 2 ) = ∂ B(0, R) × (0, R 2 ),     R G =0 when t = R 2 . One can easily see that the following lemma holds. 1.4 Green functions in different-sized cylinders relate as follows: LEMMA

G R (x, t) =

1 x t G , . R R2 R2

We are ready to apply the Green formula to b(x, t).


291

Let us first estimate b(0, R 2 /2) = B((ϕ 2 w)(0, R 2 /2), . . . , w −1 (0, R 2 /2)). Using the first property of Theorem 1.3, we get (x = (x1 , x2 ), as always) R2 R2 R 2 2 2 −1 b 0, ≤ 5Q heat (ϕ w) 0, + (ψ w ) 0, . w,2 2 2 2 Thus, ZZ 2|x|2 R2 5 2 2 (ϕ w)(x) exp − d x1 d x2 b 0, ≤ Q heat 2 π w,2 R2 R2 ZZ 2|x|2 5 heat 2 2 −1 + Q w,2 (ψ w )(x) exp − d x1 d x2 . π R2 R2 Now by the Green formula in C(R, R 2 ), ZZZ R2 ∂ b 0, =− + 1 G R (x, t)b(x, t) d x1 d x2 dt 2 C(R,R 2 )∩{t>ε} ∂t ZZZ ∂ = G R (x, t) − 1 b(x, t) d x1 d x2 dt ∂t C(R,R 2 )∩{t>ε} ZZ + b(x, ε)G R (x, ε) d x1 d x2 B(0,R)

∂b ∂G R GR − b ds dt + ∂n outer ∂ 0 C(R,R 2 )∩{t>ε} ∂n outer ZZZ ∂ = G R (x, t) − 1 b(x, t) d x1 d x2 dt ∂t C(R,R 2 )∩{t>ε} ZZ ZZ ∂G R R + b(x, ε)G (x, ε) d x1 d x2 + b ds dt B(0,R) ∂ 0 C(R,R 2 )∩{t>ε} ∂n inner ZZZ ∂ ≥ G R (x, t) − 1 b(x, t) d x1 d x2 dt. ∂t C(R,R 2 )∩{t>ε} ZZ

The last inequality is clear: the double integrals are both nonnegative because b is nonnegative and because G R is nonnegative and vanishes on the side boundary. Let us combine estimates of b(0, R 2 /2) into ZZ ZZ ZZZ ∂ 10 2 2 −1 R ϕ w + ψ w . G (x, t) − 1 b(x, t) ≤ ∂t π R2 C(R,R 2 )∩{t>ε} (1.6) Fix R and ε, and choose the compact set M = (x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R 2 . The next calculation is simple, but it is key to the proof.

292


LEMMA

1.5

Let v = (ϕ 2 w)(x, t), (ψ 2 w−1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w−1 (x, t) , where all entries are heat extensions. If (x, t) ∈ M, then ∂ ∂v ∂v ∂v ∂v 2 − 1 b(x, t) = (−d 2 B) , + (−d B) , . ∂t ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6 Proof We compute the derivative with respect to time and the Laplacian of b using the chain rule and obtain the following: ∂v ∂ b = ∇ B, , ∂t ∂t R6 ∂v ∂v ∂v ∂v 2 1b = (d 2 B) , + (d B) , + (∇ B, 1v)R6 . ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6 Combining the two, we get ∂ ∂v ∂v ∂v ∂v ∂v 2 − 1 b = ∇ B, − 1v 6 − (d 2 B) , − (d B) , . R ∂t ∂t ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6 However, the first term is zero because all entries of the vector v are solutions of the heat equation. By Theorem 1.3 in M, −d 2 B ≥ |d x||dy|. For (x, t) ∈ M, Lemma 1.5 gives ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ − 1 b(x, t) ≥ + . ∂t ∂ x1 ∂ x1 ∂ x2 ∂ x2

(1.7)

Combining (1.6) and (1.7), we get ZZ ∂ϕ ∂ψ ∂ϕ ∂ψ 10Q heat Z Z w,2 2 2 −1 G (x, t) ϕ w + ψ w . + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 π R2 M (1.8) Now it is time to use Lemma 1.4. So (1.8) implies ZZZ x t ∂ϕ ∂ψ ∂ϕ ∂ψ , 2 G + R ∂ x ∂ x ∂ x ∂ x R 1 1 2 2 M ZZ ZZ 10Q heat w,2 ≤ ϕ2w + ψ 2 w−1 , (1.9) π

ZZZ

R


293

where M = {(x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R 2 }. Let us fix any compact M0 in R3+ and choose R and ε in such a way that M0 ⊂ M. Restrict the integration in (1.9) to M0 , and let R → ∞. Taking into account that G(0, 0) = a > 0, we obtain Z Z Z ZZ 10Q heat Z Z ∂ϕ ∂ψ ∂ϕ ∂ψ w,2 ϕ2w + ψ 2 w−1 . + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 πa M0 But M0 is an arbitrary compact set in the upper half-space. Therefore, ZZZ

ZZ ∂ϕ ∂ψ ∂ϕ ∂ψ 10Q heat Z Z w,2 ϕ2w + ψ 2 w−1 . (1.10) + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 πa R3+

This immediately implies that ZZZ

∂ϕ ∂ψ ∂ϕ ∂ψ 20Q heat w,2 k f k L 2 (w) kψk L 2 (w−1 ) + ≤ 3 ∂ x ∂ x ∂ x ∂ x πa 1 1 2 2 R+

(1.11)

by substituting ϕ/t, tψ into (1.10) and minimizing over t. This proves Theorem 1.2.

Now we couple inequality (1.11) with Lemma 1.1, getting the right weighted estimates for R12 , R22 , R1 R2 , T . In particular, we have proved Theorem 0.6 for the case p = 2 up to the proof of Theorem 1.3. Proof of Theorem 1.3 We start with a much simpler “model” operator, Tσ . The logic is the following. We want to get a sharp weighted estimate of kTσ k L 2 (w)→L 2 (w) via the A2 -characteristic of w. Strangely enough, we first consider a two-weight estimate of kTσ k L 2 (u)→L 2 (v) with different weights u, v. In the paper of Nazarov, S. Treil, A. Volberg [26], one can find that this norm is attained on some “simple” test functions—and that this holds for every pair u, v. Thus, the same is true for u = v = w. However, on the family T of test functions, one can compute Nw,2 (Tσ ) := sup{kTσ tk L 2 (w) : t ∈ T , ktk L 2 (w) = 1}. It turns out that Nw,2 (Tσ ) ≈ Q class w,2 . J. Wittwer does this in [31] (see also [28]). Thus, we get kTσ k L 2 (w)→L 2 (w) = Nw,2 (Tσ ) ≈ Q class w,2 . This sharp one-weight estimate for the model operator (notice that its proof used a fact proved in two-weight theory) is used to construct the Bellman function for our operator. An application of the Green formula to this Bellman function is the content of the proof of Theorem 1.2.

294


So let us show what the model operator is, what its sharp weighted estimate is, and how one obtains a special function (the Bellman function) from this estimate. Consider the family of dyadic singular operators Tσ : Tσ f = 6 I ∈D σ (I )( f, h I )h I . Here D is a dyadic lattice on R, h I is a Haar function associated with the dyadic interval I (h I is normalized in L 2 (R, d x)), and σ (I ) = ±1. We call the family Tσ the martingale transform. It is a dyadic analog of a Calderón-Zygmund operator. The following are important questions about Tσ , the first one about two-weight estimates and the second one about one-weight estimates. (1) What are necessary and sufficient conditions for supσ kTσ k L 2 (u)→L 2 (v) < ∞? (2) What is the sharp bound on supσ kTσ k L 2 (w)→L 2 (w) in terms of w? How can one compute supσ kTσ k L 2 (w)→L 2 (w) ? These questions are dyadic analogs of notoriously difficult questions about classical Calderón-Zygmund operators like the Hilbert transform, the Riesz transforms, and the Ahlfors-Beurling transform. The dyadic model is supposed to be easier than the continuous one. This turns out to be true. The answers to the questions above appeared in [26] and [31]. Moreover, these answers are key to answering questions about classical Calderón-Zygmund operators. Strangely enough, the answer to the second question (which seems to be easier because it is about one-weight) seems to require the ideas from the two-weight case. Here is our explanation of this phenomenom. The necessary and sufficient conditions on (u, v) to answer the first question were given in [26]. They amount to the fact that supσ kTσ k L 2 (u)→L 2 (v) is almost attained on the family of simple test functions. This fact has beautiful consequences in the one-weight case. For then supσ kTσ k L 2 (w)→L 2 (w) is attainable (almost) on the family of simple test functions. One may try to compute supσ kTσ tk L 2 (w) for every element of this test family, thus getting a good estimate for the norm supσ kTσ k L 2 (w)→L 2 (w) . Test functions are rather simple, so this program can be carried out. This has been done in [31]. Here is the result. Recall that dyadic Q w,2 := sup hwi I hw−1 i I . I ∈D

THEOREM 1.6 For any A2 -weight w, we have the inequality dyadic

sup kTσ k L 2 (w)→L 2 (w) ≤ AQ w,2 . σ

Let us rewrite Theorem 1.6 as follows: dyadic sup 6 I ∈D σ (I )( f, h I )(g, h I ) ≤ AQ w,2 k f k L 2 (w) kgk L 2 (w−1 ) , σ (I )=±1


295

or dyadic 6 I ∈D ( f, h I ) (g, h I ) ≤ AQ w,2 k f k L 2 (w) kgk L 2 (w−1 ) . This inequality is scaleless, so we write it as J ∈ D,

1 6 I ∈D ,I ⊂J h f i I− − h f i I+ hgi I− − hgi I+ |I | |J | dyadic

1/2

1/2

≤ AQ w,2 h f 2 wi J hg 2 w−1 i J . (1.12) Here I− , I+ are the left and the right halves of I , and h·il means averaging over l, as usual. Given a fixed J ∈ D and a number Q > 1, we wish to introduce the Bellman function of (1.12), B(X, Y, x, y, r, s) n 1 = sup 6 I ∈D ,I ⊂J h f i I− − h f i I+ hgi I− − hgi I+ |I | : |J | h f i J = x, hgi J = y, hwi J = r, hw−1 i J = s, dyadic

h f 2 wi J = X, hg 2 w−1 i J = Y, w ∈ A2

dyadic

, Q w,2

o ≤Q .

Obviously, the function B does not depend on J , but it does depend on Q. Its domain of definition is the following: R Q := 0 ≤ (X, Y, x, y, r, s), x 2 ≤ X s, y 2 ≤ Y r, 1 ≤ r s ≤ Q . By (1.12) it satisfies 0 ≤ B ≤ AQ X 1/2 Y 1/2 .

(1.13)

We prove that it also satisfies the following differential inequality. Denote v := (X, Y, x, y, r, s),

v− = (X − , Y− , x− , y− , r− , s− ),

v+ = (X + , Y+ , x+ , y+ , r+ , s+ ), and let v, v+ , v− lie in R Q such that v = 1/2(v− + v+ ). Then B(v) −

1 B(v+ ) + B(v− ) ≥ |x+ − x− ||y+ − y− |. 2

(1.14)

In fact, let f, g, w almost maximize B(v) (on the interval J ), let f + , g+ , w+ do this for B(v+ ), and let f − , g− , w− do this for B(v− ). The scalelessness of B allows us to put f + , g+ , w+ on J+ and f − , g− , w− on J− . Then we have “gargoyle” functions ( ( ( f + on J+ , g+ on J+ , w+ on J+ , F= G= W = f − on J− , g− on J− , w− on J− .

296


Obviously, hFi J = 1/2(x+ + x− ) = x, hGi J = y, hW i J = r, hW −1 i J = s, hF 2 W i J = X, hG 2 W =1 i J = Y . These numbers together form the vector v. In other words, F, G, W compete with f, g, w in definition (1.12) of Bellman function B(v). By this definition, B(v) ≥

1 6 I ∈D ,I ⊂J hFi I− − hFi I+ hGi I− − hGi I+ |I |. |J |

But the almost optimality of f + , g+ , w+ on J+ and f − , g− , w− on J− gives us B(v+ ) ≥ −ε +

1 6 I ∈D ,I ⊂J+ hFi I− − hFi I+ hGi I− − hGi I+ |I |, |J+ |

B(v− ) ≥ −ε +

1 6 I ∈D ,I ⊂J− hFi I− − hFi I+ hGi I− − hGi I+ |I | |J− |

and

(recall that F = f ± on J± , G = g± on J± ). Combining these, we get 1 B(v+ ) + B(v− ) ≥ −ε + hFi J− − hFi J+ hGi J− − hGi J+ 2 = −ε + h f − i J− − h f + i J+ hg− i J− − hg+ i J+ = −ε + x− − x+ y− − y+ .

B(v) −

We are done with (1.14) because ε is an arbitrary positive number. Therefore, our B is a very concave function. We modify B to have its Hessian satisfy the conclusion of Theorem 1.3. To do that we fix a compact K in the interior of R Q , and we choose ε such that 100ε < dist(K , ∂ R Q ). Consider the convolution of B with (1/ε6 )ϕ(v/ε), v ∈ R6 , where ϕ is a bell-shaped infinitely differentiable function with support in the unit ball of R6 . It is now very easy to see that this convolution (we call it B K ,Q ) satisfies the following inequalities: 0 ≤ B K ,Q ≤ 6Q(X + Y ),

(1.15)

and for any vector ξ = (ξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 ) ∈ R6 , −(d 2 B K ,Q ξ, ξ )R6 ≥ 2|ξ2 ||ξ3 |.

(1.16)

The factor 2 appears because B(v) − (1/2)(B(v+ ) + B(v− )) in (1.14) corresponds to −(1/2)d 2 B. Theorem 1.3 is completely proved.


297

2. The sharp weighted estimate for the Ahlfors-Beurling operator in L p (w d A) for p > 2 Here again T is the Ahlfors-Beurling transform on the plane, and d A denotes Lebesgue measure on the plane. Now we prove that kT k L p (w d A)→L p (w d A) ≤ C( p)Q heat w, p .

(2.1)

In Section 3, it is shown how this implies kT k L p (w d A)→L p (w d A) ≤ C( p)Q class w, p .

(2.2)

Let us recall that for the weights on R, we can introduce dyadic

Q w, p

p−1

:= sup hwi I hw−(1/( p−1)) i I I ∈D

.

The weight ρ := w−(1/( p−1)) is called the dual weight, and dyadic

dyadic

0

(Q ρ, p0 ) p = (Q w, p ) p .

(2.3)

We need the following result of S. Buckley. (We formulate its dyadic version.) Let us denote M d f (x) = supx∈I,I ∈D h| f |i I . This is the dyadic maximal function. THEOREM 2.1 For any weight on the line R, dyadic

kM d k L p (w d x)→L p (w d x) ≤ C( p)Q ρ, p0 ,

1 < p < ∞.

We use it now to extend Theorem 1.6 to p > 2. THEOREM 2.2 For any weight on the line R, dyadic

sup kTσ k L p (w d x)→L p (w d x) ≤ C( p)Q w, p , σ

2 < p < ∞.

The proof that follows is word for word the extrapolation proof of Garcia-Cuerva and Rubio de Francia in [12]. We only need to be absolutely precise with exponents. Proof dyadic Let w ∈ A p . Let T stand for any of the Tσ ; let t = ( p − 2)/( p − 1). Then Z Z Z 2/ p p 2 |T f | w d x = sup |T f | uw d x ≤ sup |T f |2 vw d x. u≥0,kuk

≤1 0 L p /t (w d x)

v∈F

(2.4)

298


Here F is a family of functions v of the following type: v = v(u) := u +

S2u S3u Su + + + ··· , 2kSk 4kSk2 8kSk3

where u runs over the unit ball of L p /t (w d x), the nonlinear operator S is given by the formula t S(u) := w−1 M d (u 1/t w) , 0

and kSk denotes kSk L p0 /t (w d x)→L p0 /t (w d x) . Now let us prove the following: (1) u ≤ v, (2) kvk L p0 /t (w d x) ≤ 2kuk L p0 /t (w d x) , dyadic

dyadic

dyadic

(3) vw ∈ A2 , Q vw,2 ≤ Q w, p . The first inequality is clear. To prove the second inequality, we notice that if kSk is finite, then the second inequality follows immediately. So let us prove this finiteness. Now dyadic kSk ≤ C( p)(Q w, p )t . (2.5)

Z

In fact, using Theorem 2.1 we get Z p0 0 0 (Su) p /t w d x = M d (u 1/t w) w1− p d x Z Z p0 −(1/( p−1)) d 1/t = M (u w) w dx = Z 0 0 0 dyadic 0 ≤ C( p)(Q w, p ) p u p /t w p w1− p d x Z 0 dyadic 0 = C( p)(Q w, p ) p u p /t w d x. Therefore, (2.5) is proved. Let us now check that

−1 dyadic huwi I S(u)w ≤ (Q w, p )1−t , I

p0 M d (u 1/t w) ρ d x

∀I ∈ D , u ≥ 0.

(2.6)

In fact, (1 − t)(1 − p) = 1. Using this and the pointwise estimate on (M d (u 1/t w))(x) ≥ hu 1/t wi I , ∀x ∈ I , we get

−t −(1/( p−1)) huwi I M d (u 1/t w) w−(1/( p−1)) I ≤ hu 1/t witI hwi1−t · hu 1/t wi−t iI . I I hw Therefore,

−t −(1/( p−1)) (1−t)(1− p) huwi I M d (u 1/t w) w−(1/( p−1)) I ≤ hwi1−t iI I hw dyadic

≤ (Q w, p )1−t .


299

We are done with (2.6). Now at last we have our third estimate as a combination of (2.5), (2.6), and the fact that by its definition, S(v) ≤ 2kSkv: dyadic

∀v ∈ F ,

dyadic

Q vw,2 ≤ C( p)Q w, p .

(2.7)

Now using (2.7) and inequality kvk L p0 /t (w d x) ≤ 2kuk L p0 /t (w d x) ≤ 2, we get Z

|T f | p w d x

2/ p

Z

Z dyadic |T f |2 vw d x ≤ A(Q vw,2 )2 | f |2 vw d x v∈F Z 2/ p dyadic ≤ AC( p)(Q w, p )2 | f |pw d x kvk L p0 /t (w d x) ≤ sup

dyadic

≤ 2AC( p)(Q w, p )2 k f k2L p (w d x) . In the second inequality we used Theorem 1.6. Thus, dyadic kTσ k L p (w d x)→L p (w d x) ≤ 2AC( p)Q w, p , and Theorem 2.2 is completely proved. Theorem 1.3 has the following ( p > 2)-version. 2.3 p For any Q > 1, p > 2, define the domain D Q := {0 < (X, Y, x, y, r, s) : x p < THEOREM

0

p

0

X s p−1 , y p < Y r p −1 , 1 < r s p−1 < Q}. Let K be any compact subset of D Q . Then ( p)

there exists a function B = B Q,K (X, Y, x, y, r, s) infinitely differentiable in the small neighborhood of K such that 0 (1) 0 ≤ B ≤ C( p)Q X 1/ p Y 1/ p , (2) −d 2 B ≥ |d x||dy|. Proof The proof repeats verbatim the proof of Theorem 1.3. But now we base the considerations on Theorem 2.2 rather than Theorem 1.6. Now we are ready to prove our main estimate (2.1). The proof repeats the one given in the previous section for p = 2, but instead of b(x, t) we consider b( p) (x, t) := B ( p) v(x, t), where v(x, t) is the 6-vector 0 ( f p w)(x, t), (g p w−(1/( p−1)) )(x, t), f (x, t), g(x, t), w(x, t), w−(1/( p−1)) (x, t) .

300


The rest is identical, but the use of b( p) explains why one needs to choose Q at −(1/( p−1)) (x, t)) p−1 . least as large as Q heat w, p = sup(x,t)∈R3+ w(x, t)(w Inequality (2.1) is completely proved. We have also proved our main Theorem 0.6. 3. The comparison of classical and heat A p -characteristics In this section we give an easy proof of the following theorem. 3.1 There exists a finite absolute constant b such that THEOREM

class Q heat w, p ≤ bQ w, p .

Proof Constants are denoted by the letters c, C; they may vary from line to line and even within the same line. We introduce the following notation. Bk denotes B(0, 2k ), k = R 0, 1, 2, . . . , h f i B stands for the average (1/|B|) B f d A, and f (B) stands for R RR h 2 B f d A. If B = B(0, r ), then h f i B stands for (1/(πr )) R2 f (x) exp − (kxk2 /(r 2 )) d x1 d x2 . LEMMA 3.2 Suppose that f and g, positive functions on the plane, are such that sup B h f i B hgi B = A. Then there exists a finite absolute constant c such that

h f i B hgihB ≤ c A for any disc B. Proof Scale invariance allows us to prove this for only one disc, B = B(0, 1). We start the estimate: A h f i B hgihB ≤ ch f i B 6k 22k exp(−22k−2 ) . h f i Bk On the other hand, h f i Bk > ch f i Bk−1 > · · · > ck h f i B . (Recall that B is the unit disc.) Plugging this into the inequality above, we get h f i B hgihB ≤ ch f i B 6k C k exp(−22k−2 )

A . h f iB

In other words, h f i B hgihB ≤ c A6k C k exp(−22k−2 ) = c A, and the lemma is proved.


301

Now we want to prove Theorem 3.1. Fix B. Again, by scale invariance it is enough to consider B = B(0, 1). By the previous lemma, we know that h f i Bk hgihBk ≤ c A

(3.1)

for any k. Now h f ihB hgihB ≤ chgihB 622k exp(−22k−2 )h f i Bk ≤ chgihB 622k exp(−22k−2 )

cA hgihBk

.

The last inequality uses (3.1). On the other hand, hgihBk > chgihBk−1 > · · · > ck hgi B . (Recall that B is the unit disc.) Plugging this into the inequality above, we get h f ihB hgihB ≤ chgihB 6k C k exp(−22k−2 )

cA hgihB

.

In other words, h f ihB hgihB ≤ c2 A6k C k exp(−22k−2 ) = c2 A. Theorem 3.1 is completely proved. 4. Injectivity at the critical exponent and regularity of solutions of the Beltrami equation Theorem 0.4 was proved in [2]. It reduces the question of the injectivity of I − µT , I − T µ in L 1+k (C), k = kµk∞ < 1, to estimate (0.1): kT k L p (ω d A)→L p (ω d A) ≤

C , 1 + (1/k) − p

where ω = | f z ◦ f −1 | p−2 , f is a ((1 + k)/(1 − k))-quasiconformal homeomorphism, and p ∈ [2, 1 + (1/k)). This estimate has been proved twice in the present paper. We first proved the heat class estimate of the norm kT k via AQ heat w, p , then noticed that Q w, p ≈ Q w, p , and finally class noticed that (see [2]) Q w, p ≤ C/(1+(1/k)− p) for special ω’s that are certain powers of Jacobians of quasiconformal homeomorphisms. The second time, we started with heat the estimate of the norm kT k via AQ heat w, p . Then we directly computed Q w, p for ω’s that are certain powers of Jacobians of quasiconformal homeomorphism and again saw that Q heat w, p ≤ C/(1 + (1/k) − p) (see Theorem 0.7). Thus, we can be sure that injectivity holds. This fact is immediately applicable to so-called weakly quasiregular maps. Recall (see [2]) that the solution of Fz¯ − µFz = 0

in ⊂ C

302

PETERMICHL and VOLBERG 1,q

can be called q-weakly quasiregular if it belongs to Wloc . Here µ ∈ L ∞ (), kµk∞ = k < 1. It is known (see [2]) that if q > 1 + k, then a q-weakly quasiregu1,2+ε lar map is actually in Wloc for a certain positive . In particular, it is quasiregular. Therefore, it is continuous, open, discrete, and so on. However, the proof of this “selfimprovement” fact is very subtle. It is based on Astala’s sharp estimates of smoothness of quasiconformal homeomorphisms, which were obtained as a solution of a famous problem of F. Gehring and E. Reich. In [2], one can find an easy example that shows this is no longer true for q < 1+k. Thus, the remaining question is about the critical exponent q = 1 + k. THEOREM

Let kµk∞

4.1 = k < 1. Then (1 + k)-weakly quasiregular maps are also quasiregular.

We repeat the argument from [2]. This is done for the sake of the convenience of the reader. Proof Choose ϕ ∈ C∞ 0 (). Set G = Fϕ. Then G z¯ − µG z = (ϕz¯ − µϕz )F.

(4.1)

Obviously, G is a Cauchy transform of the compactly supported function ψ := G z¯ . Then (4.1) can be rewritten as (F0 := (ϕz¯ − µϕz )F): (I − µT )ψ = F0 ∈ L 2 (C).

