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DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1, THE SPECTRAL BOUND AND PRINCIPAL ¨ EIGENVALUES OF SCHRODINGER OPERAT...

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DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1,

THE SPECTRAL BOUND AND PRINCIPAL ¨ EIGENVALUES OF SCHRODINGER OPERATORS ON RIEMANNIAN MANIFOLDS EL MAATI OUHABAZ

Abstract Given a complete Riemannian manifold M and a Schrödinger operator −1 + m acting on L p (M), we study two related problems on the spectrum of −1+m. The first one concerns the positivity of the L 2 -spectral lower bound s(−1 + m). We prove that if M satisfies L 2 -Poincaré inequalities and a local doubling property, then s(−1 + m) > 0, provided that m satisfies the mean condition Z 1 inf m(x) d x > 0 p∈M |B( p, r )| B( p,r ) for some r > 0. We also show that this condition is necessary under some additional geometrical assumptions on M. The second problem concerns the existence of an L p -principal eigenvalue, that is, a constant λ ≥ 0 such that the eigenvalue problem 1u = λmu has a positive solution u ∈ L p (M). We give conditions in terms of the growth of the potential m and the geometry of the manifold M which imply the existence of L p -principal eigenvalues. Finally, we show other results in the cases of recurrent and compact manifolds. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Poincaré inequalities, the mean condition, and the spectral bound . 2.1. Schrödinger operators with nonnegative potentials . . . . . 2.2. Potentials with positive and negative parts . . . . . . . . . 2.3. Necessity of the mean condition . . . . . . . . . . . . . . 3. Principal eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 3.1. Stability of the essential spectrum . . . . . . . . . . . . . 3.2. Large potentials: Existence of an L 2 -principal eigenfunction 3.3. Small potentials: Existence of an L p -principal eigenfunction DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1, Received 2 December 1999. Revision received 20 October 2000. 2000 Mathematics Subject Classification. Primary 35P05; Secondary 47F05, 58J05. 1

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

2 6 7 12 15 18 18 20 23

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EL MAATI OUHABAZ

3.4. Recurrent manifolds: Nonexistence of L p -principal eigenfunctions 4. Principal eigenvalues on compact manifolds . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

28 31 32

1. Introduction Let M be a complete Riemannian manifold of dimension d M . By ∇ we denote the Riemannian gradient, and by −1 we denote the (positive) Laplace-Beltrami operator on M. It is well known that 1 is self-adjoint on L 2 (M) and has on each L p (M), 1 ≤ p < ∞, a realization 1 p (12 = 1) which generates a strongly continuous semigroup. Let m ∈ L ∞ (M) be a real-valued function. By well-known perturbation arguments, the operator 1 p − m, with domain D(1 p ), is well defined and generates a strongly continuous semigroup. In addition, 1 − m is self-adjoint on L 2 (M). Let s(−1 + m) := inf σ (−1 + m) be the spectral bound of the Schrödinger operator −1 + m. In this paper we investigate the following two questions. Question 1 On which noncompact Riemannian manifolds M can one characterize nonnegative potentials m for which the spectral bound of the Schrödinger operator −1 + m satisfies s(−1 + m) > 0? Question 2 For which Riemannian manifolds M and potentials m does the linear elliptic eigenvalue problem ( 1u = λmu on M, (EVP p ) u > 0, u ∈ D(1 p ), and kuk p = 1 have a solution (λ, u) ∈ [0, ∞) × D(1 p )? These two problems are in some sense related (see Sections 3.2 and 3.4) but are of independent interest. The fact that the spectral bound satisfies s(−1 + m) > 0 gives an exponential decay in time of the solution to the Schrödinger equation    ∂u(t, ·) = 1u(t, ·) − mu(t, ·), t > 0, ∂t  u(0, ·) ∈ L 2 (M). Concerning Question 2, it is known from a fundamental work of P. Hess and T. Kato [HK] that the existence of a solution to the eigenvalue problem gives information on bifurcation of solutions to some nonlinear problems, where (EVP) is obtained by

¨ SCHR ODINGER OPERATORS ON MANIFOLDS

3

linearization. Hess and Kato studied the eigenvalue problem on smooth domains of Rd . Both questions have been studied in the Euclidean case, and one of the aims of the present paper is to find the most general possible class of Riemannian manifolds on which similar results as in the Euclidean case are still valid. First, consider Question 1. We would like to know what quantities control the spectral bound s(−1 + m) and how these quantities depend on both m and M. We show that Poincaré inequalities play an important role in the study of spectral bounds of Schrödinger operators. More precisely, we make the following assumptions on M. The manifold M satisfies the L 2 -Poincaré inequalities (PI2 (R)) (for some R > 0), that is, Z Z 2 2 |u(x) − u B(x0 ,r ) | d x ≤ C(R)r |∇u|2 d x (PI2 (R)) B(x0 ,r )

B(x0 ,r )

for all u ∈ C ∞ (B(x0 , r )) and all 0 ≤ r ≤ R. Here u B(x0 ,r ) is the average of u on the ball B(x0 , r ) (see Section 2). We also assume that the following local doubling property holds: |B(x, 2r )| ≤ C(R)|B(x, r )|,

∀x ∈ M, ∀r ∈ [0, R],

(D(R))

where |B(x, r )| denotes the Riemannian volume of the ball B(x, r ) and C(R) is a constant that might depend on R. The following is a type of results we show in connection to Question 1. THEOREM

Assume that the complete manifold M satisfies (PI2 (R)) and (D(R)) for some R > 0. If m is a nonnegative bounded potential that satisfies the mean condition Z 1 inf m(x) d x > 0, (IC(R)) p∈M |B( p, R)| B( p,R) then s(−1 + m) > 0. If m changes sign with positive part m + that satisfies (IC(R)) and negative part − m that vanishes at infinity, then s(−1 + λm) > 0 for all λ > 0 and small enough. We also study the necessity of the mean condition (IC(R)) in this result. Under some assumption on the manifold, we show that for nonnegative potentials, the property s(−1 + m) > 0 implies that m satisfies (IC(R)) for some R > 0. Our assumption on the manifold holds if, for example, it has nonnegative Ricci curvature (see Section 2.3). We do not know, however, whether the previous result holds without assuming Poincaré inequalities (PI2 (R)).

4

EL MAATI OUHABAZ

Let us mention that our assumptions (PI2 (R)) and (D(R)) are satisfied for a wide class of manifolds. For example, manifolds with Ricci curvature bounded from below satisfy (PI2 (R)) and (D(R)) for all R > 0. The validity of (D(R)) in this situation follows from J. Cheeger, M. Gromov, and M. Taylor [CGT], and (PI2 (R)) follows from P. Buser [Bu] and L. Saloff-Coste [Sa2]. In the Euclidean case where M = Rd (endowed with the flat metric), it is shown by W. Arendt and C. Batty [AB1] that for bounded nonnegative potentials m, s(−1+ m) > 0 is equivalent to the condition Z inf m(x) d x > 0 p∈Rd

B( p,R)

for some R > 0. Our condition (IC(R)) coincides with this condition since |B( p, R)| is independent of p in this particular case. Note also that some related results with different conditions on m are given in G. Metafune and D. Pallara [MePa] in the Euclidean case and in P. Li and S.-T. Yau [LY, Cors. 1.1 and 1.2] in the case of Riemannian manifolds with Ricci curvature bounded from below. The results in [LY] give estimates for s(−1 + m) in terms of inf m(x) and are different from ours. In particular, it is assumed there that 1m and |∇m| satisfy precise growth conditions. Our proof is completely different from the one given in [AB1]. The proof given in this reference consists of estimating the L ∞ -norm of e−t (−1+m) 1 which gives the growth of the semigroup e−t (−1+m) in L ∞ (Rd ) and then uses the fact that this growth is the same as in L 2 (Rd ), a result that was shown by B. Simon [Si] (see also R. Hempel and J. Voigt [HV] for a more general result). In the more general setting of the present paper, one cannot apply this strategy because it may happen that the spectral bound of −1 on L p (M) depends on p (see K.-T. Sturm [Stu] and the references there). The key idea in our proof is the following. We show that if the manifold satisfies (PI2 (R)) and the potential m satisfies (IC(R)), then the operator −1 + m, subject to the Neumann boundary conditions on balls B(y, R), has a spectral bound that is uniformly bounded from below (with respect to y ∈ M). This uniform lower bound, together with a finite covering property of the manifold, provides a lower bound for s(−1 + m). Concerning Question 2, there are several works on this question in the Euclidean case M = Rd and smooth domains of Rd . Hess and Kato [HK] studied the case of the Laplacian with Dirichlet boundary conditions on a smooth bounded domain of Rd and considered the eigenvalue problem on the space of continuous functions. K. Brown, C. Cosner, and J. Fleckinger [BCF], Brown, D. Daners and J. López-Gómez [BDL], and Daners [Da] (see also the references in these papers) considered the eigenvalue problem in Rd when d ≥ 3 and showed that if the potential has fast decay at infinity, then for each p ≥ 2d/(d − 2) there exists a solution (λ, u) ∈ (0, ∞) × L p (Rd ). The situation where d ≤ 2 is different. It is shown in [BCF] that for a class of potentials, the eigenvalue problem has no solution. More recent results in R2 can be found in G.


5

Rozenblum and M. Solomyak [RoS]. Note that more general conditions on m which imply the existence of a solution (λ, u) with bounded and continuous u are given in [Da], but the eigenfunction u is not in L p (Rd ), in general. In [AB2], Arendt and Batty studied the eigenvalue problem on L 2 (Rd ). They showed the existence of a solution (λ, u) to (EVP2 ), provided that the positive part m + of m satisfies s(−1 + m + ) > 0. This is the condition that we investigate in Question 1. We extend all of these results to a more general situation of Riemannian manifolds. We show the existence and uniqueness of a solution to (EVP2 ) if the manifold satisfies (PI2 (R)) and (D(R)) and if m has large positive part and nontrivial negative part. We show that the existence result of [BCF] and [BDL] holds for all manifolds that satisfy a Sobolev inequality Z |∇u|2 d x ≥ ckuk22d/(d−2) M

and potentials m ∈ L d/2 (M) (for some d > 2). We summarize some of our results in the following. THEOREM

(i)

(ii)

Assume that the manifold M has infinite volume and satisfies (PI2 (R)) and (D(R)) for some R > 0. If the positive part m + of m satisfies (IC(R)) and m − vanishes at infinity, then (EVP2 ) has a solution and this solution is unique. If M satisfies a Sobolev inequality (with some d > 2) and m ∈ L d/2 (M), then for all p ∈ [2d/(d − 2), ∞) the problem (EVP p ) has a solution.

We also show a nonexistence result if the manifold is recurrent (see Section 3.4). Finally, we give a complete treatment of the elliptic eigenvalue problem when the manifold M is compact (see Section 4). We want to emphasize that most of our results remain valid, with the same proofs, for more general elliptic operators in divergence form. We have chosen to present them in the case of the Laplacian 1 for the sake of simplicity. Notation. As mentioned above, B( p, r ) denotes the open ball of center p and radius R r . Its Riemannian volume is denoted by |B( p, r )| = B( p,r ) d x. The semigroups generated by 1 and 1 − m are written et1 and e−t (−1+m) . These semigroups are initially defined on L 2 (M) but extend to all L p (M), 1 ≤ p < ∞. We write, if necessary to be specified, the corresponding generators on L p as 1 p and 1 p − m (12 = 1). The norm in L p (M, d x) is written k · k p . By D (M), we denote the space of C ∞ -functions with compact support in M, and H 1 (G) := W 1,2 (G) is the classical L 2 -Sobolev space on a domain G of M. For a given function f acting on M, the notation f 6≡ 0 means that f is a nontrivial function.

6

EL MAATI OUHABAZ

2. Poincaré inequalities, the mean condition, and the spectral bound Let M be a complete Riemannian manifold. Our main assumption on M is the following L 2 -Poincaré inequalities on balls. Let R > 0. We assume that there exists a constant C(R) depending only on R such that for all r ∈ (0, R], all x0 ∈ M, and all u ∈ C ∞ (B(x0 , r )), Z Z |u(x) − u B(x0 ,r ) |2 d x ≤ C(R)r 2 |∇u|2 d x. (PI2 (R)) B(x0 ,r )

B(x0 ,r )

Here u B(x0 ,r ) denotes the average of u on the ball B(x0 , r ), that is, Z 1 u(x) d x. u B(x0 ,r ) := |B(x0 , r )| B(x0 ,r ) We also need the following local doubling property. There exist a radius R > 0 and a positive constant C(R) such that |B(x, 2r )| ≤ C(R)|B(x, r )|,

∀x ∈ M, ∀r ∈ [0, R].

(D(R))

Poincaré inequalities on Riemannian manifolds have attracted the attention of several authors in recent years. It is now well established that Poincaré inequalities and a local doubling property are equivalent to Harnack inequalities (see Saloff-Coste [Sa1], [Sa2] and A. Grigor’yan [Gr2]). Moreover, (PI2 (R)) and (D(R)) imply a family of Sobolev-Poincaré inequalities (see Saloff-Coste [Sa1], [Sa2], P. Hajłasz and P. Koskela [HaKo], P. Maheux and Saloff-Coste [MS], and the references therein). Note also that D. Jerison [Je] has shown that weak Poincaré inequalities (the integral in the right-hand side of (PI2 (R)) is taken over the ball B(x0 , 2r )) and the doubling condition (D(R)) imply (PI2 (R)). In this section we show how to apply Poincaré inequalities to estimate (from below) the spectral bound of a given Schrödinger operator −1 + m. Let −1 be the positive Laplace-Beltrami operator on the manifold M, and consider a potential m = m + − m − , where 0 ≤ m + , m − ∈ L ∞ (M). We consider on M the Schrödinger operator −1 + λm, where λ ∈ R is a coupling constant. This operator is well defined on the domain D(1). We search for conditions on the potential m which imply that the spectral bound s(−1 + λm) := inf σ (−1 + λm) satisfies s(−1 + λm) > 0 for λ > 0 close to zero. As mentioned in the introduction, we show that if the manifold satisfies (PI2 (R)) and (D(R)), and if m is a bounded nonnegative potential, then the mean condition Z 1 inf m(x) d x > 0 (IC(R)) p∈M |B( p, R)| B( p,R) implies the property s(−1 + m) > 0. The converse holds under some additional assumptions on the manifold (see Section 2.3). We also study the case where the


7

potential m changes sign on M and show that if m + satisfies (IC(R)) and if m − is small at infinity, then s(−1 + λm) > 0 for all small λ > 0 (see Section 2.2). 2.1. Schrödinger operators with nonnegative potentials We first consider the case where m = m + . We have the following theorem. THEOREM 1 Suppose that m is a bounded nonnegative potential that satisfies (IC(R)) for some radius R > 0. Assume that the manifold M satisfies (PI2 (R)) and (D(R)). Then s(−1 + m) > 0.

Remark Note that if M satisfies (PI2 (R0 )) and (D(R0 )) for some fixed R0 > 0, then it satisfies (PI2 (R)) and (D(R)) for any fixed R > R0 . This follows from the fact that (PI2 (R0 )) and (D(R0 )) are equivalent to parabolic Harnack inequalities (see [Sa1], [Sa2], [Gr2]). We could then assume in Theorem 1 that M satisfies (PI2 (R0 )) and (D(R0 )) and that m satisfies (IC(R)) for some R0 > 0 and R > 0. In order to prove Theorem 1, we need some auxiliary results. We first recall the following well-known fact. 2 Assume that M satisfies (D(R)) for some R > 0. Then M has the following finite covering property: for each r ∈ (0, R], there exist a sequence (xi )i∈N of points in M and a positive number N (depending on R) such that M = ∪i≥0 B(xi , r ), the balls B(xi , r/2) are disjoint, and each x ∈ M is contained in at most N balls B(xi , r ). PROPOSITION

Proof The proof is classical, but we give details for the reader’s convenience. First, note that if (D(R)) is satisfied for a fixed R > 0, then (D(R 0 )) holds for any given larger R 0 . In order to prove this, it is enough to prove that (D(R)) implies (D(2R)). That is, we need to prove that |B(x, 4r )| ≤ C 0 (R)|B(x, 2r )|,

∀x ∈ M, ∀r ∈ [0, R].

Fix r ∈ [0, R] and x ∈ M. Consider the set N := {(xi ) ∈ M, d(xi , x j ) ≥ r for i 6 = j and d(xi , x) ≤ 3r }.

Using the relation of inclusion, one finds a maximal element (xi ) of N . We have B(x, 4r ) \ B(x, 2r ) ⊆ ∪i B(xi , 2r ).

8

EL MAATI OUHABAZ

Indeed, assume that y ∈ B(x, 4r ) \ B(x, 2r ) and d(xi , y) ≥ 2r for all i. Consider y0 such that d(x, y0 ) = 3r and d(y, y0 ) ≤ r . (This is possible because of the completeness of the manifold; see [GHL, Chap. 2].) We have d(y0 , xi ) ≥ d(y, xi ) − d(y, y0 ) ≥ r. This implies that {y0 , (xi )} ∈ N and contradicts the fact that (xi ) is a maximal element of N . This proves the desired inclusion. Note that by compactness of closed balls we can choose (xi ) to be finite. Using (D(R)), we can write X |B(x, 4r )| ≤ |B(x, 2r )| + |B(xi , 2r )| i

r X r 2 ≤ C(R) B x, + B xi , . 2 2 i

The balls B(xi , r/2) are disjoint, and hence r X r B xi , = ∪i B xi , 2 2 i r ≤ B x, 3r + . 2 We have then shown that 7r |B(x, 4r )| ≤ C 00 (R) B x, . 2 Iterating this, one obtains |B(x, 4r )| ≤ C 0 (R)|B(x, 2r )|. Now we prove the assertions of the proposition. Consider the set M := {(xi ) ∈ M, d(xi , x j ) ≥ r for i 6 = j}.

Using again the relation of inclusion, we obtain a maximal element (xi )i∈I of M . It is clear that B(xi , r/2) are disjoint and M = ∪i∈I B(xi , r ). We show that we can choose (xi ) to be a sequence. Let k ∈ N, and fix x ∈ M. Since the closed ball B(x, k) is compact, there exist xi1,k , . . . , xi N (k),k ∈ {xi , i ∈ I } N (k)

such that B(x, k) ⊂ ∪ j=1 B(xi j,k , r ). This and the equality M = ∪∞ k=1 B(x, k) show that we can choose (xi ) to be a sequence. Now, let Ir (x) = {i ∈ N, x ∈ B(xi , r )}. We want to show that the number of elements of Ir (x) can be estimated uniformly in x. Fix i ∈ Ir (x). One has B(xi , r ) ⊂ B(x, 2r ) ⊂ B(xi , 3r ).


9

Moreover, one has r B xi , ≥ C 0 (R)|B(xi , 3r )| ≥ C 0 (R)|B(x, 2r )|. 2 The first inequality is a consequence of (D(2R)) (which follows from (D(R)) as shown above), and the second one follows from the inclusion B(x, 2r ) ⊂ B(xi , 3r ). Now we have X r X |B(x, 2r )| ≥ |B(x, 2r )|, B xi , ≥ C 0 (R) 2 i∈Ir (x)

i∈Ir (x)

which gives the desired conclusion. In Lemma 3 we prove an estimate for the spectral bound of the Schrödinger operator on balls. Lemma 3 is the core idea in the proof of Theorem 1. Consider a ball B(y, R), and denote by 1 y the Laplacian with Neumann boundary conditions on B(y, R), that is, the operator associated with the minimal closure of the form Z a(u, v) = ∇u · ∇v d x, u, v ∈ C ∞ (B(y, R)). B(y,R)

We now let s y (λ) := s(−1 y + λm) be the spectral bound of the operator −1 y + λm acting on L 2 (B(y, R)). We have the following lemma. 3 Assume that M satisfies (PI2 (R)) and (D(R)) for some R > 0. Let m = m + − m − ∈ L ∞ (M) and satisfy (IC(R)), that is, Z 1 inf m(x) d x > 0. p∈M |B( p, R)| B( p,R) LEMMA

Then there exist two constants δ > 0, > 0 such that for all y ∈ M and all λ ∈ (0, ) we have s y (λ) ≥ δλ. Proof Let us simplify notation and write B y = B(y, R). Note that under (PI2 (R)) and (D(R)), the Neumann Laplacian 1 y has compact resolvent (see Maheux and SaloffCoste [MS, Secs. 5 and 6]). This is equivalent to saying that the embedding D(1 y ) ⊂ L 2 (B y ) is compact.

(2.1)

From the equality D(−1 y + λm) = D(−1) and (2.1), it follows that −1 y + λm has compact resolvent too. Now, let u λ,y ∈ L 2 (B y ) be a nonnegative eigenvector

10

EL MAATI OUHABAZ

of the operator −1 y + λm, which is associated with the eigenvalue s y (λ). (Note that the existence of a nonnegative eigenvector associated with the first eigenvalue s y (λ) follows from the positivity preserving of the semigroup e−t (−1 y +λm) ; this is a classical fact.) We can normalize u λ,y such that u λ,y ≥ 0

on B y

and

(2.2)

ku λ,y k1 = |B y |.

Here and in the rest of this proof, ku λ,y k p denotes the norm of u λ,y in L p (B y ). We know that for all λ ≥ 0 and all y, −1 y u λ,y + λmu λ,y = s y (λ)u λ,y .

(2.3)

Taking the scalar product with the constant function 1 yields Z Z λ mu λ,y = s y (λ) u λ,y = s y (λ)|B y |. By

(2.4)

By

R We see from this equality that we have to estimate B y mu λ,y . In order to do this, we apply the Poincaré inequality (PI2 (R)) to u λ,y . (Note that by (2.2), the average of u λ,y on B y is 1.) We have Z Z Z mu λ,y − m ≤ kmk∞ |u λ,y − 1| By By By "Z #1/2 p ≤ kmk∞ |B y | |u λ,y − 1|2 By

q

≤ kmk∞ |B y

| · C(R)R 2

#1/2

"Z

2

By

|∇u λ,y |

.

p Let us put M(R) = kmk∞ C(R)R 2 . We have shown that Z "Z #1/2 Z p 2 mu λ,k − m ≤ M(R) |B y | |∇u λ,y | . By By By Now, take the scalar product with u λ,y in (2.3) to obtain Z Z 2 |∇u λ,y | + λ m|u λ,y |2 = s y (λ)ku λ,y k22 . By

By

In particular, we have k∇u λ,y k22 ≤ s y (λ)ku λ,y k22 + λkmk∞ ku λ,y k22 .

(2.5)


11

Using this inequality and the fact that s y (λ) ≤ λkmk∞ (which follows immediately from (2.4)), we obtain k∇u λ,y k22 ≤ 2λkmk∞ ku λ,y k22 .

(2.6)

But ku λ,y k22 ≤ 2 ku λ,y − 1k22 + |B y | ≤ 2 C(R)R 2 k∇u λ,y k22 + |B y | . We insert this in (2.6) and see that if 4λC(R)R 2 kmk∞ < 1, then k∇u λ,y k22 ≤

4λkmk∞ |B y | . 1 − 4λC(R)R 2 kmk∞

Now this inequality and (2.5) imply that Z

s

Z By

mu λ,y ≥

m − M(R)|B y | By

4λkmk∞ . 1 − 4λC(R)R 2 kmk∞

Using (IC(R)), we can write that for some constant δ 0 > 0, Z m ≥ δ 0 |B y |, ∀y ∈ M. By

We obtain from the last two inequalities that there exist two constants δ 00 , ε > 0 independent of y (but depending on R) such that Z mu λ,y ≥ δ 00 |B y |, ∀λ ∈ (0, ε). By

We now apply (2.4) to finish the proof of the lemma. Proof of Theorem 1 Let m = m + ∈ L ∞ (M) and satisfy the condition (IC(R)). By Proposition 2, the manifold M has a finite covering by balls B(xk , R), k ∈ N. That is, there exist a sequence (xk )k ∈ M and a number N such that M = ∪k≥0 B(xk , R) and each x ∈ M is contained in at most N balls Bk := B(xk , R). This implies, in particular, that for all u ∈ L 2 (M), Z Z XZ 2 2 |u| ≤ |u| ≤ N |u|2 . (2.7) M

k≥0

Bk

M

Let us denote by 1k the Laplacian with Neumann boundary conditions on the ball Bk . It follows from Lemma 3 that there exist two constants , δ > 0 such that sk (λ) := s(−1k + λm) ≥ λδ,

∀k ≥ 0 and ∀λ ∈ (0, ).

(2.8)

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EL MAATI OUHABAZ

This implies that for λ ∈ (0, ) and all u ∈ D (M), Z Z Z |∇u|2 + λ m|u|2 ≥ λδ Bk

Bk

|u|2 .

(2.9)

Bk

Now taking the sum over k and using (2.7), we obtain that for all u ∈ D (M) and all λ ∈ (0, ), Z Z Z δ |∇u|2 + λ m|u|2 ≥ λ |u|2 . N M M M This inequality extends to all u ∈ H 1 (M) because of the density of D (M) in H 1 (M) (see [Au, Th. 2.6] or [Dav, Th. 5.2.3]). An application of the min-max principle gives the conclusion of Theorem 1. As mentioned in the introduction, the Poincaré-type inequalities (PI2 (R)) and doubling condition (D(R)) are satisfied for all R > 0 if the manifold has Ricci curvature bounded from below (see [Bu], [Sa2], [CGT]). One can then apply Theorem 1 in this situation in order to study the positivity of the spectral bound s(−1 + m) for bounded nonnegative potentials m. We also mentioned above that Theorem 1 was shown in [AB1] in the particular case where M = Rd (endowed with the flat metric). 2.2. Potentials with positive and negative parts In view of Lemma 3 it seems likely that when the potential m satisfies (IC(R)) but changes sign on M, then one can show that s(−1 + λm) > 0 for all λ > 0 small enough. We could, however, show this only under some additional assumption on the negative part of m. Let us recall that a function f , defined on a noncompact manifold M, vanishes at infinity if ∀ > 0 there exist R0 > 0 and x0 ∈ M s.t. | f (x)| < for a.e. x ∈ / B(x0 , R0 ). (2.10) 4 Assume that the manifold M has infinite volume and satisfies (PI2 (R)) and (D(R)) for some R > 0. Let m = m + − m − ∈ L ∞ (M). If m satisfies one of the next two conditions, then there exists > 0 such that THEOREM

s(λ) := s(−1 + λm) > 0 (i) (ii)

for all λ ∈ (0, ).

The potential m + − N m − satisfies (IC(R)), where N is a constant satisfying (2.7). The positive part m + satisfies (IC(R)), and m − vanishes at infinity.


13

Proof Assume that (i) is satisfied. By Lemma 3, there exist ε0 > 0 and δ > 0 such that Z Z Z |∇u|2 + λ (m + − N m − )|u|2 ≥ δλ |u|2 , ∀λ ∈ (0, ε0 ). By

By

By

Now applying Proposition 2, we can write, as in the proof of Theorem 1, that for all u ∈ D (M), Z Z Z Z 1 X |∇u|2 + λ m|u|2 ≥ |∇u|2 + λ (m + − N m − )|u|2 N M M Bk k≥0 Bk Z δ |u|2 ≥λ N M for all λ ∈ (0, ε0 ). We now assume that (ii) is satisfied. Since the result is trivially true when s(−1) > 0, we can assume that s(−1) = 0. Fix x0 ∈ M, ε > 0, and R0 > 0 such that m − (x) < ε for x ∈ / B(x0 , R0 ). Let − B(R0 ) := B(x0 , R0 ) and V R0 := m · χ M\B(R0 ) . It follows from the equality Z s(λ) = inf [|∇u|2 + m|u|2 ] d x, u ∈ H 1 (M), kuk2 = 1 M

that s(λ) ≥

1 1 s − 1 + λ(m + − 2m − · χ B(R0 ) ) + s − 1 + λ(m + − 2V R0 ) . (2.11) 2 2

We first show that the second term in the right-hand side is greater than zero for 0 < λ close to zero. Indeed, since V R0 ≤ ε, then s − 1 + λ(m + − 2V R0 ) ≥ s(−1 + λm + ) − 2λε. This implies that d d s − 1 + λ(m + − 2V R0 ) |λ=0+ ≥ s(−1 + λm + )|λ=0+ − 2ε. dλ dλ Note that the function λ → s(−1 + λm + ) is concave (as the infimum of affine functions). This property and the fact that s(−1 + λm + ) > 0, ∀λ > 0, by Theorem 1, imply that d s(−1 + λm + )|λ=0+ > 0. dλ We now choose ε > 0 such that 2ε < (d/dλ)s(−1 + λm + )|λ=0+ , and we obtain that d s − 1 + λ(m + − 2V R0 ) |λ=0+ > 0. dλ

(2.12)

14

EL MAATI OUHABAZ

This implies that there exists η > 0 such that s − 1 + λ(m + − 2V R0 ) > 0,

∀λ ∈ (0, η).

Let us now show that s(−1 + λ(m + − 2m − · χ B(R0 ) )) ≥ 0 for small λ > 0. Let ( pk ) be a sequence of elements of M such that the balls B( pk , R) are disjoint and M = ∪k≥0 B( pk , 2R) (see the proof of Proposition 2). Since M has infinite P volume, then k≥0 |B( pk , 2R)| = ∞ and because of the assumption (D(R)) we have X |B( pk , R)| = ∞. (2.13) k≥0

Let n be large enough, and let G ⊂ M be a bounded domain with smooth boundary such that n [ B(R0 ) ⊂ B( pk , 2R) ⊂ G. k=0

Since

m+

satisfies (IC(R)), there exists δ > 0 such that Z n Z n X X m + (x) d x ≥ m + (x) d x ≥ δ |B( pk , R)|. G

k=0

B( pk ,R)

k=0

Using (2.13), we see that for n large enough, Z G

(m + − 2m − · χ B(R0 ) )(x) d x ≥ δ

n X

|B( pk , R)| − 2km − k∞ |B(R0 )| > 0. (2.14)

k=0

Let us now denote by 1G the Laplacian with Neumann boundary conditions on G, and let sG (λ) := s − 1G + λ(m + − 2m − · χ B(R0 ) ) . Since B(R0 ) ⊂ G, we have Z

|∇u| + λ 2

M

Z

(m + − 2m − · χ B(R0 ) )|u|2 Z Z Z 2 + − 2 ≥ |∇u| + λ (m − 2m · χ B(R0 ) )|u| ≥ sG (λ) |u|2 .

M

G

G

G

From this we see that the proof is finished if we show that sG (λ) ≥ 0 for all λ > 0 small enough. Since G is bounded and has smooth boundary, then the embedding H 1 (G) ⊂ L 2 (G) is compact (see [Au, p. 55]). This implies, in particular, that 1G has compact resolvent. Using this, it is not difficult to show that Z 1 d + sG (0 ) = (m + − 2m − · χ B(R0 ) ) d x dλ |G| G (see the end of the proof of Theorem 17 or [Ka, p. 405]). We conclude from (2.14) that (d/dλ)sG (0+ ) > 0, which gives the desired conclusion.


15

COROLLARY 5 Assume that M has infinite volume and satisfies (PI2 (R)) and (D(R)) for some R > 0. If m is a bounded potential that satisfies

lim

d(x,x0 )→∞

m(x) exists and is greater than zero

for some fixed point x0 ∈ M, then s(−1 + λm) > 0 for λ > 0 small enough. Proof The assumption on m implies that there exist α > 0 and r > 0 such that m + (x) ≥ α,

∀x ∈ / B(x0 , r ).

Put m 0 = m − + α · χ B(x0 ,r ) . Since m 0 vanishes at infinity, Theorem 4 implies that s(−1 + λ(α − m 0 )) > 0 for all λ ∈ (0, ). Now the inequality m ≥ α − m 0 implies the corollary. 2.3. Necessity of the mean condition Now we turn to the converse of Theorem 1. A natural question is to ask whether (IC(R)) is a necessary condition for m in order to have s(−1 + m) > 0. Of course, for general manifolds this question has a negative answer. The reason is that there are noncompact manifolds for which s(−1) > 0. (This is the case for the hyperbolic space; see [Dav, Sec. 5.7]; see also [Stu] and the McKean result in [Str].) Note also that the previous inequality holds when the manifold is compact with boundary and when −1 is subject to the Dirichlet boundary conditions. We show that condition (IC(R)) is necessary in Theorem 1 if the volume of balls satisfies the condition inf sup

r ≥1 p∈M

|B( p, r + 1)| − |B( p, r )| = 0. |B( p, r )|

(2.15)

Note that this condition implies, in particular, that the volume grows uniformly subexponentially, and hence, by [Br] or [Dav, Th. 5.2.10], the spectral bound satisfies s(−1) = 0. This condition also implies that the spectrum of −1 in L p (M) is independent of p ∈ [1, ∞) (see [Sh], [Stu]). We do not, however, use these results, but our proof is similar to that of [Dav, Th. 5.2.10]. THEOREM 6 Assume that condition (2.15) holds. Let m be a nonnegative bounded potential acting on M. If s(−1 + m) > 0, then m satisfies (IC(R)) for some R > 0.

16

EL MAATI OUHABAZ

Proof For a given p ∈ M and r ≥ 1, let us define   if x ∈ B( p, r ),  1 φ p,r (x) = 1 + r − d(x, p) if r ≤ d(x, p) ≤ r + 1,   0 if d(x, p) ≥ r + 1. Clearly, φ p,r ∈ H 1 (M), and its L 2 -norm satisfies kφ p,r k22 ≥ |B( p, r )|. Let us put 9( p, r ) :=

|B( p, r + 1)| − |B( p, r )| . |B( p, r )|

We have the following estimates: 1

Z

kφ p,r k22

|∇φ p,r (x)|2 ≤ 9( p, r )

(2.16)

M

and 1 kφ p,r k22

Z

1 m(x)|φ p,r (x)| ≤ |B( p, r )| M 2

Z B( p,r )

m(x) d x + kmk∞ 9( p, r ). (2.17)

Now, let ε > 0. By (2.15), there exists r0 ≥ 1 such that for all p ∈ M, 9( p, r0 )(1 + kmk∞ ) < ε. If m does not satisfy (IC(R)) for any R, then there exists p0 ∈ M such that Z m(x) d x ≤ ε|B( p0 , r0 )|. B( p0 ,r0 )

(2.18)

(2.19)

Using (2.18) and (2.19), it follows from (2.16) and (2.17) that the infimum of Z Z 1 1 2 |∇φ p,r (x)| + m(x)|φ p,r (x)|2 kφ p,r k22 M kφ p,r k22 M is zero. This means that s(−1 + m) = 0. COROLLARY 7 The conclusion of Theorem 6 holds if manifold M satisfies one of the following conditions: (i) M has nonnegative Ricci curvature;


(ii)

17

for all p ∈ M and all r > 0, |B( p, r )| := v(r ), where the function v(·) grows subexponentially.

Proof If the manifold has nonnegative Ricci curvature, then by [CGT, Prop. 4.1], or [Dav, p. 167], condition (2.15) holds. Now, assume that (ii) holds. We have inf

r ≥1

v(r + 1) − v(r ) = 0. v(r )

Indeed, if v(r + 1) − v(r ) /v(r ) ≥ λ for all r ≥ 1, then v(n + 1) ≥ (1 + λ)v(n) ≥ (1 + λ)n v(1),

∀n ∈ N, n ≥ 1.

This implies that v(n) ≥ ewn v(1) with w = log(1 + λ). By assumption, v(·) grows subexponentially; hence λ = 0. We finish this section by showing that in the case of compact manifolds, a stronger result holds. Assume that 0 ≤ m ∈ L 1loc (M). The operator −1 + m is understood as the operator associated with the minimal closure of the symmetric form Z Z a(u, v) := ∇u · ∇v d x + muv d x, D(a) = D (M). M

M

We have the following proposition. PROPOSITION 8 Assume that −1 + m has a compact resolvent. (This is the case if, in particular, M is compact.) If m 6≡ 0, then s(−1 + m) > s(−1).

Proof Assume that s(−1 + m) = s(−1), and let u be a nonnegative eigenvector associated with the eigenvalue s(−1 + m). (A nonnegative u exists because of the positivity preserving of the semigroup e−t (−1+m) . The latter follows by applying the well-known Beurling-Deny criteria.) We have (−1 + m)u = s(−1 + m)u = s(−1)u. Since m is nonnegative, we have the following pointwise inequality: e−t·s(−1) u = e−t (−1+m) u ≤ et1 u.

(2.20)

18

EL MAATI OUHABAZ

On the other hand, ket1 uk2 ≤ e−t·s(−1) kuk2 , and we then conclude that et1 u = e−t·s(−1) u. Using this and (2.20), we obtain m · u = 0. It is known that et1 is irreducible, that is, that et1 f (x) > 0 for a.e. x ∈ M for all f ∈ L 2 (M), f ≥ 0, f 6≡ 0 (see [Dav, Th. 5.2.1]). Thus, the equality et1 u = e−t·s(−1) u implies that u(x) > 0 for a.e. x ∈ M. We then conclude from m · u = 0 that m = 0. 3. Principal eigenvalues Let M be a complete Riemannian manifold. By the well-known Beurling-Deny criteria, the semigroup et1 acts on L p (M) for all p ∈ [1, ∞). We denote by D(1 p ) the domain of the corresponding generator on L p (M). Now, let m = m + − m − be a bounded potential acting on M. We consider on L p (M) the following elliptic eigenvalue problem: ( 1u = λmu on M, (EVP p ) u > 0, u ∈ D(1 p ), and kuk p = 1. When there exists a couple (λ, u) with λ ≥ 0 and u satisfying (EVP p ), we say that λ is a principal eigenvalue and that u is a corresponding principal eigenvector (or a principal eigenfunction). Note that if u ∈ L p (M) is a nonnegative and nontrivial solution of 1u = λmu, then u > 0 a.e. on M. This follows from the fact that e−t (−1+λm) u = u and the irreducibility of the semigroup e−t (−1+λm) . The irreducibility of the Schrödinger semigroup e−t (−1+λm) can be shown by using the irreducibility of et1 and the pointwise inequality (3.1) et1 f ≤ e−t (−1+λm−kmk∞ ) f = etkmk∞ e−t (−1+λm) f, which is valid for all 0 ≤ f ∈ L 2 (M) and all t ≥ 0. 3.1. Stability of the essential spectrum Before showing the existence of a principal eigenvector in L 2 , we need some preparation. In Propositions 9 and 10, we give some conditions on a given potential m such that the operators −1 + m and −1 have the same essential spectrum. In the first result we need the following diagonal upper bound on the heat kernel pt (x, y) of −1: pt (x, x) ≤

Cewt √ , |B(x, t)|

∀t > 0, x ∈ M.

(3.2)


19

Here w and C are some constants. Note that this estimate holds under various geometrical assumptions on the manifold (see, e.g., [Dav], [Gr2], [LY], [Sa2], [VSC], and the references therein). For example, it is known that if the Ricci curvature of the manifold is bounded from below, then the above estimate is satisfied. PROPOSITION p 9

Let vt (x) :=

|B(x,

√ t)|. Assume that for all t > 0, vt ∈ L ∞ (M) and pt (x, x) L∞

satisfies (3.2). If for all t > 0, m/vt ∈ L 2 ∩ L ∞ (the closure in L ∞ (M) of L 2 ∩ ∞ t1 2 L ), then the operator me is compact on L (M). Moreover, the operators −1 + m and −1 have the same essential spectra (as operators acting on L 2 (M)). Proof First, note that by the semigroup property and (3.2), we have Z Ce2wt | pt (x, y)|2 dy = p2t (x, x) ≤ . √ |B(x, 2t)| M Now, assume that m/vt ∈ L 2 (M). The kernel m(x) pt (x, y). Using (3.20 ), we have Z Z Z dx |m(x)|2 | pt (x, y)|2 dy ≤ Ce2wt M

of the operator met1 is given by

m(x) 2 d x < ∞, v (x) 2t M

M

(3.20 )

∀t > 0.

This implies that met1 is a Hilbert-Schmidt operator. Hence it is, in particular, compact. L∞

Now if m/vt ∈ L 2 ∩ L ∞ , then there exists a sequence γn,t ∈ L 2 (M)∩L ∞ (M) which converges in L ∞ to m/vt . The assumption that vt ∈ L ∞ implies that the sequence m n,t := γn,t vt converges in L ∞ to m. The inequality km n,t et1 − met1 kL (L 2 ) ≤ km n,t − mk∞ implies that met1 is a uniform limit of compact operators and hence that it is compact. In order to show the second statement, it is sufficient, by using the well-known Weyl’s theorem (cf. [Sc, Th. 4.7]), to show that for some λ > 0, the operator (λ − 1 + m)−1 − (λ − 1)−1 is compact on L 2 (M). This difference of the resolvents can be written as −(λ − 1 + m)−1 m(λ − 1)−1 . Then it suffices to show that for some λ > 0, the operator m(λ − 1)−1 is compact. In order to do this, we write the Laplace transform Z ∞ −1 m(λ − 1) = e−λt met1 dt. 0

But for all t > is a compact operator on L 2 (M), and it is easy to R 1/ε show that the operator in the right-hand side is the limit in norm of ε e−λt met1 dt as ε → 0. This gives the compactness of the desired operator. 0 we know that met1

20

EL MAATI OUHABAZ

In the particular case where the potential m vanishes at infinity, no condition on the heat kernel is needed. More precisely, we have the following proposition. PROPOSITION 10 Assume that m is a bounded potential that vanishes at infinity. Then the operators m(λ − 1)−1 and (λ − 1 + m)−1 − (λ − 1)−1 are compact on L 2 (M). In particular, −1 + m and −1 have the same essential spectrum.

Proof As in the previous proof, it is enough to show that the operator m(λ−1)−1 is compact on L 2 (M). Let u n be a sequence of elements in L 2 (M) which converges weakly to zero. Let vn := (λ − 1)−1 u n and ε > 0. Since m vanishes at infinity, then there exists a bounded open set G with smooth boundary such that |m(x)| < ε for x ∈ / G. Hence we have Z Z Z |mvn |2 ≤ kmk2∞ |vn |2 + ε2 |vn |2 . M

G

M\G

The L 2 -norm of vn is bounded, and vn converges weakly to zero in H 1 (G). From the fact that G has smooth boundary, it follows that the embedding of H 1 (G) into L 2 (G) R is compact (see [Au, p. 55]). This implies that G |vn |2 converges to zero as n → ∞. We then obtain that Z lim sup |mvn |2 ≤ ε2 sup kvn k22 . M

n≥0

This is true for all ε > 0; hence we obtain the desired conclusion. There are more general results on stability of essential spectrum for Schrödinger operators on the Euclidean space (see [ReSi], [Vo], [MePa], and the references therein). A result that describes relative compact perturbations of the Laplcian in L 1 (Rd ) is given in [Vo]. It seems likely that one can show a similar result for more general manifolds. We do not pursue this investigation here. The two results given in this section are sufficient for purposes on principal eigenvalues. Note also that results on essential spectrum of Schrödinger-type operators are given in [Wa]. The so-called super-Poincaré inequalities are introduced there, and it is shown that such inequalities characterize the bottom of the essential spectrum. 3.2. Large potentials: Existence of an L 2 -principal eigenfunction We study in this section the eigenvalue problem in L 2 (M). It is expected that a positive L 2 -eigenfunction exists only at the bottom of the spectrum (see Proposition 11). Thus, when looking for a principal eigenvalue, we should first search for a positive λ1 that satisfies s(−1 + λ1 m) = 0. The remaining work is to show that zero is an eigenvalue of the operator −1 + λ1 m in L 2 (M). We show that if the positive part m +


21

of m is large (in the sense that it satisfies the condition (IC(R))) and if the manifold satisfies Poincaré inequalities, then (EVP2 ) has a solution and this solution is unique. This result was shown in [AB2] in the particular case where M = Rd (endowed with the flat metric). As above, we assume that M is a complete manifold. PROPOSITION 11 Assume that the eigenvalue problem (EVP2 ) has a solution (λ1 , u) such that u is bounded and continuous on M. Then s(−1 + λ1 m) = 0.

Proof Let (λ1 , u) be as in the proposition, and let v ∈ D (M). Write v = u · ψ. It is easy to see that Z Z Z |∇(uψ)|2 d x = ∇u · ∇(ψ 2 u) d x + u 2 |∇ψ|2 d x. (3.3) M

M

M

In addition, ψ 2 · u ∈ H 1 (M). It follows from the definition of 1 that Z Z ∇u · ∇(ψ 2 u) d x = (−1u)uψ 2 d x. M

M

This, together with (3.3) and the fact that 1u = λ1 mu, implies that Z Z |∇(uψ)|2 d x + λ1 m(uψ)2 d x ≥ 0. M

M

We have shown that Z

|∇v|2 d x + λ1 M

Z

mv 2 d x ≥ 0. M

This inequality extends to all v ∈ H 1 (M). (Remember that D (M) is dense in H 1 (M).) Consequently, s(−1 + λ1 m) ≥ 0. But the existence of u ∈ L 2 (M) which satisfies 1u = λ1 mu implies s(−1 + λ1 m) = 0. The argument of using formula (3.3) in the previous proof was mentioned to the author by Gilles Carron. 12 Suppose that M has infinite volume and satisfies (PI2 (R)) and (D(R)) for some R > 0. Let m = m + − m − ∈ L ∞ (M) be such that (i) m + satisfies (IC(R)), (ii) m − 6 ≡ 0 and vanishes at infinity. THEOREM

22

EL MAATI OUHABAZ

Then there exists a unique principal eigenvalue with an eigenfunction u ∈ L 2 (M). That is, there exists a unique λ ≥ 0 such that (EVP2 ) has a solution and this solution is unique. If s(−1) > 0, then the same conclusion holds without assuming (i). Proof (a) By Theorem 4, there exists ε > 0 such that s(λ) := s(−1 + λm) > 0

for all λ ∈ (0, ).

(3.4)

(b) Since m − 6≡ 0, then there exists u ∈ D (M) such that kuk2 = 1, u · m + = 0, and u · m − 6≡ 0. Hence Z Z 2 s(λ) ≤ |∇u| − λ m − |u|2 . M

M

This implies that s(λ) → −∞ as λ → +∞. On the other hand, the function λ → s(λ) is a concave function. It follows from these properties and (3.4) that there exists a unique λ1 > 0 such that s(λ1 ) = 0. (c) We show that λ1 is a principal eigenvalue. It follows from the second resolvent equation that (λ − 1 + λ1 m + )−1 − (λ − 1)−1 = −(λ − 1)−1 λ1 m + (λ − 1 + λ1 m + )−1 and that for all λ > 0 and f ∈ L 2 (M), f ≥ 0, we have 0 ≤ (λ − 1 + λ1 m + )−1 f ≤ (λ − 1)−1 f. Proposition 10 asserts that m − (λ − 1)−1 is a compact operator. The last inequality then implies that the operator m − (λ − 1 + λ1 m + )−1 is compact too. In particular, the operators −1 + λ1 m + − λ1 m − and −1 + λ1 m + have the same essential spectrum. Since s(−1 + λ1 m + ) > 0 and s(−1 + λ1 m) = 0, it follows that zero is an eigenvalue of the operator −1 + λ1 m as an operator acting on L 2 (M). Then there exists a nontrivial u ∈ L 2 (M) such that 1u = λ1 mu. This is equivalent to e−t (−1+λ1 m) u = u for all t ≥ 0. From the positivity preserving of the semigroup e−t (−1+λ1 m) , it follows that |u| ≤ e−t (−1+λ1 m) |u|. Now the self-adjointness of e−t (−1+λ1 m) and the fact that s(−1 + λ1 m) = 0 imply that ke−t (−1+λ1 m) |u|k2 ≤ kuk2 . This inequality, together with the previous one, implies that e−t (−1+λ1 m) |u| = |u|.


23

Finally, the fact that |u(x)| > 0 on M follows from the irreducibility of the Schrödinger semigroup e−t (−1+λ1 m) , as explained in the beginning of this section. Note also that the irreducibility of e−t (−1+λ1 m) implies the uniqueness of positive normalized eigenfunction (see, e.g., [Dav, Prop. 1.4.3]). (d) We show the uniqueness of the principal eigenvalue. Because of step (b), we only have to show that if λ ≥ 0 and if (λ, v) is a solution to (EVP2 ), then s(λ) = 0. For such λ, we have s(λ) ≤ 0. Assume that s(λ) < 0. As explained in (c), Proposition 10 implies that σess (−1 + λm) = σess (−1 + λm + ). Hence s(λ) is an eigenvalue of −1 + λm. There exists a corresponding strictly positive eigenvector φ ∈ L 2 (M). We have Z Z Z s(λ) φ·v = (−1 + λm)φ · v = φ · (−1 + λm)v = 0, M

M

M

which is not possible because both v and φ are strictly positive. This finishes the proof of the theorem. Note that if the semigroup et1 is bounded from L 2 (M) into L ∞ (M), then we obtain a principal eigenvector in L p (M) for all [2, ∞) and hence a solution to (EVP p ). This follows from the fact that e−t (1+λm) is also bounded from L 2 (M) into L ∞ (M) (remember the pointwise inequality e−t (−1+λm) f ≤ eλtkmk∞ et1 f, f ≥ 0) and the fact that the principal eigenfunction u satisfies u = e−t (−1+λm) u. 3.3. Small potentials: Existence of an L p -principal eigenfunction We gave in Section 3.2 a condition on the potential which guarantees the existence of a principal eigenvector u ∈ L 2 (M). In this section we study the case of “small” potentials. (The positive part m + does not satisfy (IC(R)).) We obtain a principal eigenvector in L p (M) for some values of p. Before stating our result, let us mention that it was shown in [BCF] and [BDL] that in the particular case where M = Rd with d ≥ 3 and m satisfies |m(x)| ≤

C (1 + |x|2 )α

(3.5)

for some constants α > 1 and C > 0, there exists a principal eigenvalue with an eigenvector which belongs to L 2d/(d−2) (Rd ). We extend this result by showing that if the manifold M satisfies a Sobolev inequality and m ∈ L d/2 , then there exists a principal eigenvector u ∈ L 2d/(d−2) (M). Here we say that M satisfies a Sobolev inequality if there exists a constant d > 2 such that Z |∇u|2 ≥ Ckuk22d/(d−2) , ∀u ∈ H 1 (M), (3.6) M

24

EL MAATI OUHABAZ

where C is a positive constant. It is well known that (3.6) is equivalent to the following estimate on the heat kernel pt (x, y) of −1: pt (x, y) ≤ Ct −d/2 ,

∀t > 0, x, y ∈ M

(3.7)

(see [Dav], [VSC]). The Sobolev inequality plays a fundamental role in our next result. THEOREM 13 Suppose that M satisfies the Sobolev inequality (3.6) with some d > 2. Suppose that the potential m ∈ L ∞ (M) ∩ L d/2 (M) with negative part m − 6≡ 0. Then there exists a principal eigenvalue λ1 > 0 with a corresponding eigenvector u ∈ L 2d/(d−2) (M) ∩ L ∞ (M). In particular, for all p ∈ [2d/(d − 2), ∞), the problem (EVP p ) has a solution (λ1 , u) ∈ (0, ∞) × L p (M).

Proof In this proof we follow similar ideas as in [BCF] and [BDL]. In these papers the potential m is assumed to satisfy (3.5) in order to apply Hardy’s inequality. Here we show that under the Sobolev inequality (3.6), the condition m ∈ L d/2 suffices to apply the strategy of proof given in these papers. (a) Let B(xk , 1), k ≥ 1, be such that M = ∪k≥1 B(xk , 1). Define on M the function X 1 f (x) = χ B(xk ,1) , k 2 |B(xk , 1)|2/d k≥1

where χ B(xk ,1) denotes the characteristic function of the ball B(xk , 1). It is easy to see that f ∈ L d/2 (M, d x), f > 0 on M, and 1/ f ∈ L ∞ loc (M, d x). Now, put φ := |m|+ f . Hence φ satisfies the same properties as f , namely, 0 < φ ∈ L d/2 (M, d x) and 1/φ ∈ L ∞ loc (M, d x). Let H := L 2 (M, φ(x) d x), and define on H the symmetric form Z a(u, v) := ∇u · ∇v d x, D(a) = H 1 (M). (3.8) M

The form a is closable. In order to prove this, we have to show that if u k ∈ H 1 (M), u k → 0 in H (as k → ∞) and a(u k − u j , u k − u j ) → 0 (as k, j → ∞), then a(u k , u k ) → 0 as k → ∞ (see [Ka, Th. 1.17, Chap. 6]). Fix B any ball in M. It follows from the fact that 1/φ ∈ L ∞ loc (M, d x) that there exists a constant c B such that Z Z 2 |u k | d x ≤ c B |u k |2 φ(x) d x → 0, k → ∞. (3.9) B

M


25

On the other hand, the sequence ∇u k is a Cauchy sequence in L 2 (M, d x); then it converges to v ∈ L 2 (M, d x). This implies that the sequence ∇u k|B is a Cauchy sequence in L 2 (B, d x). It follows from this and (3.9) that the sequence u k (restricted to B) is a Cauchy sequence in H 1 (B). Using (3.9) again, we conclude that v = 0 on B. Since B is an arbitrary ball, this implies that v = 0. This means that a(u k , u k ) → 0 as k → ∞, and the claim is then shown. Denote again by a, with domain V , the closure of the previous form. By definition we have u ∈ V ⇔ ∃(u k )k ∈ H 1 (M) s.t. u k → u in H and a(u k − u j , u k − u j ) → 0, k, j → ∞. We need the following properties of V : 1 u ∈ V ⇒ u ∈ Hloc (M)

(3.10)

and a(u, u) ≥

Ckuk22d/(d−2)

≥C

0

Z

|u| φ(x) d x ≥ C 2

0

Z

M

|u|2 |m(x)| d x

(3.11)

M

for all u ∈ V . Here C and C 0 are positive constants. The first inequality in (3.11) follows from the Sobolev inequality (3.6). The second one follows easily by applying Hölder’s inequality since φ ∈ L d/2 (M, d x). We now show (3.10). Let u k ∈ H 1 (M), u k → u in H , and ∇u k is a Cauchy sequence in L 2 (M, d x). The condition 1/φ ∈ L ∞ loc (M, d x) implies that for any ball 2 B ⊂ M, u k|B is a Cauchy sequence in L (B, d x). This, together with the fact that ∇u k|B is also a Cauchy sequence in L 2 (B, d x), implies that u k|B is convergent in H 1 (B). We then obtain that u |B ∈ H 1 (B) and hence obtain (3.10). The space V is complete with respect to the norm kukV := a(u, u) + kuk2H

1/2

Z

|∇u|2 d x +

:= M

Z

|u|2 φ(x) d x

Z

|∇u|2 d x

1/2

,

√ a(·, ·). Because of this,

∀u ∈ V.

M

(b) Define the symmetric form b(u, v) := −

Z muv d x, M

.

M

As a consequence of (3.11), the norm k · kV is equivalent to we can change our notation and write kukV :=

1/2

D(b) = V.

(3.12)

26

EL MAATI OUHABAZ

The form b is well defined and is continuous. Indeed, by Cauchy-Schwarz inequality, Z 1/2 Z 1/2 2 2 |b(u, v)| ≤ |m||u| |m||v| , M

M

and by Hölder’s inequality (or (3.11)), Z |m||u|2 ≤ kmkd/2 kuk22d/(d−2) . M

We conclude from this and (3.11) that |b(u, v)| ≤ CkukV kvkV .

(3.13)

Hence b is a bounded symmetric and continuous form on V . It follows that there exists a bounded operator T acting on V such that b(u, v) = (T u, v)V ,

∀u, v ∈ V,

(3.14)

where (u, v)V := a(u, v) is the scalar product of V . (Here we use (3.12).) (c) We show that T is a compact operator on V . Let (u k ) be a bounded sequence in V . Fix G, an open bounded smooth domain of M. As in the proof of (3.10), the sequence (u k|G ) is a bounded sequence in the Sobolev space H 1 (G). The embedding H 1 (G) into L 2 (G, d x) is compact (see [Au]). Hence after extracting a subsequence we can assume that u k|G is convergent in L 2 (G, d x). In particular, this sequence is a Cauchy sequence in L 2 (G, |m| d x). (Remember that m ∈ L ∞ (M, d x).) We now show that u k is a Cauchy sequence in L 2 (M \ G, |m| d x). We have Z |u k − u j |2 |m| d x ≤ kmk L d/2 (M\G) ku k − u j k22d/(d−2) . M\G

By the Sobolev inequality (3.6) and the fact that the sequence u k is bounded in V , we obtain that for some constant C, Z |u k − u j |2 |m| d x ≤ Ckmk L d/2 (M\G) , ∀k, j ≥ 0. M\G

But the term in the right-hand side vanishes as G increases to M. We have then shown that u k is a Cauchy sequence in L 2 (M, |m| d x). In order to finish the proof of the compactness of T , we write kT u k − T u j kV = T (u k − u j ), T (u k − u j ) V = b u k − u j , T (u k − u j ) Z =− m(x)(u k − u j )T (u k − u j ) d x M

Z

|m||u k − u j |2

≤ M

1/2 Z

|m||T (u k − u j )|2 M

1/2

.


27

R It follows from (3.11) that the term M |m||T (u k −u j )|2 can be estimated by the norm in V of T (u k − u j ) which is then bounded in k and j. The previous inequality and the fact that u k is a Cauchy sequence in L 2 (M, |m| d x) imply that (T u k ) is a Cauchy sequence in V . This proves that T is a compact operator on V . (d) The largest eigenvalue of T is given by R − M m|u|2 b(u, u) = sup , µ = sup a(u, u) u∈V,u6 =0 u∈V,u6 =0 a(u, u) and since by assumption m − 6≡ 0, it follows that µ > 0. It is well known that |u| ∈ H 1 (M) for all u ∈ H 1 (M) and ∇|u| = |∇u|. Using this and the definition of V , it is not hard to see that |u| ∈ V for all u ∈ V and a(|u|, |u|) = a(u, u). Now, let u ∈ V be a nontrivial eigenvector of T associated with the eigenvalue µ. The above equality a(|u|, |u|) = a(u, u) implies that R R − M m|u|2 − M m|u|2 = . a(u, u) a(|u|, |u|) This implies that |u| is also an eigenvector of T with eigenvalue µ. (e) We finish the proof of the theorem by showing that v := |u| is a principal eigenvector with a principal eigenvalue λ1 := 1/µ. First, note that by (3.11), v ∈ L 2d/(d−2) (M, d x). Moreover, it satisfies Z Z ∇v · ∇ψ d x + λ1 mvψ d x = 0, ∀ψ ∈ V. (3.15) M

M

In particular, (3.15) holds for all ψ ∈ D (M). Denote, as previously, by 1q the realization of 1 in L q (M, d x). For 1 < q < ∞, the space D (M) is a core of −1q (see [Str, Th. 3.5 and its proof]). This means that ∀u ∈ D(1q ), ∃u n ∈ D (M) : u n → u in L q (M, d x) and 1u n → 1q u.

(3.16)

Let p = 2d/(d − 2), and let q be its conjugate number. Fix u ∈ D(−1q + λ1 m) = D(−1q ), and let u n be as in (3.16). We have from (3.15), Z v · (−1q + λ1 m)u n d x = 0. M

But m ∈

L ∞ (M, d x);

hence when n → ∞, Z v(−1q + λ1 m)u d x = 0. M

Since this is true for all u ∈ D(−1q + λ1 m), we obtain v ∈ D(−1 p + λ1 m)

and

(−1 p + λ1 m)v = 0.

(3.17)

28

EL MAATI OUHABAZ

It follows from this that for all t ≥ 0, e−t (−1 p +λ1 m) v = v, and again by irreducibility of the semigroup e−t (−1 p +λ1 m) we obtain v > 0 on M. Thus we have shown that λ1 is a principal eigenvalue with a principal eigenvector v ∈ L 2d/(d−2) (M, d x). Finally, as mentioned above, the Sobolev inequality (3.6) implies that the semigroup et (1 p ) maps L p (M, d x) into L ∞ (M, d x), and this property holds also for the semigroup e−t (−1 p +λ1 m) because m ∈ L ∞ (M, d x). It then follows that v ∈ L 2d/(d−2) (M, d x) ∩ L ∞ (M, d x). Consequently, there exists a principal eigenvector in L p (M, d x) for all p ∈ [2d/(d − 2), ∞). Remarks (1) One can give a different proof to the previous result by showing that the selfadjoint operator (−1)−1/2 m(−1)−1/2 is compact on L 2 (M). The compactness of this operator is shown in [Car]. I owe this observation to Gilles Carron. (2) Continuity of the principal eigenfunction. Assume that (λ1 , u) is a solution of (EVP p ) for some p ∈ [1, ∞). If the semigroup et1 satisfies et1 1 = 1 and u ∈ L ∞ (M), then u ∈ Cb (M) (i.e., u is bounded and continuous on M). Indeed, the assumption et1 1 = 1 implies that et1 L ∞ (M) ⊂ Cb (M) (see [Dav, Cor. 5.2.7]). Now [OSSV, Th. 4.1] asserts that e−t (−1+λ1 m) L ∞ (M) ⊂ Cb (M). The continuity of u follows from this and the fact that u = e−t (−1+λ1 m) u and u ∈ L ∞ (M). (3) Principal eigenfunctions in L ∞ (M). The problem (EVP∞ ), in which 1∞ denotes the adjoint operator of 11 , is very different from the case of a finite p. In fact, if et1 1 = 1, then for a large class of potentials m = m + − m − , every λ ≥ 0 is a principal eigenvalue with eigenvector in L ∞ (M). This follows in a more general situation from [MO, Ths. 3.3. and 4.1]. 3.4. Recurrent manifolds: Nonexistence of L p -principal eigenfunctions In this section we say that the manifold M (or the semigroup et1 ) is recurrent if Z ∞ et1 f (x) dt = ∞, a.e. x ∈ M, for some f ∈ L 1 (M, d x), f ≥ 0, f 6 ≡ 0. 0

(3.18) Note that by irreducibility of the semigroup et1 , if (3.18) holds for some f , then it holds for all f ∈ L 1 (M)+ L ∞ (M), f ≥ 0, f 6 ≡ 0 (see, e.g., [FOT]). As an example, it is well known that the Gaussian semigroup et1 acting on L 2 (Rn ) is recurrent if n ≤ 2. In Riemannian manifolds there are known conditions (in terms of the growth of the volume) which imply the recurrence property (see [Gr1]). Our aim here is to show that if the manifold is recurrent, then for a large class of potentials m, there exists no principal eigenvector in any L p -space for p < ∞. We first prove the following result.


29

PROPOSITION 14 Assume that the manifold is recurrent, and let m ∈ L 1 (M) ∩ L ∞ (M), m 6≡ 0. If R M m(x) d x ≤ 0, then s(−1 + λm) < 0 for all λ > 0. In particular, there is no principal eigenvalue with eigenfunction u ∈ L 2 (M) ∩ ∞ L (M).

Proof The proof of the first assertion is taken from [Ou]. First, the recurrence property implies that there exists a sequence u n ∈ H 1 (M) such that 0 ≤ u n ≤ 1 (a.e.) and Z u n → 1 (a.e.) and |∇u n |2 d x → 0 M

(see [FOT]). Assume that for some λ > 0, s(−1 + λm) = 0. Hence the quadratic form Z Z a(u, u) = |∇u|2 d x + λ m|u|2 d x M

M

is nonnegative. The Cauchy-Schwarz inequality implies that |a(u n , φ)|2 ≤ a(u n , u n )a(φ, φ). R The property of u n and the condition M m(x) d x ≤ 0 imply that a(u n , u n ) → 0. Hence a(u n , φ) → 0 for all φ ∈ H 1 (M). Using again the property of the sequence R u n , we obtain M mφ d x = 0. Since φ is arbitrary, this gives m ≡ 0, which is a contradiction. In order to prove the last assertion, let us note that the recurrence property implies the conservation property et1 1 = 1 (see [FOT]). If there exits a principal eigenfunction u ∈ L 2 ∩ L ∞ , then u ∈ Cb (M) (see the remark at the end of Section 3.3) and we conclude, using Proposition 11, that s(−1 + λ1 m) = 0 for some λ1 > 0. The first assertion shows that this cannot hold. 15 R Assume that M is recurrent and M m(x) d x ≤ 0. Assume in addition that et1 is bounded from L 2 (M) into L ∞ (M) for some t > 0. Then for all p ∈ [1, 2), there exists no principal eigenvalue with eigenvector in L p (M). COROLLARY

This corollary is a simple application of Proposition 14 since if (λ1 , u) is a solution of (EVP p ), then e−t (1+λ1 m) u is a principal eigenvector that belongs to L 2 (M)∩L ∞ (M). If we make some more assumptions on m, then we can show that there is no principal eigenvector in L p (M) for p ∈ [2, ∞). More precisely, we assume that for

30

EL MAATI OUHABAZ

some λ > 0, (λ − 1 + m)−1 − (λ − 1)−1 is a compact operator on L 2 (M).

(3.19)

Conditions that imply this property are given in Section 2.1. In particular, (3.19) is satisfied if m ∈ L 2 (M) ∩ L ∞ (M) and the Sobolev inequality (3.6) is satisfied. It is easy to see by interpolation that the difference of the resolvents in (3.19) is a compact operator in L p (M) for 1 < p < ∞. PROPOSITION 16 Assume that M is recurrent. Let m ∈ L 1 (M) ∩ L ∞ (M) which satisfies (3.19), and R let M m(x) d x ≤ 0. Then for all p ∈ [2, ∞), there exists no principal eigenvector in L p (M). In other words, the eigenvalue problem (EVP p ) has no solution.

This result was shown in [BCF] in the Euclidean case where M = Rn with n ≤ 2, R under the condition M m(x) d x < 0. Our method is different and can be applied to more general operators than −1. Proof Assume that for some p ∈ [2, ∞), there exists (λ1 , u) a solution to (EVP p ). Hence u ∈ D(−1 p ) and −1u + λ1 mu = 0. Let q denote the conjugate number of p. By Proposition 14, we have s(−1 + λ1 m) < 0. This implies that s(−1q + λ1 m) < 0. Indeed, if s(−1q + λ1 m) ≥ 0, then for all ε > 0 there exists a constant Cε such that ke−t (−1q +λ1 m) kL (L q ) ≤ Cε eεt . The same estimate holds for the adjoint operator, and by the Riesz-Thorin interpolation theorem, this estimate holds on L 2 . This means that s(−1 + λ1 m) ≥ 0, which is not the case. Now assumption (3.19) implies that the operators −1q + λ1 m and −1q have the same essential spectra (see [Sc, Th. 4.7]). Since s(−1q ) ≥ 0, it follows that s(−1q + λ1 m) is not in the essential spectrum σess (−1q + λ1 m) of the operator −1q + λ1 m. We claim that s(−1q + λ1 m) is an eigenvalue of −1q + λ1 m and has a positive eigenvector. In order to show this claim, let µn ∈ (−∞, s(−1q + λ1 m)) which converges to s(−1q + λ1 m), and let φ ∈ L q (M) such that k(µn + 1q − λ1 m)−1 φkq → ∞ as n → ∞.

(3.20)

This is possible because s(−1q + λ1 m) ∈ σ (−1q + λ1 m), and hence the sequence k(µn + 1q − λ1 m)−1 kL (L q ) → ∞. Such φ exists then by the uniform boundedness theorem. The positivity preserving of the operator (−µn − 1q + λ1 m)−1 implies (−µn − 1q + λ1 m)−1 φ ≤ (−µn − 1q + λ1 m)−1 |φ|.


31

This implies (3.20) with |φ| in place of φ. Now a simple calculation shows that the sequence f n := (−µn − 1q + λ1 m)−1 |φ| /k(−µn − 1q + λ1 m)−1 |φ|kq satisfies (s(−1q + λ1 m) − (−1q + λ1 m)) f n → 0 as n → ∞. But the sequence f n has a convergent subsequence, otherwise s(−1q + λ1 m) ∈ σess (−1q + λ1 m) (see [Sc, Th. 4.4]). If f denotes the limit of this subsequence, then k f kq = 1 and f > 0 (a.e.) on M and (−1q + λ1 m) f = s(−1q + λ1 m) f . This shows the claim. We can now finish the proof of the proposition. Let u and f be as above. We have Z Z s(−1q + λ1 m) f (x)u(x) d x = (−1q + λ1 m) f (x)u(x) d x M M Z = f (x)(−1 p + λ1 m)u(x) d x = 0. M

But both f and u are positive on M and s(−1q + λ1 m) < 0. Hence the last equality cannot hold. This shows the proposition. 4. Principal eigenvalues on compact manifolds We assume in this section that the manifold M is compact. If the manifold M has boundary, then 1 is subject to Dirichlet or Neumann boundary conditions. In the first case s(−1) > 0, and in the second one s(−1) = 0. Let m ∈ L ∞ (M) be a nontrivial potential. We study the existence of a solution (λ1 , u) of the eigenvalue problem (EVP2 ). Note that if (λ1 , u) is a solution of (EVP2 ), then (λ1 , e−t (−1+λ1 m) u) is a solution to (EVP p ). The reason for that is the fact that for compact manifolds, the Sobolev inequality (3.6) holds and this implies that the semigroup et1 is bounded from L 2 (M) into L ∞ (M) (see [Dav], [VSC]). The same holds for e−t (−1+λ1 m) because m ∈ L ∞ (M). In particular, the eigenfunction u is in L 2 (M) ∩ L ∞ (M). For the same reason and because of the fact that M has finite volume, if (EVP p ) has a solution for some p ∈ [1, ∞), then (EVP2 ) has a solution too. We then obtain that a principal eigenfunction u of (EVP p ) for some p ∈ [1, ∞) satisfies u ∈ L 1 (M) ∩ L ∞ (M). Because of this, we only have to study principal eigenvalues with corresponding eigenvector in L 2 (M). Note that in the case where s(−1) = 0, zero is a principal eigenvalue with a corresponding principal eigenvector a constant function. We show that in this case it is sometimes possible to find another principal eigenvalue λ1 > 0. The following result describes all that can happen in the case of compact manifolds. 17 Assume that s(−1) > 0. Then there exists a principal eigenvalue λ1 > 0 if and only if m − 6≡ 0. Assume now that s(−1) = 0.

THEOREM

(i) (ii)

32

EL MAATI OUHABAZ

(1) (2)

R If M m(x) d x ≤ 0, then there is no principal eigenvalue in (0, ∞). R If M m(x) d x > 0, then there exists a principal eigenvalue in (0, ∞) if and only if m − 6≡ 0.

Proof (i) We assume that s(−1) > 0. If m − 6 ≡ 0, then as we have seen previously, s(λ) := s(−1 + λm) → −∞ as λ → +∞. Hence there exists a unique λ1 > 0 such that s(λ1 ) = 0. Since M is compact, it follows that the operator −1 + λm has compact resolvent. Hence zero is an eigenvalue of −1 + λ1 m. The existence of a positive principal eigenfunction can be shown as in Section 3. If m − = 0, then s(λ) > 0 for all λ > 0 and it follows that there exists no principal eigenvalue. (ii) Assume in the remainder of this proof that s(−1) = 0. It follows then that M R∞ is recurrent (because 0 et1 f dt = ∞ with the constant function f = 1 ∈ L 1 (M)). R If M m(x) d x ≤ 0, then Proposition 14 asserts that there is no principal eigenvalue in (0, ∞). R Assume now that M m(x) d x > 0. Then there exists > 0 such that s(λ) > 0,

∀λ ∈ (0, ).

(4.1)

Indeed, let u λ be a normalized eigenvector of −1 + λm with eigenvalue s(λ), and write for all λ > 0, Z Z Z s(λ) uλ d x = (−1 + λm)u λ d x = λ mu λ d x. (4.2) M

M

M

From the compactness of the embedding of D(−1) into L 2 (M), we can find a sequence u λn that converges in L 2 (M). Let v with kvk2 = 1 be the limit of this sequence. Since s(λ) → 0 as λ → 0, it follows easily that v = |M|−1/2 . Hence when λ → 0 in (4.2), we obtain that the right derivative of s(·) at zero satisfies R s 0 (0+ ) = |M|−1 M m(x) d x. This shows that s 0 (0+ ) > 0 and gives (4.1). Now if m − = 0, then s(λ) > 0 for all λ > 0 and it follows that there is no principal eigenvalue in (0, ∞). If m − 6≡ 0, then there exists a unique λ1 > 0 such that s(λ1 ) = 0. As we have seen previously, this implies that λ1 is a principal eigenvalue. Acknowledgments. I wish to express my thanks to Daniel Daners for explaining to me some of his results on principal eigenvalues and to Gilles Carron and Thierry Coulhon for their comments and help during the preparation of this paper. References [Al]

W. ALLEGRETTO, Principal eigenvalues for indefinite-weight elliptic problems in Rn ,

Proc. Amer. Math. Soc. 116 (1992), 701–706. MR 93a:35114


[AB1]

[AB2] [Au] [Br] [BCF]

[BDL]

[Bu] [Car] [CGT]

[CS] [Da]

[Dav]

[Doy] [DR]

[FOT]

[GHL] [Gr1] [Gr2]

33

W. ARENDT and C. J. K. BATTY, Exponential stability of a diffusion equation with

absorption, Differential Integral Equations 6 (1993), 1009–1024. MR 94k:35038 4, 12 , The spectral function and principal eigenvalues for Schrödinger operators, Potential Anal. 7 (1997), 415–436. MR 98g:35148 5, 21 T. AUBIN, Nonlinear Analysis on Manifolds: Monge-Ampère Equations, Grundlehren Math. Wiss. 252, Springer, New York, 1982. MR 85j:58002 12, 14, 20, 26 R. BROOKS, A relation between growth and the spectrum of the Laplacian, Math. Z. 178 (1981), 501–508. MR 83a:58089 15 K. J. BROWN, C. COSNER, and J. FLECKINGER, Principal eigenvalues for problems with indefinite weight function on R N , Proc. Amer. Math. Soc. 109 (1990), 147–155. MR 90m:35140 4, 5, 23, 24, 30 ´ ´ K. J. BROWN, D. DANERS, and J. LOPEZ-G OMEZ , Change of stability for Schrödinger semigroups, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 827–846. MR 96h:35035 4, 5, 23, 24 ´ P. BUSER, A note on the isoperimetric constant, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 213–230. MR 84e:58076 4, 12 G. CARRON, L 2 -cohomologie et inégalités de Sobolev, Math. Ann. 314 (1999), 613–639. MR 2000f:53045 28 J. CHEEGER, M. GROMOV, and M. TAYLOR, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15–53. MR 84b:58109 4, 12, 17 T. COULHON and L. SALOFF-COSTE, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), 293–314. MR 94g:58263 D. DANERS, Principal eigenvalues for some periodic-parabolic operators on R N and related topics, J. Differential Equations 121 (1995), 293–313. MR 96h:35080 4, 5 E. B. DAVIES, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge Univ. Press, Cambridge, 1989. MR 90e:35123 12, 15, 17, 18, 19, 23, 24, 28, 31 P. G. DOYLE, On the bass note of a Schottky group, Acta Math. 160 (1988), 249–284. MR 90b:30053 X. T. DUONG and D. W. ROBINSON, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), 89–128. MR 97j:47056 ¯ M. FUKUSHIMA, Y. OSHIMA, and M. TAKEDA, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math. 19, de Gruyter, Berlin, 1994. MR 96f:60126 28, 29 S. GALLOT, D. HULIN, and J. LAFONTAINE, Riemannian Geometry, 2d ed., Universitext, Springer, Berlin, 1990. MR 91j:53001 8 A. A. GRIGOR’YAN, Stochastically complete manifolds and summable harmonic functions, Math. USSR-Izv. 33 (1989), 425–432. MR 90a:58189 28 , The heat equation on noncompact Riemannian manifolds, Math. USSR-Sb. 72 (1992), 47–77. MR 92h:58189 6, 7, 19

34

EL MAATI OUHABAZ

[HaKo] P. HAJŁASZ and P. KOSKELA, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1211–1215. MR 96f:46062 6 [HV] R. HEMPEL and J. VOIGT, The spectrum of a Schrödinger operator in L p (Rν ) is p-independent, Comm. Math. Phys. 104 (1986), 243–250. MR 87h:35247 4 [HK] P. HESS and T. KATO, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), 999–1030. MR 81m:35102 2, 4 [Je] D. JERISON, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), 503–523. MR 87i:35027 6 [Ka] T. KATO, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995. MR 96a:47025 14, 24 [LY] P. LI and S.-T. YAU, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153–201. MR 87f:58156 4, 19 [MS] P. MAHEUX and L. SALOFF-COSTE, Analyse sur les boules d’un opérateur sous-elliptique, Math. Ann. 303 (1995), 713–740. MR 96m:35049 6, 9 [MO] I. MCGILLIVRAY and E.-M. OUHABAZ, “Existence of bounded invariant solutions for absorption semigroups” in Differential Equations, Asymptotic Analysis, and Mathematical Physics (Potsdam, Germany, 1996), Math. Res. 100, Akademie, Berlin, 1997, 226–241. MR 98d:47102 28 [MePa] G. METAFUNE and D. PALLARA, On the location of the essential spectrum of Schrödinger operators, preprint, 2000. 4, 20 [Ou] E. M. OUHABAZ, On the spectral function of some higher order elliptic or degenerate-elliptic operators, Semigroup Forum 57 (1998), 305–314. MR 99f:35143 29 [OS] E. M. OUHABAZ and P. STOLLMANN, Stability of the essential spectrum of second-order complex elliptic operators, J. Reine Angew. Math. 500 (1998), 113–126. MR 99f:35142 [OSSV] E.-M. OUHABAZ, P. STOLLMANN, K.-T. STURM, and J. VOIGT, The Feller property for absorption semigroups, J. Funct. Anal. 138 (1996), 351–378. MR 97j:47061 28 [Pin] Y. PINCHOVER, On criticality and ground states of second order elliptic equations, II, J. Differential Equations 87 (1990), 353–364. MR 91j:35023 [ReSi] M. REED and B. SIMON, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. MR 58:12429c 20 [RoS] G. ROZENBLUM and M. SOLOMYAK, On principal eigenvalues for indefinite problems in Euclidean space, Math. Nachr. 192 (1998), 205–223. MR 99e:35162 5 [Sa1] L. SALOFF-COSTE, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 1992, 27–38. MR 93d:58158 6, 7 [Sa2] , Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), 417–450. MR 93m:58122 4, 6, 7, 12, 19 [Sc] M. SCHECHTER, Spectra of Partial Differential Operators, 2d ed., North-Holland Ser. Appl. Math. Mech. 14, North-Holland, Amsterdam, 1986. MR 88h:35085 19, 30, 31 [Sh] M. A. SHUBIN, “Spectral theory of elliptic operators on noncompact manifolds” in Méthodes semi-classiques (Nantes, France, 1991), Vol. 1, Astérisque 207, Soc.


35

Math. France, Montrouge, 1992, 5, 35–108. MR 94h:58175 15 [Si]

[Str] [Stu] [Te] [VSC]

[Vo] [Wa]

B. SIMON, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526,

MR 86b:81001a; Erratum, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 426. MR 86b:81001b 4 R. S. STRICHARTZ, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48–79. MR 84m:58138 15, 27 K.-T. STURM, On the L p -spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal. 118 (1993), 442–453. MR 94m:58227 4, 15 A. TERTIKAS, Critical phenomena in linear elliptic problems, J. Funct. Anal. 154 (1998), 42–66. MR 99d:35039 N. TH. VAROPOULOS, L. SALOFF-COSTE, and T. COULHON, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, Cambridge, 1992. MR 95f:43008 19, 24, 31 J. VOIGT, Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal. 67 (1986), 167–205. MR 88a:81036 20 F.-Y. WANG, Functional inequalities for empty essential spectrum, J. Funct. Anal. 170 (2000), 219–245. MR 2001a:58043 20

Institut de Mathématiques, Université de Bordeaux I, 351 cours de la Libération, 33405 Talence, France; [email protected]


UNIFORMLY LEVI DEGENERATE CR MANIFOLDS: THE 5-DIMENSIONAL CASE PETER EBENFELT

Abstract In this paper, we consider real hypersurfaces M in C3 (or, more generally, 5dimensional CR (Cauchy-Riemann) manifolds of hypersurface type) at uniformly Levi degenerate points, that is, Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require that the hypersurface satisfy a certain second-order nondegeneracy condition (called 2-nondegeneracy) at the point. One of our main results is the construction, near any point p0 ∈ M satisfying the above conditions, of a principal bundle P → M and a Rdim P -valued 1-form ω, uniquely determined by the CR structure on M, which defines an absolute parallelism on P. If M is real-analytic, then covariant derivatives of ω yield a complete set of local biholomorphic invariants for M. This solves the biholomorphic equivalence problem for uniformly Levi degenerate hypersurfaces in C3 at 2-nondegenerate points.

0. Introduction 0.1. A brief history and basic concepts: Motivation for the present work A fundamental problem in the study of real submanifolds in complex space is the biholomorphic equivalence problem, which in its most general form asks for (intrinsic) conditions on two submanifolds M, M 0 ⊂ C N at distinguished points p0 ∈ M, p00 ∈ M 0 which guarantee that there exists a local biholomorphism H : C N → C N defined near p0 such that H ( p0 ) = p00 and H (M ∩ U ) = M 0 ∩ U 0 , for some open neighborhoods U, U 0 ⊂ C N of p0 and p00 , respectively. When M and M 0 are realanalytic, an equivalent formulation is to ask for a real-analytic local CR diffeomorphism f : M → M 0 defined near p0 ∈ M with f ( p0 ) = p00 . (For standard definitions and results on real submanifolds in complex space and abstract CR structures, the reader is referred to [BER], for example). DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1, Received 29 September 1999. Revision received 26 October 2000. 2000 Mathematics Subject Classification. Primary 32V05, 32V40; Secondary 32V20. Author’s work supported in part by a grant from the Swedish Natural Science Research Council. 37

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The case where M and M 0 are real-analytic and Levi nondegenerate hypersurfaces was solved by E. Cartan [C1], [C2] in C2 , and by N. Tanaka [T1], [T2] and S.-S. Chern and J. Moser [CM] in C N , N ≥ 2. The solution consists of producing a fiber bundle Y → M, for any given Levi nondegenerate hypersurface M ⊂ C N , and a 1-form ω on Y , valued in Rdim Y , which at every y ∈ Y gives an isomorphism between Ty Y and Rdim Y (an absolute parallelism or {1}-structure on Y ; see, e.g., [KN] or [K]) such that the following holds. If there exists a CR diffeomorphism f : M → M 0 , then there exists a diffeomorphism F : Y → Y 0 (where corresponding objects for M 0 are denoted with 0 ) such that F ∗ ω0 = ω and the following diagram commutes: F

Y −−−−→   πy

Y0   0 yπ

(0.1.1)

M −−−−→ M 0 f

Conversely, if there exists a diffeomorphism F : Y → Y 0 such that F ∗ ω0 = ω, then there exists a CR diffeomorphism f : M → M 0 such that (0.1.1) commutes. Suppose that such a bundle Y → M and such an Rdim Y -valued 1-form ω can be constructed for every M in some given class of manifolds. Then we say that the bundle Y → M with 1-form ω reduces the CR structure on M to a parallelism (in this class). The construction of a bundle Y → M which reduces the CR structure on M to a parallelism for a class of CR submanifolds in C N reduces the biholomorphic equivalence problem for the real-analytic manifolds in this class to the equivalence problem for {1}-structures. The latter problem was solved by Cartan and is well understood (see, e.g., [G] or [Ste]). The bundle Y → M constructed in [CM] is in fact a principal fiber bundle with group G 0 , where G 0 is the isotropy subgroup of SU( p + 1, q + 1), p + q = N − 1, and p, q are the number of positive and negative eigenvalues, respectively, of the Levi form. The authors of [CM] also construct a Cartan connection 5 valued in the Lie algebra su( p + 1, q + 1) which defines the same parallelism as ω (given a suitable identification of Rdim Y with su( p + 1, q + 1)). Covariant differentiation of the curvature := d5 − 5 ∧ 5 produces a complete set of invariants for a real-analytic Levi nondegenerate hypersurface. In particular, it follows that a real-analytic strongly pseudoconvex hypersurface in C N is locally biholomorphic to a piece of the (2N −1)sphere (the “standard model” for such hypersurfaces) if and only if the curvature is identically zero (i.e., the connection is flat). The reader is also referred to the work of D. Burns and S. Shnider [BS] and S. Webster [We1] for further discussion in the Levi nondegenerate case. More recently, CR manifolds of higher codimension whose Levi forms are suitably nondegenerate have been studied by several authors. Since the main focus in

UNIFORMLY LEVI DEGENERATE CR MANIFOLDS

39

ˇ and H. the present paper is on hypersurfaces, we mention only the papers by A. Cap Schichl [CS], V. Ezhov, A. Isaev, and G. Schmalz [EIS], T. Garrity and R. Mizner [GM], and Schmalz and J. Slovák [SS], and we refer the interested reader to these papers for further information about the higher codimensional case. In this paper, we consider real hypersurfaces (and, more generally, CR manifolds of hypersurface type) which have degenerate Levi forms. Before describing our main results, we should mention that another approach to the biholomorphic equivalence problem is via normal forms. Normal forms for certain types of Levi degeneracies were studied by the author in [E3] and [E4]; another class of Levi degeneracies in C2 was considered by P. Wong [Wo]. However, at least to the best of the author’s knowledge, the geometric approach as described above has not been previously studied for CR manifolds with degenerate Levi forms, except in the case where the manifold is locally biholomorphically equivalent to one of the form M˜ × C p with M˜ Levi nondegenerate. The latter situation was studied by M. Freeman [F1]. The situation in the present paper is quite the opposite; the manifolds considered here are not locally equivalent to M˜ × C p for any manifold M˜ and p > 0. The idea in this paper is to use the cubic form as a complement to the degenerate Levi form. We restrict our attention to 5-dimensional manifolds. As the number of dimensions increases, the algebraic complexity of the resulting tensors and the number of cases that need to be distinguished greatly increases. The paper is organized as follows. Our main results, and some applications including a unique continuation principle, are explained in §0.2. §1 is devoted to preliminary material including basic definitions and properties of Levi uniform CR manifolds. The necessary constructions for the main results are given in §2–3, and in §4 a discussion and characterization of the tube over the light cone is given. A proof of the unique continuation principle is given in §5. In §6, some examples of everywhere Levi degenerate hypersurfaces that arise in partial differential equation (PDE) theory are given. 0.2. The main results Our main results concern real hypersurfaces M in C3 (or, more generally, 5dimensional CR manifolds of hypersurface type), which are uniformly Levi degenerate in the sense that the Levi form has one nonzero and one zero eigenvalue in a neighborhood of a distinguished point p0 ∈ M (Levi uniform of rank 1 according to Definition 1.5). Such hypersurfaces M ⊂ C3 are foliated by complex curves (cf. Proposition 1.12), but in general this foliation cannot be (even locally) biholomorphically straightened; that is, M is not locally equivalent to M˜ × C for any hypersurface M˜ ⊂ C2 . We should point out that obstructions to a local straightening of a complex foliation of a CR manifold were investigated by Freeman [F1]. In the present paper

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we require that M be 2-nondegenerate at p0 (see §1 or [BER, Chapter 11]). The latter condition guarantees that M is holomorphically nondegenerate (see Stanton [Sta1], [Sta2]; cf. also [BER, Chapter 11]) and, in particular, that the foliation of M cannot be locally biholomorphically straightened. In fact, if M ⊂ C3 is real-analytic, connected, and everywhere Levi degenerate, then there is a proper real-analytic subvariety V ⊂ M such that either M is locally equivalent to M˜ × C, for M˜ ⊂ C2 , near each p ∈ M \ V (locally biholomorphically straight) or M is Levi uniform of rank 1 and 2-nondegenerate at every p ∈ M \ V . The most important example (indeed, the “standard model”) of a hypersurface of the type considered here is the tube in C3 over the light cone in R3 . Example 0.2.1 The tube 0C in C3 over the light cone in R3 , that is, the variety defined by (Re Z 1 )2 + (Re Z 2 )2 − (Re Z 3 )2 = 0,

(0.2.2)

is Levi uniform of rank 1 at every nonsingular point, that is, at every point where it is a real submanifold. In particular, 0C is Levi degenerate at every point, but it cannot be locally biholomorphically straightened. The latter fact was first proved by A. Sergeev and V. Vladimirov (see [SV]) by calculating the Freeman obstructions mentioned above. The reader can also verify that 0C is 2-nondegenerate at every nonsingular point. This example is discussed in greater detail in §4. (See also [E4], where 0C is further discussed in connection with a normal form for 2-nondegenerate hypersurfaces in C3 .) Let us also point out that the biholomorphically invariant geometry of the tube over the light cone in C4 plays an important role in, for example, axiomatic quantum field theory since 0C bounds the so-called past and future tubes (see, e.g., Sergeev and Vladimirov [SV]). Other examples of everywhere Levi degenerate hypersurfaces that arise naturally are given, for motivation, in §6. For the class of hypersurfaces described above, we define a new CR invariant kˆ (see (3.8)). One of our main results is the following. We refer the reader to §1 for relevant definitions. 1 Let M be a 5-dimensional CR manifold of hypersurface type which is 2-nondegenerate and Levi uniform of rank 1 at p0 ∈ M. ˆ p0 ) 6 = 0, then there exist an open neighborhood M0 of p0 in M, a prin(a) If k( cipal fiber bundle P0 → M0 with a 2-dimensional group G 0 ⊂ GL(R3 ), and a 1-form ω0 on P0 which defines an isomorphism between Tu P0 and R7 for every u ∈ P0 and reduces the CR structure on M0 to a parallelism. THEOREM


(b)

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ˆ p0 )| 6= 2, then there exist an open neighborhood M1 of p0 in M, a If |k( principal fiber bundle P1 → M1 with a 1-dimensional group G 1 ⊂ GL(R3 ), and a 1-form ω1 on P1 which defines an isomorphism between Tu P1 and R6 for every u ∈ P1 and reduces the CR structure on M1 to a parallelism.