(4.2)

1,1+k In fact, F ∈ Wloc , so by Sobolev’s theorem (ϕz¯ −µϕz )F ∈ L ((2+2k)/(1−k)) (C), and (ϕz¯ − µϕz )F has compact support. So it is in L 2+ε (C). Looking at (4.2), we can see that µ = 0 outside of supp ϕ. Our equation (4.2) has an obvious solution as a Neumann series. It converges in L 2+ε (C) (we use the fact that kT k in L p is close to 1 when p is close to 2) and has its support in supp ϕ. So it is in L 1+k (C) as well. Let us call it ψ0 . Now we have two solutions of (4.2) in L 1+k (C) − ψ and ψ0 . However, we have already proved the injectivity of I − µT in L 1+k (C). Therefore, ψ = ψ0 . Thus, G z¯ = ψ ∈ L 2+ε (C). Therefore, the compactly supported function G is in W 1,2+ε (C). This means that G is quasiregular and, in particular, continuous, open, discrete, and so on. Theorem 4.1 is completely proved.

Acknowledgments. We are deeply grateful to Kari Astala, Eero Saksman, Michael Frazier, Tadeusz Iwaniec, Fedor Nazarov, and Jacob Plotkin for discussions and criticism.


303

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K. ASTALA, T. IWANIEC, and E. SAKSMAN, Beltrami operators, preprint, 1999. 281,

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Z. Anal. Anwendungen 1 (1982), 1 – 16. MR 85g:30027 282 , The best constant in a BMO-inequality for the Beurling-Ahlfors transform, Michigan Math. J. 33 (1986), 387 – 394. MR 88b:42024 , Hilbert transform in the complex plane and the area inequalities for certain quadratic differentials, Michigan Math. J. 34 (1987), 407 – 434. MR 89a:42025 , “L p -theory of quasiregular mappings” in Quasiconformal Space Mappings, ed. Matti Vuorinen, Lecture Notes in Math. 1508, Springer, Berlin, 1992. CMP 1 187 088 282 T. IWANIEC and G. MARTIN, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29 – 81. MR 94m:30046 282 , Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25 – 57. MR 97k:42033 282 O. LEHTO, Remarks on the integrability of the derivatives of quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 371 (1965), 3 – 8. MR 31:5975 , “Quasiconformal mappings and singular integrals” in Convegno sulle Transformazioni Quasiconformi e Questioni Connesse (Rome, 1974), Sympos. Math. 18, Academic Press, London, 1976, 429 – 453. MR 58:11387 283 O. LEHTO and K. I. VIRTANEN, Quasiconformal Mappings in the Plane, 2d ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973. MR 49:9202 282, 283 F. NAZAROV, S. TREIL, and A. VOLBERG, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), 909 – 928. MR 2000k:42009 281, 293, 294 F. NAZAROV and A. VOLBERG, Heating of the Ahlfors-Beurling operator and the estimates of its norms, preprint, 2000, http://math.msu.edu/˜volberg/papers/heating1/piminusone.html 283 S. PETERMICHL and J. WITTWER, A sharp estimate for the weighted Hilbert transform via Bellman functions, preprint, 2000, http://williams.edu/mathematics/jwittwer 293 E. STEIN, Harmonic Analysis: Real-variable Methods, Orthogonality, And Oscillatory Integrals, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993. MR 95c:42002 285, 287 G. WANG, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522 – 551. MR 96b:60120 283 J. WITTWER, A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7 (2000), 1 – 12. MR 2001e:42022 293, 294

Petermichl Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA; [email protected]


305

Volberg Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA; [email protected]; current: Equipe d’Analyse, Université Pierre et Marie Curie-Paris 6, 4 Place Jussieu, 75252 Paris CEDEX 05, France; [email protected]


IRREDUCIBILITY CRITERION FOR TENSOR PRODUCTS OF YANGIAN EVALUATION MODULES A. I. MOLEV

Abstract The evaluation homomorphisms from the Yangian Y(gln ) to the universal enveloping algebra U(gln ) allow one to regard the irreducible finite-dimensional representations of gln as Yangian modules. We give necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules. 1. Introduction The Yangian Y(gln ) for the general linear Lie algebra gln is a deformation of the universal enveloping algebra U(gln [x]) in the class of Hopf algebras (see V. Drinfeld [6]). A theorem of Drinfeld (see [8]; see also V. Tarasov [24]) provides a complete description of finite-dimensional irreducible representations of Y(gln ) in terms of their highest weights. Recently, Arakawa [2] has found a character formula for each of these representations with the use of the Kazhdan-Lusztig polynomials (see also Vasserot [25] for the case of quantum affine algebras). M. Nazarov and Tarasov [22] (see also I. Cherednik [5]) have given an explicit construction for a class of so-called tame representations. However, the structure of the general finite-dimensional irreducible Y(gln )-module (with n ≥ 3) still remains unknown. In this paper we establish an irreducibility criterion for tensor products of the Yangian evaluation modules which thus solves this problem for a wide class of representations of Y(gln ). Let λ = (λ1 , . . . , λn ) be an n-tuple of complex numbers such that λi − λi+1 is a nonnegative integer for each i. Denote by L(λ) the irreducible finite-dimensional representation of the Lie algebra gln with the highest weight λ. For each a ∈ C, there is an evaluation homomorphism ϕa from the Yangian Y(gln ) to the universal enveloping algebra U(gln ) (see Section 2 for the definitions). Using ϕa , we make L(λ) into a Yangian module and denote it by L a (λ). We keep the notation L(λ) for the evaluation module L a (λ) with a = 0. The Hopf algebra structure on Y(gln ) allows DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 26 September 2000. Revision received 10 April 2001. 2000 Mathematics Subject Classification. Primary 17B37. Author’s work supported by the Australian Research Council.

307

308

A. I. MOLEV

one to regard tensor products of the type L a1 (λ(1) ) ⊗ L a2 (λ(2) ) ⊗ · · · ⊗ L ak (λ(k) )

(1.1)

as Yangian modules. Our main result is a criterion of irreducibility of these modules. To formulate it we first note that the problem can be reduced to the particular case where all the parameters ai in (1.1) are equal to zero. This is done by using the composition of the module (1.1) with an appropriate automorphism of the Yangian (see Proposition 2.4). We give first an irreducibility criterion for the tensor product L(λ) ⊗ L(µ) of two evaluation modules. It is well known (see, e.g., [10] and Theorem 3.1) that this module is irreducible if the differences λi − µ j are not integers. Furthermore, for any c ∈ C, the simultaneous shifts λi 7 → λi + c and µ j 7→ µ j + c for all i and j do not affect the irreducibility of L(λ) ⊗ L(µ) (see Proposition 2.5). Thus, we may assume without loss of generality that all the entries of λ and µ are integers. We use the following definition. Two disjoint finite subsets A and B of Z are crossing if there exist elements a1 , a2 ∈ A and b1 , b2 ∈ B such that either a1 < b1 < a2 < b2 or b1 < a1 < b2 < a2 . Otherwise, A and B are called noncrossing. Given a highest weight λ with integer entries, introduce the following subset of Z: Aλ = {λ1 , λ2 − 1, . . . , λn − n + 1}. THEOREM 1.1 The module L(λ) ⊗ L(µ) is irreducible if and only if the sets Aλ \ Aµ and Aµ \ Aλ are noncrossing.

Using the argument of N. Kitanine, J. Maillet, and V. Terras [11], [15], Nazarov and Tarasov [23, Theorem 4.9] demonstrated that the irreducibility criterion for the multiple tensor product (1.1) can be obtained from the particular case of k = 2 tensor factors. Namely, the following binary property holds. Here λ(1) , . . . , λ(k) are n-tuples ( p) ( p) of complex numbers such that λi − λi+1 is a nonnegative integer for each i and p. BINARY PROPERTY

The module

L(λ(1) ) ⊗ L(λ(2) ) ⊗ · · · ⊗ L(λ(k) )

(1.2)

is irreducible if and only if all the modules L(λ( p) ) ⊗ L(λ(q) ) with p < q are irreducible. Note that the “only if” part of this assertion is implied by the well-known Proposition 2.6 (see Section 2). If the module (1.2) is irreducible, its highest weight is easy to

IRREDUCIBILITY CRITERION FOR YANGIAN MODULES

309

find. Therefore, together with Theorem 1.1, the binary property allows one to determine whether a given irreducible Y(gln )-module can be realized in a tensor product (1.2). For the proof of Theorem 1.1, we use the Gelfand-Tsetlin bases of the gln modules L(λ) and L(µ). The key role is played by the formulas for the action of the Drinfeld generators of Y(gln ) in these bases, as well as by the quantum minor formulas for the Yangian lowering operators [16] (cf. Nazarov and Tarasov [20], [22]). In the case of the Yangian Y(gl2 ), our criterion coincides with the one obtained by V. Chari and A. Pressley [3], and it is also implicitly contained in Tarasov’s paper [24] (see also [17]). Nazarov and Tarasov [21] found a criterion of irreducibility of (1.1) in the case where each highest weight λ( p) has the form (α, . . . , α, β, . . . , β) with α − β ∈ Z+ . This generalized earlier results by T. Akasaka and M. Kashiwara [1], and A. Zelevinsky [26]. B. Leclerc, Nazarov, and J.-Y. Thibon [12] have found an irreducibility criterion for the induction products of evaluation modules over the affine Hecke algebras of type A with the use of the canonical bases (see also Leclerc and Thibon [13], Leclerc and Zelevinsky [14]). The application of the Drinfeld functor (see [7]; see also [2]) leads to an irreducibility criterion (equivalent to ours) for the Yangian modules (1.1), when the highest weights λ( p) satisfy some extra conditions. Namely, assuming that the λ( p) are partitions (we may do this without loss of generality), one should require that for all p < q the sum of the lengths of λ( p) and λ(q) does not exceed n. 2. Preliminaries We refer the reader to the expository papers [17] and [19], where the results on the structure theory and representations of the Yangians are collected. The Yangian Y(n) = Y(gln ) (see [6], [8]) is the complex associative algebra with (1) (2) the generators ti j , ti j , . . . , where 1 ≤ i, j ≤ n, and the defining relations ti j (u), tkl (v) = where

(1)

1 tk j (u)til (v) − tk j (v)til (u) , u−v

(2.1)

(2)

ti j (u) = δi j + ti j u −1 + ti j u −2 + · · · ∈ Y(n)[[u −1 ]] and u is a formal (commutative) variable. The Yangian Y(n) is a Hopf algebra with the coproduct 1 : Y(n) → Y(n) ⊗ Y(n) defined by n X 1 ti j (u) = tia (u) ⊗ ta j (u).

(2.2)

a=1

Given sequences a1 , . . . , ar and b1 , . . . , br of elements of {1, . . . , n}, the corresponding quantum minor of the matrix [ti j (u)] is defined by the following equivalent

310

A. I. MOLEV

formulas: ar t ab11 ··· ··· br (u) =

=

X

sgn σ · taσ (1) b1 (u) · · · taσ (r ) br (u − r + 1)

(2.3)

sgn σ · ta1 bσ (1) (u − r + 1) · · · tar bσ (r ) (u).

(2.4)

σ ∈Sr

X σ ∈Sr

ar The series t ab11 ··· ··· br (u) is skew-symmetric under permutations of the indices ai or bi . The Poincaré-Birkhoff-Witt theorem for the Yangian Y(n) (see, e.g., [19, Corollary 1.23]) implies that, given a subset of indices {a1 , . . . , ar } ⊆ {1, . . . , n}, the coefficients of the series tai a j (u) with i, j = 1, . . . , r generate a subalgebra of Y(n) isomorphic to Y(r ). The mapping 1··· b j ···n ti j (u) 7 → (−1)i+ j t 1···bi ··· n (−u) (2.5)

defines an algebra automorphism of Y(n); the hats indicate the indices to be omitted. The following proposition is proved in [18, Proposition 1.1] by using the Rmatrix form of the defining relations (2.1). PROPOSITION 2.1 We have the relations min{k,l} X a1 ··· ak c1 ···cl t b1 ···bk (u), t d1 ···dl (v) =

(−1) p−1 p! (u − v − k + 1) · · · (u − v − k + p) p=1 X a1 ··· c j ··· c j p ··· ak c1 ··· ai ··· ai ··· cl × t b1 ··· bk1 (u)t d1 ··· dl1 p (v) i 1 < ··· 3, and we denote this character by q : G Q → F× p.

GALOIS REPRESENTATIONS AND COHOMOLOGY

525

We also refer to the character 4 , which is ramified only at 2 and cuts out the field √ Q( −1). 2.3. Irreducible GLn (F p )-modules The natural generalization of the weight in Serre’s conjecture is an irreducible GLn (F p )-module. To see this, we note that the Eichler-Shimura theorem (see [23]) relates the space of modular forms of weight k to cohomology with coefficients in Symg (C2 ) with g = k − 2. Hence, an eigenform f of level N , nebentype , and weight k gives rise to a collection of Hecke eigenvalues which, when reduced modulo p, also occurs in H 1 00 (N ), Vg () , where Vg ∼ = Symg (F¯ 2p ) is the space of two-variable homogeneous polynomials of degree g over F¯ p with the natural action of SL2 (F¯ p ). Ash and G. Stevens have shown in [5] that any system of Hecke eigenvalues occurring in the cohomology of 00 (N ) with coefficients in some GLn (F p )-module also occurs in the cohomology with coefficients in at least one irreducible GLn (F p )-module occurring in a composition series of the original module. Hence, there is some irreducible GLn (F p )-module W such that the system of eigenvalues coming from f also occurs in H 1 (00 (N ), W ()). Given this fact, it is natural to ask which irreducible modules give rise to the system of eigenvalues. We may parameterize the complete set of irreducible GLn (F p )-modules as in [10]. Definition 2.3 We say that an n-tuple of integers (b1 , . . . , bn ) is p-restricted if 0 ≤ bi − bi+1 ≤ p − 1,

1 ≤ i ≤ n − 1,

and 0 ≤ bn < p − 1. PROPOSITION 2.4 The set of irreducible GLn (F p )-modules is in one-to-one correspondence with the collection of all p-restricted n-tuples.

The one-to-one correspondence in this proposition may be described explicitly as follows: the module F(b1 , . . . , bn ) corresponding to the p-restricted n-tuple

526

ASH, DOUD, and POLLACK

(b1 , . . . , bn ) is the unique simple submodule of the dual Weyl module W (b1 , . . . , bn ) with highest weight (b1 , . . . , bn ). Theorem 8.1 gives an explicit model for the module F(b1 , b2 , b3 ) in the case n = 3, but for larger n no general computational models are known to the authors. In dealing with Galois representations, it often becomes necessary to associate an irreducible module to an n-tuple that is not p-restricted. We do this via the following definition. Definition 2.5 Let (a1 , . . . , an ) be any n-tuple of integers. Define F(a1 , . . . , an )0 = F(b1 , . . . , bn ), where (b1 , . . . , bn ) is a p-restricted n-tuple for which ai ≡ bi

(mod p − 1).

We note that in certain cases (namely, when some ai ≡ ai+1 (mod p − 1)) the module F(a1 , . . . , an )0 may not be well defined. In this case we interpret any statement concerning F(a1 , . . . , an )0 to mean that the statement is true for some choice of F(b1 , . . . , bn ) as in the definition. For example, if p = 5, a statement concerning F(1, 0, 0)0 is true if the statement is true for either F(1, 0, 0) or F(5, 4, 0) (or both). When dealing with modules defined by the prime notation, we say that a module F(a1 , . . . , an )0 is determined unambiguously if there is a unique p-restricted sequence congruent to (a1 , . . . , an ) modulo p − 1. 2.4. The strict parity condition We modify slightly the statement of the strict parity condition in [4] for ease of exposition, but our formulation is logically equivalent to that in [4]. Definition 2.6 Let V = F¯ np be an n-dimensional space with the standard action of GLn (F¯ p ). A Levi subgroup L of GLn (F¯ p ) is the simultaneous stabilizer of a collection D1 , . . . , Dk of L subspaces such that V = i Di . If each Di has a basis consisting of standard basis vectors for V , then L is called a standard Levi subgroup. Example 2.7 The standard Levi subgroups of GL2 (F¯ p ) are the subgroup of diagonal matrices and the whole of GL2 (F¯ p ).


527

Example 2.8 The standard Levi subgroups of GL3 (F¯ p ) are the subgroup of diagonal matrices, the whole of GL3 (F¯ p ), and the three subgroups   ∗ 0 0 0 ∗ ∗ , 0 ∗ ∗

  ∗ 0 ∗ 0 ∗ 0 , ∗ 0 ∗



 ∗ ∗ 0 ∗ ∗ 0  . 0 0 ∗

Definition 2.9 Let ρ : G Q → GLn (F¯ p ) be a continuous representation. A standard Levi subgroup L of GLn (F¯ p ) is said to be ρ-minimal if L is minimal among all standard Levi subgroups that contain some conjugate of the image of ρ. Definition 2.10 A semisimple continuous representation ρ : G Q → GLn (F¯ p ) satisfies the strict parity condition with Levi subgroup L if it has the following properties: (1) L is ρ-minimal; (2) the image of complex conjugation is conjugate inside L to a matrix   1   ±  −1  .. . with strictly alternating signs on the diagonal. Example 2.11 Any odd irreducible two-dimensional (resp., three-dimensional) representation satisfies strict parity, with L = GL2 (F¯ p ) (resp., L = GL3 (F¯ p )). Example 2.12 Let ρ be the direct sum of a two-dimensional odd irreducible representation and a one-dimensional representation, with image contained inside   ∗ 0 0 L = 0 ∗ ∗ 0 ∗ ∗

or

  ∗ ∗ 0 L = ∗ ∗ 0 . 0 0 ∗

Then ρ satisfies the strict parity condition, with Levi subgroup L. Example 2.13 Let ρ be the direct sum of a two-dimensional even irreducible representation and a

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one-dimensional representation, with the image of ρ contained inside   ∗ ∗ L =  ∗ . ∗ ∗ Then ρ satisfies strict parity with this Levi subgroup exactly when ρ is odd. Remark 2.14 Note that any odd three-dimensional Galois representation is conjugate to a representation that satisfies the strict parity condition for some standard Levi subgroup L. More generally, if ρ is an n-dimensional representation where the number of +1 eigenvalues and the number of −1 eigenvalues of complex conjugation differ by at most one, then ρ satisfies the strict parity condition for some standard Levi subgroup L. Definition 2.15 If ρ : G Q → GLn (F¯ p ) lands inside a Levi subgroup L, and σ : G Q → GLn (F¯ p ) is another representation of G Q , we say that ρ ∼L σ if there is a matrix M ∈ L such that Mρ(g)M −1 = σ (g) for all g ∈ G Q . If L = GLn (F¯ p ), then we may write ρ ∼ σ. 2.5. Weights We now begin to predict the weights (or irreducible modules) for which we expect to find cohomology eigenclasses with ρ attached. Following the example of Serre’s conjecture, we expect these weights to be determined by the restriction of ρ to a decomposition group at p, so we are interested in studying representations of the decomposition group G p . For convenience we denote the inertia group G p,0 by I p and the wild ramification group G p,1 by Iw . We begin by considering simple representations of G p . LEMMA 2.16 Let V be a simple n-dimensional F¯ p [G p ]-module, with the action of G p given by a


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representation ρ : G p → GL(V ). Then we may choose a basis for V such that   ϕ1   .. ρ| I p =  , . ϕn

m , . . . , ψ m for some with the characters ϕ1 , . . . , ϕn equal to some permutation of ψn,1 n,n m ∈ Z.

Proof This proof is almost identical to the proof in [13] for two-dimensional representations. We first note that ρ has finite image, so that we may actually realize it over a finite extension of F p . Hence, we may find an F pm [G p ]-module V 0 such that V = V 0 ⊗ F¯ p . We note that Iw must act trivially on V 0 since the invariants V 0 Iw are a nontrivial G p submodule of the simple module V 0 (since the image of Iw under ρ is a p-group). Hence, we may diagonalize ρ| I p . Since the Frobenius acts on the tame inertia as pth powers, we see that the set of diagonal characters must be stable under taking pth powers. Finally, since V is simple, the Frobenius must permute the diagonal characters transitively, resulting in the characterization given above. Remark 2.17 Note that for a given V , Lemma 2.16 yields n distinct values of m modulo ( p n − 1). If m 0 is one of them, the others are congruent to pm 0 , p 2 m 0 , . . . , p n−1 m 0 modulo ( p n − 1). Definition 2.18 Let V be a simple G p -module, diagonalized as in Lemma 2.16 with some choice of exponent m. If possible, write m as m = a1 + a2 p + · · · + an p n−1 , with 0 ≤ ai − an ≤ p − 1 for all i. Suppose that (b1 , . . . , bn ) satisfies bi ≥ bi+1 for all i < n and is obtained by permuting the entries of (a1 , . . . , an ). Then we say that (b1 , . . . , bn ) is derived from V . If the action of G p on V is given by a representation ρ, we say that the n-tuple is derived from ρ. Remark 2.19 Note that not all values of m have an expansion of the form given here. For example, if p = 5, n = 3, m = 30, there is no expansion satisfying the above properties. It is a simple exercise to see that every simple module has at least one derived n-tuple and that a given value of m yields a unique n-tuple if it yields any. Hence, a simple

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n-dimensional G p -module may have at most n n-tuples derived from it, but it can have fewer. Now let V be any n-dimensional G p -module, with the action of G p given by ρ : G p → GL(V ). We may find a composition series {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vk = V. Let each composition factor Vi /Vi−1 have dimension di , and set d0 = 0. By diagonalizing ρ on each simple composition factor, we may find a basis (e1 , . . . , en ) of V such that ρ is upper triangular, with diagonal characters (ϕ1,1 , . . . , ϕ1,d1 , ϕ2,1 , . . . , ϕ2,d2 , . . . , ϕk,1 , . . . , ϕk,dk ), where the first d1 characters come from the action on V1 /V0 , the next d2 from the action on V2 /V1 , and so on. For each composition factor, choose m i such that for some i j, ψdmi ,1 = ϕi, j , and such that m i yields a di -tuple derived from Vi /Vi−1 . Concatenating these di -tuples gives us an n-tuple (a1 , . . . , an ). We wish to preserve the order of the integers in our n-tuple which come from an individual composition factor, so we make the following definition. Definition 2.20 A permutation σ of the integers {1, . . . , n} is compatible with the filtration 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vk = V given above if for 0 ≤ s < k and a, b ∈ [1 + we have σ (a) < σ (b).

Ps

j=0 d j , ds+1

+

Ps

j=0 d j ]

with a < b,

Definition 2.21 Let V be an n-dimensional G p -module with chosen basis {e1 , . . . , en } with respect to which the action of G p is upper triangularized, and let (a1 , . . . , an ) be an ntuple obtained as above. If σ is a permutation of the integers {1, . . . , n} compatible with the filtration above and such that the action of G p with respect to the ordered basis {eσ (1) , . . . , eσ (n) } remains upper triangular, then we say that the n-tuple (aσ (1) , . . . , aσ (n) ) is derived from V . Remark 2.22 Note that there is at least one (and possibly more) n-tuple derived from V , namely, the original n-tuple (a1 , . . . , an ). In addition, even the choice of this original n-tuple is not unique, so that there usually are many n-tuples derived from a given V .


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Definition 2.23 Let ρ : G Q → GLn (F¯ p ) be a semisimple continuous representation, conjugated to land in a ρ-minimal standard Levi subgroup L. Let D1 , . . . , Dk be the subspaces of F¯ np given in the definition of L. Then we have representations ρi : G Q → GL(Di ), which make each Di into a G Q -module. Let G p be a decomposition group above p, and consider each Di as a G p -module. Let di = dim Di , and let (a1 , . . . , adi ) be a di -tuple derived from Di as above. If the standard basis elements of F¯ np which span Di are e jr , 1 ≤ r ≤ di , with jr < js for r < s, then set b jr = ar for r = 1, . . . , di . Doing this for each Di produces an n-tuple (b1 , . . . , bn ). Such an n-tuple is said to be derived from ρ, with Levi subgroup L. Remark 2.24 Note that the above discussion may (in many cases) be summarized more informally as follows. Given a representation ρ : G Q → GLn (F¯ p ) which lands inside a ρminimal standard Levi subgroup L, we may upper triangularize its restriction to inertia by conjugating by an element of L. This gives a sequence of characters of the tame inertia group on the diagonal. Group these characters together into niveau d collections. (A niveau d collection is set of d characters, each a power of a different fundamental character of niveau d with the same exponent m and all appearing in the same “Levi block”.) For a given niveau d collection, write the exponent m as a1 + a2 p + · · · + ad p d−1 , with 0 ≤ ai − ad ≤ p − 1 for all i, and let (b1 , . . . , bd ) be the ordered (decreasing) d-tuple with the same components as (a1 , . . . , ad ). Then construct an n-tuple (c1 , . . . , cn ) as follows: if the ith character in the niveau d collection is in the kth diagonal position in the image of ρ, set ck = bi . (Note that the order of the bi should be preserved in the n-tuple.) This procedure gives the same derived n-tuples as above, except when there is a combination of wild ramification and multiple niveau d collections containing the same characters, in which case the more complicated procedure described above is needed.