Theorem 1 is a consequence of the more detailed Theorems 3.1.37 and 3.2.9. ˆ p0 ) 6 = 0 and |k( ˆ p0 )| 6= 2, then there are two The reader should observe that if k( different bundles over a neighborhood of p0 . Both P0 and P1 are submanifolds of a larger principal G-bundle P → M, which is defined in a more general context in §2. In general, P1 , although of lower dimension, is not a submanifold of P0 . The groups G 0 and G 1 are subgroups of G. The bundle P1 is a reduction of P, whereas P0 is in general not; the action of G 0 on P0 differs slightly from the action of G 0 on P. The bundles P0 , P1 are uniquely defined by intrinsic conditions on M. Hence, P0 extends as a bundle over the (open) subset of points on M where kˆ 6 = 0, and P1 extends as a ˆ 6 = 2. bundle over the subset of points where |k| We should also mention that the groups G 0 , G 1 , and hence the bundles P0 , P1 , in Theorem 1 are disconnected and have two components. In order to obtain a connected bundle, we have to choose an orientation for the CR structure at p0 as explained in §3 (see Theorems 3.1.37 and 3.2.9). As mentioned above, the most important example is the tube 0C over the light cone for which the invariant kˆ ≡ 2i. We now characterize 0C among all M, as in Theorem 1, by a curvature condition in the spirit of the characterization of the sphere among strongly pseudoconvex hypersurfaces as described §0.1. There is a subgroup H of GL(C4 ) and a subgroup H0 of H such that H can be viewed as a principal fiber bundle over 0C with group H0 . The matrix-valued Maurer-Cartan forms 5 of H define a Cartan connection on 0C valued in h, the Lie algebra of H , with vanishing curvature = d5 − 5 ∧ 5. (All this is explained in detail in §4.) Now, since kˆ is an invariant which is ≡ 2i for 0C , a necessary condition for a CR manifold M, as in Theorem 1, to be locally equivalent to 0C is kˆ ≡ 2i for M. For such M we can identify the group G 0 with the group H0 and construct, using ω, an h-valued 1-form 5 which, unfortunately, in general is not a Cartan connection. However, we have the following result, which is a consequence of the more detailed Theorem 4.31. THEOREM 2 Let M be a real-analytic CR manifold satisfying the conditions in Theorem 1 with kˆ ≡ 2i near p0 ∈ M. Then there exists an h-valued 1-form 5 on the principal bundle P0 → M0 , where M0 ⊂ M and P0 are given by Theorem 1, which gives an isomorphism between Tu P0 and h for every u ∈ P0 and has the following property. There exists a real-analytic CR diffeomorphism f : M → 0C , defined near p0 ∈ M,

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if and only if the curvature := d5 − 5 ∧ 5

(0.2.3)

vanishes identically near p0 . Remark. Examples of real-analytic hypersurfaces that satisfy the conditions in Theorem 2 at generic points include tubes over homogeneous algebraic varieties in R3 (see Proposition 6.13) and tubes over surfaces in R3 which are everywhere characteristic for the wave operator (see Proposition 6.27). An example of a real-analytic hypersurface satisfying the conditions in the theorem (at generic points) for which the curvature 6 ≡ 0 is given by Freeman’s hypersurface {Z ∈ C3 : (Re Z 1 )3 + (Re Z 2 )3 − (Re Z 3 )3 + 0}. This fact is a consequence of Proposition 6.36. We conclude this section by giving two applications of Theorem 1. We use the notation Aut(M, p0 ) for the stability group of a CR manifold M at p0 ∈ M, that is, the group of germs at p0 of local smooth CR diffeomorphisms f : (M, p0 ) → (M, p0 ). Suppose that M satisfies the conditions of Theorem 1 at p0 . Pick any point u ∈ Pp0 , where P → M is the principal G 0 bundle given by Theorem 1 and Pp denotes the fiber over p ∈ M. By [K, Theorem 3.2] and Theorem 1, the group Aut(M, p0 ) embeds as a closed submanifold of the fiber Pp0 ∼ = G 0 via the mapping Aut(M, p0 ) 3 f 7 → F(u) ∈ Pp0 ,

(0.2.4)

where F : P → P is the lift of f as in diagram (0.1.1). Thus, dim Aut(M, p0 ) is at ˆ p0 )| = 2 and at most 1 if |k( ˆ p0 )| 6 = 2. We formulate this as most 2 if the invariant |k( follows. COROLLARY 3 Let M be a 5-dimensional CR manifold of hypersurface type which is 2-nondegenerate and Levi uniform of rank 1 at p0 ∈ M. Then dim Aut(M, p0 ) ≤ 2.

The bound in Corollary 3 cannot be improved since dim Aut(0C , p) = 2 for any p ∈ 0C , as is shown in §4. We should mention that in [Er] it was shown that the bound Aut(M, p0 ) ≤ 3 holds for the class of all real-analytic 2-nondegenerate hypersurfaces in C3 ; observe that when M is real-analytic, then it follows from the reflection principle in [BJT] that every f ∈ Aut(M, p0 ) is real-analytic. As our final application we give a strong unique continuation principle for CR mappings. We use the notation f : (M, p0 ) → (M 0 , p00 ) to denote the fact that f is a smooth mapping defined in a neighborhood of p0 in M, sending this neighborhood into M 0 and f ( p0 ) = p00 .


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THEOREM 4 Let M, M 0 be 5-dimensional CR manifolds of hypersurface type, and assume that both M and M 0 are Levi uniform and 2-nondegenerate at p0 ∈ M and p00 ∈ M 0 , respectively. Let f 1 , f 2 : (M, p0 ) → (M 0 , p00 ) be smooth CR diffeomorphisms. If j p10 f 0 = j p10 f 1 , where j p1 f denotes the first-order jet at p ∈ M of a mapping f : M → M 0 , then f 1 ≡ f 2 near p0 .

A similar unique continuation principle follows from the work of Chern and Moser [CM] for Levi nondegenerate CR manifolds. We should point out that Theorem 4 is new even for embedded real hypersurfaces. In addition, for embedded real hypersurfaces one can relax the condition that the mappings be diffeomorphisms. To state a more general result in this case, we need a definition. If M is minimal at p0 (see [BER, Chapter 1] for the definition; we mention here only the fact that if a hypersurface M is 2-nondegenerate, then it is minimal), then any smooth CR mapping f : (M, p0 ) → (M 0 , p00 ) attains its maximal rank on a dense open subset of any sufficiently small neighborhood of p0 (see [BR3]). Following [BR3], we call this integer the generic rank of f near p0 . THEOREM 5 Let M, M 0 ⊂ C3 be smooth real hypersurfaces, and assume that both M and M 0 are Levi uniform and 2-nondegenerate at p0 ∈ M and p00 ∈ M 0 , respectively. Let f 0 : (M, p0 ) → (M 0 , p00 ) be a smooth CR mapping such that its generic rank near p0 is at least 3. If f 1 : (M, p0 ) → (M 0 , p00 ) is a smooth CR mapping with j p10 f 0 = j p10 f 1 , then f 0 ≡ f 1 near p0 . Moreover, f 0 is a local CR diffeomorphism near p0 .

Theorems 4 and 5 are proved in §5. Remarks. (i) An inspection of the proofs of Theorems 4 and 5 shows that it is not necessary for the mappings to be C ∞ for the conclusion to hold. Indeed, it suffices to assume that the mappings are C 2 . On the other hand, even if we assume only C 2 -regularity of the mappings, the proof (in combination with standard result on regularity for solutions of PDEs) actually yields C ∞ -smoothness. Moreover, if M and M 0 in addition are assumed to be real-analytic, then the conclusion is that the mappings are real-analytic (cf. also [Ha], [Hay]). Hence, the method of proof also gives regularity results, and a reflection principle in the real-analytic case. The reflection principle, however, follows from the more general one in [BJT] (as mentioned above) once it is shown that the mappings are local diffeomorphisms (i.e., the last conclusion in Theorem 5). Since this part of Theorem 5 essentially follows from previous results due to M. Baouendi and L. Rothschild (see the proof), this reflection principle should

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be regarded as a known result. (ii) The generic rank condition imposed on f 0 in Theorem 5 is necessary for the conclusion to hold. In the example in §5.3, we show that there are real hypersurfaces M, M 0 ⊂ C3 with points p0 ∈ M and p00 ∈ M 0 as in Theorem 5, and a smooth CR mapping f 0 : (M, p0 ) → (M 0 , p00 ) with generic rank 2 near p0 such that j pk0 ( f 0 ) equals the k-jet of the constant mapping for any nonnegative integer k, but f 0 is not constant in any neighborhood of p0 . 1. Preliminaries Let M be a CR manifold with CR bundle V . Recall that this means that V is a subbundle of the complexified tangent bundle CT M such that V p ∩ V¯p = {0} for every p ∈ M, and V is formally integrable; that is, any commutator between sections of V is again a section of V . Sections of V are henceforth called CR vector fields. We denote the CR dimension of M, that is, the (complex) dimension of the fibers V p for p ∈ M, by n. We assume that M is of hypersurface type; that is, the complex dimension of T p0 M := (V p ⊕ V¯p )⊥ ⊂ CT p∗ M, for p ∈ M, is 1. In particular, the dimension of M is 2n + 1. For the remainder of this paper, unless explicitly stated otherwise, all CR manifolds are of hypersurface type. The bundle T 0 M ⊂ CT ∗ M is called the characteristic bundle, and real sections of T 0 M are called characteristic forms. The subbundle T 0 M ⊂ CT ∗ M, defined at p ∈ M by T p0 M := V p⊥ , is called the holomorphic cotangent bundle. The formal integrability of V is equivalent to the following property of T 0 M: If ω is a section of T 0 M, then dω is a section of the ideal generated by T 0 M in the exterior algebra of CT M. Let L 1¯ , . . . , L n¯ be a basis for the CR vector fields near some distinguished point p0 ∈ M. Also, let θ be a nonvanishing characteristic form near the same point p0 . We define a linear operator T A¯ on the holomorphic 1-forms on M, that is, the sections of T 0 M, near p0 by 1 T A¯ ω := L A¯ y dω, (1.1) 2i where y denotes the usual contraction by a vector field. Note that, for a holomorphic 1-form ω, T A¯ ω is again a holomorphic 1-form which can be thought of as the Lie derivative of ω along the vector field L A¯ . For p ∈ M near p0 and positive integers k, we define the subspace E k, p ⊂ T p0 M as the (complex) linear span of θ p and (T A¯ j · · · T A¯ 1 θ) p , for all 1 ≤ j ≤ k and all j-tuples (A1 , . . . , A j ) ∈ {1, . . . , n} j . We define E 0, p to be T p0 M. The CR manifold M is said to be finitely nondegenerate at p of E k,0 = T p0 M

(1.2)

for some integer k ≥ 1, and k0 -nondegenerate at p if k0 is the smallest integer k for which (1.2) holds. It was shown in [E2] (see also [BER, Chapter 11]) that this


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definition is consistent with the one for real hypersurfaces of Cn+1 given in [BHR]. (These notions can also be extended to CR manifolds of arbitrary codimension; see, e.g., [BER].) The Levi form, Lθ , of M at p and θ p is a Hermitian form on V p which, relative to the basis L 1, ¯ p , . . . , L n, ¯ p , is represented by a Hermitian (n × n)-matrix g AB ( p) , where ¯ 1≤A,B,≤n g AB ¯ ( p) :=

1

θ p , [L B , L A¯ ] p . 2i

(1.3)

We use here and throughout this paper the convention that L B = L B¯ . The Levi form can also be expressed using the Lie derivative as follows:

g AB (1.4) ¯ ( p) = (T A¯ θ) p , L B, p . From this observation we see that 1-nondegeneracy of M at p is equivalent to the classical notion of Levi nondegeneracy of M at p. Definition 1.5 A CR manifold M of hypersurface type is 1-uniform or Levi uniform (of rank r ) at p ∈ M if the rank of the Levi form is constant (and equal to r ) in a neighborhood of p. Observe that the extremal cases of CR manifolds which are Levi uniform of rank zero or n at a point p are precisely those which are Levi flat or Levi nondegenerate, respectively, at p. Such CR manifolds, which, in addition, are real-analytic, are by now fairly well understood: a real-analytic Levi flat CR manifold is locally CR equivalent to the real hyperplane Im Z n+1 = 0 in Cn+1 , and a theory for real-analytic Levi nondegenerate CR manifolds was developed by Cartan [C1], [C2], Tanaka [T1], [T2], and Chern and Moser [CM]. The reader should also observe that any real-analytic CR manifold is Levi uniform outside a proper real-analytic subvariety (in particular, on a dense open subset). Let us point out that the tube over the light cone (see Example 0.2.1) is an example of a Levi uniform CR manifold which is neither Levi nondegenerate nor Levi flat. We denote by N p ⊂ V¯p the Levi nullspace at p, that is, those vectors X p ∈ V¯p for which the linear form Y p 7→ Lθ (Y p , X¯ p ) on V p is zero. If M is Levi uniform (of rank r ) at p0 ∈ M, then the subspaces N p for p near p0 form a (rank n − r ) subbundle N of V¯ . From now on, we assume that M is Levi uniform of rank r , with 0 < r < n, in a neighborhood of p0 (to which we restrict our attention). We may arrange our basis for the CR vector fields L 1¯ , . . . , L n¯ so that L r +1 , . . . , L n is a basis

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for the sections of N near that point. The Levi form for p near p0 then takes the form gαβ 0 ¯ (g AB , (1.6) ¯ )= 0 0 where (gαβ ¯ )1≤α,β≤r is an r × r nondegenerate Hermitian matrix of smooth functions. In what follows we use the summation convention and also the convention that capital roman indices A, B, . . . run over the integers {1, . . . , n} and Greek indices α, β, . . . run over {1, . . . , r }. The cubic tensor (see, e.g., Freeman [F1] and Webster [We2]) can now be represented near p0 by n − r (n × n)-matrices (h A¯ Bk ¯ )1≤A,B≤n , k = r + 1, . . . , n, of smooth functions, where

h A¯ Bk (1.7) ¯ := T B¯ T A¯ θ, L k . The cubic tensor is usually defined using commutators of three vector fields (see [We2]), but for a Levi uniform manifold it is not difficult to see (cf. the computation in the proof of Proposition 1.8) that the commutator definition coincides with the one using Lie derivatives given here. We should point out that higher-order tensors, or Levi forms, have also been studied in the literature (see, e.g., [F1], [We2], [E3]). We do not discuss these further in this work. PROPOSITION 1.8 Assume that M is Levi uniform of rank r at p0 . Then, in the notation introduced above, for every k = r + 1, . . . , n and all p in a neighborhood of p0 , it holds that h A¯ Bk ¯ =0 whenever A or B belongs to {r + 1, . . . , n}.

Proof Using a well-known identity (see, e.g., [He, Chapter 1.2]; see also the remark concerning our normalization of the pairing h·, ·i in [E3]), we have

h A¯ Bk ¯ : = T B¯ T A¯ θ, L k = d(T A¯ θ), L B¯ ∧ L k

= L B¯ T A¯ θ, L k − L k T A¯ θ, L B¯ − T A¯ θ, [L B¯ , L k ]

= − T A¯ θ, [L B¯ , L k ] , (1.9)

where the last identity follows from the fact that T A¯ θ, L k ≡ 0 and T A¯ θ, L B¯ ≡ 0. It is not difficult to see that the matrices h A¯ Bk ¯ are symmetric (cf. [E3]), so to prove the proposition it suffices to show that h A¯ lk = 0 in a neighborhood of p0 for k, l ≥ r + 1; ¯

that is, T A¯ θ, [L l¯, L k ] = 0 in view of (1.9). To this end, note, using the fact that L k, p and L l, p are null vectors for the Levi form at every p near p0 , that

T A¯ θ, [L l¯, L k ] = − θ, [L A¯ , [L l¯, L k ]] . (1.10)


Thus, by also using the Jacobi identity, we obtain

T A¯ θ, [L l¯, L k ] = θ, [L l¯, [L k , L A¯ ]] + θ, [L k , [L A¯ , L l¯]] .

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(1.11)

The second term on the right-hand side of (1.11) vanishes since [L A¯ , L l¯] is a CR vector field by the formal integrability of V and L k is a null vector field for the Levi form. To show that the first term also vanishes, we must show that [L k , L A¯ ] is a section of V ⊕ V¯ . This fact follows again from the fact that L k is a null vector

field for the Levi form since the latter is equivalent to θ, [L k , L A¯ ] = 0 for every A = 1, . . . , n. The proof of Proposition 1.8 is complete. Let us digress briefly to note the following result which, although of no importance for the remainder of this paper, follows from (the proof of) Proposition 1.8 above. The result is most certainly known, but we include the brief proof for the convenience of the reader. PROPOSITION 1.12 If M ⊂ Cn+1 is a real hypersurface that is Levi uniform of rank r < n at p0 ∈ M, then M is foliated by complex manifolds of dimension n − r in a neighborhood of p0 .

Remark 1.13 As mentioned in the introduction, the foliation given by Proposition 1.12, even when M is real-analytic, can in general not be locally straightened; that is, it is in general not true that M, as in the proposition, is CR equivalent to a real hypersurface of the form M˜ ×Cr ⊂ Cn+1 , where M˜ is a real hypersurface in Cn+1−r and r > 0. Indeed, if M is CR equivalent to M˜ ×Cr (which is a holomorphically degenerate hypersurface), then it cannot be finitely nondegenerate at any point (see, e.g., [BER, Chapter 11]). Obstructions to local straightening were also studied by Freeman [F1]. Proof of Proposition 1.12 An immediate consequence of Proposition 1.8 and (1.9) is that, for k, l ≥ r + 1, [L k , L l¯] =

n X

(akml¯ L m + bkm¯l¯ L m¯ ),

(1.14)

m=r +1

where the akml¯ and bkm¯l¯ are smooth functions satisfying almk¯ + bkm¯l¯ = 0. A similar argument shows that N is formally integrable, that is, the commutator [L k , L l ], for k, l ≥ r +1, is also a section of N. Thus, by the Frobenius theorem, M is foliated near p0 by 2(n − r ) -dimensional integral manifolds of Re L k , Im L k , k = r + 1, . . . , n. By the Newlander-Nirenberg theorem, these manifolds are (n − r )-dimensional complex submanifolds in Cn+1 .

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PETER EBENFELT

Returning to the cubic tensor, we observe that Proposition 1.8 shows that the matrices (h A¯ Bk ¯ ) representing it are of the form ! h α¯ βk 0 ¯ (h A¯ Bk . (1.15) ¯ )= 0 0 We conclude this section with the following observation, whose proof is immediate and left to the reader, characterizing 2-nondegeneracy for Levi uniform CR manifolds of rank r in terms of the (r × r )-matrices (h α¯ βk ¯ ) in (1.15). PROPOSITION 1.16 Assume that M is Levi uniform of rank r at p0 . Then in the notation introduced above, M is 2-nondegenerate at p0 if and only if the symmetric matrices h α¯ βk ¯ ( p0 ) 1≤α,β≤r , k = r + 1, . . . , n, are linearly independent over C.

2. A G-structure for Levi uniform CR manifolds of rank n − 1 We keep the notation from the previous section. However, from now on we restrict ourselves to the case r = n − 1, that is, the case where the rank of the Levi form near p0 is n − 1. Thus, we assume that M is a smooth CR manifold (of hypersurface type) of CR dimension n which is Levi uniform of rank n − 1 at a distinguished point p0 ∈ M. We restrict our attention to a small neighborhood of p0 . In what follows, M denotes a sufficiently small neighborhood of p0 . We have the following invariant subbundles of the cotangent bundle CT ∗ M: T 0 M ⊂ T 00 M ⊂ T 0 M, where T 0 M and T 0 M were introduced in §1 and where T 00 M is defined by

T p00 M = ω p ∈ T p0 M : ω p , L p = 0, ∀L p ∈ N p .

(2.1)

(2.2)

Observe that dim T p0 M = 1,

dim T p00 M = n,

dim T p0 M = n + 1.

(2.3)

Let θ, θ 1 , . . . , θ n be a basis for the holomorphic 1-forms (i.e., sections of T 0 M) with the additional properties that θ is real and a basis for the sections of T 0 M and θ, θ 1 , . . . , θ n−1 is a basis for the sections of T 00 M. Any other such basis θ˜ , θ˜ 1 , . . . , θ˜ n is related to θ, θ 1 , . . . , θ n by    ˜  u 0 0 θ θ α u α 0  β  θ˜ α  =  , (2.4) θ  u β θn u n u nβ v θ˜ n


49

where the coefficients in the (n + 1) × (n + 1) -matrix in (2.4) are smooth functions (all complex valued except u, which is real valued); also, recall that we are using the summation convention and that Greek indices run over the set {1, . . . , n − 1} since r = 1 here. Observe that T p00 M ∩T p00 M = T p0 M and T 00 M ∪T 00 M is a rank 2n−1 subbundle of ∗ CT M. The 1-forms θ, θ α , θ α¯ , where θ α¯ = θ α as mentioned in §1, yield a coframe for the bundle T 00 M ∪ T 00 M. Consider the bundle Y → M consisting of all such coframes (ω, ωα , ωα¯ )τ ,      u 0 0 θ ω   β α uα ωα  =  0 θ (2.5) u   , β ¯ β ωα¯ u α 0 u αβ θ where (u, u α , u αβ ) ∈ (R \ {0}) × Cn−1 × GL(Cn−1 ). If we let G ⊂ GL(C2n−1 ) denote the group consisting of matrices of the form   u 0 0   (2.6) S = u α u αβ 0  , (u, u α , u αβ ) ∈ (R \ {0}) × Cn−1 × GL(Cn−1 ), u α 0 u αβ then Y → M is a principal fiber bundle over M with group G; (2.5) gives a trivialization of Y in which (u, u α , u αβ ), or S given by (2.6), are (global) coordinates of Y . We denote by g the Lie algebra of G, that is, the space of matrices   v 0 0   T = v α vβα 0  , (v, v α , vβα ) ∈ R × Cn−1 × M (Cn−1 ), (2.7) v α 0 vβα where M (Cn−1 ) denotes the space of all (n − 1) × (n − 1) -matrices. If we pull ¯ back the forms θ, θ A , θ A (where capital roman indices run over {1, . . . , n}) to Y , still ¯ denoting the pulled back forms by θ, θ A , θ A , then (2.5) defines 1-forms ω, ωα , ωα¯ on Y . The reader can verify that the latter 1-forms are invariantly defined on Y , that is, independent of the initial choice of θ, θ α , θ α¯ above. Differentiating the 1-forms ω, ωα , ωα¯ , we obtain       ω θ ω     d ωα  = d SS −1 ∧ ωβ  + Sd θ β  , (2.8) ¯ ¯ α ¯ β β ω ω θ where S ∈ G is given by (2.6). The elements of the matrix-valued 1-form d SS −1 are Maurer-Cartan forms for the Lie group G (see, e.g., [G]). Let L , L A , L A¯ denote a dual ¯ basis relative to θ, θ A , θ A . Thus, the L A¯ form a basis for the CR vector fields, L A =

50

PETER EBENFELT

L A¯ , and L n spans the Levi null bundle N. We use the notation and representations introduced in §1 for the Levi form and cobic tensor (relative to the bases chosen). Since M is Levi uniform of rank n − 1 near p0 , the Levi form (g AB ¯ ) satisfies (1.6) with r = n − 1, and the cubic tensor (h A¯ Bn ¯ ) satisfies, by Proposition 1.8, h n¯ Bn ¯ = h A¯ nn = 0 in a neighborhood of p . Moreover, the matrix (h ) is symmetric. The ¯ 0 α¯ βn ¯ αβ ¯ ); that is, matrix (gαβ ) is invertible and Hermitian, and we denote its inverse by (g ¯ αγ ¯ = δ γ . Using the formal integrability of V and the fact that θ is real, we obtain gαβ ¯ g β α¯ β dθ = igαβ ¯ θ ∧ θ + φ ∧ θ,

(2.9)

where φ is a real 1-form. Similarly, by the formal integrability we can write dθ α = ηα ∧ θ + ηβα ∧ θ β + h αB¯ θ B ∧ θ n , ¯

(2.10)

for some 1-forms ηα and ηβα , and some matrix (h αB¯ ). 2.11 In the notation introduced above, we have LEMMA

β

h n¯ = 0,

β

¯ 2ig αβ h α¯ γ¯ n = h γ¯ .

(2.12)

Proof Recall the operators T A¯ introduced in §1. Observe that, by the definition of T 00 M and the fact that L n spans N, the 1-forms θ and Tα¯ θ form a basis for the sections of T 00 M, and Tn¯ θ = cθ for some smooth function c. Indeed, we have ¯ θ β = g αβ Tα¯ θ + cβ θ

(2.13)

for some smooth functions cβ , as is straightforward to verify. In view of (2.13), a direct calculation using the definition of the cubic tensor shows

¯ h α¯ Bn (2.14) L B¯ ydθ β , L n = 2ig αβ ¯ . On the other hand, a calculation using (2.10) shows

β L B¯ ydθ β , L n = h B¯ ,

(2.15)

which completes the proof. Thus, we rewrite (2.10) as follows: dθ α = ηα ∧ θ + ηβα ∧ θ β + h αβ¯ θ β ∧ θ n , ¯

α 2ig ν¯ α h ν¯ βn ¯ = h β¯ .

(2.16)


In view of (2.8), we can write    1 0 ω 1α 1α α   d ω = β 1α¯ 0 ωα¯

51

     µ¯ ν i gˆ µν 0 ¯ ω ∧ω ω    α µ¯ n  0   ∧ ωβ  +  hˆ µ¯ ω ∧ θ  , ¯ 1αβ¯¯ ωβ hˆ α ωµ ∧ θ n¯

(2.17)

µ¯

where 1, 1α , 1α¯ := 1α , 1αβ , 1αβ¯¯ := 1αβ are Maurer-Cartan forms for G modulo α ∼ ω, ωα , ωα¯ , θ n , θ n¯ and where gˆ αβ ¯ and hˆ are functions on Y = M × G satisfying the β¯

following: α β u −1 gˆ αβ ¯ ( p, ST )u µ u ν = gˆ µν ¯ ( p, T ),

β hˆ αβ¯ ( p, ST )u µ = hˆ νµ¯ ( p, T )u αν ,

(2.18)

where S, T ∈ G with S given by (2.6) and p ∈ M. The last action of G in (2.18) can µ also be described using the cubic tensor hˆ α¯ βn ¯ = gˆ αµ ¯ h α¯ /2i by α β u −1 hˆ α¯ βn ¯ ( p, ST )u µ u ν = hˆ µ¯ ¯ ν n ( p, T ).

(2.19)

3. The 5-dimensional case We now proceed under the assumption that n = 2. Thus, M is a 5-dimensional CR manifold which is Levi uniform of rank 1; Greek indices α, β, . . . now run over the single integer 1. In what follows we also assume that M is 2-nondegenerate at p0 , which in this case simply amounts to requiring that hˆ 1¯ 12 ¯ ( p0 ) 6 = 0. In view of the first identity in (2.18) and (2.19), we can make an initial choice of the θ, θ 1 so that (the (1 × 1)-matrix) g11 ¯ is constant with g11 ¯ = 1 and (the (1 × 1)2 matrix) h 1¯ 12 ¯ is positive near p0 . By a suitable initial choice of θ , we may further assume that h 1¯ 12 ¯ is constantly equal to 1 near p0 . It follows from (2.18) and (2.19), as the reader can verify, that on the bundle Y the equations gˆ 11 ¯ = hˆ 1¯ 12 ¯ =1

(3.1)

define a reduction (see, e.g., [Ste, Chapter 7]) Y10 of Y to a principal bundle over a neighborhood of p0 (which we identify with M) with group G 01 consisting of those matrices S, as in (2.6) with n = 2, for which u > 0 and u 11 = ±u 1/2 . Hence, the bundle Y10 has two connected components over M and for each component the covectors ω1 ( p0 , u, u 11 ), given by the points on this component, define a linear form on V¯p0 which is invariantly defined up to multiplication by a positive number. We refer to a choice of this linear form (up to multiplication by a positive real number) as a choice of orientation for the normalized CR structure at p0 . Thus, a choice of orientation determines a component Y1 of Y10 which is a principal bundle over M with group G 1 ⊂ G, where G 1 consists of those matrices S, as in (2.6) with n = 2, for which u > 0 and u −1/2 u 11 = 1. In what follows we assume that such a choice of

52

PETER EBENFELT

orientation has been made. We use the fiber coordinates (u, u 1 ) ∈ R+ × C in the trivialization of Y1 given by a choice of θ, θ 1 , θ 2 as described above. It is easy to compute the Lie algebra g1 of G 1 . We have T ∈ g1 if and only if   2v 0 0   (3.2) T =  v 1 v 0 , (v, v 1 ) ∈ R × C. 1 v 0 v ¯

¯

Taking the pullbacks of all the forms ω, ω1 , ω1 , θ 2 , θ 2 , 1, 11 , 111 to Y1 , we see from formula (2.8) (with S ∈ G 1 now) that, pulled back to Y1 , the matrix   1 0 0   1 = 11 111 0  (3.3) ¯ ¯ 11 0 111¯ ¯

¯

is valued in g1 modulo (ω, ω1 , ω1 , θ 2 , θ 2 ), that is, ¯

111 = 111¯ =

1 ¯ ¯ 1 mod (ω, ω1 , ω1 , θ 2 , θ 2 ). 2

Let us therefore rewrite (2.17) as follows:      1¯    iω ∧ ω1 1 0 0 ω ω    tˆ1 ω1¯ ∧ ω1      d ω1  = 11 21 1 0  ∧ ω1  +  ¯  11  ¯ ¯ ¯ 1 1 1 1 ¯ 1 1 1 ω ω 1 1 0 tˆ11 2 ¯ ω ∧ω   0   ¯ ¯ 2iω1 ∧ θ 2 + rˆ11 ω1 ∧ θ 2 + sˆ11 θ 2 ∧ ω1  , +   ¯ ¯ ¯ ¯ −2iω1 ∧ θ 2 + rˆ11 ω1 ∧ θ 2 + sˆ11 θ 2 ∧ ω1

(3.4)

(3.5)

where, by a slight abuse of notation, all the forms in (3.5) denote the pulled back forms on Y1 . The 1-form 1 is uniquely determined by the form of the structure equation (3.5) up to transformations ˜ = 1 + aω, 1 (3.6) ˜ given by (3.6) preserves where a is a smooth function on Y1 ; that is, replacing 1 by 1 (3.5), and this is the only transformation doing so, as is easily verified (cf. also [G, Lecture 3]). In fact, a direct calculation shows that ¯ 1 = u −1 du + iu −1/2 u 1 ω1 − u 1 ω1 + φ mod ω, where φ is the pullback of a real 1-form on M. The 1-form θ 2 on Y1 is determined by the condition hˆ 1¯ 12 ¯ ≡ 1 up to transformations θ˜ 2 = θ 2 + c1 ω1 + cω,

(3.7)


53

where c, c1 are smooth functions on Y1 . The 1-form 11 is determined modulo ω, ¯ ¯ ω1 , ω1 , θ 2 , θ 2 . The precise form of the indeterminacy in 11 is not important at this point. By using the integrability of V and the fact that hˆ 1¯ 12 ¯ is constant on Y1 , we deduce that we can write d θ˜ 2 = dθ 2 + dc1 ∧ ω1 + c1 dω1 + dc ∧ ω + cdω ¯ ¯ = η ∧ ω + η1 ∧ ω1 + k˜1¯ ω1 ∧ θ˜ 2 + kˆ θ˜ 2 ∧ θ˜ 2 ,

(3.8)

where η and η1 are 1-forms (depending also on c1 and c), (3.9)

ˆ k˜1¯ = kˆ1¯ + 2ic1 − c1 k,

and kˆ1¯ , kˆ are smooth functions on Y1 . Moreover, kˆ is constant on the fibers Y1 → M, whereas kˆ1 ( p, ST ) = u −1/2 kˆ1¯ ( p, T ) for S, T ∈ G 1 and S of the form (2.6). The reader can easily verify that the function kˆ is also uniquely determined, that is, independent of the choice of 1, 11 , θ 2 . It is also easy to check that the functions rˆ11 , sˆ11 in (3.5) are uniquely determined and that they are constant on the fibers Y1 → ˆ p), rˆ 1 ( p), sˆ 1 ( p) are invariants of the CR structure on M at p ∈ M. M. Hence, k( 1 1 However, they are not independent, as the following proposition shows. PROPOSITION 3.10 We have the following identities:

sˆ11 = rˆ11 =

1ˆ k. 2

(3.11)

Proof Differentiating the first row in the structure equation (3.5), using the fact that d 2 ω = 0, we obtain ¯ ¯ 0 = d1 ∧ ω − 1 ∧ dω + i(dω1 ∧ ω1 − ω1 ∧ dω1 ). (3.12) Applying equation (3.5) again, we obtain ¯ ¯ ¯ ¯ 0 = d1 + i(11 ∧ ω1 − 11 ∧ ω1 ) ∧ ω + i sˆ11 − rˆ11 θ 2 ∧ ω1 ∧ ω1 .

(3.13)

The first identity in (3.11) follows immediately from (3.13). To prove the second identity, we differentiate the formula for dω1 given by (3.5). We obtain 1 1 1¯ 1 0 = d11 ∧ ω − 11 ∧ dω + (d1 ∧ ω1 − 1 ∧ dω1 ) + d tˆ11 ¯ ∧ω ∧ω 2 1 1¯ 1 1¯ 1 1¯ 2 1¯ 2 1 1 2 + tˆ11 ¯ (dω ∧ ω − ω ∧ dω ) + 2i(dω ∧ θ − ω ∧ dθ ) + d rˆ1 ∧ ω ∧ θ ¯

¯

¯

+ rˆ11 (dω1 ∧ θ 2 − ω1 ∧ dθ 2 ) + d sˆ11 ∧ θ 2 ∧ ω1 + sˆ11 (dθ 2 ∧ ω1 − θ 2 ∧ dω1 ). (3.14)

54

PETER EBENFELT ¯

¯

If we substitute (3.5) and (3.9) in (3.14) and collect the (ω1 ∧ θ 2 ∧ θ 2 )-terms, we obtain 2i kˆ = 2i rˆ11 + 2i sˆ11 , (3.15) which proves the second identity in (3.11). 1 For future reference, let us remark that the function tˆ11 ¯ in (3.5) depends on the choice 1 of θ 2 . For such a fixed choice, the function tˆ11 ¯ satisfies the following on Y1 : 1 tˆ11 ¯ ( p, ST ) =

1 tˆ11 3 u1 ¯ ( p, T ) , +i √ 2 u u

(3.16)

where S, T ∈ G 1 and S is given by (2.6). Let us also observe that replacing θ 2 by θ˜ 2 , 1 as given by (3.7), the function tˆ11 ¯ changes by 1 1 1 tˆ11 ¯ → tˆ11 ¯ − (2ic1 + sˆ1 c1 ).

(3.17)

We distinguish two cases and begin by handling the most difficult one, which also contains the tube over the light cone as given by Example 0.2.1. ˆ p0 ) 6 = 0 3.1. The case k( ˆ First, since k( p0 ) 6 = 0, there exists a small open neighborhood of p0 in which kˆ 6= 0. We identify M with this neighborhood of p0 in this subsection. Recall the formula in ¯ ˆ p0 )| could equal 2, it may not (3.9) for the coefficient k˜1¯ of ω1 ∧ θ˜ 2 in d θ˜ 2 . Since |k( be possible to solve the equation k˜1¯ = 0, that is, kˆ1¯ + 2ic1 − c1 kˆ = 0

(3.1.1)

for c1 in a neighborhood of p0 . Let us write kˆ = 2ir eit ,

(3.1.2)

where r > 0 and t are real-valued functions on Y1 which are constant on the fibers of Y1 → M. We seek c1 in the form c1 = eit/2 (ρ1 + iρ2 ),

(3.1.3)

where ρ1 , ρ2 ∈ R. We can choose ρ2 uniquely so that k˜1¯ = ir 0 eit/2 for some real-valued function r 0 on Y1 which is constant on the fibers of Y1 → M. The function ρ2 satisfies the same transformation rule as kˆ1¯ ; that is, ρ2 ( p, ST ) = u −1/2 ρ2 ( p, T )

(3.1.4)


55

for S, T ∈ G 1 and S of the form in (2.6). We can express the above by saying that Im e−it/2 c1 is uniquely determined by the condition (3.1.5)

Re e−it/2 kˆ1¯ ≡ 0.

Now, with Im e−it/2 c1 determined by (3.1.5), equation (3.17) implies that the function 1 tˆ11 ¯ is determined up to 1 1 it/2 tˆ11 + sˆ11 ρ1 e−it/2 , ¯ → tˆ11 ¯ − 2iρ1 e or equivalently, in view of Proposition 3.10 and (3.1.2), 1 ˆ −it/2 ˆt ¯1 → tˆ¯1 − 2iρ1 eit/2 + kρ 1e 11 11 2 1 1 it/2 = tˆ11 1+ r . ¯ − 2iρ1 e 2

(3.1.6)

Let us write u 1 = eit/2 (x + i y). It follows from (3.16) that there is a submanifold Y2 ⊂ Y1 , which is defined (uniquely in view of (3.1.6)) by the equation 1 Re(e−it/2 tˆ11 ¯ ) = 0.

(3.1.7)

The manifold Y2 can be viewed as a principal fiber bundle Y2 → M with group G 2 ⊂ G 1 , where G 2 is defined by the equation Im e−it ( p0 )/2 u 1 = 0, if we let G 2 act on Y2 as follows:    ω u α it/2    g ω := e x ωα¯ e−it/2 x for g ∈ G 2 ⊂ G 1 of the form 

u

 eit ( p0 )/2 x

e−it ( p0 )/2 x

0 √ u 0

0 √ u 0

  0 ω   0 ωα  √ u ωα¯

 0 0 . √ u

(3.1.8)

(3.1.9)

(3.1.10)

Observe that the principal G 2 bundle Y2 is a reduction of the principal G 1 bundle Y1 1 00 it/2 , for only if the phase function t is constant. By definition, we have tˆ11 ¯ = ir e 00 some real-valued function r , on the bundle Y2 . Hence, we can determine ρ1 uniquely in (3.1.6) so that 1 tˆ11 (3.1.11) ¯ ≡0

56

PETER EBENFELT

on Y2 . It follows from (3.16) and (3.1.6) that we have ρ1 ( p, ST ) =

x ρ1 ( p, T ) 3 + √ 2(2 + r ) u u

(3.1.12)

for S, T ∈ G 2 and S of the form (2.6) with u 1 = eit/2 x and x ∈ R. Thus, θ 2 is determined on Y2 up to θ˜ 2 = θ 2 + cω

(3.1.13)

by (3.1.5) and (3.1.11). Observe that on Y2 , where u 1 = eit/2 x for x ∈ R, we have 11 = eit/2 ξ

¯

¯

mod ω, ω1 , ω1 , θ 2 , θ 2 ,

(3.1.14)

where ξ is a real 1-form on Y2 which is not uniquely determined. We can write the structure equation for dω1 on Y2 as follows: 1 ¯ ¯ dω1 = eit/2 ξ ∧ ω + 1 ∧ ω1 + 2iω1 ∧ θ 2 + rˆ11 ω1 ∧ θ 2 + sˆ11 θ 2 ∧ ω1 2 ¯ ¯ 1¯ ∧ ω + eθ 2 ∧ ω + eθ + ieit/2 (bω1 ∧ ω + bω ¯ 2 ∧ ω) (3.1.15) for some functions b and e on Y2 . Using (3.6) and (3.1.13), we obtain 1 ˜ ∧ ω1 + 2iω1¯ ∧ θ˜ 2 + rˆ11 ω1 ∧ θ˜ 2 + sˆ11 θ˜ 2¯ ∧ ω1 dω1 = eit/2 ξ˜ ∧ ω + 1 2 ˜ 1¯ ∧ ω + eθ˜ 2 ∧ ω + e¯θ˜ 2¯ ∧ ω), ˜ 1 ∧ ω + bω (3.1.16) + ieit/2 (bω where 1 b˜ = b + 2

1 −it/2 ae + cˆ ¯s11 e−it/2 − 2i ce ¯ it/2 − crˆ11 e−it/2 2

(3.1.17)

and 1 1 −it/2 1 ¯ ¯ (ae ω + ae ¯ it/2 ω1 ) + cˆ ¯s11 e−it/2 ω1 + cˆs11 eit/2 ω1 2 2 1 ¯ ¯ − −2i ce ¯ it/2 ω1 + 2ice−it/2 ω1 + crˆ11 e−it/2 ω1 + c¯rˆ11 eit/2 ω1 + qω. 2 (3.1.18)

ξ˜ = ξ +

Here q is an arbitrary real-valued function on Y2 , and a and c are as in (3.6) and (3.1.13), respectively. We deduce that ξ is determined by the structure equation


57

(3.1.16) up to transformations given by (3.1.18). Let us rewrite (3.1.17) and (3.1.18) using Proposition 3.10 and (3.1.2) as follows: ˜b = b + 1 e−it/2 1 a + i (r − 2) ζ¯ + ir ζ , (3.1.19) 2 2 and 1 1 ¯ ˜ξ = ξ + e−it/2 a + i (r + 2) ζ + ir ζ ω1 2 2 1 it/2 1 ¯ ¯ + e a − i (r + 2) ζ − ir ζ ω1 + qω, 2 2

(3.1.20)

where we have used the notation ζ = e−it c. By (3.13), we have on Y1 , ¯

¯

d1 = i(11 ∧ ω1 − 11 ∧ ω1 ) + 8 ∧ ω,

(3.1.21)

for some real 1-form 8. Hence, on Y2 we have, by (3.1.14), ¯

¯

d1 = ie−it/2 ξ ∧ ω1 − ieit/2 ξ ∧ ω1 + 9 ∧ ω + 91 ∧ ω1 + 91¯ ∧ ω1 ,

(3.1.22) ¯

¯

for some 1-forms 9, 91 , 91¯ = 91 on Y2 such that 91 = 0 modulo ω1 , ω1 , θ 2 , θ 2 . Recall that 1 is determined up to transformations (3.6), θ 2 up to transformations (3.1.13), and ξ up to transformations (3.1.20). Substituting in (3.1.22), we obtain ˜ = d1 + da ∧ ω + a dω d1 ¯ ¯ = ie−it/2 ξ˜ ∧ ω1 − ieit/2 ξ˜ ∧ ω1 + i f˜ω1 ∧ ω1 + . . . ,

(3.1.23)

˜ and where . . . signify the remaining terms in the expansion of d 1 1 f˜ = f − a + iζ − i ζ¯ , 2

(3.1.24)

for some real-valued function f on Y2 . Hence, a is uniquely determined as a function of ζ by the condition f˜ = 0, (3.1.25) and we have a = 2i(ζ − ζ¯ ).

(3.1.26)

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PETER EBENFELT

In view of (3.1.3), (3.1.4), and (3.1.12), we have dc1 = eit/2

3 ξ 2(2 + r )

¯

¯

mod 1, ω, ω1 , ω1 , θ 2 , θ 2 .

(3.1.27)

It follows, by using (3.8), (3.1.20), and (3.1.24), that d θ˜ 2 = eit/2

3 ¯ ξ˜ ∧ ω1 + mω ˜ 1 ∧ ω1 + . . . , 2(2 + r )

(3.1.28)

where . . . signify the remaining terms in the expansion of d θ˜ 2 and m˜ is given by 3ieit (r + 1) ζ + ζ¯ + ic 4 (r + 2) ieit (7r + 11)ζ + 3(r + 1)ζ¯ =m+ 4(r + 2)

m˜ = m +

(3.1.29)

for some function m on Y2 ; in the last line of (3.1.29), we have used c = eit ζ . Since r > 0, we can determine ζ uniquely by the condition m˜ = 0.

(3.1.30)

Let us summarize our efforts so far. We have determined 1 and θ 2 uniquely on Y2 . The 1-form ξ is determined up to ξ˜ = ξ + qω,

(3.1.31)

for some real-valued function q on Y2 . We conclude the construction in this section by defining a unique choice of ξ . In view of (3.1.22), we have ¯ ˜ 1¯ ∧ ω + . . . , ˜ 1 ∧ ω + lω d1 = ie−it/2 ξ˜ ∧ ω1 − ieit/2 ξ˜ ∧ ω1 + lω

(3.1.32)

where l˜ = l + ie−it/2 q

(3.1.33)

for some function l on Y2 . Hence, we may determine q uniquely by the condition Im eit/2l˜ = 0.

(3.1.34)

The uniquely determined, linearly independent 1-forms ω, Re ω1 , Im ω1 , Re θ 2 , Im θ 2 , 1, ξ

(3.1.35)

on Y2 form a global coframe for T ∗ Y2 and hence define an absolute parallelism on Y2 . The following result is a consequence of the construction above (see [G] or [CM]).


We use the notation introduced previously, and also,   ω Re ω1    Im ω1      ω :=  Re θ 2  .    Im θ 2     1  ξ

59

(3.1.36)

THEOREM 3.1.37 Let M be a 5-dimensional CR manifold of hypersurface type which is 2-nondegenerate ˆ p0 ) 6 = 0 and that an orientation and Levi uniform of rank 1 at p0 ∈ M. Suppose that k( is chosen for the normalized CR structure at p0 (as explained in §3). Then there exist an open neighborhood M0 of p0 in M, a principal fiber bundle Y2 → M0 with a 2-dimensional structure group G 2 ⊂ GL(C3 ), and a 1-form ω on Y2 which defines an isomorphism between Ty Y2 and R7 for every y ∈ Y2 and such that the following holds. Let M 0 be a 5-dimensional CR manifold of hypersurface type which is 2-nondegenerate and Levi uniform of rank 1 at p00 ∈ M. Suppose that kˆ 0 ( p0 ) 6= 0 (where corresponding objects for M 0 are denoted with 0 ) and that an orientation is chosen for the normalized CR structure of M 0 at p00 . Then if there exists a local CR diffeomorphism f : (M, p0 ) → (M 0 , p00 ) preserving the oriented CR structures of M and M 0 at p0 and p00 , respectively, there exists a diffeomorphism F : Y2 → Y20 with π 0 ◦ F(π −1 ( p0 )) = { p00 } such that F ∗ ω0 = ω and the following diagram commutes: F

Y2 −−−−→   πy

Y20   0 yπ

(3.1.38)

M −−−−→ M 0 f

Conversely, if there exists a diffeomorphism F : Y2 → Y20 with π 0 ◦ F(π −1 ( p0 )) = p00 such that F ∗ ω0 = ω, then there exists a CR diffeomorphism f : (M, p0 ) → (M 0 , p00 ) preserving the oriented CR structures of M and M 0 at p0 and p00 , respectively, such that (3.1.38) commutes. The 1-form ω is given by (3.1.36) and is uniquely determined by (3.1.5), (3.1.11), (3.1.25), (3.1.30), and (3.1.34). The group G 2 is defined by (3.1.10). ˆ p0 )| 6= 2 3.2. The case |k( ˆ ˆ 6 = 2. In this Since |k( p0 )| 6= 2, there is a small neighborhood of p0 in M in which |k| neighborhood, which we identify with M in this section, we can solve uniquely for c1

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PETER EBENFELT

in the equation (3.2.1)

k˜1¯ = 0,

where k˜1¯ is given by equation (3.1.1). Thus, c1 is uniquely determined by condition (3.2.1), and hence θ 2 is determined up to transformations of the form in (3.1.13). Moreover, it follows from (3.16) that there exists a uniquely determined submanifold Y3 ⊂ Y1 defined by 1 tˆ11 (3.2.2) ¯ = 0. The submanifold Y3 is a subbundle (a reduction) of the principal G 1 -bundle Y1 with group G 3 , where G 3 is the subgroup of G 1 defined by (3.2.3)

u 1 = 0. ¯

¯

Thus, on Y3 we have 11 = 0 modulo ω, ω1 , ω1 , θ 2 , θ 2 . It follows that the structure equation for dω1 on Y3 can be written dω1 =

1 ˜ ∧ ω1 + 2iω1¯ ∧ θ˜ 2 + rˆ11 ω1 ∧ θ˜ 2 + sˆ11 θ˜ 2¯ ∧ ω1 1 2 ˜ 1 ∧ ω + 2 bω˜ 1¯ ∧ ω + 1 eθ 2 ∧ ω + 2 eθ 2¯ ∧ ω, + 1 bω

(3.2.4)

where

1 2˜ b := 1 b − rˆ11 c + sˆ11 c¯ + a, b := 2 b − 2ic (3.2.5) 2 for some functions 1 b, 2 b, 1 e, and 2 e on Y3 . We can determine c, and hence θ 2 , uniquely on Y3 by the condition 2˜ b = 0. (3.2.6) 1˜

We then determine a, and hence 1, uniquely on Y3 by the condition Re1 b˜ = 0.

(3.2.7)

The uniquely determined 1-forms ω, Re ω1 , Im ω1 , Re θ 2 , Im θ 2 , 1 on Y3 form a global coframe for T ∗ Y3 and hence define an absolute parallelism on Y3 . As in 3.1, we have the following result. We use the notation introduced above, and also,   ω Re ω1      Im ω1  ω :=  (3.2.8) .  Re θ 2     Im θ 2  1


61

THEOREM 3.2.9 Let M be a 5-dimensional CR manifold of hypersurface type which is 2-nondegenerate ˆ p0 )| 6= 2 and that an oriand Levi uniform of rank 1 at p0 ∈ M. Suppose that |k( entation is chosen for the normalized CR structure at p0 (as explained in §3). Then there exist an open neighborhood M0 of p0 in M, a principal fiber bundle Y3 → M0 with a 1-dimensional structure group G 3 ⊂ GL(C3 ), and a 1-form ω on Y3 which defines an isomorphism between Ty Y3 and R6 for every y ∈ Y3 and such that the following holds. Let M 0 be a 5-dimensional CR manifold of hypersurface type which is 2-nondegenerate and Levi uniform of rank 1 at p00 ∈ M. Suppose that the invariant |kˆ 0 ( p0 )| 6 = 2 (where corresponding objects for M 0 are denoted with 0 ) and that an orientation is chosen for the normalized CR structure of M 0 at p00 . Then if there exists a local CR diffeomorphism f : (M, p0 ) → (M 0 , p00 ) preserving the oriented CR structures of M and M 0 at p0 and p00 , respectively, there exists a diffeomorphism F : Y3 → Y30 with π 0 ◦ F(π −1 ( p0 )) = { p00 } such that F ∗ ω0 = ω and the following diagram commutes F

Y3 −−−−→   πy

Y30   0 yπ

(3.2.10)

M −−−−→ M 0 f

Conversely, if there exists a diffeomorphism F : Y3 → Y30 with π 0 ◦ F(π −1 ( p0 )) = p00 such that F ∗ ω0 = ω, then there exists a CR diffeomorphism f : (M, p0 ) → (M 0 , p00 ) preserving the oriented CR structures of M and M 0 , such that (3.2.10) commutes. The 1-form ω is given by (3.2.8) and is uniquely determined by (3.2.1), (3.2.6), and (3.2.7). The group G 3 is defined by (3.2.3), and on Y3 , (3.2.2) holds. 4. A curvature characterization of the tube over the light cone in C3 In this section, we continue to consider only the case n = 2. We change slightly the convention from §2 that capital roman indices A, B, etc., run over the set {1, 2} (i.e., {1, . . . , n} with n = 2) and instead let them run over the set {1, 2, 3}. We also use the convention that small roman indices a, b, etc., run over the set {0, 1, 2, 3}. Denote by 0 the light cone in R3 , that is, the zero locus of the quadratic form {x, x} where {·, ·} denotes the bilinear form which in the standard coordinates of R3 is given by {x, y} := x 1 y 1 + x 2 y 2 − x 3 y 3 . (4.1) We denote by 0C the tube in C3 over 0 (as in Example 0.2.1). Hence, 0C is the zero locus of {Z , Z }C , where {z, w}C for complex vectors z, w ∈ C3 is defined by {z, w}C := {Re z, Re w}.

(4.2)

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PETER EBENFELT

We call a frame for 0C a 4-tuple (Z 0 , Z 1 , Z 2 , Z 3 ) of 4-vectors, where Z 0 = t (1, z 0 ),

Z A = (0, x A ),

t ∈ R, z 0 ∈ C3 , x A ∈ R3 ,

(4.3)

which satisfy the following conditions. The real vectors x1 , x2 , x3 ∈ R3 satisfy {x1 , x1 } = {x3 , x3 } = {x1 , x2 } = {x2 , x3 } = 0, {x1 , x3 } = −1,

{x2 , x2 } = 1

(4.4)

and also Re t z 0 = x1 .

(4.5)

Observe that conditions (4.4) are equivalent to the fact that the (symmetric) matrix representation of the bilinear form {·, ·} relative to the basis x1 , x2 , x3 is by the matrix 3, where   0 0 −1 (4.6) 3 :=  0 1 0  . −1 0 0 We also write 3 = (λ AB ).

(4.7)

The set of all frames for 0C can be viewed as a real subgroup of the complex Lie group GL(C4 ) as follows. If (Z 0 , Z A ) is a given frame, then any other frame (Z 00 , Z 0A ) is obtained as v 0 0 0 (Z 0 , Z A ) = (Z 0 , Z B ) , (4.8) (k1B − vδ1B ) + iv B k AB where v, v B ∈ R and the (3 × 3)-matrix (k AB ) satisfies k CA k BD λC D = λ AB

(4.9)

det(k BA ) = 1.

(4.10)

and also We denote by H 0 the subgroup of GL(C4 ) consisting of all matrices of the form v 0 M= (4.11) (k1B − vδ1B ) + iv B k AB which satisfy (4.9) and (4.10). We also denote by K the subgroup of GL(R3 ) which consists of (k BA ) satisfying (4.9) and (4.10). The group K is isomorphic to the Lorenz group SO(2, 1). Indeed, if O denotes the orthogonal transformation for which O τ 3O equals the diagonal matrix with 1, 1, −1 on the diagonal, then K = O(SO(2, 1))O τ .


63

Note that (4.2), (4.4), and (4.5) imply that Z 0 , considered as the affine point z 0 ∈ C3 where Z 0 and z 0 are as in (4.3), can be viewed as a point on 0C . We denote by H00 the subgroup of H 0 consisting of those matrices which preserve Z 0 as an affine point on 0C , that is, the group of matrices of the form v 0 N= , (4.12) 0 k AB where (k BA ) ∈ K with k1A x A = vx1 . A must be of the form  v (k BA ) = 0 0

straightforward calculation shows that (k BA ) a 1 0

1 2 −1  2a v av −1  ,

(4.13)

v −1

for some a ∈ R. Thus, the group of frames for 0C (which, given a fixed frame, can be identified with the group H 0 via (4.8)) is a principal fiber bundle P 0 → 0C with group H00 . Let us now choose an orientation, as explained in §3, for the normalized CR structure of 0C and denote by P the group of frames consistent with this orientation. Then, as is easily verified, P is isomorphic to the group H , where H is the subgroup of H 0 consisting of matrices of the form (4.11) with v > 0. If we also denote by H0 the subgroup of H00 consisting of those matrices of the form (4.12) for which v > 0, then P → M is a principal fiber bundle with group H0 . The reader should note that 0C has two connected components. This is reflected on the bundle P by the fact that the Lorenz group SO(2, 1) has two components. A choice for the (4 × 4)-matrix 5 = (πba ) of Maurer-Cartan forms for the group H is given by (4.14) d Z a = πab Z b . The Maurer-Cartan equations of structure then become dπab = πac ∧ πcb ,

(4.15)

which follows directly from differentiating (4.14). Due to the form (4.3) of the frames (Z 0 , Z A ), we have that π00 = dv/v, π A0 = 0, and the πab are real. By differentiating defining equations (4.5) and again using these equations, we deduce that the (3 × 3)matrix (π AB ) (which is a Maurer-Cartan matrix for the group K ) is given by  1  π1 π21 0 (4.16) (π AB ) = π12 0 π21  , 2 0 π1 −π11 for some real 1-forms π11 , π21 , π12 . By differentiating Z 0 = v(1, z 0 ), we obtain d Z0 =

dv dv (v, vz 0 ) + v(0, dz 0 ) = Z 0 + (0, vdz 0 ). v v

(4.17)

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PETER EBENFELT

Thus, by equation (4.5), we have 1 dv x1 + (vdz 0 + vd z¯ 0 ) = d x1 , v 2

(4.18)

and hence, also using (4.14), 1 A dv A A A (π + π¯ 0 )x A = π1 − δ x A. 2 0 v 1 Using (4.16), we obtain the equations  1   π11 − π00 = (π01 + π¯ 01 ),   2  1 π 2 = (π02 + π¯ 02 ),   2  1   0 = π03 + π¯ 03 .

(4.19)

(4.20)

The last formula in (4.20) implies that π03 is a purely imaginary form. The matrix of 1-forms 5 = (πba ) is valued in the Lie algebra h of H , and if we change frame using (4.8), then the corresponding matrix of 1-forms 50 for the new frame (Z 00 , Z 0A ) is related to 5 by 50 = ad(M−1 )5 := M−1 5M, (4.21) where M is the matrix given by (4.11). It follows that 5 is a Cartan connection on P with group H (see, e.g., [K, Chapter 4] and also [CM]), which is flat (i.e., with vanishing curvature form) by (4.15). Let us relate the above to the results in previous sections. It is straightforward to verify that we can set  i  ω = − π03 ,    2    1    ω1 = π02 ,   2i  1 2 θ = π01 ,   4i     1   1 = π00 + π11 = 2π00 + (π01 + π¯ 01 ),    2   1 1 ξ(= 1 ) = −π2

(4.22)

for the forms given by Theorem 3.1.37. Indeed, using (4.15) and (4.20), we obtain the


65

equations  1¯ 1    dω = 1 ∧ ω + iω ∧ ω ,      dω1 = ξ ∧ ω + 1 1 ∧ ω1 − iω1 ∧ θ 2 + iθ 2¯ ∧ ω1 ) + ω1¯ ∧ θ 2 ,    2   1 ¯ 2 1 dθ = ξ ∧ ω + 2iθ 2 ∧ θ 2 ,  2    ¯   d1 = iξ ∧ ω1 − iξ ∧ ω1 ,      1   dξ = − (1 + 2iθ 2 − 2iθ 2¯ ) ∧ ξ, 2

(4.23)

ˆ defined in which satisfy the conditions of Theorem 3.1.37. Note that the invariant k, ˆ §3, satisfies k ≡ 2i. Thus, in what follows, the phase function t and the modulus r as defined by (3.1.2) are identically zero and 1, respectively. Recall that the group of frames is a principal fiber bundle P → 0C with group H0 . For any N ∈ H0 , where N ∈ H0 is given by (4.12) and (k BA ) by (4.13), a change of frame (Z 00 , Z 0A ) = (Z 0 , Z B )N results in the change of connection form 5 = ad(N 0

−1

π00 )5 = vl AB π0A

0

l AB k CD πCA

,

(4.24)

where (l CD ) denotes the inverse of (k BA ). Using (4.22) and calculating the inverse of (k AB ) given by (4.13), we deduce that (4.24) yields the corresponding transformation  0 ω = v 2 ω,       (ω1 )0 = −avω + vω1 ,      2 0 1 1 (θ ) = a 2 ω − aω1 + θ 2 , 4 2    0 −1 1¯  1 = iav (ω − ω1 ) + 1,        ξ 0 = 1 ia 2 v −1 (ω1 − ω1¯ ) − iav −1 (θ 2 − θ 2¯ ) − 1 av −1 1 + v −1 ξ. 2 2 In particular, we obtain 

  ω0 u  1 0  (ω )  = x ¯ x (ω1 )0

0 √ u 0

  ω 0  1  0 ω  , √ ¯ u ω1

(4.25)

(4.26)

where u = v2,

x = −av.

(4.27)

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PETER EBENFELT

Hence, we have defined an isomorphism φ : H0 → G 2 , where G 2 is as defined in §3.1 with t ≡ 1, defined by  v 0 0 0 v a φ 0 0 1 0 0 0

  2 0 v 1 2 −1  2 a v  = −av av −1  −av v −1

 0 0 v 0 . 0 v

(4.28)

Let M be any 5-dimensional CR manifold of hypersurface type which satisfies the conditions in Theorem 3.1.37. Furthermore, we assume that the invariant kˆ ≡ 2i in a neighborhood of p0 . Let ω, ω1 , θ 2 , 1, ξ be the forms given by Theorem 3.1.37 such that ω is given by (3.1.36). Define the h-valued 1-form 5 = (πab ) by  0 π0 π 1 0 5 :=  π 2 0 π03

0 π11 π12 0

0 π21 0 π12

 0 0  , π21  −π11

(4.29)

where π00 , π01 , π02 , π03 , π11 , π12 , π21 are obtained by solving (4.22), also using the first two equations of (4.20). Clearly, 5 defines an isomorphism between Ty Y2 , where Y2 → M is the principal bundle given by Theorem 3.1.37, and h for every y ∈ Y2 . However, it is not difficult to verify that 5 is in general not a Cartan connection, that is, it does not transform according to (4.21). Nevertheless, when we define the curvature = d5 − 5 ∧ 5,

(4.30)

a direct consequence of Cartan’s solution of the equivalence problem for {1}- structures (see, e.g., [G]) is the following characterization of the tube over the light cone. THEOREM 4.31 Let M be a 5-dimensional real-analytic CR manifold of hypersurface type which is 2-nondegenerate and Levi uniform of rank 1 at p0 ∈ M. Assume that kˆ ≡ 2i. Choose an orientation for the normalized CR structure at p0 (as explained in §3), and denote by Y2 → M0 the principal bundle over a neighborhood M0 of p0 , with 1-form ω,


67

given by Theorem 3.1.37. Then the h-valued 1-form 5 defined by (4.29), where  1 ¯   π00 := (1 − 2iθ 2 + 2iθ 2 ),   2     1 ¯   π11 := (1 + 2iθ 2 − 2iθ 2 ),    2     π01 := 4iθ 2 , (4.32)  π02 := 2iω1 ,      π03 := 2iω,      ¯   π12 := i(ω1 − ω1 ),    1  π2 := −ξ ∼ h for every y ∈ Y2 , and ω is given by (3.1.36), defines an isomorphism Ty Y2 = with the following property. There exists a local real-analytic CR diffeomorphism f : M → 0C near p0 if and only if the curvature given by (4.30) vanishes identically. We conclude the discussion of the tube over the light cone by computing the dimension of the stability group Aut(0C , p0 ) of 0C at a point p0 ∈ 0C . Observe that, given a frame (Z 0 , Z A ) in P, the manifold 0C can be viewed as the quotient group H/H0 via the identification P ∼ = H provided by (4.8). Let us denote the affine point on 0C corresponding to Z 0 by p0 . Then, under the identification 0C ∼ = H/H0 , p0 corresponds to the coset eH0 , where e ∈ H denotes the identity matrix. The group H0 acts on the left on H/H0 , and each homomorphism a H0 7 → ba H0 , for b ∈ H0 , preserves the point p0 ∼ = eH0 . It is straightforward to verify that the action is effective; that is, if for b ∈ H0 the homomorphism a H0 7 → ba H0 is the identity, then b = e. Let us denote by f b : (0C , p0 ) → (0C , p0 ) the mapping corresponding to the homomorphism a H0 7→ ba H0 . Each f b is a CR diffeomorphism. (Indeed, it is not difficult to compute f b in coordinates and see that f b is induced by an invertible linear transformation of C3 .) Thus, b 7→ f b embeds H0 as a subgroup of Aut(0C , p0 ). Since dim H0 = 2, we conclude that dim Aut(0C , p0 ) ≥ 2. On the other hand, by Theorem 3.1.37 and [K, Theorem 3.2], it follows (as in the introduction) that the subgroup of Aut(0C , p0 ) consisting of those CR diffeomorphisms that preserve the orientation of the CR structure chosen above embeds as a closed submanifold of Pp0 ∼ = H0 . Hence, we have dim Aut(0C , p0 ) = 2.

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5. Proofs of Theorems 4 and 5, and an example 5.1. Proof of Theorem 4 First of all, since M, M 0 are assumed to be 2-nondegenerate and Levi uniform at p0 ∈ M, p00 ∈ M 0 , it follows (as the reader can verify) that they must in fact be Levi uniform of rank 1 and not of rank 2 (in which case they would be Levi nondegenerate) or zero (in which case they would be Levi flat). Thus, we may apply the main results of the present paper. We keep the notation used in §3 and the convention that M and M 0 are assumed to be sufficiently small neighborhoods of p0 and p00 , respectively. We ˆ p0 ) 6 = 0 and |k( ˆ p0 )| 6= 2. We give here only have two cases to consider, namely, k( the proof in the former case and leave the other case (which is completely analogous) to the reader. For brevity we denote by G the group G 2 and by P → M, P 0 → M 0 the principal G-bundles Y2 → M, Y20 → M given by Theorem 3.1.37 (where some choice of orientations for the normalized CR structures of M and M 0 at p0 and p00 , respectively, 0 0 have been made). Let θ, θ 1 , θ 2 and θ 0 , θ 1 , θ 2 be 1-forms on M and M 0 near p0 and p00 , respectively, such that the principal bundles P → M, P 0 → M 0 are locally trivialized: P ∼ = M × G, P 0 ∼ = M 0 × G. Now, suppose that f : (M, p0 ) → (M 0 , p00 ) is a CR diffeomorphism. (There is no loss of generality in assuming that f preserves the orientations of the CR structures chosen above.) For the convenience of the reader, we give here an explicit description of the diffeomorphism F : P → P 0 provided by Theorem 3.1.37. Since f is a CR diffeomorphism, the 1-forms θ˜ := f ∗ (θ 0 ), θ˜ α := f ∗ (θ α )0 , α = 1, 2, (5.1.1) yield a basis for the sections of T 0 M. Moreover, θ˜ , θ˜ 1 yield a basis for the sections of T 00 M and θ˜ is a characteristic form near p0 . In addition, there exists a smooth function γ near p0 , valued in G, such that     θ θ˜  1  ˜1 θ θ (5.1.2)   = γ  . ¯ ¯ θ1 θ˜ 1 The mapping F : P → P 0 is defined by F( p, S) := ( f ( p), Sγ ( p)−1 ),

( p, S) ∈ M × G,

(5.1.3)

in the local trivializations P ∼ = M × G, P 0 ∼ = M 0 × G (cf. also [G]). From this it follows that, for example, F( p0 , e), where e ∈ G denotes the identity, is completely determined by j p10 ( f ).


69

We now complete the proof of Theorem 4. Let f 0 , f 1 : (M, p0 ) → (M 0 , p00 ) be CR mappings as in the theorem. Denote by F 0 , F 1 : P → P 0 the lifts of f 0 and f 1 , respectively. As mentioned above, the assumption that j p10 ( f 0 ) = j p10 ( f 1 ) implies that F 0 ( p0 , e) = F 1 ( p0 , e). The conclusion of Theorem 4 is now a consequence of the following simple uniqueness result, whose proof is left to the reader. LEMMA 5.1.4 Let η1 , . . . , ηk and η10 , . . . , ηk0 be 1-forms near the origin in Rk such that η1 (0), . . . , ηk (0) as well as η10 (0), . . . , ηk0 (0) span T0 Rk . Assume that f 1 , f 2 : (Rk , 0) → (Rk , 0) are smooth mappings such that

( f i )∗ (η0j ) = η j ,

i = 1, 2, j = 1, . . . , k.

(5.1.5)

Then f 1 ≡ f 2 near zero. This completes the proof of Theorem 4. 5.2. Proof of Theorem 5 In view of Theorem 4, it suffices to prove the last statement in Theorem 5, namely, that f 0 is a diffeomorphism. To prove this we apply results about geometric properties of CR mappings which impose conditions, not previously introduced in this paper, on the manifolds. We refer the reader to [BER] for relevant definitions and basic results. For the remainder of this section, we denote f 0 simply by f . We first claim that the generic rank of f must in fact be 5 (= dimR M). Indeed, since M is 2nondegenerate at p0 , there is no formal holomorphic vector field tangent to it at p0 (see [BER, Theorem 11.7.5]). Moreover, since M is in addition minimal at p0 , [BR3, Theorem 2] implies that the generic rank of f is either full (i.e., 5) or even, and hence, by the generic rank assumption on f , the generic rank is either 4 or 5. It is shown in [BR3] that if the generic rank of f is 4, then M 0 contains 2-dimensional complex submanifolds (i.e., complex hypersurfaces) through points p 0 ∈ M 0 arbitrarily close to p00 . Hence, in this case M 0 is not minimal in any neighborhood of p0 . This contradicts the assumption that M 0 is 2-nondegenerate at p00 , since finite nondegeneracy implies minimality and finite nondegeneracy is an open condition. This proves the claim above. We now choose local coordinates Z = (z, w) ∈ C2 × C vanishing at p0 , and 0 Z = (z 0 , w0 ) ∈ C2 × C vanishing at p00 such that M and M 0 are defined, respectively, near p0 = (0, 0) and p00 = (0, 0) by Im w = φ(z, z¯ , Re w)

and

Im w0 = φ 0 (z 0 , z¯ 0 , Re w0 ),

(5.2.1)

where φ(0) = φ 0 (0, 0) = 0 and dφ(0) = dφ 0 (0) = 0. Let H (Z ) denote the formal power series in Z = (Z 1 , Z 2 , Z 3 ) associated with the smooth CR map-

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PETER EBENFELT

ping f at p0 (see [BER, Proposition 1.7.4]). To show that f is a local CR diffeomorphism at p0 = 0, it suffices to show that the formal mapping H is invertible at zero. In the coordinates (z, w), and (z 0 , w0 ) introduced above, we can write H (z, w) = F (z, w), G (z, w) , where F is valued in C2 and G is valued in C. The fact that f maps M into M 0 implies that G z, s + iφ(z, z¯ , s) ∼ φ 0 F z, s + iφ(z, z¯ , s) , F¯ z¯ , s − iφ(z, z¯ , s) , Re G z, s + iφ(z, z¯ , s) , (5.2.2) where ∼ denotes equality of power series. By applying the two vector fields L j :=

φz¯ j ∂ ∂ − 2i , ∂ z¯ j 1 + iφs ∂ w¯

j = 1, 2,

(5.2.3)

which are tangent to M and form a basis for the CR vector fields near zero, to (5.2.2) and setting (z, z¯ , s) = (0, 0, 0), we first conclude that ∂ G /∂z j (0) = 0 for j = 1, 2. We prove that H is invertible by showing that ∂ G /∂w(0) 6= 0 and det(∂ F j /∂z k (0)) 6= 0, where j, k = 1, 2. Recall from the proof of Theorem 4 above that both M and M 0 are Levi uniform of rank 1 at p0 and p00 , respectively. It follows that there is a smooth choice of Levi form in a neighborhood of p0 in M which is positive semidefinite in that neighborhood and hence that M is pseudoconvex near p0 . A similar argument shows that M 0 is also pseudoconvex near p00 . Since M 0 is both minimal and pseudoconvex at p00 = 0, it is also minimally convex (see [BR4] for the definition of minimal convexity and this particular statement). Since f : (M, p0 ) → (M 0 , p00 ), where M is minimal at p0 and M 0 is minimally convex at p00 , has generic full rank, [BR4, Theorem 2] implies that ∂ G /∂w(0, 0) 6 = 0. (This fact can also be deduced by combining [Be, Theorem 3] with [BER, Proposition 9.4.10].) To show that det(∂ F j /∂z k (0)) 6= 0, where j, k = 1, 2, we point out that finite nondegeneracy implies essential finiteness. Indeed, finite nondegeneracy at a point is equivalent to essential finiteness and to essential type 1 at that point (see [BER, Proposition 11.8.27]; the reader is also referred to [BER, Chapter 11] for the definitions of essential finiteness and essential type). By [BR2, Theorem 2] and [BR1, Theorem 3], we have dimC C[[z]]/I F1 (z, 0), F2 (z, 0) = 1, (5.2.4) where C[[z]] denotes the ring of formal power series in z = (z 1 , z 2 ) and the nota tion I F1 (z, 0), F2 (z, 0) stands for the ideal in C[[z]] generated by F1 (z, 0) and F2 (z, 0). It is not difficult to see that (5.2.4) is equivalent to det ∂ F j /∂z k (0) 6 = 0. This completes the proof of the fact that f is a local CR diffeomorphism near p0 and hence the proof of Theorem 5.


71

5.3. An example showing necessity of the generic rank condition in Theorem 5 Recall that 0C ⊂ C3 denotes the smooth locus of the tube over the light cone in R3 (see Example 0.2.1 and §4). Observe that 0C is geometrically convex. Let us fix a point, say, p0 = (1, 0, 1) ∈ 0C . The convexity of 0C near p0 implies that the real linear form Re `(Z ), where `(Z ) := Re Z 3 − Re Z 1 , is nonnegative for all p ∈ 0C near p0 . (This is of course easy to check directly as well.) Define, for Z ∈ C3 in the half-space S := {Re `(Z ) ≥ 0}, p h(Z ) := `(Z ), (5.3.1) where the square root in (5.3.1) is the branch in the right half-plane which preserves the point ζ = 1. The reader can verify that the function J (Z ) := e−1/ h(Z )

(5.3.2)

is holomorphic in the open half-plane D, is C ∞ in its closure D, and vanishes to infinite order on the boundary ∂ D (see also [BER, Example 1.4.9]). Since 0C is locally contained in D near p0 , meets D in a dense open subset near p0 , and meets ∂ D at p0 (indeed, along a complex line contained in 0C ), we deduce that the restriction j of J to a neighborhood of p0 in 0C is a smooth CR function with generic (real) rank 2 which vanishes to infinite order at p0 . Let M 0 ⊂ C3 be any smooth real hypersurface that is Levi uniform of rank 1 at 0 p0 ∈ M 0 . This implies, in particular, that M 0 is foliated near p00 by complex curves (see, e.g., Proposition 1.12). Let γ : (C, 0) → (C3 , p00 ) be a parametrization of the complex curve through p00 . The CR mapping f 0 : (0C , p0 ) → (M 0 , p00 ) defined by f 0 ( p) := γ J ( p) , for p ∈ 0C near p0 , is then smooth, of generic rank 2, and vanishes to infinite order at p0 . Hence, if we also choose M to be 2-nondegenerate at p00 , all conditions of Theorem 5 except the generic rank condition on f 0 are satisfied but the conclusion fails. 6. Examples of everywhere Levi degenerate CR manifolds and computations of kˆ Trivial examples of real hypersurfaces in Cn+1 which are everywhere Levi degenerate can be obtained by taking any hypersurface of the form M˜ × C, where M˜ is a real hypersurface in Cn . Such hypersurfaces, as mentioned in the introduction, are never 2-nondegenerate and hence are not of interest to us in the present paper. We give here two (from our viewpoint) more interesting situations where everywhere Levi degenerate hypersurfaces in Cn+1 arise naturally. We also exhibit two classes of such manifolds for which the invariant kˆ satisfies kˆ ≡ 2i and hence for which Theorem 2 is applicable (see Propositions 6.13 and 6.27). We conclude with a result showing that the curvature in Theorem 2 is a nontrivial invariant.

72

PETER EBENFELT

Example 6.1 (Everywhere characteristic hypersurfaces) Let p(Z , ∂) := p(Z , ∂/∂ Z ) be a homogeneous holomorphic partial differential operator; that is, p(Z , ζ ) is a holomorphic function of (Z , ζ ) in some domain U ×Cn+1 ⊂ Cn+1 × Cn+1 and a homogeneous polynomial in ζ , and ∂/∂ Z = (∂/∂ Z 1 , . . . , ∂/∂ Z n+1 )

(6.2)

( p(Z , ζ ) is called the principal symbol of the operator p(Z , ∂)). A real hypersurface M ⊂ Cn+1 is called characteristic at p0 ∈ M ∩ U for p(Z , ∂) (or, more generally, for any holomorphic partial differential operator whose principal symbol is p(Z , ζ )) if p p0 , ∂ρ( p0 , p¯ 0 )/∂ Z = 0, (6.3) where ρ(Z , Z¯ ) = 0 is a defining equation for M near p0 , and M is called everywhere characteristic if M ∩ U is characteristic at every point. Observe that if p(Z , ∂) is homogeneous of degree 1 (i.e., it is a holomorphic vector field), then M is everywhere characteristic for p(Z , ∂) if and only if the holomorphic vector field p(Z , ∂) is tangent to M in U . Hence, being everywhere characteristic for a holomorphic partial differential operator can be viewed as a higher-degree analogue of holomorphic degeneracy (see [BER, Chapter 11]). Everywhere characteristic hypersurfaces, relative to a given operator p(Z , ∂), arise as natural boundaries for the holomorphic continuation of (holomorphic) solutions of p(Z , ∂)u = 0 (see, e.g., [Ho, Chapter 9.4]). A concrete example is given by the so-called Lie ball defined by the equation  n+1 2 1/2 X |Z |2 + |Z |4 − Z k2  < 1. (6.4) k=1

The Lie ball is the maximal domain in Cn+1 to which every harmonic function in the unit ball of Rn+1 can be holomorphically continued (see, e.g., [A]; cf. also [E1]). The boundary of the Lie ball is everywhere characteristic (at every smooth point) for the “Laplace operator” n+1 X (∂/∂ Z k )2 . (6.5) j=1

Another example is the tube over the light cone (see Example 0.2.1) which is everywhere characteristic for the “wave operator” n X (∂/∂ Z k )2 − (∂/∂ Z n+1 )2 . j=1

We have the following.

(6.6)


73

PROPOSITION 6.7 Let M ⊂ Cn+1 be a real hypersurface that is everywhere characteristic for a homogeneous partial differential operator p(Z , ∂). Then M is everywhere Levi degenerate.

Proof Pick p0 ∈ M, and let ρ(Z , Z¯ ) = 0 be a defining equation for M near p0 ∈ M. We first claim that the CR vector field L :=

n+1 X k=1

∂ pζk Z , ρ Z (Z , Z¯ ) , ∂ Z¯ k

(6.8)

where ρ Z := ∂ρ/∂ Z and pζ = ∂ p/∂ζ , is tangent to M. Indeed, since M is everywhere characteristic for p(∂), we have p Z , ρ Z (Z , Z¯ ) = a(Z , Z¯ )ρ(Z , Z¯ ) (6.9) for some function a. The claim now follows from Euler’s formula. By differentiating (6.9) with respect to Z¯ , it is straightforward (and left to the reader) to verify that L is a null vector for the Levi form at every p ∈ M near p0 . This proves the proposition. Example 6.10 (Tubes over homogeneous algebraic varieties) Let p(x) be a homogeneous polynomial in x = (x1 , . . . , xn+1 ) with real coefficients, and assume that ∂ p/∂ x is not identically zero along the variety VR := {x ∈ Rn+1 : p(x) = 0}. Then the tube VC over VR in Cn+1 , VC := {Z ∈ Cn+1 : p(Re Z ) = 0},

(6.11)

is a real hypersurface (outside a lower-dimensional real algebraic subvariety) which we denote by M. The “radial” CR vector field L=

n+1 X

Re Z j ∂/∂ Z¯ j

(6.12)

j=1

is tangent to M, and the reader can easily verify that L is a null vector for the Levi form of M at every p ∈ M. A concrete example is again the tube over the light cone (Example 0.2.1). Another example is the cubic defined by (6.11) with p(x) = x13 + x23 − x33 which was given by Freeman [F2] as an example of a manifold foliated by complex curves but not locally biholomorphic to a manifold of the form M˜ × C. We compute the invariant kˆ for two classes of real hypersurfaces M ⊂ C3 that share some interesting properties with the tube over the light cone, and show that Theorem 2 is applicable for manifolds from these classes. First, we consider hypersurfaces M ⊂ C3 which are tubes over homogeneous algebraic varieties in R3 as in Example 6.10 above . For convenience we replace Re Z above by Z + Z¯ .

74

PETER EBENFELT

PROPOSITION 6.13 Let M ⊂ C3 be a real hypersurface that is contained in the real algebraic variety

V := {Z ∈ C3 : p (Z + Z¯ ) = 0},

(6.14)

where p(x) is a real-valued homogeneous polynomial. Assume that M is Levi uniform and 2-nondegenerate at Z 0 ∈ M. Then the invariant kˆ ≡ 2i in a neighborhood of Z 0 . Proof For simplicity we write x = Z + Z¯ . There is no loss of generality in assuming that x10 6= 0 and px3 (x 0 ) 6= 0. As a basis for the CR vector fields on M near Z 0 we choose L1 =

1 ( px3 ∂/∂ Z 2 − px2 ∂/∂ Z 3 ), N1

L2 =

3 1 X x j ∂/∂ Z j , N2

(6.15)

j=1

where N1 , N2 are factors to be chosen. As mentioned above, L 2 spans the Levi null bundle N near Z 0 . For θ, θ 1 , θ 2 we choose (the pullbacks to M of ) θ =i

3 X

(6.16)

px j d Z j

j=1

and θ1 =

N1 (x1 d Z 2 − x2 d Z 1 ), x 1 px3

θ2 =

N2 d Z 1. x1

(6.17)

The reader can verify that θ spans T 0 M, that θ , θ 1 span T 00 M, and that θ, θ 1 , θ 2 span

T 0 M, as defined in §2. Moreover, we have θ α , L β = δβα , and a calculation shows that the nonzero entry g11 ¯ of the Levi form is given by g11 ¯ :=

1

1 dθ, L 1¯ ∧ L 1 = 2 Q(x), i N1

(6.18)

where Q(x) denotes the quadratic form Q(x) = px23 px2 x2 − 2 px2 px3 px2 x3 + px22 px3 x3 .

(6.19)

Since M is assumed Levi uniform and 2-nondegenerate (and hence Levi uniform of rank 1; see the proof of Theorem 5), we have Q(x) 6= 0 near x 0 . We may assume that Q(x) > 0 in a neighborhood of x 0 since otherwise we could replace p(x) by − p(x). So, we choose p (6.20) N1 (x) = Q(x) 0 and conclude that g11 ¯ ≡ 1 near x .


75

To compute the only nonzero element h 1¯ 12 ¯ in the third-order tensor, we use Lemma 2.12, equation (2.16), and the fact that g11 ¯ = 1. After a straightforward calculation, we obtain

2i h 1¯ 12 ¯

√ √ ∂ Q ∂ x2 Q px3 x2 − x1 √ ∂ x 2 px3 ∂ x 2 x 1 px3 N2 Q √ √ px2 ∂ Q ∂ x2 Q − − x1 . (6.21) x2 √ ∂ x 3 px3 ∂ x 3 x 1 px3 N2 Q

= dθ 1 , L 1¯ ∧ L 2 =

Let us simplify the expression √ √ px ∂ Q ∂ x2 Q E(x) := √ 3 x2 − x1 ∂ x 2 px3 ∂ x 2 x 1 px3 Q √ √ px2 ∂ Q ∂ x2 Q −√ − x1 x2 ∂ x 3 px3 ∂ x 3 x 1 px3 Q appearing in (6.21). Observe that, for j = 2, 3, √ √ √ ∂ x2 Q 1 Q x2 ∂ Q = δ2 j + . ∂ x j x 1 px3 x 1 px3 x 1 ∂ x j px3

(6.22)

(6.23)

Substituting this in (6.22), we find E(x) = 1.

(6.24)

Thus, (6.21) can be simplified to 2i h 1¯ 12 ¯ = 1/N2 , and hence we choose N2 (x) =

1 , 2i

(6.25)

so that h 1¯ 12 ¯ = 1. With these choices of N1 , N2 , as is shown in §3, the invariant kˆ is computed from the formula

kˆ = dθ 2 , L 2¯ ∧ L 2 =−

3 X N2 1 ¯ ¯ j ∧ ∂/∂ Z k d Z ∧ d Z , x x ∂/∂ Z j k 1 1 x12 |N2 |2 j,k=1

=−

1 = 2i. ¯ N2

(6.26)

This completes the proof of Proposition 6.13. We also compute kˆ for tubes over hypersurfaces in R3 which are everywhere characteristic for the wave operator. Observe that if {x ∈ R3 : f (x) = 0} is a hypersurface in R3 which is everywhere characteristic for the wave operator (∂/∂ x1 )2

76

PETER EBENFELT

+(∂/∂ x2 )2 − (∂/∂ x3 )2 , then the real hypersurface M ⊂ C3 defined by M = {Z ∈ C3 : f (Re Z ) = 0} is everywhere characteristic for the complex wave operator (∂/∂ Z 1 )2 + (∂/∂ Z 2 )2 − (∂/∂ Z 3 )2 and hence is of the type considered in Example 6.1. As above, we use Z + Z¯ instead of Re Z . PROPOSITION 6.27 Let M ⊂ C3 be a

real smooth hypersurface, defined locally near Z 0 ∈ M by an equation of the form f (Z + Z¯ ) = 0. Assume that the hypersurface { f (x) = 0} ⊂ R3 is everywhere characteristic for the wave operator near x 0 = Z 0 + Z¯ 0 , that is, f x21 + f x22 − f x23 = 0,

(6.28)

and that M is Levi uniform and 2-nondegenerate at Z 0 . Then kˆ ≡ 2i near Z 0 . Proof As in the proof of Proposition 6.13, we use x = Z¯ + Z¯ . We observe that we may choose 1 L2 = f x1 ∂/∂ Z 1 + f x2 ∂/∂ Z 2 − f x3 ∂/∂ Z 3 (6.29) N2 as a vector field spanning the Levi null bundle N near Z 0 . Also, take θ1 =

N1 f x1 f x3

f x1 d Z 2 − f x2 d Z 1 ,

(6.30)

and choose L 1 , θ, and θ 2 as in the proof of Proposition 6.13 (replacing p by f , of course). By repeating the calculations carried out in the proof of Proposition 6.13 and choosing N1 , N2 accordingly, we deduce that if f x1 (x 0 ) f x3 (x 0 ) 6= 0 (which we may assume without loss of generality), then ˆ ) = −2i f x1 (x) X R k(Z (x), (6.31) f x1 R(x)2 where R=

∂ f x2 f x2 f x1 ∂ f x2 − f x1 f x3 ∂ x 3 f x1 ∂ x 2 f x1

and X = f x1

∂ ∂ ∂ + f x2 − f x3 . ∂ x1 ∂ x2 ∂ x3

(6.32)

(6.33)

The details of these calculations are left to the reader. Pick Z 1 ∈ M near Z 0 . After a translation, and applying a real Lorentz transformation if necessary, we may assume that Z 1 = 0, − f x1 (0) = f x3 (0) = 1, and f x2 (0) = 0. By solving for x3 in the equation


77

f (x) = 0, we may also assume that f (x) = x3 − φ(x1 , x2 ), where φ(x1 , x2 ) satisfies φ(0, 0) = φx2 (0, 0) = 0, φx1 (0, 0) = 1, and φx21 + φx22 = 1.

(6.34)

A straightforward calculation shows that M is Levi uniform of rank 1 and 2-nondegenerate at zero if and only if φx2 x2 (0, 0) 6 = 0. By differentiating (6.34), we find that φx1 x j (0, 0) = 0,

φx1 x j xk (0, 0) = −φx2 x j (0, 0)φx2 xk (0, 0),

(6.35)

for j, k = 1, 2. By substituting f (x) = x3 − φ(x1 , x2 ) in (6.31) and using (6.35), we ˆ 1 ) = 2i. The proof of Proposition 6.27 is now complete since Z 1 is conclude that k(Z arbitrary. We conclude the paper with the following result which implies that in general the curvature in Theorem 2 is nontrivial. 6.36 For an integer k ≥ 2, let PROPOSITION

Vk := {Z ∈ C3 : (Z 1 + Z¯ 1 )k + (Z 2 + Z¯ 2 )k − (Z 3 + Z¯ 3 )k = 0}.