3. Conjecture CONJECTURE 3.1 Let ρ : G Q → GLn (F¯ p ) be a continuous semisimple Galois representation. Suppose that ρ satisfies the strict parity condition with Levi subgroup L. Let (a1 , . . . , an ) be an n-tuple derived from ρ with the Levi subgroup L, and let V = F(a1 −(n −1), a2 − (n − 2), . . . , an − 0)0 . Further, let N = N (ρ) be the level of ρ, and let = (ρ) be

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the nebentype character of ρ. Then ρ is attached to a cohomology eigenclass in H ∗ 00 (N ), V () . Remark 3.2 We note that in the case of two-dimensional Galois representations, we may take ∗ to be 0 or 1, and in fact, for irreducible two-dimensional representations, we may take ∗ to be 1. In the case of three-dimensional Galois representations, we may take ∗ to be at most 3, and for irreducible Galois representations (or sums of an even twodimensional representation with a one-dimensional representation) we may take ∗ to be equal to 3, as explained in [4]. As mentioned previously, any odd two-dimensional or three-dimensional representation is conjugate to a representation that satisfies strict parity for some standard Levi subgroup L. In our computations we test the conjecture for three-dimensional representations by computing H 3 . In cases where ρ is the sum of three characters or the sum of an odd two-dimensional representation and a character, we are thus actually testing a stronger assertion than Conjecture 3.1, namely, that the cohomology class exists in H 3 (see, e.g., Tables 4 and 9). We did not test any ρ that are sums of three characters in this paper, but several examples of such may be found in [3] and [1]. In addition, we do not present computational examples for p = 2 as this would involve rewriting portions of our computer programs. In addition, for p = 2 and p = 3, our computational techniques (based on those in [1]) do not always compute the whole of H 3 . Nevertheless, we have no reason to doubt our conjecture for these primes. In particular, problems with the weight and nebentype that occur when p = 2 or p = 3 for Serre’s original conjecture involving classical modular forms modulo p should not occur for our conjecture, which involves mod p cohomology. Remark 3.3 Note that Conjecture 3.1 applies to Galois representations of arbitrary dimension, but that we have no computational evidence for dimension higher than 3. Forthcoming work of Ash with P. Gunnells and M. McConnell touches on the case of certain fourdimensional representations. Remark 3.4 Note that the conjecture makes no claim of predicting all possible weights that yield an eigenclass with ρ attached. In fact, we have three types of computational examples in which additional weights (not predicted by the conjecture) do yield eigenclasses that appear to have ρ attached. The first type of additional weight occurs if ρ is attached to a quasi-cuspidal


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eigenclass (e.g., if ρ is either irreducible or reducible as a sum of an even twodimensional representation and a character). In this case, for certain weights, we may define an extra weight as follows. Definition 3.5 Let F(a, b, c) be an irreducible GLn (F p )-module, with a − c < p − 2. Then we may define ( F p − 2 + c, b, a − ( p − 2) if a ≥ p − 2, M = F(d, e, f ) = F 2( p − 2) + c + 1, b + ( p − 1), a + 1 if a < p − 2. Then we say that M is the extra weight associated to F(a, b, c). Applying [10, Proposition 2.11], it is easy to see that if F(d, e, f ) is the extra weight associated to F(a, b, c), there is an exact sequence 0 → F(d, e, f ) → W (d, e, f ) → F(a, b, c) → 0. Now, suppose that ρ is attached to a quasi-cuspidal homology eigenclass in weight F(a, b, c). Examining the long exact homology sequence associated to this short exact sequence, we find that a quasi-cuspidal eigenclass α in H3 (00 (N ), F(a, b, c)()) (in particular, any eigenclass corresponding to an irreducible Galois representation) either is the image of an eigenclass in H3 (00 (N ), W (d, e, f )()) or has nonzero image β in H2 (00 (N ), F(d, e, f )()). In the second case, β is an eigenclass, and using Theorem 3.10 and Lefschetz duality, we find that there is an eigenclass γ in H3 (00 (N ), F(d, e, f )()) which has the same eigenvalues as α. Hence, for each quasi-cuspidal eigenclass in an appropriate weight there are two possibilities: either the eigenclass lifts to the dual Weyl module, or the eigenclass gives rise to another eigenclass with the same eigenvalues in the extra weight. Our experimental evidence supports the hypothesis that in all such cases a quasi-cuspidal eigenclass gives rise to another eigenclass with the same eigenvalues in the extra weight. The second class of additional weights which we have observed consists of certain weights which would be predicted by our conjecture if we eliminated the strict parity condition. These additional weights have been observed only for representations ρ that are either the sum of three characters or the sum of an odd twodimensional representation and a character. These additional weights seem to occur fairly rarely and sporadically and may be related to the occurrence of eigenclasses in H 2 which have ρ attached. A full investigation of them would require new computational techniques, beyond those developed in this paper. The third class of additional weights consists of extra weights associated to weights that would be predicted by our conjecture but for the strict parity condi-

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tion. As in the second case, these additional weights occur only rarely, and only for reducible ρ. Before beginning to present computational evidence for Conjecture 3.1, we begin by proving several facts about the conjecture. THEOREM 3.6 If Conjecture 3.1 is true for a representation ρ, then it is true for the representation ρ ⊗ ωs , where ω is the cyclotomic character modulo p.

Proof First, note that twisting by ωs does not affect the predicted level or nebentype in any way. Denote the level of ρ by N and the nebentype of ρ by . If ρ has niveau 1, then this is just [4, Proposition 2.6]. For higher niveau representations, we note that twisting by ωs changes the value of m coming from a niveau d character by s(1 + p + · · · + p d−1 ); hence, it changes all the values of ai arising from m by s. Following this change through the permutations involved in deriving an n-tuple, we find that twisting a representation ρ by ωs adds s to each element of a derived n-tuple. This change is then reflected in the predicted weight, and we have that the set of predicted weights for ρ ⊗ ωs is precisely the set of twists by dets of the predicted weights of ρ. Finally, if an eigenclass v shows up in weight V and has ρ attached, then we may consider v as lying in cohomology with weight V ⊗ dets , and we see easily (as in [4]) that in this new cohomology group v has ρ ⊗ ωs attached. Hence, if ρ is attached to a cohomology class in each of the weights predicted by Conjecture 3.1, then ρ ⊗ ωs satisfies the conjecture as well. We now note that there is a correspondence between systems of Hecke eigenvalues arising from modular forms and systems of eigenvalues arising from arithmetic cohomology in characteristic p, similar to that given by the Eichler-Shimura isomorphism in characteristic zero. In particular, we note that by [6, Proposition 2.5], for p > 3, any system of Hecke eigenvalues comes from the mod p reduction of an eigenform of level N , nebentype , and weight k = g + 2 if and only if it comes from a Hecke eigenclass in H 1 (00 (N ), Vg (F¯ p )()), where Vg (F¯ p ) is the gth symmetric power of the standard representation of GL2 (F p ). THEOREM 3.7 If p > 3, Serre’s conjecture implies Conjecture 3.1 for n = 2.


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Proof For a complete description of Serre’s conjecture, including Serre’s prediction of the weight, see [22] or [13]. There are two cases: where ρ is niveau 1 and where ρ is niveau 2. In either case we note that the level and nebentype predicted by Serre’s conjecture are identical to those predicted by Conjecture 3.1, so that we need only deal with the weight. Suppose that ρ : G Q → GL2 (F¯ p ) is odd, semisimple, and has niveau 1. If ρ is reducible, Conjecture 3.1 is true (see [4, Proposition 2.7]), so we may assume that ρ is irreducible. If ρ is tamely ramified, we have a ω 1 , ρ| I p ∼ ω a2 with 0 ≤ a1 , a2 ≤ p − 2. Conjecture 3.1 predicts a weight of F(a1 − 1, a2 )0 . If a2 < a1 , then a −a ω1 2 ρ ⊗ ω−a2 | I p ∼ , ω0 and Serre’s conjecture claims that ρ ⊗ ω−a2 corresponds to a modular form of weight 1 + a1 − a2 or (via [6, Proposition 2.5]) that ρ ⊗ ω−a2 corresponds to a cohomology class with coefficients in F(a1 − a2 − 1, 0). Twisting by ωa2 (which corresponds to twisting the weight by deta2 ), we find that ρ corresponds to a cohomology class with coefficients in F(a1 − 1, a2 ), exactly as predicted by Conjecture 3.1. If a2 ≥ a1 , then p−1+a −a 1 2 ω p−1−a2 ρ⊗ω |I p ∼ , ω0 and by Serre’s conjecture (together with [6, Proposition 2.5]), ρ ⊗ ω p−1−a2 corresponds to a cohomology class with coefficients in F( p − 2 + a1 − a2 , 0). Twisting by ωa2 as before, we find that ρ has weight F(a1 − 1 + ( p − 1), a2 ), exactly as predicted. Now if ρ is wildly ramified at p, then β ω ∗ ρ| I p ∼ , ωα with 0 ≤ α ≤ p − 2 and 1 ≤ β ≤ p − 1, and Conjecture 3.1 predicts a weight of F(β − 1, α)0 . Before applying Serre’s conjecture, we twist ρ by ω−α to obtain β−α ω ∗ −α ρ⊗ω ∼ . ω0 Applying Serre’s conjecture to this representation, we find that it has weight

536


1 + (β − α) (i.e., F(β − α − 1, 0)) if β > α + 1, 2 (i.e., F(0, 0)) if β = α + 1 and ρ ⊗ ω−α is peu ramifiée, p + 1 (i.e., F( p − 1, 0)) if β = α + 1 and ρ ⊗ ω−α is très ramifiée, 1 + (β − α) + ( p − 1) (i.e., F(β − α + ( p − 1) − 1, 0) if β ≤ α. Twisting each of these weights by detα , we find that ρ corresponds to a cohomology class in weight F(β − 1, α)0 in every case. (Note that when β − 1 ≤ α we may add p − 1 to β − 1 to obtain a p-restricted pair.) This proves the theorem in the case when ρ has niveau 1. Suppose that m ψ ρ|I p ∼ , ψ 0m (1) (2) (3) (4)

where m = a + bp, and 0 < a − b ≤ p − 1. (Note that if a = b, we are really in niveau 1.) For simplicity we use the fact that ψ and ψ 0 have order p 2 − 1 to reduce to the case where 0 < m < p 2 − 1, so that b < p − 1. The weight predicted by Conjecture 3.1 is then F(a − 1, b)0 . Now, ! ψ a−b −b ρ ⊗ ω ∼L , ψ 0 a−b so that by Serre’s conjecture ρ ⊗ ω−b corresponds to a cohomology class with coefficients in F(a − b − 1, 0). Twisting by ωb , we see that ρ then corresponds to a cohomology class with coefficients in F(a − 1, b), exactly as predicted. Hence, Serre’s conjecture implies Conjecture 3.1 for n = 2. We now prove a partial converse to Theorem 3.7, which shows that in certain cases Conjecture 3.1 is actually equivalent to Serre’s conjecture. THEOREM 3.8 Assume Conjecture 3.1. Let p > 3, and let ρ : G Q → GL2 (F¯ p ) be a semisimple continuous odd Galois representation. If each weight predicted by Conjecture 3.1 is defined unambiguously, then Serre’s conjecture is true for ρ.

Proof We may clearly assume that ρ : G Q → GL2 (F¯ p ) is irreducible since Serre’s conjecture says nothing about reducible representations. First, note that ρ cannot be attached to any class in H 0 since, according to [2, Theorem 4.1.4], any class in H 0 is a twist of a punctual class, and a punctual class corresponds to a reducible representation by [2, Lemma 4.1.2]. Conjecture 3.1 implies that ρ is attached to an eigenclass in H 1 (00 (N ), V ()), where N , V , and are as predicted in the conjecture. We note that the level and


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nebentype predicted by Conjecture 3.1 are exactly the same as those predicted by Serre’s conjecture. If ρ is tamely ramified and has niveau 1, then we have a ω ρ| I p ∼ , ωb and we may further conjugate ρ so that 0 ≤ b ≤ a < p − 1. The weights predicted by Conjecture 3.1 are then F(a − 1, b)0 and (permuting the diagonal characters) F(b − 1, a)0 . These are defined unambiguously exactly when a 6= b + 1. For a > b + 1, we have that F(a − 1, b) embeds in Va−1+bp , so Conjecture 3.1 predicts that ρ is attached to a cohomology eigenclass in weight Va−1+bp since any system of eigenvalues occurring in a submodule occurs in the containing module (see [4]). This implies (by [6, Proposition 2.5]) that ρ is attached to an eigenform of weight 1+a+bp, which is exactly the weight predicted by Serre’s conjecture. For a = b = 0, the predicted weights for Conjecture 3.1 and Serre’s conjecture are both F( p − 2, 0). For a = b 6= 0, Conjecture 3.1 predicts a weight of F(a − 1 + p − 1, b), while Serre’s conjecture predicts a weight of Vb−1+ pa . Using [10, Table 1] (specifically, the last line, as b 6 = 0), we see that F(a − 1 + p − 1, b) is a subquotient of Vb−1+ pa . Hence, we are finished if we can show that the system of eigenvalues corresponding to ρ in weight F(a − 1 + p − 1, b) also shows up in weight Vb−1+ pa . Lemma 3.9 shows that for GL2 , systems of eigenvalues of eigenclasses that are not twists of punctual classes are inherited from subquotients, so that we are finished. For a tamely ramified niveau 2 representation, the proof is essentially identical— one of the weights predicted in Conjecture 3.1 embeds in the module corresponding to the weight predicted by Serre’s conjecture. If ρ is wildly ramified, then we have α ω ∗ ρ= . ωβ Conjecture 3.1 then predicts a weight of F(α−1, β)0 , which is unambiguously defined as long as α 6 ≡ β + 1 (mod p − 1). In order to apply Serre’s conjecture, we normalize so that 1 ≤ α ≤ p − 1 and 0 ≤ β ≤ p − 2. If α > β and α 6 = β + 1, then Serre’s conjecture predicts a weight of Vα−1+βp , which contains F(α − 1, β) as a submodule; hence we are finished, as before. If α ≤ β, then Serre’s conjecture predicts a weight of 1 + β + pα, and we have F(α − 1, β)0 = F(α − 1 + p − 1, β). Using [10, Table 1] as before, we find that F(α−1+ p−1, β) is a subquotient of Vβ−1+ pα , which is the module corresponding to the weight predicted by Serre’s conjecture. Hence, by Lemma 3.9, we are finished.

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LEMMA 3.9 If α is an eigenclass in H 1 (00 (N ), A), where A is a subquotient of a GL2 (F)-module B, α is not a twist of a punctual eigenclass, and p > 3, then there is an eigenclass in H 1 (00 (N ), B) with the same eigenvalues as α.

Proof Let T ⊂ S ⊂ B, with S/T ∼ = A, and examine the long exact cohomology sequence arising from the short exact sequence 0 → T → S → A → 0. Note that since p > 3, H 2 (00 (N ), T ) = 0, so that the eigenclass α must come from a class σ in H 1 (00 (N ), S). By [5], we may replace σ by an eigenclass having the same eigenvalues as α (calling the new class σ again). The long exact cohomology sequence arising from the short exact sequence 0 → S → B → B/S → 0 then shows that σ goes to a nonzero class β in H 1 (00 (N ), B) since it cannot come from H 0 (00 (N ), B/S) (as it is not a twist of a punctual class). Clearly, β has the same eigenvalues as σ . THEOREM 3.10 Assume that ρ : G Q → GLn (F¯ p ) is attached to an eigenclass α in H i (00 (N ), V ()), where N , , and V are the level, the nebentype, and a weight predicted for ρ. Then ρ ∨ = t ρ −1 is attached to a cohomology class β in H i (00 (N ), W ( −1 )), where W = V ∗ ⊗ det−(n−1) is a twist of the contragredient V ∗ of V . Further, the level, the nebentype, and a weight predicted for ρ ∨ are N , −1 , and W .

Proof The proof that there is a β in the indicated cohomology group with ρ ∨ attached is exactly the same as the proof of [4, Proposition 2.8]. For ρ of niveau 1, Ash and Sinnott also prove that the invariants of ρ ∨ are as above. The level and nebentype computations remain the same regardless of the niveau of the representation, so we need only show that W is a predicted weight for ρ ∨ . We show that if (b1 , . . . , bn ) is a derived n-tuple for ρ, then (−bn , . . . , −b1 ) is a derived n-tuple for ρ ∨ . Then, since (F(α1 , . . . , αn )0 )∗ = F(−αn , . . . , −α1 )0 , it follows that if V is a predicted weight for ρ, then W is a predicted weight for ρ ∨ . It is an easy exercise to reduce the question to simple representations of G p . Suppose that ρ is a simple representation of G p , with the n-tuple (b1 , . . . , bn ) derived


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from it. Then there must be some exponent m such that   ϕ1   .. ρ| I p =  , . ϕn

m , . . . , ψ m . Then −m is an exponent where (ϕ1 , . . . , ϕn ) is some permutation of ψn,1 n,n ∨ associated to ρ in the same way, as is any multiple of −m by a power of p. Now m = a1 + a2 p + · · · + an p n−1 , where the ai are some permutation of the decreasing n-tuple (b1 , . . . , bn ), with 0 ≤ ai − an ≤ p − 1. Let ak be the largest of the ai , which is equal to b1 . Then − p n−1−k m is congruent (modulo ( p n − 1)) to

−ak+1 − · · · − an p n−2−k − a1 p n−1−k − · · · − ak p n−1 , with 0 ≤ ai − ak ≤ p − 1, so that (−bn , . . . , −b1 ) is easily seen to be an n-tuple associated with ρ ∨ . 3.1. Heuristic for the niveau n case For the most part, we have derived our conjecture using Serre’s conjecture as a model. We can provide a suggestive heuristic for one feature of our conjecture: the weight of a niveau n representation into GLn (F¯ p ). Let ρ : G Q → GLn (F¯ p ) be given such that   ρ| I p ∼ 

ϕ1

 ..

. ϕn

 ,

where the ϕi are powers of a fundamental character of niveau n and are conjugate to each other. Let us suppose that ρ lifts to a p-adic representation 2 unramified at almost all primes. Further, suppose that 2 comes from a motive M with good reduction at p, which would conjecturally be the case were 2 attached to an automorphic representation π of cohomological type of level N prime to p (cf. [9]). Then 2 is crystalline. So by analogy it is reasonable to assume that ρ is “crystalline” in the sense of [14], that is, that it corresponds to a filtered Frobenius module for F p . n−1 Now write ϕ1 = ψ a1 +a2 p+···+an p , with 0 ≤ ai ≤ p − 1. By [14, Theorem 0.8], there is indeed a unique filtered Frobenius module 8 over F p which corresponds to a representation of G Q p into GLn (F p ) whose restriction to I p is equivalent to ρ| I p . This is our motivation for choosing the ai in the given range. Assuming again that 2 and M exist, the Hodge numbers of M would be the same as the Hodge-Tate numbers of 2|G Q p , and these in turn would be the same as

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the jumps in the filtration of the filtered Frobenius module associated to 2|G Q p . If we take the latter to be the same as the jumps in 8, they are a1 , . . . , an . Now, suppose we are in the generic case; that is, suppose that |ai − a j | > 1 for i 6 = j. Let {b1 , . . . , bn } = {a1 , . . . , an }, with b1 > b2 > · · · > bn . Assuming the general picture of L. Clozel (following R. Langlands) of the relationship between automorphic representations and motives, as found in [9], especially Chapters 3 and 4, the motive M predicts the existence of an automorphic representation π attached to M such that π∞ ⊗ W has (g, K )-cohomology, where W is the irreducible representation of GLn (C) with highest weight (b1 − (n − 1), b2 − (n − 2), . . . , bn ). By analogy, we conjecture that ρ will be attached to a cohomology class with weight V = F(b1 − (n − 1), b2 − (n − 2), . . . , bn ). After all, ρ is the reduction of 2 modulo p, and W mod p (or, more precisely, the reduction modulo p of a model for W over Z p ) has V as a composition factor. If we now require our conjecture to be closed under twisting by powers of ω, a simple exercise yields the weights predicted by Conjecture 3.1 for niveau n, dimension n, in the generic case. By “continuity” we extend the heuristic to the nongeneric case. 4. Symmetric squares Using work of Ash and P. Tiep [7], who proved that certain Galois representations are in fact attached to cohomology eigenclasses, we are able to verify certain special cases of Conjecture 3.1. THEOREM 4.1 Let σ : G Q → GL2 (F p ) be a continuous irreducible odd Galois representation ramified only at p. Assume that Serre’s conjecture is true for σ , and let k be the weight predicted by Serre’s conjecture. Then if 2 < k < ( p+3)/2 and Sym2 σ is irreducible, Sym2 σ is attached to a cohomology eigenclass in weight F(2(k − 2), k − 2, 0), and this weight is predicted by Conjecture 3.1.

Proof By [6, Proposition 2.5], we see that σ is attached to a cohomology eigenclass in H 1 (SL2 (Z), Uh (F¯ p )), where h = k − 2 and Uh (F¯ p ) = Symh (F¯ 2p ), with the standard action of GL2 (F¯ p ) on F¯ 2p . Then, by [7, Corollary 5.3], Sym2 σ is attached to a cohomology eigenclass in H 3 (SL3 (Z), F(2h, h, 0)). Hence, we need only show that the weight F(2h, h, 0) is predicted by Conjecture 3.1. If σ has niveau 1, this is trivial since we must have k−1 ω ∗ σ |I p ∼ , 1


so that Sym2 σ | I p

541

 2(k−1) ω  ∼

 ∗ ∗ . 1

∗ ωk−1

If σ has niveau 2, then we must have σ |I p ∼

!

ψ k−1

with 1 ≤ k − 1 ≤ ( p − 1)/2, so that  ψ 2(k−1)  2 Sym σ | I p ∼ 

ψ 0 k−1

,

 ωk−1 ψ 0 2(k−1)

 ,

with 2 ≤ 2(k − 1) ≤ p − 1. Clearly, a predicted weight for this representation is F(2(k − 2), k − 2, 0). Example 4.2 Let K be a totally complex S4 -extension of Q, such that the quartic subfield of K has discriminant p 3 , where p is a prime congruent to 5 mod 8 (for examples of such fields, see [11]). The unique three-dimensional irreducible unimodular mod p representation of S4 gives rise to an irreducible unimodular representation ρ : G Q → GL3 (F p ) which is ramified only at p. This representation is (up to a twist by a power of the cyclotomic character) the symmetric square of a two-dimensional irreducible representation σ : G Q → GL2 (F¯ p ) with projective image isomorphic to S4 and image of order 96 (see [20]). Serre’s conjecture is true for σ since σ has a lift to a twodimensional irreducible complex Galois representation with solvable image to which we apply the theorem of Langlands and J. Tunnell [24]. Hence, σ is modular and so, by the -conjecture, Serre’s conjecture holds for σ (see [11] for more details). One easily checks that σ has niveau 1 and that the weight predicted by Serre’s conjecture for σ is ( p + 3)/4, so that Theorem 4.1 applies. Hence, at least one of the weights predicted for ρ by Conjecture 3.1 yields an eigenclass with ρ attached. In fact, this weight is F(( p − 5)/2, ( p − 5)/4, 0) ⊗ det3( p−1)/4 . Example 4.3 Let K be a totally complex S4 -extension of Q such that the quartic subfield of K has discriminant − p, where p is a prime congruent to 3 mod 8. Let ρ be the unimodular irreducible three-dimensional Galois representation associated to K as above. Again, there is a two-dimensional irreducible representation σ : G Q → GL2 (F p ), with projective image isomorphic to S4 , such that σ is ramified only at p (see [20]; this time

542


the image of σ has order 48) and (again up to a twist by a power of the cyclotomic character) ρ is the symmetric square of σ . (Note that up to twisting, the symmetric square depends only on the projectivization of a representation.) One checks easily that Serre’s conjecture predicts a weight of ( p + 1)/2 for σ (again σ has niveau 1) and that (just as above) Serre’s conjecture is true for σ . Hence, one of the weights predicted by Conjecture 3.1 does in fact contain an eigenclass with ρ attached. In this case, the weight is F( p − 3, ( p − 3)/2, 0) ⊗ det( p−1)/2 . Example 4.4 Let K be a complex S4 -extension of Q with K ramified at only one prime p, with p congruent to 3 modulo 8, and with ramification index at p equal to 4 (for examples of such extensions, see [11]). Let ρ be the unique unimodular irreducible three-dimensional mod p Galois representation with image isomorphic to S4 and such that the fixed field of the kernel of ρ is K . Then, up to twisting, ρ is the symmetric square of a representation σ : G Q → GL2 (F p ) with image isomorphic to S˜4 (i.e., isomorphic to GL2 (F3 )). In this case, σ has niveau 2, Serre’s conjecture is true for σ and its twists, and a twist of σ has weight ( p + 5)/4 (see [11]), so that Theorem 4.1 applies. Hence, one of the weights predicted for ρ gives a cohomology group that contains an eigenclass predicted for ρ. In this case, the weight that works is F(( p − 3)/2, ( p − 3)/4, 0) ⊗ det(3 p−5)/4 .