(6.37)

If k ≥ 3, then for any p ∈ Vk , there is no local biholomorphism near p sending Vk into V2 (which is the tube over the light cone). Proof Suppose, in order to reach a contradiction, that there exist a p ∈ Vk and a local biholomorphism Z 0 = H (Z ) near p with H (Vk ) ⊂ V2 . There is no loss of generality in assuming that p is a smooth point of Vk ∩R3 and that H ( p) is a smooth point of V2 . Since V2 is everywhere characteristic for the complex wave operator (6.6), Vk must be everywhere characteristic for the homogeneous second-order partial differential operator with principal symbol ! 3 0 0 0 X ∂ Z i0 ∂ Z j ∂ Z i0 ∂ Z j ∂ Z i0 ∂ Z j p(Z ; ζ ) = + − ζi ζ j (6.38) ∂ Z1 ∂ Z1 ∂ Z2 ∂ Z2 ∂ Z3 ∂ Z3 i, j=1

(cf., e.g., [Ho, Chapter 6]); that is, we must have p Z ; (x1k−1 , x2k−1 , −x3k−1 ) = a(Z , Z¯ )(x1k + x2k − x3k ),

(6.39)

where, as above, x = Z + Z¯ . By restricting to real values of Z and identifying the homogeneous terms of order 2k − 2, we get Q(x1k−1 , x2k−1 , −x3k−1 ) = b(x)(x1k + x2k − x3k ),

(6.40)

78

PETER EBENFELT

for some nondegenerate quadratic form Q(ζ ) and some homogeneous polynomial b(x) of degree k − 2. It is not difficult to see, and left to the reader to verify, that this cannot happen unless k = 2. This completes the proof of Proposition 6.36. Acknowledgment. The author would like to thank the anonymous referee for many comments that helped to improve the presentation of this work. References [A]

V. AVANISSIAN, Cellule d’harmonicité et prolongement analytique complexe, Travaux

[BER]

M. S. BAOUENDI, P. EBENFELT, and L. P. ROTHSCHILD, Real Submanifolds in Complex

en Cours, Hermann, Paris, 1985. MR 87j:31009 72

[BHR]

[BJT] [BR1] [BR2] [BR3] [BR4] [Be] [BS]

[CS] [C1]

[C2]

[CM]

Space and Their Mappings, Princeton Math. Ser. 47, Princeton Univ. Press, Princeton, 1999. MR 2000b:32066 37, 40, 43, 44, 45, 47, 69, 70, 71, 72 M. S. BAOUENDI, X. HUANG, and L. P. ROTHSCHILD, Regularity of CR mappings between algebraic hypersurfaces, Invent. Math. 125 (1996), 13–36. MR 97c:32028 45 M. S. BAOUENDI, H. JACOBOWITZ, and F. TREVES, On the analyticity of CR mappings, Ann. of Math. (2) 122 (1985), 365–400. MR 87f:32044 42, 43 M. S. BAOUENDI and L. P. ROTHSCHILD, Germs of CR maps between real analytic hypersurfaces, Invent. Math. 93 (1988), 481–500. MR 90a:32036 70 , Geometric properties of mappings between hypersurfaces in complex space, J. Differential Geom. 31 (1990), 473–499. MR 91g:32032 70 , Remarks on the generic rank of a CR mapping, J. Geom. Anal. 2 (1992), 1–9. MR 92m:32025 43, 69 , A generalized complex Hopf lemma and its applications to CR mappings, Invent. Math. 111 (1993), 331–348. MR 95c:32017 70 S. BELL, Local regularity of CR homeomorphisms, Duke Math. J. 57 (1988), 295–300. MR 89f:32026 70 D. BURNS, JR. and S. SHNIDER, “Real hypersurfaces in complex manifolds” in Several Complex Variables (Williamstown, Mass., 1975), Proc. Sympos. Pure Math. 30, Part 2, Amer. Math. Soc., Providence, 1977, 141–168. MR 56:8896 38 ˇ A. CAP and H. SCHICHL, Parabolic geometries and canonical Cartan connections, Hokkaido Math. J. 29 (2000), 453–505. MR CMP 1 795 487 39 E. CARTAN, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I, Ann. Math. Pura Appl. 11 (1932), 17–90 (or in Oeuvres complètes, Part 2, Vol. 2, Gauthier-Villars, Paris, 1953, 1231–1304). MR 15:383o 38, 45 , Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, II, Ann. Scoula Norm. Sup. Pisa 1 (1932), 333–354 (or in Oeuvres complètes, Part 3, Vol. 2, Gauthier-Villars, Paris, 1955, 1217–1238). MR 17:697u 38, 45 S.-S. CHERN and J. K. MOSER, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 54:13112 38, 43, 45, 58, 64


[E1]

[E2]

[E3]

[E4] [Er] [EIS]

[F1] [F2]

[G]

[GM] [Ha] [Hay] [He] [Ho]

[K] [KN]

[SS]

79

P. EBENFELT, Holomorphic extension of solutions of elliptic partial differential

equations and a complex Huygens’ principle, J. London Math. Soc. (2) 55 (1997), 87–104. MR 98g:35029 72 , “Nondegeneracy conditions and normal forms for real hypersurfaces in ´ complex space” in Journées “Equations aux Dérivées Partielles” ´ (Saint-Jean-de-Monts, 1997), Ecole Polytech., Palaiseau, 1997, exp. no. 7. MR 2000b:32074 44 , New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy, J. Differential Geom. 50 (1998), 207–247. MR 2000a:32077 39, 46 , Normal forms and biholomorphic equivalence of real hypersurfaces in C3 , Indiana Univ. Math. J. 47 (1998), 311–366. MR 2000h:32053 39, 40 A. E. ERSHOVA, Automorphisms of 2-nondegenerate hypersurfaces in C3 (in Russian), Mat. Zametki 69 (2001), 214-222. MR CMP 1 830 221 42 V. V. EZHOV, A. V. ISAEV, and G. SCHMALZ, Invariants of elliptic and hyperbolic CR-structures of codimension 2, Internat. J. Math. 10 (1999), 1–52. MR 2000d:32052 39 M. FREEMAN, Local biholomorphic straightening of real submanifolds, Ann. of Math. (2) 106 (1977), 319–352. MR 57:3429 39, 46, 47 , “Real submanifolds with degenerate Levi form” in Several Complex Variables (Williamstown, Mass., 1975), Proc. Sympos. Pure Math. 30, Part 1, Amer. Math. Soc., Providence, 1977, 141–147. MR 56:15971 73 R. B. GARDNER, The Method of Equivalence and Its Applications, CBMS-NSF Regional Conf. Ser. Appl. Math. 58, Soc. Indust. Appl. Math., Philadelphia, 1989. MR 91j:58007 38, 49, 52, 58, 66, 68 T. GARRITY and R. MIZNER, The equivalence problem for higher-codimensional CR structures, Pacific J. Math. 177 (1997), 211–235. MR 99b:32007 39 C.-K. HAN, Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms, Math. Ann. 309 (1997), 401–409. MR 99b:32024 43 A. HAYASHIMOTO, On the complete system of finite order for CR mappings and its application, Osaka J. Math. 35 (1998), 617–628. MR 2000b:32071 43 S. HELGASON, Differential Geometry and Symmetric Spaces, Pure Appl. Math. 12, Academic Press, New York, 1962. MR 26:2986 46 ¨ L. HORMANDER , The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis, Grundlehren Math. Wiss. 256, Springer, Berlin, 1983. MR 85g:35002a 72, 77 S. KOBAYASHI, Transformation Groups in Differential Geometry, Ergeb. Math. Grenzgeb. 70, Springer, New York, 1972. MR 50:8360 38, 42, 64, 67 S. KOBAYASHI and K. NOMIZU, Foundations of Differential Geometry, I, Interscience Tracts in Pure Appl. Math. 15, Vol. 1, Interscience, New York, 1963, MR 27:2945; II, Interscience Tracts in Pure Appl. Math. 15, Vol. 2, Interscience, New York, 1969. MR 38:6501 38 ´ , The geometry of hyperbolic and elliptic CR-manifolds of G. SCHMALZ and J. SLOVAK codimension two, Asian J. Math. 4 (2000), 565–597. MR CMP 1 796 695 39

80

[SV]

[Si] [Sta1] [Sta2] [Ste] [T1] [T2] [We1] [We2]

[Wo]

PETER EBENFELT

A. G. SERGEEV and V. S. VLADIMIROV, “Complex analysis in the future tube” in

Several Complex Variables, II, Encyclopaedia Math. Sci. 8, Springer, Berlin, 1994, 179–253. MR 95e:32001 40 C. L. SIEGEL, Symplectic geometry, Amer. J. Math. 65 (1943), 1–86. MR 4:242b N. STANTON, Infinitesimal CR automorphisms of rigid hypersurfaces, Amer. J. Math. 117 (1995), 141–167. MR 96a:32036 40 , Infinitesimal CR automorphisms of real hypersurfaces, Amer. J. Math. 118 (1996), 209–233. MR 97h:32027 40 S. STERNBERG, Lectures on Differential Geometry, 2d ed., Chelsea, New York, 1983. MR 88f:58001 38, 51 N. TANAKA, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14 (1962), 397–429. MR 26:3086 38, 45 , On generalized graded Lie algebras and geometric structures, I, J. Math. Soc. Japan 19 (1967), 215–254. MR 36:4470 38, 45 S. M. WEBSTER, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41. MR 80e:32015 38 , “The holomorphic contact geometry of a real hypersurface” in Modern Methods in Complex Analysis, Ann. of Math. Stud. 137, Princeton Univ. Press, Princeton, 1995, 327–342. MR 96K:32035 46 P. P. WONG, A construction of normal forms for weakly pseudoconvex CR manifolds in C2 , Invent. Math. 69 (1982), 311–329. MR 84e:32020a 39

Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden; [email protected]; current: Department of Mathematics, University of California at San Diego, La Jolla, California 92093, USA; [email protected]


SUR LA NATURE NON-CYCLOTOMIQUE DES POINTS D’ORDRE FINI DES COURBES ELLIPTIQUES LOÏC MEREL

Appendice de E. Kowalski et P. Michel Résumé Nous e´ tudions le corps K p engendré par les points d’ordre premier p d’une courbe elliptique sur un corps de nombres. Avec l’aide de Kowalski et Michel, nous démontrons que pour presque tout nombre premier p, toute courbe elliptique sur le corps cyclotomique Q(µ p ) possédant un sous-groupe cyclique Q(µ p )-rationnel d’ordre p a potentiellement bonne réduction en caractéristique p. Nos méthodes s’appliquent aussi pour e´ tudier, nombre premier par nombre premier, les courbes elliptiques n’ayant pas potentiellement bonne réduction en p. En particulier, nous démontrons qu’on a K p 6 = Q(µ p ) pour 5 < p < 1000 et p ≡ 1 (mod 4). Nous faisons un usage crucial des travaux de Kato sur la conjecture de Birch et Swinnerton-Dyer. 0. Introduction Lorsque E est une courbe elliptique sur un corps de nombres, le corps K p (E) engendré par les points de p-division de E contient un corps cyclotomique Q(µ p ) engendré par les racines p-ièmes de l’unité — Cela est une conséquence e´ vidente de la structure fournie par les accouplements de Weil. Cet article vise a` e´ tablir que K p (E) ne peut guère eˆ tre aussi petit que Q(µ p ). Notons S l’ensemble des nombres premiers p tels qu’il existe une telle courbe elliptique avec Q(µ p ) = K p (E). Il est connu que l’ensemble S contient les nombres 2, 3 et 5 (puisque la courbe modulaire paramétrant le problème de module correspondant est de genre nul et possède au moins un point Q(µ p )-rationnel). E. Halberstadt semble avoir e´ tabli récemment que S ne contient pas 7 (voir [6]). Nous sommes incapable d’établir la finitude de S (les techniques de [21] sont de peu de secours). Pour aborder cette dernière question, il semble naturel de séparer le problème en trois DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1, Received 19 May 2000. Revision received 18 September 2000. 2000 Mathematics Subject Classification. Primary 11F, 11G, 11M, 14G.

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classes de courbes elliptiques sur Q(µ p ), selon la géométrie de la fibre en l’idéal P au dessus de p du modèle de Néron sur Z[µ p ] : — Le cas p-cuspidal, c’est-à-dire le cas des courbes elliptiques n’ayant pas potentiellement bonne réduction en P . Nous obtenons beaucoup d’informations dans ce cas. — Le cas p-ordinaire, c’est-à-dire le cas des courbes elliptiques ayant potentiellement bonne réduction ordinaire en P . Dans le cas ordinaire on peut distinguer le cas (non déterminé par la géométrie) des anomalies, c’est-à-dire le cas où la réduction modulo P de la courbe elliptique possède un point d’ordre p. Une e´ tude facile montre que les courbes elliptiques ordinaires donnant lieu aux e´ léments de S présentent des anomalies. — Le cas p-supersingulier, qui concerne, comme on s’en doute, les courbes elliptiques ayant réduction potentiellement supersingulière en caractéristique p. J. Oesterlé m’a convaincu que ces courbes ne donnent naissance a` aucun e´ lément p > 3 de S. Faute de pouvoir montrer la finitude de S, nous nous proposons d’aller dans la direction générale suivante : e´ tudier les ensembles S0,c , S0,o et S0,s de nombres premiers pour lesquels il existe une courbe elliptique sur Q(µ p ) p-cuspidale, p-ordinaire ou psupersingulière respectivement munie d’un sous-groupe C d’ordre p qui est Q(µ p )rationnel. Ajoutons que notre méthode semble indiquer que l’énumération des trois cas dénote une difficulté croissante. Dans le cas cuspidal, nous obtenons le résultat suivant (conséquence de la proposition 5, du corollaire 3 de la proposition 6 et de l’appendice). ´ OR E` ME TH E L’ensemble S0,c est fini. Les techniques utilisées dans notre preuve font appel de façon centrale aux idées vieilles de vingt ans de B. Mazur [17] et aux résultats récents de K. Kato (heureusement rédigés pour l’essentiel par A. Scholl [26]) en direction de la conjecture de Birch et Swinnerton-Dyer. Nous procédons en deux e´ tapes : d’abord nous nous efforçons de montrer que les points Q(µ p )-rationnels p-cuspidaux (en le sens ci-dessus, mais notre méthode s’applique aussi parfois aux points p-ordinaires) de la courbe modulaire X 0 ( p) sont quadratiques réels. Nous concluons en appliquant les méthodes de S. Kamienny d’étude des points quadratiques de X 0 ( p). Soient p un nombre premier et χ un caractère de Dirichlet (Z/ pZ)∗ −→ C∗ . On résumera par H p (χ) l’assertion suivante : il existe une forme modulaire parabolique P n f = ere L( f, χ, s) qui n an q de poids 2 pour 00 ( p) telle que la fonction enti`

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P −s ne s’annule pas en s = 1. Nous prolonge la série de Dirichlet ∞ n=1 an χ(n)n démontrons que les nombres premiers appartenant a` S0,c ne vérifient pas H p (χ ) pour au moins un caractère χ : (Z/ pZ)∗ −→ C∗ non quadratique pair. L’étude de H p (χ ) est donc essentielle a` notre démonstration. E. Kowalski et P. Michel démontrent que tout nombre premier p > 1025 vérifie H p (χ ) pour χ non quadratique pair (voir [13]). L’une de leurs démonstrations, qui est e´ lémentaire, est donnée en appendice. On est tenté de ne pas se satisfaire de leur borne. C’est pourquoi nous faisons quelques efforts, indépendamment de Kowalski et Michel, visant a` e´ tablir H p (χ) dans la deuxième partie. Nos résultats sont partiels, mais nous espérons qu’ils jettent une lumière intéressante sur l’hypothèse H p (χ). En particulier, nous démontrons que H p (χ) est vérifiée dans les trois cas suivants : lorsque p est un nombre premier congru a` 11 ou 19 modulo 20 et χ est impair, lorsque χ est injectif et p ∈ / {2, 3, 5, 7, 13, 1487} et lorsque χ est un caractère d’ordre une puissance d’un nombre premier ` > 3. L’étude du dernier cas fait intervenir l’idéal d’Eisenstein de l’algèbre de Hecke de J0 ( p). Cela retiendra peut-être l’attention du lecteur familier avec [16]. Ajoutons que l’hypothèse H p (χ) est fausse lorsque p ∈ {2, 3, 5, 7, 13}, et lorsque χ est quadratique pair. W. Stein a vérifié qu’elle n’est fausse dans aucun autre cas pour p < 1000. En se fondant sur une variante de la proposition 16, et avec l’aide de Stein, nous avons démontré qu’aucun nombre premier p vérifiant 7 < p < 1000 n’appartient a` S, sauf, peut-être, 13. Les détails relatifs a` ces calculs seront publiés séparément.

1. Géométrie et arithmétique de X 0 ( p) et J0 ( p) 1.1. Un lemme e´ lémentaire de géométrie arithmétique Soit p un nombre premier. Conformément a` l’usage, notons X 0 ( p) la courbe sur Q qui classe grossièrement les courbes elliptiques généralisées munies d’un sousgroupe cyclique d’ordre p (voir [1]). Notons X0 ( p) le modèle régulier minimal de X 0 ( p) sur Z. Notons J0 ( p) la variété jacobienne de X 0 ( p). Notons T le sous-anneau (commutatif) de EndQ J0 ( p) engendré par les opérateurs de Hecke et l’involution W p . (Nos notations co¨ıncident notamment avec celles de [16].) Notons J0 ( p) le modèle de Néron sur Z de J0 ( p) (dont la composante neutre n’est autre que Pic0 (X0 ( p)).) Notons X0 ( p)` la partie lisse de X0 ( p) : elle est obtenue en oˆ tant les sections qui sont supersingulières dans la fibre en p (voir [1]). Pour e´ tudier l’existence de points Q(µ p )-rationnels sur X 0 ( p) nous ferons usage du résultat suivant. Lorsque χ est un caractère a` valeurs complexes, on notera Z[χ] le sous-anneau de C engendré par les valeurs de χ.

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PROPOSITION 1 Soit p un nombre premier. Soient K et L deux extensions finies de Q p vérifiant les inclusions Q p ⊂ K ⊂ L ⊂ Q p (µ p ) et d’anneaux des entiers O K et O L respectivement. Posons S0 = spec O K et S = spec O L . Soit X un schéma séparé et noetherien sur S0 de fibre spéciale X¯ . Soient s1 et s2 des sections du morphisme canonique X × S0 S −→ S dont les restrictions a` la fibre générique (les “points” L-rationnels) sont notées P1 et P2 . Supposons que les restrictions de ces sections a` la fibre spéciale de S co¨ıncident avec une section s¯ : spec F p −→ X¯ . Supposons que pour tout caractère χ : Gal(L/K ) −→ C∗ il existe un morphisme de S0 -schémas φχ : X −→ A , où A est le modèle de Néron sur S0 d’une variété abélienne A sur Q p et tχ ∈ End K A ⊗ Z[χ ] vérifiant les deux conditions suivantes : (i) L’homomorphisme de Z[χ]-modules φχ∗ ◦ tχ∗ (déduit des morphismes φχ et tχ par passage aux espaces cotangents et extension des scalaires) : Cotgφχ (¯s ) A ⊗ Z[χ] −→ Cotgs¯ X ⊗ Z[χ ] est surjectif (on dira alors que le couple (tχ , φχ ) constitue une pseudo-immersion formelle au point s¯ ). (ii) On a dans A(L) ⊗ Z[χ] : X X χ(σ )tχ σ φχ (P1 ) = χ(σ )tχ σ φχ (P2 ) . σ ∈Gal(L/K )

σ ∈Gal(L/K )

Alors les sections s1 et s2 (et donc P1 et P2 ) co¨ıncident. Démonstration Notons P l’idéal de Z p [µ p ] au-dessus de pZ p . Notons PX (resp. PA ) l’idéal maximal du complété formel de X (resp. A ) le long de s¯ (resp. φχ (¯s )). Supposons que s1 et s2 ne co¨ıncident pas. Par passage aux espaces cotangents, 2 −→ P /P 2 qui ces sections définissent des homomorphismes s1∗ et s2∗ : PX /PX ne co¨ıncident pas par le lemme de Nakayama. Pour χ caractère de Gal(L/K ) notons P eχ = (1/[L : K ]) σ ∈Gal(L/K ) χ(σ ¯ )σ l’idempotent de Z[χ, 1/( p − 1)][Gal(L/K )] qui projette sur la composante χ -isotypique. Le nombre p e´ tant premier a` [L : K ], ces idempotents définissent une décomposition en somme directe du F p -espace vectoriel P /P 2 . Il existe donc un charactère χ : Gal(L/K ) −→ Z[χ ]∗ tel que eχ ◦ s1∗ et eχ ◦ s2∗ soient distincts. Comme l’application cotangente φχ∗ ◦ tχ∗ est surjective, les applications eχ ◦ s1∗ ◦ φχ∗ ◦ tχ∗ et eχ ◦ s2∗ ◦ φχ∗ ◦ tχ∗ sont distinctes. Or cela contredit la formule dans A (O L ) ⊗ Z[χ ] déduite de l’hypothèse (ii) par extension au modèle de Néron et application a` l’espace cotangent. Remarque. L’hypothèse (i) est vérifiée lorsque tχ ∈ End K A et tχ ◦ φχ est une immersion formelle (au sens habituel) au point s¯ , d’où notre terminologie. L’hypothèse (ii) est vérifiée par exemple lorsque tχ = 1 et lorsque A, tχ , P1 et P2 sont M-rationnels


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pour un sous-corps M de L tel que tχ A(M) = 0. Mais ces hypothèses plus faibles ne nous suffisent pas. 1.2. Points d’ordre p de J0 ( p) L’assertion suivante est une conséquence facile des travaux de Mazur [16]. PROPOSITION 2 La variété abélienne J0 ( p) ne possède pas de point Q(µ p )-rationnel d’ordre p.

Démonstration L’énoncé est e´ vident et sans intérêt si p = 2. Supposons donc que p > 2. Soit P ∈ J0 ( p)(Q(µ p ))[ p]. Si P est non nul, quitte a` multiplier par un e´ lément de T approprié on peut supposer que l’annulateur de P dans T est un idéal maximal M qui est nécessairement de caractéristique résiduelle p. Considérons le sous-schéma N en T/M-vectoriel engendré par P. C’est un sousschéma T/M-vectoriel de J0 ( p)[M]. Notons i la dimension de l’espace vectoriel sur ¯ T/M sous-jacent. Comme J0 ( p)(Q)[M] est un espace vectoriel de dimension 2 sur T/M (voir [16, proposition II.14.2] qui s’applique car p > 2), on a i = 1 ou 2. Supposons qu’on ait i = 1. D’après [16, proposition II.14.1], l’idéal M est un idéal premier d’Eisenstein. Cela n’est possible que lorsque p divise le numérateur de ( p − 1)/12, c’est-à-dire jamais. Cela exclut le cas i = 1. ´ Etudions maintenant le cas i = 2. On obtient alors un homomorphisme de groupes ¯ Gal Q(µ p )/Q ' (Z/ pZ)∗ −→ Aut J0 ( p)(Q)[M] ' GL2 (T/M). Comme T/M est un corps fini de caractéristique p > 2, une e´ tude facile nous indique que l’image de cet homomorphisme est conjuguée (dans GL2 (T/M)) a` un sous-groupe diagonal. On obtient donc l’existence d’un sous T/M[Gal(Q(µ p )/Q)]module de dimension 1 sur T/M. L’étude du cas i = 1 exclut précisément cette situation. On a donc P = 0. COROLLAIRE

Soit P un point d’ordre fini n de J0 ( p)(Q(µ p )). L’extension de P au modèle de Néron de J0 ( p) est d’ordre n dans la fibre en p. Démonstration On sait que n est premier a` p d’après la proposition précédente. Un lemme de spécialisation bien connu (voir par exemple [17]) permet de conclure : dans un schéma

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en groupe fini et plat sur l’anneau des entiers O d’une extension finie de Q p , l’ordre d’un point d’ordre premier a` p est déterminé dans la fibre spéciale. Or la variété abélienne J0 ( p) possède bonne réduction en dehors de p, si bien que J0 ( p)[n] s’étend en un schéma en groupe fini et plat sur spec Z[1/ p]. Remarque. Ce dernier corollaire pourrait se déduire simplement du fait que le modèle de Néron sur Z de J0 ( p) est une variété abélienne semi-stable et ne possède pas de sous-schéma en groupe de type µ d’ordre p d’après [16, théorème 2]. 1.3. Rappels sur la géométrie de la section cuspidale Considérons les pointes ∞ et 0 de X0 ( p) qui sont des sections Spec Z −→ X0 ( p). Ces notations désignent encore, par abus, toutes les sections qui s’en déduisent par changement de base (comme dans [16, II.1]). Considérons le morphisme (sur Q) φ : X 0 ( p) −→ J0 ( p) qui a` P associe la classe du diviseur (P) − (∞). Il s’étend en un morphisme sur Spec Z encore noté φ : X0 ( p)` −→ J0 ( p). Soit t ∈ T. L’action de t s’étend en un endomorphisme sur Spec Z de J0 ( p). Notons φt le morphisme de ´ Z-schémas obtenu en composant φ avec t. Etudions la géométrie de φt dans la fibre en p, en reprenant la méthode de [17]. 3 Supposons qu’on ait t ∈ / pT. Le morphisme φt est une immersion formelle le long de la fibre spéciale en p de la pointe ∞. PROPOSITION

Démonstration Pour démontrer cela il suffit de vérifier d’une part que l’application déduite de φt sur les corps résiduels des complétés formels est bijective (ce qui est e´ vident) et d’autre part que l’application φt∗ déduite de φt sur les espaces cotangents en ∞ et 0 est surjective. L’étude de ces espaces cotangents est menée a` bien dans [17] a` l’aide de la structure fournie par la courbe de Tate. D’une part, le complété formel de X0 ( p)/F p en ∞ s’identifie a` F p [[q]] (voir [1]), d’où l’isomorphisme de F p -espaces vectoriels Cot∞ (X0 ( p)/F p ) ' F p . D’autre part, Cot0 J0 ( p)/F p s’identifie, par application de la dualité de Grothendieck, a` H0 (X0 ( p)/F p , ), lequel s’identifie a` son tour a` Hom(T, F p ), par la théorie des q-développements (voir [17, 2.e])). En utilisant ces identifications dans le diagramme commutatif suivant, l’application φ ∗ est décrite


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ainsi (voir [17, démonstration de la proposition 3.1]) : Cot0 J0 ( p)/F p −→ Cot∞ X0 ( p)/F p ↓

↓

Hom(T, F p ) −→

Fp

ψ

7→

ψ(T1 ).

L’application cotangente déduite de l’action de t sur J0 ( p) est l’endomorphisme dual de t dans Cot0 (J0 ( p)/F p ). En composant les applications cotangentes, et en utilisant la compatibilité fournie par [17, lemme 2.1], on obtient que l’application cotangente φt∗ : Hom(T, F p ) ' Cot0 J0 ( p)/F p −→ Cot∞ X0 ( p)/F p ' F p déduite de φt est donnée par ψ 7 → ψ(t). Cette application cotangente est e´ videmment surjective si et seulement si t ∈ / pT. Remarque. Contrairement a` [17], nous travaillons ici avec des sous-variétés abéliennes de J0 ( p) et non des variétés abéliennes quotient. Cela nous dispense d’étudier les problèmes techniques posés par le comportement des suites exactes courtes de variétés abéliennes après passage aux espaces cotangents. Il serait plus délicat de prouver que φt en tant que morphisme a` valeurs dans la variété abélienne t J0 ( p) est une immersion formelle en caractéristique p. Ce point de vue a e´ galement e´ té adopté par Parent dans sa thèse. Soit χ un caractère de Dirichlet (Z/ pZ)∗ −→ C∗ . Posons T[χ] = T ⊗ Z[χ ] et F p [χ ] = F p ⊗ Z[χ ]. COROLLAIRE

Soit tχ ∈ T[χ] dont l’image dans T[χ ]/ pT[χ ] engendre un F p [χ ]-module libre. Le couple (tχ , φ) constitue une pseudo-immersion formelle au point ∞/F p . Démonstration Cela se déduit du diagramme figurant dans la démonstration de la proposition 3 en e´ tendant les scalaires a` Z[χ ]. 1.4. La géométrie des sections ordinaires Notons JS l’ensemble des invariants modulaires des courbes elliptiques supersingulières en caractéristique p. C’est un sous-ensemble de P1 (F p2 ) déduit du sousSpec F p -schéma fini de P1 défini par le polynôme supersingulier. Rappelons quelle est la structure de la fibre spéciale en p de X0 ( p) (voir [1, V.1]) : elle possède deux composantes irréductibles isomorphes a` X0 (1) (qui, via

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l’invariant modulaire j, est canoniquement isomorphe a` P1 ) qui se croisent transversalement en les points supersinguliers. L’involution W p e´ change ces composantes et change un point supersingulier en son conjugué par Gal(F p2 /F p ). Considérons la partie lisse X0 ( p)`/F p de X0 ( p)/F p . Tout point de X0 ( p)`/F p est dans l’image de l’une des deux immersions ouvertes α1 et α p : P1 −JS ' X0 (1)− j −1 (JS ) −→ X0 ( p)`/F p . Notons 1 S le groupe des diviseurs de degré zéro a` support dans les courbes elliptiques supersingulières en caractéristique p a` isomorphisme près. Rappelons que la composante neutre de la fibre en p du modèle de Néron de J0 ( p) est un tore sur F p dont le groupe des caractères F¯ p -rationnels s’identifie a` 1 S (voir [23]). Voici comP ment est obtenue cette identification. Soit E n E [E] ∈ 1 S . Soit F un faisceau inversible sur X0 ( p)/F p . Sa restriction aux composantes irréductibles de X0 ( p)/F p , qui sont isomorphes a` P1/F p , est trivialisable. Notons s0 et s∞ des sections jamais nulles de ces restrictions aux composantes contenant les pointes 0 et ∞ respectivement. Le P caractère λ de J0 ( p)0/F p correspondant a` E [E] ∈ 1 S est donné par la formule λ(F¯ ) =

Y

n s∞ (PE )/s0 (PE ) E ,

E

où PE désigne le point de X0 ( p)/F p (F¯ p ) correspondant a` la courbe elliptique supersingulière E et où F¯ désigne la classe de F dans le groupe de Picard. Cette identification munit 1 S d’une structure de T-module qui a e´ té e´ tudiée en détail, d’un point de vue théorique et d’un point de vue expérimental, par J.-F. Mestre et Oesterlé [22], [23], par B. Gross et S. Kudla [5] et par Gross [4]. Ajoutons que le groupe des composantes connexes de J0 ( p)/F p est d’ordre e´ gal au numérateur de ( p − 1)/12 d’après [16, appendice]. Considérons une situation plus générale que celle e´ tudiée dans la section 1.3. Soit d un entier plus que ou e´ gal a` 1 (nous n’utilisons dans le reste de cet article que les cas d = 1 et d = 2 ; mais il nous semble que le cas général sera utile tôt ou tard). Notons par l’indice supérieur (d) le passage a` la puissance symétrique d-ième. Considérons le morphisme φ (d) : X 0 ( p)(d) −→ J0 ( p), normalisé par le fait que la puissance symétrique d-ième de la pointe ∞ est envoyée sur zéro. Par propriété universelle des modèles de Néron, il s’étend en un morphisme (d) sur Z encore noté φ (d) : X0 ( p)` −→ J0 ( p). Pour t ∈ T, la composition de φ (d) avec la multiplication par t dans J0 ( p), donne un morphisme sur Z (d)

φt (1)

Bien entendu, on a φt

= φt .

(d)

: X0 ( p)` −→ J0 ( p).


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Quitte a` composer φ (d) avec la multiplication par p − 1, ce qui est sans conséquence pour les propriétés de finitude de points rationnels et d’immersion formelle en caractéristique p, on peut supposer, compte-tenu de ce qui vient d’être (d) (d) dit sur la nature du groupe des composantes, que φt (X0 ( p)/F p `) est contenu dans J0 ( p)0/F p . Soit un caractère λ du tore J0 ( p)0/F¯ correspondant au diviseur p P u E parcourt les courbes elliptiques supersingulières. Soit E n E [E] ∈ 1 S , o` (d) ( j1 , j2 , . . . , jd ) ∈ (P1 (F¯ p ) − JS )d . Posons Pd = α1 ( j1 , j2 , . . . , jd ). On a λ ◦ φ (d) (Pd ) =

d Y Y

n ji − j (E) E .

i=1 E

Cette formule a e´ té obtenue pour d = 1 par Mestre et Oesterlé [23] ; la formule générale s’en déduit en remarquant que φ (d) (P1 , . . . , Pd ) = φ(P1 ) + · · · + φ(Pd ). Pour u ∈ {1, 2, . . . , d}, notons ιu l’homomorphisme de groupes 1 S −→ F¯ p donné par la formule X X nE ιu n E [E] = u . j − j (E) E E Soit j ∈ P1 (F¯ p ) − JS . Posons P = α1 ( j) ∈ X0 ( p)` /F p (F¯ p ). Notons P (d) la puissance symétrique d-ième de P. 4 Soit t ∈ T. S’il existe δ1 , δ2 ,. . . ,δd ∈ 1 S tels que le déterminant de la matrice (ιu (tδi ))u,i∈{1,2,...,d} soit non nul dans F p , la restriction a` la fibre spéciale en p du (d) morphisme φt est une immersion formelle au point P (d) . PROPOSITION

Démonstration P Posons tδi = E m E,i [E]. Notons λi le caractère du tore J0 ( p)0/F¯ correspondant p a` tδi par l’identification mentionnée ci-dessus. Notons λ le morphisme de groupes algébriques (λ1 , . . . , λd ) : J0 ( p)0/F¯ −→ Gdm /F¯ p . p

(d)∗

Il faut donc e´ tablir que l’homomorphisme d’anneaux locaux φt sur les (d) complétés formels déduit de φt est surjectif. Il suffit pour cela de prouver que l’homomorphisme d’anneaux φ (d)∗ ◦ λ∗ est surjectif, c’est-à-dire que le morphisme (d) λ ◦ φ (d) : X0 ( p)`/F −→ Gdm/F est une immersion formelle au point P (d) . p

p

(d)

Composons encore par l’immersion ouverte α1 . Il suffit de démontrer que le (d) morphisme λ ◦ φ (d) ◦ α1 : (P1 − S )(d) −→ Gdm est une immersion formelle en (d) j . Notons σu le u-ième polynôme symétrique e´ lémentaire en j1 , . . . , jd . La i-ième

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coordonnée du morphisme λ ◦ φ ◦ α1 est donnée par la fraction rationnelle Fi : ( j1 , . . . , jd ) 7→ Fi ( j1 , . . . , jd ) =

=

d Y Y

m jk − j (E) E,i

k=1 E d YX

(−1)u σd−u j (E)u

E

m E,i

.

u=0

L’espace cotangent en j (d) de (P1 − S )(d) a pour base la famille dσ1 , . . . , dσd . Un calcul e´ vident de différentielle logarithmique donne d−1 X X d Fi m E,i (−1)u j (E)u dσd−u . = Qd Fi j − j (E) k=1 k E u=0 (d)

Le morphisme λ ◦ φ ◦ α1 est une immersion formelle en ( j1 , . . . , jd ) si et seulement la famille (d Fi ) est linéairement indépendante dans l’espace cotangent en ( j1 , . . . , jd ) de (P1 − S )(d) . Comme on examine la situation en ( j1 , . . . , jd ) = ( j, . . . , j), cela revient a` vérifier que la matrice carrée d’ordre d de terme (i, u) e´ gal a` P u−1 /( j − j (E))d est inversible. Une manipulation e ´ lémentaire monE m E,i j (E) tre que cette matrice est de déterminant e´ gal a` celui de la matrice (ιu (tδi ))u,i∈{1,2,...,d} . Cela achève notre démonstration. Remarque. Cette condition d’indépendance linéaire est a` comparer a` celle obtenue par Kamienny dans [9], pour P = ∞ ; néanmoins nous ne faisons pas intervenir la suite des opérateurs de Hecke T1 , T2 , . . . . Comme nous le verrons dans la troisième partie, la proposition 4 est utile a` l’étude au cas par cas de la non-existence de points p-ordinaires de X 0 ( p). Pour e´ tudier de façon uniforme ces points, il faudrait savoir comparer les T-modules 1 S et H1 (X 0 ( p)(C), Z) de façon a` comprendre la relation entre les valeurs de fonctions L et la structure de T-module de 1 S . Les meilleures informations dont on dispose dans cette direction semblent eˆ tre contenues dans [5] et [4]. Notons encore ιu l’homomorphisme de Z[χ ]-modules : 1 S ⊗Z[χ ] −→ F p [χ ] obtenu a` partir de ιu par extension des scalaires par Z[χ]. COROLLAIRE

Soit tχ ∈ T ⊗ Z[χ ]. S’il existe δ1 , δ2 ,. . . ,δd ∈ 1 S ⊗ Z[χ ] tels que le déterminant de la matrice (ιu (tχ δi ))u,i∈{1,2,...,d} soit inversible dans F p [χ], le couple (tχ , φ (d) ) est une pseudo-immersion formelle au point P (d) .


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Démonstration Cela se déduit de la proposition 4 par extension des scalaires par Z[χ ]. 1.5. Discussion de la littérature sur les travaux de Kato Soit χ un caractère de Dirichlet (Z/ pZ)∗ −→ C∗ , identifié par la théorie du corps de classes a` un caractère de Gal(Q(µ p )/Q). Soit tχ un e´ lément de T[χ] engendrant un sous-anneau intègre de T[χ ] et tel que tχ g = 0 dès lors que g ∈ S2 (00 ( p)) vérifie L(g, χ , 1) = 0. Posons K = tχ T[χ ] ⊗ Q ; c’est un sous-corps de nombres de T[χ ] ⊗ Q. Nous avons besoin de l’énoncé suivant, que nous comprenons comme une conséquence des résultats annoncés par Kato (série d’exposés a` l’IAS (Institute for Advanced Study), Princeton, automne 1995), qui résulte de la conjecture de Birch et Swinnerton-Dyer pour les variétés abéliennes sur Q et qui ne dépend que de p, χ et K. La composante χ -isotypique du Gal Q(µ p )/Q -module tχ J0 ( p) Q(µ p ) ⊗ Z[χ] est finie. Signalons que cela ne semble pas résulter de [12]. En attendant que Kato publie ses travaux, sans prétendre donner une démonstration, essayons d’orienter le lecteur dans la lecture des textes de Scholl [26] et K. Rubin [25] pour obtenir le résultat dont nous avons besoin. Soit f ∈ S0 (00 ( p)) une forme primitive telle que tχ f 6= 0 ; on a donc L( f, χ , 1) 6 = 0. Considérons la forme modulaire f χ obtenue en tordant la forme f par le caractère χ. C’est une forme primitive pour 01 ( p 2 ) lorsque χ 6= 1. La théorie de G. Shimura lui associe une variété abélienne Aχ simple sur Q qui est un quotient de J1 ( p 2 ). Rappelons quels sont les liens entre Aχ et la variété abélienne A associée a` f par la théorie de Shimura [27] : Aχ est un facteur de la décomposition a` isogénie près de la variété abélienne obtenue a` partir de A/Q(µ p ) par restriction des scalaires de Q(µ p ) a` Q. De plus la série L de Aχ est donnée par la formule : Y L(Aχ , s) = L( f 0 , s), f0

où f 0 parcourt les formes primitives conjuguées de la forme modulaire f χ . Il s’ensuit que la non nullité de L( f, χ, 1) e´ quivaut a` la non nullité de L(Aχ , 1) et donc a` la non nullité de L( f χ , 1). Par ailleurs la finitude de Aχ (Q) e´ quivaut a` la finitude de la composante χ -isotypique de tχ J0 ( p)(Q(µ p ))⊗Z[χ]. On se ramène ainsi a` démontrer que Aχ (Q) est un groupe fini. La variété abélienne Aχ possède des multiplications par un ordre O du corps de nombres K . Soit λ un idéal maximal de O . Notons l la caractéristique résiduelle

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correspondante. Pour rester dans un cadre aussi proche que possible de celui de [25], nous imposons a` l d’être distinct de p et totalement décomposé dans K (cela est loisible sans eˆ tre nécessaire). On a donc K λ = Ql . Le module de Tate λ-adique Tλ de Aχ fournit une représentation l-adique V de dimension 2 et continue : ¯ ρ : Gal(Q/Q) −→ GL(Tλ ⊗ Ql ) ' GL2 (K λ ) ' GL2 (Ql ) ' GL(V ). Cette représentation l-adique entre dans le cadre de l’étude de [25, section 1]. Pour prouver la finitude de Aχ (Q), il suffit de prouver la finitude du groupe de Selmer associé (voir [25]). Pour cela on applique le formalisme des systèmes d’Euler comme dans [25, sections 2 et 3] (les preuves détaillées figurent dans les références indiquées dans [25]). Pour appliquer ce formalisme on vérifie que V est bien muni d’un système d’Euler, en adaptant [26, section 5.2] au cas où Aχ n’est pas nécessairement une courbe elliptique. Plus précisément, on utilise que J1 ( p 2 ) et donc Aχ est un quotient de J ( p 2 ) ; or dans [26] est e´ tabli l’existence d’un système d’Euler, noté (ξr ∈ H1 (Q(µr ), Tl (J (N )))r ∈R (où R est l’ensemble des entiers sans facteur carré premiers a` p), relatif au module de Tate Tl (J (N )) l-adique de J ( p 2 ) qui par projection sur la variété abélienne Aχ donne lieu a` un système d’Euler (ξr (Aχ ))r ∈R relatif a` V , que l’on peut utiliser pour appliquer les théorèmes de [25]. La non nullité de ξ1 (Aχ ) résulte de la non nullité de L( f χ , 1) (nous n’avons pas vérifié ce point essentiel, qui est démontré dans [25, théorème 5.2.7], lorsque Aχ est un quotient elliptique). Pour appliquer [25, théorème 3.1], et conclure a` la finitude du groupe de Selmer il faut encore vérifier que l’hypothèse Hyp(Q∞ , V ) de [25, section 3] est satisfaite. ¯ C’est-à-dire l’existence de τ ∈ Gal(Q/Q) agissant trivialement sur les racines de l’unité d’ordre une puissance de l et tel que l’endomorphisme ρ(τ ) soit non-scalaire et possède 1 comme valeur propre. Comme J0 ( p) (et donc aussi A) a réduction purement multiplicative en p, il ex¯ iste τ0 ∈ Gal(Q/Q) (plus précisément dans un groupe de décomposition en p) agissant de façon unipotente et non triviale sur le module de Tate l-adique de A. En raison des accouplements de Weil, il en résulte que τ0 agit trivialement sur les racines d’ordre p−1 ¯ une puissance de l. Posons τ = τ0 ∈ Gal(Q/Q). Cet e´ lément opère de façon unipotente et non triviale sur le module de Tate l-adique de A et trivialement sur les racines d’ordre une puissance l. Puisque Aχ est obtenue par restriction des scalaires de Q(µ p ) a` Q, et que τ opère trivialement sur Q(µ p ), ρ(τ ) est unipotent et non trivial. Il satisfait donc les conditions cherchées. Cela prouve l’hypothèse Hyp(Q∞ , V ). 1.6. Points quadratiques p-cuspidaux La question des points quadratiques se traite grâce aux méthodes de Kamienny et ` noter que la question des points Q(√± p)-rationnels de X 1 ( p) a e´ té Mazur [11]. A e´ tudiée dans [8] et [7]. Par un point p-cuspidal Q(µ p )-rationnel de Y0 ( p), nous entendons un point Q(µ p )-rationnel de X 0 ( p) distinct d’une pointe et d’invariant mod-


93

ulaire qui est de valuation p-adique strictement négative (de façon e´ quivalente c’est un point dont l’extension en une spec Z[µ p ]-section co¨ıncide avec une pointe dans la fibre en p mais pas dans la fibre générique). PROPOSITION 5 La courbe modulaire Y0 ( p) ne possède pas de points quadratiques Q(µ p )-rationnels qui sont p-cuspidaux lorsque p > 19 et p 6 = 37.

Démonstration Supposons que la courbe modulaire X 0 ( p) possède un tel point P (qui serait bien √ √ entendu Q( p) ou Q( − p) rationnel). Il en résulterait l’existence d’un point Qrationnel du carré symétrique X 0 ( p)(2) de X 0 ( p). Un tel point s’étend en un point sur Z de X0 ( p)(2) . Comme ce point est déduit d’un point cuspidal Q(µ p )-rationnel et de son conjugué, sa réduction modulo p co¨ıncide avec le carré symétrique de l’une des deux pointes de la fibre en p de X0 ( p). Quitte a` appliquer l’involution W p , on peut supposer que cette pointe est ∞. (2) Considérons maintenant le morphisme φ (2) : X0 ( p)` −→ J0 ( p) (voir section 1.4). Composons-le avec le morphisme canonique de schémas en groupes sur Z, J0 ( p) −→ J˜ ,

où J˜ est le modèle de Néron sur Z du quotient d’Eisenstein J˜ de J0 ( p) (voir [16]). Le fait que J˜ ne possède qu’un nombre fini de points Q-rationnels (voir [16]) permet d’utiliser le critère d’immersion formelle de Kamienny [10] : si les opérateurs k T1 et T2 sont F p -linéairement indépendants dans T/( pT + ∩∞ u I désigne k=1 I ) (o` l’idéal d’Eisenstein de T, et où T1 et T2 sont les e´ léments de T déduits des correspondances de Hecke T1 et T2 ) alors P co¨ıncide avec la pointe ∞. Prenons note que Kamienny n’a utilisé son critère d’immersion formelle qu’en caractéristiques distinctes de 2 et p ; en réalité l’argument fonctionne de façon identique en caractéristique p si on tient compte du fait que J0 ( p) ne possède pas de point Q-rationnel d’ordre p, ce qui est prouvé dans [16]. Supposons que les opérateurs T1 et T2 soient linéairement dépendants. On a alors T2 ∈ pT + Z. L’inégalité de Ramanujan-Petersson impose qu’on a T2 ∈ Z ; en effet, si α et β sont deux valeurs propres distinctes de T2 , le nombre (α − β)/ p est un entier alg´ √ebrique non nul √ dont toutes les valeurs absolues archimédiennes sont bornées par 4 2/ p, et on a 4 2/ p < 1 lorsque p > 7. Kamienny [9], a démontré que T2 n’est pas un scalaire dans le quotient d’Eisenstein lorsque p > 61. Un examen des tables de formes modulaires montre que cela est encore le cas lorsque p > 19 et p 6 = 37. Remarque. Au vu de la section 1.7, seuls nous intéressent les cas où p ≡ 1 (mod 4).

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√ Y a-t-il des points p-cuspidaux et Q( p)-rationnels de Y0 ( p) pour p = 17 et p = 37 ? 1.7. Preuve du théorème principal Par point p-supersingulier Q(µ p )-rationnel de X 0 ( p) on entendra un point Q(µ p )rationnel dont l’extension en une spec Z[µ p ]-section est supersingulière dans la fibre en p. PROPOSITION

6

Soient P1 et P2 deux points Q(µ p )-rationnels de X 0 ( p) dont les extensions s1 et s2 a` X0 ( p)` co¨ıncident sans eˆ tre supersingulières dans la fibre en p. Supposons que pour tout caractère de Dirichlet χ de Gal(Q(µ p )/Q), il existe tχ ∈ T ⊗ Z[χ] tel que la classe du diviseur tχ ((P1 ) − (P2 )) soit d’ordre fini dans la composante χisotypique de J0 ( p)(Q(µ p )) ⊗ Z[χ ] et tel que le couple (tχ , φ) soit une pseudoimmersion formelle en caractéristique p au point P1/F p = P2/F p . On a alors P1 = P2 . Démonstration C’est une application facile de la proposition 1 avec X = X0 ( p), A = J0 ( p), φχ = φ, L = Q p (µ p ) et K = Q p . L’hypothèse (i) est satisfaite. Considérons P l’élément σ ∈Gal(Q(µ p )/Q) tχ χ(σ )σ (φχ ◦ s1 − φχ ◦ s2 ) de J0 ( p)(Z[µ p ]) ⊗ Z[χ]. Il est nul dans la fibre en p et d’ordre fini dans la fibre générique. Il est donc nul d’après le corollaire de la proposition 2 et par platitude de Z[χ]. Cela prouve que (ii) est vérifié puisque Gal(Q(µ p )/Q) ' Gal(Q p (µ p )/Q p ). COROLLAIRE 1 Soit K un sous-corps de Q(µ p ) d’anneau des entiers O K . Soit P un point Q(µ p )rationnel de Y0 ( p) non p-supersingulier. Notons P/F p le point F p -rationnel obtenu en restreignant a` la fibre en p l’extension en une spec Z[µ p ]-section de P. Supposons que pour tout caractère de Gal(Q(µ p )/Q) qui est non trivial sur Gal(Q(µ p )/K ), il existe tχ ∈ T ⊗ Z[χ] tel que la composante χ-isotypique de tχ J0 ( p)(Q(µ p )) ⊗ Z[χ] soit finie et que le couple (tχ , φ) soit une pseudo-immersion formelle au point P/F p . Alors P est K -rationnel.

Démonstration Il suffit e´ videmment de prouver qu’on a P1 = P2 pour P1 = P et P2 = P σ0 , où σ0 engendre le groupe cyclique Gal(Q(µ p )/K ). Comme l’extension Q(µ p )|Q est totalement ramifiée en p, les extensions s1 et s2 de P1 et P2 a` X0 ( p) co¨ıncident dans la fibre en p. Si χ est non trivial sur Gal(Q(µ p )/K ), la classe de tχ ((P1 ) − (P2 )) est


95

d’ordre fini dans la composante χ -isotypique de J0 ( p)(Q(µ p )). Sinon, on a, puisque σ P2 = P1 0 , X χ(σ )(P1σ − P2σ ) = 0. σ ∈Gal(Q(µ p )/K )

On peut donc appliquer la proposition 6 a` P1 et P2 , d’où P1 = P2 . COROLLAIRE 2 Soit P un point p-cuspidal Q(µ p )-rationnel de Y0 ( p). Si on a H p (χ ) pour tout caractère de Dirichlet de (Z/ pZ)∗ non quadratique pair, alors P est un point quadratique réel.

Démonstration Notons K le plus grand sous-corps totalement réel de Q(µ p ) de degré 1 ou 2. Appliquons le corollaire 1 de la proposition 6 pour prouver que P est K -rationnel. Observons d’abord qu’il suffit de prouver que pour tout caractère χ 6= 1 de Gal(Q(µ p )/K ) il existe tχ ∈ T ⊗ Z[χ ], tχ 6 = 0, tel que la composante χ-isotypique de tχ J0 ( p)(Q(µ p )) soit finie. (Cette dernière propriété est inchangée si on remplace tχ par un e´ lément de Q[χ ]tχ .) D’après le lemme d’approximation dans les anneaux de Dedekind, il existe a ∈ Q[χ ] tel que atχ ∈ T ⊗ Z[χ ] et tel que l’image de atχ dans T[χ]/ pT[χ] soit un F p [χ]-module libre. D’après le corollaire de la proposition 3, le couple (tχ , φχ ) est une pseudo-immersion formelle au point ∞/F p = P/F p . Cela permet de conclure par application du corollaire 1 de la proposition 6. Il reste a` e´ tablir que H p (χ) entraˆıne l’existence de tχ . Soit f ∈ S2 (00 ( p) telle ¯ que L( f, χ , 1) 6= 0. Toute forme primitive f 0 conjuguée de f par Gal(Q/Q[χ]) vérifie e´ galement L( f 0 , χ, 1) 6= 0 (cela se voit par exemple en utilisant l’intégration sur les classe d’homologie comme dans la section 2.1). Soit tχ ∈ T[χ] tel que tχ ¯ annule toute forme primitive f 0 qui n’est pas conjuguée de f par Gal(Q/Q[χ ]). Le sous-anneau de T[χ ] engendré par tχ est alors intègre. La finitude de la composantes χ -isotypique de tχ J0 ( p)(Q(µ p )) résulte des travaux de Kato (voir section 1.5). Remarque. Nous ne prétendons pas que L( f, χ , 1) 6= 0 entraˆıne L(g, χ , 1) 6= 0 ¯ lorsque f et g sont des formes primitives conjuguées par Gal(Q/Q). Comme on le verra dans la proposition 7, l’hypothèse H p (χ ) ne dépend que de la classe de conjugaison de χ . COROLLAIRE 3 Si p est congru a` 1 modulo 4, p > 17 et p 6 = 37, supposons qu’on ait H p (χ ) pour tout caractère χ non quadratique. Alors Y0 ( p) ne possède pas de points Q(µ p )-rationnels p-cuspidaux.

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Si p est congru a` −1 modulo 4, et p ∈ / {3, 7}, supposons qu’on ait H p (χ ) pour tout caractère χ. Alors Y0 ( p) ne possède pas de points Q(µ p )-rationnels pcuspidaux. Démonstration Soit P un point Q(µ p )-rationnel et p-cuspidal de Y0 ( p). Si p est congru a` −1 modulo 4, le corollaire 3 entraˆıne que P est Q-rationnel. D’après [16, corollaire 4.4], Y0 ( p) ne possède pas de point Q-rationnel p-cuspidal lorsque p ∈ / {2, 3, 5, 7, 13}. Si p est congru a` 1 modulo 4, le point P est quadratique d’après le corollaire 2. Mais cela est exclu d’après la proposition 5. Remarques. William Stein a vérifié l’hypothèse H p (χ) lorsque χ n’est pas quadratique pair et p ∈ / {2, 3, 5, 7, 13} et p < 1000. Cela prouve que l’ensemble S0,c ne contient aucun e´ lément p < 1000 en dehors de l’ensemble {2, 3, 5, 7, 13, 17, 37}. Lorsque K est un sous-corps de Q(µ p ), on peut démontrer l’inexistence de points K -rationnels p-cuspidaux comme ci-dessus en se contentant de l’hypothèse H p (χ ) pour les caractères χ qui se factorisent par le groupe quotient de (Z/ pZ)∗ correspondant a` Gal(K /Q). Une démonstration analogue a` celle que nous venons de proposer, et ses applications aux points rationnels de X 0 ( p) comme ci-dessus, mais sous l’hypothèse plus forte que le morphisme φχ peut eˆ tre choisi indépendamment de χ et que tχ = 1 figure implicitement dans [17] (voir la formalisation figurant dans [17, corollaire 4.3]).

2. Méthodes non-analytiques et séries de Dirichlet 2.1. Rappels sur les symboles modulaires Dans cette deuxième partie la courbe modulaire X 0 ( p) sera exclusivement vue comme la surface de Riemann compacte et connexe X 0 ( p)(C) = 00 ( p)\H ∪ P1 (Q), où H est le demi-plan de Poincaré. On notera ptes l’ensemble 00 ( p)\P1 (Q) de ses pointes. Soit (α, β) ∈ P1 (Q)2 . Notons {α, β} la classe d’homologie, dite symbole modulaire, dans H1 (X 0 ( p)(C), ptes; Z) définie par la classe de l’image dans X 0 ( p) d’un chemin continu reliant α a` β dans H . Observons que, lorsque α et β ont des dénominateurs premiers a` p (resp. divisibles par p), ils sont conjugués par 00 ( p) et {α, β} appartient au sous-groupe H1 (X 0 ( p)(C), Z) de H1 (X 0 ( p)(C), ptes; Z). Lorsque g = ac db ∈ SL2 (Z), le symbole modulaire {g0, g∞} ne dépend que de 00 ( p)g, c’est-á-dire que de la classe de (c, d) dans P1 (Z/ pZ). Notons le ξ(c, d),


97

ou encore, en notation inhomogène, ξ(c/d). Lorsque c et d parcourent les entiers premiers a` p, les classes ξ(c/d) engendrent H1 (X 0 ( p)(C); Z). Lorsque a ∈ Z, on a ξ(a) = {0, 1/a}. On a ξ(1) = 0. L’involution W p de X 0 ( p) est déduite de z 7→ −1/ pz sur H . On a donc la formule W p (ξ(a)) = −{−a/ p, ∞}. L’espace H1 (X 0 ( p)(C), ptes; Z) est muni de l’involution donnée par la conjugaison complexe. Cette involution est déduite de l’involution z 7→ −¯z de H . Elle agit donc par ξ(u) 7→ −ξ(1/u) = ξ(−u) sur les symboles modulaires. Elle induit une involution sur H1 (X 0 ( p)(C); Z). On notera H1 (X 0 ( p)(C); Z)+ et H1 (X 0 ( p)(C); Z)− les parties invariantes et anti-invariantes de H1 (X 0 ( p)(C); Z) par la conjugaison complexe. Rappelons que H1 (X 0 ( p)(C); Z) est un module acyclique sous l’action de la conjugaison complexe (voir [20, section 1.3]). 2.2. Reformulation topologique Dans le reste de cette seconde partie, par caractère on entendra caractère de Dirichlet (Z/ pZ)∗ −→ C∗ . 7 Soit χ un caractère différent de 1. On a H p (χ) si et seulement si le symbole modulaire PROPOSITION

θχ =

p−1 X a=1

na o χ(a) , ∞ ∈ H1 X 0 ( p)(C), Z[χ ] p

est non nul. Démonstration P p−1 Notons τ (χ ) la somme de Gauss a=0 χ(a)e−2iπa/ p . On a H p (χ ) si et seulement si l’expression Z ∞ p−1 χ(−1)2πiτ (χ) X L( f, χ, 1) = − χ(a) ¯ f (z) dz p a/ p a=1

est non nulle pour une forme modulaire f de poids 2 pour 00 ( p). Notons ω f la forme différentielle sur X 0 ( p) déduite de f (z) dz par passage au quotient. On a alors Z χ(−1)2πiτ (χ) L( f, χ, 1) = − ωf. p θχ¯ Observons que θχ est invariant (resp. anti-invariant) par la conjugaison complexe lorsque χ est pair (resp. impair). En effet, cette conjugaison est déduite de l’involution

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z 7→ −¯z sur H ; elle transforme donc le symbole modulaire {α, β} en {−α, −β} et donc θχ en χ (−1)θχ . Comme a/ p (pour a entier premier ‘a p) et ∞ sont conjugués par 00 ( p), le symbole modulaire ∞, ap , et donc θχ , sont e´ léments de H1 X 0 ( p)(C), Z[χ] . L’intégration des formes différentielles holomorphes fournit un accouplement parfait entre H0 (X 0 ( p)(C), 1 ) et la partie invariante (resp. anti-invariante) par la conjugaison complexe de H1 (X 0 ( p)(C), C). Comme toute forme différentielle holomorphe sur X 0 ( p) s’obtient de façon unique comme image d’une forme modulaire par f 7→ ω f , l’hypothèse H p (χ ) e´ quivaut a` la nullité de θχ¯ , qui elle-même e´ quivaut a` la nullité de θχ . 2.3. Reformulation e´ lémentaire Commençons par traiter le cas du caractère trivial. La proposition suivante est bien connue au moins depuis les travaux de Mazur [16] (i.e., le quotient d’Eisenstein de J0 ( p) est non trivial si et seulement si le numérateur de ( p − 1)/12 est différent de 1). Nous en donnons une preuve facile. PROPOSITION 8 On a H p (1) pour tout nombre premier p ∈ / {2, 3, 5, 7, 13}.

Démonstration D’après [21], il suffit de prouver que l’élément d’enroulement e ∈ H1 (X 0 ( p)(C), Q) est non nul. Rappelons que cet e´ lément d’enroulement est défini de la façon suivante. Posons V = HomC (H0 (X 0 ( p), 1 ), C). Notons L le plongement dans V de R H1 (X 0 ( p)(C), Z) par l’application qui a` c associe ω 7 → c ω. Cette image est un réseau de V et J0 ( p)(C) s’identifie canoniquement a` V /L par la théorie d’Abel. Soit ce un 1-cycle donné par l’image dans X 0 ( p)(C) d’un chemin reliant zéro a` i∞ dans H . Il a pour bord le diviseur (∞) − (0) de X 0 ( p). L’élément d’enroulement e est, R par définition, l’image de ω 7→ ce ω par l’isomorphisme d’espaces vectoriels réels R V ' H1 (X 0 ( p)(C), R). Il est donc nul si et seulement si ω 7 → ce ω est non nul. R Cette application ω 7→ ce ω est un e´ lément de V dont l’image dans V /L ' J0 ( p)(C) est la classe du diviseur (∞) − (0). La classe d’un diviseur formé par la différence de deux points distincts est non nulle dans la jacobienne d’une courbe de R genre non nul. L’application ω 7→ ce ω et donc a fortiori e sont non nuls lorsque le genre de X 0 ( p) est non nul. Or le genre de X 0 ( p) est nul si et seulement si on a p ∈ {2, 3, 5, 7, 13}.


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Pour k ∈ {1, 2, . . . , p − 1}, notons k∗ l’unique e´ lément de {1, 2, . . . , p − 1}, tel que kk∗ ≡ −1 (mod p). Pour u ∈ (Z/ pZ)∗ et pour χ caractère différent de 1, posons F(χ, u) =

−1/u X

0

χ(x) − χ(−1/x)

x=u

où le symbole 0 signifie que la somme ne tient pas compte des termes pour lesquels x ∈ pZ et par abus de notation les bornes sont des représentants quelconques de u et −1/u dans Z. PROPOSITION 9 Soit χ un caractère différent de 1. On a H p (χ) si et seulement si il existe u ∈ (Z/ pZ)∗ tel que F(χ , u) 6= 0.

Démonstration D’après la proposition 7, il suffit d’étudier la nullité de θχ , ou, ce qui en revient au même puisque W p est une involution, de −χ(−1)W p θχ . Or, on a −χ(−1)W p θχ =

p−1 X

χ(a)ξ(a).

k=1

Utilisons les produits d’intersection sur H1 (X 0 ( p)(C), Z). Ils constituent un accouplement bilinéaire non dégénéré a` valeurs dans Z noté •. Comme on l’a rappelé dans la section 2.1, les e´ léments ξ(k) engendrent H1 (X 0 ( p)(C), Z) lorsque k parcourt {1, 2, . . . , p − 1}. On a donc W p θχ = 0 si et seulement si W p θχ • ξ(k) = 0 pour tout k ∈ {1, 2, . . . , p − 1}. Rappelons [21, lemme 4] (lemme des cordes). On désigne par corde Ck le segment de droite orienté de C reliant e2πik∗ / p a` e2πik/ p . Soient k et k 0 deux e´ léments de {1, . . . , p − 1} tels que k 6= k 0 , k 6 = k∗0 . Le produit d’intersection ξ(k) • ξ(k 0 ) est e´ gal au nombre d’intersection (égal a` −1, 0 ou 1) des cordes Ck 0 et Ck . On en déduit immédiatement la formule W p θχ • ξ(k) =

−1/u X

0

χ(x) − χ(−1/x) = F(χ , u).

x=u

Remarque. Au vu de la proposition 9, la condition H p (χ) n’est jamais satisfaite lorsque χ est un caractère quadratique pair. Si on admet la conjecture de Birch et √ Swinnerton-Dyer, cela entraine que J0 ( p)(Q( p)) contient un T-module libre de rang 1. Peut-on en mettre en e´ vidence un générateur ?

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2.4. Congruences e´ lémentaires PROPOSITION 10 Supposons que p soit un nombre premier congru a` 11 ou 19 modulo 20 et que χ soit un caractère impair. Alors on a H p (χ). Démonstration Puisque p est congru a` 1 ou −1 modulo 5, 5 est un carré modulo p, par la loi de réciprocité quadratique. L’équation x 2 + 3x + 1 = 0 possède donc des solutions dans Z/ pZ. Notons u 1 et u 2 ces solutions. On a −1/u 1 = u 1 + 3 et (u 1 + 1)(u 1 + 2) = 1. Appliquons le critère e´ lémentaire de la proposition 9 a` u = u 1 . Cela donne −1/u X1

0

χ (x)−χ (−1/x) = χ(u 1 +1)−χ −1/(u 1 +1) +χ(u 1 +2)+χ −1/(u 1 +2) .

x=u 1

En utilisant les relations −1/u 1 = u 1 + 3 et (u 1 + 1)(u 1 + 2) = 1 et l’imparité de χ , on obtient −1/u X1

0

χ (x) − χ(−1/x) = 2χ(u 1 + 1) + 2χ(u ¯ 1 + 1) = 2χ(u ¯ 1 + 1) 1 + χ (u 1 + 1)2 .

x=u 1

Comme p est congru a` −1 modulo 4, il n’y a pas de racine primitive quatrième de P 1 0 l’unité dans Z/ pZ et donc χ(u 1 + 1)2 6 = −1. La somme −1/u x=u 1 χ (x) − χ (−1/x) n’est donc pas nulle et on a H p (χ). PROPOSITION 11 Supposons que χ soit un caractère quadratique impair et que p ∈ / {3, 7}. Alors on a H p (χ ).

Démonstration On a alors p ≡ −1 (mod 4). Comme χ est quadratique et impair on a F(χ, u) = 2

−1/u X

0

χ(x).

x=u

Le caractère χ est a` valeurs dans {−1, +1}. Il suffit donc d’établir l’existence de a ∈ {1, 2, . . . , p − 1} avec a et a∗ de même parité pour conclure. En effet dans ce cas la somme qui définit F(χ, u) possède un nombre impair de termes e´ gaux a` 1 ou −1, ce qui entraˆıne la non nullité de F(χ, a + pZ). Supposons que pour tout a ∈ {1, 2, . . . , p−1}, les entiers a et a∗ soient de parités opposées. Comme p est congru a` −1 modulo 4, on a 4∗ = (3 p − 1)/4. Soient i et j deux entiers plus que zéro tels que (3 p − 1)/4 = i j. On a (2i)∗ = 2 j sauf si i ou j


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est e´ gal a` 1. On en conclut que (3 p − 1)/4 est premier et donc impair sauf si p = 3, cas désormais exclu. Par conséquent (7 p − 1)/8 est entier. Par un raisonnement analogue a` celui qui précède on obtient que (7 p − 1)/8 est un nombre premier sauf si p < 8, cas que l’on peut désormais e´ carter. Cela nous donne que (7 p − 1)/8 est premier a` 3. C’est pourquoi on a 3|(2 p − 1) et donc 3∗ = (2 p − 1)/3 qui est un nombre impair. Cela est absurde. 12 Supposons que χ soit un caractère injectif. Alors on a H p (χ ) pour tout nombre premier p ∈ / {2, 3, 5, 7, 13, 1487}. PROPOSITION

Démonstration Le nombre premier p est totalement décomposé dans le corps Q[χ ] = Q(µ p−1 ). Soit P un idéal de Q[χ ] au-dessus de p. Le caractère χ˜ , obtenu en composant χ avec la réduction modulo P co¨ıncide donc avec l’élévation a` une certaine puissance k dans (Z/ pZ)∗ . Comme χ est injectif, l’entier k est premier a` p − 1. Quitte a` remplacer P par un idéal conjugué, on peut supposer qu’on a k = 1. Notons F¯ la réduction modulo P de F. On a donc dans Z/ pZ, ¯ F(χ, u) =

−1/u X

0

x + 1/x.

x=u

L’un des trois nombres 3, 5 et 15 est un carré modulo p. En d’autres termes, l’une des trois e´ quations x 2 + 4x + 1 = 0, x 2 + 3x + 1 = 0 et x 2 + 8x + 1 = 0 admet des solutions dans Z/ pZ. Il existe donc u ∈ (Z/ pZ)∗ tel que −1/u soit e´ gal a` u + 4, u + 3 ou u + 8. ¯ Calculons F(χ, u) lorsqu’on a Pd (u) = u 2 + du + 1 ≡ 0 (mod p). Soit i un entier vérifiant 0 < i < d. On a, en utilisant la relation (u +i)(u +d −i) ≡ i(d −i)−1 (mod p), u+i +

1 1 i(d − i) i(d − i) +u+d −i + ≡ + u+i u+d −i u+d −i u+i i(d − i)(2u + d) ≡ (mod p). i(d − i) − 1

Notons qu’on a Pd0 (u) = 2u + d et que le polynôme Pd n’a pas de racine multiple pour p 6 | d 2 − 4. Plus précisément les polynômes P3 , P5 et P15 sont sans racine multiple pour p distinct de 2, 3 et 5. On a alors Pd0 (u) 6= 0. Lorsque u + i 6= 0, pour

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0 < i < d, on obtient la relation X 1 0 i(d − i) ¯ , F(χ, u) = Pd0 (u) 2 i(d − i) − 1 0 3. COROLLAIRE

Soit E une courbe elliptique sur Q p (µ p ) telle que K p (E) = Q p (µ p ) et j (E) ∈ Z p [µ p ]. Alors la fibre spéciale du modèle de Néron de E n’a pas potentiellement bonne réduction supersingulière et possède un point F p -rationnel d’ordre p. Démonstration

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Le groupe E(Q p (µ p )) possède un sous-groupe C isomorphe a` (Z/ pZ)2 car j (E) ∈ Z p [µ p ]. Comme p > 3, ce sous-groupe est contenu dans E 0 (Q p (µ p )). D’après la proposition 15, on a p − 1 ≥ |(C − 0) ∩ E 1 (Q p (µ p ))|. Cela interdit le cas supersingulier, car on a alors C ⊂ E 1 (Q p (µ p )). On est donc dans le cas potentiellement ordinaire, ce qui entraˆıne que l’ordre de C ∩ E 1 (Q p (µ p )) est p. Le groupe E 0 (Q p (µ p ))/E 1 (Q p (µ p )) ' E(F p ) contient donc un e´ lément d’ordre p. 3.2. Quelques données dues a` William Stein Par abus de notation, nous dirons qu’un e´ lément j ∈ P1 (F p ) présente une anomalie s’il existe une courbe elliptique sur F p d’invariant modulaire j et possédant un point F p -rationnel d’ordre p. PROPOSITION 16 Soit p un nombre premier congru a` −1 modulo 4. Supposons que pour tout j ∈ P1 (F p ) présentant une anomalie et tout caractère de Dirichlet χ : (Z/ pZ)∗ −→ C, il existe tχ ∈ T[χ ] et δ ∈ 1 S tels que L(tχ J0 ( p), χ, 1) 6 = 0 et ι1 (tχ δ) 6= 0. Alors on ap∈ / S.

Démonstration La condition L(tχ J0 ( p), 1) 6 = 0 impose H p (χ) (et donc p > 7) et, par le théorème de Kato, la finitude de la composante χ -isotypique de tχ J0 ( p)(Q(µ p )). Si p ∈ S, on a p ∈ So ( p ∈ Sc et p ∈ Ss sont exclus d’après le corollaire 3 de la proposition 6 et le corollaire de la proposition 15 respectivement). Soit E une courbe elliptique sur Q(µ p ) possédant p 2 −1 points Q(µ p )-rationnels d’ordre p. Elle a donc bonne réduction en l’idéal au dessus de p et présente une anomalie (i.e., elle possède un point F p -rationnel d’ordre p) d’après le corollaire 3 de la proposition 6 et le corollaire de la proposition 15. Elle définit un point Q(µ p )-rationnel p-ordinaire de Y0 ( p). Ce point est Q-rationnel par application du corollaire 1 de la proposition 6 (le critère de pseudo-immersion formelle est vérifié pour d = 1). Dans ces conditions, d’après [17], Y0 ( p) n’a pas de point Q-rationnel pour p ∈ / {3, 7, 11, 19, 43, 67, 163} et pas de point Q-rationnel sans multiplications complexes et donc pas de point pordinaire pour p ∈ / {3, 7, 11}. Le cas p = 11, qui revient a` examiner trois courbes el¯ liptiques (à Q-isomorphisme près), a e´ té e´ tudié en détail par G. Ligozat dans [14] (voir démonstration de [14, proposition 5.6.1]) qui démontre en examinant les réductions modulo 23 que ces courbes ne donnent pas lieu a` un e´ lément de S. William Stein a constaté que, pour p < 1000, p > 7 et p ≡ −1 (mod 4), les hypothèses de la proposition 16 sont vérifiées.


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Remerciements. J’exprime ma gratitude envers ceux qui ont pris la peine d’écouter et parfois de répondre a` mes questions : Jean-Marc Fontaine, Benedict Gross, Emmanuel Halberstadt, Emmanuel Kowalski, Alain Kraus, Barry Mazur, Jean-François Mestre, Philippe Michel, Joseph Oesterlé, Marusia Rebolledo, David Rohrlich, Anthony Scholl, William Stein. Les résultats de ce cet article ont e´ té partiellement exposés lors du colloque en l’honneur de Barry Mazur pour son soixantième anniversaire. La présente version présente des modifications par rapport a` une version préliminaire et incomplète qui a diffusée sous forme de prépublication de l’Institut de mathématiques de Jussieu. Ajoutons que cet article, dont les prémices datent de l’automne 1995 passé a` l’été Institute for Advanced Studies (Princeton), a e´ té essentiellement rédigé lors d’une visite a` l’Université de Pondichéry. References [1]

[2]

[3]

[4]

[5] [6] [7] [8] [9] [10] [11]

P. DELIGNE et M. RAPOPORT, “Les schémas de modules de courbes elliptiques” dans

Modular Functions of One Variable (Antwerp, 1972), II, Lecture Notes in Math. 349, Springer, Berlin, 1973, 143–316, MR 49:2762; “Correction” dans Modular Functions of One Variable (Antwerp, 1972), IV, Lecture Notes in Math. 476, Springer, Berlin, 1975, 149. MR 52:3177 83, 86, 87 M. FLEXOR et J. OESTERLE´ , “Sur les points de torsion des courbes elliptiques” dans Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988), Astérisque 183, Soc. Math. France, Montrouge, 1990, 25–36. MR 91g:11057 107 L. FOURQUAUX, Produits de Petersson de formes modulaires associées aux valeurs de ´ fonctions L, Mémoire de Diplome d’Etudes Approfondies, Université Pierre et Marie Curie, 2000. 106 B. H. GROSS, “Heights and the special values of L-series” dans Number Theory (Montréal, 1985), CMS Conf. Proc. 7, Amer. Math. Soc., Providence, 1987, 115–187. MR 89c:11082 88, 90 B. H. GROSS et S. S. KUDLA, Heights and the central critical values of triple product L-functions, Compositio Math. 81 (1992), 143–209. MR 93g:11047 88, 90 E. HALBERSTADT, lettre du 8 octobre 1998. 81 √ S. KAMIENNY, Points of order p on elliptic curves over Q( p), Math. Ann. 261 (1982), 413–424. MR 84g:14047 92 , p-torsion in elliptic curves over subfields of Q(µ p ), Math. Ann. 280 (1988), 513–519. MR 90a:11061 92 , Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math. 109 (1992), 221–229. MR 93h:11054 90, 93 , Torsion points on elliptic curves over fields of higher degree, Internat. Math. Res. Notices 1992, 129–133. MR 93e:11072 93 S. KAMIENNY et B. MAZUR, “Rational torsion of prime order in elliptic curves over number fields ; appendix by A. Granville” dans Columbia University Number Theory Seminar (New York, 1992), Astérisque 228, Soc. Math. France,

LO ÏC MEREL

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Montrouge, 1995, 3, 81–100. MR 96c:11058 92 [12]

[13] [14]

[15] [16]

[17] [18] [19] [20]

[21] [22]

[23] [24] [25]

[26]

[27]

¨ , Finiteness of the group of rational points for V. A. KOLYVAGIN et D. YU. LOGACHEV

some abelian modular varieties, Leningrad Math. J. 1 (1990), 1229–1253. MR 91c:11032 91 E. KOWALSKI et P. MICHEL, Deux théorèmes de non-annulation de valeurs spéciales de fonctions L, Manuscripta Math. 104 (2001), 1–19. MR 1 820 726 83 G. LIGOZAT, “Courbes modulaires de niveau 11” dans Modular Functions of One Variable (Bonn, 1976), V, Lecture Notes in Math. 601, Springer, Berlin, 1977, 149–237. MR 57:3079 108 YU. I. MANIN, Parabolic points and zeta function of modular curves, Math. USSR-Izv. 6 (1972), 19–64. MR 47:3396 ´ B. MAZUR, Modular curves and the Eisenstein ideal, Inst. Hautes Etudes Sci. Publ. Math. 47 (1977), 33–186. MR 80c:14015 83, 85, 86, 88, 93, 96, 98, 102, 103, 104 , Rational isogenies of prime degree, Invent. Math. 44 (1978), 129–162. MR 80h:14022 82, 85, 86, 87, 96, 108 , On the arithmetic of special values of L-functions, Invent. Math. 55 (1979), 207–240. MR 82e:14033 105 B. MAZUR et J. TATE, Refined conjectures of the “Birch and Swinnerton-Dyer type,” Duke Math. J. 54 (1987), 711–750. MR 88k:11039 104 L. MEREL, L’accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de J0 ( p), J. Reine Angew. Math. 477 (1996), 71–115. MR 97f:11045 97, 102, 103 , Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449. MR 96i:11057 81, 98, 99, 105 J.-F. MESTRE, “La méthode des graphes: Exemples et applications” dans Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields (Katata, 1986), Nagoya Univ., Nagoya, 1986, 217–242. MR 88e:11025 88 J.-F. MESTRE et J. OESTERLE´ , Courbes elliptiques de conducteur premier, manuscrit non publié. 88, 89 H. RADEMACHER, Zur Theorie der Modulfunktionen, J. Reine Angew. Math. 167 (1931), 312–336. 105 K. RUBIN, “Euler systems and modular elliptic curves” dans Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 351–367. MR 2001a:11106 91, 92 A. J. SCHOLL, “An introduction to Kato’s Euler systems” dans Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 379–460. MR 2000g:11057 82, 91, 92 G. SHIMURA, Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures 1, Iwanami Shoten, Tokyo; Publ. Math. Soc. Japan 11, Princeton Univ. Press, Princeton, 1971. MR 47:3318 91


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Merel Théorie des nombres, case 247, Institut de mathématiques de Jussieu, 4 place Jussieu, F-75252 Paris CEDEX 05, France ; [email protected] ; Unité de Formation et de Recherche de mathématiques, case 7012, Université Denis Diderot, 2 place Jussieu, F-75251 Paris CEDEX 05, France

Appendice Vérification de l’hypothèse H p (χ) pour p grand E. KOWALSKI

et

P. MICHEL∗

Dans le texte qui précède, Merel a introduit l’hypothèse H p (χ), pour p premier et χ un caractère de Dirichlet primitif modulo p. Il s’agit d’un problème de nonannulation pour certaines fonctions L automorphes. Il a e´ galement fourni l’assertion e´ lémentaire suivante qui est e´ quivalente a` H p (χ ) : Il existe u modulo p, u 6= 0, tel que x=− Xu¯ χ(x) − χ (−1)χ(x) ¯ 6 = 0. x=u

La somme est prise entre des représentants entiers quelconques de u et de −u, ¯ où u¯ est l’inverse de u modulo p. On voit que si χ est quadratique et pair, l’expression en question est identiquement nulle. Nous montrons ici que réciproquement, si χ n’est pas quadratique pair, et p assez grand ( p > B, pour une constante absolue et effective B), cette assertion est vraie, et donc H p (χ) l’est e´ galement. Dans la note [KM] nous e´ tablissons directement le théorème de non-annulation (sous une forme plus forte) et e´ tudions le cas restant de χ quadratique pair (c‘est a` dire la valeur centrale de la dérivée des fonctions L correspondantes). 1. Préliminaires Dans tout ce qui suit, p est un nombre premier fixé et χ 6 = 1 est un caractère modulo p e´ galement fixé. On e´ tend comme d’habitude χ a` Z/ pZ (et Z) en posant χ(0) = 0. La somme de Gauss associée a` χ est a X τ (χ) = χ(a)e (1) p a mod p

et on a τ (χ )τ (χ¯ ) = χ(−1) p. On note W (χ) le “signe” de la somme de Gauss, √ c’est a` dire le nombre complexe de module 1 défini par W (χ ) = τ (χ)/ p. On a ∗ Kowalski

a bénéficié de la bourse National Science Foundation DMS-9202022.

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W (χ )W (χ¯ ) = χ(−1), W (χ)2 W (χ) ¯ 2 = 1. De plus, pour tout x modulo p on a χ(x) =

ax X 1 χ(a)e ¯ . τ (χ) ¯ p

(2)

a mod p

1 Le caractère χ vérifie W (χ)2 = 1 si et seulement si χ est quadratique et pair. LEMME

Démonstration En effet, soit σα , pour α ∈ (Z/ pZ)× , l’automorphisme du corps cyclotomique Q(µ p( p−1) ) donné par e(1/ p) 7→ e(α/ p) et fixant e(1/( p −1)). La somme de Gauss, donc aussi W (χ)2 , sont dans ce corps et on a σα τ (χ)2 2 = χ(α) ¯ 2 W (χ)2 σα W (χ) = p donc pour avoir W (χ)2 = 1, il faut que χ(α)2 = 1 pour tout α ∈ Z/ pZ, α 6 = 0. Cela signifie que χ is quadratique. Mais alors W (χ)2 = W (χ)W (χ) ¯ = χ(−1), donc W (χ )2 = 1 requiert aussi que χ soit pair. La réciproque est e´ vidente. 2. Réductions On pose bχ (a) = χ(a) ¯ − W (χ) ¯ 2 χ(a) pour a ∈ Z/ pZ, et F(χ, u) =

x=− Xu¯

χ(x) − χ(−1)χ(x) ¯

x=u

pour u ∈ Z/ pZ, u 6= 0. Le problème a` résoudre est de montrer que si F(χ , ·) est identiquement nulle, alors χ est quadratique et pair. LEMME 2 On a pour tout u 6= 0,

a(1 − u) 1 X bχ (a) au ¯ F(χ , u) = e −e . τ (χ) ¯ 1 − e(a/ p) p p a6 =0

Démonstration Soit f (χ, u) =

x=− Xu¯ x=u

χ(x) ;


113

on a F(χ , u) = f (χ, u) − χ(−1) f (χ, ¯ u). On utilise (2) pour e´ crire χ(x) en terme de caractères additifs. Choisissons le représentant u 1 de u tel que 0 < u 1 < p, et un représentant u 2 > u 1 de −u. ¯ Alors, (2) donne x=u X2 ax 1 X f (χ , u) = χ(a) ¯ e τ (χ) ¯ p x=u a6 =0

1

et la somme intérieure est une progression géométrique, x=u X2

e

x=u 1

ax p

au 1 − e (u 2 − u 1 + 1)a/ p 1 =e p 1 − e(a/ p)

d’où, réduisant u 1 et u 2 modulo p de nouveau a(1 − u) 1 X χ(a) ¯ au ¯ f (χ , u) = e −e . τ (χ) ¯ 1 − e(a/ p) p p

(3)

a6 =0

Appliqué a` χ¯ au lieu de χ , cela donne a(1 − u) χ(−1) X χ(a) au ¯ χ (−1) f (χ¯ , u) = e −e . τ (χ) 1 − e(a/ p) p p

(4)

a6 =0

Puisque τ (χ )τ (χ¯ ) = χ (−1) p, χ (−1) τ (χ) ¯ 1 τ (χ) ¯ 2 1 = = = W (χ¯ )2 τ (χ ) p τ (χ) ¯ p τ (χ) ¯ donc le lemme découle des deux formules (3) et (4) pour f (χ, u) et f (χ, ¯ u). Définissons au bχ (a) e , 1 − e(a/ p) p a6 =0 a −au X bχ (a) H (u) = e e . 1 − e(a/ p) p p G(u) =

X

(5) (6)

a6 =0

Le lemme s’écrit donc F(χ, u) = G(u) − H (u). ¯

(7)

On va maintenant appliquer l’analyse de Fourier sur le groupe multiplicatif (Z/ pZ)× , considérant l’identité hypothétique F(χ, ·) = 0 comme une relation de “modularité” entre G et H que l’on analyse par “transformation de Mellin”. Ce n’est que l’un de plusieurs choix possibles ici, d’autres solutions sont sans doute disponibles.

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Soit X le groupe des caractères multiplicatifs de Z/ pZ. On définit la transformée fˆ d’une fonction f définie sur (Z/ pZ)× , X fˆ(ψ) = f (u)ψ(u). (8) u6 =0

Par (7), on a ˆ ˆ ¯ F(χ, ψ) = G(ψ) − Hˆ (ψ). LEMME

(9)

3

On a ˆ G(ψ) = τ (ψ)

X a6 =0

bχ (a) ¯ ψ(a), 1 − e(a/ p)

Hˆ (ψ) = ψ(−1)τ (ψ)

X a6 =0

a bχ (a) ¯ e ψ(a). 1 − e(a/ p) p

(10) (11)

Démonstration Cela découle immédiatement des définitions et de (2). L’équation (9) peut s’écrire a X bχ (a) p X bχ (a) ˆ , ψ) = τ (ψ) ¯ ψ(a) − e ψ(a) F(χ 1 − e(a/ p) τ (ψ) 1 − e(a/ p) p a6 =0

a6 =0

ou bien a¯ X bχ (a) X bχ (a) ¯ τ (ψ) ˆ ¯ ¯ F(χ , ψ) = W (ψ)2 ψ(a) − e ψ(a). (12) p 1 − e(a/ p) 1 − e(a/ ¯ p) p a6 =0

a6=0

LEMME 4 Soit f une fonction sur X de la forme X ¯ f (ψ) = c(a)W (ψ)2 ψ(a) a6 =0

pour des nombres complexes c(a). Alors pour tout b modulo p, b 6= 0, on a 1 X 1X ¯ ψ(b) f (ψ) = c(a)S(a, b; p) p−1 p ψ∈X

où S(a, b; p) =

a6 =0

X ax + b x¯ e p x6=0

est la somme de Kloosterman.


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Démonstration Il suffit d’appliquer le lemme suivant, bien connu, et d’inverser l’ordre de sommation.

LEMME 5 Pour tout x modulo p, x 6= 0, on a

X

¯ W (ψ)2 ψ(x) =

ψ∈X

p−1 S(1, x; p). p

Démonstration On calcule, en développant le carré de la somme de Gauss : X ψ∈X

y + z X 1X ¯ ψ(x) ψ(y)ψ(z)e p p y,z ψ 1 X y + zX e ψ(x¯ yz) = p y,z p ψ p − 1 X y + z p−1 = e = S(1, x, p). p yz=x p p

¯ W (ψ)2 ψ(x) =

Soit Fˆ1 (χ , ψ) le membre de gauche de (12). On calcule 1 X ¯ Fˆ1 (χ, ψ) ψ(b) p−1 ψ∈X

et on obtient par le lemme et par orthogonalité des caractères, pour tout b 6 = 0, b X bχ (a) bχ (b) 1 X ¯ Fˆ1 (χ , ψ) = 1 ψ(b) S(a, b; p) − e . p−1 p 1 − e(a/ p) 1 − e(b/ p) p ψ∈X

a6 =0

(13) 3. Fin de la preuve On peut maintenant prouver la proposition suivante. 1 Il existe une constante absolue P telle que si p > P, et χ n’est pas quadratique pair, alors F(χ , ·) n’est pas identiquement nulle. PROPOSITION

LEMME

6

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On a pour 0 ≤ x ≤ π, r

2 (2π x) ≤ |1 − e(x)| ≤ 2π x. 5

LEMME 7 Pour tout a 6= 0, on a

2√10 √ 1 X bχ (a) √ S(a, b; p) ≤ p log 2( p − 1) ≤ 2.02 p log 2( p − 1) . p 1 − e(a/ p) π a6=0

Démonstration On a |bχ (a)| ≤ 2, et de plus l’estimation de Weil pour les sommes de Kloosterman, pour tout ab 6= 0, √ S(a, b; p) ≤ 2 p donc 1 X bχ (a) 4 X 1 S(a, b; p) ≤ √ p 1 − e(a/ p) p |1 − e(a/ p)| a6=0

a6 =0

X 8 1 =√ p |1 − e(a/ p)| 0 0, la constante implicite du O ne dépendant que de ε. Démonstration Soit w = W (χ). ¯ On a X X |bχ (a)|2 = 2 − w2 χ(a)2 − w¯ 2 χ(a) ¯ 2 a≤A

a≤A

= 2A − w2

X a≤A

χ(a)2 − w¯ 2

X a≤A

χ¯ (a)2 .


117

Comme χ n’est pas quadratique, χ 2 est un caractère non-trivial. D’après D. Burgess [Bur], [FI], on a alors X X χ (a)2 A1−δ , χ(a) ¯ 2 A1−δ a≤A

a≤A

pour un certain δ = δ(ε) > 0 ; c’est là que l’hypothèse A = p 1/4+ε avec ε > 0 intervient. Notons que si l’on admet l’hypothèse de Riemann pour les fonctions L de caractères de Dirichlet, on peut raffiner considérablement cette inégalité. Démonstration de la Proposition 1 Soit χ un caractère tel que F(χ, ·) = 0. Alors, d’après (13), on a pour tout b 6 = 0 l’égalité b bχ (b) 1 X bχ (a) e = S(a, b; p). (14) 1 − e(b/ p) p p 1 − e(a/ p) a6 =0

L’idée est qu’une telle identité n’est pas possible car les deux membres ne sont pas du même ordre de grandeur. Le membre de droite, d’après le Lemme 8, est borné par 1 X bχ (a) √ S(a, b; p) ≤ 2.02 p log 2( p − 1) . (15) p 1 − e(a/ p) a6 =0

Ainsi pour b < p/2, on déduit l’inégalité p |bχ (b)| √ ≤ 2.02 p log 2( p − 1) . b 2π

(16)

En particulier, si χ est quadratique pair, |bχ (b)| = 0 et le membre de gauche est toujours nul. Si χ est quadratique impair, |bχ (b)| = 2 donc on obtient, en prenant √ b = 1, l’inégalité p ≤ 2.02π log(2( p − 1)) qui est impossible dès que p ≥ 3067. Si χ n’est pas quadratique, on peut appliquer le Lemme 8 : celui ci implique que, si l’on fixe√ε > 0 quelconque, alors si p > P il existe b ≤ A = p 1/4+ε tel que |bχ (b)| ≥ 2. Procédant comme ci-dessus avec ce b dans (16) on trouve que cette inégalité impliquerait √ 2 3/4−ε √ p ≤ 2.02 p log 2( p − 1) . 2π Cela est impossible pour p assez grand si ε a e´ té choisi moins que 1/4. Remarque. Pour obtenir une constante explicite B telle que p > B est suffisant pour la validité de cet argument, il suffit d’avoir une forme explicite du lemme 8, ce qui revient a` avoir une forme explicite de l’estimation de Burgess (ou, en fait, de n’importe

LO ÏC MEREL

118

quelle estimation de sommes de caractères S(χ, A) =

X

χ(a)

a≤A

meilleure que l’inégalité de Polya-Vinogradov √ |S(χ, A)| ≤ 4 p log( p) √ car il faut trouver b < p(log p)−1 tel que bχ (b) ne soit pas trop petit). Malheureusement, il ne semble pas que des constantes possibles aient e´ té explicitées. De plus, dans l’inégalité de Burgess, le δ(ε) est en général très petit, d’autant plus que ε est petit ; par exemple, pour ε = 1/16, donc 1/4 + ε = 5/16, on a δ = 1/256. Cela signifie que si la constante implicite est mauvaise dans l’inégalité de Burgess, on aura le lemme seulement pour p très grand. Remarque. En fait, on peut e´ tablir l’hypothèse H p (χ) de manière complètement e´ lémentaire (i.e., sans avoir recours a` l’inégalité de Burgess qui utilise l’hypothèse de Riemann pour les courbes sur les corps finis). Supposons que χ n’est pas quadratique et que |W (χ )2 −1| ≤ p −1/4 (dans le cas contraire, on conclut a` nouveau par (16) appliquée a` b = 1), on applique alors le lemme suivant, au caractère (non-trivial) χ 2 (c’est une variante de la célèbre majoration de Vinogradov du plus petit premier nonrésidu quadratique ; sa preuve est e´ lémentaire et n’utilise que des arguments simples de crible ; cf. [LK, Thm. 7.7.6]). LEMME 9 Soit p un nombre premier assez grand et χ un caractère non-trivial modulo p, alors √ 1/2 e log2 p tel que |χ(`) − 1| ≥ 1/ log3 p. il existe un premier 2 ≤ ` ≤ p

On obtient alors la majoration √

p (1−1/

e)/2

≤ 4.04π log5 p log 2( p − 1) qui est impossible si p est assez grand. References [Bur]

D. A. BURGESS, On character sums and L-series, II, Proc. London Math. Soc. (3) 13

[FI]

J. FRIEDLANDER et H. IWANIEC, A mean-value theorem for character sums, Michigan

(1963), 524–536. MR 26:6133 117 Math. J. 39 (1992), 153–159. MR 92k:11084 117


119

[KM]

E. KOWALSKI et P. MICHEL, Deux théorèmes de non-annulation de valeurs spéciales

[LK]

L. K. HUA [HUA LOO KENG], Introduction to Number Theory, Springer, Berlin, 1982.

de fonctions L, Manuscripta Math. 104 (2001), 1–19. MR 1 820 726 111 MR 83f:10001 118

Kowalski Université Bordeaux I-A2X, 351, cours de la Libération, 33405 Talence CEDEX, France; [email protected] Michel L’Institut Universitaire de France ; Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier CEDEX 05, France ; [email protected]


MULTIPLETS OF REPRESENTATIONS AND KOSTANT’S DIRAC OPERATOR FOR EQUAL RANK LOOP GROUPS GREGORY D. LANDWEBER

Abstract Let g be a semisimple Lie algebra, and let h be a reductive subalgebra of maximal rank in g. Given any irreducible representation of g, consider its tensor product with the spin representation associated to the orthogonal complement of h in g. Recently, B. Gross, B. Kostant, P. Ramond, and S. Sternberg [2] proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of h, called a multiplet. Kostant [7] then constructed a formal h-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of h-representations corresponding to the given representation of g. We reproduce these results in the Kac-Moody setting for the extended loop alge˜ and Lh. ˜ We prove a homogeneous generalization of the Weyl-Kac character bras Lg formula, which now yields a multiplet of irreducible positive energy representations of Lh associated to any irreducible positive energy representation of Lg. We construct an Lh-equivariant operator, analogous to Kostant’s Dirac operator, on the tensor product of a representation of Lg with the spin representation associated to the complement of Lh in Lg. We then prove that the kernel of this operator gives the Lh-multiplet corresponding to the original representation of Lg. Contents 0. Introduction . . . . . . . . . . . . . . . . 1. Loop groups and their representations . . . 1.1. Loop groups . . . . . . . . . . . . . 1.2. The central extension . . . . . . . . 1.3. Affine roots and the affine Weyl group 1.4. Positive energy representations . . . .

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DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1, Received 8 May 2000. Revision received 31 October 2000. 2000 Mathematics Subject Classification. Primary 17B67; Secondary 17B35, 22E46, 81R10.

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121 124 124 124 126 128

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2. The spin representation . . . . . . . 3. The homogeneous Weyl-Kac formula 4. The Clifford algebra Cl(g) . . . . . 5. The Dirac operator on g . . . . . . 6. The Dirac operator on g/h . . . . . 7. The Clifford algebra Cl(Lg) . . . . 8. The Dirac operator on Lg . . . . . 9. The Dirac operator on Lg/Lh . . . 10. The kernel of the Dirac operator . . References . . . . . . . . . . . . . . .

GREGORY D. LANDWEBER

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129 132 136 140 142 146 151 155 157 159

0. Introduction Although this paper is chiefly concerned with representations of Lie groups and loop groups, the motivation for these results originally comes from M-theory. In physics the Lie group Spin(9) arises as the little group for massive particles in 10-dimensional superstring theories and as the little group for massless particles in 11-dimensional supergravity. Recently, T. Pengpan and P. Ramond noticed that the irreducible representations of Spin(9) come in triples, with the Casimir operator taking the same value on all three representations, and where the dimensions of two such representations sum to the dimension of the third. Ramond brought this curious fact to the attention of Sternberg, who in collaboration with Gross and Kostant then showed that these triples of representations of B4 = Spin(9) actually correspond to representations of the exceptional Lie group F4 , which contains B4 as an equal rank subgroup. In fact, this is not an isolated phenomenon. In [2] Gross, Kostant, Ramond, and Sternberg consider the general case where h is a reductive Lie algebra that is a maximal rank subalgebra of some semisimple Lie algebra g. Letting G and H denote the compact, simply-connected Lie groups with Lie algebras g and h, respectively, associated to any irreducible representation of G is a set of χ(G/H ) irreducible representations of H , where χ (G/H ) is the Euler number of the homogeneous space G/H . We refer to such a set of H -representations as a multiplet. As in the case of B4 ⊂ F4 , all of the representations in a multiplet share the same value of the Casimir operator, and the alternating sum of the dimensions of these representations vanishes. The relation between a representation of G and the H -representations in the corresponding multiplet is given by the following homogeneous generalization of the Weyl character formula, viewed as an identity in the representation ring R(H ): X Vλ ⊗ S+ − Vλ ⊗ S− = (−1)c Uc(λ+ρG )−ρ H , (1) c∈C

where Vλ and Uµ denote the irreducible representations of G and H with highest weights λ and µ, respectively, S = S+ ⊕ S− is the spin representation associated

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to the complement of h in g, the subset C ⊂ WG of the Weyl group of G has one representative from each coset of W H , and (−1)c is the sign of the element c. In representation theory the Casimir operator of a Lie algebra is analogous to the Laplacian. Using the spin representation, we can also consider operators analogous to the Dirac operator. Furthermore, we can choose a particular Dirac operator such that its square is the Casimir operator shifted by a constant, giving a representation theory version of the Weitzenböck formula. Such a Dirac operator was introduced in a more formal setting by A. Alekseev and E. Meinrenken in [1], and the geometric version of this Dirac operator is examined in [12]. Since the Casimir operator takes the same value on all of the representations in a multiplet, it follows that this Dirac operator likewise takes a constant value, up to sign, on each multiplet. In the homogeneous case, for any linear operator 6 ∂ : Vλ ⊗ S+ → Vλ ⊗ S− , since both the domain and the range are finite-dimensional, the index of 6 ∂ must be given by (1). This prompted Kostant to search for a Dirac operator whose kernel and cokernel are precisely those representations on the right-hand side of (1). In [7] Kostant constructs a Dirac operator 6 ∂ g/h on Vλ ⊗ S with a cubic term associated to the fundamental 3-form on g. The kernel of Kostant’s Dirac operator is M Ker 6 ∂ g/h = Uc(λ+ρG )−ρ H , (2) c∈C

and the signs (−1)c on the right side of (1) can be recovered by decomposing the operator 6 ∂ g/h according to the positive and negative half-spin representations. Taking the kernel of Kostant’s Dirac operator therefore gives an explicit construction of the multiplet of H -representations corresponding to a given representation of G. This paper takes the results discussed above and reformulates them in the KacMoody setting, replacing the equal rank Lie groups H ⊂ G with their corresponding loop groups L H ⊂ LG. After briefly reviewing the representation theory of loop groups in §1, we introduce the positive energy spin representation S Lg associated to a loop group in §2, using it to reformulate the Weyl-Kac character formula. In §3 we prove the following homogeneous version of the Weyl-Kac character formula: X + − Hλ ⊗ S Lg/Lh − Hλ ⊗ S Lg/Lh = (−1)c Uc(λ−ρ g )+ρ h , c∈C

where Hλ and Uµ denote the positive energy representations of the central extensions ˜ and L˜ H with lowest weights λ and µ, respectively, the subset C ⊂ WG now lives LG in the affine Weyl group of G, and −ρ g and −ρ h are the lowest weights of the spin representations S Lg and S Lh .

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In §§4–6 we return to the case of compact Lie groups, reviewing various results of [1] and [7]. There we construct Kostant’s Dirac operator, compute its square, and prove that its kernel has the form given by (2). Our approach here differs slightly from Kostant’s, which views the Lie algebra g as an orthogonal extension of h. Instead we first consider the Dirac operator on g and a twisted Dirac operator on h and then construct Kostant’s Dirac operator as their difference, an idea borrowed from [4] and [5]. In addition, we avoid working with a basis for g wherever possible, which greatly simplifies the computations and hopefully elucidates their meanings. These sections can stand alone as an alternative exposition on Kostant’s Dirac operator, and they provide an outline of the more advanced material in the subsequent sections. The remaining sections reprise these results for the loop group case. In §7 we examine the Clifford algebra associated to a loop group, which builds on the treatment of infinite dimensional Clifford algebras given in [8]. We then introduce the Dirac and Casimir operators associated to a loop group in §8, and we construct the loop group analogue 6 ∂ Lg/Lh of Kostant’s Dirac operator in §9. These Dirac and Casimir operators appear in the physics literature in [3], [4], and [5] as the odd and even zero-mode generators for the N = 1 superconformal algebras associated to current (Lie group) and coset space (homogeneous space) models. In contrast, our treatment builds these operators on a mathematical foundation, viewing them as canonical objects rather than working in terms of a basis. Finally, we compute the square of the Dirac operator 6 ∂ Lg/Lh , and in §10 we prove that its kernel is M Ker 6 ∂ Lg/Lh = Uc(λ−ρ g )+ρ h , c∈C

just as for compact Lie groups. So once again, taking the kernel of this Dirac operator provides an explicit construction for the multiplet of representations of L˜ H ˜ corresponding to any given representation of LG. Note Anthony Wassermann, who has independently obtained results similar to those in this paper, pointed out to me that with only minor modifications, the arguments presented here provide a quick proof of the Weyl-Kac character formula.

1. Loop groups and their representations 1.1. Loop groups Let G be a compact connected Lie group, and let LG denote the group of free loops on G, that is, the space of smooth maps from S 1 to G, where the product of two loops


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is taken pointwise. The Lie algebra of the loop group LG is simply the vector space Lg of loops on the Lie algebra g of G, with brackets again taken pointwise. The group Diff(S 1 ) of diffeomorphisms of the circle acts on loop spaces by reparameterizing the loops, and in particular the subgroup S 1 of rigid rotations of the circle acts on LG and Lg. This circle action induces a Z-grading on the complexified Lie algebra LgC , L which is the closure of the direct sum of the Fourier components k∈Z gC z k , where gC z k denotes loops of the form z 7 → X z k for X ∈ gC . We are interested in those representations of LG which likewise admit a Z-grading intertwining with the S 1 action on LG, or, in other words, representations of the semidirect product S 1 n LG. L Such a representation E then decomposes into eigenspaces k∈Z E (k) according to the S 1 -weight k, called the energy. (This terminology comes from an analogy with quantum mechanics, where the energies are eigenvalues of the Hamiltonian operator, which generates time translation.) 1.2. The central extension The representations that we consider are actually projective representations of LG. ˜ of To realize them as true representations, we must introduce a central extension LG 1 LG by S . This is analogous to taking the universal cover of a compact Lie group, except that here we lift to a circle bundle rather than a finite cover. The corresponding central extension of the Lie algebra, which is called a Kac˜ = Lg ⊕ RI , where I is the infinitesimal generator of the Moody algebra, is Lg 1 ˜ is determined by a central S subgroup. The Lie bracket on the central extension Lg choice of ad-invariant inner product on Lg. Any ad-invariant inner product on the Lie algebra g induces an inner product on Lg by averaging the pointwise inner products. For loops ξ, η ∈ Lg, this gives Z 2π

1 ξ(θ ), η(θ) dθ, (3) hξ, ηi = 2π 0 ˜ we must actually go which is ad-invariant on Lg. To extend this inner product to Lg, ˜ ˜ Lg, where R is generated by one step further and extend it to the semidirect sum R ⊕ the infinitesimal rotation ∂θ , and we define the inner product by ha ∂θ + ξ + x I, b ∂θ + η + y I i = hξ, ηi − ay − bx for a, b, x, y ∈ R and ξ, η ∈ Lg. This inner product is ad-invariant on the extended ˜ provided that the Lie bracket on the central extension Lg ˜ is ˜ Lg, Lie algebra R ⊕ [ξ, η] Lg ˜ = [ξ, η] Lg + hξ, ∂θ ηi I.

(4)

Although this central extension depends on the original choice of inner product on g, there is a unique ad-invariant inner product on g (up to scaling) if g is simple. In

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this case the universal central extension corresponds to the smallest possible scaling ˜ exponentiates to give a central extension LG ˜ of the loop for which the Lie algebra Lg group LG. This smallest inner product on g is the basic inner product, which is scaled so that the highest root αmax of g satisfies kαmax k2 = 2. If G is not simple but only semisimple, then a given projective representation of LG can still be lifted to a true representation of some S 1 -extension of LG. However, the universal central extension of LG is no longer a circle bundle but rather an extension by the torus T d , where d counts the number of simple components of G. At the Lie algebra level, an ad-invariant inner product on g can be scaled separately on each of the simple components, and the central term in the Lie bracket (4) now becomes d separate terms corresponding to the basic inner products for each of these components. Remark. Let G be simply connected. Topologically, the invariant inner products on g correspond to elements of the Lie algebra cohomology H 3 (g) ∼ = H 3 (G; R) by associ3 ∗ ating to any inner product its fundamental 3-form ω ∈ 3 (g ) given by ω(X, Y, Z ) = hX, [Y, Z ]i for X, Y, Z ∈ g. The possible central extensions of the Lie algebra Lg by a circle thus correspond to elements of the real cohomology H 3 (G; R), and the universal central extension of Lg is then an extension by the dual space K = H3 (G; R). On the other hand, the central extensions of the loop group LG correspond to circle bundles, which are classified by their Chern classes c1 ∈ H 2 (LG; Z) ∼ = H 3 (G; Z) 3 in the integral lattice of H (G; R). Writing L = H3 (G; Z) for the dual lattice in K , ˜ is an extension of LG by the torus K /L. Using the universal central extension LG the cohomology spectral sequence for this extension and noting that H 1 (LG; Z) = H 2 (G; Z) = 0, we obtain the exact sequence d2 ˜ Z) → H 1 (K /L; Z) → ˜ Z) → 0. 0 → H 1 ( LG; H 2 (LG; Z) → H 2 ( LG;

Now, by our construction of the torus K /L, we have a canonical isomorphism H 1 (K /L; Z) ∼ = H 3 (G; Z), and we also have a canonical isomorphism H 2 (LG; Z) ∼ = 3 H (G; Z). The map d2 is therefore a homomorphism d2 : H 3 (G; Z) → H 3 (G; Z), and the universality condition requires that d2 be the identity map. In particular, if ˜ is the universal central extension, then d2 must be an isomorphism, and it follows LG ˜ Z) = H 2 ( LG; ˜ Z) = 0, which in terms of homotopy implies that LG ˜ is that H 1 ( LG; 2-connected. So, whereas taking the universal cover of a compact Lie group G kills the obstruction π1 (G), the loop group LG is already simply connected, but taking its universal central extension kills the obstruction π2 (LG). ˜ introduced above is the Lie algebra of the semidirect ˜ Lg The semidirect sum R ⊕ 1 ˜ and from here on we refer to representations of S 1 n LG ˜ as repproduct S n LG, resentations of LG. Given such a representation, we call the weight of the central


127

˜ the level or central charge, and since this circle by definition commutes S 1 in LG with the rest of the loop group, it follows that the level is constant on each irreducible representation. Unless stated otherwise, from here on we assume that G is simply ˜ denote the connected and simple, we use the basic inner product on g, and we let LG universal central extension. However, the following discussion can be generalized to the semisimple case by treating the d simple components separately and viewing the level as a d-vector. 1.3. Affine roots and the affine Weyl group Let T be a maximal torus of G. When considering the representation theory of loop groups, rather than taking the abelian subgroup L T as the maximal torus of LG, we ˜ instead use the maximal torus S 1 × T × S 1 of S 1 n LG. Here the first S 1 -factor corresponds to rotation of loops, while the second comes from the central extension. The Cartan subalgebra is then R ⊕ t ⊕ R, and the weights of LG are of the form λ = (m, λ, h), where m is the energy, λ is a weight of G, and h is the level. In this notation the roots of LG, also called the affine roots of G, consist of the weights (m, α, 0) with m ∈ Z and α a root of G as well as the weights (m, 0, 0) for nonzero m, counted with multiplicity rank G = dim t. Given a system of positive roots for G, we take the positive roots of LG to be the roots (0, α, 0) for α > 0 as well as all roots (m, α, 0) with m > 0, including roots of the form (m, 0, 0). If {αi } is a set of simple roots for G, then the corresponding simple affine roots for LG are (0, αi , 0) as well as the root (1, −αmax , 0), where αmax is the highest root of G. The affine Weyl group WG of G is the group generated by the reflections through the hyperplanes corresponding to the affine roots of G. In terms of loop groups, given ˜ generany root α = (k, α, 0) of LG, there is a corresponding su(2) subalgebra of Lg k −k ated by the loops E α z and E −α z and the coroot 2 1 Hk,α = E α z k , E −α z −k Lg ˜ = Hα + 2 ikkHα k I, where {E α , E −α , Hα } span the su(2) subalgebra of g associated to the root α. Note that these elements are normalized so that hE α , E −α i = (1/2)kHα k2 = 2hα, αi−1 . The reflection of a weight λ = (m, λ, h) through the hyperplane orthogonal to α is then sk,α (λ) = λ − λ(Hk,α )α = m − λ(Hα )k + 12 hkHα k2 k 2 , λ − λ(Hα )α + 12 hkHα k2 kα, h .

(5)

Furthermore, these sk,α are generated by the reflections s0,α , which act solely on the t∗ component and generate the usual Weyl group WG , as well as the transformations tα (λ) = s1,α s0,α (λ) = m + λ(Hα ) + 21 hkHα k2 , λ + h Hα , h ,

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where we use the inner product to identify the coroot Hα ∈ t with the weight (1/2)kHα k2 α in t∗ . Restricting to t∗ , the tα are simply translations by the coroots, which generate the coweight lattice L ⊂ t. We therefore have WG ∼ = WG n L. Note that under the action of the affine Weyl group, the level h is fixed, while the energy m is shifted so as to preserve the inner product (m 1 , λ1 , h 1 ) · (m 2 , λ2 , h 2 ) = hλ1 , λ2 i − m 1 h 2 − m 2 h 1

(6)

on R ⊕ t∗ ⊕ R. Thus, at any given level h, the affine Weyl action is completely determined by its restriction to t∗ . In particular, the element sk,α corresponds to the reflection through the hyperplane given by the equation hλ, αi = hk. These hyperplanes divide t∗ into connected components called alcoves, and the affine Weyl group acts simply transitively on these alcoves. Given a positive root system for LG, the corresponding fundamental alcove is the unique alcove satisfying λ · α ≤ 0 for all α > 0. This alcove is bounded by the hyperplanes corresponding to the negatives of the simple affine roots, or in other words, a weight λ = (m, λ, h) lies in the fundamental alcove if and only if −λ is in the positive Weyl chamber for G and hλ, −αmax i ≤ h. 1.4. Positive energy representations A representation H of LG is a positive energy representation if H (k) = 0 for all k < m for some fixed integer m, or in other words, there is a minimum energy when H is decomposed into its constant energy eigenspaces. In the literature, positive energy representations are often normalized so that this minimum energy is zero. However, we consider positive energy representations with the full spectrum of minimum energies. When restricted to the positive energy representations, the representation theory of loop groups behaves quite analogously to the representation theory of compact Lie groups. In particular, the positive energy representations satisfy the following fundamental properties (for a complete discussion, see [11]). (i) A positive energy representation is completely reducible into a direct sum of (possibly infinitely many) irreducible positive energy representations. (ii) An irreducible positive energy representation H is of finite type: each of the constant energy subspaces H (k) is a finite-dimensional representation of G. (iii) Every irreducible positive energy representation H has a unique lowest weight λ = (m, λ, h), in the sense that λ−α is not a weight of H for any positive root α of LG. The lowest weight space is 1-dimensional and generates H. (iv) A weight λ = (m, λ, h) is antidominant for LG if it lies in the fundamental Weyl alcove described at the end of §1.3 above. The lowest weight of a positive energy representation is antidominant, and every antidominant weight is realized as the lowest weight of some positive energy representation.


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As a consequence of (iii), an irreducible positive energy representation H is completely characterized by its minimum energy m, its minimum energy subspace H (m) ∼ = V−λ , and its level h. Property (iv) implies that for a positive energy representation, the level h is always nonnegative and is zero only for the trivial representation. Also, for a fixed minimum energy m, there are only finitely many positive energy representations at each level h, but as the level tends to infinity, the representation theory of LG resembles that of G. If Hλ is the irreducible positive energy representation with lowest weight λ = (0, λ, h), then Hλ also contains all the weights in the orbit of λ under the affine Weyl group WG . Recalling that the affine Weyl group action preserves the inner product (6), it turns out that the orbit of λ consists of all weights µ = (m, µ, h) at level h satisfying λ · λ = µ · µ, or, equivalently, kµk2 − 2mh = kλk2 . This equation sweeps out a paraboloid, and the weights of Hλ all lie in its interior. (As the level h tends to infinity, this paraboloid flattens into a cone.) For an example, see Figure 1 at the end of §3, which gives the weights of the irreducible representation of LSU(2) with lowest weight (0, −1, 2). 2. The spin representation If V is a finite-dimensional vector space with an inner product and V = W ⊕ W ∗ is a polarization of V into a maximal isotropic subspace W and its dual, then the spin representation of the Clifford algebra Cl(V ) can be written in the form SV = 3∗ (W ) ⊗ (det W )−1/2 ,

(7)

where det W denotes the top exterior power of W . The resulting spin representation SV is independent of the choice of polarization, which is accounted for by the factor of (det W )−1/2 . On the other hand, if V is infinite-dimensional, then this determinant factor does not make sense, and so we can no longer use (7) to define the spin representation. Without this determinant factor to correct for the choice of polarization, different polarizations give rise to distinct spin representations. For a general discussion of infinite-dimensional Clifford algebras and their spin representations, see [8]. For our purposes, consider the Lie algebra Lg with the inner product (3) induced by the basic inner product on g. If we complexify Lg, then the orthogonal complement of the Cartan subalgebra tC in LgC decomposes into the sum of the positive and negative root spaces, each of which is isotropic with respect to the inner product on LgC . We can therefore use this polarization to define a positive energy spin representation associated to the complement of t in Lg: M O S Lg/t := Sg/t ⊗ 3∗ gC z k = Sg/t ⊗ 3∗ gC z k , (8) k>0

k>0

where we have explicitly factored out the contribution Sg/t coming from the constant

130


loops (or zero modes). Here we have used the expression (7) for the spin representation, except that we have dropped the portion of the (det W )−1/2 factor coming from the positive energy modes. If we were to include that factor, it would contribute an overall anomalous energy shift of Y −1/2 P z k dim g = z −(1/2) k>0 k dim g = z (1/24) dim g , (9) k>0

where in the last equality we use the Riemann zeta function trick to write the infinite P sum as k>0 k = ζ (−1) = −1/12. Fortunately, by normalizing the spin representation to have minimum energy zero, we can safely ignore this factor. For the moment we are interested only in the character of the spin representation. The restriction of the character of S Lg/t to S 1 × T is completely determined by the description (8) of the spin representation. However, in correcting for the infinite determinant factor, the spin representation acquires a nonzero central charge. THEOREM 1 If G is simple, then the central charge of the spin representation S Lg/t is the value of the quadratic Casimir operator of g in the adjoint representation: g

cG = 1ad = −

1 X (ad X i )2 = hρG , αmax i + 1, 2 i

where ρG is half the sum of the positive roots, αmax is the highest root of G, and {X i } is an orthonormal basis for g. Proof To compute the central charge of the spin representation S Lg/t , we extend it to obtain the spin representation associated to the entire Lie algebra Lg. Since the construction of spin representations is multiplicative, we have S Lg ∼ = St ⊗ S Lg/t .

These two spin representations have the same central charge since they differ only by the finite-dimensional factor St . However, the extended spin representation S Lg ˜ and in fact, S Lg is the direct sum of admits an action of the full Lie algebra Lg, dim St copies of an irreducible positive energy representation of Lg. Examining the structure of this representation, the first three energy levels of S Lg are as follows: S Lg (0) = Sg , S Lg (1) = Sg ⊗ gC , S Lg (2) = Sg ⊗ gC ⊕ Sg ⊗ 32 (gC ).


131

Letting α denote the highest root of g, and c the central charge of S Lg , the highest weights of S Lg (0) and S Lg (1) are then (0, ρ, c) and (1, ρ +α, c), respectively, while the weight (2, ρ + 2α, c) is not present in S Lg (2). Therefore, the weights (0, ρ, c) and (1, ρ + α, c) form a complete string of weights for the root α = (1, α, 0), and so they must be related to each other by the affine Weyl element s1,α , the reflection through the hyperplane orthogonal to α. By (5) the difference of these weights is (1, ρ + α, c) − (0, ρ, c) = α = −(0, ρ, c)(H1,α )α, so we obtain −1 = (0, ρ, c)(H1,α ) = ρ(Hα ) − 21 kHα k2 c = hρ, αi − c, where (1/2)kHα k2 = 1 and ρ(Hα ) = hρ, αi in the basic inner product since α is the highest root. The central charge of the spin representation is thus c = hρ, αi + 1. The quadratic Casimir operator of a Lie algebra does not depend on the choice of orthonormal basis, and it commutes with the action of the Lie algebra. It therefore acts by a constant times the identity on each irreducible representation. On the irreducible representation of highest weight α, the value of the Casimir operator is (1/2)kαk2 + hα, ρi. In particular, if G is simple, then the adjoint representation is irreducible, and taking α to be the highest root of G, which satisfies kαk2 = 2 in the basic inner product, we again obtain the value hρ, αi + 1, as desired. We can now compute the character of S Lg/t directly from the decomposition (8) and Theorem 1. Written in terms of the affine roots α = (k, α, 0), the character is Y Y Y χ (S Lg/t ) = u cG eiα/2 + e−iα/2 1 + eiα z k = e−iρ G (1 + eiα ), α>0

k>0, α

α>0

˜ and ρ G = (0, ρG , −cG ). where u is a parameter on the central S 1 -extension in LG, Here −ρ G is the lowest weight of S Lg/t , which corresponds to the square root of the determinant in (7). This weight is the loop group version of ρG , half the sum of the positive roots of G, which is also characterized by the identity ρG (Hα ) = 1 for each of the simple roots α of G. In the loop group case, the identity ρ G (Hα ) = 1 must hold as α ranges over the simple affine roots, including the additional root (1, −αmax , 0). However, in our proof of Theorem 1, the condition ρ G (H1,−αmax ) = 1 is the same equation (up to sign) that we used to compute the central charge cG . + − The spin representation decomposes as S Lg/t = S Lg/t ⊕ S Lg/t into the sum of two half-spin representations. In particular, since the complement of t in g is evendimensional, the zero mode factor Sg/t of S Lg/t splits into half-spin representations, and the exterior algebra in (8) splits into its even and odd degree components. The difference of the characters of these half-spin representations is Y + − χ S Lg/t − χ S Lg/t = e−iρ G 1 − eiα , (10) α>0

132


which can be viewed either as a supertrace on S Lg/t or as the character of the virtual + − representation S Lg/t − S Lg/t . Using the notation of spin representations, the WeylKac character formula becomes the following. THEOREM 2 (Weyl-Kac character formula) If G is simply connected and simple, then the character of the irreducible positive ˜ with lowest weight λ is given by the quotient energy representation Hλ of LG P w iw(λ−ρ G ) w∈WG (−1) e χ(Hλ ) = (11) + − , χ S Lg/t − χ S Lg/t

where WG is the affine Weyl group of G and ρ G = (0, ρG , −cG ). Note that as an immediate consequence of the Weyl-Kac character formula, if we consider the trivial representation with λ = 0, we obtain the identity X + − χ S Lg/t − χ S Lg/t = (−1)w e−iw(ρ G ) , w∈WG

which gives an alternative expression for the signed character (10) of the spin representation appearing in the denominator of (11). Remark. If G is semisimple, then we recall that the universal central extension of LG is an extension not by a circle but rather by the torus T d , where d counts the number of simple components. In this case the central charge of the spin representation is the d-vector cG = (cG 1 , . . . , cG d ), where G 1 , . . . , G d are the simple components of G. If we work with the universal central extension of LG and define ρ G = (0, ρG , −cG ), then the Weyl-Kac character formula still holds as written. In fact, using the appropriate universal central extension, the Weyl-Kac character formula continues to hold for an arbitrary compact Lie group G. 3. The homogeneous Weyl-Kac formula Let g be a compact, semisimple Lie algebra, and let h be a reductive subalgebra of maximal rank in g. In [2] Gross, Kostant, Ramond, and Sternberg prove a homogeneous generalization of the Weyl character formula, associating to each grepresentation a set of h-representations with similar properties, called a multiplet. THEOREM 3 (Homogeneous Weyl formula) Let Vλ and Uµ denote the irreducible representations of g and h with highest weights λ and µ, respectively. The following identity holds in the representation ring R(h): X − Vλ ⊗ S+ (−1)c Uc(λ+ρg )−ρh , (12) g/h − Vλ ⊗ Sg/h = c∈C


133

where the sum is taken over the subset C of elements c ∈ Wg of the Weyl group of g for which c(λ + ρg ) − ρh are dominant weights of h. Note that if h = t is a Cartan subalgebra of g, then C is the full Weyl group Wg , and (12) becomes the Weyl character formula. Also, note that by stating this result in terms of the Lie algebras h ⊂ g rather than their corresponding Lie groups H ⊂ G, we bypass the issue of whether the spin representation Sg/h exponentiates to give a true representation of H . Geometrically this is equivalent to the condition that G/H be a spin manifold. Theorem 3 has an immediate analogue for loop groups. The only complication is that simply working at the level of Lie algebras is no longer sufficient to avoid the geometric obstruction, which in this case is the condition that G/H admit a string structure (see [10]). Rather, we must work with the universal central extensions. Given g ˜ be the universal central extension of Lg, and let Lh ˜ and h as described above, let Lg ˜ ˜ be the restriction of Lg to Lh. Note that Lh is not in general the universal central ˆ Rather, Lh ˜ is a quotient of Lh. ˆ Since h extension of Lh, which we denote by Lh. has the same rank as g, if t is a Cartan subalgebra of h, then it is likewise a Cartan ˆ and R ⊕ ˜ are then R ⊕ t ⊕ Rdh ˜ Lh ˜ Lg subalgebra of g. The Cartan subalgebras of R ⊕ d g and R ⊕ t ⊕ R , respectively, where dg is the number of simple components of g and dh ≥ dg . In other words, we have the commutative diagram quotient

R ⊕ t ⊕ Rdh −−−−→ R ⊕ t ⊕ Rdg     y y ˆ ˜ Lh R⊕

quotient

−−−−→

˜ ˜ Lh R⊕

R ⊕ t ⊕ Rdg   y inclusion

−−−−−→

˜ ˜ Lg R⊕

where the vertical arrows are inclusions of Cartan subalgebras. The weights of Lh and Lg live in the spaces dual to their Cartan subalgebras, and dual to the quotient map we have an inclusion R ⊕ t∗ ⊕ Rdg −→ R ⊕ t∗ ⊕ Rdh . We may therefore view the weight lattice of Lg as a subset of the weight lattice of Lh. On the other hand, if we ignore the central extension (i.e., restrict to weights of level zero), then the weight lattices are identical, and the roots of Lh are a subset of the roots of Lg. Consequently, the affine Weyl group Wh of h, which is generated by the reflections through the hyperplanes orthogonal to the roots of Lh, is a subgroup of the affine Weyl group Wg of g. 4 (Homogeneous Weyl-Kac formula) ˜ and Lh ˆ Let Hλ and Uµ denote the irreducible positive energy representations of Lg THEOREM

134


with lowest weights λ and µ, respectively. We then have the following identity for ˆ virtual representations of Lh: X + − Hλ ⊗ S Lg/Lh − Hλ ⊗ S Lg/Lh = (−1)c Uc(λ−ρ g )+ρ h , (13) c∈C

where the sum is taken over the subset C of elements c ∈ Wg of the affine Weyl group ˆ of g for which c(λ − ρ g ) + ρ h are antidominant weights of Lh. Proof We first note that the construction of the spin representation is multiplicative, provided that the underlying vector spaces are even-dimensional. In our case the positive and negative energy subspaces pair off, while for the zero modes the maximal rank condition implies that the complement of h in g and the complement of t in h are even-dimensional, so we have + − + − + − S Lg/t − S Lg/t = S Lg/Lh − S Lg/Lh ⊗ S Lh/t − S Lh/t . (14) Applying the Weyl-Kac character formula (11) to the left side of (13), and factoring the Weyl-Kac denominator using (14), we obtain P w iw(λ−ρ g ) w∈Wg (−1) e + − (15) χ Hλ ⊗ S Lg/Lh − χ Hλ ⊗ S Lg/Lh = + − . χ S Lh/t − χ S Lh/t Recall that the affine Weyl group acts simply transitively on the Weyl alcoves. Due to the ρ g shift, the weight λ − ρ g lies in the interior of the fundamental Weyl alcove for g, and thus for any w ∈ Wg , the weight w(λ − ρ g ) likewise lies in the interior of some Weyl alcove. Furthermore, the Weyl alcoves for g are completely contained inside the Weyl alcoves for h, and so there exists a unique element w0 ∈ Wh such that w0 w(λ − ρ g ) lies in the interior of the fundamental Weyl alcove for h. Shifting ˆ Putting back by ρ h , we see that the weight w0 w(λ − ρ g ) + ρ h is antidominant for Lh. c = w0 w, we can write w = (w0 )−1 c, and, more generally, we have Wg = Wh C . Using this decomposition to rewrite the numerator on the right side of (15), we have P P w iwc(λ−ρ g ) w iw(λ−ρ g ) X w∈Wh (−1) e w∈Wg (−1) e c (−1) + − = + − χ S Lh/t − χ S Lh/t χ S Lh/t − χ S Lh/t c∈C X (−1)c χ Uc (λ−ρ g )+ρ h , = c∈C

where the second line follows by applying the Weyl-Kac character formula (11) for ˆ This proves the character form of identity (13). Lh.


135

The subset C ⊂ Wg appearing in Theorem 4 does not depend on the weight λ. Rather, it consists of all elements of the affine Weyl group of g which map the fundamental Weyl alcove for g into the fundamental Weyl alcove for h. Since the affine Weyl group acts simply transitively on the Weyl alcoves, it follows that the cardinality of C is the ratio of the volumes of the fundamental alcoves for g and h. Equivalently, the elements of C are representatives of the cosets of Wh in Wg , so the cardinality of C is the index of Wh in Wg . In particular, the sum appearing in (13) is finite if and only if h is semisimple. In such cases, |C | is the index of Wh in Wg , which is also the Euler number of the corresponding homogeneous space G/H . Examples of pairs h ⊂ g with both h and g semisimple include Dn ⊂ Bn with |C | = 2 as well as the case B4 ⊂ F4 with |C | = 3 that prompted [2]. On the other hand, for pairs h ⊂ g corresponding to complex homogeneous spaces G/H , the group H must contain a U(1) component, and so (13) is an infinite sum. We note that in the physics literature (see [4], [5]), the N = 1 superconformal coset models on G/H possess an additional N = 2 symmetry precisely when C is infinite. At the other extreme, if h = t is a Cartan subalgebra of g, then ρ h vanishes, C is the full affine Weyl group Wg , and the homogeneous Weyl-Kac formula becomes X + − Hλ ⊗ S Lg/Lt − Hλ ⊗ S Lg/Lt = (−1)w Uw(λ−ρ g ) . (16) w∈Wg

This identity is equivalent to the Weyl-Kac character formula (11), but it is expressed slightly differently. Since t is abelian, the irreducible positive energy representation ˜ takes the particularly simple form Uµ of Lt M O Uµ = Sym∗ tC z k = Sym∗ (tC z k ), k>0

k>0

∗

where Sym is the symmetric algebra, and the character of this representation is Y Y χ (Uµ ) = eiµ (1 + z k + z 2k + · · · )dim t = eiµ (1 − z k )− dim t . (17) k>0

k>0

On the other hand, the signed character of the spin representation on Lt/t is Y + − χ S Lt/t − χ S Lt/t = (1 − z k )dim t

(18)

k>0

since the product in (10) is taken over the positive roots (k, 0, 0), each counted with multiplicity dim t. In particular, the products in the characters (17) and (18) cancel each other, yielding the Weyl-Kac character formula for L T : + − χ Uµ ⊗ S Lt/t − χ Uµ ⊗ S Lt/t = eiµ . So, multiplying formula (16) by the character (18), we recover the usual form of the Weyl-Kac character formula (11) for LG.

136


Example Take g = su(2), and let h = u(1) be the Cartan subalgebra of diagonal elements. In this particular case, we can use the homogeneous Weyl-Kac formula to explicitly compute the character of the entire spin representation S Lg/Lh , not just the difference of the two half-spin representations. Here we have ρ g = (0, ρg , −cg ) = (0, 1, −2), and so the lowest weight of the spin representation is −ρ g = (0, −1, 2). The halfspin representation S + (resp., S − ) is obtained by acting on a lowest weight vector by an even (resp., odd) number of Clifford multiplications by the positive generators E + and E ± z n for n > 0 of Lg/Lh. Since each of these generators shifts the su(2) weight by ±2, the su(2) weights of all elements in S + must be of the form 4n − 1, while the weights for S − are all of the form 4n + 1. Hence the weights of S + and S − are distinct, and thus there is no cancellation when we take their difference. Applying (16) for the case of the trivial representation with λ = 0, we obtain X X + S Lg/Lh = Uw(0,−1,2) = U(2n 2 −n,4n−1,2) , − S Lg/Lh =

w∈Wg+

n∈Z

X

X

Uw(0,−1,2) =

w∈Wg−

U(2n 2 +n,4n+1,2) ,

n∈Z

where we have explicitly written out the action of Wsu(2) ∼ = Z2 n Z: wn± (m, λ, h) = (m ± λn + hn 2 , ±λ + 2hn, h). Using (17) for χ (Uµ ), the characters of the half-spin representations are X Y 2 + (z, w, u) = u 2 w4n−1 z 2n −n (1 − z k )−1 , χ S Lg/Lh k>0

n∈Z

χ

− S Lg/Lh (z, w, u)

=u

2

X n∈Z

w

4n+1 2n 2 +n

z

Y

(1 − z k )−1 ,

k>0

where the powers of z, w, and u correspond to the energy, su(2) weight, and level, respectively. Combining these half-spin representations, the total spin representation has character X Y χ S Lg/Lh (z, w, u) = u 2 w2n−1 z (1/2)n(n−1) (1 − z k )−1 . n∈Z

k>0

The orbit of the lowest weight −ρ g = (0, −1, 2) under the affine Weyl group consists of all weights (m, λ, 2) with λ odd and m = (1/8)(λ2 − 1). This equation sweeps out a parabola, and the remaining weights live inside it, satisfying m > (1/8)(λ2 − 1). The weights of S Lg/Lh are shown in Figure 1, with the orbit of −ρ g drawn as open circles. The multiplicity of any such weight can be derived from (17) and is given by the number of partitions of m − (1/8)(λ2 − 1) into positive integers.


m

10 9 8 7 6 5 4 3 2 1 0

◦

• • • • ◦

• • • • • • • ◦

137

• • • • • • • • • ◦

• • • • • ◦ • • • • • • • • • • • • • • • • • • • ◦ • • • • • • • • • • • ◦ • • • • • ◦ ◦ ◦ −9 −7 −5 −3 −1 +1 +3 +5 +7 +9

λ

Figure 1. The weights of the spin representation on LSU(2)/LU(1)

4. The Clifford algebra Cl(g) Let g be a finite-dimensional Lie algebra with an ad-invariant inner product. Recall that the Clifford algebra Cl(g) is generated by the elements of g subject to the anticommutator relation {X, Y } = X · Y + Y · X = 2 hX, Y i for all X, Y ∈ g. There is a natural Clifford action on the exterior algebra 3∗ (g∗ ), which is given on the generators X ∈ g by c(X ) = ι X + ε X ∗ , where ι X is interior contraction by X ∈ g and ε X ∗ is exterior multiplication by the dual element X ∗ ∈ g∗ satisfying X ∗ (Y ) = hX, Y i. Using the distinguished element 1 of the exterior algebra, the map x 7→ c(x)1 gives an isomorphism Cl(g) → 3∗ (g∗ ) of left Cl(g)-modules, called the Chevalley identification. We may therefore view the Clifford algebra as the exterior algebra 3∗ (g∗ ) with the alternative multiplication X ∗ · η = X ∗ ∧ η + ιX η

(19)

for X ∈ g and η ∈ 3∗ (g∗ ). Consider the graded Lie superalgebra gˆ = g−1 ⊕ g0 ⊕ R1 , where the subscript denotes the integer grading. The exterior algebra 3∗ (g∗ ) is a representation of this Lie superalgebra gˆ , with g−1 acting by interior contraction, g0 acting by the coadjoint action, and the generator d ∈ R1 acting as the exterior derivative. On the generators ξ ∈ g∗ , these operators are given by ι X ξ = ξ(X ), (ad∗X ξ )(Y ) = −ξ(ad X Y ), (dξ )(X, Y ) = − 12 ξ([X, Y ]),

ι X : 3k (g∗ ) → 3k−1 (g∗ ), ad∗X : 3k (g∗ ) → 3k+0 (g∗ ), d : 3k (g∗ ) → 3k+1 (g∗ )

138


for X, Y ∈ g. These operators then extend as super-derivations to the full exterior algebra, and they satisfy the identities [ad∗X , ιY ] = ι[X,Y ] and {d, ι X } = ad∗X . If we perturb this action by taking d 0 = d − ι∗ , where is a closed g-invariant form of odd degree, then the commutation relations on gˆ are unchanged. 5 Using the Chevalley identification, the action of gˆ = g−1 ⊕ g0 ⊕ R1 on 3∗ (g∗ ) can be expressed in terms of the adjoint action of the Clifford algebra as THEOREM

ad X ∗ = 2 ι X , ∗

ad d X =

2 ad∗X ,

X ∗ ∈ 31 (g∗ ),

(20)

d X ∈ 3 (g ),

(21)

∈ 33 (g∗ ),

(22)

ad = 2 d − 2 ι∗ ,

∗

2

∗

where is the fundamental 3-form given by (X, Y, Z ) = −(1/6)hX, [Y, Z ]i. Proof First, we show that the operators ι X and ad∗X are super-derivations with respect to the Clifford product (19). For X, Y ∈ g and η ∈ 3∗ (g∗ ), we have ι X (Y ∗ · η) = ι X (Y ∗ ∧ η) + ι X ιY η = (ι X Y ∗ ) ∧ η − Y ∗ ∧ ι X η − ιY ι X η = (ι X Y ∗ ) · η − Y ∗ · ι X η, ad∗X (Y ∗ · η) = ad∗X (Y ∗ ∧ η) + ad∗X ιY η = (ad∗X Y ∗ ) ∧ η + Y ∗ ∧ ad∗X η + ιY ad∗X η + ι[X,Y ] η = (ad∗X Y ∗ ) · η + Y ∗ · ad∗X η. Now, to prove the identities (20) and (21), we need only verify them for the generators g∗ = 31 (g∗ ), but it follows from the definition of the Clifford algebra that {X ∗ , Y ∗ } = 2 hX, Y i = 2 ι X Y ∗ , and by applying (20) and the identity {d, ι X } = ad∗X , we obtain [d X ∗ , Y ∗ ] = −2 ιY d X ∗ = −2 ad∗Y X ∗ = 2 ad∗X Y ∗ . To prove (22) we first verify that it holds when acting on a generator X ∗ ∈ g∗ : {, X ∗ }(Y, Z ) = (2 ι X )(Y, Z ) = −hX, [Y, Z ]i = (2 d X ∗ )(Y, Z ).


139

Finally, we show that d 0 = d − ι∗ is a super-derivation for Clifford multiplication: d 0 (X ∗ · η) = d(X ∗ ∧ η) − ι∗ (X ∗ ∧ η) + d ι X η − ι∗ ι X η = (d X ∗ ) ∧ η − X ∗ ∧ dη − ι(d X ∗ )∗ η + X ∗ ∧ ι∗ η − ι X dη + ad∗X η + ι X ι∗ η = (d 0 X ∗ ) · η − X ∗ · d 0 η, where we use the expansion (d 0 X ∗ )·η = (d X ∗ )·η = (d X ∗ )∧η+ad∗X η−ι(d X ∗ )∗ η. Although the Clifford algebra Cl(g) does not admit an integer grading, it does have the distinguished subspaces g and spin(g), which correspond via the Chevalley identification to the first two degrees of the exterior algebra: 31 (g∗ ) ←→ g ⊂ Cl(g),

32 (g∗ ) ←→ spin(g) ⊂ Cl(g).

Since Spin(g) is the double cover of SO(g), there is a Lie algebra isomorphism spin(g) ∼ = so(g), and given any element a ∈ so(g), the corresponding element a˜ ∈ spin(g) is uniquely determined by the identity [a, ˜ X ∗ ] = (a X )∗ for all X ∈ g. In particular, the adjoint action ad : g → so(g) lifts to a Lie algebra homomorphism aed : g → spin(g) satisfying [e ad X, Y ∗ ] = (ad X Y )∗ = ad∗X Y ∗ . However, from identity (21) we see that the spin lift of the adjoint action must be aed X = (1/2) d X ∗ . Let {X i } be a basis of g, and let {X i∗ } denote the corresponding dual basis of g satisfying hX i∗ , X j i = δi j . In terms of this basis, the map aed : X 7→ (1/2) d X ∗ is 1 X ∗ aed X = − X i · [X, X i ], (23) 4 i

while the element γ = (1/4) corresponding to the fundamental 3-form is given by 1 X ∗ 1 X ∗ γ =− X i · X ∗j · [X i , X j ] = X i · aed X i . (24) 24 6 i, j

i

Rewriting Theorem 5 in terms of this new notation, we obtain the following. COROLLARY 6 The elements 1, X , aed X , γ for X ∈ g span a Lie superalgebra R+ ⊕ g− ⊕ g+ ⊕ R− in the Clifford algebra Cl(g) with the commutation relations

[e ad X, Y ] = [X, Y ],

[e ad X, aed Y ] = aed[X, Y ],

[e ad X, γ ] = 0, g

1 trg 1ad , {X, Y } = 2 hX, Y i, {γ , X } = aed X, {γ , γ } = − 24 P g where 1ad = −(1/2) i ad X i∗ ad X i is the quadratic Casimir operator.

140


Proof All of the commutation relations follow immediately from Theorem 5 and the above discussion with the exception of that for {γ , γ }. For example, we derive ad∗X dY ∗ = 12 d ad∗X Y ∗ = aed[X, Y ], 1 [e ad X, γ ] = − 4 , 12 d X ∗ = − 14 dd X ∗ = 0.

[e ad X, aed Y ] =

1 2

To compute {γ , γ }, we note that the fundamental 3-form is closed, so we have {γ , γ } =

1 4

, 14 =

1 8

d −

1 ∗ 8 ι

= − 18 h, i.

Written in terms of an orthonormal basis {X i } for g, the fundamental 3-form is X

=− X i , [X j , X k ] X i∗ ∧ X ∗j ∧ X k∗ , i< j0 (eiα/2 + e−iα/2 ), and so the highest weight ρg = (1/2) α>0 α of Sg/t appears with multiplicity 1. The highest weight of the tensor product Vλ ⊗ Sg/t is then λ + ρg , appearing with multiplicity 1, and likewise the weights w(λ + ρg ) for w ∈ Wg all have multiplicity 1 in Vλ ⊗ Sg/t . Choosing a common Cartan subalgebra t ⊂ h ⊂ g, the spin representation factors as Sg/t ∼ = Sp ⊗ Sh/t . As we noted above, the weight ρh appears with multiplicity 1 in the second factor Sh/t . It follows that the weights w•λ for w ∈ Wg can appear at most once in the tensor product Vλ ⊗ Sp , as each such weight contributes one weight of the form (w • λ) + ρh = w(λ + ρg ) to the tensor product Vλ ⊗ Sp ⊗ Sh/t ∼ = Vλ ⊗ Sg/t . On the other hand, we see from the homogeneous Weyl formula (12) that the irreducible representations Uc•λ for c ∈ C appear at least once in the decomposition of the tensor product Vλ ⊗ Sg/h . We therefore conclude that the representations Uc•λ for c ∈ C each occur exactly once in Vλ ⊗ Sp . LEMMA 10 If µ is a weight of Vλ ⊗ Sp satisfying kµ + ρh k2 = kλ + ρg k2 , then there exists a unique Weyl element w ∈ Wg such that µ + ρh = w(λ + ρg ).

Proof If µ is a weight of Vλ ⊗ Sp , then µ + ρh is a weight of the product representation Vλ ⊗ Sp ⊗ Sh/t ∼ = Vλ ⊗ Sg/t . Since the Weyl group acts simply transitively on the Weyl chambers, there exists an element w ∈ Wg such that w−1 (µ + ρh ) is dominant, where we recall that a weight ν is dominant if and only if hν, αi ≥ 0 for all positive roots α. Note that every weight of Sg/t can be obtained from its highest weight ρg by subtracting a sum of positive roots. Likewise, for the tensor product Vλ ⊗ Sg/t , the difference (λ + ρg ) − w−1 (µ + ρh ) is a sum of positive roots, and it follows that kλ + ρg k2 ≥ kw−1 (µ + ρh )k2 , with equality holding only when (λ + ρg ) − w−1 (µ + ρh ) = 0. As for the uniqueness of w, if λ is dominant, then the weight λ + ρg lies in the interior of the positive Weyl chamber for g, and thus the weights w(λ + ρg ) for w ∈ Wg are distinct.

146


Theorem 8 now follows immediately from the above two lemmas. We can actually be slightly more specific about the kernel of the Dirac operator, recovering the signs appearing in the homogeneous Weyl formula (12). Recall that the spin representation − decomposes as Sp = S+ p ⊕ Sp into two half-spin representations. Since the Dirac operator is an odd element of the nonabelian Weil algebra U (g) ⊗ Cl(g), it interchanges − S+ p and Sp . Restricting the domain of the Dirac operator to the positive half-spin representation, we obtain an operator + − 6 ∂+ g/h : Vλ ⊗ Sp → Vλ ⊗ Sp .

Furthermore, since the Dirac operator is formally self-adjoint, its adjoint is − + 6 ∂− g/h : Vλ ⊗ Sp → Vλ ⊗ Sp ,

the restriction of the Dirac operator to the negative half-spin representation. Since these Dirac operators are acting on finite-dimensional vector spaces, the index is the difference of the domain and the range, so we have − − + Ker 6 ∂ + g/h − Ker 6 ∂ g/h = Vλ ⊗ Sp − Vλ ⊗ Sp ,

(30)

which is given by the homogeneous Weyl formula (12). Comparing this with the − kernel of 6 ∂ g/h = 6 ∂ + g/h ⊕ 6 ∂ g/h given in Theorem 8, we therefore obtain Ker 6 ∂ + g/h =

M (−1)c =+1

Uc•λ ,

Ker 6 ∂ − g/h =

M

Uc•λ .

(−1)c =−1

In other words, there is no cancellation on the left-hand side of equation (30), and the signed kernel of this Dirac operator picks out precisely those representations, with sign, appearing on the right-hand side of the homogeneous Weyl formula (12). 7. The Clifford algebra Cl(Lg) In Section 4 we examined the Clifford algebra associated to a finite-dimensional Lie algebra with an invariant inner product. The infinite-dimensional case is more complicated, and the general theory of such infinite-dimensional Clifford algebras and their spin representations is developed in the mathematical literature by Kostant and Sternberg in [8]. Here we consider the Clifford algebra associated to the Lie algebra Lg of smooth maps from S 1 to a finite-dimensional Lie algebra g, where we restrict to the dense subspace of loops with finite Fourier expansions. This finiteness condition ensures that the Lie algebra Lg has a countable basis, and the complexification of this L loop space is then LgC = k∈Z gC z k = gC [z, z −1 ], the Lie algebra of finite Laurent series with values in gC . Averaging the pointwise inner products over the loop, we obtain an invariant inner product on Lg given by (3).


147

The Clifford algebra Cl(Lg) is spanned by finite sums of products of the form ξ1 · · · ξn for loops ξi ∈ Lg, subject to the relation {ξ, η} = 2hξ, ηi. However, the loop space analogues of the elements aed X and γ introduced in §4 are in fact infinite sums, so we must instead work with a formal completion of the Clifford algebra. Unfortunately, the product of two such infinite sums does not necessarily converge. On the other hand, given a spin representation S Lg of the Clifford algebra Cl(Lg), we can view End(S Lg ) as a completion of Cl(Lg) with a well-defined product given by the composition of endomorphisms. As we discussed in §2, to define the spin representation we must first choose a polarization. With respect to the action of the infinitesimal generator ∂θ of rotations, the complexified loop space LgC decomposes + into its negative, zero, and positive energy subspaces LgC = Lg− C ⊕ gC ⊕ LgC , + − where LgC and LgC are isotropic subspaces that are dual to each other with respect to the inner product. The spin representation corresponding to this polarization is S Lg := Sg ⊗ 3∗ (Lg+ C ), and the Clifford action c : Cl(LgC ) → End(S Lg ) is given by  1 ⊗ ε(ξ ) for ξ ∈ Lg+  C,  c(ξ ) =

1 ⊗ ι(ξ )   c(ξ ) ⊗ (−1) F

for ξ ∈ Lg− C, for ξ ∈ gC ,

where ε and ι are exterior multiplication and interior contraction, respectively, and F is the degree operator on the exterior algebra. If {ηi } is a basis for Cl(Lg− C ), then when applied to a specific element of the spin representation S Lg , all but finitely P many of the operators c(ηi ) = ι(ηi ) vanish. Formal infinite sums i c(ωi )c(ηi ) with coefficients ωi ∈ Cl(gC ⊕ Lg+ C ) therefore yield well-defined operators on the spin representation, and in fact all elements of End(S Lg ) can be expressed in this form. The exterior algebra that we consider here is not 3∗ (Lg∗ ) but rather the algebra 3∗ (Lg)∗ of skew-symmetric multilinear forms on Lg. Such forms can be expressed as formal infinite sums of basic products of the form ξ1∗ ∧ · · · ∧ ξn∗ for ξi ∈ Lg. In infinite dimensions, the Chevalley map ch : Cl(Lg) → 3∗ (Lg)∗ is no longer surjective; its image consists of all forms given by finite sums of the basic wedge products. Although the Chevalley map fails to converge if we attempt to extend it to the completion End(S Lg ) of Cl(Lg), we can perturb it by terms of lower degree to remove the infinite contributions. Separating the Clifford algebra into its positive and negative energy factors, we define the normal ordering map n : Cl(LgC ) → 3∗ (LgC )∗ by n(ω+ · ω− ) = ch(ω+ ) ∧ ch(ω− ), − − where ω+ ∈ Cl(gC ⊕ Lg+ C ) and ω ∈ Cl(LgC ). The normal ordering map extends to the completion End(S Lg ) of the Clifford algebra, and its image is the subspace

148


3∗ (LgC )+ ⊂ 3∗ (LgC )∗ given by − ∗ 3∗ (LgC )+ = ω ∈ 3∗ (LgC )∗ (ιη ω)+ ∈ 3∗ (Lg− C ) for all η ∈ 3 (LgC ) , ∗ where (·)+ denotes the projection of 3∗ (LgC )∗ onto 3∗ (Lg+ C ) , and we identify + ∗ ∗ 3∗ (Lg− C ) with a subspace of 3 (LgC ) via the inner product. In terms of a ba∗ − sis {ηi } for 3 (Lg ), we may write elements of 3∗ (LgC )+ as formal infinite sums P ∗ + ∗ ∗ i ωi ∧ ηi , with ωi ∈ 3 (gC ⊕ LgC ) in the zero and positive energy components.

L Remark. Decomposing LgC = k∈Z gC z k in terms of its energy grading, we define a secondary degree on LgC counting only the negative energy contribution ( 0 for k ≥ 0, k sdeg X z = k for k < 0, where X ∈ gC and X z k is the loop z 7→ X z k for |z| = 1. Let Lg∗C = g∗C [z, z −1 ] denote the reduced dual of LgC . Extending sdeg to the exterior algebra 3∗ (Lg∗C ), we note that 3∗ (LgC )+ is the completion of 3∗ (Lg∗C ) with respect to sdeg. In other P words, 3∗ (LgC )+ consists of all formal infinite sums i ωi of sdeg-homogeneous elements ωi ∈ 3∗ (Lg∗C ) with sdeg ωi → ∞. We can now use the normal ordering identification n : End(S Lg ) → 3∗ (LgC )+ to define product and bracket structures on 3∗ (LgC )+ . The normal ordered product ω1 ·n ω2 = n(n −1 ω1 · n −1 ω2 ) on the exterior algebra differs from the product induced by the Chevalley identification by terms of lower degree. However, many of the supercommutators remain unchanged. In particular, the normal ordered bracket with the dual ξ ∗ ∈ Lg∗C ∼ = 31 (LgC )+ of a loop ξ ∈ LgC is still given by [ξ ∗ , ω+ ∧ ω− ]n = n n −1 (ξ ∗ ), n −1 (ω+ ∧ ω− ) = n[ξ, ch−1 ω+ · ch−1 ω− ] = n [ξ, ch−1 ω+ ] · ch−1 ω− ± ch−1 ω+ · [ξ, ch−1 ω− ] = 2ιξ ω+ ∧ ω− ± ω+ ∧ 2ιξ ω− = 2 ιξ (ω+ ∧ ω− ), − ∗ ∗ − ∗ for ω+ ∈ 3∗ (gC ⊕ Lg+ C ) and ω ∈ 3 (LgC ) of homogeneous degree. Thus

[ξ ∗ , ω]n = 2 ιξ ω for ξ ∈ LgC and ω ∈ 3∗ (LgC )+ .

(31)

Reprising the discussion of §4, for any ξ ∈ Lg, consider the 2-form dξ ∗ given by dξ ∗ (η, ζ ) = −(1/2)hξ, [η, ζ ]i for all η, ζ ∈ Lg. Although ∂θ is not an element of Lg, we can nevertheless define an analogous 2-form d∂θ∗ by d∂θ∗ (ξ, η) := (1/2)hξ, ∂θ ηi. Note that d∂θ∗ is closed but not exact, so it defines a cohomology class in H 2 (Lg). Finally, the fundamental 3-form is given by (ξ, η, ζ ) = −(1/6)hξ, [η, ζ ]i for


149

ξ, η, ζ ∈ Lg. These elements all lie in 3∗ (LgC )+ , and they satisfy the identities ιξ dη∗ = [ξ, η]∗ ,

ιξ d∂θ∗ = −(∂θ ξ )∗ ,

ιξ = dξ ∗ .

(32)

Using the normal ordered product and bracket on 3∗ (LgC )+ coming from End(S Lg ), we obtain the loop space version of Corollary 6. 11 ˜ Lg, If g is simple, then the elements 1, ξ ∗ for ξ ∈ Lg, aed ξ = (1/2) dξ ∗ for ξ ∈ R ⊕ ∗ + and γ = (1/4) span a Lie superalgebra in 3 (Lg) ⊂ End(S Lg ) satisfying THEOREM

{ξ ∗ , η∗ } = 2hξ, ηi, [e ad ξ, η∗ ] = [ξ, η]∗ , [e ad ∂θ , ξ ∗ ] = (∂θ ξ )∗ , {γ , ξ } = aed ξ, ∗

[γ , aed ∂θ ] = 0,

[e ad ξ, aed η] = aed [ξ, η] + icg hξ, ∂θ ηi, [e ad ∂θ , aed ξ ] = aed(∂θ ξ ), [γ , aed ξ ] =

1 2

icg (∂θ ξ )∗ ,

{γ , γ } = icg aed ∂θ −

1 24 cg dim g,

where cg is the value of the Casimir operator of g in the adjoint representation. Proof The bracket {ξ ∗ , η∗ } = 2hξ, ηi is simply the definition of the Clifford algebra, while the brackets [e ad ξ, η∗ ] = [ξ, η]∗ , [e ad ∂θ , ξ ∗ ] = (∂θ ξ )∗ , and {γ , ξ ∗ } = aed ξ follow ˜ Lg and immediately from (31) and (32). By the Jacobi identity, for any ξ, η ∈ R ⊕ ζ ∈ Lg we have [e ad ξ, aed η], ζ ∗ = aed ξ, [e ad η, ζ ∗ ] − aed η, [e ad ξ, ζ ∗ ] ∗ ∗ ∗ = ξ, [η, ζ ] − η, [ξ, ζ ] = [ξ, η], ζ = aed [ξ, η], ζ ∗ , ˜ Lg on S Lg . In Theorem 1 which shows that aed is a projective representation of R ⊕ we established that this spin representation has central charge cg , which gives us the brackets [e ad ξ, aed η] = aed [ξ, η] + icg hξ, ∂θ ηi and [e ad ∂θ , aed ξ ] = aed ∂θ ξ . To compute γ 2 , we write it as the sum γ 2 = (γ 2 )0 + (γ 2 )2 of homogeneous forms of degrees zero and 2. (We shall see that γ 2 has no components of degrees 4 or 6.) Since ιξ = ad ξ ∗ for ξ ∈ Lg is a derivation with respect to the bracket, we have ιξ γ 2 = [ιξ γ , γ ] =

1 2

[e ad ξ, γ ].

Taking one further interior contraction, we obtain ιξ ιη γ 2 = − 41 [e ad ξ, aed η] − aed [ξ, η] = − 14 icg hξ, ∂θ ηi,

150


which is a constant. It follows that γ 2 has no components of degree higher than 2 and that (γ 2 )2 is the 2-cocycle determining the central extension of Lg for the spin representation aed. In fact, this 2-cocycle is a multiple of aed ∂θ , and we have (γ 2 )2 (ξ, η) = − 21 ιξ ιη γ 2 = 18 icg hξ, ∂θ ηi = 12 icg (e ad ∂θ )(ξ, η). Going back up one level, we see that [e ad ξ, γ ] = 2 ιξ γ 2 = icg ιξ aed ∂θ = − 21 icg (∂θ ξ )∗ . Finally, the value of the constant (γ 2 )0 is the value of γ 2 acting on the minimum energy subspace S Lg (0) of the spin representation since aed ∂θ vanishes there. However, all the terms in γ 2 vanish on S Lg (0) except the contribution from the constant loops, g and thus (γ 2 )0 = (1/48) trg 1ad = (1/48) cg dim g, as we proved in Corollary 6. Taking a slightly different view of this theorem, the commutation relations given in Theorem 11 determine a Lie superalgebra (with subscripts denoting the grading) ˜ Lg)even ⊕ Rodd , Reven ⊕ Lgodd ⊕ (R ⊕ and the identification of 3∗ (Lg)+ with its image in End(S Lg ) gives a representation of this Lie superalgebra on the spin representation S Lg . Actually, we can extend this Lie superalgebra further. The component Reven ⊕ Lgodd ⊕ Lgeven is called a super Kac-Moody algebra, and using superspace notation, its complexification is a central extension of the polynomial algebra g ⊗ C[z, z −1 , 2], where 2 is an odd variable (i.e., 22 = 0). The super Virasoro algebra SVir is the universal central extension of the Lie algebra of derivations of C[z, z −1 , 2]. (Note that the even derivations are just the vector fields on the circle.) The super Virasoro algebra therefore acts on the super Kac-Moody algebra, and their semidirect sum is referred to as the N = 1 superconformal current algebra (see [3]): ˜ (Reven ⊕ Lgodd ⊕ Lgeven ). SVir ⊕ In our case, the elements aed ∂θ and γ span the even and odd zero-mode subspaces of the super Virasoro algebra, with commutator {γ , γ } = icg aed ∂θ + (1/24) i dim g . Here, the additional (1/24) dim g term, which is sometimes incorporated into the definition of aed ∂θ , corresponds to the anomalous energy shift we encountered in (9). Given an orthonormal basis {X i } for g, the loops X in = X i z n for n ∈ Z form a basis for LgC satisfying hX in , X mj i = δi, j δn,−m . In terms of this basis, we have aed ξ = −

1 X −k X i · [ξ, X ik ], 4 i,k

aed ∂θ =

1 X ik X kj · X −k j , 2 j, k>0

1 X −k 1 X −k k+l γ =− X i · X −l · [X , X ] = X i · aed X ik . i j j 24 6 i, j,k,l

i,k


151

Note that in the expressions for aed ξ and γ , the ordering of the factors does not matter (up to sign) since they are orthogonal and therefore anticommute with each other. However, in the expression for aed ∂θ , we have {X ik , X i−k } = 2, so changing the order of the factors shifts the operator by a constant. Here we see normal ordering in action, forcing us to write factors X ik with k positive on the left and factors X i−k with −k negative on the right. In physics notation this would be written as P k −1 aed ∂θ = −(1/4) j, k∈Z ik : X −k j X j : , where : ξ η : = n (n ξ ∧ n η) denotes the normal ordered product in the Clifford algebra. (This colon notation is misleading as it is not a map on the Clifford algebra but rather an instruction to replace all Clifford products between the colons with normal ordered products.) If the Lie algebra g is not simple, then Theorem 11 still holds, albeit with slightly modified commutation relations. For a general finite-dimensional Lie algebra g, the P g Casimir operator 1ad = −(1/2) i (ad X i )2 no longer takes a constant value cg . In this case, the role of the quadratic element aed ∂θ is played by the 2-cocycle ωaed for the projective spin representation aed, given on ξ, η ∈ Lg by g

ωaed (ξ, η) := [e ad ξ, aed η] − aed [ξ, η] = i hξ, 1ad ∂θ ηi, g

where the Casimir operator 1ad acts pointwise on the loop space Lg. Viewing ωaed as an element of the Clifford algebra, we have the commutator ∗ g [ωaed , ξ ∗ ] = −2 ιξ ωaed = 4i 1ad ∂θ ξ , so we may also view the projective cocycle as ωaed = 4i aed(1g ∂θ ), where 1g is the formal Casimir operator in the universal enveloping algebra of g. We therefore have g [ωaed , aed ξ ] = 4i [e ad(1g ∂θ ), aed ξ ] = 4i aed 1ad ∂θ ξ , and the adjoint action of γ in Theorem 11 then becomes ∗ g [γ , aed ξ ] = 12 i 1ad ∂θ ξ , {γ , γ } =

1 4

ωaed −

1 24

g

trg 1ad ,

with the other commutation relations remaining unchanged. Alternatively, the projective cocycle ωaed can be viewed as the 2-form component of the Casimir operator g

Lg

1aed = −2i aed(1g ∂θ ) + 1aed = − 12 ωaed +

1 8

g

trg 1ad

for the spin representation aed of Lg, which we discuss in Theorem 12 below. 8. The Dirac operator on Lg Following our discussion in Section 5, given an arbitrary positive energy representa˜ → End(H ), we construct a Dirac operator tion r : Lg 6 ∂ r := rˆ + 1 ⊗

1 2

Lg ∈ End(H ⊗ S Lg ),

152


where rˆ is the tautological End(H )-valued 1-form on Lg given by rˆ (ξ ) = r (ξ ) for all ξ ∈ Lg, and Lg is the fundamental 3-form given on ξ, η, ζ ∈ Lg by Lg (ξ, η, ζ ) = −(1/6)hξ, [η, ζ ]i for ξ, η, ζ ∈ Lg. As in the previous section, we implicitly identify 3∗ (Lg)+ with its image in End(S Lg ). Written in terms of a basis {X in } of Lg satisfying hX in , X mj i = δi, j δn,−m , this Dirac operator is 1 X −n X i · X −m · [X i , X j ]n+m j 12 i, j,m,n i,n X 1 = X i−n r (X in ) + aed X in . 3

6 ∂r =

X

X i−n r (X in ) −

i,n

Note that all of the individual factors in this expression (anti) commute with each other, so 6 ∂ r does indeed give a well-defined operator on the tensor product H ⊗ S Lg without requiring normal ordering or dealing with any infinite constants. In its most general form, if we take the representation r to be the canonical inclusion r : Lg ,→ U (Lg) of Lg into its universal enveloping algebra U (Lg), then the corresponding universal Dirac operator is an element of the formal completion of the ˜ ⊗ Cl(Lg). (Alternatively, we may view A as nonabelian Weil algebra A = U ( Lg) ˜ even ⊕ Lgodd .) the universal enveloping algebra of the super Kac-Moody algebra Lg As we saw in the previous section, the product of two such infinite formal sums does not necessarily converge. However, keeping in mind that we are really working with operators on Hilbert spaces, we can indeed extend multiplication to a suitable subspace A + of the formal completion, which we define as the largest subspace for which the homomorphism A → End(H ⊗ S Lg ) extends to A + for any positive ˜ In particular, if H is a faithful representation of energy representation H of Lg. ˜ U ( Lg)—we can construct such a representation by taking the Hilbert space direct ˜ sum of countably many irreducible positive energy representations of Lg—then the homomorphism A + ,→ End(H ⊗ S Lg ) induces a product structure on A + . Fortunately we can perform all of our computations here using the techniques of the ˜ ˜ previous section, working with U ( Lg)-valued forms on Lg. Using this extended multiplication, the square of the Dirac operator is 6 ∂ 2 = rˆ 2 + {ˆr , 12 Lg } +

1 4

2Lg .

Since rˆ is an End(H )-valued 1-form on Lg, its square is a sum rˆ 2 = (ˆr 2 )0 + (ˆr 2 )2 of forms of homogeneous degrees zero and 2. For the degree 2 component, we have (ˆr 2 )2 = rˆ ∧ rˆ , and the “curvature” d rˆ + rˆ ∧ rˆ of the representation r is given by (d rˆ + rˆ ∧ rˆ )(ξ, η) = 21 [r (ξ ), r (η)] − r ([ξ, η]) = 12 ωr (ξ, η), where ωr ∈ 32 (Lg)+ is the 2-cocycle corresponding to the projective representation r . If g is simple and I is the generator of the universal central extension of Lg, then ωr = 4 r (I ) aed ∂θ . The degree zero component of rˆ 2 is given by the following.


153

THEOREM 12 Lg The operator 1r := −(1/2) rˆ 2 0 is called the Casimir operator for the loop group Lg, and if g is simple, then the Casimir operator acting on the irreducible positive energy representation Hλ with lowest weight λ = (m, −λ, h) is given by g

1rLg = −i (h + cg ) (r (∂θ ) − im) + 1λ = −i (h + cg ) r (∂θ ) +

1 2

kλ − ρ g k2 − kρ g k2 ,

(33)

g

where ρ g = (0, ρg , −cg ) and cg = 1ad is the value of the quadratic Casimir operator of g acting on the adjoint representation, and the inner product is given by (6). Proof In order to simply our calculations, we first note the following identities: [ˆr , aed ξ ](η) = [r (η), aed ξ ] − {ˆr , 12 [η, ξ ]∗ } = r ([ξ, η]), [ˆr , r (ξ )](η) = [r (η), r (ξ )] = r ([η, ξ ]) + hη, ∂θ ξ i r (I ) = [e ad ξ, rˆ ] + r (I )(∂θ ξ )∗ (η), and [e ad ξ, rˆ 2 ]0 = −[e ad ξ, (ˆr 2 )2 ]0 = −[e ad ξ, d rˆ ]0 = 2 (e ad I ) rˆ (∂θ ξ ) = 2 (e ad I ) r (∂θ ξ ). Lg

We now compute that the commutator of 1r

with an element ξ ∈ Lg is [1rLg , r (ξ )] = − 21 [ˆr 2 , r (ξ )]0 = − 12 rˆ , [ˆr , r (ξ )] 0 = 12 rˆ , [ˆr , aed ξ ] 0 − 21 rˆ , r (I )(∂θ ξ )∗ 0 = −(e ad I ) r (∂θ ξ ) − r (I ) r (∂θ ξ ) = − (r (I ) + aed I ) r (∂θ ), r (ξ ) .

˜ rLg := 1rLg + (r (I ) + aed I ) r (∂θ ) commutes with the It follows that the operator 1 action of Lg and therefore takes a constant value on each irreducible representation. Acting on the minimum energy subspace Hλ (m) of Hλ , the only terms contributing Lg to 1r are those coming from the constant loops, and thus this constant is g ˜ Lg = 1rLg 1 λ H (m) + i (h + cg ) r (∂θ ) H (m) = 1λ − (h + cg ) m. λ

λ

The desired result then follows immediately. By definition, the zero-form component of rˆ 2 acts as the identity operator on S Lg . Lg To compute the action of 1r , we can therefore restrict it to the minimum energy

154


subspace S Lg (0) of the spin representation. In terms of a basis {X in }, we have X 1 X r (X in ) X i−n r (X mj ) X −m j H ⊗S (0) 2 Lg i,n j,m X 1 X −n n =− r (X i ) r (X j ) X i · X j + r (X in ) r (X −n ) X · X j j i 2

1rLg = −

i, j

= 1rg −

n>0

X

r (X in ) r (X i−n ),

i, n>0

which is the usual definition of the Casimir operator for a loop group. The Casimir operator can be used to define the energy operator r (∂θ ) in terms of the action of Lg. g The constant term 1λ is sometimes incorporated into r (∂θ ), in which case it is viewed as an anomalous energy shift due to the degeneracy of the vacuum. Returning to our computation of the square of the Dirac operator, we note that the cross term is given by the anticommutator {ˆr , (1/2) Lg } = d rˆ , and we obtain 6 ∂ 2 = −2 1rLg +

1 2

ω% −

1 12

g

trg 1ad ,

where ω% is the 2-cocycle corresponding to the diagonal action % = r ⊗ 1 + 1 ⊗ aed on the tensor product H ⊗ S Lg . If g is simple, then this 2-cocycle is ω% = 4 %(I ) aed ∂θ . Furthermore, if Hλ is the irreducible positive energy representation of Lg with lowest weight λ = (m, −λ, h), then, using (33) for the Casimir operator, we have 6 ∂ 2λ = 2i (h + cg ) %(∂θ ) − im − kλ + ρg k2 (34) = 2 %(I ) %(∂θ ) − kλ − ρ g k2 . Note that unlike the finite-dimensional case discussed in §5, the square of the Dirac operator for Lg does not take a constant value on each irreducible representation. Here the Dirac operator fails to commute with the diagonal action % of Lg on H ⊗ S Lg . Since the Dirac operator satisfies the identity ιξ 6 ∂ = %(ξ ), we have [%(ξ ), 6 ∂] = ιξ 6 ∂ 2 =

1 2 ιξ

ω% = 2 %(I ) ιξ aed ∂θ = −%(I )(∂θ ξ )∗ ,

˜ ˜ Lg. and thus 6 ∂ commutes only with the subalgebra R ⊕ g ⊕ R of R ⊕ If the Lie algebra g is reductive but not simple, then expression (34) for the square of the Dirac operator still holds, provided that the central extension satisfies ω% = 4 %(I ) aed ∂θ . In other words, the invariant inner product on g must satisfy [%(ξ ), %(η)] − %([ξ, η]) = %(I )hξ, ∂θ ηi for some pure imaginary constant %(I ). Given any irreducible positive energy projective representation H of Lg, we can always choose an invariant inner product on g


155

˜ such that H ⊗ S Lg is a true representation of the corresponding central extension Lg. However, this choice of inner product depends on the representation, so this approach does not give a universal expression for the Dirac operator. 9. The Dirac operator on Lg/Lh As in §6, let h be a Lie subalgebra of g, and let p denote the orthogonal complement of h with respect to the invariant inner product on g. This orthogonal decomposition extends to the loop Lie algebra Lg = Lh ⊕ Lp, and the Clifford algebra decomposes as Cl(Lg) ∼ = Cl(Lh) ⊗ Cl(Lp). If p is even-dimensional, as is the case when h has the same rank as g, then we can also factor the spin representation as S Lg ∼ = S Lh ⊗ S Lp , ˜ of levels ch and cg − ch , respectively, where S Lh and S Lp are representations of Lh and the action of Lh on S Lp is aed Lp : Lh → 32 (Lp)+ ,→ End(S Lp ) ζ 7 → (e ad Lp ζ )(ξ, η) =

1 4

hξ, [ζ, η]i

for ζ ∈ Lh and ξ, η ∈ Lp. ˜ on a Hilbert space H , its Given any positive energy representation r Lg of Lg ˜ on H . Now, consider the diagonal reprerestriction gives a representation r Lh of Lh 0 =r ˜ on the tensor product H ⊗ S Lp . Using sentation r Lh Lh ⊗ 1 + 1 ⊗ aed Lp of Lh the construction of the previous section, we build the twisted Dirac operator 0 + 6 ∂ 0Lh = rˆLh

1 2

Lh ∈ End(H ⊗ S Lp ⊗ S Lh ) ∼ = End(H ⊗ S Lg ).

Noting that the diagonal action %0Lh = r 0 ⊗ 1 + 1 ⊗ aed Lh on H ⊗ S Lg is simply the restriction of the action % Lg = r ⊗ 1 + 1 ⊗ aed Lg to Lh, we obtain the identities ιζ 6 ∂ 0Lh = %0Lh (ζ ) = % Lg (ζ ), [% Lg (ζ ), 6 ∂ 0Lh ] =

1 2 ιζ

Lh

ω%0 =

1 2 ιζ

ω%Lg ,

for ζ ∈ Lh. The difference 6 ∂ Lg/Lh := 6 ∂ Lg − 6 ∂ 0Lh is basic with respect to Lh; that is, ιζ 6 ∂ Lg/Lh = 0,

[% Lg (ζ ), 6 ∂ Lg/Lh ] = 0,

for all ζ ∈ Lh, and thus it can be written as the Lh-equivariant operator 6 ∂ Lg/Lh = rˆLp +

1 2

Lp ∈ End(H ⊗ S Lp ) Lh ,

(35)

where rˆLp is the End(H )-valued 1-form on Lp given by rˆ (ξ ) = r (ξ ) for ξ ∈ Lg and Lp is the fundamental 3-form given by Lp (ξ, η, ζ ) = −(1/6)hξ, [η, ζ ]i for ξ, η, ζ ∈ Lp. Writing this Dirac operator in terms of a basis {X in } of Lp satisfying hX in , X mj i = δi, j δn,−m , we have 6 ∂ Lg/Lh =

X i,n

X i−n r (X in ) −

1 X −n X i · X −m · [X i , X j ]n+m , p j 12 i, j,m,n

156


where [X, Y ]p denotes the projection of [X, Y ] onto p. As we saw in the finite-dimensional case, the two Dirac operators 6 ∂ 0Lh and 6 ∂ Lg/Lh are decoupled, or, in other words, they anticommute with each other: 0 0 6 ∂ Lh , 6 ∂ Lg/Lh = rˆLh , 6 ∂ Lg/Lh + 12 Lh , 6 ∂ Lg/Lh = 0, where the first summand vanishes since for all ζ ∈ Lh we have 0 rˆLh , 6 ∂ Lg/Lh (ζ ) = [r 0 (ζ ), 6 ∂ Lg/Lh ] = 0, and the second summand vanishes as the odd operators (1/2) Lh and 6 ∂ Lg/Lh act on distinct representations S Lh and H ⊗ S Lp and therefore anticommute. Since these two operators are decoupled, the square of the Dirac operator on Lg/Lh is 6 ∂ 2Lg/Lh = 6 ∂ Lg

2

− 6 ∂ 0Lh

2

Lh

= −2 1rLg − 1r 0

+

1 2

Lh

ω%Lg − ω%0

+

1 12

g h trg 1ad − trh 1ad .

Now, consider the case where g is simple, h is reductive, and Hλ is the irreducible positive energy representation of Lg with lowest weight λ. Since 6 ∂ Lg/Lh is an Lhequivariant operator on Hλ ⊗ S Lp , it is a constant on each of the irreducible sub˜ representations of Lh. If Uµ is the irreducible positive energy representation of Lh with lowest weight µ, then, using (34), we see that the square of the Dirac operator takes the value (6 ∂ Lg/Lh )2 µ = 2 % Lg (I ) % Lg (∂θ ) − kλ − ρ g k2 − 2 %0Lh (I ) %0Lh (∂θ ) + kµ − ρ h k2 = −kλ − ρ g k2 + kµ − ρ h k2 (36) on the Uµ components of Hλ ⊗ S Lp . Note that the nonconstant terms vanish since ˜ ˜ Lh. % Lg and %0Lh agree on R ⊕ Note that in the above construction we are using the invariant inner product on h obtained by restricting our invariant inner product on g. When g is simple, we use the basic inner product on g, which is normalized so that kαmax k2 = 2, where αmax is the highest root of g. We recall that the basic inner product corresponds to the universal ˜ of Lg, which in turn restricts to give a (not necessarily universal) central extension Lg ˜ of Lh. Nevertheless, given any positive energy representation H central extension Lh ˜ of Lg, the tensor product H ⊗ S Lg is a true representation of this central extension ˜ So, if h is reductive, then the squares of the Dirac operators 6 ∂ 0 and 6 ∂ Lg/Lh are Lh. Lh indeed of the form given by (34) and (36). On the other hand, if g is not simple but rather semisimple, then the basic inner product on g is normalized so that kαi k2 = 2, where the αi are the highest roots of


157

each of the simple components of g. In this case, a projective positive energy representation of Lg is not necessarily a true representation of the corresponding central ˜ so the expression (34) for the square of the Dirac operator on Lg is not extension Lg, universal. However, if h is reductive, then the expression (36) for the square of the Dirac operator on Lg/Lh does still hold, as the nonconstant terms must vanish since the operator commutes with the action of Lh. 10. The kernel of the Dirac operator Given a linear operator d : V → W between two finite-dimensional vector spaces, the alternating sum of the dimensions in the exact sequence d

0 −→ Ker d −→ V −→ W −→ Coker d −→ 0 vanishes, and it follows that Index d = dim V − dim W . Furthermore, if V and W are G-modules and the operator d is G-equivariant, then the analogous result IndexG d = V − W holds in the representation ring R(G). In the infinite-dimensional case, this result does not necessarily hold, but for representations of loop groups, it does hold, provided that the representations are of finite type and that the operator commutes with rotating the loops. LEMMA 13 If V and W are representations of LG of finite type and D : V → W is an (S 1 nLG)equivariant linear operator, then its LG-equivariant index is the virtual representation Index LG D = V − W .

Proof Since D is S 1 -equivariant, it respects the decompositions of V and W into their constant energy subspaces, and it can be written in the block diagonal form D = L k∈Z Dk , with Dk : V (k) → W (k). If both V and W are of finite type, then each of the subspaces V (k) and W (k) is a finite-dimensional G-module, and so the (S 1 × G)equivariant index of D is given by the R(G)-valued formal power series X X X k Index S 1 ×G D = z k V (k) − W (k) = z V (k) − z k W (k). k∈Z

k∈Z

k∈Z

Since a representation of the full loop group LG is uniquely determined by its constant energy components, the LG-equivariant index must therefore be the difference of the domain and the range; hence Index LG D = V − W . Returning to the notation of the previous section, let g be semisimple, and let h be a reductive subalgebra of g with maximal rank. If we decompose the spin representation

158


+ − as S Lp = S Lp ⊕ S Lp , the Dirac operator 6 ∂ Lg/Lh interchanges the two half-spin representations and can thus be written as the sum of the operators: + − 6 ∂+ Lg/Lh : Hλ ⊗ S Lp → Hλ ⊗ S Lp , − + 6 ∂− Lg/Lh : Hλ ⊗ S Lp → Hλ ⊗ S Lp , + where 6 ∂ − Lg/Lh is the adjoint of 6 ∂ Lg/Lh . When we introduced 6 ∂ Lg/Lh in (35), we showed that it is Lh-equivariant, and all of our Dirac operators clearly commute with the generator ∂θ of rotations of the loops. The operator 6 ∂ Lg/Lh is therefore (S 1 nL H )equivariant, and since its domain and range are both of finite type, we may apply Lemma 13. The L H -equivariant index of 6 ∂ + Lg/Lh is thus the difference − + − Ker 6 ∂ + Lg/Lh − Ker 6 ∂ Lg/Lh = Hλ ⊗ S Lp − Hλ ⊗ S Lp ,

which is given by the homogeneous Weyl-Kac formula (13). − On the other hand, to compute the kernel of 6 ∂ Lg/Lh = 6 ∂ + Lg/Lh ⊕ 6 ∂ Lg/Lh we proceed as in the computation of the kernel of the finite-dimensional operator 6 ∂ g/h in §6. In fact, the proofs of Lemmas 9 and 10 apply equally well in the Kac-Moody setting using the decomposition S Lg/t ∼ = S Lp ⊗ S Lh/t , and we obtain the following. 14 ˜ with lowest weight c • λ = For each c ∈ C , the irreducible representation Uc•λ of Lh c(λ − ρ g ) + ρ h occurs exactly once in the decomposition of Hλ ⊗ S Lp . LEMMA

LEMMA 15 If µ is a weight of Hλ ⊗ S Lp satisfying kµ − ρ h k2 = kλ − ρ g k2 , then there exists a unique affine Weyl element w ∈ Wg such that µ − ρ h = w(λ − ρ g ).

Then, in light of our formula (36) for the square of the Dirac operator 6 ∂ Lg/Lh , we immediately obtain the loop group analogue of Theorem 8. 16 Let g be a semisimple Lie algebra with a maximal rank reductive Lie subalgebra h. ˜ and Lh ˜ with lowest weights Let Hλ and Uµ be the irreducible representations of Lg λ and µ. The kernel of the operator 6 ∂ Lg/Lh on Hλ ⊗ S Lp is THEOREM

Ker 6 ∂ Lg/Lh =

M

Uc•λ ,

c∈C

where c • λ = c(λ − ρ g ) + ρ h , and C ⊂ Wg is the subset of affine Weyl elements which map the fundamental Weyl alcove for g into the fundamental alcove for h.


159

Comparing this result to the homogeneous Weyl-Kac formula (13), we obtain M M Ker 6 ∂ + Uc•λ , Ker 6 ∂ − Uc•λ . Lg/Lh = Lg/Lh = (−1)c =+1

(−1)c =−1

Taking the kernels of these Dirac operators therefore gives an explicit construction for ˜ corresponding to any given irreducible the multiplet of signed representations of Lh ˜ positive energy representation of Lg. References [1]

A. ALEKSEEV and E. MEINRENKEN, The non-commutative Weil algebra, Invent. Math.

[2]

B. GROSS, B. KOSTANT, P. RAMOND, and S. STERNBERG, The Weyl character formula,

139 (2000), 135–172. MR CMP 1 728 878 122, 123, 141, 142

[3]

[4]

[5] [6] [7]

[8]

[9] [10] [11] [12] [13]

the half-spin representations, and equal rank subgroups, Proc. Natl. Acad. Sci. USA 95 (1998), 8441–8442. MR 99f:17007 121, 122, 132, 134 V. G. KAC and T. TODOROV, Superconformal current algebras and their unitary representations, Comm. Math. Phys. 102 (1985), 337–347, MR 87i:17021a; Erratum, Comm. Math. Phys. 104 (1986), 175. MR 87i:17021b 123, 150 Y. KAZAMA and H. SUZUKI, Characterization of N = 2 superconformal models generated by the coset space method, Phys. Lett. B 216 (1989), 112–116. MR 90k:81215 123, 135 , New N = 2 superconformal field theories and superstring compactification, Nuclear Phys. B 321 (1989), 232–268. MR 90g:81249 123, 135 B. KOSTANT, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 26:265 144 , A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), 447–501. MR CMP 1 719 734 121, 123 B. KOSTANT and S. STERNBERG, Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Physics 176 (1987), 49–113. MR 88m:58057 123, 129, 146 G. D. LANDWEBER, Harmonic spinors on homogeneous spaces, Represent. Theory 4 (2000), 466-473, http://ams.org/ert/home-2000.html. MR CMP 1 780 719 143 D. A. MCLAUGHLIN, Orientation and string structures on loop space, Pacific J. Math. 155 (1992), 143-156. MR 93j:57015 133 A. PRESSLEY and G. SEGAL, Loop Groups, Oxford Math. Monogr., Oxford Univ. Press, New York, 1986. MR 88i:22049 128 S. SLEBARSKI, Dirac operators on a compact Lie group, Bull. London Math. Soc. 17 (1985), 579–583. MR 87c:58115 122, 141 , The Dirac operator on homogeneous spaces and representations of reductive Lie groups, I, Amer. J. Math. 109 (1987), 283–301, MR 89a:22028; II, Amer. J. Math. 109 (1987), 499–520. MR 88g:22015 143

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Microsoft Research, One Microsoft Way, Redmond, Washington 98052, USA; current: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA; [email protected].


FRACTAL DIMENSIONS AND THE PHENOMENON OF INTERMITTENCY IN QUANTUM DYNAMICS JEAN-MARIE BARBAROUX, FRANÇOIS GERMINET, AND SERGUEI TCHEREMCHANTSEV

Abstract We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state ψ(t) = e−it H ψ and averaged in time between zero and T. These lower bounds are expressed in terms of generalized fractal dimensions Dµ±ψ 1/(1 + p/d) of the measure µψ (where d is the space dimension). This improves previous results obtained in terms of Hausdorff and Packing dimension. 1. Introduction A by now wide number of articles deals with the links between the quantum dynamics of wave packet solutions of the Schrödinger equation and the spectral properties of the associated Hamiltonian H . Actually, during the last decade an analysis originated by I. Guarneri in [15] and [16] and refined by others in [6], [8], [24], [18], and [27] established that the fractal properties of the spectral measures were relevant for the study of the spreading of wave packets. Consider a separable Hilbert space H , an orthonormal basis {en }n∈N , and a self-adjoint operator H on H . Let ψt = e−it H ψ be the solution of the Schrödinger equation ( t i ∂ψ ∂t = H ψt , ψt=0 = ψ. P With X = n nh· en i en being the position operator, we define the time-averaged moments of order p for ψ as Z Z

1 T 1 TX p

p/2 −it H 2 p hh|X | iiψ,T := ψ dt = |n| |hψt , en i|2 dt.

|X | e H T 0 T 0 n∈N (1.1) DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 1, Received 29 December 1999. Revision received 28 September 2000. 2000 Mathematics Subject Classification. Primary 81Q10, 28A80; Secondary 35J10.

161

162

BARBAROUX, GERMINET, AND TCHEREMCHANTSEV

In the specific case H = `2 (Zd ), more relevant from a physical point of view, {en } RT P p 2 is the canonical basis {δn }n∈Zd and hh|X | p iiψ,T = 1/T 0 n∈Zd |n| |ψt (n)| dt. It is now well known from a series of results (see [1], [7], [8], [15], [16], [27]) in the case H = `2 (Zd ) (extendable to H = L2 (Rd )) that if dim H (µψ ) is the Hausdorff dimension of the spectral measure µψ , then α − (ψ, p, d) := lim inf T →∞

loghh|X | p iiψ,T ≥ dim H (µψ ) p/d. log T

(1.2)

More recently, in [18] a lower bound has been proven for the upper oscillations of hh|X | p iiψ,T , namely, α + (ψ, p, d) := lim sup T →∞

loghh|X | p iiψ,T ≥ dim P (µψ ) p/d, log T

(1.3)

where dim P (µψ ) is the packing dimension of µψ . However, those results are certainly not optimal. Some 1-dimensional quantum systems with pure point spectrum can give rise to an almost ballistic motion (see [34]); that is, α + (ψ, 2, 1) = 2. Meanwhile, dim P (µψ ) = dim H (µψ ) = 0 for pure point measures. A similar phenomenon has been argued to hold for the random dimer model (see [13] [12]). In quasiperiodic models almost ballistic motion (α + (ψ, 2, 1) = 2) turns out to be a common phenomenon, actually a generic phenomenon (see [11]), even in presence of purely zero Hausdorff dimensionality of the spectral measures (see [27], [11]). These examples show how far we are from a complete understanding of “What determines the spreading of a wave packet" (see [23]). In this paper we go one step further in this undertanding and also supply (see Appendix B) a new enlightment concerning the main technique used in this field for the past ten years. We obtain new lower bounds for the growth exponents of hh|X | p iiψ,T , namely, (1.4) and (1.5). As in (1.2) and (1.3), these bounds rely only on the fractal properties of the spectral measure µψ . We point out right now that unlike the existing results, the bounds (1.4) and (1.5) we get can be nontrivial in the presence of zero dimensionality of the spectral measure (dim P (µψ ) = dim H (µψ ) = 0), even in the case of pure point spectrum (see Appendix D). Moreover, in Appendix D, Theorem D.1, we show that there is no hope to improve our result by only taking into account the fractal properties of the spectral measure encoded in its generalized fractal dimensions Dµ±ψ (q). Our result also provides a precise statement of a phenomenon discovered by recent numerical computations in some quantum models for quasicrystals (see [28], [29], [32], [38]); this phenomenon has been called intermittency. Namely, it has been suggested by physicists that the growth exponents α ± (ψ, p, d) should grow faster than linearly in p/d as proposed in (1.2) and (1.3); one should observe a more complex law for the behaviour of α ± (ψ, p, d) in the variable p/d: α ± (ψ, p, d) =

QUANTUM DYNAMICS AND MULTIFRACTAL DIMENSIONS

163

β ± (ψ, p/d) p/d, where β ± (ψ, p/d) are nondecreasing functions of p/d (and of course not smaller than dim H and dim P ). These recent numerical investigations emphasized, in this phenomenon of intermittency, the role of more refined fractal quantities, the so-called qth generalized fractal dimensions Dµ±ψ (q). The lower bounds (1.4) and (1.5) that we establish for hh|X | p iiψ,T actually appear to be intermittent lower bounds in the following sense: when Dµ± (q) is nonconstant for q ∈ (0, 1), these lower bounds grow faster than linearly in p/d. After Theorem 2.1 we discuss the application of our result to a Hamiltonian constructed with Julia matrices with self-similar spectra (see [5], [19]). For “real" Schrödinger operators, good candidates would be operators on `2 (N) or `2 (Z) with sparse or quasiperiodic (e.g., generated by substitution sequences) potential. But a careful analysis of the links between spectral properties and behaviour of the eigenfunctions would then be required, an analysis that goes, for instance, beyond the scope of [22]. Our main result (Theorem 2.1) holds for any self-adjoint operators H and for any initial state ψ such that the associated spectral measure dµψ satisfies (H)

Dµ±ψ (s) < +∞ for any s ∈ (0, 1).