5. Niveau 1 representations 5.1. Reducible representations in level 1 In [4] Ash and Sinnott dealt extensively with reducible representations ramified at only one prime. Each of their examples was a direct sum of an even two-dimensional representation with a one-dimensional representation, and they included cases where the two-dimensional representation had image isomorphic to a dihedral group or projective image isomorphic to A4 . They did not give examples in which the projective image was isomorphic to S4 or A5 . We recall their construction from [4]. Let σ : G Q → GL2 (F¯ p ) be an irreducible representation with the following properties: (1) σ is unramified outside p; (2) the image of σ has order relatively prime to p; (3) σ (Frob∞ ) is central, where Frob∞ is a complex conjugation in G Q ; (4) σ (G p,0 ) has order dividing p − 1.


543

Then, choosing integers j and k appropriately, we find that the representation ρ = (σ ⊗ ω j ) ⊕ ωk is three-dimensional and odd, and by adjusting j and k, we may adjust the predicted weight of ρ to some extent. In particular, we need to choose j and k to have opposite parity if σ (Frob∞ ) = 1 and the same parity if σ (Frob∞ ) = −1. In addition, we choose j and k to give the simplest possible weight. The reducibility of these representations makes it possible to reduce the weight to calculable levels; however, in the examples that we consider here the weight is still quite high. Hence, rather than being able to calculate many Hecke eigenvalues, we found it impractical to calculate more eigenvalues than those at 2 and 3, due to time constraints. We begin by specifying the fixed field of the kernel of the projective image of σ , which is a totally real number field. 5.1.1. Representations of type A4 In [4] Ash and Sinnott presented several examples of reducible Galois representations that are sums of one-dimensional characters with even two-dimensional representations having projective image isomorphic to A4 . Using the same computational techniques as in [1], we have been able to find other A4 -extensions for which we can compute the predicted quasi-cuspidal homology classes. These examples are given in Table 1. We begin with a quartic polynomial f that has four real roots and whose splitting field K is an A4 -extension of Q ramified only at one prime p. We know (by [4, Lemma 4.1]) that K sits inside an Aˆ 4 -extension Kˆ of Q, with Kˆ /Q ramified only at p. In fact, there are two possibilities for Kˆ ; following [4], we take Kˆ to be the one that has ramification index 3 at p. Let K 4 be the quartic extension of Q defined by f . We note that K 4 must be contained in an octic subextension K 8 of Kˆ , with K 8 /K 4 unramified at all finite primes. Since K 8 has Kˆ as its Galois closure, we may determine whether Kˆ is totally real or totally complex by comparing the two-ranks of the class group and the narrow class group of K 4 . For instance, when p = 1009, the class number of K 4 is two, and the narrow class group is cyclic of order four. Thus, the two class groups have the same two-rank, so Kˆ must be real (since K 8 and all its conjugates must be real). If Kˆ is totally real, we write its sign as 1; otherwise its sign is −1. Now Aˆ 4 has a unique two-dimensional irreducible unimodular mod p representation σ : G Q → GL2 (F p ). We see easily that σ | I p = ωd ⊕ ω−d with d = ( p − 1)/3. We now take ρ = (σ ⊗ ω j ) ⊕ ωk with j = 2d and k = 1 if the sign of Kˆ is 1, and k = 2 otherwise. We note that ρ satisfies the conditions of the construction of [4] and has a predicted weight of F(( p − 1)/3 − 2, 0, 0) if k = 1 and F(( p − 1)/3 − 2, 1, 0) if k = 2. For each of the examples in Table 1, we have calculated the interior homology of

544


Table 1. Reducible representations of type A4

Polynomial x 4 − 2x 3 − 13x 2 − 9x + 4 x 4 − x 3 − 16x 2 + 3x + 1 x 4 − x 3 − 10x 2 + 3x + 20 x 4 − x 3 − 13x 2 + 12x + 16 x 4 − 2x 3 − 19x 2 + 29x + 1 x 4 − 2x 3 − 31x 2 − 51x − 4 x 4 − 2x 3 − 39x 2 + x + 125 x 4 − 2x 3 − 51x 2 + 100x + 83 x 4 − 2x 3 − 51x 2 + 32x + 192 x 4 − 2x 3 − 37x 2 + 10x + 29 x 4 − 2x 3 − 43x 2 + 127x − 55

Sign −1 1 −1 −1 −1 1 1 1 1 1 1

p 163 277 349 397 547 607 1009 1399 1699 1777 1951

k 2 1 2 2 2 1 1 1 1 1 1

Weight F(52, 1, 0) F(90, 0, 0) F(114, 1, 0) F(130, 1, 0) F(180, 1, 0) F(200, 0, 0) F(334, 0, 0) F(464, 0, 0) F(564, 0, 0) F(590, 0, 0) F(648, 0, 0)

SL3 (Z) in the given weight using the techniques described in [1] and found it to be one-dimensional. We have also calculated the Hecke eigenvalues at 2 and 3 and found that they exactly match the values predicted from the characteristic polynomial of the image of Frobenius under ρ by Conjecture 3.1. 5.1.2. Representations of type S4 Totally real S4 -extensions ramified at only one prime can have two types of ramification; either the ramification index is 2, or the ramification index is 4. For our purposes, the extensions with ramification index 4 are better (since they yield lower weights), although they are more difficult to find. They can, however, be found by application of explicit class field theory, and many such examples are known. Only the two below yield predicted weights that are feasible for computation. Example 5.1 Let K be the splitting field of the polynomial x 4 −x 3 −1017x 2 +9665x +60608. Then K is a totally real S4 -extension of Q, ramified only at p = 2713, with ramification index e = 4. Let S˜4 be the central extension of S4 by Z/2Z which is isomorphic to GL2 (F3 ). Then K embeds in an S˜4 -extension K˜ of Q (by [4, Lemma 4.1]), and K˜ /K must further ramify at p (as described in [11]), so that in K˜ , p = 2713 has e = 8. We need to determine whether K˜ is totally real or totally complex. To do this, we note that S˜4 has three conjugacy classes of subgroups of order 6 and that each subgroup of order 12 contains exactly one subgroup of order 6 from each conjugacy class. In terms of field extensions then, each subfield of K˜ of degree 4 has exactly three


545

quadratic extensions lying in K˜ . Hence, if K˜ is totally real, the degree 4 subfield K 4 of K must have a Klein four extension contained inside K˜ , hence ramified only at p (in particular, not ramified at infinity). Such an extension would lie inside the ray class field of K 4 modulo pm (where p is the unique prime of K 4 lying over p). However, the two-part of the ray class group of K 4 modulo pm is cyclic for every m (see [17]). Hence, K˜ must be totally complex. Now, we let σ be the two-dimensional representation σ : G Q → GL2 (F¯ p ), with image isomorphic to S˜4 and kernel equal to G K˜ , chosen such that σ |I p =

3( p−1)/8 ω

ω( p−1)/8

=

3(339) ω

ω339

.

Taking j = −339, k = 1 (with the same parity since σ (Frob∞ ) = −1), we see that ρ = (σ ⊗ ω j ) ⊕ ωk has  678  ω , ρ| I p ∼ L  ω1 0 ω where we take   ∗ ∗ L =  ∗ . ∗ ∗ Then the weight predicted by Conjecture 3.1 is F(678 − 2, 1 − 1, 0)0 = F(676, 0, 0), the level is 1, and the nebentype is trivial. Computations using the techniques of [1] show that the interior cohomology is in fact one-dimensional. The Hecke eigenvalues at 2 and 3 correspond exactly to σ , as predicted by Conjecture 3.1. A similar construction can be performed with the splitting field K of the polynomial x 4 − 6668x 3 + 16598046x 2 − 18278822428x + 7514424150025, which is a totally real S4 -extension of Q ramified only at p = 3137. In this case, K˜ is totally real, and the predicted weight is F(782, 0, 0). Again, the homology is one-dimensional and the eigenvalues at 2 match ρ. The image of the Frobenius at 3 is of order 8, however, and presents some difficulty. We have determined σ (and hence also ρ) by a local condition at p, namely, its restriction to inertia at I p . Determining the Frobenius at 3 is a local condition at 3, and combining these two determinations in order to determine Tr(ρ(Frob3 )) is a global problem that involves calculations in a large number field. We thus have not distinguished between two possibilities for eigenvalues at 3 which would correspond to ρ. One of these possibilities does in fact occur in the predicted cohomology.

546


5.1.3. Representations of type A5 Ash and Sinnott’s construction works best with A5 -extensions if the ramification index of the unique ramified prime is as large as possible. However, totally real A5 extensions of Q ramified at only one prime with ramification index 5 are quite difficult to find. D. Doud thanks S. Harding for showing him the second example below with p = 3821. Example 5.2 Let K be the splitting field of the polynomial f = x 5 − 7402x 3 − 3701x 2 + 14804x + 11103. Then K is a totally real Galois extension of Q, with Galois group A5 , ramified only at p = 3701. K must lie inside an extension Kˆ of Q with Galois group Aˆ 5 (the unique nonsplit central extension of A5 by Z/2Z). In fact, K lies inside two such extensions, one in which primes above p ramify further, and one in which primes above p do not ramify further. Let Kˆ be an Aˆ 5 -extension of Q containing K , in which p has ramification index 5. Let H be a subgroup of Aˆ 5 of order 20. Using the computer algebra system Magma, one can see that H has a quotient group that is cyclic of order 4. Hence, the degree 6 subextension of K must have a cyclic quartic extension contained in Kˆ which is unramified at all finite primes. A defining polynomial for the degree 6 subextension of K may be found as the minimal polynomial of the element α1 α2 + α2 α3 + α3 α4 + α4 α5 + α5 α1 , where αi , 1 ≤ i ≤ 5, are the roots of f . Using PARI/GP (see [18]) to compute the ideal class group and the narrow class group, we find that both are cyclic of order 4. Hence, the only possible cyclic quartic extension of the degree 6 subfield of K which is unramified at all finite primes is also unramified at infinity, so that Kˆ is totally real. Now Aˆ 5 has two two-dimensional mod p representations. Call them σ and σ 0 . On inertia at p = 3701, we may choose σ and σ 0 such that 3( p−1)/5 ( p−1)/5 ω ω 0 σ |I p ∼ and σ |I p ∼ . ω2( p−1)/5 ω−( p−1)/5 If we let ρ = (σ ⊗ω−2( p−1)/5 )⊕ω, then ρ is an odd three-dimensional representation, and if it is conjugated to land inside   ∗ ∗ L =  ∗ , ∗ ∗


then it satisfies strict parity. We then have  ( p−1)/5 ω ρ| I p ∼ L 

547

 ,

ω 1

giving a predicted weight of F(( p − 1)/5 − 2, 0, 0) = F(738, 0, 0). We may calculate the quasi-cuspidal homology in this weight and find that it is one-dimensional and has the appropriate eigenvalues at 2 and 3 to correspond to ρ. In this case, there is an ambiguity similar to that in the preceding example, in that we have not determined which of the two conjugacy classes of order 5 contains the Frobenius at 2. The computed eigenvalues at 2 are in fact one of the two possible pairs of values. A similar calculation may be carried out for the A5 -extension defined by the polynomial x 5 − 3821x 3 − 3821x 2 + 3821x + 3821 and ramified only at p = 3821. In this case Kˆ is again totally real. Hence, as above, we get a predicted weight for ρ of F(( p − 1)/5 − 2, 0, 0) = F(762, 0, 0). Calculating the quasi-cuspidal homology in this weight yields a one-dimensional space, which has appropriate eigenvalues at 2 and 3 to correspond to ρ, with the ambiguity that we have not determined the conjugacy class of elements of order 10 (resp., 5) containing the Frobenius at 2 (resp., 3), just as in the previous examples. 5.2. Reducible representations in higher level With the introduction of levels higher than one, we gain immensely in reducing the weight of the representations that we can find. In particular, we find that we can actually compute “companion forms,” or classes with different weights, attached to the same representation. These offer important examples of Conjecture 3.1. We work out one interesting example in full detail and describe others in a table format. Example 5.3 Let K be the S3 -extension of Q given as the splitting field of the polynomial x 3 − x 2 − 3x + 1. Then K is ramified only at p = 37 (with ramification index 2) and at q = 2 (with ramification index 3). Since S3 has a two-dimensional mod 37 representation, we obtain a two-dimensional Galois representation σ : G Q → GL2 (F37 ), with the fixed field of the kernel of σ equal to K . Let ω be the cyclotomic character modulo 37, and let ρ = σ ⊕ ω. We note that σ is an even representation since K is totally real, so we want to conjugate ρ to land inside the Levi subgroup   ∗ ∗ L =  ∗ . ∗ ∗

548


Table 2 Eigenvalues a(`, 1) a(`, 2)

2 * *

3 2 24

5 5 22

7 6 15

11 10 26

13 13 17

17 17 13

19 19 35

23 23 8

29 29 29

31 31 31

37 1 1

41 3 27

43 6 6

47 9 25

Now inside L, the image of complex conjugation is conjugate to the matrix  1  −1

 , 1

so that ρ satisfies strict parity. One sees easily that the level of ρ is equal to the level of σ , which is 22 (since the ramification at 2 is tame and the image of inertia at 2 under σ does not fix a subspace). The nebentype of ρ is trivial since the determinant is just ω19 . Finally, if we examine the restriction of ρ to inertia at 37, we find that  18  ω . ρ| I37 ∼ L  ω1 ω0 Thus, the weight predicted by Conjecture 3.1 is F(18 − 2, 1 − 1, 0)0 = F(16, 0, 0). When we compute the cohomology in this weight, in level 4 with trivial nebentype, we obtain a fifteen-dimensional space containing a one-dimensional eigenspace with eigenvalues given by Table 2. We now compute the trace Tr(ρ(Frob` )) and T2 (ρ(Frob` )) (the sum of products of pairs of eigenvalues) for ` between 2 and 47. To do this, we note that the characteristic polynomial of ρ(Frob` ) is det I − xρ(Frob` ) = det I − xσ (Frob` ) 1 − xω(Frob` ) = 1 − Tr(σ (Frob` ))x + det(σ (Frob` ))x 2 (1 − `x) = 1 − Tr(σ (Frob` )) + ` x + det(σ (Frob` )) + Tr(σ (Frob` ))` x 2 − det σ (Frob` ) `x 3 ,

so that the trace of ρ(Frob` ) is Tr(σ (Frob` ))+` and T2 (ρ(Frob` )) = det(σ (Frob` ))+ Tr(σ (Frob` ))`. Using PARI/GP, we may calculate these two values for ` from 2 to 47 (excluding the ramified primes 2 and 37), and we find that they exactly match the values of a(`, 1) and à(`, 2) calculated above.


549

Other reducible examples are easily computed just as above. In each row of Table 3, we give a polynomial whose splitting field K is a totally real Galois extension of Q with Galois group G, such that G has a unique two-dimensional representation σ modulo p. We also give the predicted weight(s), level, and nebentype of the cohomology classes corresponding to ρ = σ ⊕ ω. Several examples have more than one predicted weight, coming from multiple orderings of the diagonal characters. Such predictions actually occur in all of these examples, but most are too large for us to calculate and hence do not appear in this table. For all of the examples in this table, all Hecke eigenvalues up to ` = 47 coincide exactly with the coefficients of the characteristic polynomial of the image of Frobenius, as predicted by Conjecture 3.1. We may also apply Conjecture 3.1 to reducible representations that are the sum of an odd two-dimensional representation and a character. In order to satisfy strict parity, such a representation must land inside a Levi subgroup of the form     ∗ ∗ ∗ L =  ∗ ∗ or L = ∗ ∗  . ∗ ∗ ∗ For each such three-dimensional ρ, we thus have four predicted weights, two from each choice of Levi subgroup. In Table 4, for each example we give a polynomial f that has Galois group G = S3 or D4 , together with a prime p and the ramification index of p in the splitting field K of f . If we let σ be the unique two-dimensional mod p Galois representation arising from K , and ρ = σ ⊕ ω0 , we also give the level N and nebentype associated to ρ, and the set of predicted weights arising from Conjecture 3.1. In this case we are able to compute with all the predicted weights, and we find that in every case an eigenclass with the correct eigenvalues (up to ` = 47) appears in every predicted weight. The last examples in the table, in which σ has image isomorphic to D4 (the dihedral group with 8 elements), are interesting in that fewer than four weights are predicted. In these cases the four predicted weights are not distinct, so that the total number of weights in which we expect to find eigenvalues with ρ attached is less than four. For instance, in the last example in Table 4, in which p = 5, the image of inertia at 5 is contained in the center of D4 , so that the restriction of σ to inertia at 5 has diagonal characters ω2 and ω2 . The coincidence of these diagonal characters results in the fact that only two distinct weights are predicted. 5.3. Irreducible representations in higher level In order to find irreducible three-dimensional Galois representations, it is necessary to find Galois groups that have irreducible three-dimensional mod p representations. For p larger than 3 this is easily done: the groups A4 , S4 , and A5 all have threedimensional irreducible mod p representations. We thus concentrate primarily on rep-

550


Table 3. Reducible higher-level niveau 1 examples (even two-dimensional plus ω1 )

Polynomial x 3 − x 2 − 3x + 1 x 3 − x 2 − 4x + 2 x 3 − x 2 − 5x − 1 x 3 − x 2 − 4x + 1 x 3 − x 2 − 5x + 4 x 3 − 5x − 1 x 3 − 7x − 5

G S3 S3 S3 S3 S3 S3 S3 S3

x 3 − x 2 − 6x + 5 x 3 − 7x − 1 x 3 − x 2 − 9x + 8 x 4 − x 3 − 3x 2 + x + 1

S3 S3 S3 D4

x 4 − x 3 − 5x 2 + 2x + 4

D4

x 4 − 2x 3 − 4x 2 + 5x + 5

D4

x 4 − x 3 − 7x 2 + 3x + 9

D4

x 4 − 2x 3 − 4x 2 + 5x + 2

D4

x 4 − x 3 − 6x 2 + 8x − 1

D4

x 4 − x 3 − 5x 2 + x + 1

D4

p 37 79 101 107 67 43 11 41 17 5 5 7 5 29 5 89 5 101 5 181 17 53 13 61 13 53

Weight(s) F(16, 0, 0) F(37, 0, 0) F(48, 0, 0) F(51, 0, 0) F(31, 0, 0) F(19, 0, 0) F(3, 0, 0) F(18, 0, 0) F(6, 0, 0) F(0, 0, 0) F(0, 0, 0) F(8, 6, 2), F(6, 6, 4) F(0, 0, 0), F(6, 4, 2) F(12, 0, 0) F(0, 0, 0), F(6, 4, 2) F(42, 0, 0) F(0, 0, 0), F(6, 4, 2) F(48, 0, 0) F(0, 0, 0) F(88, 0, 0) F(6, 0, 0) F(24, 0, 0) F(4, 0, 0) F(28, 0, 0) F(4, 0, 0) F(24, 0, 0)

Level 4 4 4 3 7 11 43 17 41 157 269 53 29 5 89 5 101 5 181 5 53 17 61 13 53 13

1 4 1 3 7 11 43 17 41 157 269 53 29 5 89 5 101 5 181 5 53 17 61 13 53 13


551

Table 4. Reducible higher-level niveau 1 examples (odd two-dimensional plus ω0 )

Galois representation p = 7, e = 3 N = 19, = 19 G = S3 x 3 − x 2 + 5x − 6 p = 7, e = 3 N = 47, = 47 G = S3 x 3 − x 2 − 2x − 27 p = 7, e = 3 N = 59, = 59 G = S3 x 3 − x 2 + 5x + 8 p = 7, e = 3 N = 59, = 59 G = S3 x 3 − x 2 − 9x + 36 p = 7, e = 3 N = 59, = 59 G = S3 x 3 − x 2 − 2x − 20 p = 19, e = 3 N = 3, = 3 G = S3 x 3 − x 2 − 6x − 12 p = 13, e = 3 N = 43, = 43 G = S3 x 3 − x 2 − 17x + 38 p = 3, e = 2 N = 13, = 13 G = D4 x4 + x2 − 3 p = 3, e = 2 N = 37, = 37 G = D4 x 4 + 5x 2 − 3 p = 3, e = 2 N = 61, = 61 G = D4 x 4 − 7x 2 − 3 p = 3, e = 2 N = 73, = 73 G = D4 x 4 + 34x 2 − 3 p = 5, e = 2 N = 39, = 3 13 G = D4 x 4 − x 3 − 8x − 1

Weights F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(10, 5, 0), F(16, 11, 6) F(22, 11, 0), F(34, 23, 12) F(6, 3, 0), F(10, 7, 4) F(14, 7, 0), F(22, 15, 8) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(6, 5, 2) F(4, 1, 0)

552


Table 5

Triple (2( p − 1)/3, ( p − 1)/3, 0) (( p − 1)/3, 0, 2( p − 1)/3) (0, 2( p − 1)/3, ( p − 1)/3) (( p − 1)/3, 2( p − 1)/3, 0) (2( p − 1)/3, 0, ( p − 1)/3) (0, ( p − 1)/3, 2( p − 1)/3)

Weight F(2( p − 4)/3, ( p − 4)/3, 0) F(2( p − 4)/3, ( p − 4)/3, 0) ⊗ det2( p−1)/3 F(2( p − 4)/3, ( p − 4)/3, 0) ⊗ det( p−1)/3 F(2(2 p − 5)/3, (2 p − 5)/3, 0) F(2(2 p − 5)/3, (2 p − 5)/3, 0) ⊗ det( p−1)/3 F(2(2 p − 5)/3, (2 p − 5)/3, 0) ⊗ det2( p−1)/3

resentations (up to a twist) whose images are isomorphic to one of these groups. Of course, we deal only with odd representations. For all the irreducible niveau 1 representations presented in this section, the three-dimensional Galois representation is a symmetric square of an odd two-dimensional representation; hence the correspondences presented here are not native three-dimensional phenomena. 5.3.1. Representations of type A4 Suppose that p is a prime congruent to 1 mod 3 and that K is a totally complex A4 -extension ramified at p, with ramification index 3. There may be other ramified primes, which would then contribute to the level. Since A4 has an irreducible 3-dimensional mod p representation, we obtain an irreducible three-dimensional representation ρ : G Q → GL3 (F p ). We observe that the restriction of ρ to inertia at p is  a  ω , ρ| I p =  ωb c ω where (a, b, c) is some permutation of (2( p − 1)/3, ( p − 1)/3, 0). The six permutations of (2( p − 1)/3, ( p − 1)/3, 0) then give six predicted weights for ρ. The six weights are displayed in Table 5. Hence, we expect to find three cohomology eigenclasses, each with one of ρ, ρ ⊗ ω( p−1)/3 , and ρ ⊗ ω2( p−1)/3 attached, in each of the two weights F(2( p − 4)/3, ( p − 4)/3, 0) and F(2(2 p − 5)/3, (2 p − 5)/3, 0). In fact, however, since ρ ⊗ ω( p−1)/3 ∼ ρ, the three eigenclasses may coincide, and there may actually be only one such eigenclass in each weight. In practice, in order to compute the cohomology associated to a representation as above, we often have to twist by a character that is unramified at p in order to reduce the level. We illustrate with an example. Example 5.4 Let K be the splitting field of the polynomial x 4 −x 3 +5x 2 −4x +3, which is ramified


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Table 6. Irreducible higher level niveau 1 examples Galois representation p = 7, e = 3 N = 13, = 13 G = A4 , χ = 13 x 4 − x 3 + 5x 2 − 4x + 3 p = 7, e = 3 N = 29, = 29 G = A4 , χ = 29 x 4 − x 3 + 5x 2 − 6x + 7 p = 7, e = 3 N = 26 , = 1 G = A4 , χ = 1 x 4 − 2x 3 + 2x 2 + 2 p = 13, e = 3 N = 5, = 5 G = A4 , χ = 5 x 4 − x 3 − 3x + 4 p = 13, e = 3 N = 52 , = 1 G = A4 , χ = 1 x 4 − x 3 − 3x + 4 p = 19, e = 3 N = 7, = 7 G = A4 , χ = 7 x 4 + 3x 2 − 7x + 4 p = 19, e = 3 N = 11, = 11 G = A4 , χ = 11 x 4 + 15x 2 − 11x + 81 p = 7, e = 3 N = 53, = 53 G = S4 , χ = 1 x 4 − x 3 + 4x 2 + 1 p = 13, e = 4 N = 19, = 19 G = S4 , χ = 1 x 4 − x 3 + 2x 2 + 4x − 88 p = 7, e = 3 N = 73, = 73 G = A5 , χ = 73 x 5 − 5x 3 − x 2 + 9x + 7

Predicted weights F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(6, 3, 0), F(10, 7, 4), F(14, 11, 8) F(14, 7, 0), F(18, 11, 4), F(22, 15, 8) F(6, 3, 0), F(10, 7, 4), F(14, 11, 8) F(14, 7, 0), F(18, 11, 4), F(22, 15, 8) F(10, 5, 0), F(16, 11, 6), F(22, 17, 12) F(22, 11, 0), F(28, 17, 6), F(34, 23, 12) F(10, 5, 0), F(16, 11, 6), F(22, 17, 12) F(22, 11, 0), F(28, 17, 6), F(34, 23, 12) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(7, 5, 3), F(13, 8, 6), F(16, 14, 9) F(16, 8, 3), F(19, 14, 6), F(25, 17, 9) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4)

at p = 7 (with e = 3) and at 13 (with e = 2). The predicted weights are F(2, 1, 0) and F(6, 3, 0). The level of ρ is 132 , and the nebentype is trivial. Unfortunately, this level is too large for us to use in computations. However, ρ ⊗13 is easily seen to have level 13 and nebentype 13 . Thus, we predict the existence of cohomology eigenclasses in weights F(2, 1, 0) and F(6, 3, 0), level 13, and nebentype 13 , which are attached to ρ ⊗ . Direct computation shows that these eigenclasses do in fact exist and that the eigenvalues match, at least up to ` = 47. Other A4 -extensions that give rise to computable cohomology classes are shown in Table 6. Each example in this table gives a polynomial f with Galois group G. The prime p is given, together with its ramification index e in the splitting field of f . When G equals A4 , ρ is the twist by the character χ of the unique irreducible threedimensional mod p representation of G Q cutting out the splitting field of f . The level N , nebentype , and predicted weights for ρ are indicated in the table. In all cases, we have computationally verified the existence of an eigenclass in the predicted weight, level, and character, with the correct eigenvalues (up to ` = 47) to have ρ attached.