The result reads then as follows: α (ψ, p, d) ≥ −

Dµ−ψ

and α + (ψ, p, d) ≥ Dµ+ψ

1 1 + p/d

1 1 + p/d

p/d

(1.4)

p/d,

(1.5)

where Dµ−ψ (q) and Dµ+ψ (q) are the lower and upper qth generalized fractal dimensions (see Definition 2.2). In Appendix C we discuss the validity of Hypothesis (H), which, we note, holds for any compactly supported measures. In particular, Theorem 2.1 applies to the examples where the intermittency phenomenon has been argued to hold. To achieve this we first derive a ( p/d)-dependent lower bound L ψ (T ) for hh|X | p iiψ,T (see Theorem 3.1). Then we establish via Theorem 4.2 the connection between L ψ (T ) and the generalized fractal dimensions, by adjusting for each single T the “thin" part of µψ that supplies the faster dynamical travel. Concerning the connection between L ψ (T ) and Dµ±ψ (q), we moreover obtain a kind of optimality, in the sense that this quantity L ψ (T ) that minors hh|X | p iiψ,T is shown to have its growth exponents exactly equal to Dµ±ψ 1/(1 + p/d) p/d. We point out that Dµ±ψ 1/(1 + p/d) are increasing functions of p/d and are, respectively, not smaller than dim H (µψ ) and dim P (µψ ) (for all p/d > 0). Therefore (1.4) and (1.5) do improve the bounds (1.2) and (1.3) above.

164


These new bounds are a consequence of a double improvement of the approaches of Guarneri [15], [16], J.-M. Combes [8], and Y. Last [27] and of J.-M. Barbaroux and S. Tcheremchantsev [6]. The first improvement is due, after a decomposition ψ = φ + χ, φ ⊥ χ , to a better control of the key quantity Z 1 ∞ X |he−it H ψ, en i|2 h(t/T ) dt Bψ (T, N ) := T 0 |n|≤N Z ∞ X 2 ≤ Bχ (T, N ) + Re he−it H φ, en ihe−it H ψ, en ih(t/T ) dt, T 0 |n|≤N

(1.6) and, more particularly, of the last term (the crossed term). We stress right now that this better control of the crossed term is essential, meaning that using the former available estimates (see [27], [6]) does not lead to the right fractal dimensions Dµ±ψ 1/(1 + β) , as illustrated in Appendix B. R1 Here h(z) is some positive function in C0∞ ([0, 1]) such that 0 h(z) dz = 1. Afterwards, and this is Theorem 3.1, one is able to obtain a constant C(ψ, p, h) > 0 such that for all T > 0, ( ) 2+2β |hφ, ψi| hh|X | p iiψ,T ≥ C(ψ, p, h)L ψ (T ), with L ψ (T ) = sup β , φ∈Hψ kφk2 Uφ,ψ (T ) where Hψ is the cyclic subspace spanned by ψ and H , Z Uφ,ψ (T ) = dµψ (x) dµφ (y)R T (x − y) , R2

and R is some bounded fast-decaying function defined by (3.5) in Section 3. (One 2 should think of R(w) as of the Gaussian e−w /4 .) Thus, as in [6] and [18], one can choose, for each T , a T -dependent vector φ in the decomposition ψ = φ + χ, which contains enough spectral information to approximate the supremum in L ψ (T ). The second improvement then consists of the way one chooses this particular vector φ. We show (see Theorem 4.1) that a judicious choice enables one to connect, up to a logarithmic factor, the quantity L ψ (T ) to the integral Z h i−( p/d)/(1+ p/d) 1 Iµψ , T −1 = dµψ (x) µψ x − T −1 , x + T −1 1 + p/d R which defines the generalized fractal exponents Dµ±ψ 1/(1 + p/d) . Our method also applies to the previous approaches in [15], [16], [8], [27], and [6], where the crossed term in (1.6) was not treated well. It yields, respectively, as stated in Appendix B, the fractal dimensions Dµ±ψ (1 + β)/(1 + 2β) and Dµ±ψ (1 +


165

2β)/(1 + 3β) . Since the functions Dµ±ψ (q) are nonincreasing functions of q, our Theorem 2.1 gives a better lower bound. This means that a better estimate of the crossed term in (1.6) does provide an improvement (see Theorem B.3). Finally, using extra assumptions on the decay of the generalized eigenfunctions u(n, x) of H (in the spirit of [23], [25]), it is possible to improve the above bounds. In particular, suppose that there exists a constant C such that for µψ a.e. x, P 2 d ± ± |n| 0, Iµ takes the value +∞. Remark 2.2 (i) For our purpose it is sufficient to discuss the case q ∈ (−∞, 1) (see, e.g., [10] for the general case). (ii) There actually exists a wide number of generalized fractal dimensions. For example, they can be defined with the help of the so-called “singularity spectrum function” f µ of the measure µ (see [20], [30]), or as a solution of an implicit equation (see [20, Formula 2.8], [30]). The resulting dimensions coincide with each other in certain very specific cases, like, for example, cookie-cutter measures in R (see [30]).


167

In order to state our results, we also define the following integrals that could be considered as approximations of the quantities Iµ (q, ε). The function R is a bounded even function with fast decay properties at ±∞ and is precisely defined in (3.5): K µ (q, ε) =

Z

Z supp µ

dµ(y)R (x − y)/ε

dµ(x)

q−1

.

R

In Lemma 4.3 we prove that for any measure µ verifying the condition (H) of our theorem, taking K µ (q, ε) in (2.2) instead of Iµ (q, ε) leads to the same values for the generalized fractal dimensions. We review below some properties of the fractal dimension numbers Dµ± (q) that are of interest for us. 2.1 Let µ be a Borel probability measure. (i) Dµ− (q) and Dµ+ (q) are nonincreasing functions of q ∈ (−∞, 1). (ii) For all q ∈ (−∞, 1), Dµ− (q) ≥ dim H (µ). (iii) For all q ∈ (−∞, 1), Dµ+ (q) ≥ dim P (µ). (iv) If µ has a bounded support, then for all q ∈ (0, 1), 0 ≤ Dµ− (q) ≤ Dµ+ (q) ≤ 1. PROPOSITION

Proof Statement (i) is already known (see, e.g., [10]). This is a straightforward consequence of concave and convex Jensen inequalities (for the proof of (ii) and (iii), see Appendix A). Statement (iv) follows from Corollary C.1. Note that (iv) does not necessarily hold any more if one lets q vary in (−∞, 1) (see Appendix D). From now on, each time we refer to the exponent β it should be understood that β = p if H is a general separable Hilbert space with basis {en }n∈N , and β = p/d in the specific case H = `2 (Zd ) equipped with its canonical basis {δn }n∈Zd . Our main result is the following one. THEOREM 2.1 Let H be a self-adjoint operator on a separable Hilbert space H , and let ψ be a vector in H , kψkH = 1. Assume that the spectral measure µψ associated to ψ is such that (H) Dµ±ψ (s) < +∞ for any s ∈ (0, 1).

Then for hh|X | p iiψ,T defined by (1.1) and with β as described just above, the following holds: loghh|X | p iiψ,T 1 − lim inf ≥ β Dµψ , T →∞ log T 1+β

168

and


loghh|X | p iiψ,T lim sup ≥ β Dµ+ψ log T T →∞

1 1+β

.

Remark 2.3 (i) As a consequence of Proposition 2.1(i)–(iii), this result does improve previous known bounds of [27], [1], and [18]. (ii) As we show in Appendix D, Theorem D.1, for any β = p/d > 0 and any δ > 0 there exist a bounded self-adjoint operator H and a vector ψ such that limT →∞ loghh|X | p iiψ,T / log T = 0, but at the same time Dµ±ψ ((1/(1 + β)) − δ) > 0. Therefore one cannot hope to obtain a general (i.e., without additional assumptions on µψ or on generalized eigenfunctions) lower bound of the form β Dµ±ψ q(β) with some q(β) < 1/(1 + β) like, for example, Dµ±ψ (1 − β). Proof of Theorem 2.1 The proof of Theorem 2.1 is the combination of Theorems 3.1 and 4.2, which are proved, respectively, in Sections 3 and 4. We now discuss a model of Jacobi matrices, the Julia matrices. In this case, the upper bound derived in [5] together with Theorem 2.1 above enable one to prove for small p that the increasing exponents of the moments of order p are entirely controlled by the generalized fractal dimensions. Julia matrices There exists a class of models for which nontrivial (i.e., nonballistic) upper bounds for the moments of order p are derived in terms of generalized fractal dimensions: the Julia matrices. They are constructed by considering polynomials, disjoint iterated function systems (IFS), giving rise to a real hyperbolic Julia set J (see, e.g., [5], [19], and references therein for details). Given such an IFS, one considers the balanced measure µ of maximal entropy on J and then constructs a Hamiltonian H (H corresponds to the Jacobi matrix associated to µ) on `2 (N) as follows. Let Pn , n ≥ 0, denote the orthogonal and normalized polynomials associated to µ. The family (Pn )n∈N forms a Hilbert basis in L2 (R, µ) and satisfies a three terms recurrence relation E Pn (E) = tn+1 Pn+1 (E) + vn Pn (E) + tn Pn−1 (E), n ≥ 0, where vn ∈ R and tn ≥ 0 are bounded sequences, and P−1 = 0. Therefore the isomorphism of L2 (R, µ) onto `2 (N) associated with the basis (Pn )n∈N carries the operator of multiplication by E in L2 (R, µ) into the self-adjoint finite difference operator H defined


169

on `2 (N) by H ψ(n) =

tn+1 ψ(n + 1) + tn ψ(n − 1) + vn ψ(n), t1 ψ(1) + v0 ψ(0),

n ≥ 1, n = 0.

(2.3)

Then µ is the spectral measure of H associated to the state δ0 located at the origin. For this model the upper and lower fractal dimensions Dµ± (q) are equal (:=Dµ (q)), and continuous for q ∈ (0, 1); furthermore, we have Dµ 1/(1 + p) = Dµ (1 − p) + O ( p 2 ) (see [5] and references therein). It is established in [5] that there exists a critical value pc ≥ 2 such that for all p ∈ (0, pc ), α + (δ0 , p, 1) ≤ Dµ (1 − p). Therefore, putting together Theorem 2.1 and [5, Theorem 1], we get, for the exponents of any moments of order p ∈ (0, pc ) and for the initial state δ0 , that 1 Dµ ≤ α ± (δ0 , p, 1) ≤ Dµ (1 − p), 1+ p and thus for small p > 0, Dµ (1 − p) + O ( p 2 ) ≤ α ± (δ0 , p, 1) ≤ Dµ (1 − p). This is the first model of a Schrödinger-like operator treated rigorously for which such bounds are derived. If Dµ (q) were known to be strictly decreasing in some interval (0, δ), one would get intermittency for hh|X | p iiψ (T ), with small p. However, to our best knowledge, this fact is only emphasized by numerics on the generalized fractal dimension (see, e.g., [28], [29]), and no rigorous results are provided. 3. A general lower bound This section can be regarded as the first part of the proof of our main result, Theorem 2.1. Let H be a self-adjoint operator in Hilbert space H , and let {en } be an orthonormal basis in H labelled by n ∈ N or by n ∈ Zd . Let ψ be some vector in H such that kψk = 1. We are interested in lower bounds for the moments of the abstract position operator associated to the basis {en }, defined as X p |X |ψ (t) = |n| p |he−it H ψ, en i|2 . (3.1) n

`2 (Zd ),

In particular, if H = one can take the canonical basis en (k) = δnk , n, k ∈ Zd , to obtain the moments of the usual position operator. Our results can also be extended to the case H = L2 (Rd ) (see [3]) and Z p |X |ψ (t) = |x| p |ψ(t, x)|2 dx, ψ(t, x) = (e−it H ψ)(x). Rd

170


It can be done using Theorem 3.2 in the same manner as in [6] (see [6, Corollary 2.4 and Theorem 2.5]; see also [27]). However, for the sake of simplicity, we consider in this paper only |X | p given by (3.1). We derive a lower bound for the time-averaged moments of position operator in terms of an abstract quantity in the spirit of [6], namely, |hφ, ψi|2+2β , kφk2 Uφ,ψ (T )β (3.2) where Hψ is the cyclic subspace spanned by ψ and H . The exponent β = p if n ∈ N, and β = p/d if n ∈ Zd . Notice that if φ = χ (H )ψ, where is a Borel set, then L ψ (φ, T ) reads as kφk2+4β L ψ (φ, T ) = . (3.3) Uφ,ψ (T )β L ψ (T ) = sup L ψ (φ, T ), φ ∈ Hψ , hψ, φi 6 = 0 ,

L ψ (φ, T ) =

The quantity Uφ,ψ (T ) is defined as follows: Z Z Uφ,ψ (T ) = dµφ (x) dµψ (y)R (x − y)T ,

(3.4)

R R

where R(w) is a bounded and fast-decaying function defined in (3.5). Throughout the paper we use the notation (ε playing the role of 1/T ) Z x−y (R) b (x, ε) := dµψ (y)R , ε R R so that Uφ,ψ (T ) = R dµφ (x)b(R) (x, 1/T ). The quantity Uφ,ψ (T ) is crucial since one may consider that it codes the determinating part of the spectral information that is involved in the dynamical behaviour of the considered quantum system. The quantity in (3.4) should be compared to other quantities such as Z Z Z 2 2 dµψ (x) dµψ (y) or dµψ (x) dµψ (y)e−(x−y) T /4 , |x−y|≤1/T

R R

which already appear, in the limit ε = T −1 → 0, as key quantities in order to discuss the nature of the spectrum (see [26], [27], [34], [6]). The main result of this section is the following. THEOREM 3.1 Let H be a self-adjoint operator on H , and let ψ be a vector in H , kψk = 1. Let {en } be some orthonormal basis in H and Z 1 T X p −it H p hh|X | iiψ (T ) = |n| |he ψ, en i|2 dt. T 0 n


171

R1 Let h ∈ C0∞ ([0, 1]) be any positive function such that 0 h(z) dz = 1, and define ( 1 if |w| ≤ 1, R(w) = (3.5) 2 ˆ |h(w)| if |w| > 1, where hˆ stands for the Fourier transform of h. Then, for all p > 0 and with L ψ (T ) as defined in (3.2), there exists a constant C(ψ, p, h) such that for all T > 0, hh|X | p iiψ (T ) ≥ C(ψ, p, h)L ψ (T ). Remark 3.1 We point out that h seems to be the necessary trick to take into account averaging on time between [0, T ] only, instead of [−T, T ]. This trick actually allows one to deal with the crossed term in (1.6) and to recover a function R with fast decay at ±∞. 2 Replacing h by the usual Gaussian e−z /4 in (3.15) and thus in (3.6) leads to the same p result, but with |X |ψ (t) averaged over [−T, T ]. As is now well known (see [15], [16] [8], [27]), a key point in the proof of lower bounds for hh|X | p iiψ (T ) is a good control on the behaviour of the wave packet inside a ball of radius N , namely, Z 1 ∞ X Bψ (T, N ) = |he−it H ψ, en i|2 h(t/T ) dt. (3.6) T 0 |n|≤N

To that end, we need Theorem 3.2, which is a generalization of [6, Theorem 2.1]. In order to state it, we first recall well-known facts involving the spectral theorem for the self-adjoint operator H and the chosen vector ψ (see, e.g., [33] and references therein). Namely, there exists a unitary map Wψ from the cyclic subspace Hψ into the space L2 (R, dµψ ) such that Wψ (ψ) = 1 and Wψ (e−it H ψ) = e−it x . We denote by Pψ the orthogonal projection on Hψ . The map Wψ has a kernel u(n, x) defined by u(n, ·) = Wψ (Pψ en )(·), so that Z −i H t (3.7) he ψ, en i = e−i xt u(n, x) dµψ (x), and, more generally, for any vector ξ ∈ H , one has Z hPψ ξ, en i = Wψ Pψ ξ (x)u(n, x) dµψ (x).

(3.8)

In the case H = `2 (Zd ) and en (k) = δnk , for each fixed x ∈ R the vector u(n, x)n∈Zd may be seen as a generalized eigenfunction of H (i.e., in a distributional sense). This observation is of interest for some applications (see [26] and Theorem 4.3).

172


The expansions (3.7) and (3.8) are of course also possible with any vector φ ∈ H and corresponding kernel v(n, y) = Wφ (Pφ en )(y). (Actually, if φ = f (H )ψ, with f ∈ L2 (R, dµψ ), then one checks that v(n, y) = f (y)u(n, y).) We are now ready to formulate Theorem 3.2. 3.2 Let h(z) be some function in L1 (R), let H be a self-adjoint operator acting on H , and let A be a Hilbert-Schmidt operator in H . For any couple of vectors ψ, φ from H , define the quantity Z 1 +∞ (h) hAe−it H φ, e−it H ψih(t/T ) dt. Dφ,ψ (T ) = T −∞ THEOREM

Let (h) Uφ,ψ (T )

Z Z

ˆ dµφ (x) dµψ (y)|h((x − y)T )|2 .

= R R

The following estimate holds: 1/2 (h) (h) |Dφ,ψ (T )| ≤ kAk2 Uφ,ψ (T ) , where kAk2 is the Hilbert-Schmidt norm of A. P In the special case A = |n|≤N h·, en ien , one has kAk2 ≤ C N d/2 , and therefore Z +∞ X 1 (h) |Dφ,ψ (T, N )| = he−it H φ, en ihen , e−it H ψih(t/T ) dt T −∞ |n|≤N

1/2 (h) ≤ C N d/2 Uφ,ψ (T ) . Here d = 1 if one considers the abstract position operator associated with the base {en } labelled by n ∈ N, and d ≥ 1 in the case H = `2 (Zd ) equipped with the canonical basis en = δn . Proof Since A is Hilbert-Schmidt, there exist two orthonormal bases { f n }n∈N and {gn }n∈N of P 2 H and a monotonely decreasing sequence {E n }n∈N , E n ≥ 0, such that ∞ n=1 E n = P ∞ 2 kAk2 < +∞ and A = n=1 E n h·, f n ign . Therefore (h)

Dφ,ψ (T ) =

Z

1 T

∞ +∞ X

E n he−it H φ, f n ihgn , e−it H ψih(t/T ) dt.

(3.9)

−∞ n=1

Then (3.7) reads as −it H

he

φ, f n i =

Z

dµφ (x)e−it x u(n, x), R

(3.10)


173

where u(n) = Wφ (Pφ f n ) ∈ L2 (R, dµφ ). The similar formula holds for ψ with v(n) = Wψ (Pψ gn ) ∈ L2 (R, dµψ ). One obtains from (3.9) and (3.10) that Z Z (h) dµφ (x) dµψ (y)hˆ (x − y)T S(x, y), (3.11) Dφ,ψ (T ) = R R

where S(x, y) =

∞ X

E n u(n, x)v(n, y).

n=1

The sum converges in L2 (R2 , µφ × µψ ). Applying the Cauchy-Schwarz inequality to (3.11), one gets (h)

(h)

|Dφ,ψ (T )|2 ≤ Uφ,ψ (T )kSk2L2 (R2 ,dµ

φ ×dµψ )

.

(3.12)

One can easily see that kSk2L2 (R2 ,dµ

φ ×dµψ )

∞ X

=

E n E k ank bnk ,

n,k=1

where Z ank = R

dµφ (x)u(k, x)u(n, x) = Wφ (Pφ f k ), Wφ (Pφ f n ) L2 (R,dµ

φ)

= hPφ f k , f n iH ,

(3.13)

where we used an analog of (3.8), and in the same manner bnk = hPψ gk , Pψ gn iH = hgk , Pψ gn iH . We have also used the fact that both Pφ and Pψ are orthogonal projections. By Parseval equality, ∞ X

∞ X

|ank |2 = kPφ f k k2 ,

n=1

|bnk |2 = kPψ gn k2 .

k=1

Therefore, as k f k k = kgn k = 1 for all k, n, kSk4L2 (R2 ,dµ ×dµ ) φ ψ

≤

∞ X

E k2 kPφ f k k2

k=1

∞ X

E n2 kPψ gn k2

n=1

≤

X ∞

E n2

2

= kAk42 .

n=1

(3.14) The first statement of the theorem follows from (3.12) and (3.14). The proof of the second part is essentially the same. The only difference is that in the case H = `2 (Zd ) the sums are taken over n ∈ Zd : |n| ≤ N . In particular, estimate (3.14) reads X X kSk4L2 (R2 ,dµ ×dµ ) ≤ kPφ ek k2 kPψ en k2 ≤ C N 2d . φ

ψ

|k|≤N

This ends the proof.

|n|≤N

174


One should stress that the proof we presented here is simpler than the proof of [6, Theorem 2.1] because we do not use the product space H ⊗ H . In the case ψ = φ and h(z) = exp(−z 2 /4), the result of Theorem 3.2 is equivalent to that of [6, Theorem 2.1]. Proof of Theorem 3.1 R1 Pick a positive function h(z) ∈ C0∞ ([0, 1]) such that 0 h(z) dz = 1. The role of 2 . Note that one trivially has, for any ˆ h is to supply a fast-decaying function |h(w)| z ∈ [0, 1], h(z) ≤ khk∞ χ[0,1] (z). Then one verifies that Z ∞X 1 dt p hhX iiψ (T ) ≥ |n| p |he−it H ψ, en i|2 h(t/T ) khk∞ 0 T n ≥

Np kψk2 − Bψ (T, N ) , khk∞

(3.15)

with Bψ (T, N ) defined in line (3.6). As usual, one needs a control of this quantity Bψ (T, N ) which represents the behaviour of the wave packet in a ball of radius N . Decompose the vector ψ as φ + χ, with hφ, χi = 0 and φ 6= 0. (One should think of φ = χ (H )ψ.) Thus Bψ (T, N ) = Bφ (T, N ) + Bχ (T, N ) Z ∞ X 2 + Re he−it H φ, en ihe−it H χ, en ih(t/T ) dt T 0 |n|≤N

= −Bφ (T, N ) + Bχ (T, N ) Z ∞ X 2 Re he−it H φ, en ihe−it H ψ, en ih(t/T ) dt. + T 0 |n|≤N

R∞ Then, taking into account that 1/T 0 h(t/T ) dt = 1 and h(z) ≥ 0, we have P Bχ (T, N ) ≤ kχ k2 = kψk2 − kφk2 . Let A = |n|≤N h·, en ien . Then (h)

Bψ (T, N ) ≤ kψk2 − kφk2 + 2 Re Dφ,ψ (T, N ), (h)

where Dφ,ψ (T, N ) was defined in Theorem 3.2. The second statement of this theorem immediately gives 1/2 (h) Bψ (T, N ) ≤ kψk2 − kφk2 + C N d/2 Uφ,ψ (T ) , (3.16) where, as in Theorem 3.2, (h)

Uφ,ψ (T ) =

Z Z R R

ˆ dµφ (x) dµψ (y)|h((x − y)T )|2 .


175

2 ˆ ˆ As |h(w)| ≤ 1 for all w, and by definition (3.5) of R, we clearly have R(w) ≥ |h(w)| (h) for all w. Therefore Uφ,ψ (T ) ≤ Uφ,ψ (T ), and estimate (3.16) is valid with Uφ,ψ (T ) ˆ 2. defined by (3.4), that is, with the function R instead of |h|

We are now in position to finish the proof of Theorem 3.1. The basic strategy is standard: let N be the largest integer such that C N d/2 Uφ,ψ (T )1/2 ≤ kφk2 /2; it yields kφk2 . 2 Inequalities (3.17) and (3.15) yield, with some positive constant C(ψ, p, h), Bψ (T, N ) ≤ kψk2 −

hh|X | p iiψ,T ≥ C(ψ, p, h)

kφk2+4β = C(ψ, p, h)L ψ (φ, T ), Uφ,ψ (T )β

β=

p . d

(3.17)

(3.18)

One recovers L ψ (φ, T ) as given in line (3.3). It is this latter lower bound that is used in the proof of the lower bound in Theorem 4.1. However, in the more general case where φ is any function of H with hψ, φi 6= 0, one gets the bound with L ψ (φ, T ) :=

|hφ, ψi|2+2β . kφk2 Uφ,ψ (T )β

(3.19)

Indeed, take such a φ, and then define φ˜ = (hφ, ψikφk−2 )φ as in [6]; one thus ˜ then hφ, ˜ χi = 0 and one is able to apply the result checks that if χ = ψ − φ, ˜ Taking into account that kφk ˜ = |hφ, ψi|kφk−1 and that line (3.3) to ψ and φ. 2 −4 Uφ,ψ ˜ (T ) = |hφ, ψi| kφk Uφ,ψ (T ), one finds the announced expression (3.19). To optimize the lower bound, we should take the supremum of L ψ (φ, T ) for a given T over all possible φ. One can show in the same manner as in [6, Lemma 3.1] that it is sufficient to take φ only from the cyclic subspace Hψ . This gives us L ψ (T ) defined in line (3.2). 4. Toward the fractal dimensions This section deals with the connection between the dynamic quantity L ψ (T ) introduced in the previous section and the fractal dimensions defined in Section 2, called the generalized fractal dimensions. We prove the following. THEOREM 4.1 Let H be a self-adjoint operator on H , and let ψ be a vector in H , kψkH = 1. Assume (H), namely, the spectral measure µψ associated to ψ is such that

Dµ±ψ (s) < +∞

for any s ∈ (0, 1).

176


Then, for all β > 0, there exists a constant C1 > 0 such that, for all ε > 0: 1+β 1+β C1 1 1 −1 K , ε ≤ L (ε ) ≤ K , ε , µψ ψ µψ 1+β 1+β | log ε|1+β

(4.1)

where K µψ

Z Z −β/(1+β) 1 ,ε = dµψ (x) dµψ (y)R((x − y)/ε) . 1+β supp µψ R

This actually implies the following. 4.2 Under the same hypothesis (H) as previously stated, one has log L ψ (T ) log L ψ (T ) 1 1 − + = β Dµψ , lim sup = β Dµψ . lim inf T →∞ log T 1+β log T 1+β T →∞ THEOREM

Remark 4.1 (i) We point out that the first inequality in (4.1), that is, the lower bound on L ψ (ε−1 ), is sufficient to prove our main result, Theorem 2.1. However, the right side of (4.1), that is, the upper bound, is of interest too. In particular, it says that once one derives the lower bound L ψ (T ), one cannot hope for a better result than the one we stated. We also take advantage of the right part of (4.1) in Appendix B, Theorem B.3. (ii) Note that the proof of the right-hand side inequality in (4.1), that is, the upper bound, is true for any measures µψ . (iii) In Appendix C we show that (H) holds for all measures verifying the condition Z |x|λ dµψ (x) < +∞ for any λ > 0. (4.2) R

This is true in particular if µψ has a compact support. Moreover, if (4.2) holds, Dµ±ψ (s) ∈ [0, 1] for any s ∈ (0, 1). We start by providing a proof for Theorem 4.1. Throughout this section, integrations must be systematically understood on the support of the measure µψ or µφ we consider. Proof of the upper bound in Theorem 4.1 The upper bound of (4.1) in Theorem 4.1 is proved if, taking any function f ∈ L2 (R, dµψ ), one shows 1+β Z 1+β 1 L ψ ( f (H )ψ, ε−1 ) ≤ K µψ ,ε = dµψ (x)b(R) (x, ε)−β/(1+β) , 1+β (4.3)


177

R where b(R) (x, ε) = dµψ (y)R((x − y)/ε). This result follows from CauchySchwarz and Hölder inequalities. Pick φ = f (H )ψ, with f ∈ L2 (R, dµψ ). Therefore Z dµφ (x) = | f (x)|2 dµψ (x) and hφ, ψi = dµψ (x) f (x). R Remember also the equality Uφ,ψ (T ) = dµφ (x)b(R) (x, ε). Then, rewriting −1 L ψ (( f (H )ψ, ε ) as given in (3.2), one gets R dµψ (x) f (x) 2+2β −1 L ψ ( f (H )ψ, ε ) = R R β . (4.4) dµψ (x)| f (x)|2 dµψ (x)| f (x)|2 b(R) (x, ε) One starts with a Cauchy-Schwarz inequality applied to the numerator and to the functions b(R) (x, ε)−β/(2(1+β)) and f (x)b(R) (x, ε)β/(2(1+β)) . It yields Z 2+2β Z 1+β (R) −β/(1+β) dµψ (x) f (x) ≤ dµ (x)b (x, ε) ψ Z ×

dµψ (x)b

(R)

β/(1+β)

(x, ε)

| f (x)|

2

1+β

= K µψ (1/(1 + β), ε)1+β Z 1+β (R) β/(1+β) 2 × dµψ (x)b (x, ε) | f (x)| .

(4.5)

A Hölder inequality applied to the last term, and with the coefficients p = 1 + β and p 0 = (1 + β)/β, leads to Z 1+β (R) β/(1+β) 2 dµψ (x)b (x, ε) | f (x)| Z

dµψ (x)(| f (x)| )

Z

dµψ (x)| f (x)|2

= ≤

2 1/(1+β)

Z

β/(1+β) 1+β 2 (R) | f (x)| b (x, ε)

β dµψ (x)| f (x)|2 b(R) (x, ε) .

(4.6)

One thus recovers exactly the denominator term, and (4.3) holds. Remark 4.2 We stress the very strong link that comes out between L ψ (ε−1 ) and the integral K µψ (q, ε) with the particular value q = 1/(1+β). The same appears to hold with the other lower bounds L 1 (T ) and L 2 (T ) defined in Appendix B and coming from the former approaches (see [15], [16], [8], [27], [6]). This shows how relevant the fractal dimensions Dµ± (q) are with regard to the time behaviour of the quantities that have been studied for many years in quantum dynamics.

178


We now turn to the second and main part of Theorem 4.1, that is, the lower bound. Our basic strategy to get the lower bound in (4.1) is to estimate the quantity L ψ (φ, T ) in (3.19), with a vector φ = χ(r ) (H )ψ. And (r ) is a “thin" set of the form (r ) = {x ∈ supp µψ | εr +A/N < b(R) (x, ε) ≤ εr }, but which supports, roughly speaking, a significant part of the mass of the integral K µψ (1/(1 + β), ε). The constant A(q) is fixed by Lemma 4.1. The integer N stands for the integer part of log T . Before going on with the proof of Theorem 4.1, we need the following two lemmas. 4.1 ± Let q ∈ (0, 1), and suppose that (H) holds for µψ ; that is, Dµψ (s) < +∞ for any R −1 s ∈ (0, 1). Define b(x, ε) = dµψ (x)g (x − y)ε , with g(w) = χ[−1,1] (w) or R(w). Then there exist A = A(q) and ε0 (q) > 0 such that for all ε ∈ (0, ε0 ), Z b(x, ε)q−1 dµψ (x) ≤ ε. (4.7) LEMMA

{x∈supp µ | b(x,ε)≤ε A }

LEMMA 4.2 Let µψ , q, and A = A(q) be as in the previous lemma. Let N > 0 be an integer. Then, for b(x, ε) defined as in Lemma 4.1, there exist an r0 and a set (r0 ) = {x ∈ supp µψ | εr0 +A/N < b(x, ε) ≤ εr0 } such that, for all ε small enough, Z Z 1 q−1 b(x, ε)q−1 dµψ (x). b(x, ε) dµψ (x) ≥ 2N (r0 )

Lemma 4.2 is the key lemma to get Theorem 4.1 since it is this lemma that supplies the set (r ), and so the vector φ = χ(r ) (H )ψ, that is needed to prove Theorem 4.1. Lemma 4.1 does not enter explicitly in the proof of Theorem 4.1 but is an important ingredient that we use twice: once while proving Lemma 4.2 and then in Lemma 4.3. Proof of the lower bound in Theorem 4.1 Let N be the integer part of − log ε. For the sake of simplicity we use N = − log ε (rather than the integer part). Take A as in Lemmas 4.1 and 4.2. Here q = 1/(1 + β), and thus q − 1 = −β/(1 + β). For fixed ε we choose φ = χ(r0 ) (H )ψ with r0 given by Lemma 4.2. Thus from the definition of (r0 ) we obtain Z −1 Uφ,ψ (ε ) = b(R) (x, ε) dµψ (x) ≤ εr0 µψ (r0 ) . (r0 )


179

Therefore, using expression (4.4) of L ψ f (H )ψ, ε−1 with f = χ(r0 ) , one has L ψ (ε

−1

1+2β Z 1+β µψ (r0 ) (Aβ)/N (R) −β/(1+β) )≥ b (x, ε) dµψ (x) β ≥ ε (r0 ) εr0 β µψ (r0 ) Z 1+β e−Aβ (R) −β/(1+β) b (x, ε) dµψ (x) ≥ (−2 log ε)1+β 1+β C1 1 K , ε , (4.8) = µψ 1+β | log ε|1+β

where in the last inequality we used Lemma 4.2 and N = − log ε. This holds for ε small enough. Remark that the fact that r0 disappears in relation (4.8) is crucial. It is of course related to the particular value q = 1/(1 + β) of the fractal dimension that enters into account. This is the place where the deep link between L ψ (T ) and Dµ±ψ (1/(1 + β)) shows up. We are left with the proofs of Lemmas 4.1 and 4.2. Proof of Lemma 4.1 For any A > 0, ε > 0, define B(A, ε) = {x ∈ supp µ | b(x, ε) ≤ ε A }. Let 0 < s < q < 1. As b(x, ε) ≥ µ([x − ε, x + ε]) whatever b(x, ε) is, we can estimate Z Z q−1 b(x, ε) dµ(x) = b(x, ε)q−s b(x, ε)s−1 dµ(x) B(A,ε) B(A,ε) Z A(q−s) ≤ε b(x, ε)s−1 dµ(x) B(A,ε) Z A(q−s) ≤ε µ([x − ε, x + ε])s−1 dµ(x) B(A,ε)

≤ε

A(q−s)

Iµ (s, ε).

(4.9)

Let us take, for example, s = q/2. As Dµ+ (s) < +∞ for any s > 0, for ε small enough one has (Dµ+ (s)+1)(1−s) 1 Iµ (s, ε) ≤ . ε Taking A = (q − s)−1 (Dµ+ (s) + 1)(1 − s) + 1 in (4.9), we obtain the result of the lemma.

180


Proof of Lemma 4.2 To simplify the notation, denote B A = {x ∈ supp µψ | b(x, ε) ≤ ε A } and also B A = {x ∈ supp µψ | b(x, ε) > ε A }. Then, using the bound of Lemma 4.1, one has for ε small enough Z Z Z b(x, ε)q−1 dµψ (x) = b(x, ε)q−1 dµψ (x) + b(x, ε)q−1 dµψ (x) BA BA Z ≤ b(x, ε)q−1 dµψ (x) + ε. BA

Remark that one always has b(x, ε) ≤ 1 (since R(w) ≤ 1). Thus one can divide the remaining set B A into N parts: (k A/N ) = {x ∈ supp µψ | ε(k+1)A/N < b(x, ε) ≤ εk A/N }, with k = 0, 1, . . . , N − 1. At least one of these N sets (k A/N ) gives rise to an integral bigger than 1/N times the integral over the whole set B A . And so k0 , r0 = Ak0 /N are picked, and thus the set (r0 ) of the lemma. R To end the proof, remark that b(x, ε)q−1 dµψ (x) ≥ 1 since q < 1. Therefore, for ε ≤ 1/2 one gets Z Z 1 q−1 q−1 b(x, ε) dµψ (x) − ε b(x, ε) dµψ (x) ≥ N (r0 ) Z 1 b(x, ε)q−1 dµψ (x). ≥ 2N We now turn to the proof of Theorem 4.2, which is actually a consequence of Theorem 4.1 and of the following Lemma 4.3 that relates the integrals K µψ (q, ε) to the integrals Iµψ (q, ε) that enter into account in the definition of the generalized fractal dimension Dµ±ψ (q). More precisely, Lemma 4.3 says that under assumption (H) on µψ , both K µψ (q, ε) and Iµψ (q, ε) have the same growth exponents Dµ±ψ (q). This is of course because of the fast decay properties of the function R we have chosen (via the choice of h ∈ C0∞ ([0, 1])). 4.3 Let q ∈ (0, 1). Suppose that (H) holds for µψ . Then for all ν ∈ (0, 1), LEMMA

1 Iµ (q, ε1−ν ) ≤ K µψ (q, ε) ≤ Iµψ (q, ε), (4.10) 22−q ψ where the left inequality holds for ε small enough (ε ≤ ε(ν)). As a consequence, log K µψ (q, ε) 1 lim inf = Dµ−ψ (q) 1 − q ε→0 − log ε and log K µψ (q, ε) 1 lim sup = Dµ+ψ (q). 1 − q ε→0 − log ε

(4.11)


181

Remark 4.3 We strongly believe that (4.11) holds in full generality for any µψ . Proof of Lemma 4.3 Throughout this proof we denote B(x, ε) = [x − ε, x + ε]. One has Z q−1 Iµψ (q, ε) = µψ B(x, ε) dµψ (x) Z =

dµψ (x)

Z

χ[−1,1]

x−y ε

q−1 dµψ (x) .

Since, by (3.5), χ[−1,1] (w) ≤ R(w) and q − 1 < 0, one has K µψ (q, ε) ≤ Iµψ (q, ε). So we need to get the lower bound of (4.10). First, notice that Z x−y (R) b (x, ε) = dµψ (y)R ε Z Z x−y x−y = dµψ (y)R + dµψ (y)R ε ε |(x−y)/ε| 0 as small as one wants. Proof of Theorem 4.2 Theorem 4.2 is a direct consequence of Theorem 4.1 and of Lemma 4.3. As already mentioned at the end of Section 2, our main theorem, namely, Theorem 2.1, is a direct consequence of Theorem 3.1 (see Section 3) and of Theorem 4.2. We end this section with a few words concerning the case where, as in [23] and [25], one assumes some further properties on the spatial behaviour of the kernel u(n, x). Assume that for some constant C independent of the energy x, and for γ ≤ d, one knows that for µψ a.e. x, X |u(n, x)|2 ≤ C N γ , (Hypothesis (HS)); |n|≤N

then one can not only get a better result taking β = p/γ instead of β = p/d, but it turns out that one can push our approach one step further to reach the dimension numbers Dµ±ψ (1 − β). The gain takes place in inequality (3.12), where a direct estimate is made possible thanks to Hypothesis (HS). This thereby enables us to avoid the Cauchy-Schwarz inequality we made in order to split the integral into a spatial part and a spectral part. This therefore leads to the lower bound hh|X | p iiψ,T ≥ C L 0ψ (T, φ) := C((kφk2+2β )/(Uφ,ψ (T )β )). Then the same technique as the one used to prove Theorem 4.1 supplies the following result. THEOREM 4.3 Suppose in addition to the hypotheses of Theorem 2.1 that the spatial hypothesis (HS) above holds for some γ and constant C. Then if β = p/γ < 1, one has

lim inf

log L 0ψ (T ) loghh|X | p iiψ,T ≥ lim inf = β Dµ−ψ (1 − β), T →∞ log T log T

lim sup

log L 0ψ (T ) loghh|X | p iiψ,T ≥ lim sup = β Dµ+ψ (1 − β). log T log T T →∞

T →∞

and T →∞


183

Appendices A. Complement of Section 2 For the reader’s convenience, we provide the proof of statements (ii) and (iii) of Proposition 2.1. Proof of Proposition 2.1(ii) We first note that from the convex Jensen inequality, and since q − 1 < 0, we have, for any set A ⊂ R, Iµ (q, ε/2)

1/(q−1)

Z

µ(x − ε/2, x + ε/2)

q−1

=

1/(q−1) dµ(x)

R

Z

µ(x − ε, x + ε)

q−1

≤

1/(q−1) dµ(x)

A

Z

µ(x − ε, x + ε) dµ(x).

≤

(A.1)

A

Consider now, for all ν ∈ (0, 1), the set dim H (µ)−ν ) A(H }. ν (ε) ≡ {x ∈ R | µ(x − ε, x + ε) < ε

We have − ) ) A(H := lim inf A(H ν (ε) ⊃ {x ∈ R | γµ (x) > dim H (µ) − ν}. ν ε→0

By the definition of dim H (µ) given in line (2.1) we get µ {x ∈ R | γµ− (x) > (H ) (H ) dim H (µ) − ν} > 0. Furthermore, lim infε→0 µ(Aν )(ε) ≥ µ(Aν ) ≥ µ {x ∈ (H ) R | γµ− (x) > dim H (µ) − ν} > 0. Thus, letting A = Aν (ε) in (A.1), we obtain Dµ− (q) = lim inf ε→0

≥ lim inf

log Iµ (q, ε/2)1/(q−1) log(ε/2) (H ) log µ Aν (ε) εdim H (µ)−ν

ε→0

log(ε/2)

= dim H (µ) − ν.

Since the above inequality is valid for all ν ∈ (0, 1), Proposition 2.1(ii) is proven. Proof of Proposition 2.1(iii) We define, for εk = e−k , dim P (µ)−ν

A(P) ν (εk ) := {x ∈ R | µ(x − εk , x + εk ) < εk

}.

184


Since limk→∞ log εk / log εk+1 = 1, we have lim supk→∞ log µ(x − εk , x + εk )/ log εk = lim supε→0 log µ(x − ε, x + ε)/ log ε. Thus we get (P) A(P) ν := lim sup Aν (εk ) k→∞

⊃ {x ∈ R | lim sup = {x ∈ R |

k→∞ γµ+ (x)

µ(x − εk , x + εk ) > dim P (µ) − ν} log εk

> dim P (µ) − ν}.

Therefore, from the definition of dim P (µ) given in line (2.1) and the above inclusions, (P) we get µ(Aν ) > 0. Using the Borel-Cantelli lemma (as done in [18]) implies that P (P) k µ Aν (εk) = ∞, and thus there exists a subsequence εk(n) & 0 of εk such that (P) (P) µ Aν (εk(n) ) ≥ k(n)−2 = (log εk(n) )−2 . Thus, letting A = Aν (εk(n) ) in (A.1), we obtain dim P (µ)−ν (P) 1/(q−1) log µ A (ε ) εk(n) ν k(n) log Iµ (q, ε) Dµ+ (q) = lim sup ≥ lim n→∞ log ε log εk(n) ε→0 = dim P (µ) − ν. Again, since the result is true for all ν ∈ (0, 1), Proposition 2.1(iii) is proven. B. Relation to other lower bounds Getting a growth exponent in terms of fractal dimensions Dµ±ψ (q) is not specific to our lower bound L ψ (T ) (3.2). It is also possible to get such relations from the lower bounds formerly derived, either directly (Barbaroux and Tcheremchantsev’s lower bound L 2 (T )) or by improving them (Guarneri, Combes, and Last’s lower bound, improved by optimizing in φ for each single T ; see L 1 (T )). We explain this point briefly in this section and thereby propose to the reader a link between the present work and the former methods. In particular, this appendix illustrates how actually deeply connected are the generalized fractal dimensions Dµ±ψ (q) to the lower bounds of hh|X | p iiψ,T studied for the past ten years, that is, since Guarneri [15], [16]. Since we focus on the relation between those lower bounds and the fractal dimensions Dµ±ψ (q), and for the sake of simplicity, we only consider vectors φ of the form χ (H )ψ instead of general φ ∈ Hψ as in (3.2). We first consider the lower bound that appears in [6]. For some constant C(ψ, p) > 0, it is shown that for all T > 0, hh|X | p iiψ,T ≥ C(ψ, p)L 2 (T ), with kφk2+8β L 2 (T ) = sup L 2 (φ, T ) := , φ = χ (H )ψ , Sφ (T )β


where Sφ (T ) =

Z R2

185

dµφ (x) dµφ (y)R T (x − y) .

Here R(w) can be chosen as the usual Gaussian e−w /4 . Then one can mimic the proof of Theorem 3.1 and relate the quantity L 2 (φ, T ) = kφk2+8β /Sφ (T )β to the integral K µψ ((1+2β)/(1+3β), T −1 ). More precisely, the same kind of Hölder inequalities as in (4.5) and (4.6) supply an upper bound for L 2 (T ), and using a set (r ) in the same spirit as in the proof of Theorem 4.1 yields the lower bound, again up to a logarithmic factor. The following theorem then holds. 2

B.1 Under the same hypotheses as in Theorem 2.1, one has log L 2 (T ) 1 + 2β − = β Dµψ lim inf T →∞ log T 1 + 3β THEOREM

and lim sup T →∞

log L 2 (T ) = β Dµ+ψ log T

1 + 2β 1 + 3β

.

The second lower bound we want to discuss in this section is the improved version, using [6], of the first kind of quantity that has been considered in order to bound from below hh|X | p iiψ,T , and that comes from Guarneri [15], [16], Combes [8], and Last [27]. Roughly, the basic idea is to take into account the function G φ (T ) := R µφ − ess sup dµφ (y)R T (x − y) where R(w) = exp(−w2 /4). (Notice that G φ (T ) is smaller than C T −α if µφ is uniformly α-Hölder continuous; see (3.10) in [27].) RT One then uses G φ (T ) in order to bound Bφ (T, N ) defined as in (3.6) but with 1/T 0 R +∞ rather than 1/T 0 h(t/T ). Using, for instance, 2|v(n, x)v(n, y)| ≤ |v(n, x)|2 + |v(n, y)|2 , Z Z X Bφ (T, N ) ≤ C dµφ (x) dµφ (y)R (x − y)T |v(n, x)v(n, y)| R R

≤ C G φ (T )

|n|≤N

Z

dµφ (x) R

X

|v(n, x)|2 ≤ C 0 G φ (T )N d .

|n|≤N

Following [27] and [6], this leads to the lower bound hh|X | p iiψ,T ≥ C(ψ, p)L 1 (T ), with kφk2+4β L 1 (T ) = sup L 1 (φ, T ) := , φ = χ (H )ψ . G φ (T )β One should compare the obtained lower bound L 1 (T ) to expression (6.10) in [27].

186


Then one can again mimic the proof of Theorem 4.1 and relate the quantity L 1 (T ) to the integral K µψ ((1 + β)/(1 + 2β), T −1 ). This leads to the following theorem. THEOREM B.2 Under the same hypotheses as in Theorem 2.1, one has log L 1 (T ) 1+β − lim inf = β Dµψ T →∞ log T 1 + 2β

and lim sup T →∞

log L 1 (T ) = β Dµ+ψ log T

1+β 1 + 2β

.

One should also compare the expression of L 1 (T ) and L 2 (T ) to the expression of L ψ (T ) given in line (3.3), that is, where as above the supremum is taken over the set of vectors φ = χ (H )ψ, namely, the supremum of kφk2+4β /Uφ,ψ (T )β . One may have the right to wonder whether there are some links between these quantities and also whether L ψ (T ) is effectively a better lower bound than L 1 (T ) and L 2 (T ). Theorem B.3 answers these questions. This theorem is actually a consequence of Theorems 4.1, B.2, and B.1, and of the nonincreasing property of the functions Dµ±ψ (q) (see Proposition 2.1(i)). THEOREM B.3 Under the same hypotheses as in Theorem 2.1, one has

lim inf

log L ψ (T ) log L 1 (T ) log L 2 (T ) ≥ lim inf ≥ lim inf , T →∞ T →∞ log T log T log T

lim sup

log L ψ (T ) log L 1 (T ) log L 2 (T ) ≥ lim sup ≥ lim sup . log T log T log T T →∞ T →∞

T →∞

and T →∞

Remark B.1 (i) It is worthwhile to point out that such a comparison is made possible thanks to the upper bounds obtained for L ψ (T ), L 1 (T ), and L 2 (T ); upper bounds that are for the first time derived for such quantities. (ii) Theorem B.3 tells us that it is worthwhile to deal with the crossed term in (1.6) while one develops Bψ (T, N ) with ψ = φ + χ . Dealing with the crossed term does lead to an improvement, with regard to the former approaches (see [15], [16], [8], [27], [6]) where this term was not treated well. (iii) One can show that L ψ (φ, T ) ≥ c1 L 1 (φ, T ) with some constant c1 > 0 uniform in T, φ. The latter is not true as one compares L ψ (φ, T ) and L 2 (φ, T ) for some φ. We confess that the inequalities involving L 2 (T ) in Theorem B.3 are quite a surprise


187

for us since we were expecting the exponents of L 2 (T ) to be bigger than those of L 1 (T ). (iv) We believe a stronger version of Theorem B.3 to be true, namely, L ψ (T ) ≥ c1 L 1 (T ) ≥ c2 L 2 (T ). C. A sufficient condition for Hypothesis (H) The following statement gives a sufficient condition for (H) to hold, which can be useful for applications. PROPOSITION C.1 Let 0 < q < 1. Assume that

1−q , one has (H1) for some λ > q

Z

|x|λ dµ(x) < +∞. R

Then 0 ≤ Dµ± (q) ≤ 1. Proof of Proposition C.1 Let b(x, ε) = µ([x − ε, x + ε]). Define (ε) = {x ∈ supp µ | b(x, ε) ≤ ε}, and I j = [( j − 1/2)ε, ( j + 1/2)ε), j ∈ Z. For any x ∈ I j , µ([x − ε, x + ε]) ≥ µ(I j ) ≥ µ I j ∩ (ε) := b j .

(C.1)

We first remark that for any j, b j ≤ ε. Indeed, if b j = 0, it is trivially true. And if b j > 0, there exists x0 ∈ I j ∩ (ε). Inequality (C.1) and the definition of (ε) imply b j ≤ µ [x0 − ε, x0 + ε] ≤ ε. Next, inequality (C.1) yields Z X Z b(x, ε)q−1 dµ(x) = (ε)

b(x, ε)q−1 dµ(x)

j:b j >0 I j ∩(ε)

≤

X

q−1 X q µ I j ∩ (ε) µ I j ∩ (ε) = bj .

j:b j >0

(C.2)

j∈Z

One can rewrite this summation as follows: X j∈Z

q

bj =

+∞ X k=0

Sk ,

Sk :=

X j∈Jk

q

bj ,

(C.3)

188


where Jk = { j ∈ Z | I j ⊂ [ek , ek+1 ) ∪ [−ek+1 , −ek )} for k > 0 and J0 = { j | I j ⊂ [−e, e)}. To be rigorous, equality (C.3) holds if one replaces in the definition of the sets Jk the quantity ek by [ek /ε]ε + ε/2, where [·] stands for the integer part, but for the sake of simplicity we use the definition of Jk given above. Let γ ∈ (0, 1). For k ≥ 0, Hölder inequality and the fact that Card Jk ≤ 2(ek+1 − k e )/ε ≤ 4ek /ε and b j ≤ ε imply Sk =

X

q bj

=

j∈Jk

X

γ q (1−γ )q bj bj

≤

X

j∈Jk

≤ ε(1−γ )q

4e ε

j∈Jk

γ q

k 1−γ q X

bj

bj

γ q X

((1−γ )q)/(1−γ q) bj

1−γ q

j∈Jk

= 4εq−1 ek(1−γ q)

X

j∈Jk

γ q bj

.

(C.4)

j∈Jk

From Assumption (H1) it is straightforward that there exists C(λ) < ∞ such that for all k, X X bj ≤ µ(I j ) ≤ C(λ)e−kλ , (C.5) j∈Jk

j∈Jk

with λ from the statement of Proposition C.1. Then one can find γ ∈ (0, 1) such that λγ q − (1 − γ q) > 0. Then (C.3)–(C.5) imply X

b j ≤ 4C γ q (λ)εq−1 q

j

+∞ X

e−k(λγ q−(1−γ q)) ≤ D(q, λ)εq−1 .

(C.6)

k=0

The proof is completed as follows. One can write Iµ (q, ε) as Z Z Iµ (q, ε) = + b(x, ε)q−1 dµ(x) =: I1 + I2 . (ε)

R\(ε)

The bounds (C.2) and (C.6) give I1 ≤ Dεq−1 . Furthermore, due to the definition of (ε) and q ∈ (0, 1), we have I2 ≤ εq−1 µ R \ (ε) ≤ εq−1 . We so obtain Iµ (q, ε) ≤ (D + 1)εq−1 and thus Dµ± (q) ≤ 1. COROLLARY C.1 Let µ be such that

Z

|x|λ dµ(x) < +∞ for any λ > 0. R

Then Dµ± (s) ∈ [0, 1] for any s ∈ (0, 1) and thus (H) holds. In particular, this is true if µ has a compact support.


189

D. An example of finite pure point measure with nontrivial generalized fractal dimensions For λ > 1 and α > 0, let an = a/n λ , xn = 1/n α , where a > 0 is a normalization P constant. We define the finite pure point probability measure µ = ∞ n=1 an δxn on R. As the measure µ has a bounded support, 0 ≤ Dµ± (q) ≤ 1 for any q ∈ (0, 1) (see Proposition 2.1(iv)). We denote by B(x, ε) the closed ball of center x and radius ε. For any given ε > 0, take N to be the integer part of ε−1/(1+α) . Thus there exists a constant c > 0 uniform in N (and ε) such that for all n ≤ cN , µ B(xn , ε) = µ({xn }) = an . Let q < 1/λ. Then we have Iµ (q, ε) =

X

cN q−1 X q−1 an µ B(xn , ε) ≥ µ B(xn , ε) an

n

=

cN X

n=1 q

an ∼ C ε−((1−qλ)/(1+α)) .

(D.1)

n=1

Therefore

1 − qλ . (1 − q)(1 + α) This implies that for 0 < q < 1/λ, one obtains strictly positive generalized fractal dimensions Dµ± (q) for the pure point measure µ. Moreover, by taking q and α small, one can render these dimensions as close to 1 as one wants. Estimate (D.1) is actually also valid for q < 0 (and one can show that the dimensions Dµ± (q) are finite for any q < 0). However, the behaviour of the fractal dimensions is rather strange: they can be greater than 1, and the bigger λ is, the bigger Dµ± (q) are as q → −∞: Dµ± (q) ≥

lim inf Dµ± (q) ≥

q→−∞

λ >1 1+α

if α is small enough. Furthermore, note that if µ is now defined with an = ae−λn , then Dµ± (q) = +∞ whenever q < 0. This shows that one should be very cautious when considering possible physical applications of Dµ± (q) with q < 0. To conclude, we give the example of a self-adjoint operator H and a state ψ, as mentioned in Remark 2.3(ii). We prove the following. THEOREM D.1 Let β > 0 and δ > 0. There exist a bounded self-adjoint operator H on H = `2 (N) and a state ψ ∈ H such that, with p = β, loghh|X | p iiψ,T 1 = 0, but Dµ±ψ − δ > 0. lim T →∞ log T 1+β

190


Proof of Theorem D.1 Let β > 0 and δ > 0 be as in the theorem, with ν > 0 to be chosen later on. Let (en )n≥1 be the canonical basis of `2 (N∗ ): en (k) = δnk , k, n ≥ 1. Define a self-adjoint operator H in H as follows: H en = xn en ,

xn = n −α ,

n ≥ 1.

It is easy to see that the spectral measure µψ associated to the vector ψ(k) = √ −λ/2 ak , λ = 1+β+ν, is exactly the measure µ defined above. (So supp µψ ⊂ [0, 1] and Theorem 2.1 applies.) Notice that ψ(t, k) = exp(−it xk )ψ(k). Then, as d = 1 one has β = p, and one checks that X hψt , |X | p ψt i = ak β−λ = C(ν) < +∞. k≥1

Therefore limT →∞ loghh|X | p iiψ,T / log T = 0. On the other hand, as we have seen above, Dµ±ψ (q) > 0 for any q < 1/λ = 1/(1 + β + ν). Taking ν small enough, for example, such that 1/(1 + β + ν) = (1/(1 + β)) − δ/2, one gets the required example. Remark D.1 One can easily construct such an example on H = `2 (Zd ) too by numbering the canonical basis (en )n∈Zd in a “spiral" way and taking β = p/d as usual. Indeed, one then gets en (k) = 1 for some k ∼ n 1/d . Note added in proof. Since this article was accepted, there have been new developments that lead to concrete applications. In [37] our lower bound is combined with results from [4] on equivalence of generalized fractal dimensions to prove intermittency for the model with sparse potential studied in [9]. Note that the belief stated in Remark 4.3 is proved in [4]. We also refer the reader to [36], where lower bounds in the spirit of ours but involving the behaviour of the generalized eigenfunctions are obtained. Acknowledgments. F. Germinet is grateful to S. De Bièvre, S. Jitomirskaya, and H. Schulz-Baldes for enjoyable and stimulating discussions on this subject matter. F. Germinet acknowledges the hospitality of the University of California, Irvine, where part of this work was done. The authors are also grateful to the referee for remarks and suggestions.


191

References [1]

[2]

[3]

[4] [5]

[6]

[7] [8]

[9]

[10]

[11] [12]

[13] [14] [15] [16]

J.-M. BARBAROUX, J.-M. COMBES, and R. MONTCHO, Remarks on the relation between

quantum dynamics and fractal spectra, J. Math. Anal. Appl. 213 (1997), 698–722. MR 98g:81028 162, 168 J.-M. BARBAROUX, F. GERMINET, and S. TCHEREMCHANTSEV, Nonlinear variation of diffusion exponents in quantum dynamics, C. R. Acad. Sci. Paris. Sér. I Math. 330 (2000), 409–414. MR 2000m:81225 165 , “Quantum diffusion and generalized Rényi dimensions of spectral measures” in Journées “Équations aux dérivées partielles” (La Chapelle sur Erdre, France, 2000), exp. no. I, Univ. Nantes, Nantes, 2000. MR 2001f:81042 165, 169 , Generalized fractal dimensions: Basic properties and equivalences, to appear in J. Math. Pures Appl. (9). 190 J.-M. BARBAROUX and H. SCHULZ-BALDES, Anomalous quantum transport in presence of self-similar spectra, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 539–559. MR 2001a:82068 163, 168, 169 J.-M. BARBAROUX and S. TCHEREMCHANTSEV, Universal lower bounds for quantum diffusion, J. Funct. Anal. 168 (1999), 327–354. MR 2000h:81060 161, 164, 169, 170, 171, 174, 175, 177, 184, 185, 186 J. BELLISSARD and H. SCHULZ-BALDES, Anomalous transport: A mathematical framework, Rev. Math. Phys. 10 (1998), 1–46. MR 99b:81046 162 J.-M. COMBES, “Connections between quantum dynamics and spectral properties of time-evolution operators” in Differential Equations with Applications to Mathematical Physics, ed. W. F. Ames, E. M. Harrel, and J. V. Herod, Math. Sci. Engrg. 192, Academic Press, Boston, 1993, 59–68. MR 94g:81205 161, 162, 163, 164, 171, 177, 185, 186 J.-M. COMBES and C. MANTICA, Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices, preprint, 2000, http://ma.utexas.edu/mp_arc-bin/mpa?yn=00-365 190 C. D. CUTLER, Some results on the behaviour and estimation of the fractal dimensions of distributions on attractors, J. Statist. Phys. 62 (1991), 651–708. MR 92b:58137 166, 167 S. DE BIÈVRE and G. FORNI, Transport properties of kicked and quasi-periodic Hamiltonians, J. Statist. Phys. 90 (1998), 1201–1223. MR 99g:82046 162 S. DE BIÈVRE and F. GERMINET, Dynamical Localization for the random dimer Schrödinger operator, J. Statist. Phys. 98 (2000), 1135–1148. MR 2000m:82053 162 D. H. DUNLAP, H.-L. WU, and P. W. PHILLIPS, Absence of localization in a random dimer model, Phys. Rev. Lett. 65 (1990), 88–91, http://prola.aps.org/PRL/v65/i1 162 K. J. FALCONER, Techniques in Fractal Geometry, Wiley, Chichester, England, 1997. MR 99f:28013 166 I. GUARNERI, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95–100. 161, 162, 163, 164, 171, 177, 184, 185, 186 , On an estimate concerning quantum diffusion in the presence of a fractal

192

[17]


spectrum, Europhys. Lett. 21 (1993), 729–733. 161, 162, 163, 164, 171, 177, 184, 185, 186 I. GUARNERI and H. SCHULZ-BALDES, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), 317–324. MR 2001c:81044 165


[18]

[19]

[20]

[21]

[22] [23]

[24]

[25]

[26] [27]

[28]

[29]

[30] [31]

[32] [33]

193

, Lower bounds on wave packet propagation by packing dimensions of spectral measures, Math. Phys. Electron. J. 5 (1999), no. 1, http://ma.utexas.edu/mpej/MPEJ.html#v5. MR 2000d:81033 161, 162, 164, 165, 166, 168, 184 , Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar spectra, Rev. Math. Phys. 11 (1999), 1249–1268. MR 2000k:81082 163, 168 T. C. HALSEY, M. JENSEN, L. KADANOV, I. PROCACCIA, and B. SCHRAIMAN, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A (3) 33 (1986), 1141–1151. MR 87h:58125 166 H. G. E. HENTSCHEL and I. PROCACCIA, The infinite number of generalized dimensions of fractals and strange attractors, Phys. D 8 (1983), 435–444. MR 85a:58064 166 S. JITOMIRSKAYA and Y. LAST, Power-law subordinacy and singular spectra, I: Half-line operators, Acta Math. 183 (1999), 171–189. MR 2001a:47033 163 R. KETZMERICK, K. KRUSE, S. KRAUT, and T. GEISEL, What determines the spreading of a wave packet?, Phys. Rev. Lett. 79 (1997), 1959–1963. MR 98g:81012 162, 165, 182 R. KETZMERICK, G. PETSCHEL, and T. GEISEL, Slow decay of temporal correlations in quantum systems with Cantor spectra, Phys. Rev. Lett. 69 (1992), 695–698, http://prola.aps.org/toc/PRL/v69/i5 161 A. KISELEV and Y. LAST, Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125–150. MR 1 741 780 165, 182 H. KUNZ and B. SOUILLARD, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), 201–246. MR 83f:39003 170, 171 Y. LAST, Quantum dynamics and decomposition of singular continuous spectra, J. Funct. Anal 142 (1996), 406–445. MR 97k:81044 161, 162, 163, 164, 168, 169, 170, 171, 177, 185, 186 G. MANTICA, “Quantum intermittency in almost periodic systems derived from their spectral properties” in Lattice Dynamics (Paris, 1995), Phys. D 103, North-Holland, Amsterdam, 1997, 576–589. 162, 169 , “Wave propagation in almost-periodic structures” in Physics and Dynamics Between Chaos, Order, and Noise (Berlin, 1996), Phys. D 109, North-Holland, Amsterdam, 1997, 113–127. 162, 169 L. OLSEN, A multifractal formalism, Adv. Math. 116 (1995), 82–196. MR 97a:28006 166 Y. B. PESIN, Dimension Theory in Dynamicals Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, 1996. MR 99b:58003 166 F. PIÉCHON, Anomalous diffusion properties of wave packet on quasiperiodic chains, Phys. Rev. Lett. 76 (1996), 4372–4375, http://prola.aps.org/toc/PRL/v76/i23 162 M. REED and B. SIMON, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972. MR 58:12429a 171

194

[34]

[35] [36] [37] [38]


R. DEL RIO, S. JITOMIRSKAYA, Y. LAST, and B. SIMON, Operators with singular

continuous spectrum, IV: Hausdorff dimensions, rank one pertubations, and localization, J. Anal. Math. 69 (1996), 153–200. MR 97m:47002 162, 170 S. J. TAYLOR and C. TRICOT, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679–699. MR 87a:28002 166 S. TCHEREMCHANTSEV, Mixed lower bounds in quantum dynamics, preprint, 2001, http://ma.utexas.edu/mp_arc-bin/mpa?yn=01-227 190 , in preparation. 190 M. WILKINSON and E. J. AUSTIN, Spectral dimension and dynamics for Harper’s equation, Phys. Rev. B 50 (1994), 1420–1429, http://prola.aps.org/search 162

Barbaroux Unité Mixte de Recherche 6629 du Centre National de la Recherche Scientifique, Unité de Formation et de Recherche de Mathématiques, Université de Nantes, 2 rue de la Houssinière, F-44072 Nantes Cédex 03, France; [email protected] Germinet Unité Mixte de Recherche 8524 du Centre National de la Recherche Scientifique, Unité de Formation et de Recherche de Mathématiques, Université de Lille 1, F-59655 Villeneuve d’Ascq Cédex, France; [email protected] Tcheremchantsev Unité Mixte de Recherche 6628 - Mathématiques, Applications et Physique Mathématique d’Orléans, Université d’Orléans, B.P. 6759, F-45067 Orléans Cédex, France; [email protected]


ON MINIMAL HYPERSURFACES WITH FINITE HARMONIC INDICES JIAQIANG MEI AND SENLIN XU

Abstract We introduce the concepts of harmonic stability and harmonic index for a complete minimal hypersurface in R n+1 (n ≥ 3) and prove that the hypersurface has only finitely many ends if its harmonic index is finite. Furthermore, the number of ends is bounded from above by 1 plus the harmonic index. Each end has a representation of nonnegative harmonic function, and these functions form a partition of unity. We also give an explicit estimate of the harmonic index for a class of special minimal hypersurfaces, namely, minimal hypersurfaces with finite total scalar curvature. It is shown that for such a submanifold the space of bounded harmonic functions is exactly generated by the representation functions of the ends. 1. Introduction A minimal submanifold M is a critical point of the volume functional, and M is said to be stable, if the second variation of its volume is always nonnegative for any normal deformation with compact support. The well-known Bernstein theorem [6] says that a complete minimal graph in R 3 must be some plane. Due to the works of W. Fleming [14], E. De Giorgi [12], F. Almgren [1], J. Simons [24], and E. Bombieri, De Giorgi, and E. Giusti [7], we know that the generalized Bernstein theorem is valid for complete minimal graphs in R n+1 only for n ≤ 7. As a natural generalization, it was proved by M. do Carmo and C. Peng [10], D. Fischer-Colbrie and R. Schoen [13], and A. Pogorelov [21] that planes are the only stable complete minimal surfaces in R 3 . For higher dimensions the corresponding generalization remains open. Recently, the work of H. Cao, Y. Shen, and S. Zhu [9] shed some new light on the structure of stable minimal hypersurfaces in R n+1 (n ≥ 3). They proved that a complete oriented stable minimal hypersurface in R n+1 (n ≥ 3) is connected at infinity, that is, has only one end. Inspired by their work, we introduce in this paper the concepts of harmonic stability and harmonic index and use them to study the connectivity at infinity of a DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 2, Received 13 March 2000. Revision received 7 November 2000. 2000 Mathematics Subject Classification. Primary 53C21, 53C42. Authors’ work supported by the National Natural Science Foundation of China and the National Education Committee Youth Science Foundation of China. 195

196

JIAQIANG MEI AND SENLIN XU

minimal hypersurface in Euclidean space. Suppose M is a minimal hypersurface in R n+1 . We define a bilinear form as follows: Z I (X, Y ) = |A|2 hX, Y i − h∇ X, ∇Y i for X, Y ∈ 0c (T M), M

where A stands for the second fundamental form of M, 0c (T M) is the set of vector fields with compact supports, and ∇ is the induced connection. The harmonic index, which is denoted by h(M), is defined as the maximal dimension of the vector spaces on which I (·, ·) is positively definite. M is called harmonic stable if h(M) = 0. Recall that M is called stable if for any compact domain ⊂ M and any smooth function f which vanishes at the boundary of , the following inequality holds: Z Z 2 |∇ f | ≥ |A|2 · f 2 .

It can be shown (see next section) that harmonic stability is a concept weaker than stability. Nevertheless, we have the following result. THEOREM (4) Let M be an oriented complete minimal hypersurface in R n+1 (n ≥ 3). Then M has only finitely many ends if its harmonic index is finite. Moreover, we have (a) the number of ends, which is denoted by e(M), satisfies

e(M) ≤ h(M) + 1; (b)

there is a nonnegative harmonic function u i related to each end E i ; these u i span an e(M)-dimensional vector space, and e(M) X

u i = 1.

i=1

As a further step one would like to investigate the explicit structure of the space of bounded harmonic functions on such a minimal hypersurface with finite harmonic index. For example, we can ask whether the ends generate this space. This is the case for a class of special minimal hypersurfaces in Euclidean spaces, namely, minimal hypersurfaces with finite total scalar curvature. For total scalar curvature, we mean R the integral M |A|n . We have the following theorem. (11) Let V be the vector space of bounded harmonic functions on an oriented complete THEOREM

ON MINIMAL HYPERSURFACES WITH FINITE HARMONIC INDICES

197

minimal hypersurface M with finite total scalar curvature in R n+1 (n ≥ 3). Then dim V = e(M), and V = span{u 1 , u 2 , . . . , u e(M) }. Minimal submanifolds with finite total scalar curvature have been studied extensively by many authors (see, for example, [18], [22], [2], [4], [5], [17], [20]). We obtain an explicit estimate of the harmonic index for hypersurfaces. (10) The following estimate for the harmonic index h(M) of an oriented complete minimal hypersurface M in R n+1 (n ≥ 3) holds: Z h(M) ≤ c(n) · |A|n , THEOREM

M

where c(n) is a constant depending only on n. Our other result is the following proposition. (9) The only oriented complete harmonic stable minimal hypersurfaces with finite total scalar curvature in R n+1 (n ≥ 3) are hyperplanes. PROPOSITION

We should mention that the above result has been proven in [23] under the stronger condition of assuming the stability of M. 2. Harmonic stability and harmonic forms We have introduced the concept of harmonic stability in Section 1. The following proposition demonstrates that stability implies harmonic stability. 1 A stable complete minimal hypersurface M in R n+1 is also harmonic stable. PROPOSITION

Proof Suppose on the contrary that M is not harmonic stable; then there exists X ∈ 0c (T M) such that Z |A|2 · |X |2 − |∇ X |2 > 0. (1) M

Let > 0 be a positive real number. Assume that supp X ⊂ B(R), where B(R) is a

198


geodesic ball with radius R. Choose a cutoff function ρ such that ρ| B(R) = 1, ρ| M\B(R+1) = 0, |∇ρ| ≤ 2. Let f = ρ · (|X |2 + )1/2 ∈ C0∞ (M). We have Z Z |A|2 · f 2 − |∇ f |2 = |A|2 ρ 2 (|X |2 + ) − (|X |2 + )|∇ρ|2 − ρ∇ρ∇|X |2 M

M

ρ2 − (|X |2 + )−1 |∇|X |2 |2 Z 4 = |A|2 |X |2 + (ρ|A|2 − |∇ρ|2 ) M

ρ2 − (|X |2 + )−1 |h∇ X, X i + hX, ∇ X i|2 Z 4 ≥ |A|2 |X |2 − |∇ρ|2 − ρ 2 (|X |2 + )−1 |X |2 |∇ X |2 ZM ≥ |A|2 |X |2 − |∇ X |2 − |∇ρ|2 . M

By (1) we know that when is sufficiently small, there holds Z |A|2 · f 2 − |∇ f |2 > 0, M

which contradicts the assumption of stability of M. Although harmonic stability is weaker than stability, we still hope some properties that hold under the condition of stability are also true under the assumption of harmonic stability. An example is the following proposition, which was first proved by B. Palmer [19] under the condition of stability. In fact, it seems that this result is more natural under the assumption of harmonic stability. PROPOSITION 2 There exists no nontrivial L 2 -harmonic 1-form on a complete harmonic stable minimal hypersurface M in R n+1 .

Before giving a proof, we make some preparations. Given p ∈ M, we can diagonalize the second fundamental form A at p, that is, we can choose the local orthonormal n frame {ei }i=1 such that at p there holds A(ei , e j ) = λi · δi j ,

1 ≤ i, j ≤ n,


199

n are the principal curvatures at p. Then the Ricci curvature of M satisfies where {λi }i=1

Ric(ei , e j ) = −λi2 · δi j . By

Pn

i=1 λi

= 0 we have λi2 =

X

λj

2

≤ (n − 1) ·

X j6 =i

j6 =i

thus λi2 ≤

λ2j ;

n n−1 X 2 n−1 · λj = · |A|2 . n n j=1

This gives us the following proposition. PROPOSITION 3 The following estimate holds for a minimal hypersurface M in R n+1 :

Ric ≥ −

n−1 · |A|2 . n

We also need another result. Suppose ϕ is a harmonic 1-form on M and ϕ # is the dual vector field of ϕ. The following Weitzenböck formula is well known: 1 1|ϕ # |2 = |∇ϕ # |2 + Ric(ϕ # , ϕ # ). 2

(2)

Proof of Proposition 2 Suppose on the contrary that ϕ is a nontrivial L 2 -harmonic 1-form on M and that its dual vector field is ϕ # . Fix p ∈ M, and let B p (r ) be the geodesic ball with center p and radius r . Choose a cutoff function ρr with compact support such that ρr | B p (r ) = 1, |∇ρr | ≤ 1. Let X r = ρr · ϕ # ∈ 0c (M). We have the following computations, where we have used Proposition 3, equation (2), and the Stokes formula: Z 0≤ |∇ X r |2 − |A|2 · |X r |2 M Z = |∇ρr |2 |ϕ # |2 + 2hρr dρr ⊗ ϕ # , ∇ϕ # i + ρr2 |∇ϕ # |2 − |A|2 ρr2 |ϕ # |2 M Z 1 = |∇ρr |2 |ϕ # |2 + ρr ∇ρr ∇|ϕ # |2 + ρr2 1|ϕ # |2 − |A|2 ρr2 |ϕ # |2 − ρr2 Ric(ϕ # , ϕ # ) 2 M

200


Z =

1 |∇ρr |2 |ϕ # |2 + ρr ∇ρr ∇|ϕ # |2 − ∇ρr2 ∇|ϕ # |2 2 M

− |A|2 ρr2 |ϕ # |2 − ρr2 Ric(ϕ # , ϕ # ) Z = |∇ρr |2 |ϕ # |2 − |A|2 ρr2 |ϕ # |2 − ρr2 Ric(ϕ # , ϕ # ) M Z Z 1 # 2 |A|2 · |ϕ # |2 . ≤ |ϕ | − n B p (r ) M\B p (r )

(3)

Letting r → ∞ and using the fact that |ϕ # | ∈ L 2 (M), we obtain Z |A|2 · |ϕ # |2 = 0. M

Since ϕ # is nontrivial, A must be vanishing on some nonempty open set. By the uniqueness of minimal hypersurfaces we know that A ≡ 0. Thus from (3) we have Z Z # 2 |∇ϕ | ≤ |ϕ # |2 . B p (r )

M\B p (r )

Letting r → ∞, we obtain |∇ϕ # | ≡ 0. This shows that ϕ # is a parallel vector field on M. In particular, |ϕ # | is constant. But |ϕ # | ∈ L 2 (M) implies that |ϕ # | ≡ 0, which is a contradiction. 3. Harmonic index and the number of ends Bombieri and Giusti [8] proved that the Harnack inequality holds on area-minimizing Euclidean hypersurfaces, which implies that there exists no nonconstant positive harmonic function on such hypersurfaces. Furthermore, this implies that in fact an areaminimizing hypersurface has only one end. Ends are topological invariants that characterize the connectivity at infinity of a noncompact manifold. By constructing bounded harmonic functions and applying the Sobolev inequality, Cao, Shen, and Zhu [9] proved that a stable minimal hypersurface in the Euclidean space also has only one end. As a particular case, the following result shows that this is also true for a harmonic stable hypersurface. 4 Let M be an oriented complete minimal hypersurface in R n+1 (n ≥ 3). Then M has only finitely many ends if its harmonic index is finite. Moreover, we have (a) the number of ends, which is denoted by e(M), satisfies THEOREM

e(M) ≤ h(M) + 1;


(b)

201

there is a nonnegative harmonic function u i related to each end E i ; these u i span an e(M)-dimensional vector space, and e(M) X

u i = 1.

i=1

In particular, if M is harmonic stable, that is, h(M) = 0, then (a) implies that M has only one end. Before giving a proof, we recall some facts. Suppose M is a complete Riemannian manifold, p ∈ M. A geodesic σ : [0, +∞) → M emanating from p is called a geodesic ray if σ |[0,t] is minimal for each t > 0. Two such geodesic rays σ, γ are called equivalent if and only if for every compact subset K ⊂ M there exists t0 ≥ 0 such that for t ≥ t0 , σ (t) and γ (t) lie in the same connected component of M\K . An equivalence class of geodesic rays is called an end of M. So for each end of M, we can choose a representative geodesic ray. Now, suppose M is an oriented complete minimal hypersurface in R n+1 (n ≥ 3) and E 1 , . . . , E m+1 are different ends of M with corresponding representative geodesic rays σ1 , . . . , σm+1 . We construct certain harmonic functions for the given ends. This method of constructing harmonic functions has been used in [15] and [9]. Our exposition follows [9] closely. S∞ We choose a smooth exhaustion of M, say, M = i=1 Di , Di ⊂ Di+1 . Here {Di } is a sequence of compact domains such that for some t0 , σ j |(t0 ,+∞) ⊂ 6 j , where 6 j (1 ≤ j ≤ m + 1) are different connected components of M\D1 . Fix a j ∈ {1, . . . , m +1}. For i ≥ 1, let u ij be the unique solution of the following Dirichlet problem on Di :   1u = 0, u| = 1,  ∂ Di ∩6 j u|∂ Di \(∂ Di ∩6 j ) = 0, where 1 is the Laplace-Beltrami operator on M. By maximum principle we have 0 ≤ u ij ≤ 1. From [9] we know that by passing to a subsequence that is still denoted by u ij , we can find a harmonic function u j on M such that lim u ij (x) = u j (x)

i→∞

for x ∈ M.

The function u j has the following property. 5 [9] The harmonic function u j is nonconstant and satisfies 0 ≤ u j ≤ 1. Also, there exists PROPOSITION

202


constant C j such that Z Di

Z

|∇u ij |2 < C j ,

∀i > 1,

|∇u j |2 < C j . M

The following Sobolev inequality for minimal hypersurfaces in R n+1 (see [16]) plays an important role in the proof of the above proposition: Z Z 2/ p φp ≤ c(n) · |∇φ|2 , (4) Di

Di

where p = 2n/(n − 2), c(n) is a constant that depends only on the dimension n, and φ is an arbitrary smooth function that vanishes on ∂ Di . We prove several further properties of the harmonic functions {u j }. These properties are crucial to the proof of Theorem 4. PROPOSITION 6 The functions {u 1 , . . . , u m+1 } form a basis of an (m + 1)-dimensional vector space of functions.

Proof We prove this by contradiction. Without loss of generality we can assume that u m+1 =

m X

ai · u i ,

ai ∈ R.

i=1

Multiplying by u m+1 , we have u 2m+1 =

m X

a j · u j · u m+1 .

(5)

j=1

Since u ij · u im+1 vanishes on ∂ Di , by (4) we obtain Z Z 2/ p i i p (u j · u m+1 ) ≤ c(n) · (u ij ∇u im+1 + u im+1 ∇u ij )2 Di Di Z ≤ 2c(n) · (|∇u im+1 |2 + |∇u ij |2 ) Di

≤ 2c(n) · (Cm+1 + C j ). Letting i → ∞, we have u j · u m+1 ∈ L p (M). By (5) we know that u m+1 ∈ L 2 p (M). But this contradicts Proposition 5 and a result of S. Yau [25], which states that a


203

nonnegative L q (q > 1)-harmonic function on a complete Riemannian manifold must be constant. This finishes our proof of Proposition 6. PROPOSITION 7 The functions {1, u 1 , . . . , u m } form a basis of an (m + 1)-dimensional vector space of functions.

Proof Again we prove this by contradiction. By Proposition 6 we can assume that 1=

m X

aj · u j,

a j ∈ R.

j=1

Multiplying by u m+1 , we have u m+1 =

m X

a j · u j · u m+1 .

j=1

The same argument as in Proposition 6 yields that u m+1 ∈ L p (M), which is impossible. Now we know that {∇u 1 , . . . , ∇u m } is a basis of an m-dimensional vector subspace of 0(T M). To obtain an m-dimensional vector subspace of 0c (T M), we use cutoff functions. Thus, let φ be a cutoff function such that supp φ is nonempty. We have the following proposition. PROPOSITION 8 The functions {φ · ∇u 1 , . . . , φ · ∇u m } form a basis of an m-dimensional vector subspace of 0c (T M).

Proof Suppose this is not the case. Then there exist constants a1 , . . . , am such that a1 · φ · ∇u 1 + a2 · φ · ∇u 2 + · · · + am · φ · ∇u m = 0.

(6)

Since supp φ is nonempty, there exists an open subset U ⊂ supp φ such that φ(x) 6= 0 for every x ∈ U . Now it follows from (6) that, on U , a1 · ∇u 1 + · · · + am · ∇u m ≡ 0. P Thus j=1 a j · u j is constant on U . But mj=1 a j · u j is a harmonic function. By the uniqueness of harmonic functions, it must be constant on M, which is impossible since we have Proposition 7. So our proof of Proposition 8 is finished. Pm

204


Now we are in a position to prove Theorem 4. Proof of Theorem 4 We prove conclusion (a) by contradiction. Thus, suppose h(M) < +∞ and e(M) > h(M) + 1. Let m = h(M) + 1, and choose different ends E 1 , . . . , E m+1 of M. Let us denote by u 1 , . . . , u m+1 the corresponding constructed harmonic functions. Let S m−1 be the standard unit sphere in R m . For (a1 , . . . , am ) ∈ S m−1 , define P u = mj=1 a j · u j . By Proposition 5 we know that u is a harmonic function on M and |∇u| ∈ L 2 (M). So as a differential form du is an L 2 -harmonic 1-form on M. Since M is not a hyperplane (otherwise e(M) = 1), |A| 6≡ 0. Choose R0 > 0 such that |A| is not identically zero on the geodesic ball B(R0 ). By Proposition 7 we know that the function S m−1 → R, Z (x1 , . . . , xm ) 7 →

m X 2 |A| · x j ∇u j 2

B(R0 )

j=1

is positive and continuous and has a positive minimum that is denoted by 0 . P Choose R > 0 such that R 2 > max{R02 , (2n/0 ) mj=1 C j }. Select a cutoff function φ on M such that φ| B(R) = 1, 1 |∇φ| < . R The harmonic 1-form du has the dual vector field ∇u. For X = φ · ∇u, we have Z I (X, X ) = |A|2 · |X |2 − |∇ X |2 M Z Z 1 1 ≥ |A|2 · |∇u|2 − 2 |∇u|2 n B(R0 ) R M Z X m 1 1 ≥ 0 − 2 |∇u j |2 n R M j=1

≥

1 1 0 − 2 n R

m X

Cj >

j=1

1 0 2n

(see the proof of Proposition 2). This yields that I (·, ·) is positively definite on the vector space span{φ∇u 1 , . . . , φ∇u m } whose dimension is m. By the definition of harmonic index we immediately obtain h(M) ≥ m.


205

This apparently contradicts the choice of m. Thus we have finished the proof of (a). By (a) we now know that M has finitely many ends. If M has only one end, (b) is certainly valid. Suppose M has more than one end. As above we can construct a harmonic function u j for each end E j (1 ≤ j ≤ e(M)). Since the number of ends is Se(M) finite, in the process of constructing u j we can assume that M\D1 = j=1 6 j . Then Pe(M)−1 u ie(M) and 1 − j=1 u ij satisfy the same equation and boundary value condition on Di . By the uniqueness we have u ie(M) = 1 −

e(M)−1 X

u ij .

j=1

Letting i → +∞, we obtain e(M) X

u j = 1.

j=1

This finishes our proof of (b). 4. Minimal submanifolds with finite total scalar curvature In this section we assume that M has finite total scalar curvature, that is, Z |A|n < ∞. M

M. Anderson [5] proved that such a minimal hypersurface with only one end must be a hyperplane. Combining with Theorem 4, we have the following proposition. PROPOSITION 9 The only oriented complete harmonic stable minimal hypersurfaces with finite total scalar curvature in R n+1 (n ≥ 3) are hyperplanes.

Under the condition of stability, the above proposition is the main theorem of [23], where the proof is different from ours. In general, M is not harmonic stable, but we prove that its harmonic index is finite and give an explicit estimate of h(M). Our method is based on [11]. In fact, [11] is concerned with submanifolds of compact manifolds, but the proof there (with minor modification) also works for minimal submanifolds in R n+1 . For completeness we provide the estimate and its proof as follows. 10 The following estimate for the harmonic index h(M) of an oriented complete minimal THEOREM

206


hypersurface M in R n+1 (n ≥ 3) holds: Z

|A|n ,

h(M) ≤ c(n) · M

where c(n) is a constant depending only on n. Proof Suppose D is a compact smooth domain on M, and consider the operator L = ∇ 2 + |A|2 on L 2 (T M| D ), where ∇ 2 is the Bochner-Laplace operator. Denote by β D the number of negative eigenvalues of the following eigenvalue problem: ( L X + λX = 0, X |∂ D = 0. Let > 0 be a positive real number. Denote by β those eigenvalues of the operator (1/ p)∇ 2 which are less than or equal to 1, where p = max{|A|2 , }. It is easily seen that β D ≤ β . Let H (x, y, t) be the heat kernel of the operator (1/ p)1−(∂/∂t) on D which sat∞ be the set of corresponding isfies the Dirichlet boundary value condition. Let {µi }i=0 eigenvalues. Define ∞ X e−2µi t . h(t) = i=0

The following inequality has been proven in [11]: β ≤ n · e2t · h(t),

∀t > 0.

On the other hand, by the definition of h(t) and the property of the heat kernel, we have Z Z h(t) = H 2 (x, y, t) p(x) p(y) dy d x. D

Differentiation with respect to t gives Z Z dh ∂H =2 H (x, y, t) p(x) p(y) (x, y, t) dy d x dt ∂t D Z Z =2 p(x) H (x, y, t) 1 y H (x, y, t) dy d x D D Z Z = −2 p(x) |∇ y H (x, y, t)|2 dy d x. D


207

The definition of the heat kernel and the Stokes formula have been used in the above computation. Repeatedly using Hölder’s inequality, we have Z Z h(t) = p(x) H 2 (x, y, t) p(y) dy d x D

D

Z

Z

H 2n/(2n−2) (x, y, t) dy

p(x)

≤ D

(n−2)/(n+2)

D

· H (x, y, t) p (n+2)/4 (y) dy Z

Z p(x)

≤

H

D

2n/(2n−2)

4/(n+2)

(x, y, t) dy

dx !n/(n+2)

(n−2)/n dx

D

2/(n+2) 2 p(x) H (x, y, t) p (n+2)/4 (y) dy d x .

Z ·

(7)

D

Define

Z

H (x, y, t) p (n+2)/4 (y) dy;

P(x, t) = D

then

  1 1 − ∂ P(x, t) = 0, x p ∂t  P(x, 0) = p (n−2)/4 (x).

Thus we have Z

d dt

P 2 (x, t) p(x) d x = 2 D

Z P(x, t) ZD

=2

∂P (x, t) p(x) d x ∂t

P(x, t)1x P(x, t) d x, D

Z

|∇x P(x, t)|2 d x ≤ 0.

= −2 D

Hence

Z

P 2 (x, t) p(x) d x ≤

Z

D

P 2 (x, 0) p(x) d x = D

Z

p n/2 d x. D

Now (7) can be rewritten as h (n+2)/n (t)

Z

p n/2 (x) d x

−2/n

D

Z

Z p(x)

≤ D

H D

2n/(n−2)

(x, y, t) dy

(n−2)/n d x.

208


By the Sobolev inequality (4) we have Z −2/n Z Z (n+2)/n n/2 h (t) p (x) d x ≤ c(n) p(x) · |∇ y H (x, y, t)|2 dy d x D

D

D

1 dh = − c(n) . 2 dt Since limt→0 h(t) = ∞, we obtain nc(n) n/2 Z h(t) ≤ · p n/2 . 4t D Hence β D ≤ β ≤ n · e2t · h(t) = n · e2t ·

nc(n) n/2 Z · p n/2 , 4t D

∀t, > 0.

Letting → 0, we get the following estimate: Z Z β D ≤ c(n) · |A|n ≤ c(n) · |A|n . D

M

By the definition of harmonic index we finally obtain Z h(M) ≤ sup β D ≤ c(n) · |A|n . D

M

From Theorem 4 we know that M has only finitely many ends, say, E 1 , E 2 , . . . , E e(M) . These ends have the harmonic representatives u 1 , u 2 , . . . , u e(M) , respectively. We have the following theorem. 11 Let V be the vector space of bounded harmonic functions on an oriented complete minimal hypersurface M with finite total scalar curvature in R n+1 (n ≥ 3). Then dim V = e(M), and V = span{u 1 , u 2 , . . . , u e(M) }. THEOREM

We need some preparations to prove this theorem. First, we need the pointwise estimate about the curvature. Such an estimate can be found in [5], [3], and [23]. The following result is basically taken from [23], where the estimate is stated in the extrinsic ball. The proof is the same as in [23], so we omit it. 12 [23] For fixed p0 ∈ M there exists an R0 > 0 such that when dist( p, p0 ) > R0 , there holds dist( p, p0 ) · |A|( p) < , PROPOSITION

where dist( p, p0 ) is the geodesic distance between p and p0 .


209

From Proposition 12 we can immediately deduce the following gradient estimate. 13 [26, p. 20] For fixed p0 ∈ M there exists an R0 > 0 such that if dist( p, p0 ) > R0 and u is a positive harmonic function on B p (2/3) dist( p, p0 ) , there holds PROPOSITION

max

B p ((1/2) dist( p, p0 ))

|∇u| c(n) ≤ , u dist( p, p0 )

where B p (r ) is the geodesic ball with center p and radius r and c(n) is a constant dependent only on n. On the other hand, the volume growth of geodesic balls can be controlled very well. PROPOSITION 14 There exist constants C1 , C2 independent of r and p such that

C1 ≤

vol B p (r ) ≤ C2 , rn

∀r > 0.

Proof The left-hand side of the inequality holds for any minimal hypersurfaces with C1 = ω(n) = vol S n−1 (see, e.g., [9]). The right-hand side has been proven by Anderson [5]. Now we have the following crucial Harnack inequality. 15 Choose p0 , R0 as in Proposition 13. We can assume that R0 is sufficiently large such that e(M) [ M n \B p0 (R0 ) = 6j, PROPOSITION

j=1 e(M)

where {6 j } j=1 are connected components corresponding to the different ends. Fix such a component, and denote it by 6. Suppose u is a positive harmonic function on 6 ∩ B p0 (R)\B p0 (R0 ) . Then there exists a constant C that is independent of u such that when 3R0 < r ≤ 3R/5 there holds sup

∂ B p0 (r )∩6

u≤C·

inf

∂ B p0 (r )∩6

u.

Proof m on ∂ B (r ) such that {B (r/4)} have pairwise Select a maximal point set {xi }i=1 p0 xi

210


empty intersections and B p0 ((5/4)r ). Hence

Sm

i=1

m X

Bxi (r/2) ⊃ ∂ B p0 (r ). Apparently,

vol Bxi

i=1

r 4

≤ vol B p0

Sm

i=1

Bxi (r/4) ⊃

5 r . 4

From Proposition 14 we know that m · C1 ·

r n 4

thus m≤

≤ C2 ·

5 n r ; 4

C2 n ·5 . C1

Notice that if xi ∈ 6, Bxi (r/2) ⊂ 6. So for arbitrary p, q ∈ ∂ B p0 (r ) ∩ 6, we can find a piecewise geodesic γ = p0 p1 ∪ p1 p2 ∪ · · · ∪ pm−1 pm˜ such that ˜ p0 = p, pm˜ = q, m˜ ≤ 2m, and pi pi+1 (0 ≤ i ≤ m˜ − 1) is a minimal geodesic with length less than or equal to r/2 contained in Bx j (r/2). By Proposition 13 we have Z Z |∇u| c(n) c(n) r ds ≤ · ds ≤ · m˜ · ≤ m · c(n). r r 2 γ u γ On the other hand, Z γ

Z Z m−1 ˜ m−1 ˜ X X d ln u |∇u| |∇u| ds ≥ ds = u pi pi+1 ds pi pi+1 u i=0

=

m−1 ˜ X

i=0

| ln u( pi+1 ) − ln u( pi )| ≥ ln

i=0

u(q) . u( p)

Therefore u(q) ≤ em·c(n) · u( p). Since p, q are arbitrary, the proof of this proposition is finished. Having this Harnack inequality, we now investigate the behavior at infinity of a harmonic function. PROPOSITION 16 Suppose u is a bounded harmonic function on M, and suppose {6 j } are as above. Then for each j, lim p∈6 j ,dist( p, p0 )→∞ u exists and is finite.

Proof Denote a j = lim R→∞ (inf6 j \B p0 (R) u). Since u is bounded, a j is finite. For any > 0,


211

we can find a sequence of Ri → ∞ such that inf6 j \B p0 (Ri ) u > a j − and a j + > inf∂ B p (Ri )∩6 j u. Consider the positive harmonic function f = u − (a j − ) on 6 j \B p0 (Ri ). From Proposition 15 one can deduce that there exists i 0 such that when i ≥ i 0 there holds sup f ≤ c(n) · inf f. ∂ B p (Ri )∩6 j

∂ B p (Ri )∩6 j

It follows immediately that sup

∂ B p (Ri )∩6 j

u ≤ 2c(n) + a j .

By the maximum principle of harmonic functions we obtain lim

sup

R→∞ 6 j \B p (R) 0

u ≤ 2c(n) + a j .

Letting → 0, by the definition of a j we have lim

p∈6 j ,dist( p, p0 )→∞

u = aj.

For each end of M we can construct bounded harmonic functions {u j } as we have done in Section 2. We have the following proposition. 17 The following equalities hold: PROPOSITION

lim

p∈6 j ,dist( p, p0 )→∞

u i = δi j ,

1 ≤ i, j ≤ e(M),

where δi j is the Kronecker delta. Proof Fix i 6= j. Let us recall the constructing of u i . Fix r > R0 , suppose R > r + 2, and consider the following Dirichlet problem: ( 1u i,R | B p0 (R) = 0, u i,R |∂ B p0 (R) ∩ 6 j = δi j . By choosing a subsequence Rk → ∞, we have lim u i,Rk = u i .

k→∞

Consider another Dirichlet problem:    1u R |6 j ∩(B p0 (R)\B p0 (r )) = 0, u R |∂ B p0 (r )∩6 j = 1,   u | = 0. R ∂ B p0 (R)

212


Since harmonic functions minimize Dirichlet integrals, there exists a constant C that is independent of R such that Z |∇u R |2 ≤ C. 6 j ∩(B p0 (R)\B p0 (r ))

Also, by the maximum principle we know that on 6 j ∩ B p0 (R)\B p0 (r ) there holds 0 ≤ u i,R ≤ u R ≤ 1. Similarly, by choosing a subsequence (still denoted by Rk ), we obtain a harmonic function u on 6 j \B p0 (r ) such that on 6 j \B p0 (r ), u Rk → u, u i ≤ u. S Choose a smooth function φ on M such that 0 ≤ φ ≤ 1, φ is zero on ( k6= j 6k )∪ B p0 (r + 1), and φ is 1 on 6 j \B p0 (r + 2). Then φ · u R is zero on the boundary of the compact domain 6 j ∩ B p0 (R)\B p0 (r + 1) . Hence by the Sobolev inequality (4) we have !(n−2)/n Z |φ · u R |2n/(n−2)

6 j ∩(B p0 (R)\B p0 (r +1))

Z ≤ c(n) ·

6 j ∩(B p0 (R)\B p0 (r +1))

|∇(φ · u R )|2

Z ≤ 2c(n) ·

6 j ∩(B p0 (R)\B p0 (r +1))

Z

|∇φ|2 + C

≤ 2c(n) ·

(|∇φ|2 + |∇u R |2 )

M

Therefore Z 6 j \B p0 (r +2)

˜ |u R |2n/(n−2) ≤ C,

Z 6 j \B p0 (r +2)

˜ |u|2n/(n−2) ≤ C.

As in Proposition 16, one can show that there exists a ≥ 0 such that lim

p∈6 j ,dist( p, p0 )→∞

u = a.

We claim that a = 0. Otherwise, by the above integral inequality we have vol 6 j < ∞. It is easily seen that this contradicts Proposition 14. Hence lim

p∈6 j ,dist( p, p0 )→∞

ui ≤

lim

p∈6 j ,dist( p, p0 )→∞

u = 0.


213

Pe(M) So for i 6= j, lim p∈6 j ,dist( p, p0 )→∞ u i = 0. On the other hand, by i=1 u i = 1 we immediately have lim p∈6i ,dist( p, p0 )→∞ u i = 1. This finishes the proof of this proposition. Now we can prove Theorem 11 easily. Proof of Theorem 11 Suppose u ∈ V , that is, u is a bounded harmonic function. From Proposition 16 we know that for each j the limit lim p∈6 j ,dist( p, p0 )→∞ u exists and is denoted by a j . Thus by Proposition 17 we have lim

dist( p, p0 )→∞

e(M) X u− a j u j = 0. j=1

By the maximum principle, u−

e(M) X

a j u j ≡ 0.

j=1

This implies V ⊂ span{u 1 , . . . , u e(M) }; hence V = span{u 1 , . . . , u e(M) }. From Proposition 6 we know that dim V = e(M). The above discussion holds for immersed submanifolds. For the embedded case, from [17] we know that one can choose the coordinates {x1 , . . . , xn+1 } of R n+1 properly such that when restricted on M, xn+1 is bounded. So xn+1 is a bounded harmonic function since M is minimal. From Theorem 11 we deduce that there exist constants {ai } such that e(M) X xn+1 = ai · u i . i=1

By Proposition 17 we have lim

p∈6 j ,dist( p, p0 )→∞

xn+1 = a j .

Thus the limit position of 6 j is the hyperplane {xn+1 = a j }. On the other hand, if M has only one end, that is, e(M) = 1, then by the above representation of xn+1 we know that M is contained in a hyperplane and therefore is the hyperplane. This gives another interpretation of Proposition 9 for the embedded case.

214


References [1]

[2] [3]

[4]

[5]

[6]

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

[12] [13]

[14] [15]

[16]

F. J. ALMGREN, JR., Some interior regularity theorems for minimal surfaces and an

extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277–292. MR 34:702 195 M. T. ANDERSON, Curvature estimates for minimal surfaces in 3-manifolds, Ann. Sci. ´ Ecole Norm. Sup. (4) 18 (1985), 89–105. MR 87e:53098 197 , “Local estimates for minimal submanifolds in dimensions greater than two” in Geometric Measure Theory and the Calculus of Variations (Arcata, Calif., 1984), Proc. Sympos. Pure. Math. 44, Amer. Math. Soc., Providence, 1986, 131–137. MR 87i:53083 208 , “Remarks on curvature integrals and minimal varieties” in Complex Differential Geometry and Nonlinear Differential Equations (Brunswick, Maine, 1984), Contemp. Math. 49, Amer. Math. Soc., Providence, 1986, 11–18. MR 87f:53063 197 , The compactification of a minimal submanifold in Euclidean space by the ´ Gauss map, Inst. Hautes Etudes Sci. Publ. Math., preprint, 1984. 197, 205, 208, 209 S. BERNSTEIN, Sur un théorème de géométrie et ses application aux e´ quations aux dérivées partielles du type elliptique, Comm. Soc. Math. Kharkov (2) 15 (1915–1917), 38–45. 195 E. BOMBIERI, E. DE GIORGI, and E. GIUSTI, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 40:3445 195 E. BOMBIERI and E. GIUSTI, Harnack’s inequality for elliptic differential equations on minimal surfaces, Invent. Math. 15 (1972), 24–46. MR 46:8057 200 H. D. CAO, Y. SHEN, and S. ZHU, The structure of stable minimal hypersurfaces in R n+1 , Math. Res. Lett. 4 (1997), 637–644. MR 99a:53037 195, 200, 201, 209 M. DO CARMO and C. K. PENG, Stable complete minimal surfaces in R 3 are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 903–906. MR 80j:53012 195 S. Y. CHENG and J. TYSK, Schrödinger operators and index bounds for minimal submanifolds, Rocky Mountain J. Math. 24 (1994), 977–996. MR 96h:58166 205, 206 E. DE GIORGI, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 79–85. MR 31:2643 195 D. FISCHER-COLBRIE and R. SCHOEN, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199–211. MR 81i:53044 195 W. FLEMING, On the oriented plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90. MR 28:499 195 PETER LI and L. F. TAM, Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set, Ann. of Math. (2) 125 (1987), 171–207. MR 88m:58039 201 J. MICHAEL and L. M. SIMON, Sobolev and mean-value inequalities on generalized submanifolds of R n , Comm. Pure Appl. Math. 26 (1973), 361–379. MR 49:9717


215

202 [17]

H. MOORE, Minimal submanifolds with finite total scalar curvature, Indiana Univ.

[18]

R. OSSERMAN, Global properties of minimal surfaces in E 3 and E n , Ann. of Math.

[19]

B. PALMER, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991),

[20]

´ J. PEREZ and A. ROS, The space of properly embedded minimal surfaces with finite

[21]

A. V. POGORELOV, On the stability of minimal surfaces (in Russian), Dokl. Akad.

[22]

R. SCHOEN, Uniqueness, symmetry, and embeddedness of minimal surfaces, J.

[23]

Y.-B. SHEN and X.-H. ZHU, On stable complete minimal hypersurfaces in R n+1 , Amer.

[24]

J. SIMONS, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968),

[25]

S. T. YAU, Some function-theoretic properties of complete Riemannian manifold and

[26]

their applications to geometry, Indiana Math. J. 25 (1976), 659–670. MR 54:5502 202 S. T. YAU and R. SCHOEN, Differential Geometry, Science Press, Beijing, 1991.

Math. J. 45 (1996), 1021–1043. MR 98e:58053 197, 213 (2) 80 (1964), 340–364. MR 31:3946 197 185–188. MR 92m:58023 198 total curvature, Indiana Univ. Math. J. 45 (1996), 177–204. MR 97k:58034 197 Nauk SSSR 260 (1981), 293–295. MR 83b:49043 195 Differential Geom. 18 (1983), 791–809. MR 85f:53011 197 J. Math. 120 (1998), 103–116. MR 99c:58040 197, 205, 208 62–105. MR 38:1617 195

209 Mei Department of Mathematics, Nanjing University, Nanjing, Jiangsu, People’s Republic of China, 210093; [email protected]; current: Department of Mathematics, National University of Singapore, 10 Kent Ridge, 2 Science Drive 2, Singapore 117543, Singapore; [email protected] Xu Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, People’s Republic of China, 230026; [email protected]


DEGENERATIONS OF MIXED HODGE STRUCTURE GREGORY J. PEARLSTEIN

Abstract We extend certain aspects of C. Simpson’s correspondence between harmonic metrics and variations of Hodge structure to the category of complex variations of mixed Hodge structure, and we prove an analog of W. Schmid’s nilpotent orbit theorem for admissible variations of graded-polarized mixed Hodge structure V → 1∗ . 1. Introduction In this paper we extend Schmid’s nilpotent orbit theorem to admissible variations of graded-polarized mixed Hodge structure V → 1∗

and derive analogs of the harmonic metric equations for variations of graded-polarized mixed Hodge structure. The original motivation for the study of such variations rests upon the following observation (see [4]). Let f : Z → S be a surjective, quasiprojective morphism. Then the sheaf V = R kf ∗ (C)

restricts to a variation of graded-polarized mixed Hodge structure over some Zariskidense open subset of S. More recently, such variations have been shown to arise in connection with the study of the monodromy representations of smooth projective varieties (see [8]) as well as certain aspects of mirror symmetry (see [5]). The basic problem of identifying a good class of abstract variations of gradedpolarized mixed Hodge structure for which one could expect to obtain analogs of Schmid’s orbit theorems was posed by P. Deligne in [4]. The accepted answer to this question was provided by J. Steenbrink and S. Zucker in [16], wherein they introduced the category of admissible variations of graded-polarized mixed Hodge structure, and they proved that every geometric variation V defined over a smooth, quasi-projective curve X is admissible, and moreover, the cohomology H k (X, V ) of such a curve X with coefficients in an admissible variation V → X carries a functorial mixed Hodge DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 2, Received 7 February 2000. Revision received 4 October 2000. 2000 Mathematics Subject Classification. Primary 14D07, 32G20. 217

218

GREGORY J. PEARLSTEIN

structure. The original question of developing analogs of Schmid’s orbit theorems for such variations has, however, remained largely unresolved. The general outline of the paper is as follows. In §2, we review the basic properties of the classifying spaces of graded-polarized mixed Hodge structures M = M W, S , {h p,q } constructed in [11] and recall how the isomorphism class of a variation of gradedpolarized mixed Hodge structure V → S may be recovered from the knowledge of its monodromy representation ρ : π1 (S, s0 ) → GL(Vs0 ),

Image(ρ) = 0,

and its period map ϕ : S → M / 0. Following [9], we then construct a natural hermitian metric h on M which is invariant under the action of the Lie group G R = g ∈ GL(VR )W | Gr(g) ∈ Aut(S , R) . Remark 1 The Lie group G R defined above acts transitively only upon the real points of M (i.e., the points F ∈ M for which the corresponding mixed Hodge structure (F, W ) is split over R). In §3, we derive analogs of the harmonic metric equations for filtered vector bundles and determine necessary and sufficient conditions for such a filtered harmonic metric to underlie a complex variation of graded-polarized mixed Hodge structure. In §4, we recall the notion of an admissible variation of graded-polarized mixed Hodge structure and present a result of Deligne [6] which shows how to construct a distinguished sl2 -representation from the limiting data of an admissible variation V → 1∗ . Making use of the material of §2 and §4, we then proceed in §5 to prove an analog of Schmid’s nilpotent orbit theorem for admissible variations of graded-polarized mixed Hodge structure V → 1∗ which gives distance estimates identical to those of the pure case. Namely, once the period mapping of such a variation is lifted to a map of the upper half-plane into M , the Hodge filtration of the given variation and that of the associated nilpotent orbit satisfy an estimate of the form dM F(z), e z N .F∞ ≤ K =(z)β exp −2π =(z) .

DEGENERATIONS OF MIXED HODGE STRUCTURE

219

2. Preliminary remarks In this section we review some background material from [16] and [11]. Definition 2.1 Let S be a complex manifold. Then a variation of graded-polarized mixed Hodge structure V → S consists of a Q-local system VQ over S equipped with (1) a rational, increasing weight filtration 0 ⊆ · · · ⊆ Wk ⊆ Wk+1 ⊆ · · · ⊆ VC of VC = VQ ⊗ C; (2) a decreasing Hodge filtration 0 ⊆ · · · ⊆ F p ⊆ F p−1 ⊆ · · · ⊆ V of V = VC ⊗ O S by holomorphic subbundles; (3) a collection of rational, nondegenerate bilinear forms W Sk : GrW k (VQ ) ⊗ Grk (VQ ) → Q

of alternating parity (−1)k ; satisfying the following mutual compatibility conditions: (a) relative to the Gauss-Manin connection of V , ∇ F p ⊆ 1S ⊗ F p−1 ; (b)

W for each index k, the triple (GrW k (VQ ), F Grk (VQ ), Sk ) defines a variation of pure, polarized Hodge structure of weight k.

As discussed in [11], the data of such a variation V → S may be effectively encoded into its monodromy representation ρ : π1 (S, s0 ) → GL(Vs0 ),

Image(ρ) = 0,

(2.2)

and its period map φ : S → M / 0.

(2.3)

To obtain such a reformulation, observe that it suffices to consider a variation V defined over a simply connected base space S. Trivialization of V relative to a fixed reference fiber V = Vs0 via parallel translation then determines the following data: (1) a rational structure VQ on V ; (2) a rational weight filtration W of V ; (3) a variable Hodge filtration F(s) of V ; (4) a collection of rational, nondegenerate bilinear forms Sk : GrkW ⊗ GrkW → C

of alternating parity (−1)k ; subject to the following restrictions:

220

(a)


the Hodge filtration F(s) is holomorphic and horizontal, that is, ∂ F p (s) ⊆ F p (s), ∂ s¯ j

∂ F p (s) ⊆ F p−1 (s) ∂s j

(2.4)

relative to any choice of holomorphic coordinates (s1 , . . . , sn ) on S; each pair (F(s), W ) is a mixed Hodge structure, graded-polarized by the bilinear forms {Sk }. Conversely, the data listed in items (1)–(4) together with the restrictions (a) and (b) suffice to determine a VGPMHS (variation of graded-polarized mixed Hodge structure) over a simply connected base. To extract from these properties an appropriate classifying space of gradedpolarized mixed Hodge structures, observe that by virtue of conditions (a) and (b), the graded Hodge numbers h p,q of V are constant. Consequently, the filtration F(s) must assume values in the set (b)

M = M (W, S , h p,q )

consisting of all decreasing filtrations F of V such that (F, W ) is a mixed Hodge structure, graded-polarized by S ; P • dimC F p GrkW = r ≥ p h r,k−r . To obtain a complex structure on M , one simply exhibits M as an open subset of an appropriate “compact dual” Mˇ. More precisely, one starts with the flag variety Fˇ consisting of all decreasing filtrations F of V such that X dim F p = f p , f p = h r,s . •

r ≥ p,s

To take account of the weight filtration W , one then defines Fˇ (W ) to be the submanifold of Fˇ consisting of all filtrations F ∈ Fˇ which have the additional property that X dim F p GrkW = h r,k−r . r≥p

As in the pure case, the appropriate “compact dual” Mˇ ⊆ Fˇ (W ) is the submanifold of Fˇ (W ) consisting of all filtrations F ∈ Fˇ (W ) which satisfy Riemann’s first bilinear relation with respect to the graded polarizations S . In particular, as shown in [11], Mˇ contains the classifying space M as a dense open subset. In order to state our next result, we recall that (see [3]) each choice of a mixed Hodge structure (F, W ) on a complex vector space V = VR ⊗ C determines a unique, functorial bigrading M V = I p,q p,q

with the following three properties:


221

(1) for each index p, F p = ⊕a≥ p I a,b ; (2) for each index k, Wk = ⊕a+b≤k I a,b ; (3) for each bi-index ( p, q), I¯ p,q = I q, p mod ⊕r α; let dM denote the distance function determined by the mixed Hodge metric h; then there exists constants K and β such that dM F(z), Fnilp (z) ≤ K =(z)β exp −2π=(z)

(iii)

for all z ∈ U with =(z) sufficiently large. Proof Observe that part (i) is a direct consequence of the admissibility of V (cf. §2). Likewise, part (ii) is a direct consequence of Schmid’s nilpotent orbit theorem, applied to the variations of pure, polarized Hodge structure carried by GrW . To prove part (iii), recall from §2 that Mˇ is a complex manifold upon which the Lie group G C = g ∈ GL(V )W | Gr(g) ∈ Aut(S , C) acts transitively. Therefore, given any element F ∈ Mˇ and a vector space decomposition Lie(G C ) = Lie(G C F ) ⊕ q, (5.2) the map u ∈ q 7→ eu .F is a biholomorphism from a neighborhood of zero in q onto a neighborhood of F in Mˇ. In particular, upon setting F = F∞ and shrinking 1 as necessary, we see that each choice of decomposition (5.2) determines a corresponding holomorphic map 0(s) : 1 → q, 0(0) = 0, via the rule

e0(s) .F∞ = ψ(s).

(5.3)

To make a choice of decomposition (5.2), we follow the methods of [11] and use the limiting mixed Hodge structure of V to define a grading of Lie(G C ). LEMMA 5.4 The limiting mixed Hodge structure (F∞ , r W ) of an admissible variation of gradedpolarized mixed Hodge structure defines a canonical, graded, nilpotent Lie algebra

q∞ =

M

℘a

(5.5)

a 0 such that u ∈ U, F ∈ S =⇒ eu .F ∈ M

d M (eu .F, F) < K |u|.

and

THEOREM 5.9 (Distance estimate) Let F(z) : U → M be a lifting of the period map ϕ of an admissible variation of graded-polarized mixed Hodge structure V → 1∗ with unipotent monodromy logarithm N and limiting mixed Hodge structure (F∞ , r W ). Then, given any G R invariant metric on M which obeys (5.8), there exist constants K and β such that =(z) 0 =⇒ dM F(z), e z N .F∞ < K y β e−2π y , z = x + i y.

Proof For simplicity of exposition, we first prove the result under the additional assumption that our limiting mixed Hodge structure (F∞ , r W ) is split over R. Having made this assumption, it then follows from the work of §4 that (1) the associate gradings r Y = Y(F∞ ,r W ) and Y = Y (F∞ , W, N ) are defined over R; (2) the endomorphism r Y − Y is an element Lie(G R ); (3) the filtration F0 = ei N .F∞ is an element of M . (This assertion is a consequence of Schmid’s SL2 orbit theorem; see [13] for details.) Now, as discussed in §4, the fact that N is a (−1, −1)-morphism of the limiting mixed Hodge structure (F∞ , r W ) implies that [r Y, N ] = −2N , and hence

rY)

ei y N = y −(1/2)(

ei N y (1/2)(

rY)

.

Consequently, because r Y preserves F∞ , rY)

ei y N .F∞ = y −(1/2)(

ei N y (1/2)(

rY)

rY)

= y −(1/2)(

rY)

ei N .F∞ = y −(1/2)(

Next, we note that by (5.3) and Lemma 5.4, F(z) = e z N .ψ(s) = e z N e0(s) .F∞ ,

.F0 .


247

and hence dM (e z N e0(s) .F∞ , e z N .F∞ ) = dM (ei y N e0(s) .F∞ , ei y N .F∞ ) rY)

= dM (y −(1/2)(

ei N y (1/2)(

rY)

e0(s) .F∞ , y −(1/2)(

rY)

ei N y (1/2)(

rY)

.F∞ ).

In particular, upon setting e0(z) = Ad(ei N ) Ad(y (1/2)(

rY)

˜

)e0(s)

and recalling that r Y preserves F∞ , it follows that dM (e z N e0(s) .F∞ , e z N .F∞ ) = dM (y −(1/2)(

rY)

e0(z) .F0 , y −(1/2)(

rY)

˜

.F0 ).

(5.10)

In addition, because [r Y, Y ] = 0, rY)

y −(1/2)(

r Y −Y )

= y −(1/2)(

y −(1/2)Y ,

and hence .F0 ) = dM (y −(1/2)Y e0(z) .F0 , y −(1/2)Y .F0 ) (5.11) r since Y − Y is an element of Lie(G R ). Therefore, by (5.8), rY)

dM (y −(1/2)(

e0(z) .F0 , y −(1/2)(

rY)

˜

˜

dM (e z N e0(s) .F∞ , e z N .F∞ ) ≤ y (1/2)(L−1) dM (e0(z) .F0 , F0 ). ˜

Consequently, by Remark 7 we have (1/2)(L−1) ˜ dM (e z N e0(s) .F∞ , e z N .F∞ ) < K |0(z)|=(z)

for some K > 0 and any fixed norm | · | on gl(V ). ˜ ˜ To further analyze 0(z), decompose 0(z) according to the eigenvalues of ad r Y , X 0[`] , [r Y, 0[`] ] = `0[`] . 0= `

Then ˜ 0(z) = ei

ad N (1/2) ad (r Y )

y

0(s) = ei

ad N (1/2) ad (r Y )

y

X

0[`] (s)

`

=

X

y `/2 ei

ad N

0[`] (s)

`

since y (1/2) ad( Y ) 0[`] = y `/2 0[`] . Therefore, denoting the maximal eigenvalue of ad (r Y ) on q∞ by c, we have r

˜ |0(z)| ≤ O(y c/2 e−2π y )

248


because 0(s) is a holomorphic function of s = e2πi z vanishing at zero. In summary, we have proven that whenever the limiting mixed Hodge structure (F∞ , r W ) of our admissible variation V is split over R and =(z) is sufficiently large, the following distance estimate holds: dM (e z N e0(s) .F∞ , e z N .F∞ ) ≤ K y β e−2π y ,

β=

1 (c + L − 1). 2

If the limiting mixed Hodge structure (F∞ , r W ) is not split over R, we may obtain the same distance estimate by first applying Deligne’s δ splitting F∞ = eiδ . Fˆ∞ and then proceeding as above. More precisely, let r Y, Y denote the gradings determined by ( Fˆ∞ , r W ) and N . Then (5.10) becomes ˜ dM (e z N e0(s) .F∞ , e z N .F∞ ) = dM y −(1/2)Y e0(z) .F0 (y), y −(1/2)Y .F0 (y) with F0 (y) = eiδ(y) ei N . Fˆ∞ ,

δ(y) = y (1/2) ad (

rY)

δ,

and δ ∈ 3−1,−1 (F∞ ,r W ) =⇒ δ(y) ∼ O(y −1 ). Thus, applying (5.8) and the remark that follows the proof of Lemma 5.7, we have dM (e z N e0(s) .F∞ , e z N .F∞ ) ≤ K y β e−2π y for =(z) sufficiently large. Thus, Theorem 5.1 is proved. With regard to analogs of Schmid’s SL2 orbit theorem for admissible variations V → 1∗ of graded-polarized mixed Hodge structure, the following result suggests that as soon as the weight filtration of V has length L ≥ 3, the resulting nilpotent orbit e z N .F∞ need not admit an approximation by an auxiliary orbit e z N . Fˆ∞ for which the limiting mixed Hodge structure ( Fˆ∞ , r W ) is split over R. THEOREM 5.12 Let (F∞ , r W ) be the limiting mixed Hodge structure associated to an admissible variation of graded-polarized mixed Hodge structure with unipotent monodromy, and let ( Fˆ∞ , r W ) be a mixed Hodge structure of the form

( Fˆ∞ , r W ) = (e−σ , r W ),

σ ∈ ker(ad N ) ∩ 3−1,−1 (F∞ ,r W ) ,


249

which is split over R. Then there exists a positive constant K such that dM (e z N . Fˆ∞ , e z N .F∞ ) ≤ K =(z)(L−3)/2 for all z ∈ U with =(z) sufficiently large. Proof Let r Y and Y denote the gradings associated to the split mixed Hodge structure ( Fˆ∞ , r W ) via the methods of §4. Then a quick review of the proof of Theorem 5.9 shows that upon setting • F0 = ei N . Fˆ∞ ∈ M , r • σ (y) = (y (1/2) ad Y )(σ ), r r we have both e z N . Fˆ∞ = e x N y −(1/2) Y .F0 and e z N .F∞ = e x N y −(1/2) Y eσ (y) .F0 . In particular, since r Y − Y ∈ Lie(G R ) and since Y is a grading of W which is defined over R, dM (e z N . Fˆ∞ , e z N .F∞ ) = dM (e x N y −(1/2) Y .F0 , e x N y −(1/2) Y eσ (y) .F0 ) r

r Y −Y )

= dM (y −(1/2)(

r

r Y −Y )

y −(1/2)Y .F0 , y −(1/2)(

y −(1/2)Y eσ (y) .F0 )

= dM (y −(1/2)Y .F0 , y −(1/2)Y eσ (y) .F0 ) ≤ y (1/2)(L−1) dM (F0 , eσ (y) .F0 ). Moreover, σ ∈ 3−1,−1 (F∞ ,r W ) =⇒ σ (y) ∼ O(y −1 ), and hence dM (F0 , eσ (y) .F0 ) ≤ K y −1 . Therefore,

dM (e z N . Fˆ∞ , e z N .F∞ ) ≤ K =(z)(L−3)/2 .

To verify that the distance estimate of Theorem 5.12 is sharp in the case where L = 3, let M ∼ = C denote the classifying space of Hodge-Tate structures constructed in Example 2.7, and let N be the nilpotent endomorphism of V = span(e2 , e0 ) defined by the rule N (e2 ) = e0 , N (e0 ) = 0. Then a short calculation shows that r W = r W (N , W ) exists and coincides with W ; • −1,−1 • at each point F ∈ M , Lie(G C ) = g(F,W ) = spanC (N ). In particular, given any point F∞ ∈ M , the map z 7 → e z N .F∞

250


is an admissible nilpotent orbit. Moreover, since M is Hodge-Tate, the resulting mixed Hodge metric on M is invariant under left translation by eλN for all λ ∈ C. Consequently, upon setting F∞ = eσ . Fˆ∞ for some element σ ∈ 3−1,−1 (F,W ) and some point Fˆ∞ ∈ MR , it then follows that dM (e z N . Fˆ∞ , e z N .F∞ ) = dM (e z N . Fˆ∞ , e z N eσ . Fˆ∞ ) = dM ( Fˆ∞ , eσ . Fˆ∞ ). Acknowledgments. As much of this work stems from the author’s thesis work, he would like to thank both his advisor Aroldo Kaplan and his committee members David Cox and Eduardo Cattani for their guidance. The author would also like to thank Ivan Mirkovic and Richard Hain for enabling his stay at Duke University during the 1998–1999 academic year, P. Deligne for both his contribution of the main lemma of §4 and his many helpful comments, and finally, to the referee for pointing out a mistake in an earlier version of Theorem 3.22. References [1]

[2]

[3] [4] [5]

[6] [7]

[8]

[9] [10]

J.-L BRYLINSKI and S. ZUCKER, “An overview of recent advances in Hodge theory” in

Several Complex Variables, VI, Encyclopaedia Math. Sci. 69, Springer, Berlin, 1990, 39–142. MR 91m:14010 E. CATTANI and A. KAPLAN, “Degenerating variations of Hodge structure” in Actes du colloque de théorie de Hodge (Luminy, 1987), Astérisque 179/180, Soc. Math. France, Montrouge, 1989, 9, 67–96. MR 91k:32019 E. CATTANI, A. KAPLAN, and W. SCHMID, Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), 457–535. MR 88a:32029 220, 224, 239, 241 ´ P. DELIGNE, La conjecture de Weil, II, Inst. Hautes Etudes Sci. Publ. Math. 52 (1980), 137–252. MR 83c:14017 217, 238 , “Local behavior of Hodge structures at infinity” in Mirror Symmetry, II, ed. B. Green and S.-T. Yau, AMS/IP Stud. Adv. Math. 1, Amer. Math. Soc., Providence, 1997, 683–699. MR 98a:14015 217 , private communication, 1995. 218, 238, 240 P. A. GRIFFITHS, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems, Bull. Amer. Math. Soc. 76 (1970), 228–296. MR 41:3470 R. M. HAIN, “The geometry of the mixed Hodge structure on the fundamental group” in Algebraic Geometry (Brunswick, Maine, 1985), ed. S. J. Bloch, Proc. Sympos. Pure Math. 46, Part 2, Amer. Math. Soc., Providence, 1987, 247–282. MR 89g:14010 217 A. KAPLAN, Notes on the moduli spaces of Hodge structures, private communication, fall 1995. 218, 221, 222 M. KASHIWARA, A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci. 22 (1986), 991–1024. MR 89i:32050


[11]

[12] [13] [14] [15] [16] [17]

251

G. J. PEARLSTEIN, Variations of mixed Hodge structure, Higgs fields and quantum

cohomology, Manuscripta Math. 102 (2000), 269–310. MR CMP 1 777 521 218, 219, 220, 221, 222, 230, 231, 232, 234, 244, 245 , The geometry of the Deligne-Hodge decomposition, Ph.D. thesis, University of Massachusetts, Amherst, 1999. W. SCHMID, Variation of Hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 52:3157 227, 242, 246 ´ C. T. SIMPSON, Higgs bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992), 5–95. MR 94d:32027 227, 228, 231, 232, 236 , Mixed twistor structures, preprint, arXiv:alg-geom/9705006. J. STEENBRINK and S. ZUCKER, Variation of mixed Hodge structure, I, Invent. Math. 80 (1985), 489–542. MR 87h:32050a 217, 219, 223, 225, 226, 238 S. USUI, Variation of mixed Hodge structure arising from family of logarithmic deformations, II: Classifying space, Duke Math. J. 51 (1984), 851–875. MR 86h:14005

Department of Mathematics, University of California, Santa Cruz, Santa Cruz, California 95064, USA; current: Department of Mathematics, University of California, Irvine, Irvine, California 92697-3875, USA; [email protected]


LOGARITHMIC DIFFERENTIAL FORMS ON p-ADIC SYMMETRIC SPACES ADRIAN IOVITA AND MICHAEL SPIESS

Abstract We give an explicit description in terms of logarithmic differential forms of the isomorphism of P. Schneider and U. Stuhler relating de Rham cohomology of p-adic symmetric spaces to boundary distributions. As an application we prove a Hodgetype decomposition for the de Rham cohomology of varieties over p-adic fields which admit a uniformization by a p-adic symmetric space. 1. Introduction Let K be a finite extension of Q p , and let X = X (d+1) be Drinfeld’s p-adic symmetric space of dimension d over K . It is a rigid analytic space, and it is defined as the complement of all K -rational hyperplanes in Pd/K . Unlike real symmetric spaces, X is not simply connected. Its cohomology groups H • (X ) for any cohomology theory satisfying certain axioms (examples are de Rham or `-adic cohomology) have been computed by Schneider and Stuhler. For 1 ≤ k ≤ d, H k (X ) is infinite-dimensional and carries a natural PGLd+1 (K )-action. In [SS] it is shown that H k (X ) is canonically and PGLd+1 (K )-equivariantly isomorphic to a certain space of locally constant functions on some flag manifold. In the following we consider the de Rham cohomology groups only. For our purposes the first computation of H k (X ) in [SS] in terms of the cohomology of a certain profinite simplicial set is important. Let H be the set of all K -rational (k) hyperplanes in Pd/K , and let Y· be the simplicial set given by r n X o Yr(k) = (H0 , . . . , Hr ) ∈ H r +1 dim K ` Hi ≤ k , i=0

with the obvious face and degeneracy maps (where ` Hi denotes a linear form defining Hi ). It is shown in [SS] that there is a natural isomorphism between H k (X ) and DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 2, Received 15 March 2000. Revision received 14 November 2000. 2000 Mathematics Subject Classification. Primary 11F85; Secondary 14F40. Authors’ work partially supported by Engineering and Physical Sciences Research Council grant number GR/M 95615. Iovita’s work partially supported by National Science Foundation grant number DMS-0070464. 253

254

ADRIAN IOVITA AND MICHAEL SPIESS

ek−1 (|Y·(k) |, K ) of the geometric realization of the dual of the reduced cohomology H (k) Y· . This can be interpreted (see Lemma 3.2) as an isomorphism between the space of distributions on H k+1 modulo a certain subspace of distributions (which we call degenerated) and H k (X ): 8 : D(H k+1 , K )/ D(H k+1 , K )deg −→ H k (X )

(1)

(see Section 4 for the definition of D(H k+1 , K )deg ). The proof in [SS] is based on the axiomatic properties of the cohomology theory and is therefore not explicit at all. Our first main result provides an explicit description of 8. For a distribution µ on H k+1 we have Z 8([µ]) = dlog (H0 , . . . , Hk ) d µ, (2) H k+1

where dlog (H0 , . . . , Hk ) := dlog (` H1 /` H0 ) ∧ · · · ∧ dlog (` Hn /` Hn−1 ). The brackets denote the class of the distribution, respectively, of the differential form. In particular, we see that the differential form Z dlog (H0 , . . . , Hk ) d µ (3) H k+1

is a canonical representative of the cohomology class 8([µ]) whenever the integral is well defined. We see in Section 4 that this is the case if µ is a measure, that is, a bounded distribution. The existence of such a canonical form representing a given de Rham cohomology class has been known previously only for the top-dimensional cohomology group H d (X ) (see [Te] for the case d = 1 and [ST1] for arbitrary d). Let klog,b (X ) be the subspace of the space of holomorphic differential k-forms k (X ) on X of type (3) (where µ is a measure). Then our result can be stated in the form that klog,b (X ) → H k (X ), ω 7→ [ω] is “almost” an isomorphism. We want to mention that logarithmic forms on X were also studied recently by E. de Shalit. In [dS] he proves that they generate a dense subspace in H k (X ). In the last section we give an application to the de Rham cohomology of padically uniformized varieties, that is, of varieties of the form X 0 := 0\X where 0 ⊆ PGLd+1 (K ) is a cocompact discrete subgroup. Their cohomolgy groups have been computed in [SS] using a covering spectral sequence E 2r,s = H r 0, H s (X ) =⇒ H r +s (X 0 ). (4) It turns out that the only interesting cohomology group is the middle one, H d (X 0 ). Denote by H d (X 0 ) = F00 ⊇ F01 ⊇ · · · ⊇ F0d+1 = 0 the filtration induced by (4). We show that it is opposite to the Hodge filtration FHi on H d (X 0 ). (This was conjectured in [SS].) As a consequence one gets a Hodge-type decomposition for H d (X 0 ): H d (X 0 ) =

d M i=0

F0i ∩ FHd−i .

LOGARITHMIC FORMS ON p-ADIC SYMMETRIC SPACES

255

We also deal with the corresponding question for the cohomology of certain local systems on X 0 induced by finite-dimensional K [0]-modules which contain a 0-stable lattice. For convenience we briefly describe the contents of the other sections. In Section 2 we prove a compatibility property between the cup product and an edge morphism of a certain spectral sequence associated to a system of closed subschemes of a variety. In Section 3 this is applied to prove the analogue of (2) for the de Rham cohomology of the complement of finitely many hyperplanes in the projective d-space. Finally, in Section 4 we show that the integrals above are well defined and prove formula (2) by reducing it to the corresponding result for the complement of finitely many hyperplanes. 2. Cup product and a spectral sequence Let K be a field of characteristic zero. For a pair (X, Y ) consisting of an algebraic K scheme X and a closed subscheme Y ⊆ X , we write H • (X ), HY• (X ) as shorthand for the de Rham cohomology of X , respectively, the de Rham cohomology of X with support in Y . (However, the results of this section are valid for any reasonable cohomology theory, e.g., `-adic or singular cohomology.) A general reference for de Rham cohomology of algebraic varieties is [Ha1]. Let X be as above, and let Y = (Y0 , . . . , Yn ) be an (n + 1)-tuple of closed subschemes of X . There exists a spectral sequence ( L ) HYsi ∩···∩Yi (X ) if 0 ≤ r ≤ n, 0 ≤ s, 0≤i e − 2

when r1 + r2 ≤ 2e.

For [X4, case IV], the above condition holds if and only if ordp (n(L 1 )) ≡ ordp (n(L 2 )) mod 2 and dp (ε2 ) > e + r1 when L 1 is not a hyperbolic plane. 1.5. The splitting numbers: A summary We summarize our results obtained in the previous local computation into the following theorem, which gives a complete solution of splitting numbers. We keep the same notation as in the previous sections.

REPRESENTATION MASSES OF SPINOR GENERA

295

THEOREM 1.5.1 Suppose rank(L) = 3. K is a splitting lattice for gen(L) if and only if the following four conditions hold: (1) det(W ) 6= −1, (2) θ A (O A (L)) ⊆ θ A (O A (W )), (3) θp (X (L p /K p )) = θp (O + (Wp )) for the ramified primes p of F in E/F, and (4) θp (O + (L nor )) ⊆ θp (O + (Wp )) = up (Fp× )2 for the inert dyadic primes p of F in E/F, where L nor is the normalization of L p with respect to K p in the sense of Definition 1.4.1.

Proof It follows from Theorem 1.1.9 and Propositions 1.2.2, 1.3.1, and 1.4.3. Remark 1.5.2 Conditions (1), (2), and (3) are necessary conditions that K be a splitting lattice for gen(L) in general high-dimensional cases. L nor in Condition (4) can be replaced by a lattice satisfying Lemma 1.4.2. 1.6. Application An immediate application of Theorem 1.5.1 gives the converse result of [SP3, Satz 3]. In [SP2, Satz 2], Schulze-Pillot defined the function ψa by the difference of two representation masses of spinor genera. In fact, this function can be defined in a more general way. Namely, we can replace a by 1-dimensional lattice K and m by fractional ideal m. Then we have ψ K a function defined on the fractional ideals of F which enjoys the same properties. By Theorem 1.5.1, one can determine exactly when ψ K equals zero. For example, if ψ K (m) 6= 0, then ψ K (mm0 ) 6 = 0 for any integral ideal m0 prime to the conductor of E/F. For [SP2, (i) in Satz 2], a better bound of h(L p , K p ) for dyadic ramified primes is given in [X4, §3], which is in fact sharp. For [SP2, (ii) in Satz 2], the bound h(L p , K p ) given there is sharp at least for nondyadic primes. We explain this point for split primes in E/F by the following example. Similar examples can be given for inert primes in E/F. Example 1.6.1 Suppose p is nondyadic and split in E/F, and suppose L p = o Fp x ⊥ py ⊥ pz

and

K p = o Fp x,

where Q(x) = 1, Q(y) = ε1 πpr1 , and Q(z) = ε2 πpr2 with ε1 , ε2 in up and r2 ≥ r1 ≥ 0.

296

FEI XU

It is clear that αp = αp+ and αp− = 0. One also has αp (pK p , L p ) = αp (K p , M), where M = o Fp x ⊥ o Fp y ⊥ o Fp z. Since O + (L p ) ⊆ O + (M)

and

O + (Wp ) ∩ O + (L p ) = O + (Wp ) ∩ O + (M),

we have αp (pK p , L p ) = O + (M) : O + (L p ) αp (K p , L p ). By some computation it can be proved that [O + (M) : O + (L p )] = N p if and only if r1 > 0. 2. Part II: Codimension less than or equal to 1 In this part we always assume that K ⊂ L with codimension less than or equal to 1 and rank(L) ≥ 3. One cannot expect to get any information from the orthogonal complement of F K in V in this case. 2.1. Representation masses We keep some notation from Part I, especially §1.1. First we need the following lemma. LEMMA 2.1.1 The cosets X A (L/K )/O A (L) are finite.

Proof Since X (L p /K p ) is compact with p-adic topology for any p ∈ Spec(o F ), we have that X (L p /K p )/O + (L p ) is finite. Let T ⊂ Spec(o F ) be finite such that T contains all dyadic primes and such that K p and L p are unimodular for any p ∈ / T. Then X A (L/K )/O A (L) ∼ =

Y

X (L p /K p )/O + (L p )

p∈T

as cosets. The lemma follows. Now we define the representation masses of classes, spinor genera, and genera.


297

Definition 2.1.2 Let m(cls+ (L)), m(spn+ (L)), and m(gen(L)) be the same as those in §1.1. We have X −1 r K , cls+ (L) = 1 m cls+ (L) , σA

where σ A runs over the cosets X A (L/K )/O A (L) with σ A L ∈ cls+ (L). We have X −1 r K , spn+ (L) = 1 m spn+ (L) , σA

where σ A runs over the cosets X A (L/K )/O A (L) with σ A L ∈ spn+ (L). We have X −1 r K , gen(L) = 1 m gen(L) , σA

where σ A runs over the cosets X A (L/K )/O A (L). This definition certainly generalizes Siegel’s original definition in [Si]. We define r (K , spn+ (L)) = 0 (resp., r (K , cls+ (L)) = 0 and r (K , gen(L)) = 0) if K is not represented by spn+ (L) (resp., cls+ (L) and gen(L)). In [HSX1], one has known which spinor genera in gen(L) represent K . The further question is, how are their representation masses? The following proposition gives some information on this subject. PROPOSITION 2.1.3 Suppose σ A ∈ X A (L/K ) with σ A X A (L/K ) = X A (L/K ). Then r (K , spn+ (L)) = r (K , spn+ (σ A L)).

Proof Since X A (L/K ) = X A (σ A L/K )σ A , the following map is bijective: X A (σ A L/K )∩O + (V )O 0A (V )O A (L)/O A (σ A L) −→ X A (L/K ) ∩ σ A O + (V )O 0A (V )O A (L)/O A (L) by sending u A O A (σ A L) to u A σ A O A (L). Since σ A−1 X A (L/K ) = X A (L/K ), one has a bijection X A (L/K )∩σ A O + (V )O 0A (V )O A (L)/O A (L) −→ X A (L/K ) ∩ O + (V )O 0A (V )O A (L)/O A (L) by sending u A O A (L) to σ A−1 u A O A (L). The proposition follows from these two bijections and Definition 2.1.2.

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FEI XU

Let ϒ A (L/K ) = σ A ∈ X A (L/K ) : σ A X A (L/K ) = X A (L/K ) . Then ϒ A (L/K ) is a group and O A (K ) ⊆ ϒ A (L/K ) where O A (K ) is the stabilizer of K with the natural action of O A (V ). COROLLARY 2.1.4 The number of different nonzero representation masses of spinor genera in gen(L) for K is at most × F θ A X A (L/K ) : F × θ A ϒ A (L/K ) θ A O A (L) .

Proof By [X3, §1], the spinor genera in gen(L) which represent K can be naturally identified as the following group: X A (L/K )O + (V )O 0A (V )/O A (L)O + (V )O 0A (V ). By Proposition 2.1.3, the number of different nonzero representation masses is at most X A (L/K )O + (V )O 0A (V ) : ϒ A (L/K )O A (L)O + (V )O 0A (V ) . The result follows from applying θ A . In practice such ϒ A (L/K ) is not so easily determined. We see that the number of different nonzero representation masses is not always 2-power. Therefore the bound given in Corollary 2.1.4 by using ϒ A (L/K ) is not reasonable. Example 2.1.5 In the proof of [X3, Th. 1.5] and [HSX1, Th. 4.2], the construction of the sublattice N of M with codimension 1 follows from [HKK, Lem. 1.6]. Therefore one has θ A X A (M/N ) = θ A O A (M) θ A O A (N ) . By Corollary 2.1.4, the spinor genera in gen(M) which represent N have the same representation mass. If N is the characteristic lattice of M in the sense of [Ki], then X A (M/N ) = ϒ A (M/N ) = O A (M). There is only one spinor genus (in fact, one class) in gen(M) which represents N . 2.2. Mass formulas for spinor genera The mass formulas involving the quadratic characters were first given in [SP2]. In that case, there is only one nontrivial quadratic character (see Remark 2.2.4). However,


299

we have more choices of quadratic characters in codimension less than or equal to 1 cases. Let χ be a homomorphism χ : J F /F × θ A O A (L) −→ {±1}. (2.2.1) Then χ =

p∈ χp ,

Q

where χp is the homomorphism Fp× /θp O + (L p ) −→ {±1}

induced by χ . For p ∈ Spec(o F ), we set X ± (L p /K p , χp ) = σ ∈ X (L p /K p ) : χp θp (σ ) = ±1 and αp± (K p , L p , χp ) = λp X ± (L p /K p , χp ) where λp is the Tamagawa measure defined in §1.1. For any M ∈ gen(L), there is σ A ∈ O A (V ) such that M = σ A L. Then we define χ(spn+ (M)) = 0 if K is not represented by spn+ (M). Otherwise, we define χ(spn+ (M)) = χ (θ A (σ A )). THEOREM 2.2.2 Suppose the number of spinor genera in gen(L) is s(gen(L)). Then X χ spn+ (M) r K , spn+ (M) spn+ (M)∈gen(L)

s gen(L) = 2

Y

αp+ (K p , L p , χp ) − αp− (K p , L p , χp )

p∈Spec(o F )

for any quadratic character in (2.2.1). Proof It is clear that we only need to consider the terms where spn+ (M) represents K . So we can assume that K ⊆ M. Write M = σ A L. Then χ spn+ (M) r K , spn+ (M) −1 = m spn+ (M) χ θ A (σ A ) ] τ A ∈ X A (M/K )/O A (M) : τ A M ∈ spn+ (M) . There is a bijection τ A ∈X A (M/K )/O A (M) : τ A M ∈ spn+ (M) −→ u A ∈ X A (L/K )/O A (L) : u A L ∈ spn+ (M)

300

FEI XU

by sending τ A to τ A σ A . Therefore −1 X χ spn+ (M) r K , spn+ (M) = m spn+ (M) χ θ A (v A ) , vA

where v A runs over X A (L/K )/O A (L) with v A L ∈ spn+ (M). It is a standard fact that Y −1 −1 s gen(L) + + m spn (M) = m spn (L) = λp O + (L p ) 2 p∈Spec(o F )

(see §1.1). Since X (L p /K p )/O + (L p ) is trivial for the infinite primes, one has X χ spn+ (M) r K , spn+ (M) spn+ (M)∈gen(L)

−1 = m spn+ (L)

X

χ θ A (v A )

v A ∈X A (L/K )/O A (L)

X Y s gen(L) χp θp (vp ) λp O + (L p ) = 2 v A ∈X A (L/K )/O A (L) p∈Spec(o F ) Y X s gen(L) χp θp (σ ) λp O + (L p ) . = 2 + p∈Spec(o F ) σ ∈X (L p /K p )/O (L p )

The theorem follows immediately. Remark 2.2.3 It is a natural question whether one can obtain the mass formula for spinor genera (see [Ki, Chap. 6, Prob. 7]). When the codimension is greater than 2, or codimension 2 but one of the conditions of (1.1.3) and (1.1.7) does not hold, then the representation masses of spinor genera in a given genus are the same by [Kn2] and [We1]. Therefore the mass formula for spinor genera follows immediately (see [Sa, Th. 5]). When the codimension is 2 with conditions (1.1.3) and (1.1.7), then Y r K , spn+ (L) = αp+ (K p , L p ) − αp− (K p , L p ) p∈Spec(o F )

+

Y

αp+ (K p , L p ) + αp− (K p , L p )

p∈Spec(o F )

by Theorem 1.1.9 and Siegel’s mass formula. When the codimension is less than 2, one has Y 1X r K , spn+ (L) = αp+ (K p , L p , χp ) − αp− (K p , L p , χp ) 2 χ p∈Spec(o F )


301

Q by Theorem 2.2.2 and the standard facts on characters, where χ = p∈ χp runs over all characters in (2.2.1). It should be mentioned that all the results in this section do not depend on the fact that the spinor genera that represent K in gen(L) have the natural group structure (see [HSX1, Th. 2.1] and [X3, §1]). One can also interpret these χ in (2.2.1) as characters of Galois group of spinor class fields in [H3] by Artin map. Remark 2.2.4 In the codimension 2 case, one has J F : F × θ A O A (W ) θ A O A (L) ≤ 2. Under conditions (1.1.3) and (1.1.7), it is clear that there is only one nontrivial character Y χ= χp : J F /F × θ A O A (W ) −→ {±1}, p∈

where χp : Fp× /θp O + (Wp ) −→ {±1} with χp (a) = (a, − det(W ))p for any a ∈ Fp× . 2.3. Application For the application one needs to restrict the characters χ in (2.2.1) to the following subgroup: F × θ A X A (L/K ) /F × θ A O A (L) (2.3.1) denoted by χres . PROPOSITION 2.3.2 There are at least two distinct nonzero representation masses of spinor genera in Q gen(L) if and only if there is χ = p∈ χp in (2.2.1) with χres 6= 1 and Y αp+ (K p , L p , χp ) − αp− (K p , L p , χp ) 6= 0. p∈Spec(o F )

Proof We first rewrite the formula in Theorem 2.2.2 as follows: X χ spn+ (M) r K , spn+ (M) spn+ (M)

s gen(L) = 2

Y p∈Spec(o F )

αp+ (K p , L p , χp ) − αp− (K p , L p , χp ) ,

302

FEI XU

where spn+ (M) runs over all spinor genera in gen(L) which represent K . Since χres 6 = 1, half of χ(spn+ (M)) is 1 and the other half is −1. If Y αp+ (K p , L p , χp ) − αp− (K p , L p , χp ) 6= 0, p∈Spec(o F )

then there are at least two spinor genera spn+ (M) and spn+ (M 0 ) in gen(L) which represent K and r K , spn+ (M) 6 = r K , spn+ (M 0 ) by the above formula. Conversely, let a(K , gen(L)) be the number of spinor genera in gen(L) which represent K . Then we can rewrite the formula in Remark 2.2.3 as follows: r K , spn+ (σ A L) X Y s gen(L) χ θ A (σ A ) αp+ (K p , L p , χp ) − αp− (K p , L p , χp ) . = 2a K , gen(L) χ p∈Spec(o ) res

F

If Y

αp+ (K p , L p , χp ) − αp− (K p , L p , χp ) = 0

p∈Spec(o F )

for all χres 6 = 1, it is clear that r K , spn+ (L) = r K , spn+ (σ A L) for all σ A ∈ X A (L/K ) by the above formula. This is a contradiction. Now we study the local factors. Let χp : Fp× /θp O + (L p ) −→ {±1} be a local quadratic character. One has the following reduction steps for determining the local factors, which are similar to Lemma 1.1.10. LEMMA 2.3.3 Suppose L 0p is an o Fp -lattice of maximal rank with K p ⊆ L p , and suppose X (L 0p /K p ) = X (L p /K p ) for some p ∈ Spec(o F ). (1) If θp (O + (L 0p )) 6 ⊆ ker(χp ), then αp+ (K p , L p , χp ) = αp− (K p , L p , χp ). (2) Otherwise, αp± (K p , L p , χp ) = αp± (K p , L 0p , χp ).

Proof (1) Since θp (O + (L 0p )) 6⊆ ker(χp ), there is w0 ∈ O + (L 0p ) such that χp θp (w0 ) = −1.


303

Then X (L p /K p ) = X (L 0p /K p ) = X (w0 L 0p /K p ) = X (L 0p /K p )w0−1 = X (L p /K p )w0−1 . This implies that X + (L p /K p , χp ) = X − (L p /K p , χp )w0 . Therefore (1) follows. (2) It is trivial. For codimension less than or equal to 1, we also have the following result. 2.3.4 Suppose there is σ ∈ X (L p /K p ) such that LEMMA

σ X (L p /K p ) = X (L p /K p )

and

χp (θp (σ )) = −1.

Then αp+ (K p , L p , χp ) = αp− (K p , L p , χp ). In particular, when θp (O + (K p )) 6 ⊆ ker(χp ), then αp+ (K p , L p , χp ) = αp− (K p , L p , χp ). Proof It follows from X + (L p /K p , χp ) = σ X − (L p /K p , χp ). Let us recall that χp is called ramified if ker(χp ) 6 ⊇ up (Fp× )2 . This implies that θp (O + (L p )) 6⊇ up (Fp× )2 . One has the similar result as Proposition 1.2.2. 2.3.5 Suppose χp is ramified. Then PROPOSITION

αp+ (K p , L p , χp ) 6= αp− (K p , L p , χp ) if and only if θp X (L p /K p ) ⊆ ker(χp ). Proof It follows from the same arguments as those in Lemma 1.2.1, Proposition 1.2.2, and Lemma 2.3.3. Finally, we consider the following example. Example 2.3.6 In the proof of [X3, Thm. 1.5], one can modify the construction of the sublattice N0 of M in codimension zero so that any two different spinor genera in gen(M) have

304

FEI XU

different representation masses. In fact, we only need to modify the localization of N0 at p = qk with 1 ≤ k ≤ t. For such qk , Mqk can be split by a hyperbolic plane, that is, Mqk = (o Fqk xk + o Fqk yk ) ⊥ Pk

with Q(xk ) = Q(yk ) = 0 and B(xk , yk ) = 1.

Pk is unimodular. Our new N0 is defined as 2hqk

(N0 )qk = (qk

2hqk

x k + qk

yk ) ⊥ Pk

for 1 ≤ k ≤ t, where qk ’s are some integers. It is clear that such new N0 satisfies all the requirements in [X3, Th. 1.5]. By a simple computation, one has X Mqk /(N0 )qk = O (Mqk ) ∪

+

2hq [k

τql k O + (Mqk )

l=1

∪

2hq [k

σql k O + (Mqk ) ,

l=1

where σqk and τqk are trivial on Pk and σqk (xk ) = πqk xk ,

σqk (yk ) = πq−1 yk k

τqk (xk ) = πq−1 xk , k

τqk (yk ) = πqk yk

and with 1 ≤ k ≤ t. Since i k in the proof of [X3, Th. 1.5] with 1 ≤ k ≤ t are the base generators of the group like (2.3.1), there are the dual characters χk corresponding to i k for 1 ≤ k ≤ t, which are the base generators of the character group. Any character χ in (2.3.1) can be written as χ = χk1 · · · χks with 1 ≤ k1 < · · · < ks ≤ t. Let Y c= λp O + (Mp ) . p∈Spec(o F )

Then we have Y

αp+ (N0 )p , Mp , χk1 · · · χks − αp− (N0 )p , Mp , χk1 · · · χks

p∈Spec(o F )

Q c tk=1 (4hqk + 1) = (4hqk1 + 1) · · · (4hqks + 1) for s = 0, 1, . . . , t. Here we assume that χk1 · · · χks = 1 when s = 0. Since any element in (2.3.1) can be written as i k1 · · · i ks with 1 ≤ k1 < · · · < ks ≤ t, there is u k1 ···ks ∈ X A (M/N0 ) such that i k1 · · · i ks = θ A (u k1 ···ks ).


305

Suppose t (1 − xk1 ) · · · (1 − xks ) Y (1 + xk ) = (1 + xk1 ) · · · (1 + xks ) k=1

X

a j1 ··· jd x j1 · · · x jd .

1≤ j1 2. Also, L i are easily shown to coincide with the orbits of the original T n−1 -action on D (given by (20)). Consider now Dt and µ| Dt . The level set 6t = µ−1 (1/2n, . . . , 1/2n) ∩ Dt also has two smooth components. Clearly, those components are in the isotopy class of L 1t and L 2t correspondingly. Now a level set of any map from a compact to a noncompact manifold carries a trivial homology class. So L 1t and L 2t (with the corresponding orientations) carry the same homology class in Dt , and we are done. Remark 8 We have constructed special Lagrangian fibrations on large parts of quintics Dt near the large complex limit quintic D. If one forgets about the special condition and studies Lagrangian fibrations, then one can say more. In fact, W.-D. Ruan has constructed in [17] Lagrangian tori fibrations on general quintics and also constructed symplectic mirrors to those fibrations. 4.3. Complete intersection of two degree 3 hypersurfaces in CP 5 In this section we want to illustrate an application of Corollary 1. Let M¯ be CP 5 . We decompose its anticanonical bundle as ¯ ' L 1 ⊗ L 2. K ( M)

(22)

Here L 1 = L 2 = (γ ∗ )⊗3 (with γ ∗ being the hyperplane bundle of CP 5 ). We have a ¯ given by T 3 -action on M, (eiθ1 , eiθ2 , eiθ3 )(z 1 , . . . , z 6 ) = (eiθ1 z 1 , eiθ2 z 2 , eiθ3 z 3 , e−i(θ1 +θ2 ) z 4 , ei(θ1 +θ2 −θ3 ) z 5 , e−i(θ1 +θ2 ) z 6 ). ¯ L 1 , L 2 . Since the linear action of T on C6 is in SL(6, C), the T T acts on K ( M), action is equivariant with respect to the isomorphism (22). We have 4 monomials g1 = z 1 z 2 z 4 , g2 = z 3 z 4 z 5 , g3 = z 1 z 2 z 6 , g4 = z 3 z 5 z 6 , which can be viewed as T -invariant sections of L 1 and L 2 . We pick η1 and η2 to be some of their linear combinations such that the conditions of Corollary 1 hold. Thus we get SLag fibrations on large parts of a complete intersection of two hypersurfaces of degree 3 in CP 5 near (η1 , η2 ).

CALIBRATED FIBRATIONS VIA GROUP ACTIONS

339

5. Group actions and coassociative submanifolds 5.1. Construction of coassociative submanifolds Let O ' R8 be the algebra of Cayley numbers. The Cayley numbers O can be viewed as pairs of quaternions, and the multiplication · on O is given by ¯ da + bc). (a, b) · (c, d) = (ac − db, ¯

(23)

We refer the reader to [10, Appendix IV.A] and to [15, p. 51] for the discussion of the basic properties of O. The algebra O is neither commutative nor associative with respect to the product ·, but it has a unit e = (1, 0) with respect to this product; and we have an orthogonal decomposition O = span(e) ⊕ Im O, where Im O ' R7 are the imaginary Cayley numbers. On Im O we have a canonical G 2 3-form ϕ0 , ϕ0 (X, Y, Z ) = hX · Y, Z i

(24)

(see [10, p. 113]). Let G 2 be the group of linear automorphisms of Im O which preserve ϕ0 . Then the elements of G 2 are orthogonal matrices on Im O. Let M0 = Im O. We have a 4-form ∗ϕ0 on M0 . This 4-form is a calibration, and a submanifold L ⊂ M0 is calibrated by ∗ϕ0 if and only if ϕ0 | L = 0 (see [10, Th. IV.1.16]). Those submanifolds are called the coassociative submanifolds. Let M 7 be a 7-manifold. A G 2 -form on M is a smooth 3-form ϕ on M such that for each point m ∈ M there is an isomorphism σ between the tangent space Tm M and Im O ' R7 such that σ ∗ (ϕ0 ) = ϕ. (25) Here ϕ0 is the standard G 2 -form on Im O as in equation (24) (see [13, pp. 293–294]). For each m ∈ M the map σ : Tm M 7→ Im O as in equation (25) is defined up to a linear transformation of Im O contained in the group G 2 . Since G 2 preserves the inner product on Im O, it is clear that various maps σ as in equation (25) give rise to a welldefined Riemannian metric on M. We are interested in the situation when the form ϕ is parallel with respect to this metric. This happens if and only if dϕ = 0 and d ∗ϕ = 0 (see [18, Lem. 16.5]). In that case the metric is Ricci flat and the holonomy of M is contained in G 2 . The existence of metrics on 7-manifolds with holonomy equal to G 2 was unknown for quite a while. R. Bryant and S. Salamon have constructed some complete metrics with holonomy G 2 in [3]. Finally, Joyce has constructed in [13] the first example of a compact simply connected Riemannian 7-manifold with holonomy equal to G 2 . Let M be a 7-manifold with a G 2 -structure (i.e., the form ϕ is parallel). From the corresponding results on Im O it follows that the 4-form ∗ϕ is a calibration on M and a 4-dimensional submanifold L ⊂ M is calibrated by ∗ϕ if and only if ϕ| L = 0. L is called a coassociative submanifold. The deformation theory of coassociative

340

EDWARD GOLDSTEIN

submanifolds was studied by McLean in [16]. He has shown in [16, Th. 4.5] that if L is a compact coassociative submanifold of a G 2 -manifold M, then the moduli space of coassociative submanifolds through L is smooth of dimension b2− (L). Similarly to the SLag geometry, very little is known about global properties of a moduli space of compact coassociative submanifolds. We show that in the noncompact case one can use structure-preserving group actions on M to construct coassociative submanifolds. LEMMA 9 Let (M, ϕ) be a (connected) 7-manifold with a G 2 -structure. Suppose H 1 (M, R) = 0. Suppose we have an effective structure-preserving action of a compact 3-dimensional Lie group G on (M, ϕ) such that the action is not regular in at least one point in M. Let M 0 be the set of regular points of the G-action, and suppose G acts freely on M 0 . Then M 0 can be covered by a family of nonintersecting coassociative manifolds, diffeomorphic to G × R.

Proof Let v1 , v2 , v3 be an orthonormal basis of the Lie algebra G of G with respect to a bi-invariant metric on G , and let X i be the flow vector field corresponding to vi on M. Let f = i X 1 i X 2 i X 3 ϕ. By Lemma 1, f is G-invariant and d f = 0; that is, f is a constant. Also, f = 0 in at least 1 point. Hence f = 0. Let α = i X 1 i X 2 i X 3 (∗ϕ). By Lemma 1, α is a closed, G-invariant 1-form. So α = dg for a G-invariant function g on M. Let v = ∇g. Let m be a regular point of the action. Then v 6= 0 at m. Also, the scalar product of v and X i is zero, so v and X i are linearly independent and span a 4-dimensional space W in the tangent space Tm M. Using some Cayley algebra, one can easily show that W is a coassociative subspace of Tm M. We assume that G acts freely on the space M 0 of regular points of the action. The complement M − M 0 corresponds precisely to the critical points of g. Let l be a nonconstant trajectory of the gradient flow of g. Then l is contained in M 0 . Let L l = G × l. Then L l is coassociative. Also, l is an embedded 1-submanifold, and g is G-invariant and increases along l. From all this we deduce that L l is an embedded 0 submanifold. If l 0 is another trajectory of ∇g in M 0 , then clearly either L l and L l coincide or they do not intersect. Thus M 0 is covered by a family of nonintersecting coassociative submanifolds, diffeomorphic to G × R. We cannot in general say anything about the set of nonregular points of the action. We do this in one example in Section 5.2.


341

5.2. Examples Let M = Im O ' R7 with the standard G 2 -form ϕ0 . There is a nontrivial Sp1 ' SU(2) linear action on M which preserves ϕ0 (see [10, Sec. IV.3]). Harvey and Lawson have also constructed a family of coassociative Sp1 -invariant subvarieties in R7 (see [10, Th. IV.3.2]). Inside the set M 0 of the regular points of the Sp1 -action those subvarieties are smooth and coincide with the submanifolds arising from Lemma 9. So in particular, those subvarieties do not intersect in M 0 , but they do intersect in R7 . So we cannot quite get a coassociative fibration even in the case M = R7 . We now demonstrate a number of other examples where Lemma 9 applies. Bryant and Salamon have constructed in [3] a number of examples of complete G 2 -metrics. Some examples are on the total space of the spin bundle over a 3-dimensional space form. Others are on total space 32− of the bundle of anti-selfdual 2-forms over a self-dual Einstein 4-manifold. Many of those 3- and 4-manifolds admit isometric actions by SO(3) and by SU(2), and those actions induce structurepreserving actions on the corresponding G 2 -manifolds. We treat one example in detail—the total space of the spin bundle over S 3 . The spin bundle is a bundle V = T S 3 ⊕ R—the direct sum of the tangent bundle to S 3 with a trivial bundle. S 3 can be viewed as a unit sphere of the quaternions. There is an S 3 -action on itself, given by q( p) = qpq −1 . Here the multiplication is the quaternionic multiplication. Obviously, this action induces an S 3 / ± 1 = SO(3)action on S 3 and on V . In [3, pp. 836–841], a G 2 -structure was constructed on the total space of V . We do not reproduce the details of the construction but only mention that the fibers of the projection of V onto S 3 are coassociative. We look for coassociative submanifolds of V , invariant under the SO(3)-action. The points ±1 ∈ S 3 are fixed by the SO(3)-action, and the linear fibers over these points are SO(3)-invariant coassociative submanifolds L ±1 ⊂ V . Take now any point m ∈ (S 3 − ±1). The stabilizer of the SO(3)-action on m is a circle. Let Nm be the orthogonal complement in the tangent space Tm S 3 to the orbit of the SO(3)-action through m. Then W = Nm ⊕ R is a subbundle of V over (S 3 − ±1). W is invariant under the SO(3)-action. Let Am be the orbit of m ∈ S 3 − ±1 under the SO(3)-action (Am is diffeomorphic to S 2 ), and let L m be the total space of W over Am . One can easily show that L m is a coassociative submanifold, invariant under the SO(3)-action. The union of all L m and of L ±1 is precisely the set of nonregular points of the action. Also, SO(3) acts freely on the set of regular points of the action. By Lemma 9 the set of regular points is covered by a family of nonintersecting coassociative submanifolds. So the whole V is covered by nonintersecting, SO(3)-invariant coassociative submanifolds. Those are a 3-dimensional family of submanifolds diffeomorphic to SO(3) × R, a 1-dimensional family of submanifolds diffeomorphic to S 2 × R2 , and 2

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submanifolds diffeomorphic to R4 . Acknowledgments. This paper was a part of the author’s work toward his Ph.D. at the Massachusetts Institute of Technology. The author wants to express his gratitude to his advisor, Tom Mrowka, for initiating him into this subject and for continuing support. He is also grateful to Gang Tian and Shing-Tung Yau for a number of useful conversations. The author thanks the referee for correcting a mistake in Section 3.3. After writing this paper, the author learned that Mark Gross has independently obtained results that are similar to some results of Sections 2 and 3.3. References [1]

M. ATIYAH, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14

[2]

V. BATYREV and E. SELIVANOVA, Einstein-Kähler metrics on symmetric toric Fano

[3]

R. BRYANT and S. SALAMON, On the construction of some complete metrics with

(1982), 1–15. MR 83e:53037 330, 332 manifolds, J. Reine Angew Math. 512 (1999), 225–236. MR 2000j:32038 325

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

exceptional holonomy, Duke Math. J. 58 (1989), 829–850. MR 90i:53055 313, 339, 341 ´ E. CALABI, Metriques kählériennes et fibrés holomorphes, Ann. Sci. Ecole Norm. Sup. (4) 12 (1979), 269–294. MR 83m:32033 312, 326 J. DUISTERMAAT, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687–706. MR 82d:58029 311, 332 E. GOLDSTEIN, Calibrated Fibrations, preprint, arXiv:math.DG/9911093 310, 311, 334, 335 , Minimal Lagrangian tori in Kähler-Einstein manifolds, preprint, arXiv:math.DG/0007135 330 M. GROSS, Special Lagrangian fibrations, II: Geometry, preprint, arXiv:math.AG/9809072 311, 332 V. GUILLEMIN and S. STERNBERG, Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge, 1984. MR 86f:58054 315, 320 R. HARVEY and H. B. LAWSON, Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 85i:53058 309, 310, 321, 339, 341 W. Y. HSIANG, On the compact homogeneous minimal submanifolds, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 5–6. MR 34:5037 328 D. JOYCE, Compact Riemannian 7-manifolds with holonomy G 2 , I, J. Differential Geom. 43 (1996), 291–328. MR 97m:53084 , Quasi-ALE metrics with holonomy SU(m) and Sp(m), Ann. Global Anal. Geom. 19 (2001), 103–132. MR 1 826 397 325, 332, 339 Y. KARSHON and S. TOLMAN, Centered complexity one Hamiltonian torus actions, preprint, arXiv:math.SG/9911189 312, 323 H. B. LAWSON and L. M. MICHELSOHN, Spin Geometry, Princeton Math. Ser. 38, Princeton Univ. Press, Princeton, 1989. MR 91g:53001 339


343

[16]

R. MCLEAN, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998),

[17]

W.-D. RUAN, Lagrangian tori fibration of toric Calabi-Yau manifold, III: Symplectic

705–747. MR 99j:53083 309, 311, 335, 340

[18]

[19] [20] [21] [22] [23]

topological SYZ mirror construction for general quintics, preprint, arXiv:math.DG/9909126 338 S. SALAMON, Riemannian Geometry and Holonomy Groups, Pitman Res. Notes Math. Ser. 201, Longman Sci. Tech., Harlow, England, 1989. MR 90g:53058 326, 331, 339 S. SALUR, Deformations of special Lagrangian submanifolds, Commun. Contemp. Math. 2 (2000), 365–372. MR CMP 1 776 986 311 A. STROMINGER, S.-T. YAU, and E. ZASLOW, Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), 243–259. MR 97j:32022 311, 331 G. TIAN and S.-T. YAU, Complete Kähler manifolds with zero Ricci curvature, II, Invent. Math. 106 (1991), 27–60. MR 92j:32028 325 ¨ , Classification of G-manifolds of complexity 1, Izv. Math. 61 (1997), D. A. TIMASHEV 363–397. MR 98h:14058 312, 323 S. T. YAU, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math 31 (1978), 339–411. MR 81d:53045 310

Department of Mathematics, Stanford University, Stanford, California 94305, USA; [email protected]


SYMMETRIC GROUPS AND THE CUP PRODUCT ON THE COHOMOLOGY OF HILBERT SCHEMES MANFRED LEHN AND CHRISTOPH SORGER

Abstract Let C (Sn ) be the Z-module of integer-valued class functions on the symmetric group Sn . We introduce a graded version of the convolution product on C (Sn ), and we show that there is a degree-preserving ring isomorphism C (Sn ) −→ H ∗ (Hilbn (A2C ); Z) to the cohomology of the Hilbert scheme of points in the complex affine plane. 1. Introduction In this paper we relate a geometric and a group-theoretic incarnation of the bosonic Fock space P = Q[ p1 , p2 , p3 , . . .]. On the geometric side this is the direct sum M H= H ∗ Hilbn (A2C ); Q n≥0

of the rational cohomology of the Hilbert schemes of generalized n-tuples in the complex affine plane, and on the group-theoretic side this is the direct sum M C = C (Sn ) ⊗Z Q n≥0

of the spaces of class functions on the symmetric groups Sn . In addition to their vertex algebra structures, both H and C carry natural ring structures on each component of fixed conformal weight: for the cohomology of the Hilbert schemes this is the ordinary topological cup product; for C it is a certain combinatorial cup product to be defined below (see (1)) and not to be confused with the usual product on the representation ring arising from the tensor product of representations. With respect to these products we can state our main theorem. THEOREM 1.1 The composite isomorphism of vertex algebras 8

9

C −→ P −→ H DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 2, Received 9 September 2000. Revision received 5 November 2000. 2000 Mathematics Subject Classification. Primary 14C05, 14C15, 20B30; Secondary 17B68, 17B69, 20C05. Lehn’s work supported by the University of Nantes. 345

346

LEHN AND SORGER

induces for each conformal weight n ∈ N0 an isomorphism of graded rings C (Sn ) −→ H ∗ Hilbn (A2C ); Z . That the composition 98 is an isomorphism of vertex algebras is by now well known (see [10] for 9 and [5] for 8). The emphasis of the theorem is on the multiplicativity of this map. Based on the geometric analysis carried out in [9], the proof is purely algebraic. Our starting point was the observation of I. Frenkel and W. Wang [6] that I. Goulden’s differential operator 1 in [7] is closely related to the operator ∂ of [9]. 2. The cup product on the group ring Z[Sn ] Let C (Sn ) denote the set of integer-valued class functions on the symmetric group Sn , that is, the set of functions Sn → Z which are constant on conjugacy classes. P Identifying a function f with the linear combination π ∈Sn f (π )π , we may think of C (Sn ) as a Z-submodule of the group ring Z[Sn ]. As such it inherits a product X ( f ∗ g)(π) = f (πσ −1 )g(σ ), σ ∈Sn

called the convolution product. Remark 2.1 The character map χ : R (Sn ) ⊗Z Q −→ C (Sn ) ⊗Z Q is a Q-linear isomorphism from the rational representation ring to the ring of rational class functions. The tensor product ring structure on R (Sn ) is quite different from the convolution product structure on C (Sn ), so that χ is not a ring homomorphism. Even though we use the identification of R (Sn ) ⊗Z Q and C (Sn ) ⊗Z Q in the definition of the vertex algebra L structure on n≥0 C (Sn ) ⊗Z Q, we never use the tensor product ring structure in this paper. An integral basis of C (Sn ) is given by the characteristic functions X χλ = π, π of type λ

where λ is a partition of n and where π runs through all permutations with cycle type λ, that is, those having a disjoint cycle decomposition with cycle lengths λ1 , λ2 , . . . , λs . For instance, the unit element of the group ring Z[Sn ]—and of C (Sn ) —is χ[1,1,...,1] . For any partition λ, let `(λ) denote the length of λ. We introduce a gradation Z[Sn ] =

n−1 M d=0

Z[Sn ](d)

CUP PRODUCT

347

as follows: a permutation π has degree deg(π) = d if it can be written as a product of d transpositions but not less. Equivalently, if π is of cycle type λ, then deg(π ) = n − `(λ). In particular, the maximal possible degree is indeed n − 1. The product in Z[Sn ] does not preserve this gradation, but it is clearly compatible with the associated filtration M F d Z[Sn ] := Z[Sn ](d 0 ); d 0 ≤d

that is, it satisfies F i Z[Sn ] ∗ F j Z[Sn ] ⊂ F i+ j Z[Sn ]. L d d−1 Z[S ] is called The induced product on Z[Sn ] = gr F Z[Sn ] = n−1 n d=0 F Z[Sn ]/F cup product and denoted by ∪. Explicitly, ( σ ∗ π if deg(σ ) + deg(π) = deg(σ π ), σ ∪π = (1) 0 else, for π, σ ∈ Sn . Clearly, the subring of class functions C (Sn ) ⊂ Z[Sn ] is generated by homogeneous elements and inherits from Z[Sn ] gradation, filtration, and, most importantly, the cup product. L Let C := n≥0 C (Sn ) ⊗Z Q. It is bigraded by conformal weight n and degree. 3. The ring of symmetric functions Let P = Q[ p1 , p2 , p3 , . . .] denote the polynomial ring in countably infinitely many variables. It is endowed with a bigrading by letting pm have conformal weight m and cohomological degree m − 1. Let Pn denote the component of conformal weight n, P that is, the subspace spanned by all monomials p1α1 · . . . · psαs with i iαi = n. Define linear maps 8n : Q[Sn ] −→ Pn by sending a permutation π of cycle type λ = (λ1 ≥ λ2 ≥ · · · ≥ λs ) to the monomial (1/n!) pλ1 · . . . · pλs . Thus for any partition λ = (1α1 2α2 . . .) we have Y 1 pi αi . 8n (χλ ) = αi ! i i

In particular, there is an isomorphism of bigraded vector spaces 8 : C −→ P . Moreover, multiplication by pm in P corresponds to linear operators rm in C which are given as follows (see [5] and the references therein): let Ind denote induction of class functions. Then rm is the map id ⊗mχ(m)

Ind

rm : C (Sn ) −−−−−−→ C (Sn ) ⊗ C (Sm ) = C (Sn × Sm ) −→ C (Sn+m ).

348

LEHN AND SORGER

The map r1 is in fact very easy to describe: let ι : Q[Sn ] → Q[Sn+1 ] be induced from the standard inclusion Sn → Sn+1 . Then r1 extends to a map Q[Sn ] → Q[Sn+1 ],

π 7→

1 X tι(π )t −1 . n! t∈Sn+1

In [7], Goulden introduces the following differential operator on P : 1 := 10 + 100 :=

1X ∂ ∂ 1X ∂ i j pi+ j + (i + j) pi p j , 2 ∂ pi ∂ p j 2 ∂ pi+ j i, j

i, j

and he proves the following. PROPOSITION 3.1 (Goulden) Let τn ∈ C (Sn ) denote the sum of all transpositions in Sn . Then 8(τn ∗ y) = 1 8(y)

for all y ∈ Q[Sn ]. Proof See [7, Prop. 3.1]. Note that 10 is of bidegree (0, 1) and that 100 is of bidegree (0, −1). Since τn is of degree 1, for the cup product introduced above, Goulden’s proposition reads as follows: 8(τn ∪ y) = 10 8(y) . (2) 4. The Hilbert scheme of points Consider the Hilbert scheme Hilbn (A2C ) of generalized n-tuples of points on the affine plane. Let us recall some basic facts: Hilbn (A2C ) is a quasi-projective manifold of dimension 2n (see [4]). The closed subset Hilbn (A2C ) O = ξ ∈ Hilbn (A2C ) | Supp(ξ ) = O ∈ A2C is a deformation retract of Hilbn (A2C ). This subvariety is (n − 1)-dimensional and irreducible (see [1]) and has a cell decomposition with p(n, n − i) cells of dimension i, where p(n, j) denotes the number of partitions of n into j parts (see [2]). In particular, the odd-dimensional cohomology vanishes, there is no torsion, and H 2i (Hilbn (A2C ); Z) = Z p(n,n−i) . In order not to worry constantly about a factor of 2, we agree to give H 2i (Hilbn (A2C ); Z) degree i.

CUP PRODUCT

349

Consider the bigraded vector space M M H := H 2i Hilbn (A2C ); Q . 0≤n 0≤i0

m>0

Then for any y ∈ P one has ch(pr1∗ O4n ) ∪ 9(y) = 9 D (y) .

(3)

Proof See [9, Th. 4.10]. The degree 1 part of the operator D is equal to −10 ; hence c1 (pr1∗ O4n ) ∪ 9(y) = −9 10 (y) .

(4)

In order to simplify notation we write ∂(y) := c1 (pr1∗ O4n ) ∪ y

for

y ∈ H ∗ Hilbn (A2C ); Q .

We can combine the identities (2) and (4) to obtain −∂98(y) = 9 10 8(y) = 98(τn ∪ y). 4.2 For each n we have PROPOSITION

(5)

350

(1) (2) (3)

LEHN AND SORGER

C (Sn ) ⊗Z Q = τn ∪ (C (Sn ) ⊗Z Q) + r1 (C (Sn−1 ) ⊗Z Q), Pn = 10 (Pn ) + p1 Pn−1 , H ∗ (Hilbn (A2C ); Q) = ∂ H ∗ (Hilbn (A2C ); Q) + q1 H ∗ (Hilbn−1 (A2C ); Q).

Proof In view of (5) and the fact that 8 and 9 are isomorphisms, the three assertions are of course equivalent. Assertion (2) follows from the identities [10 , p1 ] =

X j>0

j p j+1

∂ ∂pj

and ad([10 , p1 ])n−1 ( p1 ) = (n − 1)! pn by an easy induction. THEOREM 4.3 For all y ∈ H ∗ (Hilbn (A2C ); Q) one has

γn+1 ∪ q1 (y) − q1 (γn ∪ y) = [∂, q1 ](γn ∪ y).

(6)

Proof This is [9, Th. 4.2]. 5. The alternating character Let εn ∈ C (Sn ) denote the alternating character, that is, X εn = sgn(π)π. π∈Sn

PROPOSITION 5.1 The following identity holds for all y ∈ C (Sn ) ⊗Z Q:

εn+1 ∪ r1 (y) − r1 (εn ∪ y) = −τn+1 ∪ r1 (εn ∪ y) + r1 (τn ∪ εn ∪ y).

(7)

Proof First, note that we have the identities τn+1 − ι(τn ) =

n X (i n + 1) i=1

and

εn+1 − ι(εn ) = −

n X

(i n + 1) ∪ ι(εn ),

i=1

which together give εn+1 − ι(εn ) = −τn+1 + ι(τn ) ∪ ι(εn ).

(8)

CUP PRODUCT

351

Then n! εn+1 ∪ r1 (y) − r1 (εn ∪ y) X X = εn+1 ∪ tι(y)t −1 − tι(εn ∪ y)t −1 t

t

X = t εn+1 − ι(εn ) ι(y)t −1

since εn+1 is symmetric

t

=

X t −τn+1 + ι(τn ) ∪ ι(εn ∪ y)t −1

by (8)

t

= −τn+1

X

tι(εn ∪ y)t −1 +

t

X

tι(τn ∪ εn ∪ y)t −1

t

= n! −τn+1 ∪ r1 (εn ∪ y) + r1 (τn ∪ εn ∪ y) . 5.2 The following identities hold: PROPOSITION

X n≥0

8(εn )z n = exp

X

(−1)m−1

m>0

z m X −1 pm = 9 (γn )z n . m

(9)

n≥0

Proof The first equality can be found in [6]. The second equality is [9, Th. 4.6]. Thus under the isomorphism 98 : C → H the alternating character εn is mapped to the total Chern class γn of the tautological sheaf pr1∗ O4n . PROPOSITION 5.3 For all y ∈ C (Sn ) ⊗Z Q the following identity holds:

98(εn ∪ y) = γn ∪ 98(y).

(10)

Proof Because of Proposition 4.2 we may assume that y is of the form τn ∪ x or r1 (x). We therefore argue by induction on weight and degree, and we assume that the assertion holds for all x of either less degree or less weight than y. (The assertion is certainly trivial for the vacuum, the—up to a scalar factor—unique element of weight zero and degree zero.)

352

LEHN AND SORGER

In case y = τn ∪ x it follows from (5) and induction that 98(εn ∪ y) = 98(εn ∪ τn ∪ x) = ∂ 98(εn ∪ x) = ∂ γn ∪ 98(x) = γn ∪ ∂ 98(x) = γn ∪ 98(τn ∪ x) = γn ∪ 98(y). In case y = r1 (x) for some x ∈ C (Sn−1 ) we argue as follows: 98(εn ∪ y) = 98 εn ∪ r1 (x) = 98 r1 (εn−1 ∪ x) − τn ∪ r1 (εn−1 ∪ x) +r1 (τn−1 ∪ εn−1 ∪ x) by (7) = q1 98(εn−1 ∪ x) + ∂q1 98(εn−1 ∪ x) − q1 ∂ 98(εn−1 ∪ x) by (5) = (q1 + ∂ ◦ q1 − q1 ◦ ∂) γn−1 ∪ 98(x) by induction = γn ∪ q1 98(x) by (6) = γn ∪ 98 r1 (x) = γn ∪ 98(y). 6. Proof of Theorem 1.1 It follows from Proposition 5.3 that the assertion of Theorem 1.1 holds for rational coefficients: from [3] we know that the Chern classes of pr1∗ O4n generate the ring H ∗ (Hilbn (A2C ); Z). Hence Proposition 5.3 implies that the isomorphism 98 : C (Sn ) ⊗Z Q −→ H ∗ Hilbn (A2C ); Q preserves the cup product and that the homogeneous components of the alternating character εn ∈ C (Sn ) generate C (Sn ) ⊗Z Q. Thus in order to prove Theorem 1.1 it suffices to see that this holds as well over the integers. 6.1 The homogeneous components of the alternating character εn generate the ring C (Sn ) of integer-valued class functions with respect to the cup product. PROPOSITION

Remark 6.2 Of course, this implies the analogous statement for C (Sn ) equipped with the convolution product.

CUP PRODUCT

353

Proof Let εn (i) denote the component of εn of degree i. We must show that for each d ≥ 0 the elements Y εn (i)αi , εnλ := i≥1

(1α1 2α2

where λ = . . .) runs through all partitions of d, form a set of generators of the Z-module C (Sn )(d). Claim ρ The restriction map C (Sn ) − → C (Sn−1 ) is a surjective and degree-preserving ring homomorphism and maps εn to εn−1 . In particular, we may assume that n ≥ 2d. Indeed, surjectivity and homogeneity are obvious; hence it is enough to check that ρ is multiplicative. We have ρ(g ∪ f )(π) =

=

∼ X

g(πσ −1 ) f (σ )

σ ∈Sn ∼ X

g(πσ

σ ∈Sn−1

−1

) f (σ ) +

∼ X

g(π σ −1 ) f (σ ),

σ ∈Sn \Sn−1

∼ P where means the sum over terms satisfying the degree condition (1). The first term of the last line equals (ρ(g) ∪ ρ( f ))(π ), and it suffices to show that no summand of the second term occurs. Indeed, we may decompose any σ ∈ Sn \Sn−1 as σ = (i n)η for some transposition (i n) with i ∈ {1, . . . , n − 1} and η ∈ Sn−1 . The degree-matching condition requires deg(π) = deg πη−1 (i n) + deg (i n)η . (11)

But the right-hand side equals deg(πη−1 ) + 1 + deg(η) + 1 ≥ deg(π) + 2,

(12)

so that (11) is never fulfilled. Finally, it is clear that ρ(εn ) = εn−1 , which proves the claim. Note that ρ is not multiplicative with respect to the convolution product. Assume from now on that n ≥ 2d, and consider the Z-module M = 8(C (Sn )(d)) ⊂ P . It has a Z-basis consisting of monomials pnλ :=

Y 1 pi αi0 = 8(χλ0 ), αi0 ! i i≥1

(13)

354

LEHN AND SORGER

where as before λ = (1α1 2α2 . . .) is a partition of d and where λ0 = (1α1 2α2 . . .) is the P associated partition of n given by α10 := n − d − i>0 αi and αi0 := αi−1 for i ≥ 2. Here the assumption n ≥ 2d ensures that α10 ≥ 0. On the other hand, consider Y ci (pr1∗ O4n )αi . γnλ := 8(εnλ ) = 9 −1 (14) 0

0

i≥1

By the definition of the cup product, the elements γnλ are all contained in M and µ can therefore be expressed as linear combinations of the pn . We must show that the associated coefficient matrix is invertible over Z. This is achieved by comparison with a third, rational basis of the vector space M ⊗Z Q, provided by the elements Y (15) chλn := 9 −1 chi (pr1∗ O4n )αi . i≥1

Claim P µ Let A be the matrix defined by chλn = µ`d Aµλ γn . Then Y

| det A | =

Y

λ=(1α1 2α2 ...)`d i≥1

1 (i − 1)!

αi

.

(16)

Let < be the order on the set of partitions λ = (1α1 2α2 . . .) of d corresponding to the lexicographical order of the sequences (α1 , α2 , . . .), so that, for example, the partition [1, 1, . . . , 1] = (1d ) is the largest and [d] = (d 1 ) is the smallest. Since Chern classes and the components of the Chern character satisfy the universal identities chk =

(−1)k−1 ck + polynomials in c1 , . . . , ck−1 , (k − 1)!

it follows that chλn

=

Y (−1)i−1 αi i≥1

(i − 1)!

γnλ + linear combination of γnµ with µ > λ.

This shows that A is a lower triangular matrix with diagonal entries Aλλ =

Y (−1)i−1 αi i≥1

The claim follows directly from this.

(i − 1)!

.