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5.3.2. Representations of type S4 For p > 3, S4 has two absolutely irreducible three-dimensional representations defined over F p . Hence, by finding extensions K /Q with Galois group S4 , we may easily construct irreducible three-dimensional Galois representations that have image isomorphic to S4 . Two such examples are given in Table 6. Here the format is as in the A4 case, except that we take ρ to be the unique irreducible three-dimensional representation of G Q cutting out the splitting field of f and taking transpositions to elements of trace 1. In both of these cases, the twisting character χ is trivial. 5.3.3. Representations of type A5 The group A5 has two three-dimensional irreducible representations defined over F¯ p , for each p > 5. By composing these representations with the projection G Q → Gal(K /Q), where K is a field with Galois group A5 , we obtain irreducible threedimensional Galois representations with image isomorphic to A5 . We give one example in Table 6, which we now explain in detail. Example 5.5 Let K be the splitting field of the polynomial x 5 −5x 3 − x 2 +9x +7. Then Gal(K /Q) is isomorphic to A5 , and K is ramified only at p = 7 (with ramification index 3) and at 73 (with ramification index 2). Let ρ1 and ρ10 be the two characteristic 7 Galois representations alluded to above. Then it is easy to see that ρ1 and ρ10 are Galois conjugates of each other over the field F7 . The trace of both ρ1 and ρ10 on a generator of inertia at 73 is −1, so that both representations have level 732 and trivial nebentype. This level is too large for us to work with, so we twist both representations by the character χ = 73 to obtain ρ = ρ1 ⊗ 73 and ρ 0 = ρ10 ⊗ 73 . Now ρ and ρ 0 have level 73 and nebentype 73 . Just as in Example 5.4, the restriction of ρ (and of ρ 0 ) to inertia at 7 has diagonal characters ω0 , ω2 , and ω4 . Hence, the predicted weights are the same as in those examples, namely, F(2, 1, 0) ⊗ deta and F(6, 3, 0) ⊗ deta with a = 0, 2, 4. Computing the cohomology in each of these six weights with level 73 and nebentype 73 , we find that there is a unique eigenspace with the correct eigenvalues to correspond to ρ, and a unique eigenspace with the correct eigenvalues to correspond to ρ 0 (at least up to ` = 47). As expected, these eigenspaces are defined over F72 rather than over F7 , and they are Galois conjugates of each other over F7 . 5.3.4. Wildly ramified representations In addition to the preceding representations, we are able to calculate cohomology classes corresponding to irreducible three-dimensional representations ρ : G Q → GL3 (F¯ p ) which are wildly ramified at p. We have two types of examples of such


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representations, those having image A5 , which are wildly ramified at 5, and those having image PSL2 (F7 ), which are wildly ramified at 7. We begin our study of the type A5 representations by noting that there is a unique (up to isomorphism) injective homomorphism from A5 to GL3 (F5 ), with image generated by the three matrices       1 1 0 4 2 2 4 1 4  1 1 ,  1 2 , 0 4 1 , 1 4 2 4 2 of orders 5, 2, and 3, respectively. The fields from which we obtain our Galois representations have inertia group at 5 of order 5 or 10. In the case of representations with inertia group of order 10, we choose our representation so that the image of inertia is generated by the first two matrices above. With this choice of Galois representation, it is clear that we have  2  ω ∗ ∗ ρ| I p ∼  ω0 ∗  . ω2 Hence, we obtain a triple of (2, 0, 2) yielding a predicted weight of F(0, −1, 2)0 = F(4, 3, 2) = F(2, 1, 0) ⊗ det2 . In order to keep the level to a manageable size, we work with a twist of ρ by a character unramified at p (so that the weight is not affected). Let be the product of the characters q , where q runs through the set of primes at which ρ is ramified with ramification index 2. Then each prime q at which ρ has ramification index 2 contributes a factor of q to the level of ρ ⊗ , and each prime q at which ρ has ramification index 3 contributes a factor of q 2 to the level of ρ ⊗ . The nebentype of ρ ⊗ is easily seen to be . We have one example in which the inertia group has order 5. In this case we choose the representation so that the image of inertia is generated by the first matrix above. It is then clear that   1 ∗ ∗ ρ| I p ∼  1 ∗ , 1 yielding a predicted weight of F(−2, −1, 0)0 = F(6, 3, 0). The level of this representation is 34 (note that 3 is wildly ramified), and the nebentype is trivial. For all of these examples, we have found that the predicted eigenclass does exist in the given weight, level, and character and has the correct eigenvalues (at least up to ` = 47) to have ρ attached.

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Table 7. Wildly ramified Galois representations in niveau 1 Polynomial x 5 + 5x 3 − 10x 2 − 45 x 5 + 5x 3 − 10x 2 − 1 x 5 + 5x 3 − 10x 2 + 5 x 5 + 5x 3 − 10x 2 + 9 x 5 + 5x 3 − 10x 2 + 20 x 5 + 25x 2 − 75 x 7 − 7x 5 − 7x 4 − 7x 3 − 7x 2 − 7 x 7 + 14x 6 + 14x 5 − 14x 4 + 35 x 7 − 21x 3 + 7x − 27 x 7 − 7x 5 − 28x 2 + 7x + 4 x 7 − 7x 5 − 21x 4 − 49x 3 − 21x 2 + 1 x 7 − 14x 4 + 42x 2 − 21x − 9 x 7 + 7x 5 − 7x 4 − 49x 3 − 98x − 107

G A5 A5 A5 A5 A5 A5 PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 )

p 5 5 5 5 5 5 7 7 7 7 7 7 7

Weight F(4, 3, 2) F(4, 3, 2) F(4, 3, 2) F(4, 3, 2) F(4, 3, 2) F(6, 3, 0) F(6, 5, 4) F(8, 5, 2) F(6, 5, 4) F(8, 5, 2) F(8, 5, 2) F(6, 5, 4) F(6, 5, 4)

Level 13 31 37 41 22 · 13 34 17 52 47 26 26 34 112

13 31 37 41 13 1 17 1 47 1 1 1 1

We have also found Galois representations with image isomorphic to PSL2 (F7 ) which have low enough level that we can compute the predicted cohomology classes. The image of the representation is generated by the three matrices       1 1 0 2 3 3 1 0 0 0 1 1 , 0 1 3 , 2 6 0 , 0 0 1 0 0 4 4 0 6 of orders 7, 3, and 2, respectively. The image of inertia at 7 under the representations that we examine always has order 21 and may be chosen to be the subgroup generated by the first two matrices above. Hence, on inertia, we have  2   4  ω ∗ ∗ ω ∗ ∗ ρ| I7 ∼  or ρ| I7 ∼  ω0 ∗  ω0 ∗  . 4 ω ω2 In order to distinguish between these two possibilities, we use the action of tame ramification on wild ramification. Let K be our PSL2 (F7 )-extension, and let K p be its localization at a prime above 7. Then K p /Q p is a degree 21 extension, which is totally ramified. Denote its Galois (inertia) group by G 0 and its higher ramification subgroups by G 1 , G 2 , . . . . Clearly, there is a unique i ≥ 1 such that G i /G i+1 is nontrivial, since G 1 is simple. By [21], for a ∈ G 0 /G 1 and b ∈ G j /G j+1 , we have θ j (aba −1 ) = θ0 (a) j θ j (t), ( j)

( j+1)

where θ j : G j /G j+1 → U K /U K is the injective homomorphism sending σ to σ (π)/π, where π is a uniformizer of K p .


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2 If we consider θ0 as a map into F× p , then we see (by [19]) that θ0 = ω . We ∼ identify G i /G i+1 = G i with its image under θi . Then we have

sts −1 = t ω

2i (s)

.

However, we see easily (by matrix multiplication) that sts −1 = t 2 , so that ω2i (s) = 2, and we have  2i  ω ∗ ∗ ρ| I7 ∼  1 ∗ . ω4i Finally, analysis of the ramification groups shows that if the discriminant of the degree 7 subfield of K is exactly divisible by 78 , then i = 1, and if it is exactly divisible by 710 , then i = 2. Clearly, if i = 1, we get a predicted weight of F(0, −1, 4)0 = F(6, 5, 4), and if i = 2, we get a predicted weight of F(2, −1, 2)0 = F(8, 5, 2). The level and nebentype are easily calculated, and in the case of odd primes q that have inertia group of order 2, we twist by q to lower the level from q 2 to q. Table 7 contains information on the wildly ramified Galois representations we have studied. Each line of the table gives a polynomial whose Galois closure is a G-extension of Q (where G = A5 or PSL2 (F7 )), as well as the weight, level, and nebentype predicted by Conjecture 3.1 for the appropriate twist of ρ. In each case the representation for which the invariants were computed is actually ρ ⊗, where is the indicated nebentype (as described above, this lowers the level to a manageable size). In every example an eigenclass with the correct eigenvalues (up to ` = 47) occurs in the predicted cohomology group.

6. Niveau 2 representations 6.1. Reducible representations in higher level Each line of Table 8 gives a polynomial with splitting field a totally real S3 -extension K of Q. In each case we define σ to be the unique two-dimensional Galois representation σ : G Q → GL2 (F¯ p ) which cuts out K , and we note that σ is niveau 2. Letting ρ = σ ⊕ ωk , where k is indicated in the table, Conjecture 3.1 predicts two possible weights corresponding to ρ, as indicated in the table. We have checked that for each row of Table 8, the cohomology in the given weights, level, and nebentype does contain an eigenclass with the correct eigenvalues to correspond to ρ, at least up to ` = 47. Table 9 contains examples of Galois representations, each of which is the sum of an odd irreducible two-dimensional Galois representation and the trivial character.

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Table 8. Reducible niveau 2 representations σ ⊕ ωk with σ even Fixed field of ker(σ ) x 3 − x 2 − 8x + 7

G S3

x 3 − x 2 − 7x + 2

S3

x 3 − 11x − 11

S3

k 1 3 5 7 9 11 13 5 7 9 11 13

p 5 5 11 11 11 11 11 11 11 11 11 11

Weights F(5, 4, 1) F(4, 4, 2) F(4, 2, 2) F(5, 2, 1) F(5, 4, 3) F(22, 14, 6) F(15, 6, 3) F(12, 6, 6) F(15, 8, 3) F(12, 8, 6) F(15, 10, 3) F(12, 10, 6) F(15, 12, 3) F(12, 12, 6) F(5, 4, 3) F(22, 14, 6) F(15, 6, 3) F(12, 6, 6) F(15, 8, 3) F(12, 8, 6) F(15, 10, 3) F(12, 10, 6) F(15, 12, 3) F(12, 12, 6)

Level 73 73 13 13 13 13 13 17 17 17 17 17

73 73 13 13 13 13 13 17 17 17 17 17

In each example a polynomial is given that has Galois group G. For all but the last two examples, we let σ be the unique two-dimensional mod p representation of G, and in all cases we take ρ to be σ ⊕ 1. The ramification index e of p, and the level N and nebentype of ρ, are indicated. For each such representation, Conjecture 3.1 predicts four weights (two of the predicted weights are the same in the p = 3 cases), as indicated in the table, and in all cases we have been able to check that the predicted eigenvalues exist in the cohomology in all of the predicted weights. We explain the last two examples in Table 9 in detail in Example 6.1. Example 6.1 Let K be the splitting field of the polynomial f = x 5 − 19x 2 + 38x − 95. Then K is a totally complex D5 -extension of Q, ramified only at 7 (with ramification index 2 and residual degree 1) and 19 (with ramification index 5 and residual degree 2). The group D5 has two irreducible two-dimensional representations over F19 — we denote the corresponding Galois representations by σ and σ 0 . Let ρ (resp., ρ 0 ) be the direct sum of σ (resp., σ 0 ) with the trivial character. Then both ρ and ρ 0 are easily seen to have level 7 and nebentype 7 . We may conjugate each of ρ and ρ 0 to land in either of the standard Levi subgroups     ∗ ∗ ∗ L = ∗ ∗  or L 0 =  ∗ ∗ , ∗ ∗ ∗ and each representation satisfies strict parity with Levi subgroup L (or L 0 ), as σ and σ 0 are both odd.


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Table 9. Reducible niveau 2 representations σ ⊕ ω0 with σ odd

Galois representation p = 5, e = 3 N = 7, = 3 G = S3 x 3 − x 2 + 2x − 3 p = 5, e = 3 N = 43, = 3 G = S3 x 3 − x 2 + 2x + 12 p = 5, e = 3 N = 47, = 3 G = S3 x 3 + 5x − 5 p = 5, e = 3 N = 67, = 3 G = S3 x 3 − x 2 + 7x + 2 p = 5, e = 3 N = 83, = 3 G = S3 x 3 − 10x − 15 p = 5, e = 3 N = 83, = 3 G = S3 x 3 − x 2 + 7x − 8 p = 5, e = 3 N = 83, = 3 G = S3 x 3 − x 2 − 3x − 8 p = 17, e = 3 N = 3, = 3 G = S3 x 3 − x 2 + 6x − 12 p = 17, e = 3 N = 7, = 7 G = S3 x 3 − x 2 + 6x + 5 p = 3, e = 4 N = 7, = 7 G = D4 x 4 − 3x 2 − 3 p = 3, e = 4 N = 19, = 19 G = D4 x 4 − 30x 2 − 3 p = 3, e = 4 N = 31, = 31 G = D4 x 4 + 9x 2 − 3 p = 3, e = 4 N = 43, = 43 G = D4 x 4 − 318x 2 − 3 p = 7, e = 4 N = 11, = 11 G = D4 x 4 − 7x 2 − 7 p = 19, e = 5 N = 7, = 7 G = D5 x 5 − 19x 2 + 38x − 95 p = 19, e = 5 N = 7, = 7 G = D5 x 5 − 19x 2 + 38x − 95

Predicted weights F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(9, 4, 0), F(20, 9, 0) F(14, 10, 5), F(30, 21, 10) F(9, 4, 0), F(20, 9, 0) F(14, 10, 5), F(30, 21, 10) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(3, 0, 0), F(6, 3, 0) F(4, 4, 1), F(10, 7, 4) F(13, 2, 0), F(20, 13, 0) F(16, 14, 3), F(34, 21, 14) F(9, 6, 0), F(24, 9, 0) F(16, 10, 7), F(34, 25, 10)

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Both ρ and ρ 0 have niveau 2, but they differ on restriction to inertia at 19. We choose ρ (possibly swapping σ and σ 0 ) such that   ψ 72   ρ| I19 ∼ L  ψ 0 72 , 0 ω while ρ 0 has diagonal characters ψ 144 , ψ 0 144 , ω0 . Since 72 = 15 + 3 ∗ 19, we get a predicted weight of F(15 − 2, 3 − 1, 0)0 = F(13, 2, 0) for ρ. In addition, we may also conjugate ρ inside L, so that the diagonal characters on inertia are ψ 288 , ψ 0 288 , and ω0 . Since 288 = 22+14∗19, we also predict a weight of F(20, 13, 0) for ρ. Finally, we may conjugate ρ to land inside L 0 , yielding predicted weights of F(16, 14, 3) and F(34, 21, 14). In a similar fashion, we predict four weights for ρ 0 , namely, F(9, 6, 0), F(16, 10, 7), F(24, 9, 0), and F(34, 25, 10). In order to test whether the representations ρ and ρ 0 are attached to Hecke eigenclasses with these weights, we need to compute the characteristic polynomials of the images of Frobenius elements under ρ and ρ 0 . There is a subtlety introduced here by the fact that D5 has two conjugacy classes of order 5. On one of these classes, ρ has trace 5 and ρ 0 has trace 15, while on the other class these traces are reversed. We must determine which class contains each Frobenius element of order 5. Suppose τ ∈ G Q restricts to an element of order 5 in Gal(K /Q). Then there is some element η ∈ I19 such that τ ≡ η modulo G K . So 72 Tr ρ(τ ) = Tr ρ(η) = ψ 72 (η) + ψ 0 (η) + 1. Let P be the unique prime of K lying over p = 19, and let π be a uniformizer of P. Then ζ = ψ 72 (η) ≡ η(π )/π (mod P) is a fifth root of unity in the residue field F of P (which has order 192 ). Note that there are two possible images of ζ in F, depending on our choice of fundamental character ψ. We may, however, compute ζ + ζ p , which is in the prime field and is independent of this choice. These calculations are easily performed using PARI/GP since K is only of degree 10 over Q. A convenient uniformizer to use is a root α of the polynomial f defined above. We find, for instance, that Tr(ρ(Frob2 )) = 5, giving predicted Hecke eigenvalues a(2, 1) = 5 and a(2, 2) = 12 for the classes attached to ρ, and eigenvalues a(2, 1) = 15 and a(2, 2) = 17 for the classes attached to ρ 0 . We have computed the Hecke eigenvalues (for l ≤ 47) for cohomology classes with each of the 8 weights that arose above, and in each case we have found that the eigenvalues are exactly as predicted. 6.2. Irreducible representations in higher level In niveau 2 we again obtain several irreducible representations that are symmetric squares of odd two-dimensional representations, but we also obtain one set of exam-


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ples that are not. We begin by describing an example of the former type of representation. Example 6.2 Let p = 5, and let K be the splitting field of the polynomial f = x 4 + x 2 − x + 2. Then K has Galois group S4 and is ramified only at 5 (with ramification index 3) and at 73 (with ramification index 2). In fact, since the discriminant of f is 52 73, the quadratic subfield of K is ramified at 73, so the inertia group at 73 must be generated by a transposition. If we let ρ be the unique irreducible three-dimensional mod 5 representation of G Q cutting out K and taking transpositions to elements of trace 1, we easily determine that the level of ρ is 73 and that its nebentype is 73 . The weights predicted for ρ by Conjecture 3.1 are calculated by noting that   ψ8   ρ| I5 ∼  ψ 08 , 0 ω with 8 = 3 + 1 ∗ 5, so that we have a predicted weight of F(3 − 2, 1 − 1, 0)0 = F(1, 0, 0)0 . We may also write     ψ8 ω0     ω0 ψ8 ρ| I5 ∼  or ρ| I5 ∼   , 8 8 0 0 ψ ψ yielding predicted weights of F(3−2, 0−1, 1)0 = F(5, 3, 1) and F(0−2, 3−1, 1)0 = F(2, 2, 1)0 . In addition, we note that ψ 0 8 = ψ 16 , so we can permute the two characters of niveau 2 and write   ψ 16   ρ| I5 ∼  ψ 0 16 , ω0 with 16 = 1 + 3 ∗ 5 = 6 + 2 ∗ 5, so that we have a predicted weight of F(6 − 2, 2 − 1, 0)0 = F(4, 1, 0). We may also write     ω0 ψ 16     ω0 ψ 16 or ρ| I5 ∼  ρ| I5 ∼  ,  ψ 0 16 ψ 0 16 yielding predicted weights of F(6−2, 0−1, 2)0 = F(4, 3, 2) and F(0−2, 6−1, 2)0 = F(6, 5, 2). Calculating the cohomology in all six of these weights, we find eigenclasses with the correct Hecke eigenvalues to correspond to ρ (at least for primes up to 47). This

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Table 10. Irreducible niveau 2 representations Galois representation p = 5, e = 3 N = 73, = 73 G = S4 , χ = 1 x4 + x2 − x + 2 p = 5, e = 3 N = 144, = 1 G = S4 , χ = 1 x 4 − 2x 3 − 8x + 4 p = 7, e = 4 N = 67, = 67 G = S4 , χ = 1 x 4 − 56x + 112 p = 11, e = 3 N = 17, = 17 G = S4 , χ = 1 x 4 − x 3 + 3x + 2 p = 5, e = 3 N = 89, = 89 G = A5 , χ = 89 x 5 − 2x 3 − x 2 − 6x − 11 p = 5, e = 3 N = 151, = 151 G = A5 , χ = 151 x 5 − 3x 3 − x 2 + x − 3 p = 5, e = 3 N = 157, = 157 G = A5 , χ = 157 x 5 + 7x 3 − x 2 − 9x + 7

Predicted weights F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(6, 3, 3), F(12, 8, 4), F(7, 7, 4) F(9, 6, 3), F(3, 2, 1), F(7, 4, 1) F(5, 2, 0), F(15, 9, 3), F(8, 6, 3) F(12, 5, 0), F(12, 9, 6), F(18, 13, 6) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2)

yields a family of six “companion forms” of different weights, all of which seem to correspond to ρ. Other examples of irreducible niveau 2 representations with image isomorphic to S4 , as well as examples with image isomorphic to A5 (where the representation is the twist by χ of the unique irreducible three-dimensional mod 5 representation having image A5 and cutting out the splitting field of f ), are given in Table 10, in the same format as the examples in Table 6. In addition, examples with image of order 54 are given in [12]. These last examples have p = 5, level N = 83, with quadratic nebentype, and cannot be the symmetric square of any two-dimensional representation. 7. Niveau 3 representations We have two examples of odd niveau 3 representations, both of which support Conjecture 3.1. It is easy to see that a niveau 3 representation must be irreducible and that it cannot be the symmetric square of a two-dimensional representation. Our first example is induced from a one-dimensional representation of a subgroup of index 3 in G Q , and the second has image isomorphic to PSL2 (F7 ) in GL3 (F¯ 11 ) but is in no obvious way related to any representation of dimension less than 3. 7.1. An induced representation Let f = x 3 + 2x − 1. The Galois group of f is S3 . Let K be the splitting field of f , and let K 3 = Q(α), where α is a root of f . Then K 3 is ramified only at 59. Using


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PARI/GP, we may calculate the ray class group of K 3 modulo 7 and find that it is cyclic of order 9. If we let L be the ray class field of K 3 modulo 7, then the existence of L implies the existence of a character χ : G K 3 → µ9 ⊂ F73 of order 9, ramified only at primes above 7. If we now set G

ρ = IndG QK χ, 3

then ρ : G Q → GL3 (F¯ 7 ) must be irreducible since it has niveau 3 (as the ramification index at 7 is divisible by 9). Note that there are six choices of χ since there are six primitive ninth roots of unity in µ9 . Until we make a choice, everything we state is true for any choice of χ and hence for any ρ induced from χ . If we let M be the Galois closure of L, we see that M contains the composite field K L, which is abelian of degree 9 over K , and in fact, M is generated by the conjugates of K L over Q. We see from this that no element of Gal(M/K ) has order more than 9. Note that ρ factors through G = Gal(M/Q), so in particular, the image of inertia at 7 under ρ must be of order 9 (since inertia fixes K ). In fact, it is easy to see that the factorization of ρ through G is a faithful representation of G. Now let 2 [ GQ = gi G K 3 , i=0

where the gi are coset representatives of G K 3 in G Q , and for g ∈ G Q , note that 2 X Tr ρ(g) = χ 0 (g gi ), i=0

where χ (x) = 0

( 0

if x 6 ∈ G K 3 ,

χ(x) if x ∈ G K 3 .