CUP PRODUCT

355

Claim P µ Let B be the matrix defined by chλn = µ`d Bµλ pn . Then Y Y 1 αi | det B | = αi !. i! α1 α2

(17)

λ=(1 2 ...)`d i≥1

Recall that by Theorem 4.1 we have 9 −1 chi (pr1∗ O4n ) ∪ 9(y) = Di (y) for any polynomial y ∈ P , where the degree i component Di of the differential operator D is given by Di =

(−1)i (i + 1)!

X

pn 0 +···+n i n 0

n 0 ,...,n i >0 β

∂ ∂ · · · ni . ∂ pn 0 ∂ pn i

(18)

β

If Di is applied to a monomial p1 1 · . . . · ps s with β1 > i, then the smallest component with respect to the lexicographical order is that arising from the choice n 0 = · · · = n i = 1 in (18). More precisely,   β −i−1 i Y 1 p j β j p1 1  = (−1) (βi+1 + 1) Di  βj! j i! (β1 − i − 1)! j≥1

pi+1 βi+1 +1 1 × (βi+1 + 1)! i + 1 β j Y pj 1 × + terms of higher order. βj! j j6 =1,i+1

It follows by induction that Y α pn λ 1 chn = Di i n! i≥1 Y (−1)αi αi · pnλ + linear combinations of pnµ with µ0 > λ0 . = αi ! i! i

This shows that B is a lower triangular matrix—if we reorder the pnλ according to µ λ :⇔ µ0 > λ0 —with diagonal entries Y (−1)αi αi Bλλ = αi ! . i! i≥1

The claim follows from this.

356

LEHN AND SORGER

Claim We have

det A det B =

Y

Y i αi = 1. αi !

λ=(1α1 2α2 ...)`d i≥1

Of these two equalities the first is an immediate consequence of the two previous claims. The second is a well-known identity. In fact, it amounts to realizing that each integer k ∈ {1, . . . , d} appears both in the numerator and denominator with multiplicity p(d − k) + p(d − 2k) + p(d − 3k) + · · · , where p(s) is the number of partitions of s. Remark 6.3 We have seen that for any λ = (1α1 2α2 . . .) of d there is a polynomial rλ ∈ Z[c1 , c2 , . . .] of (weighted) degree d such that 98(χλ0 ) = rλ c1 (pr1∗ O4n ), . . . , cd (pr1∗ O4n ) whenever n ≥ 2d. On the other hand, the kernel of the restriction map ρ : C (Sn+1 ) → C (Sn ) is generated by all χλ0 , where the coefficient α10 in the presentation λ0 = (1α1 2α2 . . .) vanishes. This yields the following description of the cohomology ring in terms of generators and relations: H ∗ Hilbn (A2C ); Z = Z[c1 , c2 , . . .]/(rλ ){λ | Pi (i+1)αi >n} . 0

0

The polynomials rλ can be explicitly computed in C (Sn ) and have a direct geometric interpretation. So, for example, among the first relations that appear is r(2m ) , where m = d(n + 1)/2e, reflecting the fact that the locus of points in Hilbn (A2C ) where more than n/2 pairs of points collide is empty. Acknowledgment. Lehn gratefully acknowledges the support and the hospitality of the University of Nantes, where this paper was written. Note added in proof. E. Vasserot [11] has independently obtained a similar result using other methods. References [1]

J. BRIANC ¸ ON, Description de Hilbn C{X, Y }, Invent. Math. 41 (1977), 45–89.

MR 56:15637 348

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[2] [3] [4] [5]

[6] [7]

[8] [9]

[10] [11]

357

G. ELLINGSRUD and S. A. STRØMME, On the homology of the Hilbert scheme of points

in the plane, Invent. Math. 87 (1987), 343–352. MR 88c:14008 348 , Towards the Chow ring of the Hilbert scheme of P2 , J. Reine Angew. Math. 441 (1993), 33–44. MR 94i:14004 349, 352 J. FOGARTY, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521. MR 38:5778 348 I. B. FRENKEL, N. JING, and W. WANG, Vertex representations via finite groups and the McKay correspondence, Internat. Math. Res. Notices 2000, 195–222. MR 2001c:17042 346, 347 I. FRENKEL and W. WANG, Virasoro algebra and wreath product convolution, preprint, arXiv:math.QA/0006087. 346, 351 I. P. GOULDEN, A differential operator for symmetric functions and the combinatorics of multiplying transpositions, Trans. Amer. Math. Soc. 344 (1994), 421–440. MR 95c:20019 346, 348 I. GROJNOWSKI, Instantons and affine algebras, I: The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275–291. MR 97f:14041 349 M. LEHN, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), 157–207. MR 2000h:14003 346, 349, 350, 351 H. NAKAJIMA, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), 379–388. MR 98h:14006 346, 349 E. VASSEROT, Sur l’anneau de cohomologie du schéma de Hilbert de C2 , preprint, arXiv:math.AG/0009127. 356

Lehn Mathematisches Institut der Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany; [email protected] Sorger Mathématiques (Unité Mixte Recherche 6629 du Centre National de la Recherche Scientifique), Université de Nantes BP 92208, 2 rue de la Houssinière, F-44322 Nantes CEDEX 03, France; [email protected]


STANDARD CONJECTURES FOR THE ARITHMETIC GRASSMANNIAN G(2, N ) AND RACAH POLYNOMIALS ANDREW KRESCH AND HARRY TAMVAKIS

Abstract We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian G = G(2, N ) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding of G in projective space. The analysis of the Hodge index inequality involves estimates on values of certain Racah polynomials. 0. Introduction Let X be an arithmetic variety, by which we mean a regular, projective, and flat scheme over Spec Z, of absolute dimension d + 1. Assume that M = (M, k · k) is a hermitian line bundle on X which is arithmetically ample, in the sense of [Z] and [So, §5.2]. For each p > 0 the line bundle M defines an arithmetic Lefschetz operator p p+1 b d d L :C H (X )R −→ C H (X )R ,

α 7 −→ α · b c1 (M). ∗

d Here C H (X )R is the real arithmetic Chow ring of [GS], and b c1 (M) is the arithmetic first Chern class of M. In this setting H. Gillet and C. Soulé [GS] proposed arithmetic analogues of Grothendieck’s standard conjectures (see [Gr]) on algebraic cycles. A more precise version of the conjectures was formulated in [So, §5.3]; assuming 2 p 6 d + 1, the statement is the following. 1 (Hard Lefschetz) The map

CONJECTURE

(a)

p d+1− p b d d L d+1−2 p : C H (X )R −→ C H (X )R

DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 2, Received 4 October 2000. Revision received 12 December 2000. 2000 Mathematics Subject Classification. Primary 14G40; Secondary 14C17, 14M15, 33C45. Authors’ work partially supported by National Science Foundation Postdoctoral Research Fellowships. 359

360

(b)

KRESCH AND TAMVAKIS

is an isomorphism. p d (Hodge index) If the nonzero x ∈ C H (X )R satisfies b L d+2−2 p (x) = 0, then d xb (−1) p deg L d+1−2 p (x) > 0.

Notice that Conjecture 1 (for all p) implies that the intersection pairing p d+1− p d d C H (X )R ⊗ C H (X )R −→ R

is nondegenerate. When d = 1, Conjecture 1 follows from the Hodge index theorem for arithmetic surfaces due to G. Faltings [Fa] and P. Hriljac [Hr]. We study these conjectures when X = G(r, N ) is the arithmetic Grassmannian, parametrizing r -dimensional subspaces of an (r + N )-dimensional vector space, over any field, and M = O (1) is the very ample line bundle giving the Plücker embedding, equipped with its natural hermitian metric. The latter is the metric induced from the standard metric on complex affine space, so that the first Chern form c1 (M) on X (C) is U (r + N )-invariant and dual to the hyperplane class. Our main result is that, for G(r, N ) and O (1), Conjecture 1 holds when r = 2. For projective space (r = 1) this was shown by K. Künnemann [Ku]. Moreover, it is proved in [KM] and [Ta] that Conjecture 1 holds for G(r, N ) after a suitable scaling of the metric on O (1). To obtain the precise result for G(2, N ), we use the arithmetic Schubert calculus of [T] and linear algebra to reduce the problem to combinatorial estimates. In this case the inequality in Conjecture 1(b) asserts the positivity of a linear combination of harmonic numbers with coefficients certain Racah polynomials. The latter are a system of orthogonal polynomials in a discrete variable introduced by J. Wilson [Wi], [AW] which generalize the classical Racah coefficients or 6- j symbols (see [Ra]) of quantum physics. The results of Künnemann [Ku] show that each statement in Conjecture 1 (for p d given X , p, and M) is true if and only if it holds when C H (X )R is replaced by the Arakelov subgroup C H p (X )R associated to the Kähler form c1 (M). We therefore restrict attention to this subgroup throughout the paper. In §1 we study arithmetic Lefschetz theory for varieties that admit a cellular decomposition and derive a cohomological criterion (Corollary 1) which we use to check Conjecture 1. This criterion does not suffice to check the Hodge index inequality on more general Grassmannians. In §2 we apply classical and arithmetic Schubert calculus to reduce the conjecture for G(2, N ) to estimates for a class of Racah polynomials. The required bounds for these polynomials are established in §3. 1. Arithmetic standard conjectures on cellular spaces We study Conjecture 1 for arithmetic varieties X which have a cellular decomposition over Spec Z, in the sense of [Fu, Exam. 1.9.1]; the Grassmannian G(r, N ) is a typical

STANDARD CONJECTURES FOR ARITHMETIC GRASSMANNIANS

361

example (see [KM] for more information on these spaces and an approach to a weaker version of the conjecture). Recall that for each p the class map cl : C H p (X )R −→ H p, p (X R ) is an isomorphism of the real Chow ring C H p (X )R = C H p (X ) ⊗Z R with the space H p, p (X R ) of real harmonic differential ( p, p)-forms on X (C). We denote by L : C H p (X )R −→ C H p+1 (X )R , α 7−→ α · c1 (M) the classical Lefschetz operator associated to an ample line bundle M over X . Let us equip the holomorphic line bundle M(C) with a smooth positive hermitian metric, invariant under complex conjugation, to obtain a hermitian line bundle M. As we have indicated, to check Conjecture 1 for the operator b L(α) = α · b c1 (M) it suffices to work with the Arakelov Chow group C H p (X )R defined using the Kähler form c1 (M). Since X has a cellular decomposition, we have an exact sequence e a

ζ

0 −→ C H p−1 (X )R −→ C H p (X )R −→ C H p (X )R −→ 0

(1)

(see [KM, Prop. 6]). Here e a = a ◦ cl is the composite of the class map with the natural inclusion a : H p−1, p−1 (X R ) ,→ C H p (X )R , and ζ is the projection defined in [GS, §1]. We choose a splitting s p : C H p (X )R −→ C H p (X )R for the sequence (1) and thus arrive at a direct sum decomposition C H p (X )R ∼ = C H p (X )R ⊕ C H p−1 (X )R

(2)

for every p. Summing (1) over all p produces a sequence e a

ζ

0 −→ C H ∗−1 (X )R −→ C H ∗ (X )R −→ C H ∗ (X )R −→ 0

(3)

which is compatible with the actions of L and b L. The splitting s := ⊕ p s p of (3) does not commute with b L in general. Rather, the image of b L ◦ s − s ◦ L is contained in Ker(ζ ); hence b L ◦s −s ◦ L =e a ◦U (4) for a uniquely defined degree-preserving linear operator U on C H ∗ (X )R . We now give some conditions equivalent to the arithmetic hard Lefschetz theorem (Theorem 1). When checking these for G(2, N ), we obtain something stronger, which

362

KRESCH AND TAMVAKIS

establishes the arithmetic Hodge index theorem as well; this is quantified in Theorem 2. Recall the classical Lefschetz decomposition on C H m (X )R ' H 2m (X (C), R): M p C H m (X )R = L m− p C Hprim (X )R , p>0

where the group of primitive codimension p classes is p C Hprim (X )R = Ker L d+1−2 p : C H p (X )R → C H d+1− p (X )R . For m = d − p this decomposition induces a projection map d− p

π p : C H d− p (X )R −→ C Hcoprim (X )R , d− p

p

where C Hcoprim (X )R = L d−2 p C Hprim (X )R . THEOREM 1 Let X be an arithmetic variety of dimension d + 1 which admits a cellular decomposition. Let b L be the arithmetic Lefschetz operator associated to an ample hermitian line bundle on X . Then the following statements are equivalent. b (i) L d+1−2 p : C H p (X )R −→ C H d+1− p (X )R is an isomorphism for all p. b : C H ∗ (X )R −→ C H ∗−1 (X )R such that for every (ii) There exists a linear map 3 p b, b L]α = (d + 1 − 2 p)α. p and α ∈ C H (X )R we have [3 (iii) For some (equivalently, any) choice of splitting s of (3), with U as in (4), the map

δ p := π p

d−2 Xp

p

d− p

L d−2 p−i U L i : C Hprim (X )R −→ C Hcoprim (X )R

i=0

is an isomorphism for all p. Proof The equivalence of (i) and (ii) follows from standard Lefschetz theory; this is described, for example, in [Kl, §4]. Now consider the commutative diagram with exact rows 0

/ C H p−1 (X )

R

0

e a

L d+1−2 p

/ C H d− p (X )

R

e a

ζ

/ C H p (X )

R

b L d+1−2 p

/ C H d+1− p (X )

R

ζ

/ C H p (X )R

/0

L d+1−2 p

/ C H d+1− p (X )

R

/0


363

By classical Lefschetz theory, the left vertical map is injective and the right vertical map is surjective. The snake lemma gives an exact sequence δ

p

d− p

0 → Ker(b L d+1−2 p ) → C Hprim (X )R −→ C Hcoprim (X )R → Coker(b L d+1−2 p ) → 0. The connecting homomorphism δ is characterized by the property that e a ◦δ = b L d+1−2 p ◦ s modulo the subspace e a (Ker(π p )). Note also that (4) implies b Lk ◦ s − s ◦ Lk = e a

k−1 X

L k−1−i U L i

i=0

for all k. We deduce that δ coincides with the map δ p in (iii) and hence that statements (i) and (iii) are equivalent. This also shows that δ p does not depend on the splitting s of (3) and the associated linear operator U on C H ∗ (X )R . Remark. Assuming statement (iii), it is possible to give an explicit construction of the b in (ii) as follows. We first claim that there exists a splitting s 0 of (3), with map 3 p associated operator U 0 , such that for any p and α ∈ C Hprim (X )R we have U 0 L i α = 0 for all i < d − 2 p

and

d− p

U 0 L d−2 p α ∈ C Hcoprim (X )R .

(5)

Indeed, if we let D be the linear transformation such that s 0 − s = e a ◦ D, then U 0 = U + [L , D] and it is an exercise to check that the space of transformations [L , D] is equal to the Pd−2 p set of operators V on C H ∗ (X )R satisfying π p i=0 L d−2 p−i V L i (α) = 0 for all p α ∈ C Hprim (X )R and every p. Now choose a primitive basis for C H ∗ (X )R , and apply s 0 to get half of a basis for C H ∗ (X )R . By (iii), we may apply (L d−2 p )−1 π p U 0 L d−2 p to the basis elements p in C Hprim (X )R for each p to obtain another basis for C H ∗ (X )R , which we view (via p e a ) as the other half of our basis for C H ∗ (X )R . Let v ∈ C Hprim (X )R be one of the 0 basis elements, and let r = d − 2 p. By our conditions on s , a subset of our basis for C H ∗ (X )R consists of b v := s 0 (v), the iterates b L i (b v ) = s 0 (L i v) of b L applied to b v , the r 0 primitive element w satisfying L (w) = π p (U L r (v)), and the iterates of b L applied to w: b v, b Lb v, . . . , b Lrb v , w, Lw, . . . , L r w. (6) The action of b L is to send each element in (6) to the element on its right, except that b b by Lrb v is sent to L r w and L r w to zero. We now define 3 b (b 3 L ib v ) = i(r + 2 − i)b L i−1b v, b(L i w) = (r + 1)b 3 L ib v + i(r − i)L i−1 w.

364

KRESCH AND TAMVAKIS

b (defined this way for every basis element v) satisfies the condition of (ii). Then 3 2 p−1 Suppose the arithmetic variety X and p are such that C Hprim (X )R = 0. If, for each p nonzero α ∈ C Hprim (X )R , we have THEOREM

(−1)

p

d−2 Xp Z i=0

L d−2 p−i α ∧ U L i α > 0,

(7)

X

then the statements in the arithmetic hard Lefschetz and Hodge index conjectures are true for that X , p, and M. Proof Let (α, β) ∈ C H p (X )R be a nonzero element of the kernel of b L d+2−2 p ; the notation (α, β) refers to direct sum decomposition (2), with respect to some splitting. We claim p that α must be in C Hprim (X )R . Indeed, b L d+2−2 p (α, β) = (L d+2−2 p α, γ ) for some p−1

γ , and L d+2−2 p α = 0 implies L d+1−2 p α = 0 since C Hprim (X )R vanishes. Also, by the classical hard Lefschetz theorem, α 6 = 0. Now if h , i : C H ∗ (X )R ⊗ C H ∗ (X )R −→ R denotes the arithmetic intersection pairing, then we have X

(α, β) , b L d+1−2 p (α, β) = (α, β) , (0, L d−2 p−i U L i α + L d+1−2 p β) i

1X = 2 i

Z L

d−2 p−i

α ∧ U L i α.

X

p−1 Hence, assuming C Hprim (X )R = 0, we have b L d+1−2 p (α, β) 6= 0 for every nonzero (α, β) ∈ C H p (X )R . Moreover, if (α, β) is primitive, then the pairing of (α, β) with b L d+1−2 p (α, β) has the required sign.

COROLLARY 1 Suppose X is such that, for every p, p

C Hprim (X )R 6 = 0

implies

p−1

C Hprim (X )R = 0.

(8)

p

If condition (7) holds for every p and each nonzero α ∈ C Hprim (X )R , then both the arithmetic hard Lefschetz and Hodge index conjectures are true for X , M. Proof p−1 If C Hprim (X )R = 0, then the conjectures hold for X , M, and p by Theorem 2.


365

p−1

Assume now that C Hprim (X )R 6 = 0. It suffices to prove the Hodge index inequality for a nonzero x ∈ C H p (X )R in the kernel of b L d+2−2 p . Since, by hypothesis, we p b have C Hprim (X )R = 0, it follows that x = L(y) + e a (η) for some η ∈ C H p−1 (X )R and y ∈ C H p−1 (X )R , with y 6= 0 and b L d+4−2 p (y) = 0. Moreover, the condition d+2−2 p d+2−2 p b L (x) = 0 implies b L (e a (η)) = −b L d+3−2 p (y). Now we find

x, b L d+1−2 p (x) = − y, b L d+3−2 p (y) , and the required Hodge index inequality is a consequence of the degree p − 1 case of Theorem 2. Example We illustrate the previous results for projective space Pn and the canonical hermitian line bundle O (1) (cf. [Ku, §4]). In this case we choose the splitting C H ∗ (Pn )R =

n M

R·b ωi ⊕

i=0

n M

R · ωi ,

i=0

a (c1 (O (1))i ). Then sequence (6) is given by where b ωi = b c1 (O (1))i and ωi = e b 1, b ω, . . . , b ωn , τn , τn ω, . . . , τn ωn . P Here τn = nk=1 Hk , where each Hk = 1 + 1/2 + · · · + 1/k is a harmonic number. b for the arithmetic The remark after Theorem 1 exhibits an explicit adjoint map 3 b Lefschetz operator L(x) = b ω · x; in our example it is given by b(b 3 ωi ) = i(n + 2 − i) b ωi−1 , n+1 i b(ωi ) = b ω + i(n − i) ωi−1 . 3 τn Observe that the nonzero primitive elements of C H ∗ (Pn )R are multiples of 1 ∈ C H 0 (Pn )R ; hence Pn satisfies (8). The operator U is given by U (ωi ) = δi,n τn ωn , and condition (7) for p = 0, α = 1 becomes n Z X i=0

Pn

ω

n−i

∧ U (ω ) = τn i

Z Pn

ωn =

n X

Hk > 0.

(9)

k=1

The arithmetic Hodge index conjecture for Pn and O (1) follows by applying Corollary 1. A hermitian line bundle M = (M, k · k) on an arithmetic variety X is arithmetically ample if M is an ample invertible sheaf on X such that the first Chern form c1 (M(C))

366

KRESCH AND TAMVAKIS

is nonnegative on X (C) and for all nonempty irreducible closed subsets Y ⊂ X the height h M (Y ) is positive (see [So, §5.2]). We say that M is a limit for arithmetic ampleness if (i) (M, t k · k) is arithmetically ample for all positive scalars t < 1, and (ii) h M (Y ) = 0 for some (nonempty) irreducible closed Y ⊂ X . The line bundle O (1) on Pn considered above is a limit for arithmetic ampleness. Indeed, if Y ∈ Z 1 (Pn ) is the cycle attached to the rational point [1 : 0 : · · · : 0], then h O (1) (Y ) = 0 (see, e.g., [BGS, (3.1.6)]). Furthermore, property (i) for O (1) follows from the argument in [BGS, Prop. 3.2.4]. On the other hand, one sees that arithmetic Hodge index inequality (9) does not fail when the natural metric on O (1) is scaled by a factor t ∈ (1 − , 1 + ) for small > 0 (see also [BGS, Prop. 3.2.2]). Observe, however, that (9) becomes sharp at t = 1 if we insist that it should hold for any positive real constants Hk (following the point of view in [T, §6]). Further evidence for this statement on more general Grassmannians is given in §3. 2. The arithmetic Grassmannian G(2, N ) In this section we study Conjecture 1 for the Grassmannian of lines in projective space. For computational purposes we work with the isomorphic Grassmannian G = G(N , 2) parametrizing N -planes in (N + 2)-space throughout. Note that d = dimC G(C) = 2N . There is a universal exact sequence of vector bundles 0 −→ S −→ E −→ Q −→ 0 over G; the complex points of E and Q are metrized by giving the trivial bundle E(C) the trivial hermitian metric and the quotient bundle Q(C) the induced metric. The hermitian vector bundles that result are denoted E and Q. The real vector space C H ∗ (G)R ∼ = H 2∗ (G(C), R) decomposes as M C H ∗ (G)R = R · sa,b (Q), a,b

summed over all partitions λ = (a, b) with a 6 N , that is, whose Young diagrams are contained in the 2 × N rectangle (N , N ). Moreover, sλ (Q) = sa,b (Q) is the characteristic class coming from the Schur polynomial sa,b in the Chern roots of Q; this is dual to the class of a codimension |λ| = a + b Schubert variety in G. In the following, s1 denotes the Schur polynomial s1,0 , and it is thus just the first elementary symmetric function. The line bundle M = det(Q) giving the Plücker embedding has c1 (M) = s1 (Q); let L : C H p (G)R → C H p+1 (G)R be the associated classical Lefschetz operator. Further, for all p, let ∗ : C H p (G)R → C H 2N − p (G)R denote the Hodge star operator induced by the Kähler form s1 (Q). We then have the following.


367

PROPOSITION 1 p The space C Hprim (G)R is nonzero if and only if p = 2k 6 N . In the latter case, it is 1-dimensional and spanned by the class

αk =

k X j=0

(−1)

j

N +1− j N − 2k

N − 2k + j s2k− j, j (Q). N − 2k

Proof By computing the Betti numbers for G, one sees that dim C H p (G)R − dim C H p−1 (G)R > 0 if and only if p = 2k 6 N , and in this case the above difference equals 1. For such p we have Ker L : C H 2N −2k (G)R → C H 2N −2k+1 (G)R

k nX o = Span (−1) j s N − j,N −2k+ j (Q) . (10) j=0

One checks (10) easily using the Pieri rule: L sa,b (Q) = sa+1,b (Q) + sa,b+1 (Q), where it is understood that sc,c0 (Q) = 0 if c < c0 or c > N . From [KT] we know the action of the Hodge star operator on C H ∗ (G)R is given by (a + 1)! b! ∗ sa,b (Q) = s N −b,N −a (Q). (11) (N − a)! (N − b + 1)! Since 2k (G)R = ∗ Ker L : C H 2N −2k (G)R → C H 2N −2k+1 (G)R , C Hprim the proof is completed by applying (11) to (10) and noting that the result is proportional to αk . We now pass to the arithmetic setting, where we use the arithmetic Schubert calculus of [T, §§3, 4]. The real Arakelov Chow group C H p (G)R decomposes as M M C H p (G)R = R ·b sa,b (Q) ⊕ R · sa 0 ,b0 (Q). (12) a+b= p

a 0 +b0 = p−1

Here the indexing sets satisfy N > a > b > 0, b sa,b (Q) is an arithmetic characteristic class, and we identify the harmonic differential form sa 0 ,b0 (Q) with its image

368

KRESCH AND TAMVAKIS

in C H p (G)R . The decomposition (12) is induced by the splitting map sa,b (Q) 7−→ b sa,b (Q) which agrees with the one used in [T]. The hermitian line bundle M has b c1 (M) = b s1 (Q), and it is a limit for arithmetic ampleness; the latter property follows as in the remarks at the end of §1. We now apply the arithmetic Pieri rule of [T, §4] to compute the action of the arithmetic Lefschetz operator b L(x) = b s1 (Q) · x on the above basis elements. The induced operator U : ∗ C H (G)R → C H ∗ (G)R of (4) satisfies U (sa,b ) = 0 for a < N and U (s N ,b ) =

+1 NX

b(NX −b)/2c Hi s N ,b − (H N −b+1−i − Hi )s N −i,b+i .

i=0

(13)

i=0

Here and in the rest of this section, sa,b denotes the Schubert class sa,b (Q) ∈ C H a+b (G)R , and Hi is a harmonic number; by convention H0 = 0. Recall that the classical intersection pairing on C H ∗ (G)R satisfies Z 0 0 hsa,b , sa ,b i = sa,b ∧ sa 0 ,b0 = δ(a,b),(N −b0 ,N −a 0 ) . G

The sequence of Betti numbers for G shows that G satisfies condition (8) of Corollary 1. We proceed to check inequality (7) for all even p = 2k; this establishes Conjecture 1 for G(N , 2). In our case (7) may be written as 6(N , k) :=

Z

NX −2k

L N −2k−b αk ∧ U L N −2k+b αk > 0.

G b=0

To compute iterates of the classical Lefschetz operator L on the Schubert basis, note that X L r sµ = f λ/µ sλ . (14) λ⊃µ |λ|=|µ|+r

When λ = (λ1 , λ2 ) and µ = (µ1 , µ2 ) are partitions with at most two parts, the skew f -number in (14) satisfies |λ| − |µ| |λ| − |µ| f λ/µ = − . (15) λ1 − µ1 λ1 − µ2 + 1 This follows from the determinantal formula for f λ/µ , given, for example, in [St, Cor. 7.16.3].


369

We now apply Proposition 1 and (14), (15) to calculate L c−2k αk =

k X N +1− j N − 2k + j c−2k (−1) j L s2k− j, j N − 2k N − 2k j=0

k X X

N − 2k + j N +1− j = (−1) N − 2k N − 2k j=0 i> j c − 2k c − 2k · − sc−i,i . i−j i − 2k − 1 + j j

Therefore, L c−2k αk =

X

(−1) j

i, j

N +1− j N − 2k

N − 2k + j N − 2k

c − 2k sc−i,i . i−j

(16)

We use (16) with c = N + b to identify the coefficient of s N ,b in the expansion of L N +b−2k αk as D E X N − 2k + j N + b − 2k N +b−2k j N +1− j L αk , s N −b = (−1) N − 2k j N − 2k + j j X N − 2k + b N +1− j j b = (−1) N − 2k j N − 2k j N + 1 − b N − 2k + b = =: Cb . 2k + 1 N − 2k It now follows from (13) that U L N −2k+b αk = Cb

+1 NX

b(NX −b)/2c Hi s N ,b − Cb (H N −b+1−i − Hi )s N −i,b+i . (17)

i=0

i=0

Note also that we have the identity NX −2k b=0

2N − 2k + 2 Cb = . N +2

(18)

Now we substitute c = N − b in (16), pair with (17), and use (18) to sum over b and obtain −b)/2c N +1 NX −2k b(NX X 6(N , k) = A N ,k Hi + C Nb,i,k , (19) i=1

where

A N ,k =

N +1 N − 2k

b=0

i=0

2N − 2k + 2 N +2

370

KRESCH AND TAMVAKIS

and C Nb,i,k

=

X

(−1)

j

j

N +1− j N − 2k

N − 2k + j N − 2k

N − 2k − b Cb (Hi −H N −b+1−i ). i−j

By dividing the expression for C Nb,i,k into two sums and substituting in (19), one gets 6(N , k) = A N ,k

N +1 X

Hi +

i=1

N +1 X

B Ni ,k Hi ,

(20)

i=1

where B Ni ,k

X N − 2k + j N − 2k − b j N +1− j = (−1) Cb . N − 2k N − 2k i−j j,b

(Notice that when N − b = 2r − 1 is odd, there is a missing summand (for i = r ) X N +1− j N − 2k + j 2r − 2k − 1 (−1) j Cb Hr N − 2k N − 2k r−j j

which vanishes, as can be seen by the change of variable j 7→ 2k + 1 − j.) At this point it is convenient to introduce the variable change n = N − 2k

and

T = N + 2,

and to write (20) in the new coordinates as 6(n, T ) = An,T

T −1 X i=1

Observe that X n+ i Bn,T = (−1) j n j X n+ = (−1) j n j

Hi +

T −1 X

i Bn,T Hi .

(21)

i=1

T −1− j X n−b T −1−b n+b n i−j n−b n b j T −1− j T −1−n+i − j T +n . n i−j n−i + j j

We now substitute r = i − j, and we write the resulting sum in hypergeometric notation (see [Ro], [VK, Chap. 3]): X T −1+r −i T −1−n +r T +n i i r n +i −r (−1) Bn,T = (−1) n n r n −r r n+i T +n T −1−i −n, −i, T − n, T − i = 1 . 4 F3 −n − i, T + 1, T − n − i n n n


The Whipple transformation in [Wh, §10] applied to the above 4 F3 gives i Bn,T −n, n + 1, −i, i + 1 (−1)i = 4 F3 1 . 1, 1 + T, 1 − T An,T

371

(22)

The hypergeometric term (22) belongs to a class of orthogonal polynomials called Racah polynomials, which are studied in the following section. 3. Bounds for Racah polynomials The Racah coefficients or 6- j symbols (see [Ra]) have long been used by physicists as the transformation coefficients between two different coupling schemes of three angular momenta (see [BL] for an exposition). In mathematical language they are the entries of a change of basis matrix for the tensor product of three irreducible representations of SU(2); the two bases involved come from the associativity relation for this product (see [VK, §8.4]). It was recognized later by Wilson [Wi] that these coefficients are special cases of a class of orthogonal polynomials Rn (x; α, β, γ , δ), called Racah polynomials (see [VK, §8.5]): −n, n + α + β + 1, −s, s + γ + δ + 1 Rn s(s+γ +δ+1); α, β, γ , δ = 4 F3 1 . α + 1, β + δ + 1, γ + 1 The Racah polynomials in (22) have α = β = γ + δ = 0, with γ = T , a positive integer. We let Rn (s, T ) = Rn s(s + 1); 0, 0, T, −T . Observe that Rn (s, T ) is symmetric in n and s. The orthogonality condition (see [VK] or [AW]) reads T −1 X

(2s + 1)Rn (s, T )Rm (s, T ) =

s=0

T2 δnm . (2n + 1)

(23)

The arithmetic Hodge index inequality 6(n, T ) > 0 can be rephrased using (21) and (22) as T −1 T −1 X X s+1 (−1) Rn (s, T )Hs < Hs . (24) s=1

s=1

We give a proof of (24) which does not depend on the precise values of the harmonic numbers. Let us say that a sequence {Hk }k>1 of positive real numbers (with H0 = 0) Pk is concave increasing if Hk = i=1 h i for some monotone decreasing sequence {h i } of positive reals. THEOREM 3 Let {Hk } be any concave increasing sequence of real numbers and n, T integers with 0 6 n 6 T − 1 and T > 3. Then inequality (24) holds.

372

KRESCH AND TAMVAKIS

We believe that, in fact, (24) holds for an arbitrary sequence of positive real numbers Hk ; that is, the arithmetic standard conjectures for G(2, N ) do not depend on the relative sizes of the harmonic numbers involved. CONJECTURE 2 For any integers n, s with 0 6 n, s 6 T − 1, we have |Rn (s, T )| 6 1.

In Proposition 2 we check this conjecture for some values of n near the endpoints zero and T − 1. Computer calculations support the validity of Conjecture 2 for general n. Proof of Theorem 3 We see that (23) implies (24) except when T is exponentially large compared to n. For large T the Racah polynomials are close approximations of classical orthogonal polynomials, in this case the Legendre polynomials, and we know how to bound these. By Cauchy’s inequality, (23) gives −1 TX

|Rn (s, T )| Hs

s=0

2

6

T −1 T 2 X Hs2 . 2n + 1 2s + 1 s=0

So, (24) holds whenever T −1 X s=0

−1 1 TX 2 Hs2 < (2n + 1) Hs . 2s + 1 T

(25)

s=0

Since {Hk } is concave increasing, the average value of H0 , . . . , HT −1 is at least PT −1 HT −1 /2. As s=1 2/(2s + 1) 6 log T , the inequality (25) holds whenever 1 log T < n + . 2

(26)

To analyze the case where T is exponentially large compared to n, it is convenient to introduce the change of variable x = s(s + 1) = −1/4 + T 2 (1 + t)/2

(27)

and the rescaling pn (t) = (−1)n

n Y T 2 − i2 Rn (x; 0, 0, T, −T ). T2 i=1

(0,0)

Let Pn (t) = Pn §3.8]) that

(t) denote the nth Legendre polynomial. It is known (see [NSU, pn (t) = Pn (t) + O(1/T 2 ),


373

where the constant in the error term depends on both n and t. For our purposes we demonstrate the following. 1 Let n and T be positive integers such that 1 + 2n + 2n 2 < T 2 /10. Then

LEMMA

(a)

| pn (t) − Pn (t)| 6 (3/2) · 4n /T 2

(b)

(28)

for all t with −1 6 t 6 1. We have | pn (t) − Pn (t)| 6 1/10 whenever T > 90 and n < log T .

Proof We have the following recurrences (see [NSU]; for Pn (t) this is classical): n+1 n Pn+1 (t) + Pn−1 (t), (29) 2n + 1 2n + 1 n+1 2n 2 + 2n + 1 n 2 2 n t pn (t) = pn+1 (t) − pn−1 (t), p (t) + 1 − n 2n + 1 2T 2 T 2 2n + 1 (30)

t Pn (t) =

with initial data P0 (t) = 1 ;

P1 (t) = t,

p0 (t) = 1 ;

p1 (t) = t + 1/(2T 2 ).

(31)

Subtracting (29) from (30) leads to a recurrence in pn (t) − Pn (t). Then (28) follows by induction on n, using the known bound |Pn (t)| 6 1 for all n and all t with −1 6 t 6 1. The statement (b) is a corollary of (a). PROPOSITION 2 We have |Rn (s, T )| 6 1 when n 6 3 or n = T − 1.

Proof For n = 0 we have R0 (s, T ) = 1. When n = 1, we see from (31) that p1 (t) is an increasing linear function in t, attaining minimum when s = 0, giving R1 (0, T ) = 1, and attaining maximum when s = T − 1, giving R1 (T − 1, T ) = (1 − T )/(1 + T ); hence the inequality holds. For n = T − 1 the Pfaff-Saalschütz identity (see [Ro],

374

KRESCH AND TAMVAKIS

[VK, §8.3.3]) gives s s+ j T RT −1 (s, T ) = (−1) T+j j j j −s, s + 1, T = 3 F2 1 1, T + 1 (1 − T )(2 − T ) · · · (s − T ) = , (1 + T )(2 + T ) · · · (s + T ) X

j

so the inequality is clear. When n = 2 or n = 3, pn (t) is a quadratic or cubic polynomial, and it is a calculus exercise to check that |Rn (s, T )| 6 1 for every integer s with 0 6 s 6 T − 1. In fact, the integrality condition on s is required only when n = 3, T = 4. 2 We have |Pn (t)| 6 3/4 for t ∈ R, |t| 6 0.9, and √ n > 2. For T > 10 we have |t| 6 0.9 in (27) whenever 5/10 6 s/T 6 4/5. Qn Assume T > 90 and n < log T . Then (1/T 2n ) i=1 (T 2 − i 2 ) > 40/41.

LEMMA

(a) (b) (c)

Proof The indicated bound on Legendre polynomials is evident for n = 2, and for larger n it follows from the inequality r √ 2 , 0 6 θ 6 π. (32) sin θ |Pn (cos θ)| < πn √ One obtains (32) by using the transformed differential equation for sin θ Pn (cos θ) (see [Sz, (4.24.2)]); this is indicated in [Ho, Chap. 5, Exer. 15–16]. The proofs of (b) and (c) are routine; for the latter, one may use the inequality − log T −2n

n n Y 2 X 2 (T 2 − i 2 ) 6 2 i . T i=1

i=1

To complete the proof of Theorem 3, assume that (26) fails, so that n < log T − 1/2. If T 6 90, then n 6 3 and (24) follows from Proposition 2. (Note that the inequality in the proposition is strict unless n = 0 or s = 0.) When T > 90, we combine Lemma 1(b) with Lemma 2 to deduce the inequality (24). Indeed, (40/41)|Rn (s, T )| is bounded by 1 + 1/10 for every s, and by 3/4 + 1/10 over the middle half of the summation range. By pairing terms Hs with HT −1−s and by using the fact that Hs + HT −1−s is monotone increasing for 0 6 s 6 (T − 1)/2, we obtain (24).


375

Acknowledgments. The authors would like to thank Christophe Soulé for suggesting that we study the arithmetic standard conjectures on the Grassmannian and for general encouragement. Thanks are also due to Klaus Künnemann, Jennifer Morse, and Herb Wilf for helpful discussions, and to the anonymous referee for an improvement in the proof of Theorem 1. References [AW]

[BL]

[BGS] [Fa] [Fu] [GS]

[Gr] [Ho] [Hr] [Kl]

[Ku] [KM]

[KT]

R. ASKEY and J. WILSON, A set of orthogonal polynomials that generalize the Racah

coefficients or 6- j symbols, SIAM J. Math. Anal. 10 (1979), 1008–1016. MR 80k:33012 360, 371 L. C. BIEDENHARN and J. D. LOUCK, The Racah-Wigner Algebra in Quantum Theory, Encyclopedia Math. Appl. 9, Addison-Wesley, Reading, Mass., 1981. MR 83d:81002 371 J.-B. BOST, H. GILLET, and C. SOULE´ , Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903–1027. MR 95j:14025 366 G. FALTINGS, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), 387–424. MR 86e:14009 360 W. FULTON, Intersection Theory, 2d ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998. MR 99d:14003 360 H. GILLET and C. SOULE´ , “Arithmetic analogs of the standard conjectures” in Motives (Seattle, Wash., 1991), Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, 1994, 129–140. MR 95j:14026 359, 361 A. GROTHENDIECK, “Standard conjectures on algebraic cycles” in Algebraic Geometry (Bombay, 1968), Oxford Univ. Press, London, 1969, 193–199. MR 42:3088 359 H. HOCHSTADT, The Functions of Mathematical Physics, Pure Appl. Math. 23, Wiley-Interscience, New York, 1971. MR 58:17241 374 P. HRILJAC, Heights and Arakelov’s intersection theory, Amer. J. Math. 107 (1985), 23–38. MR 86c:14024 360 S. L. KLEIMAN, “The standard conjectures” in Motives (Seattle, Wash., 1991), Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, 1994, 3–20. MR 95k:14010 362 ¨ K. KUNNEMANN , Some remarks on the arithmetic Hodge index conjecture, Compositio Math. 99 (1995), 109–128. MR 97f:14025 360, 365 ¨ K. KUNNEMANN and V. MAILLOT, “Théorèmes de Lefschetz et de Hodge arithmétiques pour les variétés admettant une décomposition cellulaire” in Regulators in Analysis, Geometry and Number Theory, Progr. Math. 171, Birkhäuser, Boston, 2000, 197–205. MR 2000m:14030 360, 361 ¨ K. KUNNEMANN and H. TAMVAKIS, The Hodge star operator on Schubert forms, to appear in Topology. 367

376

[NSU]

[Ra] [Ro] [So]

[St] [Sz] [Ta] [T] [VK]

[Wh] [Wi] [Z]

KRESCH AND TAMVAKIS

A. F. NIKIFOROV, S. K. SUSLOV, and V. B. UVAROV, Classical Orthogonal Polynomials

of a Discrete Variable (in Russian), “Nauka,” Moscow, 1985, MR 87f:33001; English translation in Springer Ser. Comput. Phys., Springer, Berlin, 1991. MR 92m:33019 372, 373 G. RACAH, Theory of complex spectra, II, Physical Rev. 62 (1942), 438–462. 360, 371 R. ROY, Binomial identities and hypergeometric series, Amer. Math. Monthly 94 (1987), 36–46. MR 88f:05012 370, 373 C. SOULE´ , “Hermitian vector bundles on arithmetic varieties” in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62, Part 1, Amer. Math. Soc., Providence, 1997, 383–419. MR 99e:14030 359, 366 R. P. STANLEY, Enumerative Combinatorics, Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge Univ. Press, Cambridge, 1999. MR 2000k:05026 368 ¨ , Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. G. SZEGO Math. Soc., Providence, 1975. MR 51:8724 374 Y. TAKEDA, A relation between standard conjectures and their arithmetic analogues, Kodai Math. J. 21 (1998), 249–258. MR 99m:14047 360 H. TAMVAKIS, Schubert calculus on the arithmetic Grassmannian, Duke Math. J. 98 (1999), 421–443. MR 2000j:14040 360, 366, 367, 368 N. JA. VILENKIN and A. U. KLIMYK, Representation of Lie groups and Special Functions, Vol. 1: Simplest Lie Groups, Special Functions and Integral Transforms, trans. V. A. Groza and A. A. Groza, Math. Appl. (Soviet Ser.) 72, Kluwer, Dordrecht, 1991. MR 93h:33009 370, 371, 374 F. J. W. WHIPPLE, Well-poised series and other generalized hypergeometric series, Proc. London Math. Soc. (2) 25 (1926), 525–544. 371 J. A. WILSON, Hypergeometric series, recurrence relations and some new orthogonal functions, Ph.D. thesis, University of Wisconsin, Madison, 1978. 360, 371 S. ZHANG, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), 187–221. MR 95c:14020 359

Kresch Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395, USA; [email protected] Tamvakis Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395, USA; current: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110, USA; [email protected]


TORSION-FREE GENUS ZERO CONGRUENCE SUBGROUPS OF PSL2 (R) ABDELLAH SEBBAR

Abstract We study and classify all the conjugacy classes of the genus zero congruence subgroups of PSL2 (R) with no elliptic elements. We show that it suffices to classify those inside the modular group and determine them completely. We also discuss an application to modular curves. Contents 1. Introduction . . . . . . . . . . . . . . . 2. Discrete subgroups of PSL2 (R) . . . . . . 3. The torsion-free and genus zero conditions 4. A transfer theorem . . . . . . . . . . . . 5. Larcher congruence groups . . . . . . . . 6. Classification inside the modular group . . 7. Classification inside PSL2 (R) . . . . . . 8. Subgroups containing 00 (n) . . . . . . . 9. A special case . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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377 379 380 382 384 385 386 389 393 395

1. Introduction Since the appearance of Moonshine more than two decades ago, much interest has been drawn to genus zero congruence groups. In [13], J. Thompson showed that there are only finitely many conjugacy classes of particular genus zero congruence subgroups of PSL2 (R) using group-theoretic methods. Using Thompson’s results and spectral properties of automorphic functions, P. Zograf [14] showed that there are only finitely many genus zero congruence subgroups of the modular group. However, besides finiteness results and examples, nothing has been explicitly said about the number of these groups (in PSL2 (R)), and no description of their nature has been given. In this paper, we deal with an aspect of this subject; namely, we describe completely DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 2, Received 7 April 2000. Revision received 3 November 2000. 2000 Mathematics Subject Classification. Primary 11F06, 20H05; Secondary 11F03. 377

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the genus zero congruence subgroups of PSL2 (R) that contain no elliptic elements. This work was motivated by previous papers in collaboration with John McKay [6], [7] in which we studied the action of the Schwarzian derivative on automorphic functions that generate the function field of a genus zero discrete subgroup of PSL2 (R). It turns out that the Schwarzian derivative of such an automorphic function is a weight 4 automorphic form for the normalizer of the discrete subgroup in PSL2 (R). Moreover, these automorphic forms are holomorphic if and only if the group has no elliptic elements. If we specialize to genus zero congruence subgroups of PSL2 (R) which do not have elliptic elements, in other words, which are torsion-free, then the automorphic forms obtained coincide with theta-functions of some familiar rank 8 lattices. This phenomenon motivated the present classification to enable a better understanding of the situation. One of the main results of this paper is the following. THEOREM 1 There are exactly 15 conjugacy classes of torsion-free genus zero congruence subgroups of PSL2 (R).

These 15 conjugacy classes are explicitly determined in terms of classical congruence groups. This theorem is a consequence of the following. 2 Every torsion-free genus zero discrete subgroup of PSL2 (R) commensurable with the modular group is conjugate to a subgroup of PSL2 (Z). THEOREM

We then show that if a subgroup of PSL2 (R) is a congruence group, then any of its conjugates inside PSL2 (Z) (if it has any) is also a congruence group. Having transferred the problem inside the modular group, we need to classify torsion-free genus zero congruence subgroups of PSL2 (Z). We have the following. 3 Up to a modular conjugacy, there are 33 torsion-free genus zero congruence subgroups of PSL2 (Z). THEOREM

Again, these groups are described explicitly in terms of classical congruence groups. The proof of Theorem 3 is carried out by studying the cusp widths of the groups. Using work of H. Larcher [4], [5], each congruence group has a conjugate by an element of the modular group which contains a simply described group with which it shares the same cusp shape (i.e., set of cusp widths). Furthermore, if the conjugate is

TORSION-FREE GENUS ZERO CONGRUENCE SUBGROUPS

379

torsion-free and of genus zero, then it coincides with a Larcher group with which it shares the same cusp shape. This makes it easy to classify all the PSL2 (Z)-conjugacy classes of torsion-free and genus zero congruence subgroups of the modular group. We show that the 33 modular conjugacy classes of Theorem 3 are partitioned into the 15 PSL2 (R) conjugacy classes of Theorem 1. We also deal with the subgroups that contain a conjugate of 00 (n) for some n, which occur in Moonshine. Section 9 deals with a special case that leads to a geometric application concerning the modularity of P1 \ {0, 1, ∞, z}. 2. Discrete subgroups of PSL2 (R) The content of this section is common knowledge and is based generally on [9] or [12]. Let PSL2 (R) be the group of Möbius transformations τ→

aτ + b , cτ + d

a, b, c, d ∈ R, ad − bc > 0.

The group PSL2 (R) acts on H by linear fractional transformations, where H is the upper half of the complex plane, as well as on the extended real line R ∪ {∞}. We represent an element of PSL2 (R) by a matrix a b A= , ad − bc = 1, c d with the understanding that A and −A are identified. If A is not the identity element, then it has a single fixed point on R ∪ {∞} if and only if Tr(A) = 2, in which case A is called parabolic. Also, A has a fixed point in H if and only if Tr(A) < 2, in which case A is called an elliptic element. The case Tr(A) > 2 is equivalent to A having two fixed points on the real line, and A is called hyperbolic. Let us now restrict ourselves to a discrete subgroup 0 of PSL2 (R). A point τ of H is called an elliptic point of 0 if it is fixed by an elliptic element of 0. Similarly, a point of R ∪ {∞} is called a cusp of 0 if it is fixed by a parabolic element of 0. Every point in the orbit of an elliptic (resp., parabolic) point is elliptic (resp., parabolic). Moreover, the stabilizer in 0 of an elliptic point is a finite cyclic group, and the elements of finite order in 0 consist of the elliptic elements together with the identity element I . Similarly, the stabilizer in 0 of a parabolic point is an infinite cyclic group, and it consists of parabolic elements together with I . If 0 0 is a subgroup of PSL2 (R) commensurable with 0, then 0 0 is discrete and its set of cusps is the same as the set of cusps of 0. We denote by H∗ the union of H and the cusps of 0. The action of 0 on H extends to H∗ , and the quotient space H∗ / 0 is a Riemann surface. The group 0 is called a Fuchsian group of the first kind if H∗ / 0 is compact. All the discrete subgroups of

380

ABDELLAH SEBBAR

PSL2 (R) in this paper are assumed to be Fuchsian of the first kind. The genus of such a group 0 is by definition the genus of the compact Riemann surface H∗ / 0. It may occur that H/ 0 is compact; in this case 0 has no parabolic elements. An important property when H∗ / 0 is compact is that the numbers of 0-inequivalent cusps and elliptic points are finite. Let g be the genus of 0, let h be the number of inequivalent cusps, and let r be the number of inequivalent elliptic points. Let m 1 , . . . , m r be the orders of the stabilizers of all conjugacy classes of elliptic points. Then we say that 0 has signature (g; m 1 , . . . , m r ; h). The algebraic structure of the group can be determined by its signature. In fact, the group 0 has a presentation: generators: A1 , B1 , . . . , A g , Bg ; E 1 , . . . , Er ; P1 , . . . , Ph ;

(2.1)

relations: E 1m 1 = · · · = Erm r =

h Y i∈1

Pi

r Y

Ei

i=1

g Y

Ai Bi Ai−1 Bi−1 = I.

(2.2)

i=1

The generators Pi are parabolic, the E i are elliptic, and Ai and Bi are hyperbolic. 3. The torsion-free and genus zero conditions Let 0 be a Fuchsian group of the first kind. In most cases that occur in practice, 0 is a group commensurable with the modular group PSL2 (Z). The hyperbolic area of a fundamental domain for 0 acting on H is 2πχ(0), where χ (0) is the Euler characteristic of the fundamental domain given by χ(0) = 2(g − 1) + h +

r X 1 , 1− mi

(3.1)

i=1

where, as in Section 2, g is the genus, h is the number of inequivalent cusps (or the number of cusps in a fundamental domain), r is the number of inequivalent elliptic points, and m 1 , . . . , m r are their orders. This formula is a consequence of the Riemann-Hurwitz formula. If 0 is torsion-free (i.e., it has no elliptic elements) and has genus zero, then (3.1) becomes χ(0) = h − 2. (3.2) Moreover, from the presentation of 0 by generators and relations, we deduce that 0 can be generated by parabolic elements only, namely, P1 , . . . , Ph with the relation P1 · · · Ph = 1. Omitting one of the generators makes 0 a free group of rank h − 1.


381

PROPOSITION 3.1 If 0 is a torsion-free genus zero group commensurable with the modular group, then 0 is a subgroup of PSL2 (Q).

Proof The set of cusps for the modular group consists of Q ∪ {∞}, and the same is true for 0 since it is commensurable with PSL2 (Z). Assume that 0 is torsion-free and of genus zero so that it can be generated by a set of parabolic elements. Let P be any of these generators, and assume that P is represented by a matrix ac db with ad − bc = 1. The image by P of any cusp is also a cusp. In particular, P · 0, P · 1, and P · ∞ are in Q ∪ {∞}; that is, b/d, a/c, and (a + b)/(c + d) are in Q ∪ {∞}. It is not difficult to see that a, b, c, and d are rational multiples of a real number α. Since Tr(P) = ±2, we deduce also that α is rational, so that P is in PSL2 (Q). Examples of Fuchsian groups of the first kind commensurable with the modular group are the so-called congruence groups that contain, as a subgroup of finite index, a principal congruence group 0(m) which is defined for a positive integer m by 0(m) = {A ∈ PSL2 (Z), A ≡ ±I mod m} /{±I }, and the smallest such m is called the level of the group. Let us now focus on groups of this sort which are subgroups of the modular group. Examples of such groups are 1 ∗ 01 (m) = A ∈ PSL2 (Z), A ≡ ± mod m /{±I }, 0 1 a 00 (m) = c

b d

∈ PSL2 (Z), c ≡ 0 mod m /{±I }.

The indices of these groups in the modular group are given by PSL2 (Z) : 0(2) = 2 × PSL2 (Z) : 01 (2) = 6

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ABDELLAH SEBBAR

and m3 λ(m) := PSL2 (Z) : 0(m) = 2

Y 1 1− 2 , p

m > 2,

p|m p prime

m2 λ1 (m) := PSL2 (Z) : 01 (m) = 2

Y 1 1− 2 , p

m > 2,

p|m p prime

Y 1 , λ0 (m) := PSL2 (Z) : 00 (m) = m 1+ p

m ≥ 1.

p|m p prime

If 0 is subgroup of finite index of PSL2 (Z), then the hyperbolic area, 2πχ(0), of a fundamental domain for 0 satisfies 2πχ(0) = λ · 2π χ PSL2 (Z) , where λ = [PSL2 (Z) : 0]. Since the hyperbolic area of PSL2 (Z) is π/3, we deduce from the above formula that χ(0) = λ/6. Also, inside the modular group, elliptic elements have order 2 or 3, and we deduce from (3.1) that λ ν2 2ν3 = 2(g − 1) + h + + , 6 2 3

(3.3)

where νk (k = 2, 3) is the number of inequivalent elliptic elements of order k. It follows that if 0 is torsion-free and of genus zero, then λ = 6(h − 2).

(3.4)

The group 0(m) is of genus zero if and only if 1 ≤ m ≤ 5, 00 (m) is of genus zero if and only if m ∈ {1, . . . , 10, 12, 13, 16, 18, 25}, and 01 (m) is of genus zero if and only if m ∈ {1, . . . , 10, 12}. Meanwhile, 0(m) is torsion-free for m ≥ 2, 01 (m) is torsion-free for m ≥ 4, and a trace argument shows that 00 (m) is torsion-free if and only if −1 and −3 are not squares modulo m. We also mention that 00 (m) = 01 (m) for m ∈ {1, 2, 3, 4, 6}. 4. A transfer theorem The goal of this section is to show that every torsion-free genus zero subgroup of PSL2 (R) commensurable with PSL2 (Z) is conjugate to a subgroup of PSL2 (Z) of finite index. We begin by giving a description of the full normalizer of 00 (N ) inside PSL2 (R). This description was given originally in [1] and clarified in [2].


383

Let N be a positive integer, let h be the largest divisor of 24 for which h 2 | N , and set N = nh. The normalizer of 00 (N ) consists of the transformations ae b/ h A= , det(A) = e > 0 and e || n/ h, (4.1) cn de where r || s means r | s and gcd(r, s/r ) = 1 (r is called an exact divisor of s), and a, b, c, d are integers. The normalizer can also be described in terms of the Atkin-Lehner involutions that are defined as follows. Let e be an exact positive divisor of N ; then the set We of matrices ae b B= , det(B) = e, cN de is a single coset of 00 (N ). This coset, considered as an element of the normalizer quotient of 00 (N ), is called an Atkin-Lehner involution for 00 (N ). The union of all Atkin-Lehner involutions is a subgroup of the normalizer described by (4.1). The Fricke involution N0 −1 0 occurs when e = N . a b/ h Let h be as above, and let 00 (n|h) be the group of matrices cn of deterd h 0 minant 1. This group is a conjugate of 00 (n/ h) by 0 1 . The Atkin-Lehner invo lutions for 00 (n|h) are the conjugates by h0 01 of the Atkin-Lehner involutions of 00 (n/ h). The full normalizer of 00 (N ) in PSL2 (R) is obtained by adjoining to the group 00 (N |h) its Atkin-Lehner involutions. With this in mind we denote the normalizer of 00 (N ) by 00 (n|h)+; the + sign means that all the Atkin-Lehner involutions are present. If N is square-free, then the normalizer is 00 (N )+. 4.1 Every torsion-free genus zero discrete subgroup of PSL2 (R) commensurable with the modular group is conjugate to a subgroup of PSL2 (Z). THEOREM

Proof Let 0 be such a subgroup; then since the group 0 is commensurable with the modular group, using H. Helling’s theorem in [3], it is conjugate to a subgroup of 00 (m)+ for a square-free m. Since 0 is torsion-free and of genus zero, its conjugate inside 00 (m)+ is also torsion-free and of genus zero. We may therefore assume without loss of generality that 0 is inside 00 (m)+. Furthermore, 0 can be generated by parabolic elements only. Let P be such a generator. Being an element of 00 (m)+, P has the form (4.1) where h = 1 since m is square-free. If we standardize P to have determinant 1, it √ must have trace equal to ±2 since it is parabolic. This yields (a + d) e = ±2. It follows that e = 1 or e = 4. The latter is not possible since e | m and m is squarefree. Therefore P is in 00 (m). This being true for every parabolic generator of 0, we deduce that 0 is a subgroup of 00 (m).

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ABDELLAH SEBBAR

Theorem 4.1 can be seen as finding a common denominator for elements of the groups in Proposition 3.1. Also, note that the torsion-free and genus zero conditions are essential to the theorem. 5. Larcher congruence groups In [4], [5], Larcher introduced a large class of congruence subgroups of modular groups of any given level. These groups generalize classical congruence groups like 0(n), 00 (n), and 01 (n). To describe them, we follow the treatment of [5]. We first introduce the notion of a cusp width. Let 0 be a subgroup of finite index of PSL2 (Z); then the stabilizer of a cusp in 0 is a subgroup of finite index in the stabilizer of the same cusp in PSL2 (Z). This index is called the width of the cusp. Let m be a positive integer, and let d be a positive divisor of m. Write m/d = h 2 n, with n square-free. Let ε and χ be positive integers such that ε | h and χ | gcd(dε, m/dε2 ), and let τ ∈ {1, 2, . . . , χ }. Define the following: ( ! ) m 1 + εχ α dβ , γ ≡ τ α mod χ , (5.1) 0τ (m; m/d, ε, χ ) = ± m m 1 + εχ δ χγ where α, β, γ , and δ are integers. Then, with exceptions 01 (4; 2, 1, 2) and 01 (8; 8, 2, 2) which have levels 2 and 4, respectively, the groups 0τ (m; m/d, ε, χ ), which we call Larcher congruence groups, are congruence groups of level m. Moreover, d is the least cusp width and corresponds to the cusp at ∞, while m is the width of the cusp at zero. In particular, if m is square-free, then ε = χ = 1 and 0τ (m; m/d, ε, χ) = 01 (m) ∩ 0(d). The cusp widths can be determined in terms of the rational presentation of the cusps and the various data attached to these groups. More interestingly, the Larcher groups describe the cusp widths of any congruence group in the following way. Let 0 be a congruence group of level m, and let d be the least cusp width in 0. It is possible to conjugate 0 by a matrix in the modular group such that the width of ∞ becomes d and the width of zero becomes m. Note that a modular conjugacy only permutes the cusp widths; however, a nonmodular conjugacy often changes the set of cusp widths. According to [5, Sec. 3], for suitable ε, χ , and τ , the Larcher group 0τ (m; m/d, ε, χ) is a congruence group having the following properties: (1) 0τ (m; m/d, ε, χ ) is a subgroup of 0; (2) the cusp widths of 0 and 0τ (m; m/d, ε, χ) coincide. This means that up to finding the right parameters d, ε, χ, and τ , one is able to describe all the cusp widths in 0. We refer to 0τ (m; m/d, ε, χ) as the Larcher group corresponding to 0.


385

6. Classification inside the modular group In this section, we classify all the PSL2 (Z)-conjugacy classes of congruence subgroups of PSL2 (Z) following [10]. PROPOSITION 6.1 Every torsion-free genus zero congruence subgroup of the modular group is conjugate to a Larcher congruence group.

Proof Let 0 be a torsion-free genus zero congruence subgroup of level m. Then 0 has a set of parabolic generators. Up to a modular conjugacy, we can assume that the least cusp width d corresponds to ∞, and that the cusp zero has width m. Let 0τ (m; m/d, ε, χ ) be the corresponding Larcher group. According to Section 5, the stabilizers of each cusp with respect to both groups are the same. It follows that every parabolic generator of 0, which is a generator of the stabilizer of the corresponding cusp, is also in 0τ (m; m/d, ε, χ ). Therefore 0 = 0τ (m; m/d, ε, χ). PROPOSITION 6.2 If 0τ (m; m/d, ε, χ) is of genus zero, then

dε ≤ 5,

md ≤ 25, χ

m ≤ 12. εχ

(6.1)

Proof It is clear that 0τ (m; m/d, ε, χ) ⊆ 00 (m/χ) ∩ 01 (m/εχ ) ∩ 0(d) since d divides m/εχ. Also, 00 (m/χ) ∩ 0(d) is conjugate to 00 (md/χ) ∩ 01 (d), and 00 (m/χ ) ∩ 01 (m/εχ )∩0(d) is conjugate to 01 (m/εχ)∩0(dε) since dε divides m/εχ. A necessary condition to have 0τ (m; m/d, ε, χ) of genus zero is that the groups 00 (md/χ ), 01 (m/εχ ), and 0(dε) are all of genus zero. This yields (6.1). From Proposition 6.2, we see that 1 ≤ d ≤ 5, and for each d there are few values of ε such that dε ≤ 5; namely, ε = 1 if d = 3, 4, or 5, ε = 1 or 2 if d = 2, and 1 ≤ ε ≤ 5 if d = 1. This provides us with a (short) list of possible values of χ and τ in each case, and therefore with a (short) list of Larcher groups which contains the genus zero Larcher groups. All the genus zero groups that appear are conjugate to a 0(m), 00 (m), or 01 (m), except 01 (8) ∩ 0(2) which can be checked to be of genus zero by finding its signature. The groups obtained which are not of genus zero are easily seen to be so because some conjugate of them is contained in a group of the form 0(m), 00 (m), or 01 (m) which is clearly not of genus zero (see [10] for more details). We have the following.

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ABDELLAH SEBBAR

THEOREM 6.3 Up to modular conjugacy, there are exactly 33 congruence subgroups of the modular group which are torsion-free and of genus zero, all of which are given in Table 1.

Table 1 Index 6

Level 2 4 3 4 5 6 8 9 4 6 7 8 12 16

Group 0(2) 00 (4) 0(3) 00 (4) ∩ 0(2) 01 (5) 00 (6) 00 (8) 00 (9) 0(4) 00 (3) ∩ 0(2) 01 (7) 2b 01 (8), 00 (8) ∩ 0(2), ± 1+4a 4c 1+4d , a ≡ c 00 (12) b 00 (16), ± 1+4a 8c 1+4d , a ≡ c mod 2

36

6 9 10 18 27

48

8

32

00 (2) ∩ 0(3) 3b 01 (9), ± 1+3a 3c 1+3d , a ≡ c mod 3 01 (10) 00 (18) 1+3a b ± 9c 1+3d , a ≡ c mod 3 4b 01 (8) ∩ 0(2), ± 1+4a 4c 1+4d , a ≡ c mod 2 2b 01 (12), ± 1+6a 6c 1+6d , a ≡ c mod 2 2b 00 (16) ∩ 01 (8), ± 1+4a 8c 1+4d , a ≡ c mod 2 1+6a b ± 12c 1+6d , a ≡ c mod 2 1+4a b ± 16c 1+4d , a ≡ c mod 2

5 25

0(5) 00 (25) ∩ 01 (5)

12

24

12 16 24

60

mod 2

7. Classification inside PSL2 (R) In this section, we list all the PSL2 (R)-conjugacy classes of torsion-free genus zero congruence subgroups of PSL2 (R). According to Theorem 4.1, each such group is conjugate to a subgroup of the modular group. If we establish that this conjugate


387

is also a congruence group, then it is a modular conjugate of one in the list from Theorem 6.3. PROPOSITION 7.1 If 0 is a congruence subgroup of PSL2 (R) which has conjugates inside PSL2 (Z), then any of these conjugates is a congruence group.

Proof Let 0 be a congruence subgroup of PSL2 (R) of level n. Let C = such that C −1 0C ⊆ PSL2 (Z). Since 0(n) ⊆ 0, the two matrices 1 − abn −b2 n 1 0 C −1 C= n 1 a2n 1 + abn and C

−1

1 n 1 + cdn C= 0 1 −c2 n

d 2n 1 − cdn

a b c d

, ad − bc = 1,

have integer entries. We deduce that a 2 n, b2 n, c2 n, d 2 n, abn, and cdn are in Z. Using this fact and multiplying ad − bc = 1 by adn 2 , we obtain adn 2 ∈ Z. Similarly, we have acn 2 , bcn 2 , bdn 2 ∈ Z. We are going to show that 0(n 3 ) ⊆ C −1 0C. Let 1 + n3 x n3 y ∈ 0(n 3 ); A= n3 z 1 + n3t then C AC −1 = XZ TY , where X = 1 + n 3 (ad x + bdz − acy − bct), Y = n 3 (a 2 y + abt − abx − b2 z), Z = n 3 (cd x + d 2 z − c2 y − cdt), T = 1 + n 3 (acy + adt − bcx − bdz). It is clear that C AC −1 ∈ 0(n) ⊆ 0. This is also true if A ≡ −I mod n 3 . Remark 7.1 The same calculations show that if 0 contains 01 (n) (rather than 0(n)), then any of its conjugates inside PSL2 (Z) contains 0(n 2 ), and if 0 contains 00 (n), then its conjugates in PSL2 (Z) contain 0(n). In view of Proposition 7.1, any torsion-free genus zero congruence subgroup of PSL2 (R) is conjugate to a group from the list of Theorem 6.3. Thus, we need only find the PSL2 (R)-conjugacy classes among the groups of Theorem 6.3. Also, we need only look at each index since subgroups of different indices cannot be conjugate.

388

ABDELLAH SEBBAR

Index 6. The groups 0(2) and 00 (4) are conjugate by Index 12. The groups 0(3) and 00 (9) are conjugate by 0(2) and 00 (8) are conjugate by 20 01 .

20 01

30 01

. , and the groups 00 (4) ∩

Index 24. The groups 00 (3) ∩ 0(2) and 00 (12) are conjugate by 20 01 since 00 (3) ∩ 0(2) = 00 (6) ∩ 0(2). The group 0(4) is conjugate by 40 01 to 00 (16), and it is conjugate by 20 01 to 00 (8) ∩ 0(2). The group 01 (8) is conjugate by 10 1/2 to the 1 group 1 + 4a b ± , a ≡ c mod 2 8c 1 + 4d which is conjugate by 10 02 to 1 + 4a 2b ± , a ≡ c mod 2 . 4c 1 + 4d Index 36. The group 00 (2)∩0(3) is conjugate to 00 (18). The group 01 (9) is conjugate by 10 11 30 01 = 30 11 to 1 + 3a 3b ± , a ≡ c mod 3 , 3c 1 + 3d and it is conjugate by 10 τ/3 , τ = ±1, to 1 1 + 3a b ± , a ≡ τ c mod 3 . 9c 1 + 3d Index 48. The group 01 (8) ∩ 0(2) is conjugate by 20 01 , 20 11 , respectively, to 00 (16) ∩ 01 (8), 1 + 4a 4b ± , a ≡ c mod 2 , 4c 1 + 4d 1 + 4a 2b ± , a ≡ c mod 2 , 8c 1 + 4d and 1 + 4a ± 16c

b , a ≡ c mod 2 . 1 + 4d The group 01 (12) is conjugate by 20 11 to 1 + 6a 2b ± , a ≡ c mod 2 6c 1 + 6d

1 1/2 0 1

, and

1 1/4 0 1

,


and by

1 1/2 0 1

389

to 1 + 6a ± 12c

b , a ≡ c mod 2 . 1 + 6d

Index 60. The group 0(5) is conjugate by

50 01

to 00 (25) ∩ 01 (5).

THEOREM 7.2 There are 15 PSL2 (R)-conjugacy classes of torsion-free genus zero congruence subgroups of PSL2 (R). Representatives for these classes are 0(5), 01 (8) ∩ 0(2), 00 (n) for n = 4, 6, 8, 9, 12, 16, 18, and 01 (n) for n = 5, 7, 8, 9, 10, 12.

Proof In view of the above discussion, the congruence groups listed in this theorem form a set of representatives for the PSL2 (R)-conjugacy classes of torsion-free genus zero congruence subgroups. We need only check that no two of them are conjugate. Since all these groups are in PSL2 (Z), the proof of Proposition 7.1 provides us with a necessary condition for two such groups to be PSL2 (R)-conjugate; namely, if one contains 0(n) for some n, then the other one must contain 0(n 3 ). For the groups listed in the theorem, this would imply that if two of them are conjugate, then their levels have the same set of prime divisors. This allows us to see that each two groups of the same index in the list are not conjugate. 8. Subgroups containing 00 (n) In this section, as an application of the above, we investigate the groups given in Theorem 7.2 that have a conjugate containing 00 (n). These groups are of interest in Moonshine and the theory of replicable functions, and they were studied in [6], [7]. Ignoring the groups 00 (n), we have to deal with the groups 01 (n), in addition to 01 (8) ∩ 0(2) and 0(5). Let N be a positive integer such that 01 (N ) contains a conjugate of 00 (n) for some n. According to Remark 7.1, we must have 0(n) ⊆ 01 (N ). It follows that N | n. Let C = ac db , ad −bc = 1 (a, b, c, d real numbers), such that C00 (n)C −1 ⊆ 01 (N ). We have 1 1 1 − ac a2 −1 C C = ∈ 01 (N ) 0 1 −c2 1 + ac and C

1 0 1 + bdn C −1 = n 1 d 2n

−b2 n ∈ 01 (N ). 1 − bdn

390

ABDELLAH SEBBAR

It follows that ac ≡ c2 ≡ bdn ≡ d 2 n ≡ 0

mod N

and

a 2 , b2 n ∈ Z.

(8.1)

Let α ∈ Z such that gcd(α, n) = 1. There exist β and γ such that αγ − βn = 1, so that A = αn γβ ∈ 00 (n). The product C AC −1 = XZ YT , where X = adα + bdn − acβ − bcγ , Y = a 2 β + abγ − abα − b2 n, Z = cdα + d 2 n − c2 β − cdγ , T = acβ + adγ − bcα − bdn must be in 01 (N ), so that X ≡ T ≡ ±1 mod N , Z ≡ 0 mod N , and Y ∈ Z. Using (8.1) we deduce from the expressions of X and T that adα − bcγ ≡ adγ − bcα ≡ ±1

mod N .

(8.2)

Since ad − bc = 1, these equations sum to α + γ ≡ ±2

mod N .

Since det(A) = 1, we have αγ ≡ 1 mod n and hence mod N since N | n. We deduce that α 2 + 1 ≡ ±2α mod N . Meanwhile, because α is an arbitrary element of (Z/nZ)× and N | n, α can be any element of (Z/N Z)× since the projection (Z/nZ)× → (Z/N Z)× is surjective. We have shown the following. PROPOSITION 8.1 If 01 (N ) contains a conjugate of 00 (n) for some n, then

∀a ∈ (Z/N Z)× ,

(a ± 1)2 ≡ 0

mod N .

(8.3)

PROPOSITION 8.2 The only positive integers N for which (8.3) holds are the divisors of 16 and 36.

Proof Let N be such an integer, and write N = 2k N 0 with gcd(2, N 0 ) = 1; then the integer a = 2 + N 0 is relatively prime to N , so that (a ± 1)2 ≡ 0 mod N and hence mod N 0 . This implies N 0 | 9. Hence, only 2 and 3 may divide N . It follows that 5 is relatively prime to N , so that (5 ± 1)2 ≡ 0 mod N , yielding N | 16 or N | 36. Conversely, each positive divisor of 16 or 36 satisfies (8.3).


391

Remark 8.1 If we require that 01 (N ) contain some 00 (n), then a necessary condition is ∀a ∈ (Z/N Z)× ,

a±1≡0

mod N ,

which holds if and only if N is a divisor of 4 or 6. Conversely, for each positive divisor N of 4 or 6, we have 01 (N ) = 00 (N ), a fact that was mentioned at the end of Section 3. Remark 8.2 For the remainder of this section, we implicitly use the following fact: the positive divisors of 24 are the only positive integers r for which ∀x, y, x y ≡ 1 mod r =⇒ x ≡ y mod r. This property is also essential in describing the normalizer of 00 (n) as in Section 4. Using Proposition 8.2, the groups 01 (N ) for N = 5, 7, 10 do not contain any con 9 −2 jugate of a 00 (n). This is also true for 0(5) ⊂ 01 (5). If P = 27 −3 , then P00 (81)P −1 ⊂ 01 (9); this uses the simple fact that if ad ≡ 1 mod 9, then 2a − d ≡ 2d − a ≡ ±1 mod 9. For the group 01 (8) ∩ 0(2), we proceed in the following way: the invariance group of the Hauptmodul η(τ )/η(4τ ), where η is the Dedekind eta-function, is a modular subgroup with index 48 and level 32. (It can be easily checked that it contains 0(32), which is enough for our purposes.) It must be a conjugate of one of the two representatives of the index 48 subgroups, namely, 01 (8) ∩ 0(2) and 01 (12). The latter is excluded since its level is divisible by 3. Moreover, the invariance group of η(8τ )/η(32τ ) (the conjugate by τ 7→ 8τ of η(τ )/η(4τ )) contains 00 (28 ). Therefore 01 (8) ∩ 0(2) contains a conjugate of 00 (28 ), and so does 16 −1 . 01 (8). A conjugating matrix is 32 2 Remark 8.3 The same argument works for 01 (9) since it is conjugate to the invariance group of η(τ )/η(9τ ), and η(3τ )/η(27τ ) is invariant under 00 (81). It was conjectured in [7] that no other genus zero torsion-free group besides those conjugated to the above ones contains a conjugate of some 00 (n). To prove this conjecture, we need to eliminate the case of 01 (12). PROPOSITION 8.3 Let n be a positive integer divisible by 12, and let k be an integer. If the equation

x(x + k) ≡ 1

mod n

(8.4)

392

ABDELLAH SEBBAR

has a solution, then there are four solutions that are distinct modulo 12. Proof Assume that the equation (8.4) has a solution x0 ; then −x0 − k is also a solution. Also, x0 + k is an inverse of x modulo n and hence modulo 12, and therefore k is a multiple of 12 by Remark 8.2. Write n = 2α 3β n 0 with gcd(6, n 0 ) = 1, and consider the following four systems: x ≡ x0 x ≡ x1 x ≡ x2

mod n 0 , mod 2α , mod 3β ,

where x1 and x2 take values in {x0 , −x0 − k}. By the Chinese remainder theorem, each system provides a solution to equation (8.4). Meanwhile, x0 and −x0 − k are distinct modulo 4 and modulo 3 since k ≡ 0 mod 12. It follows that the four systems yield four solutions to (8.4) that are distinct modulo 12. COROLLARY 8.4 There is no positive integer n for which 00 (n) has a conjugate inside 01 (12).

Proof Assume there exists a positive integer n and a real matrix C = ac db with ad − bc = 1 such that C00 (n)C −1 ⊆ 01 (12). By Remark 7.1, 12 | n. Since the projection (Z/nZ)× → (Z/12Z)× is surjective, we can choose x0 ∈ Z with gcd(x0 , n) = 1 and x0 ≡ 1 mod 12. Let y0 and z 0 be such that x0 z 0 − ny0 = 1. Set k = z 0 − x0 so that x0 is a solution to the equation x(x + k) ≡ 1 mod n. Using Proposition 8.3, there exists x1 such that x1 (x1 + k) ≡ 1 mod n and x1 ≡ 7 mod 12. Hence, if we set z 1 = x1 + k, there exists y1 such that x1 z 1 − ny1 = 1. The matrices xn0 zy00 and x1 y1 n z 1 are in 00 (n) and hence are conjugated by C into 01 (12). Using the relations (8.2) with (α, γ ) = (xi , z i ), i = 0, 1, and taking into account that both matrices have traces congruent to 2 modulo 12 because of the choices of x0 and x1 , we have ad x0 − bcz 0 ≡ 1

mod 12,

ad x1 − bcz 1 ≡ 1

mod 12.

Since ad − bc = 1 and z 0 − x0 = z 1 − x1 = k, taking the difference of these two congruences yields x0 − x1 ≡ 0 mod 12, which is a contradiction since x0 and x1 were chosen such that x0 − x1 ≡ ±6 mod 12. The corollary follows. THEOREM 8.5 Any torsion-free genus zero group containing some 00 (n) with finite index is conjugate to one of the following groups:

01 (8), 01 (9), 01 (8) ∩ 0(2), 00 (n) for n = 4, 6, 8, 9, 12, 16, 18.


393

9. A special case In this section, we study some properties of the index 12 groups that were found in the previous sections. More interesting properties (geometric and analytic) and details can be found in [11]. The groups described in this paper are all congruence groups. If we drop the congruence condition, then there are infinitely many torsion-free genus zero subgroups of the modular group. In fact, for each λ and h satisfying λ = 6(h − 2), there is a torsion-free genus zero subgroup of the modular group with index λ and h cusps (see [8]). However, if the number of cusps is arbitrary, this is not true of the cusp widths. The situation is as follows: let X be a set of λ letters, and consider pairings (x, y) of permutations x and y acting on X satisfying x 2 = y 3 = 1 and such that the group generated by x and y is transitive on X . We define the equivalence classes (x, y) modulo a conjugation of x and y by a permutation in Sλ . Then there is a oneto-one correspondence between conjugacy classes of subgroups of finite index λ in the modular group and equivalence classes of pairings (x, y) (see [8]). The subgroup is torsion-free if and only if x and y are fixed point free, and it is of genus zero if and only if the total number of disjoint cycles of x, y, and x y is λ + 2. Moreover, the subgroup has cusp widths n 1 , n 2 , . . . , n h , where h is the number of cusps, if and only if the permutation x y consists of h disjoint cycles of lengths n 1 , n 2 , . . . , n h . If we look at the case λ = 12, we want x and y acting fixed point freely on a set of 12 elements such that hx, yi is transitive on these points and such that x y decomposes P into 4 cycles of lengths n 1 , n 2 , n 3 , n 4 with n i = 12. It is not difficult to check (by computer or simply using graph theory; see [11]) that the only partitions of 12 into 4 positive integers which are realized are those listed in Table 2 with the corresponding groups. These also account for all the equivalence classes of pairings (x, y). Notice that the cusp widths of Table 2 give all the possible quadruples of positive integers whose sum is 12 and whose product is a square. This fact has an explanation related to the theory of modular elliptic surfaces (see [11]). We deduce the following from Table 2. 9.1 The six congruence subgroups of index 12 given in Table 1 account for all the torsionfree genus zero subgroups of index 12 in PSL2 (Z). PROPOSITION

If a subgroup of PSL2 (R) is conjugate to a torsion-free genus zero subgroup of index 12 in PSL2 (Z), then this group has only four cusps or, equivalently, a fundamental domain for this group has hyperbolic area 4π. Proposition 9.1, together with Theorem 4.1, shows that there are only four conjugacy classes of torsion-free genus zero

394

ABDELLAH SEBBAR

Table 2

0(3)

3−3−3−3

00 (4) ∩ 0(2) 4 − 4 − 2 − 2 01 (5)

5−5−1−1

00 (6)

6−3−2−1

00 (8)

8−2−1−1

00 (9)

9−1−1−1

subgroups of PSL2 (R) having hyperbolic area 4π which can be represented by only four congruence subgroups of the modular group. Because these groups are torsionfree and of genus zero, the quotient of the upper half-plane H by one of these groups is just the projective line P1 minus four points. To determine these four points, we need to find a Hauptmodul for each group as well as its values at the cusps. For the four PSL2 (R)-conjugacy classes representing the six modular subgroups of index 12, we have Table 3. Table 3

Group

Hauptmodul η(τ/3) η(3τ )

3

0(3)

00 (4) ∩ 0(2)

η(2τ )12 η(τ )4 η(4τ )8

01 (5)

1 q

00 (6)

η(τ )5 η(3τ ) η(2τ )η(6τ )5

Q∞

n=1 (1 − q

Values at the cusps 3, z 2 + 3z + 9, ∞ 4, −4, 0, ∞

n )−5( n5 )

In the expression of the Hauptmodul for 01 (5),

z 2 − 11z − 1, 0, ∞ 5, −4, −3, ∞

n 5

denotes the Legendre symbol


395

and q = exp(2πiτ ). The quadratic polynomials indicate that their roots are values for the Hauptmoduls, and the order in which these values are listed follows the ordering of the cusp widths given in Table 2. The Hauptmoduls are chosen so that their Fourier expansion in q has the form q −1/m +O(q 1/m ), where m is the cusp width at ∞, except for 01 (5) which has the form 1/q + 5 + O(q). The expressions for the Hauptmoduls and their values are found in the Moonshine tables of [2], except for 01 (5) whose Hauptmodul f is deduced from 0(5) and whose values are deduced from 00 (5), namely, that the Hauptmodul for 01 (5) is simply the fifth power of the one for 0(5). If g is a Hauptmodul for 00 (5) of the form 1/q + O(q), then g takes the values 6 and ∞ at the two cusps of 00 (5). By desymmetrizing (01 (5) has index 2 in 00 (5)), we have 1 g = f −5+ f which leads to the values of f given in Table 3. Theorem 9.2 determines up to isomorphism all the modular curves, that is, quotients of the upper half-plane H by a modular subgroup, which are given by P1 minus four points. Applying a linear fractional transformation to the four values for each group in Table 3 so that each triple is sent to 0, 1, ∞, we obtain 17 different values for the fourth cusp. Denoting exp(2πi/3) by ω and the roots of z 2 − 125z + 125 by α and β, we have the following. THEOREM 9.2 The curve P1 \ {0, 1, ∞, z} is a modular curve if and only if z or 1/z is a member of 9 α . −8, −1, , 2, 9, −ω, α, β, − 8 β

Using Theorem 4.1 and Proposition 9.1, we deduce the following. 9.3 The only values of z for which P1 \ {0, 1, ∞, z} is a quotient of the upper half-plane by a Fuchsian group commensurable with the modular group are those given by Theorem 9.2. COROLLARY

Acknowledgment. I thank John McKay for his support and insights. I also thank Oliver Atkin for helpful discussions. References [1]

A. O. L. ATKIN and J. LEHNER, Hecke operators on 00 (m), Math. Ann. 185 (1970),

134–160. MR 42:3022 382

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ABDELLAH SEBBAR

[2]

J. H. CONWAY and S. P. NORTON, Monstrous moonshine, Bull. London Math. Soc. 11

[3]

H. HELLING, On the commensurability class of the rational modular group, J. London

[4]

H. LARCHER, The cusp amplitudes of the congruence subgroups of the classical

(1979), 308–339. MR 81j:20028 382, 395 Math. Soc. (2) 2 (1970), 67–72. MR 43:3353 383

[5] [6] [7] [8] [9] [10] [11]

[12]

[13]

[14]

modular group, Illinois J. Math. 26 (1982), 164–172. MR 83a:10040 378, 384 , The cusp amplitudes of the congruence subgroups of the classical modular group, II, Illinois J. Math. 28 (1984), 312–338. MR 85i:11034 378, 384 J. MCKAY and A. SEBBAR, Fuchsian groups, Schwarzians, and theta functions, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 343–348. MR 2000a:11059 378, 389 , Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), 255–275. MR CMP 1 795 562 378, 389, 391 M. H. MILLINGTON, Subgroups of the classical modular group, J. London Math. Soc. (2) 1 (1969), 351–357. MR 39:5477 393 R. A. RANKIN, Modular Forms and Functions, Cambridge Univ. Press, Cambridge, 1977. MR 58:16518 379 A. SEBBAR, Classification of torsion-free genus zero congruence groups, Proc. Amer. Math. Soc. 129 (2001), 2517–2527. MR CMP1 838 872 385 , Modular subgroups, forms, curves, and surfaces, to appear in Canad. Math. Bull. http://journals.cms.math.ca/cgi-bin/vault/viewprepub/sebbar8056.prepub 393 G. SHIMURA, Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures 1, Iwanami Shoten, Tokyo; Publ. Math. Soc. Japan 11, Princeton Univ. Press, Princeton, 1971. MR 47:3318 379 J. G. THOMPSON, “A finiteness theorem for subgroups of PSL(2, R) which are commensurable with PSL(2, Z)” in The Santa Cruz Conference on Finite Groups (Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math. 37, Amer. Math. Soc., Providence, 1980, 533–555. MR 82b:20067 377 P. ZOGRAF, A spectral proof of Rademacher’s conjecture for congruence subgroups of the modular group, J. Reine Angew. Math. 414 (1991), 113–116. MR 92d:11041 377

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada; [email protected]


THE DIHEDRAL LIE ALGEBRAS AND GALOIS (l) SYMMETRIES OF π1 P1 − ({0, ∞} ∪ µ N ) A. B. GONCHAROV

Abstract We describe the image of the absolute Galois group acting on the pro-l completion of the fundamental group of the Gm minus N th roots of unity. We relate the structure of the image with geometry and topology of modular varieties for the congruence subgroups 01 (m; N ) of GLm (Z) for m = 1, 2, 3 . . . . Contents 1. An outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Formulations of the main results . . . . . . . . . . . . . . . . 3. The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The dihedral Lie coalgebra . . . . . . . . . . . . . . . . . . . 5. The dihedral Lie algebras and special equivariant derivations . . 6. Cohomology of some discrete subgroups of GL2 (Z) and GL3 (Z) 7. Proofs of the theorems from Section 2 . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

397 403 416 426 443 458 468 484

1. An outline This is the next in the series of papers [G2], [G4], [G5] devoted to the study of higher cyclotomy, understood as motivic theory of multiple polylogarithms at roots of unity, and its relationship with modular varieties for GLm /Q , for all m ≥ 1. Let µ N be the group of N th roots of unity. Our main objective is a mysterious link between the structure of the motivic fundamental group of X N := P1 − {0, ∞} ∪ µ N = Gm − µ N and the geometry and topology of the following modular varieties for GLm /Q : 01 (m; N )\ GLm (R)/R∗+ · Om

for m > 1,

(1)

DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 3, Recieved 14 June 2000. Revision received 9 January 2001. 2000 Mathematics Subject Classification. Primary 11G55, 11F67, 11R32, 20F34. Author’s work supported by the Max Planck Institute, National Science Foundation grant numbers DMS´ 9500010 and DMS-9800998, and the Institute des Hautes Etudes Scientifiques. 397

398

A. B. GONCHAROV

where 01 (m; N ) ⊂ GLm (Z) is the subgroup stabilizing the vector (0, . . . , 0, 1) mod N . For m = 1 it is S N := Spec Z[ζ N ][1/N ] . An approach via adeles provides a coherent description for all m (see Section 2.2). In the present paper we turn to the Galois side of the story. However, to keep the motivic perspective, let us recall the following. According to P. Deligne [D1] the motivic fundamental group of a variety is not just a group but rather a Lie algebra object in the category of mixed motives. It can be viewed as a pronilpotent completion of the topological fundamental group of the corresponding complex variety equipped with lots of additional structures of analytic, geometric, and arithmetic nature. Mixed motives are algebro-geometric objects. They can be seen through their realizations. The two most popular realizations that hypothetically capture all the information about the category of the mixed motives are the Hodge realization provided by analysis and algebraic geometry, and the l-adic realization provided by arithmetic and algebraic geometry. The l-adic realization of the motivic fundamental group is obtained from the (l) action of the absolute Galois group Gal(Q/Q) on the pro-l completion π1 (X ) of the fundamental group of the variety X . That is why in this paper we study the action of (l) the Galois group on π1 (X N ). In the case N = 1 this problem has been addressed by A. Grothendieck [Gr], Deligne [D1], Y. Ihara [I1]–[I4], V. Drinfeld [Dr], and others (see the wonderful survey by Ihara [I3]). For N > 1 it has not been investigated. Our point of view is that we should study the problem for all N , penetrating the structures independent of N (like modular complexes; see Section 2.5), and then go to the limit using the natural maps X N M −→ X N given by z 7−→ z M . The last step will be discussed elsewhere. Briefly, our approach is this. Let G be a commutative group. In [G4] we construct a bigraded Lie algebra D•• (G), called the dihedral Lie algebra of G (see Section 3 below). We relate D•• (µ N ) to the Lie algebra of the image of Gal(Q/Q) in (l) Aut π1 (X N ). On the other hand, in [G2], [G4], [G6], the structure of the Lie algebra D•• (µ N ) is related to the geometry of modular varieties (1). Using these results, we (l) study the action of Gal(Q/Q) on π1 (X N ). In particular, we obtain new results about (l) 1 the action on π1 (P − {0, 1, ∞}). Here is a more detailed account. Let X be a regular curve over Q. Let X be the corresponding projective curve, and let vx be a nonzero tangent vector at a point x ∈ X . Then according to Deligne [D1] one can define the geometric profinite fundamental group b π1 (X, vx ) based at the vector vx . If X , x, and vx are defined over a (l) number field F ⊂ Q, then the group Gal(Q/F) acts by automorphisms of b π1 (X, vx ). If X = X N , there is a tangent vector v∞ at ∞ corresponding to the inverse t −1 of the canonical coordinate t on P1 − ({0, ∞} ∪ µ N ). (l) Since any finite l-group is nilpotent, the pro-l group π1 (X N , v∞ ) is pronilpotent.

GALOIS SYMMETRIES OF FUNDAMENTAL GROUPS

399

(l)

Let L N = L(l) (X N , v∞ ) be the l-adic pro-Lie algebra corresponding via the Maltsev (l) theory to π1 (X N , v∞ ) (see [D1, Chapter 9]). It is a free pronilpotent Lie algebra with generators corresponding to the loops around zero and all N th roots of unity. We call it the l-adic fundamental Lie algebra of X N . The Galois group acts by its automorphisms. S Let ζn be a primitive nth root of unity, and let Q(ζl ∞ N ) := Q(ζl n N ). For the reasons explained in Section 3.1 we restrict the action of the Galois group to the subgroup Gal(Q/Q(ζl ∞ N )), picking up the homomorphism (l) Gal Q/Q(ζl ∞ N ) −→ Aut L N . (l)

(l)

Let Der L N be the Lie algebra of all derivations of the Lie algebra L N . Linearizing, (l) as explained in Sections 2.1 or 3.2, the above map, we get the Lie subalgebra G N (l) sitting inside of the Lie algebra Der L N : (l)

(l)

G N ,→ Der L N .