Using this description of ρ, we may calculate values of Tr(ρ(g)) in terms of χ for various g, given that we know the order of π(g), where π : G Q → S3 is the natural projection onto the Galois group of K . Let g 0 ∈ G K 3 be a conjugate by some gi of g if such a conjugate exists; for π(g) of order 2, ρ(g) has trace χ (g 0 ), and for π(g) of order 3, ρ(g) has trace zero (since no conjugate of g is in G K 3 ). In fact, we may go even further and compute the values of χ (g 0 ) using class field theory. Class field theory shows the existence of an isomorphism between the ray class group J of K 3 modulo 7 and the group µ9 of ninth roots of unity. We fix this isomorphism by setting the image of the ideal p above 2 in K 3 with inertial degree 1 to have image χ(Frobp ) = ζ9 , where Frobp is a Frobenius above p (note that p has order 9 in the ray class group). Given any ideal of K 3 , we may then compute its image in J

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Table 11 p O(Frob p ) O(π(Frob p )) χ (Frob0p ) Tr(ρ(Frob p ))

2 18 2 ζ9 ζ9

3 9 3 * 0

5 9 3 * 0

7 * * * *

11 18 2 ζ95 ζ95

13 6 2 ζ96 ζ96

17 9 1 ζ9 0

19 9 3 * 0

23 18 2 ζ9 ζ9

29 3 3 * 0

31 18 2 ζ92 ζ92

37 18 2 ζ97 ζ97

41 3 3 * 0

43 2 2 1 1

47 18 2 ζ9 ζ9

in terms of the image of the ideal above 2 and hence find the image of any Frobenius element under χ . The ray class computations are easily done using PARI/GP since the degree of K 3 is only 3. Using these techniques, we find the values in Table 11. The only value that has not yet been explained is the trace of Frobenius at 17. This trace is zero since 17 splits completely in K (hence also in K 3 ). Hence, there are g three distinct conjugates Frob17i of Frob17 , all in G K 3 , and their images under χ are ζ9 , ζ94 , and ζ97 , so that the trace of ρ(Frob17 ) is zero. Direct computation in the ray class group shows that if p ≤ 47 is a rational prime with π(Frob p ) having order 2, and g is any Frobenius element for p, then χ (g 2 ) = χ(g 0 )2 . Since this is true for any conjugate of g 2 , we have Tr(ρ(g 2 )) = 3χ (g 0 )2 = 3 Tr(ρ(g))2 . Using this fact, a simple computation (using Magma) shows that the eigenvalues of ρ(g) must be ξ , ξ , and −ξ , where ξ = Tr(ρ(g)). Hence, the characteristic polynomial det(1 − ρ(g)X ) is equal to 1 − ξ X − ξ 2 X 2 + ξ 3 X 3. In particular, we use the fact that det ρ(g) = −ξ 3 = −(Tr ρ(g))3 . We now compute the level and character of ρ. The prime 59 has ramification index 2 in the fixed field of ρ, and if g is a generator of inertia at 59, then π(g) has order 2 (since 59 has ramification index 2 in K ). In addition, χ(g 0 ) must be simultaneously a ninth root and a square root of 1, hence equal to 1. Then Tr(ρ(g)) = χ (g 0 ) = 1, so the three eigenvalues of g are 1, 1, and −1, and the level of ρ must be 59, with nebentype 59 . Finally, we calculate the predicted weights for ρ. These weights in fact depend on the choice of χ . We recall that the fundamental characters of niveau 3 are denoted by θ , θ 0 , and θ 00 . Since 7 has ramification index 9 in M, we know that ρ must have niveau 3. In fact, we have that either     θ 76 θ 38     or ρ| I7 ∼  ρ| I7 ∼  θ 0 76 θ 0 38 .  76 38 00 00 θ θ


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Note that det ρ = ω38 59 = ω2 59 in the first case, while det ρ = ω76 59 = ω4 59 in the second. Thus, in order to obtain the first case, we choose χ (and hence ζ9 = χ (Frob2 )) such that 3 −ζ93 = − Tr ρ(Frob2 ) = det ρ(Frob2 ) = ω2 (Frob2 )59 (Frob2 ) = −4, and in order to get the second, we choose χ (and hence ζ9 ) such that −ζ93 = −2. Note that each of the two possibilities comes from three choices of χ . Hence, we should expect three eigenclasses in each predicted weight—one for each choice of χ . Considering the first case, m = 38 = 3 + 5 ∗ 7 + 0 ∗ 72 , so we get a triple (a, b, c) = (5, 3, 0), yielding predicted weight F(5 − 2, 3 − 1, 0) = F(3, 2, 0). We may also permute the characters on the diagonal, which has the effect of multiplying m by 7 or 72 modulo 73 − 1, yielding predicted triples and weights as follows. For 7 ∗ m = 266 = 7 + 9 ∗ 7 + 4 ∗ 72 , we get predicted weight F(9 − 2, 7 − 1, 4) = F(3, 2, 0) ⊗ det4 . For 49 ∗ m ≡ 152 = 5 + 7 ∗ 7 + 2 ∗ 72 , we get predicted weight F(7 − 2, 5 − 1, 2) = F(3, 2, 0) ⊗ det2 . We may similarly calculate weights for the second possibility and find the following predicted weights: F(3, 1, 0) ⊗ det1 ,

F(3, 1, 0) ⊗ det3 ,

and

F(3, 1, 0) ⊗ det5 .

Computations show that cohomology classes with the correct eigenvalues (up to ` = 47) exist in all of these weights. In each weight there is a triple of eigenclasses, defined over F73 and conjugate over F7 , each corresponding to a choice of χ as above. 7.2. A representation with image PSL2 (F7 ) We begin by noting that the irreducible polynomial f 1 = x 7 − 11x 5 − 22x 4 + 33x 2 + 33x + 11 has Galois group PSL2 (F7 ), as reported by both PARI/GP and Magma. If L = Q(α), where α is a root of f , we find that the discriminant of L is 116 312 . Since 11 is tamely ramified, we may conclude that the ramification index of 11 in the

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Table 12. Character table of PSL2 (F7 )

Class Size Order χ1 χ2 χ3 χ4 χ5 χ6

1 1 1 1 3 3 6 7 8

2 21 2 1 −1 −1 2 −1 0

3 56 3 1 0 0 0 1 −1

4 42 4 1 1 1 0 −1 0

5 24 7 1 α α¯ −1 0 1

6 24 7 1 α¯ α −1 0 1

splitting field K of f is e = 7. Using the main result of [8], we see easily that the ramification index of 31 in K is 2. √ The character table of PSL2 (F7 ) is given in Table 12, where α = (−1 + −7)/2 √ and α¯ = (−1 − −7)/2. Over F¯ 11 , we have that α and α¯ are equal to 4 and 6, with √ the order depending on our choice of −7. The existence of the PSL2 (F7 )-extension K gives rise to two irreducible threedimensional Galois representations defined over F¯ 11 . The image of inertia at 11 under both representations has order 7, so they are both niveau 3. We choose σ to be the representation which, when restricted to inertia at 11, has diagonal characters θ 190 , θ 0 190 , and θ 00 190 , and we choose σ 0 to be the other (with diagonal characters on inertia equal to θ 570 , θ 0 570 , θ 00 570 ). We note that the level of σ (and of σ 0 ) is 312 since the elements of order 2 are mapped to matrices of trace −1. This level is too large for convenient calculation, so we investigate ρ = σ ⊗ 31 and ρ 0 = σ 0 ⊗ 31 , which are easily seen to have level 31 and nebentype 31 . In order to calculate the predicted eigenvalues of the image of a Frobenius element of order 7 under ρ, we need to distinguish between the two conjugacy classes of order 7 in PSL2 (F7 ). In order to do this, we use a method similar to that used in Example 6.1. In this case, the method needs to be modified slightly since we are dealing with much larger fields. We begin by using Magma to determine the Galois group G ∼ = PSL2 (F7 ) of f as a permutation group acting on the roots of f . We note that each root of f is a uniformizer for all primes lying above 11 in K (since 11 is tamely ramified, and all the ramification occurs in L/Q). Let α be a root of f , and let τ be an element of order 7 in G. Then we easily compute a complex approximation to β = τ (α)/α. If P is the prime of K lying over 11 and having inertia group generated by τ , then the


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image of β in the residue field of P is a Galois conjugate of the primitive seventh root of unity θ 190 (τ ). Hence, the trace of σ (τ ) is equal to β + β 11 + β 121 mod P. We actually compute a complex approximation to γ = β + β 2 + β 4 , which is equal to this trace modulo P. Knowing that this trace is congruent to either 4 or 6 modulo P, we compute δ1 = γ − 4, and δ2 = γ − 6. Exactly one of δ1 and δ2 should lie in P. We note that if K 8 is the unique degree 8 subfield of K fixed by hτ i (so that K 8 is the decomposition field of P), then there is a unique degree 1 prime p in K 8 , and P lies over p. Hence, we may determine whether δi lies in P by determining whether the norm of δi (from K to K 8 ) lies in p. We compute a complex approximation to this norm (and all of its Galois conjugates) and then easily find a complex approximation to the minimal polynomial of this norm. This polynomial should have rational integer coefficients, so after examining the polynomial to see that this is true to many decimal places, we round off. We then calculate the valuation of the norm of δi at the unique degree 1 prime in K 8 (using PARI/GP). For our choice of τ , we find that δ1 ∈ / p, while δ2 ∈ p. Hence, Tr(σ (τ )) = 6. Then, using similar techniques, we determine that τ is a Frobenius element for the prime 7, but not for the primes 2, 13, or 23. Hence, for example, we predict that Tr ρ(Frob2 ) = Tr σ (Frob2 ) 31 (Frob2 ) = 4 · (1) = 4 and Tr ρ(Frob13 ) = Tr σ (Frob13 ) 31 (Frob13 ) = 4 · (−1) = 7. Returning to our study of ρ, we have  θ 190  ρ| I p ∼  θ 0 190

 θ 00 190

 .

Note that m = 190 = 3 + 6 ∗ 11 + 1 ∗ 112 . Hence, one weight predicted by the conjecture for ρ is F(6 − 2, 3 − 1, 1)0 = F(4, 2, 1) = F(3, 1, 0) ⊗ det1 . We may also take m = 11·190 or m = 112 ·190, which yield predicted weights of F(6, 6, 0)⊗det5 and F(8, 3, 0) ⊗ det2 . Computing the cohomology in weight F(3, 1, 0) ⊗ det1 , we find a one-dimensional eigenspace with the eigenvalues indicated in Table 13. These eigenvalues are exactly what Conjecture 3.1 predicts, in order for ρ to be attached to this eigenclass. The same system of eigenvalues (up to ` = 47) also occurs in the other two weights predicted for ρ. Similarly, the predicted weights for ρ 0 are F(6, 0, 0) ⊗ det7 , F(3, 2, 0) ⊗ det6 , and F(8, 5, 0) ⊗ det8 . Each of these weights yields an eigenclass with the correct eigenvalues to have ρ 0 attached (at least for ` ≤ 47).

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Table 13. Orders of ρ(Frob` ) and eigenvalues in weight F(4, 2, 1) ` O(ρ(Fr` )) a(`, 1) a(`, 2)

2 7 4 3

3 3 0 0

5 4 1 9

7 7 6 10

11 * 0 0

13 7 7 3

17 4 10 2

19 4 1 7

23 7 7 6

29 3 0 0

31 * * *

37 2 1 8

41 3 0 0

43 3 0 0

47 3 0 0

8. Computational techniques We now give an overview of our methods for computing the various Hecke eigenclasses on which we have reported in this paper. We begin by noting that we do not, in fact, do any direct calculations of cohomology. Instead we compute with homology, exploiting the natural duality, as in [7, Section 3]. Let p and N be positive integers with p prime, and let V be a representation of the semigroup generated by S pN and 00 (N ). Then we wish to calculate H3 (00 (N ), V ) along with the action of various Hecke operators. The groups H3 are easier to calculate than H1 or H2 since the virtual homological dimension of SL3 (Z) is 3 (see [1]). In addition, one can show that for many classes of three-dimensional Galois representations, if the representation is attached to any homology class, then it is attached to a class in H3 (cf. [4]). By Shapiro’s lemma, SL3 (Z) H3 00 (N ), V ∼ = H3 SL3 (Z), Ind00 (N ) V , and by [5, Lemma 1.1.4] this isomorphism is compatible with the action of the Hecke operators away from pN . This reduces our problem to computing the homology of the full group SL3 (Z) as long as we are willing to consider sufficiently general weights. The broad outline of our calculations follows that of [1]. In particular, we first use a slight modification of their [1, Theorem 1] to identify H3 (SL3 (Z), V ) with the subspace of all v ∈ V such that (1) v · d = v for all diagonal (but not necessarily scalar) matrices d ∈ SL3 (Z); (2) v · z = −v for all monomial matrices of order 2 in SL3 (Z); (3) v + v · h + v · (h 2 ) = 0, where   0 −1 0 h =  1 −1 0  . 0 0 1 We refer to conditions 1 and 2 as the semi-invariant condition and to condition 3 as the h-condition. Given a sufficiently concrete realization of V , computing the subspace satisfying these conditions is simply an exercise in linear algebra. In Section 8.2.2 we discuss some optimizations we have employed in carrying out the calculation. Once


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we have this subspace in hand, we then use [1, Lemma 3] to compute the actions of various Hecke operators with respect to a basis of this space. The main difference between our calculations and those in [1] is our use of more general coefficient modules. We describe below our construction of the modSL3 (Z) ules Ind00 (N ) F(a, b, c) for a p-reduced triple (a, b, c). Another significant difference is a sharp increase in efficiency and hence in the complexity of the calculations we can tackle. This increase is due partly to better algorithms (as described below) and partly to having the entire calculation done using C++ code rather than relying on Mathematica. 8.1. Models for weights We have performed our calculations with a variety of weight modules. Our basic strategy has been to build more complicated weights up from simpler ones. In this subsection we describe the GL3 (F p )-modules with which we have worked, giving in particular a model for F(a, b, c) for a general p-reduced triple (a, b, c). Details of the implementation of these representations and of the process of inducing from 00 (N ) are left to Section 8.2. To begin, we view F¯ p3 as the standard 3-dimensional (right) F¯ p [GL3 (F p )]-module on which S pN acts via reduction modulo p. Then Symg (F¯ p3 ) is the space of homogeneous polynomials over F¯ p of total degree g in three variables x, y, z. An element m of GL3 (F p ) acts on f ∈ Symg (F¯ p3 ) by f (x) · m = f (mx), where x is the column vector t (x, y, z). Note that for a ≤ p − 1, the representation Syma (F¯ p3 ) is irreducible and is in fact isomorphic to F(a, 0, 0) = W (a, 0, 0). Note also that this action is the contragredient of the standard action used in the statement of the conjecture, as required by the duality between homology and cohomology. Next we look at the module F(a, b, 0) for p-restricted (a, b, 0). THEOREM 8.1 Let (a, b, 0) be a p-restricted triple. Then the GL3 (F p )-submodule of Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ) generated by

v=

b X i=0

(−1)i

b i

(x a−i y i ⊗ x i y b−i )

is isomorphic to F(a, b, 0). Proof Recall that for any nonincreasing triple (α, β, γ ) of integers, both W (α, β, γ ) and

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F(α, β, γ ) are modules over GL3 (F¯ p ) and not just over GL3 (F p ). We prove that the GL3 (F¯ p )-module generated by v is isomorphic to F(a, b, 0). Since (a, b, 0) is assumed to be p-restricted, F(a, b, 0) remains irreducible when viewed as a representation of GL3 (F p ). We may then conclude that the GL3 (F p )-module generated by v is isomorphic to F(a, b, 0). Since we are now looking at representations of GL3 of an algebraically closed field, we may employ the theory of highest weights in representations of algebraic groups (cf. [15, Section 31]). In particular, if we work with respect to the standard diagonal torus and the upper triangular Borel, we note that the nonincreasing triples (n 1 , n 2 , n 3 ) correspond to the dominant weights   t1 0 0  0 t2 0  7 → t n 1 t n 2 t n 3 . 1 2 3 0 0 t3 Then F(n 1 , n 2 , n 3 ) is the unique irreducible representation of GL3 (F¯ p ) with highest weight (n 1 , n 2 , n 3 ). Now, Young’s rule (see [16, p. 129]) gives us that W (a, 0, 0) ⊗ W (b, 0, 0) has a filtration W0 ⊃ W1 ⊃ · · · ⊃ Wb+1 = 0, with the quotients Wi /Wi+1 isomorphic to the modules W (a + b, 0, 0), . . . , W (a + b − i, i, 0), . . . , W (a, b, 0) in the given order (so that W (a +b, 0, 0) is a quotient and W (a, b, 0) is a submodule). Since (a, b, 0) is p-restricted, each W (a +b−i, i, 0) is irreducible if a +b−i ≤ p −2 or a + b − 2i = p − 1, and otherwise has F(a + b − i, i, 0) and F( p − 2, i, a + b − i − p + 2) as composition factors (see [10, Proposition 2.11]). We see then that F(a, b, 0) appears only once as a composition factor of W (a, 0, 0) ⊗ W (b, 0, 0) and that it appears as a submodule and not just a subquotient. It follows that W (a, 0, 0) ⊗ W (b, 0, 0) has a unique highest weight vector w of weight (a, b, 0) and that the GL3 (F¯ p )-module generated by this vector is isomorphic to F(a, b, 0). The lemma below shows that v is such a vector, and hence the GL3 (F¯ p )module generated by v is isomorphic to F(a, b, 0). LEMMA 8.2 The vector

v=

b X b (−1)i (x a−i y i ⊗ x i y b−i ) i i=0

in Sym is a highest weight vector of weight (a, b, 0). Here “highest” refers to the usual lexicographic ordering of the weights. a

(F¯ p3 ) ⊗ Symb (F¯ p3 )


Proof It is clear that v is a weight vector of weight images of v under the operators    1 0 0 1 0 g1 =  1 1 0  , g2 =  0 1 0 0 1 0 1

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(a, b, 0). We need only show that the  0 0 , 1



 1 0 0 g3 =  0 1 0  1 0 1

are all equal to v plus something in the span S of vectors of weight strictly less than (a, b, 0). Clearly, v · g2 and v · g3 are both equal to v modulo S. For v · g1 , we calculate v · g1 =

=

b X i=0 b X

(−1)

i

(−1)

i

b i

b i

X b−i i X

i=0

=

a X

b X

x a−i (x + y)i ⊗ x i (x + y)(b−i)

k=0 j=0

X v

u=a−b v=a−u i=a−u u a−u v b−v

·x y

⊗x y

(−1)

i

i k

b i

b−i j

x a−i+k y i−k ⊗ x i+ j y b−i− j

i u−a+i

b−i v−i

.

Setting α = i − (a − u), expanding the binomial coefficients, and canceling equal terms, the inner sum becomes ±

u+v−a X

(−1)α

α=0

b! (b − v)!(u + v − a − α)!α!(a − u)!

u+v−a X b! 1 (−1)α (b − v)!(a − u)! α!(u + v − a − α)! α=0 u+v−a X b! u+v−a (−1)α , =± α (b − v)!(a − u)!(u + v − a)!

=±

α=0

which is zero if u + v > a. Thus, the only terms x u y a−u ⊗ x v y b−v that appear in v · g1 with nonzero coefficient have u + v = a. It is now easy to see that v · g1 is in fact exactly equal to v. For arbitrary p-restricted (a, b, c), we note that F(a, b, c) ∼ = F(a −c, b−c, 0)⊗detc . In practice, we did all of our calculations with F(a − c, b − c, 0) and simply scaled by detc at the end. We have also made use of representations of the form Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ), Syma (F¯ p3 ) ⊗ Symb (F¯ p3 )∗ and subquotients of Syma (F¯ p3 ) for a larger than p − 1. By

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keeping track of the irreducible constituents of these representations, we were sometimes able to show that certain systems of Hecke eigenvalues come from a specific irreducible module (see [12] for more details). 8.2. Implementation The implementation of our algorithms has two very distinct parts. On the one hand, we need to do calculations involving various GL3 (Z/ pN Z)-modules V . This includes the basic vector space operations as well as multiplying an element in V by an element of GL3 (Z/ pN Z). Further, we need to identify a basis of V and be able to decompose elements of V with respect to that basis. For efficiency reasons it is also important to be able to determine the coefficient of a given basis element in some product v · g without computing all of v · g. On the other hand, we need to carry out various higher-level computations, such as finding the solutions to the h-condition above in order to compute homology with coefficients in V . These calculations can be described in terms of the basic operations of the previous paragraph without any specific knowledge about the module V . We have made use of object-oriented programming techniques to keep these two computational issues strictly separated. This allows us to switch from computing with one module to computing with another without having to rewrite any of the code describing the higher-level algorithms. 8.2.1. Coefficient modules We now look at a few of the implementation details behind some of our coefficient modules. As we stated above, the basic building block for all of our representations is Symg (F¯ p3 ), the space of homogeneous polynomials of degree g in three variables. The monomials form a natural basis of this space, and it is a simple matter to compute the coefficient of any given monomial in a product v · g. We have optimized this code to work especially well when many of g’s entries are zero. This is the case for the element h above as well as for many of the matrices arising in our Hecke operator calculations. The representations Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ) again have natural bases coming from the monomial bases of Syma (F¯ p3 ) and Symb (F¯ p3 ), and all operations on the tensor product can be carried out in terms of those on each factor. We denote by Bab = {wi } this basis of Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ), and we let h·, ·i be the bilinear form with hwi , w j i = δi j . The subspace F(a, b, 0) of Syma ⊗ Symb does not come equipped with a canonical basis. For ease of computation we choose a basis in which each basis vector has a distinguished leading term. In other words, we choose a basis {vi } such that for each i there is an element wi ∈ Bab with hwi , vi i = 1 and hwi , v j i = 0 for j 6= i. We then


573

let h·, ·i F be the bilinear form with hvi , v j i F = δi j . Then for v ∈ F(a, b, 0), we have hv j , vi F = hw j , vi, and so we can compute coordinates with respect to this basis of F(a, b, 0) in terms of those with respect to the basis Bab . The final step in obtaining our general weights is to induce a representation W from 00 (N ) to SL3 (Z). The W we use are of the form F(a, b, 0) ⊗ for some a SL3 (Z) character of (Z/N Z)× . We view V = Ind00 (N ) W as the space of functions V = f : SL3 (Z) → W : f (xg) = f (x) · g for g ∈ 00 (N ) with SL3 (Z) acting by left translation. We let {ri } be a set of representatives for SL3 (Z)/ 00 (N ), and we let {wa } be a basis for W . We again choose a bilinear form h, i on W with hwa , wb i = δab . Then we define φri ,wa : SL3 (Z) → W by ( wa · ri−1 x if x ∈ ri 00 (N ), φri ,wa (x) = 0 otherwise. It is clear that the functions φri ,wa comprise a basis of V . In order to express the action of SL3 (Z) on V with respect to this basis, we need to introduce a bit of notation. For x ∈ SL3 (Z), let {x} be the unique representative ri in x00 (N ). Then (φri ,wa g)(x) = φri ,wa (gx) ( wa · ri−1 gx if gx ∈ ri 00 (N ), = 0 otherwise ( wa · ri−1 g{g −1ri }{g −1ri }−1 x if x ∈ g −1ri 00 (N ), = 0 otherwise X

wa · ri−1 g{g −1ri }, wb φ{g−1 ri },wb (x). = b

Note that in order to compute the actions of Hecke operators on H3 (SL3 (Z), V ), we also need to know how elements of S = m ∈ M3 (Z) : det(m) is positive and prime to pN act on V . Let 6 be the semigroup generated by 00 (N ) and S pN . Then S = SL3 (Z)6 and 00 (N ) = SL3 (Z) ∩ 6. (This is part of what it means for the Hecke pair (00 (N ), 6) to be compatible with (SL3 (Z), S).) Thus, if m ∈ S, we have m = ns

574


for some n ∈ SL3 (Z) and s ∈ 6. Moreover, n is determined modulo 00 (N ) and so the coset representative {n} depends only on m. If we extend our notation to write {m} = {n}, the formula above for the action of g on φri ,wa makes sense for any g ∈ S. This action of S on V described by the formula induces the correct action of H ( pN ) on H3 (SL3 (Z), V ) (i.e., the one compatible with the action on H3 (00 (N ), W )). The ri may be chosen so that each is congruent to the identity modulo p, which greatly speeds up some of the calculations. Note that SL3 (Z)/ 00 (N ) ∼ = P2 (Z/N ) and so is easy to work with. Also, note that our formula shows at once how to compute the coefficients of a basis element φr j ,wb in v · g for v ∈ V and g ∈ S. 8.2.2. Finding homology Now we move on to the general algorithms we have used to compute the homology of SL3 (Z) with coefficients in some representation V . While this is a simple exercise in linear algebra, we have found it useful to tailor certain optimizations to our situation to allow us to work with larger examples. A typical instance of finding the solutions to the h-condition, for example, involves finding the kernel of a 700000 × 30000 matrix. These optimizations have been largely heuristic. We make no claim of having optimal algorithms. Let V be a 6-module of dimension d, with basis {vi }, and let h·, ·i be the bilinear form with hvi , v j i = δi j . Let K be the 24-element group of monomial matrices in SL3 (Z). Then for p > 3, the space of semi-invariants in V is the image of the operator X P= (g)g, g∈K

where (g) is the sign of the permutation on three letters induced by g. Our computations do not include examples for which p = 2, and for p = 3 only a minor adjustment is needed. Computing the action of P on each vi is not computationally intensive since we have specially optimized all of our coefficient modules with regard to the operation of monomial matrices. We then use column reduction to find a basis for V · P. We note that the dimension dsemi of V · P is approximately d/24. The more serious stage of the calculation is finding the solutions of the hcondition on V · P. We describe our algorithm for finding the solutions of the h-condition on any c-dimensional subspace W of V · P with basis {bi }. We are looking for the nullspace of the (d × c)-matrix M = (m i j ), where m i j = hvi , b j · (1 + h + h 2 )i is the coordinate of vi in b j · (1 + h + h 2 ). Simply computing this matrix and performing Gaussian elimination would theoretically allow us to find the nullspace but is hopelessly inefficient in both space and time. Although the matrix M is quite sparse, it becomes much denser as the elimination proceeds. Since we work with very large d (d on the order of 7 × 105 is not uncommon), we would


575

rapidly run out of memory. We touch on four optimizations we have made to speed up the calculation and to reduce the memory requirements. First, we note that the rows of M are highly redundant as there are at most about 1/24th as many columns as rows. We exploit this by computing the rows of M one at a time and only storing those that yield new information about the kernel. Recall that we have set up our coefficient modules so that we can individually compute the entries hvi , b j · (1 + h + h 2 )i in the ith row of M without having to compute all of b j · (1 + h + h 2 ). We discuss below another optimization that makes this separate computation especially efficient. As we find a new row R, we continue our elimination process by subtracting from R the appropriate multiples of all the previously stored rows. If we are left with a nonzero row, we append it to our stored matrix, which remains in row-echelon form. If we are left with the zero row, then R did not add any constraints on the kernel of M and we may discard it and move on to the next row. This guarantees that we never waste space by storing redundant rows and caps the maximum number of rows we will ever store at c ≤ dsemi ≈ d/24. We denote by E the matrix that we are building up row by row in this process. Our second optimization is motivated by the fact that most of the information about the kernel of M can be obtained from M’s early rows. At each stage in our calculation, we clearly have ker M ⊂ ker E. Since E is in row-echelon form, we can immediately read off the dimension of ker E. In practice, we find that the dimension of ker E drops below 1 or 2 percent of dsemi after we run through as few as one fifth or so of the rows of M. Once this happens, we pause our calculation and compute (a basis for) the kernel of E, which is relatively easy to do since E is already in rowechelon form. We have now reduced our problem to finding the kernel of 1 + h + h 2 not on W but on the much smaller space ker E. We then start the algorithm over, replacing W by ker E. Our new choice of W guarantees that the initial rows of the new matrix M will all be zero, and so we can resume our calculation with the row at which we had paused. It is crucial here that we have not computed M all at once and thus do not have to make any time-consuming adjustments to account for our new basis. Indeed, it is now much easier to compute the new M, as it has far fewer columns. In practice the calculation very rapidly proceeds through the remaining rows of M and then computes the kernel of the new E, which is equal to the kernel of M. Our choice of a cutoff on the dimension of ker E is entirely heuristic, and we adjust it based on the size of V . Both of the optimizations above rely on the efficiency of the calculation of the coefficients hvi , v · (1 + h + h 2 )i of each vi in v · (1 + h + h 2 ) for v ∈ V . Although our modules allow for the calculation of hvi , v · gi for any v and g without computing all of v · g, there is still a great deal of work duplicated if we separately perform this calculation for all of the vi . For our calculations of the Hecke operators (see below)

576


this is not necessary, but as described above, we must do this in the cases g = h and g = h 2 . We have optimized for this by storing some of the common pieces of SL3 (Z) these calculations. For example, when V = Ind00 (N ) W , we begin by computing and storing the entire matrices describing the actions of h and h 2 on W , and also the permutations induced by h and h 2 on P2 (Z/N ). Since the dimension of W is small compared to the dimension of V (even when N = 2, the dimension of V is 7 times that of W ), this calculation is not terribly costly in space or time. These stored tables can then be used to compute the action of h and h 2 on elements of V very quickly. We have implemented similar strategies when V is not induced but is the tensor product of two smaller representations. Finally, we have increased our available memory by making use of disk space and swapping pieces of our matrix in and out of memory. This requires minor modifications to the reduction algorithm described above in order to reduce the number of disk swaps. In particular, we carry out our row reduction on several (1000) new rows at once. In the end, this does not have a dramatic effect on run time, but it slashes the amount of RAM required. 8.2.3. Computing the Hecke action Our computation of the action of the Hecke operators closely mirrors that in [1], and we refer the reader to [1, Sections 3 and 8] for a discussion of modular symbols and a description of the action of Hecke operators on homology. We just summarize by noting that for v ∈ V satisfying the semi-invariant condition and the h-condition, we have X T (l, k)v = v · Mi j Bi , i, j

where 00 (N )D(l, k)00 (N ) =

a

00 (N )Bi ,

i

P the Mi j are unimodular, and the modular symbol [Bi−1 ] is homologous to j [Mi j ]. We have not recomputed the matrices Mi j but have used the files generated in the course of the calculations in [1]. If { fl } is a basis for the semi-invariants in V satisfying the h-condition, then we P P know a priori that i j f k · Mi j Bi is a linear combination l akl fl of the fl . We wish to obtain the numbers akl . To do this efficiently, we use the same trick we employed in our choice of basis for F(a, b, 0) and adjust our basis { fl } such that for each l there is a basis vector vl of V such that hvl , f m i = δlm . Then akl is the sum over i and j of hvl , f k · Mi j Bi i. As we have discussed, we are able to compute these coefficients directly. This is vastly superior to computing all of f k · Mi j Bi since the dimension of the homology space is only a tiny fraction of the dimension of V . This technique


577

was used in [1], although it could not be implemented as efficiently there due to the reliance on Mathematica’s multivariate polynomial routines. A final optimization uses the fact that the Hecke operators we are dealing with all commute and so preserve each other’s eigenspaces. The ultimate goal of our calculation is to identify simultaneous eigenvectors v of the T (l, k) attached to given Galois representations, that is, with T (l, k)v = α(l, k) for some prescribed α(l, k). If we compute the entire matrix for the action of T (2, 1) (which is very easy since T (2, 1) involves only 13 Mi j terms, whereas T (47, 1) involves 55923 such terms) and find a single eigenvector v with eigenvalue α(2, 1), we need only compute the image of the other T (l, k) on v and not on the whole homology space. Moreover, we know that v is an eigenvector of each T (l, k), and so we only need to compute a single coefficient hv` , T (l, k)i in order to determine the eigenvalue. This gives an extraordinary reduction in the time required to make the calculation. For example, we find that the dimension of the homology space at level 00 (11), weight F(22, 11)(11 ), and p = 19 is 31. We are interested in a single eigenvector in this space. In order to compute the entire matrix of a Hecke operator, we would need to find 312 = 961 coefficients of basis vectors. Instead, we reduce this to a single coefficient, giving nearly a thousandfold increase in performance. We point out that this technique was not needed in [1] as the homology spaces dealt with there were much smaller. 8.2.4. Reliability Whenever relying on a large amount of computer calculation, one hopes for a number of consistency checks on the data. Our first check is that two entirely independent programs were written to carry out the calculations on several different computers by two different authors and both programs yielded identical data where compared. The programming was done in C and C++ and compiled with gcc running on a Sparc Ultra 5, a Pentium II under Linux, and a Pentium III under Linux. We also compared our data to some of the data obtained in [1] and [4] and found everything to be consistent. Other checks include the fact that, whenever tested, the operators T (l, k) on a given homology space all commuted and that (again when tested) the Hecke operators all did preserve the space of semi-invariants in V satisfying the h-condition. Perhaps more compelling is the fact that our data meshes exactly with the Galois representations we have studied. While the correspondence is only conjectural, the agreement we observed very strongly suggests the validity of our calculations. Acknowledgments. The authors thank Warren Sinnott and Richard Taylor for many helpful conversations in the course of this work. We thank the maintainers of the Bordeaux database of number fields (available at ftp://megrez.math.u-bordeaux.fr), from which we have obtained defining polynomials for many of the fields de-

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scribed in this paper. We also thank John Jones and David Roberts, whose tables of number fields ramified only at small primes (available on the web at http://math.la.asu.edu/∼jj/numberfields/) also provided many of our defining polynomials. Finally, we thank the creators of PARI/GP (see [18]) and Magma for their software, which was used for many calculations throughout this work. References [1]

[2] [3]

[4]

[5]

[6] [7]

[8] [9]

[10]

[11] [12]

G. ALLISON, A. ASH, and E. CONRAD, Galois representations, Hecke operators, and

the mod- p cohomology of GL(3, Z) with twisted coefficients, Experiment. Math. 7 (1998), 361 – 390. MR 2000d:11068 532, 543, 544, 545, 568, 569, 576, 577 A. ASH, Galois representations attached to mod p cohomology of GL(n, Z), Duke Math. J. 65 (1992), 235 – 255. MR 93c:11036 536 A. ASH and M. MCCONNELL, Experimental indications of three-dimensional Galois representations from the cohomology of SL(3, Z), Experiment. Math. 1 (1992), 209 – 223. MR 94b:11045 532 A. ASH and W. SINNOTT, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n, Z), Duke Math. J. 105 (2000), 1 – 24. MR 2001g:11081 521, 526, 532, 534, 535, 537, 538, 542, 543, 544, 568, 577 A. ASH and G. STEVENS, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192 – 220. MR 87i:11069 525, 538, 568 , Modular forms in characteristic ` and special values of their L-functions, Duke Math. J. 53 (1986), 849 – 868. MR 88h:11036 534, 535, 537, 540 A. ASH and P. H. TIEP, Modular representations of GL(3, F p ), symmetric squares, and mod- p cohomology of GL(3, Z), J. Algebra 222 (1999), 376 – 399. MR 2001d:11058 540, 568 S. BECKMANN, On finding elements in inertia groups by reduction modulo p, J. Algebra 164 (1994), 415 – 429. MR 95h:12003 566 L. CLOZEL, “Motifs et formes automorphes: Applications du principe de fonctorialité” in Automorphic Forms, Shimura Varieties, and L-Functions (Ann Arbor, Mich., 1988), Perspect. Math. 10, Academic Press, Boston, 1990, 77 – 159. MR 91k:11042 539, 540 S. R. DOTY and G. WALKER, The composition factors of F p [x 1 , x 2 , x 3 ] as a GL(3, p)-module, J. Algebra 147 (1992), 411 – 441. MR 93h:20015 525, 533, 537, 570 D. DOUD, S4 and e S4 extensions of Q ramified at only one prime, J. Number Theory 75 (1999), 185 – 197. MR 2000b:11060 541, 542, 544 , “Three-dimensional Galois representations with conjectural connections to arithmetic cohomology” to appear in Number Theory for the Millennium (Urbana-Champaign, Ill., 2000), ed. B. Berndt et al., Peters, Boston, 2002. 562, 572


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B. EDIXHOVEN, “‘Serre’s conjecture” in Modular Forms and Fermat’s Last Theorem

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J.-M. FONTAINE and G. LAFFAILE, Construction de représentations p-adiques, Ann.

(Boston, 1995), Springer, New York, 1997, 209 – 242. CMP 1 638 480 529, 535

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´ Sci. Ecole Norm. Sup. (4) 15 (1982), 547 – 608. MR 85c:14028 539 J. E. HUMPHREYS, Linear Algebraic Groups, Grad. Texts in Math. 21, Springer, New York, 1975. MR 53:633 570 G. D. JAMES, The Representation Theory of the Symmetric Groups, Lecture Notes in Math. 682, Springer, Berlin, 1978. MR 80g:20019 570 S. LANG, Algebraic Number Theory, 2d ed., Grad. Texts in Math. 110, Springer, New York, 1994. MR 95f:11085 545 THE PARI-GROUP, PARI/GP, Version 2.1.0, available from http://gn-50uma.de/ftp/pari/00index.html 546, 578 J.-P. SERRE, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259 – 331. MR 52:8126 523, 557 , “Modular forms of weight one and Galois representations” in Algebraic Number Fields: L-Functions and Galois Properties (Durham, England, 1975), Academic Press, London, 1977, 193 – 268. MR 56:8497 541 , Local Fields, Grad. Texts. in Math. 67, Springer, New York, 1979. MR 82e:12016 556 , Sur les représentations modulaires de degré 2 de Gal(Q/Q), Duke Math. J. 54 (1987), 179 – 230. MR 88g:11022 521, 524, 535 G. SHIMURA, Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures 1, Publ. Math. Soc. Japan 11, Iwanami Shoten, Tokyo; Princeton Univ. Press, Princeton, 1971. MR 47:3318 525 J. TUNNELL, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 173 – 175. MR 82j:12015 541

Ash Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467-3806, USA; [email protected] Doud Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138, USA; current: Department of Mathematics, 292 TMCB, Brigham Young University, Provo, Utah 84602, USA; [email protected] Pollack Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, Ohio 43210, USA; [email protected]


THE TRANSCENDENTAL PART OF THE REGULATOR MAP FOR K 1 ON A MIRROR FAMILY OF K 3-SURFACES ¨ PEDRO LUIS DEL ANGEL and STEFAN J. MULLER-STACH

Abstract We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K 1 of a mirror family of quartic K 3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods, and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equations are related to Painlevé VI–type differential equations. 1. The regulator map and Picard-Fuchs equations In this paper we study the first nonclassical higher K -group K 1 (X ) for a smooth complex projective surface X . It was conjectured by H. Esnault around 1995 that certain elements in this group can be detected in the transcendental part of the Deligne cohomology group HD3 (X, Z(2)) via the regulator (Chern class) map. The transcendental part of the regulator map is defined as an Abel-Jacobi-type integral of holomorphic 2-forms over nonclosed real 2-dimensional chains in X associated to these elements. At that time it was only known that one could detect such classes in the complementary (1, 1)-part of Deligne cohomology (see, e.g., [16]). The goal of our paper is to show that Esnault’s conjecture is true by looking at the differential equations that are satisfied by the normal functions arising from such classes in a family of surfaces. It turns out that the resulting equations for Abel-Jacobi-type integrals with parameters are strongly connected to a generalization of Painlevé VI–type differential equations. The higher K -groups K 1 (X ), K 2 (X ), . . . of an algebraic variety X were defined around 1970 by D. Quillen [19]. Later S. Bloch [2] showed that on smooth quasiprojective varieties all their graded pieces with respect to the γ -filtration may be computed as grγp K n (X )Q ∼ = CH p (X, n)Q , DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3, Received 4 April 2001. Revision received 23 May 2001. 2000 Mathematics Subject Classification. Primary 14C25, 19E15; Secondary 34M55, 32Q25. Del Angel’s work supported by Consejo Nacional de Ciencia y Tecnolog´ıa grant number 00359. Müller-Stach’s work supported by a Deutsche Forschungsgemeinschaft Heisenberg fellowship. 581

¨ DEL ANGEL and MULLER-STACH

582

where CH p (X, n) are Bloch’s higher Chow groups in [2]. This isomorphism gives an explicit presentation of higher K -groups modulo torsion via algebraic cycles. Let us look more closely at the particular case of K 1 (X ) for a smooth complex projective surface X . There it is known that CH1 (X, 1) = C× and CH p (X, 1) = 0 for p ≥ 4. The remaining interesting parts of K 1 are therefore CH2 (X, 1) and CH3 (X, 1). The last group consists of zero cycles on X × P1 in good position, and therefore the map τ : CH2 (X ) ⊗ Z C× → CH3 (X, 1), x ⊗ a 7 → (x, a) is surjective. Therefore the complexity of CH3 (X, 1) is governed by the complexity of CH2 (X ), which is fairly understood by Mumford’s theorem, respectively, Bloch’s conjecture. We say that CH3 (X, 1) is decomposable. For CH2 (X, 1) the situation is quite different, and the complex geometry of X plays an essential role in the understanding of it. The natural map τ:

CH1 (X ) ⊗ Z C× → CH2 (X, 1),

D ⊗ a 7 → D × {a}

is neither surjective nor injective in general. In the literature there are several examples where the cokernel of τ is nontrivial modulo torsion and even infinite-dimensional (see [4], [11], [16], [24]). The kernel of τ is related but not equal to Pic0 (X ) ⊗ C× even modulo torsion by [20, Th. 5.2]. Note that the cokernel of τ is a birational invariant (by localization) and hence vanishes for rational surfaces and, in fact, for all surfaces with geometric genus pg (X ) = 0 and Kodaira dimension less than or equal to 1. Bloch’s conjecture would imply that it also vanishes for all surfaces of general type which satisfy pg (X ) = 0. One way to study CH2 (X, 1) is to look at the Chern class maps c2,1 :

CH2 (X, 1) → HD3 X, Z(2) =

H 2 (X, C) . F 2 H 2 (X, C)

H 2 (X, Z) +

(1.1)

The decomposable cycles (the image of τ ) are mapped to the subgroup NS(X ) ⊗Z C× ⊆

H 2 (X, C) H 2 (X, Z) + F 2 H 2 (X, C)

generated by the Néron-Severi group NS(X ) ⊂ H 2 (X, Z) of all divisors in X . It is known (see [16]) that the image of c2,1 is at most countable modulo this subgroup, so that the image of coker(τ ) in Deligne cohomology modulo NS(X ) ⊗Z C× is at most countable. One conjectures that even coker(τ ) itself is countable. P The Chern class maps c2,1 are defined as follows: let Z = a j Z j ∈ CH2 (X, 1) be a cycle. Each Z j is an integral curve and inherits a rational map f j : Z j → P1 from the projection map X × P1 → P1 . Let γ0 be a path on P1 connecting 0 with

THE TRANSCENDENTAL K 1 OF A K 3-SURFACE

583

S S −1 ∞ along the real axis; then γ := γ j := f j (γ0 ) is a closed homological 1cycle, Poincaré dual to a cohomology class in F 2 H 3 (X, Z) and therefore torsion (see [16]). If we assume that γ = 0 (e.g., if b1 (X ) = 0), then we write γ = ∂0 for a real piecewise smooth 2-chain 0. As a current, that is, as a linear functional on differentiable complex-valued 2-forms on X , the defining property of c2,1 (Z ) is Z Z 1 X c2,1 (α) = log( f j )α + α. (1.2) 2πi Z j −γ j 0 j

Now let X be a projective K 3-surface. Then pg (X ) = 1, b2 (X ) = 22, and b1 (X ) = 0. The intersection form on H 2 (X, Z) is known to be the unimodular form 2E 8 ⊕ 3H , where H is the 2-dimensional standard hyperbolic form. The Néron-Severi lattice NS(X ) ⊂ H 2 (X, Z) has an orthogonal complement T (X ) ⊂ H 2 (X, Z). In particular, there is a well-defined morphism Coker(τ ) →

T (X ) ⊗ C× . F2

If we have an arbitrary smooth family f : X → B of complex algebraic surfaces over a quasiprojective complex variety B, and an algebraic family of cycles Z b ∈ CH2 (X b , 1) for all b ∈ B, that is, Z b = Z |X b for some given cycle Z ∈ CH2 (X, 1), then we may define the normal function ν(b) := c2,1 (Z b ) ∈

T (X b ) ⊗ C× . F2

One can easily show that ν is a holomorphic section of the corresponding family of generalized tori T (X b ) ⊗ C× /F 2 . Coming back to the case of K 3-surfaces, there the canonical bundle ω X is trivial; hence the group H 0,2 (X ) = H 0 (X, 2X )∗ = C is 1-dimensional and generated by the dual of ω X . In a smooth algebraic family X b of K 3-surfaces, the composition of the regulator with the projection onto H 0,2 (X b ) Im H2 (X b , Z) produces a multivalued holomorphic function on B, denoted by ν¯ (b), which has poles at all b where the family degenerates (see proof of Lem. 3.1). It is given by the formula Z ν¯ (b) = ωXb 0b

since the integral of ω X over any effective divisor vanishes. If DPF denotes the PicardFuchs differential operator of the Gauss-Manin connection associated to the family X b of K 3-surfaces, then DPF annihilates all periods of the family. Therefore we obtain the following result.


584

LEMMA 1.1 Let B ⊂ B¯ be a smooth compactification of B. Then with the notation above, DPF (¯ν ) extends to a single-valued meromorphic function on B¯ with poles only along degeneracies of X b , and therefore it satisfies a differential equation DPF ν¯ (b) = g(b), (1.3)

¯ where g is a rational function in b ∈ B. The proof is given in the appendix. Altogether, we have obtained a map families of cycles in CH2 (X b , 1) −→ {differential equations/rational functions}. (1.4) For each such family of K 3-surfaces, it sends a family of cycles to the equation DPF ν¯ = g, respectively, the rational function g, which is the same information on a given family. One should view the resulting solutions ν¯ (b) as new transcendental functions arising from the family of K -theoretic cycles in CH2 (X b , 1). If g is a nontrivial function, then ν¯ , and hence ν, is a nonflat section of the family of Deligne cohomology groups of X b . In [16] the relationship between the infinitesimal behavior of such normal functions and the mixed Hodge structure of the total space X has already been investigated. This situation is very reminiscent of a method developed by R. Fuchs [10] in the case of the Legendre family y 2 = x(x − 1)(x − t) of elliptic curves and investigated further in the work of Yu. Manin [14, p. 134]. In particular, there is a strong connection with differential equations of a generalized form of type Painlevé VI (see [14]). There exists a formula to derive g: there is a so-called inhomogeneous PicardFuchs equation DPF ω X = drel β (1.5) before integration over 0, where β is a section of the vector bundle of (meromorphic) 1-forms in the fibers of the family f : X → B. We say that 0 does not depend on b if it can be defined as a real semialgebraic subset via flat coordinates, that is, via coordinate functions that are horizontal with respect to the Gauss-Manin connection, and such that the defining inequalities of 0 are polynomials not depending on b. This shows on the one hand that for closed 0 the periods satisfy the Picard-Fuchs equation, and on the other hand that for nonclosed 0 (not depending on b) with ∂0 = γ , we get Z Z Z g(b) = DPF ω X = drel β = β. (1.6) 0

0

γ

The last equality uses a version of Stokes’s theorem for currents since some of the differential forms involved, in general, have integrable singularities. Hence Stokes’s


585

theorem for currents (see [12, Chap. 3]) also implies that β is integrable over γ . In general, 0 depends also on b, and then there is an additional contribution from the derivatives of the boundaries of the integral. In the case of the Legendre family y 2 = x(x − 1)(x − t) of elliptic curves, β is a meromorphic function (zero-form) after a double covering y 2(x − t)2 by [10, p. 310] and [14, p. 76]. Manin has put these equations into a more formal context (so-called µ-equations), so that one can understand the sections and operators in a coordinate-free way in terms of certain locally free sheaves on B. This also plays a role in his work on the functional Mordell conjecture. Furthermore, after uniformizing the elliptic curves, the inhomogeneous Picard-Fuchs equation is equivalent to a version involving the Weierstrass p-function (see [14, p. 137]): 3 Tj d2z 1 X = α p z + ,τ , j z 2 dτ 2 (2πi)2 j=0

where α j are constants parametrizing the family of differential equations and where (T0 , . . . , T3 ) = (0, 1, τ, 1 + τ ) are the vertices of the fundamental parallelogram. In this way the transcendental aspect of the solutions and also the connection to integrable systems become apparent (see [14, p. 139]). In the future we hope to investigate further the transcendental properties of our solutions (using again uniformization) and to study the attached integrable systems. The rest of this article is devoted to a particular solution of the inhomogeneous Picard-Fuchs equation for a certain family of K 3-surfaces introduced in Section 2. In Section 3 we deduce Esnault’s conjecture from the nonvanishing of the DPF (¯ν ) in √ the special case of b = −1. In Section 4 we study a certain Shioda-Inose model of X b which has isomorphic transcendental cohomology. This leads to an explicit computation of β in this case. 2. An example: A mirror family of K 3-surfaces We study the 1-dimensional family of K 3-surfaces given by the quartic equations X b := (x, y, z, w) ∈ P3 f (x, y, z, w) = x yz(x + y + z + bw) + w4 = 0 (2.1) with b ∈ P1 . Note that this surface, for general b, is not smooth but has six singular points defining a rational singularity of type A3 . The six points are (see [17, Sec. 4]) P1 = (0, 1, −1, 0),

P2 = (1, −1, 0, 0),

P3 = (1, 0, −1, 0),

P4 = (1, 0, 0, 0),

P5 = (0, 1, 0, 0),

P6 = (0, 0, 1, 0).


586

The minimal resolution of the singularities defines a generically smooth family of K 3-surfaces. In [17] the following theorem was shown. THEOREM 2.1 (N. Narumiya and H. Shiga) The family X b has the following properties. (1) It arises as a mirror family from the dual of the simplest polytope P of dimension three. The dual mirror family is the family of all quartic K 3-surfaces. (2) The rank of Pic(X ) is greater than or equal to 19 for all b ∈ P1 \ {0, ±4, ∞} and equal to 19 for very general b (see [17, Sec. 4]). (3) T (X b ) has signature (2, 1) for b ∈ P1 \ {0, ±4, ∞}. (4) The periods of X b satisfy the Picard-Fuchs equation

(5)

(6)

3 11 3 (1 − u)23 − u22 − u2 − u = 0 2 16 32 (where 2 = u(d/du)) of the generalized Thomae hypergeometric function (see [22]) 1 2 3 F3,2 , , , 1, 1; u 4 4 4 4 and where we set u := (4/b) . In other words, the Picard-Fuchs equation is given by 3 51 3 (1 − u)u 2 8000 + 3u 1 − u 800 + 1 − u 80 − 8 = 0. (2.2) 2 16 32 The mirror map of the family X b is given by the arithmetic Thompson series T (q) of type 2A in the classification of J. Conway and S. Norton [5]: T (q) =

1 + 8 + 4372q + 96256q 2 + 124002q 3 + 10698752q 4 + . . . . q

Proof We refer to [17] for more details, but we sketch the proof of (4) and (5) since this is crucial; (1) – (3) follow from the construction. In particular, six A3 -singularities give rise to 18 independent cohomology classes of type (1,1), so that the Picard number is greater than or equal to 19. Since (5) is an easy corollary of (4), we prove (4). In [17] the periods are computed as power series in 1/b, and the differential equation in (4) follows from [22]. As in [17], we consider the new affine equation f (x, y, z) = x yz(x + y + z + 1) +

1 =0 b4

obtained by substituting w0 := bw and setting w0 = 1. The periods are integrals of the form Z 1 d x dy dz I (b) = . 2πi |x|=|y|=|z|=1/4 x yz(x + y + z + 1) + 1/b4


587

On the other hand, one has a geometric series expansion at b = ∞: ∞

X 1 (−1)n b−4n = . 4 n+1 x yz(x + y + z + 1) + 1/b (x yz) (x + y + z + 1)n+1 n=0

Changing the order of summation and integration, we obtain 1 I (b) = 2πi =

1 2πi

∞ X

Z

|x|=|y|=|z|=1/4 n=0 ∞ Z X n=0 |x|=|y|=|z|=1/4

(−1)n b−4n d x dy dz (x yz)n+1 (x + y + z + 1)n+1 (−1)n b−4n d x dy dz . (x yz)n+1 (x + y + z + 1)n+1

Now one can apply three times the residue theorem and get I (b) = (2πi)

2

∞ X (4n)! n=0

(n!)4

b−4n .

Observing the identity involving Pochhammer symbols (4n)! (1/4)n (2/4)n (3/4)n 4 n = (4 ) , (1)n (1)n (1)n (n!)4 we have shown that I (b) = (2πi) F3,2 2

4 4 1 2 3 , , , 1, 1; . 4 4 4 b

Substituting u := (4/b)4 , one gets a multiple of the functions F3,2 (1/4, 2/4, 3/4, 1, 1; u) which satisfies a differential equation of order three precisely of the type described in (4), respectively, (5), by [22]. COROLLARY 2.2 In b-coordinates, the Picard-Fuchs equation can be written as b 4 b 4 b 3 000 3 b 2 1 b b 4 3 −1 8 + 1+ 800 + − 6 80 + 8 = 0. 4 4 4 4 4 16 4 4 32 (2.3)

Proof Use the chain rule. To make the following computations easier, we follow [17] and perform the following birational coordinate change (written in affine coordinates): bx y 1 + x yz X = x y, Y =i + , Z = yz. 2 z


588

Then X, Y, Z are affine coordinates and define the family of surfaces Sb in Weierstrass form b2 1 +1 Sb : Y 2 = X X 2 + X Z + − Z 4 as an elliptic fibration over P1 in the Z -coordinate. The inverse transformation is given by x = −2

i X (1 + Z X ) , (−2Y + ibX )Z

y=

1 i Z (−2Y + ibX ) , 2 1+ ZX

z = −2

i(1 + Z X ) −2Y + ibX

in affine coordinates. The following is taken from [17] (with a slight correction). 2.3 The surfaces Sb are ramified coverings of P1 × P1 (in X, Z coordinates). In X, Y, Z coordinates, the canonical holomorphic 2-form on Sb is given up to a nonzero constant by dX dZ d X dY = 2 ω= , (2.4) YZ X (Z − 1/Z ) LEMMA

where X, Z are flat coordinates and where s p 1 b2 2 Y = Y (b) = P(X, Z ) = X X + X Z + − +1 . Z 4 Proof In the x, y, z coordinate system, the holomorphic 2-form is given up to a constant by (d x dy)/ f z in affine (x, y, z)-coordinates with w = 1. Then, using the coordinate transformations above, one computes that ω=

√ d x dy d X dY dX dZ −1 · = 2 = fz YZ X (Z − 1/Z )

since 2Y dY = X 2 (1 − 1/Z 2 ) d Z + (∂ P/∂ X ) d X . Now, if we apply the Picard-Fuchs operator DPF from Corollary 2.2 to (d X d Z )/Y Z , we get an expression of the form DPF

d Xd Z d Xd Z = K (X, Z ) · 7 , YZ Y Z

where K (X, Z ) is a polynomial function in X, Z .


589

3. The normal function and the Picard-Fuchs equation On any elliptic surface, the easiest way to find cycles in K 1 is to use fibers. However, sometimes configurations coming from Néron fibers (degenerate into a union of P1 ’s) do have trivial class in K 1 , as has already been observed by A. Be˘ılinson in [1]. But one can use one smooth fiber together with a bunch of sections (rational) and rational curves in degenerate fibers. In our example, let us take the following cycles: denote by Sb the surface defined by the equation b2 2 2 2 X+Z . (3.1) Sb : Z Y = X X Z + Z + 1 − Z 4 Let Cb be the smooth elliptic fiber over Z = 1 of this surface. Hence its defining equation is b2 X +1 . Cb : Y 2 = X X 2 + 2 − 4 The quadratic term X 2 + (2 − b2 /4)X + 1 in the right-hand side has two negative √ real roots if b is purely imaginary, for example, if b = −1. The points X = 0 and X = ∞ are rationally equivalent on Cb after taking a multiple of two since they are ramification points. The real line from 0 to ∞ does not hit the other ramification points by this observation. The surface X b in this birational model has 0 and X = ∞ as sections. The fiber over Z = 0 on Sb decomposes into three rational curves with multiplicity counted. Hence one can construct a cycle in CH2 (Sb , 1) for general b by using Cb , the two sections, and the degenerate fibers and appropriate rational functions on all curves. In X, Z coordinates the region 0 is given by the real square √ 0 ≤ Z ≤ 1, 0 ≤ X ≤ ∞. For b = −1 we make the following observation. LEMMA 3.1 √ For b = −1, all coefficients occurring in DPF ((d X d Z )/Y Z ) = K (X, Z ) · ((d X d Z )/Y 7 Z ) are positive integers, that is, all coefficients of K (X, Z ) and all coefficients of Y 2 = X (X 2 + X (Z + 1/Z − b2 /4) + 1).

Proof Here the rules of differentiating are ∂ X /∂b = ∂ Z /∂b = 0 and ∂Y /∂b = (b/4) · (1/Y 3 ). This implies that odd derivatives of 1/Y get multiplied by even powers of b. Now if we look at the coefficients of (2.3), we see that the coefficients at 9 000 and 9 0 become positive since (b/4)4 − 1 and (b/4)4 − 6 are negative rational numbers and get multiplied with (b/4)6 , respectively, (b/4)2 , which are both also negative rational numbers. The coefficients at 9 and 9 00 already involve 4th powers of b and hence are positive. Consequently, all coefficients occurring are positive. Using any computer


590

algebra program, this can be verified, and indeed one has d Xd Z d Xd Z DPF = YZ 8192 × (349951 X 3 Z 3 + 85952 X Z 3 + 85952 X 5 Z 3 + 171904 X 4 Z 4 + 171904 X 4 Z 2 + 85952 X 3 Z 5 + 171904 X 2 Z 4 + 294912 X 2 Z + 294912 X Z 2 + 98304 Z 3 + 98304 X 3 + 294912 X 4 Z + 909352 X 2 Z 3 + 909352 X 3 Z 2 + 98304 X 6 Z 3 + 294912 X 5 Z 4 + 294912 X 5 Z 2 + 909352 X 4 Z 3 + 294912 X 4 Z 5 + 909352 X 3 Z 4 + 294912 X Z 4 + 98304 X 3 Z 6 + 294912 X 2 Z 5 + 85952 X 3 Z + 171904 X 2 Z 2 ) √ /(4 X 2 Z + 4 X Z 2 + 4 X + X Z + 4 Z )7/2 X Z . This completes the proof. In particular, if we integrate over the positive region 0, we get a positive and nonzero integral. Since the boundary of 0 is defined as the rectangle 0 ≤ Z ≤ 1, 0 ≤ X ≤ ∞ and since X, Z are flat with respect to the connection, we say that 0 does not depend on b (see introduction), and this suffices to show that the normal function is nontrivial. So we have proved Esnault’s conjecture (see [16]). COROLLARY 3.2 The projected normal function ν¯ (b) does not satisfy the Picard-Fuchs equation b 4 b 4 b 3 000 3 b 2 1 b b 4 3 00 −1 8 + 1+ 8 + − 6 80 + 8 = 0. 4 4 4 4 4 16 4 4 32 (3.2) In particular, it is not a rational multiple of a period for all but a countable number of values b. For those b, the corresponding cycle Z b has no integer multiple that is decomposable modulo Pic(X b ) ⊗ C∗ .

The main open problem remains to find a 1-form β such that dβ = DPF ((d X d Z )/Y Z ). We compute such a β for the Kummer-type model of these K 3-surfaces in the following section. 4. The solution of D ω = dβ In [17] we can find the description of a 2:1 map π : Sb → Sb0 onto a Kummer surface Sb0 that has a birational model with the equation ν + 1 2 t (t − 1)(t − ν 2 ), (4.1) u 2 = s(s − 1) s − ν−1


591

where ν and b are related via the algebraic equation b2 = −4 ·

(ν 2 + 1)2 . ν(ν 2 − 1)

We prefer to use this equation for a computation of the solution of D ω = dβ since it is slightly easier, but we do not lose essential information. In this description we see that the transcendental part of H 2 (Sb ) also denoted by T (Sb ) has an Inose-Shioda structure (in the sense of [15]) and is therefore related to a variation of a family of elliptic curves. In fact, there are two isogenous elliptic curves E 1 (ν) and E 2 (ν) with equations ν + 1 2 E 1 (ν) : u 21 = s(s−1)(s−ν 2 ), E 2 (ν) : u 22 = t (t −1) t − (4.2) ν−1 together with a Nikulin involution (see [23]) on the abelian surface A = E 1 × E 2 such that the associated Kummer surface is Sb0 , and one has an isomorphism T (Sb ) ∼ = T (Sb0 ) under π∗ . In addition, this explains why the periods of Sb are squares of other hypergeometric functions related to the family E 1 (ν), respectively, E 2 (ν). More details about the birational map can be found in [17]. Further instances where InoseShioda structures and modular forms arise can be found in [8]. Let us now compute the Picard-Fuchs equation of the family E 1 (ν). If we let λ = ν 2 , we have √ ∂ν/∂λ = 1/(2ν) = 1/(2 λ), and therefore for any function 8 we have the transformation rules ∂8 ∂ν ∂8 1 ∂8 ∂8 1 = · = √ = ∂λ ∂ν ∂λ ∂ν 2 λ ∂ν 2ν and, for the second derivative, ∂ 28 1 ∂ 28 1 ∂8 . = − 3 2 2 2 ∂λ 4ν ∂ν 4ν ∂ν Plugging this into the standard hypergeometric Picard-Fuchs equation 1 λ(1 − λ)800 (λ) + (1 − 2λ)80 (λ) − 8(λ) = 0, 4 we get the new equation 1 − 3ν 2 0 8 (ν) − 8(ν) = 0 ν and the inhomogeneous variant (equality of 1-forms) p 2 1 − 3ν 2 ∂ s(s − 1)(s − ν 2 ) 2 ∂ (1 − ν ) 2 ω(s) + ω(s) − ω(s) = 2drel ν ∂ν ∂ν (s − ν 2 )2 s 2 (s − 1)2 = 2drel p , 3 s(s − 1)(s − ν 2 ) (1 − ν 2 )800 (ν) +


592

where ω(s) = p

ds s(s − 1)(s − ν 2 )

.

The rational normalized version of this equation is p 1 − 3ν 2 ∂ 1 2 s(s − 1)(s − ν 2 ) ∂2 ω(s) + ω(s) = drel . ω(s) − 2 2 2 2 ∂ν ν(1 − ν ) ∂ν 1−ν 1−ν (s − ν 2 )2 (4.3) In a similar way, we use the substitution λ = ((ν + 1)/(ν − 1))2 and get the formula ∂ν/∂λ = −(ν − 1)3 /(4(ν + 1)) and hence ∂8 ∂8 ∂ν (ν − 1)3 ∂8 = · =− , ∂λ ∂ν ∂λ 4(ν + 1) ∂ν ∂ 28 (ν − 1)6 ∂ 2 8 (ν − 1)5 (ν + 2) ∂8 = + . ∂ν ∂λ2 16(ν + 1)2 ∂ν 2 8(ν + 1)3 Combining all this, we get the equation ∂2 ν 2 − 2ν − 1 ∂ 1 ω(t) + ω(t) ω(t) + 2 2 ∂ν ν(ν − 1) ∂ν ν(ν − 1)2 p 2 t (t − 1)(t − ((ν + 1)/(ν − 1)))2 =− drel 2 ν(ν − 1) (t − ((ν + 1)/(ν − 1))2 )2 for ω(t) = p

dt t (t − 1)(t − ((ν + 1)/(ν − 1))2 )

(4.4)

.

We have to compute a sort of convolution product of these two equations in the following sense: set ω = ω(s) ∧ ω(t) = p

ds dt s(s − 1)(s

− ν 2 )t (t

− 1)(t − ((ν + 1)/(ν − 1))2 )

,

and notice that we have the product formula ∂3 ∂3 ∂2 ∂ ω(t) ω = 3 ω(s) ∧ ω(t) + 3 2 ω(s) ∧ 3 ∂ν ∂ν ∂ν ∂ν ∂2 ∂3 ∂ + 3 ω(s) ∧ 2 ω(t) + ω(s) ∧ 3 ω(t). ∂ν ∂ν ∂ν Similar formulas hold for lower derivatives. Note that s, t are flat coordinates, so that differentiating a differential form with respect to the coefficients is a well-defined procedure. Such formulas can be used to compute (∂ 3 /∂ν 3 )ω and to obtain a PicardFuchs differential operator D for ω together with a solution β of D ω = drel β.


593

LEMMA 4.1 One has the following inhomogeneous Picard-Fuchs equation involving 2-forms:

∂ 3ω 2ν + 1 ∂ 2 ω 7ν 4 − 6ν 3 − 4ν 2 + 6ν + 1 ∂ω + 3 + ν(ν + 1) ∂ν 2 ∂ν ∂ν 3 (ν − 1)2 (ν + 1)2 ν 2 ν 4 − 2ν 3 − 2ν − 1 ω = drel β, (4.5) + (ν − 1)3 (ν + 1)2 ν 2 where s(s − 1)(2ν 4 + 3ν 3 − ν 2 − 3νs + sν 2 − 2s) p ω(t) ν(s − ν 2 )2 (ν 2 − 1)2 s(s − 1)(s − ν 2 ) p 6 s(s − 1)(s − ν 2 ) 0 + ω (t) 1 − v2 (s − ν 2 )2

β = −2

t (t − 1)(2tν 4 − 7tν 3 + 7tν 2 − tν − t − 2ν 4 − 7ν 3 − 7ν 2 − ν + 1) p ω(s) ν 2 (ν − 1)4 (t − ((ν + 1)/(ν − 1))2 )2 (ν 2 − 1) t (t − 1)(t − ((ν + 1)/(ν − 1))2 ) p t (t − 1)(t − ((ν + 1)/(ν − 1))2 ) 0 +6 ω (s). (4.6) ν(ν − 1)2 (t − ((ν + 1)/(ν − 1))2 )2

+2

Proof We denote the derivative of a function (or a form in flat coordinates) f of ν by f 0 . Let us carry out the computation in a more general setting. Assume that we have two Picard-Fuchs equations, ω00 (s) − As ω0 (s) − Bs ω(s) = dβs , ω00 (t) − At ω0 (t) − Bt ω(t) = dβt , with functions βs , βt and As , Bs , At , Bt depending on ν. Now first note that ω000 (s) = As ω00 (s) + (A0s + Bs )ω0 (s) + Bs0 ω(s) +

d dβs dν

and furthermore that

d dβs = dβs0 dν by symmetry of mixed derivatives. A similar relation holds for t. By using the product formulas ω000 = ω000 (s) ∧ ω(t) + 3ω00 (s) ∧ ω0 (t) + 3ω0 (s) ∧ ω00 (t) + ω(s) ∧ ω000 (t), ω00 = ω00 (s) ∧ ω(t) + 2ω0 (s) ∧ ω0 (t) + ω(s) ∧ ω00 (t),


594

we obtain that h i 3 1 3 ω000 − (As + At )ω00 = A0s + Bs + 3Bt − A2s − As At ω0 (s) ∧ ω(t) 2 2 2 h i 1 3 + A0t + Bt + 3Bs − A2t − As At ω(s) ∧ ω0 (t) 2 2 i h 3 + As Bs + At Bt − (Bs + Bt )(As + At ) + Bs0 + Bt0 ω 2 1 3 0 + drel βs − As + At βs ω(t) + 3βs ω0 (t) 2 2 1 3 0 0 − βt − At + As βt ω(s) − 3βt ω (s) 2 2 does not involve any more terms of the form ω0 (s) ∧ ω0 (t). Now let 3 1 3 (As + At ), B := − A2s − As At + A0s + Bs + 3Bt , 2 2 2 1 3 1 C := − As Bs − (As Bt + At Bs ) − At Bt + Bs0 + Bt0 . 2 2 2

A :=

Then, assuming that we have the equality 1 3 1 3 − A2s − As At + A0s + Bs + 3Bt = − A2t − As At + A0t + Bt + 3Bs 2 2 2 2 (this condition is equivalent to the fact that the elliptic curves E 1 (ν) and E 2 (ν) are isogenous), we have the inhomogeneous Picard-Fuchs equation ω000 − Aω00 − Bω0 − Cω = drel β, where β is the 1-form 1 3 0 β := βs − As + At βs ω(t) + 3βs ω0 (t) 2 2 1 3 0 + βt − At + As βt ω(s) + 3βt ω0 (s). 2 2 In our case, As = −(1 − 3ν 2 )/(ν(1 − ν 2 )), Bs = 1/(1 − ν 2 ), At = −(ν 2 − 2ν − 1) /(ν(ν 2 − 1)), and Bt = −1/(ν(ν − 1)2 ). Therefore we get A = −3

7ν 4 − 6ν 3 − 4ν 2 + 6ν + 1 , (ν − 1)2 (ν + 1)2 ν 2 ν 4 − 2ν 3 − 2ν − 1 C =− , (ν − 1)3 (ν + 1)2 ν 2

2ν + 1 , ν(ν + 1)

B=−

and, for β, the expression above. This finishes the proof.


595

Appendix In this section we give the proof of the following lemma from Section 1. LEMMA 1.1 In the situation of Section 1, DPF (¯ν ) is a single-valued meromorphic function on B with poles only along degeneracies of X b , and therefore it satisfies a differential equation DPF (¯ν ) = g,

where g is a rational function in b ∈ B. Proof Assume that we have a family f : X¯ → B¯ of projective surfaces over a compact Rie¯ Let 6 ⊆ B¯ be the finite subset over which there are singular fibers. mann surface B. Let h : X → B be the smooth part of f . We may assume that the family is semistable (semistable reduction) and that there is a cycle Z ∈ CH2 (X, 1) such that the restriction of Z to all fibers induces the family of cycles in CH2 (X b , 1). (Both reductions perhaps require a finite cover of B, which does not, however, change the assertion.) The cycle Z has a class c3,2 (Z ) ∈ HD3 (X, Z(2)) in Deligne cohomology. By semistability, 1 := f −1 6 is a divisor with strict normal crossings and its Deligne cohomology can be computed via the logarithmic de Rham complex. Let V 2 be the sheaf of transcendental cohomology classes in R 2 h ∗ C, a local system of rank 22 − ρ(X b ) for b general. The Deligne class vanishes in F 3 ∩ H 3 (X b , Z), since b3 (X b ) = 0 in our case, and therefore induces a holomorphic normal function ν ∈ H 0 (B, V 2 ⊗ O B /F 2 ) over B. However, since the family is semistable, there is a canonical extension of ν to a holomorphic section of the sheaf R 2 f ∗ ∗X¯ / B¯ (log 1). This can be seen as follows. Let ZD ,X (2) = Cone R j∗ Z(2) → ∗X¯ (log 1)/F 2 [−1] j

be the Be˘ılinson-Deligne complex (see [9]) of X using the inclusion X ,→ X¯ , and let ZD ,h (2) = Cone R j∗ Z(2) → ∗X¯ / B¯ (log 1)/F 2 [−1] be the relative Be˘ılinson-Deligne complex of h. There is a natural surjection of complexes ZD ,X (2) → ZD ,h (2) which induces a morphism ¯ R 3 f ∗ ZD , f (2) . HD3 X, Z(2) → H 0 B, Since all fibers of h satisfy b3 (X b ) = 0, we conclude that the image of ¯ R 3 f ∗ R j∗ Z(2)) vanishes. Therefore the image of c3,2 (Z ) this element in H 0 ( B, 0 3 ¯ in H ( B, R f ∗ ZD , f (2)) is coming (at least locally because of monodromy) ¯ R 2 f ∗ ∗ (log 1)/F 2 ) and is thus an extension of ν ∈ from a class in H 0 ( B, X¯ / B¯


596

¯ By construction, it is meromorphic along 6 but still H 0 (B, R 2 f ∗ ∗X/B /F 2 ) to B. multivalued with indeterminacies in the local system of integral cohomology classes. Now we apply the Picard-Fuchs operator. This makes g(b) a single-valued complex function on B. DPF has meromorphic (rational) coefficients in b since they are the coefficients of the characteristic polynomial of the Gauss-Manin connection, which has regular singular points along 6 by P. Deligne [6]. Therefore the resulting function g(b) is holomorphic outside 6, but it can have higher order poles along 6. By Chow’s theorem, any meromorphic function on B¯ is rational. Acknowledgments. We are grateful to S. Bloch, H. Esnault, Yu. Manin, and J. Nagel for several valuable discussions in which they urged us to find the solution of the inhomogeneous equation. We also thank N. Narumiya, H. Shiga, and N. Yui for explaining their results and M. Saito for explaining his conjectures about the structure of CH2 (X, 1) and his independent work on the same problem. We thank the referee for suggestions to improve the paper. Finally, we are grateful to Universidad Autonoma de Mexico and Universität Essen for supporting this project. References [1]

[2] [3]

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del Angel Centro de Investigación en Matemáticas, A.P. 402, Guanajuato, Guanajuato C.P. 36000, Mexico; [email protected] Müller-Stach Fachbereich 6 Mathematik/Informatik, Universität Essen, D-45117 Essen, Germany; [email protected]

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