(2)

(l)

The fundamental Lie algebra L N is equipped with two filtrations preserved by the Galois action. The weight filtration can be defined on the fundamental Lie algebra of any algebraic variety over Q. In our case it coincides with the lower central series (l) of the Lie algebra L N . The depth filtration is more specific. It is given by the lower central series of the codimension one ideal I N := Ker L(l) (X N , v∞ ) −→ L(l) (Gm , v∞ ) = Ql (1) , where the map is provided by the natural inclusion X N ,→ Gm . (l) (l) These filtrations induce filtrations on Der L N and hence, via (2), on G N . The (l) associated graded for the weight and depth filtrations Gr G•• (µ N ) is a Lie algebra bigraded by negative integers −w and −m, called the weight and the depth. We call (l) it the level N Galois Lie algebra. When N = 1, it is denoted by Gr G•• . We also use (l) (l) (l) (l) shorthands G•• (µ N ) for Gr G•• (µ N ) and G•• for Gr G•• . (l) The weight filtration on L N admits a splitting, that is, is defined by a grading, compatible with the depth filtration. Such a weight splitting is provided by the eigenspaces of a Frobenius F p ∈ Gal(Q/Q), ( p 6 | N ). Therefore, taking Gr for both the weight and depth filtrations, we get an embedding (l)

(l) G•• (µ N ) ,→ Gr Der L N . (l)

The vector space Gr G−w,−m (µ N ) is nonzero only if w ≥ m ≥ 1. As a Gal(Q(ζl ∞ )/Q)-module it is isomorphic to a direct sum of copies of Ql (w).

400

A. B. GONCHAROV

For every integer m ≥ 1 we have the depth m quotient (l)

G•,≥−m (µ N ) :=

(l)

G•• (µ N ) (l)

G•, 1 the stack (7) coincides with the modular variety (1). ∗ · SO(2) = C − R and O(2)/ SO(2) acts as the When m = 2, one has GL2 (R)/R+ complex conjugation z 7 −→ z. Let us go beyond the depth 1 case. We start from the N = 1 case. The following result generalizes the first line of (6). 2.2 (l) We have dim G−w,−m = 0 if w + m is odd. THEOREM

(l) 2.3. The Galois action on π1 P1 − {0, 1, ∞}, v∞ : A description of the depth 2 quotient via the modular triangulation of the hyperbolic plane (l) The structure of the Lie algebra G•,≥−2 is completely described by the Lie commutator map (l) (l) [, ] : 32 G•,−1 −→ G•,−2 . (9) The left-hand side is known to us by (6). So to describe (9) one needs to define the right-hand side and the commutator map. Dualizing (9), we get the depth 2 piece of (l) the standard cochain complex of the Lie algebra G•,≥−2 , (l) ∨

δ

(l) ∨

G•,−2 −→ 32 G•,−1 .

(10)

To describe it we need to introduce the following two complexes: ∗ 1 2 M(2) := M(2) −→ M(2)

and

1 2 3 M∗(2) := M(2) −→ M(2) −→ M(2) .

(11)

The first one is the chain complex of the classical modular triangulation of the hyperbolic plane H2 where the central ideal triangle has vertices at 0, 1 and ∞ (see Figure 1 is the group generated by the tri2). We place it in degrees [1, 2]. For example, M(2) angles. The second complex is the chain complex of the modular triangulation of the 1 hyperbolic plane extended by cusps, that is, by P (Q). We place it in degrees [1, 3]. The group GL2 (R), acting on C − R by ac db z = (az + b)/(cz + d), commutes with z 7 −→ z. We let GL2 (R) act on the upper half-plane H2 by identifying H2 with


407

Figure 2

the quotient of C − R by complex conjugation. The action of the subgroup GL2 (Z) ∗ and M∗ are complexes of GL (Z)-modules. preserves the modular picture. So M(2) 2 (2) A digression on cohomology of a subgroup 0 of GL2 (Z) Let V be a GL2 -module. For a torsion-free subgroup 0 of GL2 (Z), the group cohomology H ∗ (0, V ) is isomorphic to the cohomology of 0\H2 with coefficients in the local system LV corresponding to V . Notice that H ∗ (0\H2 , LV ) = H ∗ (0\H2 , R j∗ LV ) where j : 0\H2 ,→ 0\H2 . For a torsion-free finite index subgroup 0 of GL2 (Z) one defines the cuspidal ∗ (0, V ) as the cohomology of 0\H with coefficients in a middle cohomology Hcusp 2 extension of the local system LV . In our case the middle extension means the sheaf j∗ LV . For any finite index subgroup 0 ,→ GL2 (Z) there is a normal torsion-free finite index subgroup e 0 ,→ 0. So if V is a Q-rational GL2 -module, one has, using the Hochshild-Serre spectral sequence, e H ∗ (0, V ) = H ∗ (e 0 , V )0/0 .

(12)

We define the cuspidal cohomology for arbitrary finite index subgroup 0 by reducing it to the torsion-free case by a formula similar to (12). 2.3 Let 0 be a finite index subgroup of GL2 (Z), and let V be a Q-rational GL2 -module. Then the complex ∗ M(2) ⊗0 V [1] (13) LEMMA

computes the cohomology H ∗ (0, V ⊗ ε2 ), and the complex M∗(2) ⊗0 V [1]

(14)

∗ (0, V ⊗ ε ). computes the cuspidal cohomology Hcusp 2

Proof The complex (13), up to a shift, is the complex of chains with coefficients in the local

408

A. B. GONCHAROV

system LV ; it is relative to the modular triangulation of 0\H2 , which has finitely many cells, with 0\H2 a union of open cells, hence giving homology groups for locally finite chains (the Borel-Moore homology). If we take the dual for the complex (13) as well as for (14), we obtain a cochain complex computing the cohomology of 0\X with coefficients in a sheaf: the sheaf is j! LV ∨ for (13), and the middle extension j∗ LV ∨ for (14). (Notice that invariants by the stabilizer of the cusp are dual to the coinvariant group.) For the complex itself, by Poincaré duality we get cohomology of 0\H2 (respectively, 0\H2 ) with value in the local system LV twisted by the orientation class, that is, LV ⊗ε2 (respectively, its middle extension). The lemma is proved. To clarify the homological algebra meaning of the modular complex, consider the (2)

∂

(2)

(2)

∗ and ∂ is dual to the differential in complex M−1 ←− M−2 , where M−∗ := M(2) (2)

∗ . Then M ∗ is a subcomplex of Hom(M the complex M(2) −∗ , Z). As a complex of (2) ∗ is the chain complex of the tree dual to the modular trianguSL2 (Z)-modules, M(2) lation, shifted by 1. As a complex of GL2 (Z)-modules it is a resolution of ε2 [1]. The stabilizers of the action of GL2 (Z) on the sets of triangles and edges of the modular triangulation are finite. Thus for any Q-rational GL2 -module V and any subgroup 0 ⊂ GL2 (Z), (2) Hom0 M−∗ , V computes H ∗−1 (0, V ).

Let us return to the description of the image of the Galois group. Let G•• be any bigraded Lie algebra. Consider the bigraded Lie algebra Gb•• := G•• ⊕ Q(−1, −1),

(15)

where Q(−1, −1) is a one-dimensional Lie algebra of the bidegree (−1, −1). The standard cochain complex of Gb•• admits a canonical decomposition ∨ ∨ ⊕ 3∗ G•• ⊗ Q(1, 1). 3∗ Gb••∨ = 3∗ G••

(16)

(l) Strangely enough, it is simpler to describe the structure of the Lie algebra Gb•,≥−2 (l) (l) than G . The Lie algebra structure of Gb is completely described by the com•,≥−2

•,≥−2

mutator map

(l)

(l)

[, ] : 32 Gb•,−1 −→ G•,−2 .

(17)

2.4 (a) The weight w part of the complex THEOREM

δ (l) (l) G•,−2 −→ 32 Gb•,−1 ∨

∨

(18)


409

dual to (17) is isomorphic to the complex ∗ M(2) ⊗GL2 (Z) S w−2 V2 ⊗ Ql .

(19)

(b) One has (l) dim G−w,−2

( =

w : odd,

0, h

w−2 6

i

, w : even.

(20)

Using part (a) and Lemma 2.3, we compute the Euler characteristic of the complex (18). Then we use (6) to get formula (20). A Hodge-theoretic version of Theorem 2.4 (l) appeared first as [G5, Theorem 7.2]. The estimate dim G−w,−2 ≥ [(w − 2)/6] has been independently obtained by Ihara [I4]. In particular, there is a canonical isomorphism ∨ = (l) (l) 1 m 1 : G−w,−2 −→ M(2) ⊗GL2 (Z) S w−2 V2 ⊗ Ql

(21)

(l)

providing a description of the vector space G−w,−2 . To describe the isomorphism of complexes from Theorem 2.4 we need to define the map (l) (l) ∨ = 2 m 2 : 32w Gb•,−1 −→ M(2) ⊗GL2 (Z) S w−2 V2 ⊗ Ql , (22) where 32w is the weight w part of 32 . We do it as follows. The stabilizer in GL2 (Z) of the geodesic from zero to i∞ on the upper half-plane is generated by 0 1 −1 0 and . −1 0 0 1 Therefore the right-hand side of (22) is identified with the space of degree w − 2 polynomials f (t1 , t2 ), skew symmetric in the variables t1 , t2 and of even degree in each of them: f (t1 , t2 ) = − f (t2 , −t1 ) = f (−t1 , t2 ). (l) (l) Let ζM (n) for n > 1 be Soulé’s generator of (G−2n+1,−1 )∨ (see Section 6). We (l)

(l)

define ζM (1) as a generator of Ql (−1, −1) = (G−1,−1 )∨ and set (l)

(l)

(l)

m 2 : ζM (2m + 1) ∧ ζM (2n + 1) 7 −→ t12m t22n − t12n t22m . (l)

(l)

The map m 2 identifies 32 (G•,−1 )∨ with the subspace of polynomials f (t1 , t2 ), skew symmetric in t1 , t2 and of positive even degree in each of them. Thus we get a precise (l) description of the Lie algebra G•,≥−2 as well. The decomposition (16) of complex (18) corresponds to the decomposition of complex (19) into a direct sum of the subcomplex computing the (truncated) cuspidal cohomology and the subcomplex, consisting of a single group in degree 2, computing the Eisenstein part of the cohomology.

410

A. B. GONCHAROV

More precisely, consider the truncated complex τ[1,2] M∗(2) ⊗GL2 (Z) V

1 2 3 := M(2) ⊗GL2 (Z) V −→ Ker M(2) ⊗GL2 (Z) V −→ M(2) ⊗GL2 (Z) V . (23) It is a subcomplex of the complex (13). THEOREM 2.5 The complex

τ[1,2] M∗(2) ⊗GL2 (Z) S w−2 V2 ⊗ Ql

(24)

is canonically isomorphic to the weight w part of the complex (10). (l) 2.4. The Galois action on π1 P1 − {0, 1, ∞}, v∞ : The depth 3 quotient (l) To describe the structure of the Lie algebra G•,≥−3 , we need only define the commutator map (l) (l) (l) [, ] : G•,−2 ⊗ G•,−1 −→ G•,−3 (25) (l)

(l)

obeying the Jacobi identity. Indeed, the commutator [, ] : 32 G•,−1 −→ G•,−2 has been described in Theorem 2.4(a). Dualizing, one sees that we need to describe the (l) depth 3 part of the standard cochain complex of G•,≥−3 , (l) ∨

(l) ∨

(l) ∨

(l) ∨

G•,−3 −→ G•,−2 ⊗ G•,−1 −→ 33 G•,−1 .

(26)

The first map is dual to the map (25). Complex (26) is described in Section 2.6 (see Theorem 2.10). Here is an important corollary. 2.6 The complex (26) computes the cuspidal cohomology i Hcusp GL3 (Z), S w−3 V3 at i = 1, 2, 3.

THEOREM

(a)

(b) (c)

The complex (26) is acyclic. We have (l) dim G−w,−3

( =

0, h

(w−3)2 −1 48

w : even, i

, w : odd.

(27)

By (b) the Euler characteristic of the complex (26) is zero. So using (6) and Theorem 2.4(b), we get formula (27). Remark. Compare Theorems 2.4 and 2.6 with [G4, Theorems 1.4 and 1.5], where the dimension of the Q-space of reduced multiple ζ -values of depth 2 and 3 is estimated


411

from above by (20) and (27). According to some standard conjectures in arithmetic algebraic geometry one should have (l)

G−w,−m = (the space of reduced multiple ζ ’s of weight −w, depth −m) ⊗Q Ql .

This supplies additional evidence for the statement that the estimates given in [G4, Theorems 1.4 and 1.5] are exact. Notice that computation of the dimension of the Q-space of reduced multiple ζ -values seems to be a transcendently difficult problem (we cannot prove that ζ (5) 6 ∈ Q), while its more sophisticated l-adic analog is easier to approach. (l)

To describe results and conjectures about the structure of the Lie algebra G•• we need to recall the definition of the modular complexes given in [G4, Section 5]. 2.5. The modular complexes ∗ Let L m be a rank m lattice. The modular complex M(m) is a complex of left Aut(L m ) = GLm (Z)-modules which sits in the degrees [1, m] and looks as follows: ∂

∂

∂

1 2 m M(m) −→ M(m) −→ · · · −→ M(m) . 1 is the trivial GL (Z)-module. When m = 2, it is isomorphic to By definition, M(1) 1 the complex on the left in (11). In general, its definition is purely combinatorial. We recall it now. (i) Extended basis and homogeneous affine basis of a lattice. We say that an extended basis of a lattice L m is an (m + 1)-tuple of vectors v1 , . . . , vm+1 of the lattice such that v1 + · · · + vm+1 = 0 and v1 , . . . , vm is a basis. Then omitting any of the vectors v1 , . . . , vm+1 , we get a basis of the lattice. The group GLm (Z) acts from the left on the set of bases of L m , considered as columns of vectors (v1 , . . . , vm ). This provides the set of extended bases with the structure of the left principal homogeneous space for GLm (Z). Let u 1 , . . . , u m+1 be elements of the lattice L m such that the set of elements {(u i , 1)} form a basis of L m ⊕ Z. The lattice L m acts on such sets by l : {(u i , 1)} 7 −→ {(u i +l, 1)}. We call the coinvariants of this action the homogeneous affine basis of L m and denote them by {u 1 : · · · : u m+1 }. Notice that {u 1 : · · · : u m+1 } is a homogeneous affine basis if and only if {u 2 − u 1 , u 3 − u 2 , . . . , u 1 − u m+1 } is an extended basis. So there is a canonical bijection

homogeneous affine basis of L m ↔ extended basis of L m , {u 1 : · · · : u m+1 } ↔ {u 2 − u 1 , u 3 − u 2 , . . . , u 1 − u m+1 }.

(28)

1 . The abelian group M 1 (ii) The group M(m) (m) is generated by the elements hv1 , . . . , vm+1 i corresponding to extended basis v1 , · · · , vm+1 of L m . To list the rela-

412

A. B. GONCHAROV

tions we need another set of the generators corresponding to the homogeneous affine basis of L m via (28): hu 1 : · · · : u m+1 i := hu 01 , u 02 , . . . , u 0m+1 i,

u i0 := u i+1 − u i .

Let 6 p,q be the set of all shuffles of the ordered sets {1, . . . , p} and { p+1, . . . , p+q}. Relations For m = 1 we have hv1 , v2 i = hv2 , v1 i. For any 1 ≤ k ≤ m one has X hvσ (1) , . . . , vσ (m) , vm+1 i = 0,

(29)

σ ∈6k,m−k

X

hu σ (1) : · · · : u σ (m) : u m+1 i = 0.

(30)

σ ∈6k,m−k

THEOREM 2.7 The double shuffle relations (29) and (30) imply the following dihedral symmetry relations for m ≥ 2:

hv1 , . . . , vm , vm+1 i = hv2 , . . . , vm+1 , v1 i = h−v1 , . . . , −vm , −vm+1 i = (−1)m+1 hvm+1 , vm , . . . , v1 i. We picture both types of the generators on the circle as shown in Figure 3.

.u

1

u 04

.

u4

u 01

.u

2

u 02

u 03

.u

3

Figure 3

Vectors of homogeneous affine basis are outside the circle, and vectors of extended basis are inside the circle (cf. Section 4.1). k . For each decomposition of L as a direct sum of k nonzero (iii) The group M(m) m i lattices L , we consider the tensor product of M 1 (L i ). We consider M 1 (L i ) as odd 0 and use the sign rule to identify ⊗M 1 (L i ) and ⊗M 1 (L i ) when the decompositions


413

k is defined to be L and L 0 differ only in the ordering of the factors. The group M(m) i the sum over all such unordered decompositions L m = ⊕L of the corresponding ⊗M 1 (L i ). In other words, it is generated by the elements hA1 i ∧ · · · ∧ hAk i where Ai is an extended basis of the sublattice L i and hAi i’s anticommute. Let us set

[v1 , . . . , vk ] := hv1 , . . . , vk , vk+1 i,

v1 + · · · + vk + vk+1 = 0.

1 −→ M 2 by setting ∂ = 0 if m = 1 and We define a homomorphism ∂ : M(m) (m)

∂ : hv1 , . . . , vm+1 i 7−→ − Cyclem+1

m−1 X

[v1 , . . . , vk ] ∧ [vk+1 , . . . , vm ] ,

k=1 • by extending ∂ where the indices are modulo m + 1. We get the differential in M(m) using the Leibniz rule: ∂ [A1 ] ∧ [A2 ] ∧ · · · := ∂ [A1 ] ∧ [A2 ] ∧ · · · − [A1 ] ∧ ∂ [A2 ] ∧ · · · + · · · .

THEOREM 2.8 The map ∂ is well defined and ∂ 2 = 0, so we get a complex.

Remark. Both the definition of the modular complex and these proofs are very similar to the definitions and basic properties of the dihedral Lie algebras discussed in detail in Section 4. Yet we do not know a general framework unifying them. Remark. Let A be an arbitrary commutative ring. Then replacing L m by a free rank m A-module, one can construct modular complexes corresponding to the ring A, recovering the construction above when A = Z. (l) 2.6. A description of the Lie algebra Gb•• via the modular complex Set Vm := L m ⊗ Q.

CONJECTURE 2.9 There exists a canonical isomorphism between the complex ∗ M(m) ⊗GLm (Z) S w−m Vm ⊗Q Ql

(31)

and the depth m, weight w part of the standard cochain complex of the Lie algebra (l) Gb•• . 1 is the trivial GL (Z)-module Z. Thus Example. The rank 1 modular complex M(1) 1 0, w even, 1 M(1) ⊗GL1 (Z) S w−1 V1 = Q, w odd.

414

A. B. GONCHAROV

(l) Thus formula (31) for m = 1 is just equivalent to (6) plus dim Gb−1,−1 = 1. For m = 2 we get the description of the depth 2 part given in Theorem 2.4.

THEOREM 2.10 There exists a canonical isomorphism between the complex ∗ M(3) ⊗GL3 (Z) S w−3 V3 ⊗Q Ql

(32)

and the depth 3, weight w part of the standard cochain complex of the Lie algebra (l) Gb•,≥−3 . THEOREM 2.11 ([G4]) The rank 3 modular complex is quasi-isomorphic to the chain complex of the Voronoi decomposition of the symmetric space H3 for GL3 (R), truncated in the degrees [1, 3].

This is [G4, Theorem 6.2]. To prove it we construct a geometric realization of the rank 3 modular complex as a subcomplex of the truncated Voronoi complex. For the geometric realization of the rank m modular complex, see [G6]–[G9]. Using it, we prove that the rank 4 modular complex, shifted by [−2], is quasi-isomorphic to the chain complex of the Voronoi decomposition of the symmetric space H4 , truncated in the degrees [3, 6]. 2.7. The diagonal Galois Lie algebras and modular complexes (l) The diagonal Lie algebra G• (µ N ) is a Lie subalgebra of the Galois Lie algebra (l) G•• (µ N ): (l) (l) G•(l) (µ N ) := ⊕w≥1 G−w,−w (µ N ) ,→ G•• (µ N ). It is graded by the weight. Equivalently, one can define the diagonal Galois Lie algebra as a Lie subalgebra of GrW G (l) (µ N ) by imposing the depth less than or equal to −w condition on each weight −w subquotient: D W Gr−w G (l) (µ N ) ,→ GrW G (l) (µ N ). G•(l) (µ N ) := ⊕w≥1 F−w

Since the Lie algebra GrW G (l) (µ N ) is noncanonically isomorphic to the Lie algebra G (l) (µ N ), the diagonal Galois Lie algebra is isomorphic, although noncanonically, to a certain Lie subalgebra of G (l) (µ N ). This is an important difference between the Galois Lie algebra and its diagonal part: the Galois Lie algebra in general is not isomorphic to G (l) (µ N ).


415

THEOREM 2.12 There is a canonical isomorphism (l)

G−1,−1 (µ N ) = HomQ (the group of the cyclotomic units in Z[ζ N ]

h1i , Ql ). N

This rather elementary result is a particular case of Theorem 2.1 for w = 1. It suggested the name “higher cyclotomy” for our story. Set E m (N ) := Z 01 (m; N )\ GLm (Z) if m > 1, E 1 (N ) := Z/N Z. CONJECTURE 2.13 Let p be a prime number. Then there exists a canonical isomorphism between the complex ∗ M(m) ⊗GLm (Z) E m ( p) ⊗ Ql (33) (l)

and the depth m part of the standard cochain complex of Gb• (µ p ). Notice that if m > 1, then ∗ ∗ M(m) ⊗01 (m;N ) Q = M(m) ⊗GLm (Z) E m (N ). (l)

The depth greater than or equal to −2 quotient of Gb• (µ p ) is described by the commutator map (l) (l) (34) [, ] : 32 Gb−1 (µ p ) −→ G−2 (µ p ). Let 01 ( p) := 01 (2; p) ∩ SL2 (Z). Consider the modular curve Y1 ( p) := 01 ( p)\H2 . Projecting the modular triangulation of the hyperbolic plane onto Y1 ( p), we get the modular triangulation of Y1 ( p). The complex involution acts on the modular curve preserving the triangulation. Consider the following complex: + the chain complex of the modular triangulation of Y1 ( p) . (35) Here + means the invariants of the action of the complex involution. 2.14 The dual to complex (34) is naturally isomorphic to complex (35). THEOREM

This is a depth 2 analog of Theorem 2.12. In particular, there is canonical isomorphism + ∨ (l) Ql triangles of the modular triangulation of Y1 ( p) = G−2 (µ p ) . (36)

416

A. B. GONCHAROV

It turns out that the subcomplex ∗ τ[1,2] M∗(2) ⊗01 (2; p) Ql ,→ M(2) ⊗01 (2; p) Ql corresponds to the maximal quotient of the Galois group acting on π (l) (X p ), of weight = depth ≥ −2, which is unramified at 1 − ζ p (see Sections 7.9 and 7.13). (l) (l) Now we turn to the depth 3 case. Let L• (µ N ) ,→ G• (µ N ) be the Lie subal(l) gebra generated by L−1 (µ N ). 2.15 (l) Conjecture 2.13 is valid for m = 3 for the Lie algebra Lb• (µ p ). So there exists a canonical isomorphism between the complex (33) and the depth 3 part of the standard cochain complex of this Lie algebra. (l) (l) Let us assume Conjecture 1.1. Then L−3 (µ p ) = G−3 (µ p ), and therefore Conjecture 2.13 is valid for m = 3.

THEOREM

(a)

(b)

(l)

There is a description of the vector space L−3 (µ p ) similar to (36) in terms of certain 4-cells on the 5-dimensional modular variety 01 (3; p)\H3 which were constructed in [G4]. See [G6]–[G9] for a construction of certain 2(m − 1)-cells in Hm which should play a similar role in general. COROLLARY 2.16 If p ≥ 5, then (l)

dim G−2 (µ p ) = (l)

( p − 1)( p − 5) , 12 (l)

dim G−3 (µ p ) ≥ dim L−3 (µ p ) =

( p − 5)( p 2 − 2 p − 11) . 48

2.8. The structure of the paper In Section 3 we explain how to linearize the action of the Galois group (l) Gal(Q/Q(ζl ∞ N )) on π1 (X N , v∞ ). We show that the Lie algebra of the image of the Galois group acts by the so-called special equivariant derivations of L(l) (X N , v∞ ), preserving the two filtrations. In Section 4 we define the dihedral Lie coalgebra D•• (G) of a commutative group G and derive its main properties. Completely similar arguments settle the basic properties of the modular complexes. In Section 5 we study the Lie algebra Der S E L(G) of special equivariant derivations of the free Lie algebra L(G) generated by the set {0}∪G. We realize the dihedral Lie algebra of G as a Lie subalgebra of Gr Der S E L(G). We prove the distribution relations in Section 5.7.


417

The depth 2 and 3 parts of the cohomology of the dihedral Lie algebra of µ N were related in [G4] to the cohomology of the groups 01 (m; N ) with coefficients in S w−m Vm for m = 2, 3. In Section 6 we compute the cohomology groups H ∗ GL3 (Z), S w−3 V3 and H ∗ 01 (3; p), Q . Finally, in Chapter 7 we apply the previous results to prove the theorems from the introduction.

3. The setup 3.1. The action of the Galois group Recall the map (l) (l) 8 N : Gal(Q/Q) −→ Aut π1 (X N , v∞ )

(37)

and the projection (l)

(l)

π1 (X N , v∞ ) −→ π1 (Gm , v∞ ) = Zl (1).

(38)

Let X be a regular curve over Q, let X be the corresponding projective curve, and let v be a tangent vector at x ∈ X . Then there is a natural map of Galois modules (see [D1]): (l) (l) Zl (1) = π1 (Tx X − 0, v) → π1 (X, v). For X = P1 − {0, ∞}, x = ∞, v = v∞ , it is an isomorphism. So for X = X N it provides a splitting of (38): (l)

(l)

I∞ : π1 (Gm , v∞ ) ,→ π1 (X N , v∞ ).

(39)

The subgroup Gal(Q/Q(ζl ∞ )) preserves the elements of the subgroup (l) I∞ Zl (1) ⊂ π1 (X N , v∞ ).

(40)

(l)

Let v, v 0 be tangent vectors at x, x 0 ∈ X . Denote by π1 (X ; v, v 0 ) the pro-l completion of the torsor of the path from v to v 0 . Let η ∈ {0} ∪ µ N ⊂ P1 . Choose a Gal(Q/Q(ζ N ))-invariant tangent vector wη at the point η. Let p be a pro-l path from v∞ to wη . The composition (l) p −1 ◦ π1 P1 − {0, η}, wη ◦ p provides a map (l)

Iη ( p) : Zl (1) −→ π1 (X N , v∞ ),

η ∈ {0} ∪ µ N .

(41)

418

A. B. GONCHAROV (l)

For a different path p 0 the map Iη ( p 0 ) is conjugated to Iη ( p) in π1 . So the conjugacy class of this map is well defined, hence stable by Gal(Q/Q(ζ N )). (l) The restriction of the map 8 N to Gal(Q/Q(ζl ∞ N )) satisfies additional constraints. The group µ N acts on X N by z 7 −→ ζ N z. Moreover, there is a natural action (l) (l) of µ N on π1 (X N , v∞ ) ⊗ Ql (regarding π1 ⊗ Ql , see [D1, Chapter 9]) commuting with the action of Gal(Q/Q(ζl ∞ N )), coming as follows. Let ξ, ζ, ζ 0 ∈ µ N . The action of ξ on Gm provides a tangent vector vξ := ξ∗ v∞ at ∞ and a map (l)

(l)

ξ∗ : π1 (Gm ; vζ , vζ 0 ) −→ π1 (Gm ; vξ ζ , vξ ζ 0 ). (l)

We have a canonical path pζ,ζ 0 in π1 (Gm ; vζ , vζ 0 ) ⊗ Ql such that pζ 0 ,ζ 00 ◦ pζ,ζ 0 = pζ,ζ 00 ,

ξ∗ pζ,ζ 0 = pξ ζ,ξ ζ 0 .

Indeed, the composition of the path provides an isomorphism of the N th power of the (l) torsor of the path from vζ to vξ ζ on Gm with π1 (Gm ) which, being abelian, does not depend on the choice of the base point/vector. It is given by com : p1 ⊗ · · · ⊗ p N −→ ξ∗N −1 p1 ◦ · · · ◦ ξ∗ p N −1 ◦ p N . Then

(l)

pζ,ξ ζ := p ◦ com( p ⊗N )−1/N ∈ π1 (Gm ; vζ , vξ ζ ) ⊗ Ql . (l)

Notice that if (N , l) = 1, then pζ,ζ 0 ∈ π1 (Gm ; vζ , vζ 0 ). There is a natural map (l) (l) π1 P1 − {0, ∞}; vζ , vζ 0 ,→ π1 (X N ; vζ , vζ 0 ) commuting with the action of the Galois group. It provides canonical path (l)

pζ,ζ 0 ∈ π1 (X N ; vζ , vζ 0 ) ⊗ Ql . (l)

We define the action of an element ξ ∈ µ N on α ∈ π1 (X N ; v∞ ) ⊗ Ql by ξ(α) := −1 p1,ξ ◦ ξ∗ (α) ◦ p1,ξ . So we see that Gal(Q/Q(ζζl ∞ N )) preserves the elements of I∞ (Zl (1)), the conjugacy classes of the “loops around zero and ζ Na ” provided by (41), and commutes with the action of the group µ N . Moreover, it is compatible in a natural sense with the maps X M N −→ X M given by z 7 −→ z, z 7 −→ z N (see Section 5.7). 3.2. Passing to the Lie algebras Let L(X N , v∞ ) be the pronilpotent Lie algebra over Q corresponding by the Maltsev theory to the pronilpotent completion of the fundamental group π1 (X N (C), v∞ ) (see [D2, Chapter 9]). We use a shorthand L N for it. L N is isomorphic, not canonically, to


419

the pronilpotent completion of a free Lie algebra with N +1 generators corresponding to the loops around zero and all N th roots of unity. (l) Let L N be the pronilpotent Lie algebra over Ql corresponding to the pro-l group (l) (l) (l) (l) π1 (X N , v∞ ). Namely, set π1 := π1 (X N , v∞ ). Recall that π1 (k) is the lower (l) (l) (l) central series for the group π1 . Then π1 /π1 (k) is an l-adic Lie group, and (l) (l) (l) bQ Ql . L N := lim Lie π1 /π1 (k) = L N ⊗ ←−

Similar to (38) and (39), there is a canonical projection p : L P1 − ({0, ∞} ∪ µ N ), v∞ −→ L P1 − {0, ∞}, v∞

(42)

and its canonical splitting i ∞ : L P1 − {0, ∞}, v∞ = L(T0 P1 , v∞ ) = Q(1) ,→ L P1 − ({0, ∞} ∪ µ N ), v∞ . (43) Just as in Section 1.2, there are well-defined conjugacy classes of “loops around zero or ζ ∈ µ N ” based at v∞ . Set X ∞ := i ∞ . LEMMA 3.1 There exist maps X 0 , X ζ : Q(1) −→ L N which belong to the conjugacy classes of P the “loops around 0, ζ ” such that X 0 + ζ ∈µ N X ζ + X ∞ = 0 and the action of µ N permutes X ζ ’s (i.e., ξ∗ X ζ = X ξ ζ ) and fixes X 0 , X ∞ .

We use the following notation. Let G be a commutative group written multiplicatively. Let L(G) be the free Lie algebra with the generators X i , where i ∈ {0} ∪ G. (We P assume 0 6∈ G.) Set X ∞ := −X 0 − g∈G X g . Denote by L(µ N ) the pronilpotent completion of the Lie algebra L(µ N ). It is isomorphic (noncanonically) to L N . Proof We follow the argument sketched by the referee. Let H be the proalgebraic group over Q consisting of all automorphisms of the Lie algebra L(µ N ) which commute with the action of µ N and preserve the conjugacy classes of the generators X 0 , X ζ , X ∞ . Let T be the H-torsor of all maps X 0 , X ζ , X ∞ : Q(1) −→ L N satisfying all the conditions of the lemma. Then T(C) is nonempty. Indeed, in the De Rham realization of L N (see [D1]), X ζ and X ∞ are dual to the basis d log(z − ζ ), d log(z) in H D1 R (P1 − ({0, ∞} ∪ P µ N ), C) and X 0 := − ζ X ζ − X ∞ . Applying the comparison theorem between the Betti and De Rham realizations, we get an element in T(C). Since H is a prounipotent algebraic group, one has H 1 (Gal(Q/Q), H(Q)) = 0, so T has a Q-point. The lemma is proved.

420

A. B. GONCHAROV

There is a homomorphism (l)

(l)

ϕ N : Gal(Q/Q) −→ Aut L N . (l)

(l)

Let Z • be the lower central series for L N . The quotient L N /Z k is a finitedimensional Lie algebra over Ql . So the image of the subgroup Gal(Q/Q(ζl ∞ N )) (l) (l) in Aut(L N /Z k ) is an l-adic Lie group. Denote by G N the projective limit (over k) of the Lie algebras of these Lie groups. It is a pro-Lie algebra over Ql . It acts by (l) derivations of the Lie algebra L N . We describe the constraints on the derivations we get using a more general setup presented below. 3.3. Special equivariant derivations A derivation D of the Lie algebra L(G) is called special if there are elements Si ∈ L(G) such that D(X i ) = [Si , X i ] for any i ∈ {0} ∪ G,

and

D(X ∞ ) = 0.

(44)

The special derivations of L(G) form a Lie algebra, denoted Der S L(G). Indeed, if D(X i ) = [Si , X i ] and D 0 (X i ) = [Si0 , X i ], then [D, D 0 ](X i ) = [Si00 , X i ],

where Si00 := D(Si0 ) − D 0 (Si ) + [Si0 , Si ].

(45)

The group G acts on the generators by h : X 0 7 −→ X 0 , X g 7−→ X hg . So it acts by automorphisms of the Lie algebra L(G). A derivation D of L(G) is called equivariant if it commutes with the action of G. Let Der S E L(G) be the Lie algebra of all special equivariant derivations of the Lie algebra L(G). 3.4. The weight and depth filtration on L N There are two increasing filtrations by ideals on the Lie algebra L N , indexed by integers n ≤ 0. The weight filtration F•W . It coincides with the lower central series for L N , W L N = F−1 LN ,

W W F−n−1 L N := [F−n L N , L N ].

The depth filtration F•D . Let I N be the kernel of projection (42). Its powers give the depth filtration F0D L N = L N ,

D F−1 LN = IN ,

D D F−n−1 L N = [I N , F−n L N ].

The weight filtration can be defined on the Lie algebra corresponding to the pronilpotent completion of the fundamental group of an arbitrary algebraic variety.


421

The weights on L N are obtained by dividing by 2 the usual weights. The weight filtration admits a splitting; that is, it is defined by a grading. These filtrations induce two filtrations on the Lie algebra Der S E L N . Taking the SE L associated graded with respect to these filtrations, we get a Lie algebra Gr Der•• N bigraded by the weight −w and depth −m. 3.5. The weight grading and depth filtration on Der S L(G) The Lie algebra L(G) is bigraded by the weight and depth. Namely, the free generators X 0 , X g are bihomogeneous: they are of weight −1, X 0 is of depth zero, and the X g ’s are of depth −1. Each of the gradings induces a filtration of L(G). The weight filtration is given by the lower central series. It goes from −∞ to −1. Let I be the kernel of the natural projection L(G) −→ Q given by X g 7 −→ 0, X 0 7 −→ 1. Its powers provide the depth filtration on L(G). It goes from −∞ to zero. The Lie algebra Der L(G) is bigraded by the weight and depth. Its Lie subalgebras Der S L(G) and Der S E L(G) are compatible with the weight grading. However they are not compatible with the depth grading. Therefore they are graded by the weight and filtered by the depth. A derivation (44) is of depth −m if each S j mod X j is of depth −m, that is, there are at least m X i ’s different from X 0 in S j mod X j . S E L(G) be the The depth filtration is compatible with the weight grading. Let Gr Der•• associated graded for the depth filtration. One of our key tools is the following theorem, proved in Section 5. THEOREM 3.2 Let G be a finite commutative group. Then there exists an injective morphism of bigraded Lie algebras SE ξG : D•• (G) ,→ Gr Der•• L(G). (l)

3.6. The problem of describing the map ϕ N There is no canonical choice of the generators X 0 , X ζ of the Lie algebra L N satisfying all the conditions of Lemma 3.1. However, their projections in L N /[L N , L N ] are independent of the choice involved. So we have a canonical isomorphism =

i : L(µ N ) −→ Gr•W L N .

(46)

It preserves the depth filtration. Remark. Let L(X, x) be the fundamental Lie algebra of X based at a point/tangent vector x. If y is another base point/vector, then there is a torsor of isomorphisms L(X N , x) −→ L(X N , y) defined up to a conjugation. However, the isomorphism of

422

A. B. GONCHAROV

the associated graded for the lower series filtration L ∗ is a canonical isomorphism i x,y : Gr L L(X, x) −→ Gr L L(X, y). If X = X N , then the lower series filtration coincides with the weight filtration. Thus the graded Lie algebra GrW L N := GrW L(X N , v∞ ) does not depend on the choice of the base point or tangent vector v∞ . So (46) is indeed a canonical isomorphism. (l)

It follows from Section 3.2 that G N acts by special equivariant derivations of the Lie (l) algebra L N , that is, (l) (l) G N ⊂ Der S E L N . (47) (l)

(l)

The map ϕ N obviously respects the two filtrations. The filtrations on Der S E L N in(l) (l) duce two filtrations on G N . Let G•• (µ N ) be the associated graded of the Lie algebra (l) (l) S E L(l) which, as was stressed by the G N . There is an inclusion G•• (µ N ) ⊂ Gr Der•• N referee, is provided by the fact that the weight filtration admits a splitting, that is, is defined by a grading, and both the depth filtration and the subspace (47) are compatible with this grading. Such a weight splitting is provided by the eigenspaces of a Frobenius F p , p 6 | N . Notice that F p normalizes the normal subgroup Gal(Q/Q(ζl ∞ N )). Let W• L be a filtration on a vector space L. A splitting ϕ : GrW L −→ L of the filtration leads to an isomorphism ϕ ∗ : End(L) −→ End(GrW L). The space End(L) inherits a natural filtration, while End(GrW L) is graded. The map ϕ ∗ respects the corresponding filtrations. The map Gr ϕ ∗ : GrW (End L) −→ End(GrW L) does not depend on the choice of the splitting. Applying this to the case L = L N , we get a canonical isomorphism ∼

GrW (Der S E L N ) = Der S E L(µ N )

(48)

respecting the weight grading. Thus there is a canonical injective morphism (l) ∼

(l) SE SE G•• (µ N ) ,→ Gr Der•• L N = Gr Der•• L(µ N ) ⊗ Ql .

(49)

The goal of this paper is to study Lie subalgebra (49). It is canonically isomorphic to the Galois Lie algebra introduced in Section 2. Indeed, the Lie algebras of images of the following two maps coincide: (l) Gal Q/Q(ζl ∞ N ) −→ Aut(π1 (X N , v∞ )[−w,−m] , (l) W D + F−m−1 ). Gal Q/Q(ζl ∞ N ) −→ Aut L N /(F−w−1


423

(l)

Why do we take the associated graded of G N for the weight and depth filtrations? (l) (l) The Lie algebra G N is isomorphic to Gr•W G N , but this isomorphism is not canonical. (l) Gr•W G N is a Lie subalgebra of GrW (Der S E L N ) ⊗ Ql . Via canonical isomorphism (48) it becomes a Lie subalgebra of Der S E L(µ N ) ⊗ Ql , which has natural generators provided by the canonical generators of L(µ N ) (see Section 5.2). This gives canonical (l) “coordinates” for the description of elements of Gr•W G N . Another reason to consider (l) the Lie algebra Gr•W G N is provided by its motivic interpretation (see Section 3.7). The surprising benefit of taking its associated graded for the depth filtration is an unexpected relation with the geometry of modular varieties for GLm , where m is the depth. Remark. Usually the associated graded for the depth filtration is not isomorphic to a subalgebra of a Lie algebra of any quotient of the Galois group. The situation is different in the following two cases when the associated graded for the depth filtration is isomorphic to the original Lie algebra. (l) (l) D GrW G (l) (i) In the depth 2 case, G•,≥−2 (µ N ) is isomorphic to GrW G N /F−3 N because there is no room for the difference. (ii) In the diagonal case, as we noticed in Section 2.7, the diagonal Lie algebra (l) defined as a Lie subalgebra of GrW G•• (µ N ) is noncanonically isomorphic to (l) a Lie subalgebra of GrW G N . (l)

3.7. Motivic interpretation of the Lie algebra Gr•W G N Let us recall the mixed Tate category of lisse l-adic sheaves over a base S. When S is the spectrum of a number field F, or the spectrum of a ring of integers in F minus a finite number of places, such a category was considered by R. Hain and M. Matsumoto in their preprint [HM]. Our exposition follows very closely the first four pages of Chapter 1 in the preprint by Beilinson and Deligne [BD] where the general S is treated. For a connected coherent scheme S over Z[1/l] such that µl ∞ 6⊂ O ∗ (S), denote by FQl (S) the Tannakian category of lisse Ql -sheaves on S. There are the Tate sheaves Ql (m) := Ql (m) S := Ql (1)⊗m S . Thanks to our condition they are mutually nonisomorphic. Call an object of FQl (S) a mixed Tate object if it admits a finite increasing filtration W , indexed by Z, such that GrkW is a direct sum of copies of Ql (−k). Let T FQl (S) be the full subcategory of mixed Tate objects in FQl (S). Then it is a Tannakian Ql -category, and obviously one has Ext1T FQ (S) Ql (0), Ql (m) = 0 for m ≤ 0. l

(This may not be so for the Ext1 in the category FQl (S).) So T FQl (S) is a mixed Tate category in the terminology of [BD]. It is easy to deduce from this that any of

424

A. B. GONCHAROV

its objects admits a unique filtration W such that GrkW is a direct sum of copies of Ql (−k), and any morphism is strictly compatible with W . There is canonical functor ω to the category of graded Ql -vector spaces W ω : X 7 −→ ⊕m Hom Ql (m), Gr−m X . It is an exact functor commuting with the ⊗-product. It is called the canonical fiber functor. Let e L T Ql (S) be the Lie algebra of all ⊗-derivations of ω. By definition a degree i element α ∈ L T Ql (S)i is a collection of natural transformations α j : ω j −→ ω j+i such that α M⊗N = α M ⊗ Idω(N ) + Idω(M) ⊗ α N and α j M(1) = α( j+1)M . Then e L T Ql (S) is a pro-Lie algebra over Ql graded by integers n ≤ 0. Let L T Ql (S) be its part graded by integers n < 0. It is a graded Lie algebra called the fundamental Lie algebra of the mixed Tate category T FQl (S). The fiber functor provides an equivalence between the category T FQl (S) and the category of finite-dimensional graded L T Ql (S)-modules. The standard Tannakian formalism in this case works as follows. Forgetting the graded structure on ω(X ), we get a fiber functor e ω. Its ⊗-automorphisms provide a group scheme over Ql which is a direct product of Gm and a prounipotent group. Then e L T Ql (S) is its Lie algebra, and L T Ql (S) is the Lie algebra of the prounipotent part. The grading on L T Ql (S) is provided by the action of Gm . From now on, let S be the spectrum of the ring of integers of a number field F punctured in a finite set S containing all primes above l. Then FQl (S) is identified with the category of finite-dimensional l-adic representations of Gal(F/F) unramified outside of S . The underlying vector space of a Galois representation provides a fiber functor on FQl (S) and thus another fiber functor on the category T FQl (S). It follows that for any Galois representation V from the category T FQl (S) the Lie algebra of (Zariski closure of) the image of the Galois group in Aut V is isomorphic to the image of e L T Ql (S) acting on e ω(V ). Notice that the image of a Galois group in a continuous representation acting in a finite-dimensional vector space V over Ql is an l-adic Lie group; indeed, it is a compact, hence a closed subgroup of Aut V , and thus by the l-adic Cartan theorem it is an l-adic Lie group. Further, any Lie subgroup of a nilpotent l-adic Lie group is an algebraic group over Ql . So we may drop “Zariski closure” above. Moreover, let GV be the Lie algebra of the image of Gal(F/F(ζl ∞ )) in Aut V . Then the Lie algebra GV is isomorphic to the image of L T Ql (S) in Der ω(V ). This isomorphism is canonical for GrW GV . As noticed in [BD, Section 1.2.3], it follows from a theorem of Soulé [So] that the l-adic regulator map provides canonical isomorphism = γ i HM S, Q(m) ⊗ Ql := grm K 2m−i (S) ⊗ Ql −→ ExtiT FQ (S) Ql (0), Ql (m) . (50) l

Indeed, let G FS be the Galois group of the maximal extension of F unramified out-


425

side S . Soulé proved that the l-adic regulator map provides isomorphism = i HM S, Q(m) ⊗ Ql −→ H i G FS , Ql (m) for i = 1, m ≥ 1, and also for i = 2, m ≥ 2 where both groups are zero. For the left one this follows from the Borel theorem, the l = 2 case (see [HW, Section B4]). By the very definitions for m ≥ 0, Ext1T FQ (S) Ql (0), Ql (m) = Ext1FQ (S) Ql (0), Ql (m) = H 1 G FS , Ql (m) , l

l

so (50) follows for the i = 1 case. Let us show that Ext2T FQ (S) Ql (0), Ql (m) ⊂ Ext2FQ (S) Ql (0), Ql (m) . l

(51)

l

Let A be an abelian category. Every class in the Yoneda Ext2A (A, C) is a product of certain classes in Ext1A (A, B) and Ext1A (B, C) for some object B. We need the following well-known fact (a reference for which we were not able to find). 3.3 1 (A, B), and let β ∈ E xt 1 (B, C). Then β◦α = 0 in E xt 2 (A, C) if and Let α ∈ E xtA A A only if there exists an object M with a three-step filtration 0 ⊂ M0 ⊂ M1 ⊂ M2 = M such that M0 = C, M1 /M0 = B, M2 /M1 = A, and LEMMA

0 −→ M0 −→ M1 −→ M1 /M0 −→ 0 represents β, 0 −→ M1 /M0 −→ M2 /M0 −→ M2 /M1 −→ 0 represents α. Proof Choose extensions p

0 −→ C −→ X −→ B −→ 0

and

0 −→ B −→ Y −→ A −→ 0

representing β and α. Set M0 := C, M1 := X . Then the existence of an object M as above is equivalent to existence of an extension 0 −→ X −→ M −→ A −→ 0 providing a class γ ∈ Ext1A (A, X ) such that its pushforward p∗ (γ ) by the map p : p

X −→ B is isomorphic to α. The exact sequence 0 −→ C −→ X −→ B −→ 0 provides a long exact sequence ∂

p∗

· · · −→ Ext1A (A, X ) −→ Ext1A (A, B) −→ Ext2A (A, C) −→ · · · . Since ∂( p∗ (α)) = β ◦ α, the statement follows. Let us prove (51). A class in Ext2T FQ (S) (Ql (0), Ql (m)) is a product of some classes l

in Ext1 (Ql (0), B) and Ext1 (B, Ql (m)). If their product is zero in Ext2FQ (S) , the fill tration on a corresponding object M provided by the lemma above can be used as

426

A. B. GONCHAROV

a weight filtration, so M belongs to the category T FQl (S), and hence the class in Ext2T FQ (S) (Ql (0), Ql (m)) is zero. l For m ≥ 2 one has Ext2FQ (S) Ql (0), Ql (m) = H 2 G FS , Ql (m) = 0. l

Combining with (51), we see that Ext2T FQ (S) (Ql (0), Ql (m)) = 0 for m ≥ 2. l

Finally, Ext2T FQ (S) (Ql (0), Ql (m)) = 0 for m < 2. Indeed, it follows from the l equivalence between the category T FQl (S) and the category of finite-dimensional graded L T Ql (S)-modules that i L T Ql (S), Ql , ExtiT FQ (S) Ql (0), Ql (m) = H(m) l

i means the degree m part of H i . Since the Lie algebra L where H(m) T Ql (S) is graded i by negative integers, H(m) (L T Ql (S), Ql ) = 0 for m < i. Therefore

Ext2T FQ (S) Ql (0), Ql (m) = 0

for all m.

l

This just means that the fundamental Lie algebra L T Ql (S) is a free negatively graded Lie algebra generated in degree −m by the dual to Ql -vector space K 2m−1 (S) ⊗ Ql . Now, let S 0 := S − {l} and S 0 be the spectrum of the ring of S 0 -integers. Then the l-adic regulator map i HM S 0 , Q(m) ⊗ Ql −→ Ext1T FQ (S) Ql (0), Ql (m) (52) l

is still an isomorphism i = 1, 2, and m ≥ 2 since each of the groups does not change when we delete a closed point from the spectrum. If i = 1, m = 1, map (52) is injective but not an isomorphism since 1 HM S 0 , Q(1) ⊗ Ql = O S∗0 ⊗ Ql ,→ Ext1T FQ (S) Ql (0), Ql (1) = E S ⊗ Ql , l

where E S is the group of S-units. Thus L T Ql (S 0 ) is a quotient of L T Ql (S), and the space of generators of these Lie algebras differ only in the degree −1. Since Gm − µ N has a good reduction outside of N , in our case S = S N ∪ {l} and (l) the Lie algebra L N is a pro-object in T FQl (S N ∪ {l}). So the Lie algebra Gr•W G N is canonically isomorphic to the image of the Lie algebra L T Ql (S N ∪ {l}) acting by derivations of the pro-object 9(L N ). However, it follows from the distribution relations (see Section 5.7) that the degree −1 component of Lie algebra L T Ql (S N ∪ {l}) always acts through its motivic quotient dual to the subspace O S∗0 ⊗ Ql . Moreover, let L T M (S) be the fundamental Lie algebra of the mixed Tate category of mixed Tate motives over S, defined in [G8]. Then L T Ql (S) = L T M (S)⊗Ql .


427

We do not use this in the present paper. See also [G3, Section 3] for the formalism of motivic Lie algebras in the general setup. (l) Deligne’s arguments show that the Lie algebra Gr•W G N is free for N = 2 (see [D2]) and N = 3, 4 (see [D3]). We prove in Section 7.10 that it is not free for sufficiently big N ; for instance, for prime p ≥ 5: the modular forms on 01 ( p) provide an obstruction to freeness.

4. The dihedral Lie coalgebra 4.1. Definitions Let G and H be two commutative groups or, more generally, two commutative group schemes. Then, generalizing a construction given in [G4], one can define a graded Lie coalgebra D• (G|H ), called the dihedral Lie algebra of G and H . In the special case when H = Spec Q[[t]] is the additive group of the formal line, it essentially coincides with a bigraded Lie coalgebra D•• (G) defined in [G4] and called the dihedral Lie algebra of G. (The second grading is related to the natural filtration on Q[[t]].) The construction of D• (G|H ) is left as an easy exercise (see the end of Section 4.5). Below we give a version of the definition of the bigraded Lie coalgebra D•• (G) = ⊕w≥m≥1 Dw,m (G)

given in [G4]. We use it only when G = µ N . The Q-vector space Dw,m (G) is generated by the symbols In 1 ,...,n m (g1 : · · · : gm+1 ),

w = n1 + · · · + nm ,

n i ≥ 1.

(53)

To define the relations we introduce the generating series {g1 : · · · : gm+1 |t1 : · · · : tm+1 } X := In 1 ,...,n m (g1 : · · · : gm+1 )(t1 − tm+1 )n 1 −1 · · · (tm − tm+1 )n m −1 . (54) n i >0

Thus {g1 : · · · : gm+1 |t1 : · · · : tm+1 } = {g1 : · · · : gm+1 |t + t1 : · · · : t + tm+1 }.

(55)

We think about the generating series (54) as a function of m + 1 pairs of variables (g1 , t1 ), . . . , (gm+1 , tm+1 ) located cyclically on an oriented circle (and called a dihedral word) as follows. The oriented circle has slots where the g’s sit and, in between the consecutive slots, dual slots where t’s sit (see Figure 4). The slots are marked by black points and the dual slots by little circles.

428

A. B. GONCHAROV

.

g1 t5 g5

t1

.g

2

.

t4

t2

.g

.

{g1 : · · · : g5 | t1 : · · · : t5 }

3

g4

t3 Figure 4

We also need two other generating series: {g1 : · · · : gm+1 |t1 , . . . , tm+1 } := {g1 : · · · : gm+1 |t1 : t1 + t2 : · · · : t1 + · · · + tm : 0}, (56) where t1 + · · · + tm+1 = 0, and {g1 , . . . , gm+1 |t1 : · · · : tm+1 } := {1 : g1 : g1 g2 : · · · : g1 · · · gm |t1 : · · · : tm+1 }, (57) where g1 · . . . · gm+1 = 1. To make these definitions more transparent, set gi0 := gi−1 gi+1 ,

ti0 := −ti−1 + ti ,

and put them on the circle together with g’s and t’s as shown in Figure 5.

.

.

.

.

g1

t1

t10

g10

.

g2

t1

t20

g20

.

g3

t30

. . .

Figure 5

Then it is easy to check that 0 {g1 : · · · : gm+1 |t1 : · · · : tm+1 } = {g1 : · · · : gm+1 |t10 , . . . , tm+1 } 0 = {g10 , . . . , gm+1 |t1 : · · · : tm+1 }.

(58)

So we have 0 {g1 : · · · : gm+1 |t1 , . . . , tm+1 } = {g10 , . . . , gm+1 |t1 : t1 + t2 : · · · : t1 + · · · + tm : 0}, (59) 0 {g1 , . . . , gm+1 |t1 : · · · : tm+1 } = {1 : g1 : g1 g2 : · · · : g1 · · · gm |t10 , t20 , . . . , tm+1 }.

(60)


429

We picture the three generating series (58) on an oriented circle. Namely, for each of them we leave on the circle only the two sets of variables among g’s, g 0 ’s, t’s, and t 0 ’s which appear in this generating series (see Figure 6).

.

t10

.

t20

.

t1

.

g1

g2 t3

t30

g3

{g1 : g2 : g3 | t10 , t20 , t30 }

g10

g30

.

.

g20 t2

{g10 , g20 , g30 | t1 : t2 : t3 }

Figure 6

The {:}-variables are outside, and the {, }-variables are inside of the circle. (One can have a fourth type of generating series, but it plays no role in our story.) For the {g : |t, }-generating series the variables sit only at slots and, for the {g, |t :}-generating series, only at dual slots. Relations (i) Homogeneity. For any g ∈ G one has {g · g1 : · · · : g · gm+1 |t1 : · · · : tm+1 } = {g1 : · · · : gm+1 |t1 : · · · : tm+1 }.

(61)

(A similar relation for t’s, when ti 7 −→ ti + t, is true by the definition (54).) (ii) The double shuffle relations ( p + q = m, p ≥ 1, q ≥ 1). We have p,q

s1 (g1 , . . . ,gm , gm+1 |t1 : · · · : tm : tm+1 ) X := {gσ (1) , . . . , gσ (m) , gm+1 |tσ (1) : · · · : tσ (m) : tm+1 } = 0, σ ∈6 p,q

(62) p,q s2 (g1

: · · · :gm : gm+1 |t1 , . . . , tm , tm+1 ) X := {gσ (1) : · · · : gσ (m) : gm+1 |tσ (1) , . . . , tσ (m) , tm+1 } = 0. σ ∈6 p,q

(63) (iii) The distribution relations. Let l ∈ Z. Suppose that the l-torsion subgroup G l of G is finite and its order is divisible by l. Then if x1 , . . . , xm are l-powers, 1 X {y1 : · · · : ym+1 |l · t1 : · · · : l · tm+1 } = 0, {x1 : · · · : xm+1 |t1 : · · · : tm+1 } − |G l | l yi =xi

430

A. B. GONCHAROV

except that a constant (in t) is allowed when m = 1 and x1 = x2 , so that one has P I1 (e : e) − yl =e I1 (y : e) is not necessarily zero. (iv) I1 (e : e) = 0. (In [G4] we did not impose this condition.) b•• (G) the bigraded vector space defined by conditions (i)–(iii) only. Then Denote by D b•• (G) = D•• (G) ⊕ Q(1,1) where Q(1,1) is generated by I1 (e : e). D Remark. If m ≥ 2, the relation (63) for g1 , . . . , gm+1 = e implies that I1,...,1 (e : · · · : e) = 0, and (iv) requires this vanishing to hold for m = 1 as well. Remark. To define a map from D•• (G) to a vector space V amounts to defining generating series {g1 : · · · : gm+1 |t1 : · · · : tm+1 } with coefficients in V , that is, in the V [[t1 , . . . , tm+1 ]], obeying (55) and (i)–(v). Remark. Altering one or both of the conditions (iii) and (iv), we can still get important Lie coalgebras. 4.2. The dihedral symmetry relations By definition they consist of the following list of relations: The cyclic symmetry relations {g1 : g2 : · · · : gm+1 |t1 : t2 : · · · : tm+1 } = {g2 : · · · : gm+1 : g1 |t2 : · · · : tm+1 : t1 }, (64) The reflection relations {g1 : · · · : gm+1 |t1 : · · · : tm+1 } = (−1)m+1 {gm+1 : · · · : g1 | − tm : · · · : −t1 : −tm+1 }, (65) The inversion relations −1 {x1−1 : · · · : xm+1 |t1 : t2 : · · · : tm+1 } = {x1 : · · · : xm+1 | − t1 : · · · : −tm+1 }. (66)

The inversion relations are precisely the distribution relations for l = −1. One can check that the reflection relations (65) for the {a, |b :}-generating series look as follows: {g1 , . . . , gm+1 |t1 : · · · : tm+1 } = (−1)m+1 {gm+1 , . . . , g1 |tm+1 : · · · : t1 }.

(67)

Using (66), one checks the same identity for the {a : |b, }-generating series. This tells the effect of changing the orientation of the circle. The dihedral symmetry for m = 1 reduces to the inversion relation.


431

THEOREM 4.1 The double shuffle relations in the case m ≥ 2 imply the dihedral symmetry relations.

Proof One has m X

(−1)k s2k,m−k (gk : gk−1 : · · · : g1 : gk+1 : · · · : gm : gm+1 |

k=0

tk , tk−1 , . . . , t1 , tk+1 , . . . , tm , tm+1 ) = 0. Indeed, let us decompose the set of all shuffles of the ordered sets {k, k −1, . . . , 1} and {k + 1, k + 2, . . . , m} into a union of two subsets, denoted Sk0 and Sk 00 . The subset Sk 00 consists of those shuffles where k + 1 appears on the left of k. After the summation 0 . This with signs the terms corresponding to Sk 00 cancel the ones corresponding to Sk+1 means that {g1 : g2 : · · · : gm : gm+1 |t1 , t2 , . . . , tm , tm+1 } = (−1)m+1 {gm : gm−1 : · · · : g1 : gm+1 |tm , tm−1 , . . . , t1 , tm+1 }. (68) Similarly, {g1 , g2 , . . . , gm , gm+1 |t1 : t2 : · · · : tm : tm+1 } = (−1)m+1 {gm , gm−1 , . . . , g1 , gm+1 |tm : tm−1 : · · · : t1 : tm+1 }. (69) To simplify the formulas below we use notation h i := gi0 = gi−1 gi+1 . Using the identities we just got, we have (58)

0 {h 1 , . . . ,h m+1 |t1 : · · · : tm+1 } = {g1 : · · · : gm+1 |t10 , . . . , tm+1 } (68)

0 0 = (−1)m+1 {gm : gm−1 : · · · : g1 : gm+1 |tm0 , tm−1 , . . . , t10 , tm+1 }

(59)

−1 −1 −1 −1 = (−1)m+1 {h −1 m−1 , h m−2 , . . . , h 1 , h m+1 , h m |

tm − tm−1 : tm − tm−2 : · · · : tm − t1 : tm − tm+1 : 0} (69)

−1 −1 −1 −1 = {h −1 m+1 , h 1 , h 2 , . . . , h m−1 , h m | − tm+1 : −t1 : −t2 : · · · : −tm−1 : −tm }.

Therefore we get {g1 , g2 , . . . , gm+1 |t1 : t2 : · · · : tm+1 } −1 −1 = {gm+1 , g1−1 , g2−1 , . . . , gm | − tm+1 : −t1 : −t2 : · · · : −tm }. (70)

432

A. B. GONCHAROV

It is not difficult to check that {g1 , . . . ,gm+1 |t1 : · · · : tm+1 }

(71)

+ s1m−1,1 (g1 , . . . , gm−1 , gm+1 , gm |t1 : · · · : tm−1 : tm+1 : tm ) − {gm+1 , g1 , . . . , gm |tm+1 : t1 : · · · : tm } X = {g1 , gσ (2) , . . . , gσ (m+1) |t1 : tσ (2) : · · · : tσ (m+1) },

(72) (73)

where the sum is over all shuffles of the sets {2, 3, . . . , m} and {m + 1}. Expression (73) does not have the form of a shuffle relation. However, applying (70), we see that it is equal to a shuffle relation. Therefore relation (71) = (72) provides the cyclic symmetry for the generators (57). Applying (70), we get inversion relations (66) for these generators. Finally, using (69) and the cyclic relations, we get reflection relation (67) for them. So we prove the dihedral symmetry for the generators (57) and thus for the generators (56) and (54). Theorem 4.1 is proved. Proof of Theorem 2.7 It is identical to the proof of Theorem 4.1 after we suppress t 0 s, rename g’s by v’s, and use the additive notations instead of the multiplicative. COROLLARY 4.2 If G is a trivial group, then Dw,m = 0 if w + m is odd.

Proof One has {e : · · · : e|t1 , . . . , tm , 0} = {e : · · · : e| − t1 , . . . , −tm , 0} by the inversion relation. This immediately leads to the statement. 4.3. Motivation: Relation with multiple polylogarithms when G = µ N There are several natural sets of the generators of the vector spaces Dw,m (G) which are reminiscent of different definitions of multiple polylogarithms. It is useful to keep them in mind in order to trace the origins of the definitions given above. (i) Let x1 · · · xm+1 = 1. We define the generators L n 1 ,...,n m (x1 , . . . , xm ) as the coefficients of the generating series (57) when tm+1 := 0: X {x1 , . . . , xm+1 |t1 : · · · : tm : 0} =: L n 1 ,...,n m (x1 , . . . , xm )t1n 1 −1 · · · tmn m −1 . n i >0

If G = µ N , they are related to multiple polylogarithms Li n 1 ,...,n m (x1 , . . . , xm ) :=

X 0 0, m ≥ 1.

We package them into the generating series X n e {g0 : · · · : gm |t0 : · · · : tm } e := In 0 ,...,n m (g0 : · · · : gm )t0 0 · · · tmn m .

(91)

n i >0

The relations are the following: the G-homogeneity for any h ∈ G is {hg0 : · · · : hgm |t0 : · · · : tm } e = {g0 : · · · : gm |t0 : · · · : tm } e ;

(92)

the cyclic symmetry (78), and in addition the constant term of (91), is zero when the g 0 s are all equal, that is, e I1,...,1 (e : · · · : e) = 0. (93) The first part of the proof of Theorem 4.3 shows that formula (81) provides a bigraded e•• (G). Lie coalgebra structure on D For A the associative algebra freely generated by a finite set S, the vector space C (A) := A/[A, A] has as its basis the cyclic words in S. We denote by C : A −→ C (A) the natural projection. The algebra A is the universal enveloping algebra of the free Lie algebra generated by S and as such has a commutative coproduct 1. If we identify A with its graded dual, using the basis afforded by words in S, this coproduct dualizes into the shuffle product X (s1 · · · sk ) ◦Sh (sk+1 · · · sk+l ) := sσ (1) · · · sσ (k+l) . (94) 6k,l

Let A(G) be the free associative algebra with the generators Y and X g , g ∈ G. The group G acts on the generators of A(G) as in Section 1.4. So it acts by automorphisms of A(G). Let Ce(A(G)) be the quotient of C (A(G)) by the subspace generated by Y n , X gn , n ≥ 0. LEMMA 4.5 There is a canonical isomorphism of Q-vector spaces

e•• (G) −→ Ce A(G) , η:D G e In 0 ,...,n m (g0 : · · · : gm ) 7 −→ C X g0 Y n 0 −1 · · · X gm−1 Y n m−1 −1 X gm Y n m −1 .

(95)

442

A. B. GONCHAROV

Proof (i) Cyclic invariance of e I corresponds to cyclic words being considered, homogeneity in g for e I corresponds to taking coinvariants, and the relation (93) corresponds to the relation X gn = 0 (n ≥ 1). We divide as well by Y n (n ≥ 0), which otherwise would have been left out of the image. e•• (G) as the image under the isomorphism η−1 (ii) Define a shuffle relation in D of C X e · (Y n 0 −1 X g1 Y n 1 −1 · · · X gk Y n k −1 ) ◦Sh (Y m 0 −1 X h 1 Y m 1 −1 · · · X hl Y m l −1 ) , (96) 0 (G) where the expressions in each of the parentheses (·) are nonempty. We define D•• e•• (G) by the subspace generated by these shuffle relations. as the quotient of D 00 (iii) Let Dw,m (G) be the Q-vector space generated by the symbols (53) subject to the relations (61), (64), (63), and Section 4.1, condition (iv). So it has the same generators as Dw,m (G), but the relations are relaxed. 4.6 00 (G) −→ D 0 There is an isomorphism i : Dw,m w,m (G) given by PROPOSITION

i : In 1 ,...,n m (g1 : · · · : gm+1 ) 7 −→ e In 1 ,...,n m ,1 (g1 : · · · : gm+1 ).

(97)

Proof Comparison of the spaces D 0 and D 00 . Defining both in terms of the {g : |t :}generating series one requires for both: G-homogeneity, cyclic invariance, nullity of the constant terms of the {e : e : · · · : et|} (this holds for D 00 for m = 1 by the definition, and for m > 1 by the {g : |t, }-shuffle relation); for D 00 : t-homogeneity, {g : |t, }-shuffle; for D 0 : shuffle (96). The map i is well defined. Observe that (96) for X e · {Y ◦sh (. . .)} gives in D 0 that ∂t {g0 : · · · : gm |t0 + t : · · · : tm + t} = 0

at t = 0,

(98)

hence the t-homogeneity. Finally, one can show that the {g : |t, }-shuffle relations (63) go under the map i precisely to the space η−1 the subspace of the shuffle relations (96) with n 0 = m 0 = 1 . (99) (In fact, in [G4] relations (63) appeared as a way to write the shuffle relations (96) with n 0 = m 0 = 1.)


443

Surjectivity of the map i. For m = 0 the t-homogeneity (98) implies {g|t} = 0, that is, e In (g0 ) = 0 in D 0 . The t-homogeneity (98) in D 0 allows us to express e In 0 ,n 1 ,...,n m (g0 : g1 : · · · : gm ) e via I1,n 1 ,...,n m (g0 : g1 : · · · : gm ) for m ≥ 1. More precisely, let S be the subspace S

of A(G)G generated by elements X e {Y ◦Sh A }. We write a = b if a − b ∈ S. Then t-homogeneity is equivalent to the formula X e Y n 0 −1 X g1 Y n 1 −1 · . . . · X gm Y n m −1 S = (−1)n 0 −1 X e X g1 (Y n 0 −1 ) ◦Sh (Y n 1 −1 X g2 Y n 2 −1 · . . . · X gm Y n m −1 ) . (100) So map i is surjective. Remark. Applying η−1 to formula (100), we get relations (77). Injectivity of map i. We need to show that general shuffle relations (96), X e (Y n 0 −1 X g1 · . . .) ◦Sh (Y m 0 −1 X h 1 · . . .) , belong to the subspace of A(G)G generated by S and the shuffle relations (96) with n 0 = m 0 = 1. Indeed, by (100), X e {Y n 0 −1 X g1 · . . .} = X e (Y ◦Sh A ) + X e {X g1 · . . .}. Since obviously X e {(Y ◦Sh A1 ) ◦Sh A2 } = X e {Y ◦Sh (A1 ◦Sh A2 )} , we see that for B := Y m 0 −1 X h 1 · (. . .) one has S X e (Y n 0 −1 X g1 · . . . ) ◦Sh B = X e (X g1 · . . .) ◦Sh B . S

Similarly, X e {B } = X e {X h 1 · . . .}; thus S X e B ◦Sh (X g1 · . . .) = X e (X h 1 · . . .) ◦Sh (X g1 · . . .) . The statement is proved. PROPOSITION 4.7 0 (G) of D e•• (G) inherits a cobracket from D e. The quotient D••

Proof Formulas (80) and (81) define a cobracket on D 00 . This cobracket, transported by i to e. For another proof, see the end of Section 5.4. D 0 , is induced from D 0 (G). The Lie coalgebra D•• (G) is a quotient of D••

444

A. B. GONCHAROV

5. The dihedral Lie algebras and special equivariant derivations 5.1. The special derivations and cyclic words (after [Dr], [K]) Let A be the free associative algebra generated by a finite set S. We define a Lie algebra Der S A of special derivations of A and describe it via cyclic words in S. A derivation D of the algebra A is called special if there are elements Bs ∈ A such that X D(s) = [Bs , s] and D s = 0. (101) s∈S

Thus a system B = {Bs } of elements of A with

P [Bs , s] = 0

(102)

defines a special derivation D B . One has [D B , DC ] = D[B,C] ,

where [B, C]s := D B (Cs ) − DC (Bs ) − [Bs , Cs ].

(103)

Formula (103) defines a Lie bracket on systems (Bs ) obeying (102). Indeed, as was pointed out to us by the referee, one can interpret them as infinitesimal automorphisms of the structure consisting of algebra A, for each s ∈ S, left module A (noted A(s)), P endomorphism a 7 −→ as of A(s), element s in A. The bracket (103) is the bracket coming from this interpretation. Define a map of Q-linear spaces Cycl : C (A) −→ A :

s1 · · · sk 7 −→

k X

si · · · si−1+k ,

1 7 −→ 0,

i=1

given on the generators by the sum of the cyclic permutations. Then set s2 · · · sk if s = s1 , ∂s : s1 · · · sk 7−→ 0 otherwise. Similarly, define the map ∂s0 by s1 · · · sk 7 −→ s1 · · · sk−1 if sk = s and by zero otherwise. Then the image of the map Cycl is precisely ∩ Ker(∂s − ∂s0 ) in the positive degree part A+ of A. If x is in this image, X X X X X [∂s x, s] = ∂s x · s − s · ∂s x = ∂s0 x · s − s · ∂s x = x − x = 0. (104) Set Ds := ∂s ◦ Cycl. Let us define a Lie bracket in C (A) by X [C1 , C2 ] := − C [Ds C1 , Ds C2 ] · s . s∈S

(105)


445

Then C (A) is the product of the central Q · 1 and of C + (A). The map κ 0 : C + (A) −→ the Lie algebra (102) (103),

κ 0 : x 7 −→ Ds (x)

is an isomorphism of Q-vector spaces. This follows from Theorem 4.2 in [K]. For the convenience of the reader we reproduce the argument. Let F 1 (A) := ⊕s∈S A ⊗ ds. There is a sequence d t 0 −→ C + (A) = A/ [A, A] + Q · 1 −→ F 1 (A) −→ [A, A] −→ 0, (106) P where d(C ) := Ds C ⊗ ds and t[a ⊗ ds] := [a, s]. It is a complex by (104), and it is clearly exact from the left and right. There is an isomorphism of graded (by the weight) Q-vector spaces A/Q · 1 −→ F 1 (A), s1 · · · sm −→ s1 · · · sm−1 ⊗ dsm . Thus the Euler characteristic of the weight ≥ −m part of (106) is zero. So the complex is exact. The map κ 0 respects the Lie brackets. In particular, this proves that (105) satisfies the Jacobi identity. Thus κ 0 is a Lie algebra isomorphism. The centralizer of s in A is Q[s]. So Ker κ 0 = ⊕s∈S Q[s]. Let Ce(A) := C (A)/ ⊕s∈S Q[s].

(107)

The map κ 0 induces a Lie algebra morphism κ : Ce(A) −→ Der S (A). 5.1 The morphism κ : Ce(A) −→ Der S (A) is an isomorphism. PROPOSITION

Proof We need only show that κ is surjective. This follows from the exactness of (106). 5.2. The Lie algebra of special equivariant derivations Now, assume S = {0} ∪ G where G is a finite commutative group. The corresponding algebra A is the algebra A(G) of 2.4, with generators Y and X g (g ∈ G). Then A(G) is graded by the weight and depth, and as before, Der S A(G) inherits a weight grading compatible with a depth filtration. On Ce(A(G)) it corresponds to the weight grading and depth filtration induced from A(G), both shifted by one. It is clear for the weight grading. For depth, if x in Ce(A) has depth m, y := Cycl (x) (in A/ ⊕s Q[s]) has the same depth. If no ∂g y has depth m − 1, it follows that y has no word of depth m except possibly X 0k in the case m = 0, which is zero in Ce(A(G)) thanks to (107). The map κ provides a linear map G κG : Ce A(G) −→ Der S A(G)G =: Der S E A(G). Proposition 5.1 implies that it is an isomorphism.

446

A. B. GONCHAROV

We identify the vector space Ce(A(G))G with its graded for the depth filtration. The graded for the depth filtration of the Lie algebra Ce(A(G))G is Ce(A(G))G with the Lie bracket given by the sum (105), with s = 0 omitted: X [C1 , C2 ] := − C [Dg C1 , Dg C2 ] · X g . g∈G

5.3. Formulation of the result Recall that Y and X g , g ∈ G, are the generators of the algebra A(G). Let X G C X hg0 Y n 0 −1 · . . . · X hgm Y n m −1 . C X g0 Y n 0 −1 · . . . · X gm Y n m −1 := h∈G

The expressions on the left are parametrized by cyclic G-equivariant words, that is, G-orbits on the set of all cyclic words. Consider the following formal expression: b ξG :=

X

G 1 e In 0 ,...,n m (g0 : · · · : gm )⊗ C X g0 Y n 0 −1 ·. . .· X gm Y n m −1 , (108) | Aut C |

where the sum is over all cyclic G-equivariant words C in X g , Y . The weight 1/| Aut C | of a given cyclic word C is the order of the automorphism group of this cyclic word, as was pointed out by the referee. Applying the map Id ⊗ Gr(κ), we get a bidegree (0, 0) element SE e•• (G)⊗ e bQ Gr Der•• ξG ∈ D A(G).

Since G is finite, Dw,m (G) is a finite-dimensional Q-vector space. Let D−w,−m (G) = Dw,m (G)∨ . Then D•• (G) := ⊕w,m≥1 D−w,−m (G) is a bigraded Lie 0 (G). e•• (G) and D•• algebra. We similarly define D We may view the element e ξG as a map between the bigraded Q-vector spaces: SE e•• (G), Gr Der•• e A(G) . (109) ξG ∈ HomQ−Vect D 0 (G) ⊂ D e•• (G). Notice that D•• (G) ⊂ D••

5.2 = S E A(G) is an isomorphism of bigraded e•• (G) −→ The map e ξG : D Gr Der•• Lie algebras. 0 (G), we get an isomorphism Restricting e ξG to D••

THEOREM

(a) (b)

=

0 SE ξG0 : D•• (G) −→ Gr Der•• L(G).

(c)

(110)

Restricting ξG0 to D•• (G), we get an injective Lie algebra morphism SE ξG : D•• (G),→ Gr Der•• L(G).

(111)


447

5.4. Proof (a) We start with a remark from linear algebra. Let L 1 , L 2 be Lie algebras, and let ζ ∈ HomQ−Vect (L 1 , L 2 ) = L ∗1 ⊗ L 2 . Denote by δ : L ∗1 −→ 32 L ∗1 the Lie cobracket on L ∗1 and by [, ] : 32 L 2 −→ L 2 the Lie bracket on L 2 . Define a symmetric product (L ∗1 ⊗ L 2 ) ⊗ (L ∗1 ⊗ L 2 ) −→ 32 L ∗1 ⊗ 32 L 2 by (a1 ⊗ a2 ) ◦ (b1 ⊗ b2 ) := (a1 ∧ b1 ) ⊗ (a2 ∧ b2 ). Then ζ is a morphism of Lie algebras if and only if 1 id ⊗[, ] (ζ ◦ ζ ). 2 P P If ζ = i Ai ⊗ Bi , then id ⊗[, ] (ζ ◦ ζ ) = i, j Ai ∧ A j ⊗ [Bi , B j ], and so the condition is X 1X δ(Ai ) ⊗ Bi = Ai ∧ A j ⊗ [Bi , B j ]. (112) 2 (δ ⊗ id)(ζ ) =

i

i, j

To check that 1/2 is needed, notice that if {ei } is a basis of a Lie algebra L, { f i } the P P dual basis, and [ei , e j ] = cikj ek , then δ f k = (1/2) i,n j=1 cikj f i ∧ f j . Let us suppose in addition that the Lie algebra bracket [, ] on L 2 is obtained by alternation of a (not necessarily associative) product ∗ on the vector space L 2 , that is, [x, y] := x ∗ y − y ∗ x. Then ζ is a Lie algebra morphism if and only if (δ ⊗ id)(ζ ) = (id ⊗∗)(ζ ◦ ζ ). If ζ =

P

i

Ai ⊗ Bi , then it looks as follows: X X δ(Ai ) ⊗ Bi = Ai ∧ A j ⊗ Bi ∗ B j . i

(113)

i, j

Let us return to our situation. The Lie bracket on Gr Ce(A(G)) is given by formula (105) where the sum is over X g only. It follows that the Lie algebra structure on the subspace Gr Ce(A(G))G is obtained by alternation of the nonassociative product ∗1 given by G C (X g0 Y n 0 −1 · . . . · X gk Y n k −1 )G ∗1 C X h 0 Y m 0 −1 · . . . · X hl Y m l −1 X := C X e Y n i −1 X g−1 g Y n i+1 −1 · . . . · X g−1 g Y n i +m j −2 i, j

i

i+1

i

· X h −1 h j

j+1

i−1

Y m j+1 −1 · . . . · X h −1 h j

j−1

Y m j−1 −1

G

(see Figure 10). It can be defined by the same formulas for any abelian group G. To visualize a single expression G δ(e In 0 ,n 1 ,...,n m )(g0 : · · · : gm ) ⊗ C X g0 Y n 0 −1 · . . . · X gm Y n m −1 (114)

448

A. B. GONCHAROV

.

gi−1

.

.

.* .

X

hj

.

gi

1

.

gi+1

.

i, j

h j−1

.

h −1 j h j+1

gi−1 gi+1

h j+1

gi−1 gi+1

. e

.

h −1 j h j−1

Figure 10

in (δ ⊗ id)(b ξG ), we proceed as follows. Take an oriented circle divided into m + 1 arcs by m + 1 black points. Label each of the black points by an element of the set {X g }, g ∈ G, and call it an X g -point. The ith arc is subdivided into n i little arcs by n i − 1 points labeled by Y (presented by little circles on Figure 11 and called Y -points). Such data correspond to a cyclic word C (X g0 Y n 0 −1 · . . . · X gm Y n m −1 ).

. .

.

cyclic word

−→

XY 3 XY 2 XY 2 XY (G = {e})

. Figure 11

Group G acts naturally on them: an element h ∈ G transforms X g -points to X hg points and leaves untouched Y -points. The orbits are called circles with (n 0 , . . . , n m )cyclic G-structure. They correspond to cyclic G-equivariant words and thus to expressions (114) as well as to the generators e In 0 ,n 1 ,...,n m (g0 : · · · : gm ) obeying the cyclic symmetry and homogeneity condition (92). Let us mark such a picture by choosing a little arc and a black point different from the ends of the arc containing the little arc. We call it a marked circle with (n 0 , . . . , n m )-cyclic G-structure. They correspond to marked cyclic G-equivariant words. It follows from formula (81) that expression (114) is a sum of the terms which are in bijective correspondence with markings of the particular circle which corresponds to C (X g0 Y n 0 −1 · · · )G . For example, the marked circle with (3, 2, 2, 1)-cyclic structure in Figure 12 corresponds to I2,1 ∧ I2,1,2 ⊗ C (X Y 2 X Y X Y 3 X Y 2 ). In general, we use the marks to cut the circle on 2 oriented semicircles and make 2 new circles by gluing the endpoints of the semicircles, adding a new black point on each of the new circles instead of the marked black point on the initial circle and using the rest of the


. .

449

. .

Figure 12. A marked circle with (3,2,2,1)-cyclic structure

points on each of the new circles. The new circles are getting natural cyclic structures. Going from a single expression (114) to the sum, weighted by 1/| Aut C |, of such expressions over all isomorphism classes of cyclic G-equivariant words, we get the sum over all marked cyclic G-equivariant words. Notice that marking a particular cyclic word C , we get a sum of marked cyclic words, each of them appearing precisely | Aut C | times. The weight makes the coefficient equal to 1. Now let us investigate how the right-hand side of (113) looks for our element b ζG . Consider the sum X 1 1 e I p ,..., pk (g0 : · · · : gk ) ∧ e Iq0 ,...,ql (h 0 : · · · : h l ) (115) | Aut C P | | Aut C Q | 0 G G ⊗ C X g0 Y p0 −1 · . . . · X gk Y pk −1 ∗1 C X h 0 Y q0 −1 · . . . · X hl Y ql −1 , (116) where the summation is over all (ordered) pairs of cyclic G-equivariant words C P = C (X g0 Y p0 −1 · . . . · X gk Y pk −1 )G and C Q = C (X h 0 Y q0 −1 · . . . · X hl Y ql −1 )G . The particular product (116) is a sum of (k +1)(l +1) cyclic G-equivariant words. Each of them corresponds to a pair {a circle with P := ( p0 , . . . , pk )-cyclic G-structure + a black point on it, a circle with Q := (q0 , . . . , ql )-cyclic G-structure + a black point on it}. We call a circle with cyclic G-structure and a choice of a black point on it a labeled circle with cyclic G-structure. It corresponds to a labeled cyclic G-equivariant word. Now going from a single expression (116) to the weighted sum over all pairs of cyclic G-equivariant words, we get the terms corresponding to pairs of labeled cyclic G-equivariant words, each with the coefficient 1 thanks to the weighting. The pairs of labeled cyclic G-equivariant words are in bijective correspondence with the marked cyclic G-equivariant words, as demonstrated in Figure 13 using labeled or marked circles instead of the corresponding words. Namely, to get a marked circle with cyclic equivariant G-structure we proceed as follows. Make a connected sum of the P- and Q-circles by connecting the chosen black points by a bridge. The orientation of the initial circles induces an orientation of their connected sum. Instead

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. .

..

.

. . .

. .

.

. .

.

.

..

.

Figure 13

of the chosen two black points on the initial circles we put a single black point on the bottom part of the bridge. We keep the rest of the points, getting a cyclic structure on the new circle. The black point on the bottom bridge together with the top part of the bridge provide the marks on the circle with the cyclic structure we get. The distinguished point on each of the three circles is chosen to be the X e -point. The procedure is reversible; the inverse map is shown on the bottom of the figure. We proved that e ξG is a morphism of bigraded Lie algebras. By Lemma 4.5 it is an isomorphism. Part (a) of the theorem is proved. Proof of part (b) 5.3 Let D ∈ Der S E A(G). Then D(X e ) ∈ L(G) D ∈ Der S E L(G). LEMMA

Proof We need the following facts. Let the free associative algebra A(S) contain the free Lie algebra L(S). Then, if x ∈ A(S) is such that [x, s] ∈ L(S), one has x ∈ L(S) + Q[s]. Indeed, by Poincaré-Birkhof-Witt it suffices to check that the kernel of ads acting on Symn (L(S)) is reduced to s n . Indeed, as a U (Q · s)-module, L(S) is the sum of Qs and of a free module, and L(S)⊗n is hence the sum of s ⊗n and a free module. We are now ready to prove the only nontrivial implication, =>, of the lemma. Since D is equivariant, D(X e ) ∈ L(G) implies that D(X g ) ∈ L(G) for any g ∈ G. P Since D( X g + Y ) = 0, we get also D(Y ) ∈ L(G). Thus, thanks to the statement we just proved, D ∈ Der S E L(G). The lemma is proved.


451

Recall that Ce(G) is a bigraded vector space and there is the isomorphism κ : Ce(G) −→ Der S A(G).

(117)

The vector space Der S A(G) admits, besides the weight grading, a depth grading, with D of degree d if all D(X g ) are of homogeneous degree d + 1. Indeed, if P [B0 , Y ] + [Bg , X g ] = 0, the same holds with the Bg (resp., B0 ) replaced by their part of degree d (resp., d + 1). The isomorphism (117) is compatible with the weight and depth grading (after a shift by one in Ce(G)). Further, Der S L(G) ,→ Der S A(G), defined by “the D(X g ) are primitive” is a bigraded subspace. The action of the group G preserves the gradings. Thus Der S E L(G) ,→ Der S E A(G) is also a bigraded subspace. Notice that the Lie algebra structure on either side of (117) does not respect the depth grading, only the depth filtration. e−w,−m (G). Then we claim that Let X ∈ D 0 b−w,−m e ξG (X ) ∈ Der S E L(G) X ∈ D (G).

(118)

Recall that there is a coproduct 1 on A(G) which dualizes to the shuffle product ◦Sh (see (94)). Let 1(Z ) := 1(Z ) − 1 ⊗ Z − Z ⊗ 1. Then L(G) = Ker 1. To prove (118), consider the expression e−w,−m (G) ⊗ A(G) ⊗ A(G) . (id ⊗1 ◦ D X e )(b ξG ) ∈ D (119) Choose two elements A, B of the natural basis in A(G). Then A ⊗ B ∈ A(G) ⊗ A(G) appears in (id ⊗1 ◦ D X e )(b ξG ) with coefficient X e (A ◦Sh B). This implies (118). So map (117) leads to an isomorphism of the bigraded vector spaces 0 b ξG0 : D•• (G) −→ Der S E L(G).

After taking the associated graded for the depth filtration, it becomes an isomorphism of bigraded Lie algebras denoted ξG0 . Summarizing, we see that shuffle relations (96) are equivalent to the condition 0 (G)) ⊂ Gr Der S E L(G). Part (b) of the theorem is proved. e ξG (D•• •• Another proof of Proposition 4.7 We just proved that ξG0 is an isomorphism of bigraded Q-vector spaces. Using (a) (and 0 (G) is a Lie the obvious fact that Der S E L(G) is a Lie algebra!), we conclude that D•• e subalgebra of D•• (G). (c) It follows from (a), (b), and Propositions 4.6 and 4.7. The theorem is proved.

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5.5. The Lie algebra of outer semispecial derivations Let X N := {0} ∪ {∞} ∪ µ N . For every point x ∈ X N there is a well-defined conjugacy class in π1 (X N , z) provided by a little lop around x. For any point y ∈ X N there is a tangent vector v y at y provided by the canonical coordinate t on X N if y 6 = ∞ and t −1 at ∞. Similarly, for the pro-l group (l) π1 (X N , v y ) there is a well-defined up to a conjugation map (l)

i x : Zl (1) −→ π1 (X N , v y ).

(120) (l)

Therefore the Galois group Gal(Q/Q(ζl ∞ N )) acts on π1 (X N , v y ) preserving the conjugacy classes of the maps i x , x ∈ X N . More precisely, define a semispe(l) cial automorphism of the group π1 (X N , v y ) as the automorphism preserving all (l) the conjugacy classes of the maps i x for x ∈ X N . Denote by Aut SS π1 (X N , v y ) (l) the group of all semispecial automorphisms of π1 (X N , v y ). We define the group (l) of outer semispecial automorphisms Out SS π1 (X N ) as the quotient of the group (l) Aut SS π1 (X N , v y ) modulo the inner automorphisms. The corresponding groups defined for different base points v y are canonically isomorphic, so we can drop the base vector from the notations. We get a canonical homomorphism (l) (121) 9 N : Gal Q/Q(ζl ∞ N ) −→ Out SS π1 (X N ). Similarly, we define the Lie algebra ODer SS L(X N ) of outer semispecial derivations of the fundamental Lie algebra of X N : ODer SS L(X N ) :=

Der SS L(X N , v y ) . InDer L(X N , v y )

(122)

Here the denominator is the Lie algebra of all inner derivations, and the numerator is the Lie algebra of semispecial derivations, that is, derivations preserving the conjugacy classes of loops around x ∈ X N . For any point x0 ∈ X N there is a Lie subalgebra Der S L(X N , vx0 ) ,→ Der SS L(X N , vx0 )

(123)

of all derivations special with respect to the point x0 . It consists of the semispecial derivations killing i x0 (Q(1)). Thus there is a natural Lie algebra homomorphism px0 : Der S L(X N , vx0 ) −→ ODer SS L(X N ). It is obviously surjective. Indeed, if D is a semispecial derivation, then there exists C ∈ L(X N , vx0 ) such that D (i x0 (Q(1))) = [C, i x0 (Q(1))]. So subtracting the inner


453

derivation X 7 −→ [C, X ], we get a derivation special with respect to x0 . Since the pronilpotent Lie algebra L(X N , vx0 ) is free, Ker px0 is one-dimensional and consists of inner derivations provided by commutator with i x0 (Q(1)). There is also the Ql -version of this story. We denote by ODer SS L(l) (X N ) the l-adic analog of (122). 5.6. A symmetric construction of the Lie algebra of special derivations Below we present a variation on the theme discussed in Section 5.5 in a bit different (and general) setup. Denote by AX the free associative algebra generated by the set X . Denote by X x the generator corresponding to x ∈ X . Let X be a finite set. Choose an element x0 ∈ X . Let S := X − {x0 }. Then there is the Lie algebra Der S (A S ) of special derivations of the algebra A S . It contains the one-dimensional subspace generated by inner derivation ∗ 7−→ [X x0 , ∗]. Our goal is to describe the structure of the quotient of the Lie algebra Der S (A S ) by this subspace in terms of the set X independently of the choice of an element x0 . We use this for X := X N to describe the corresponding quotient of the Lie algebras Der S L(X N , v y ), where y ∈ X N . As a result we get natural canonical isomorphisms between these quotients. Let AX be the quotient of AX by the ideal I (X ) generated by the element P x∈X X x . Then for any element x ∈ X there is a canonical isomorphism =

AX −→ A S .

(124)

A semispecial derivation of the algebra AX is a derivation D preserving the conjugacy classes of the generators X x ; that is, D (X x ) = [C x , X x ] for certain C x ∈ AX . The semispecial derivations form a Lie algebra Der SS (AX ). Let InDer(AX ) be the Lie subalgebra of inner derivations. The quotient ODer SS (AX ) :=

Der SS (AX ) InDer(AX )

is called the Lie algebra of outer semispecial derivations. There is a canonical surjective homomorphism px0 : Der S A S −→ ODer SS (AX ) whose kernel is the subspace generated by the inner derivations corresponding to P elements ( x∈X X x )n , n ≥ 1. There is a similar story for the Lie algebras. Namely, let L X be the free Lie algebra generated by the set X . Denote by L X its quotient by the ideal generP ated by the element x∈X X x . Then for any x ∈ X there is a canonical isomorphism L X −→ L S . We define the Lie algebra of outer semispecial derivations

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ODer SS (L X ). There is a canonical surjective morphism px0 : Der S L S −→ ODer SS (L X ) with one-dimensional kernel generated by the inner derivation given by the commuP tator with x∈X X x . Denote by C (A S ) the quotient of the Lie algebra Ce(A S ) of cyclic words in AX P by the subspace generated by the elements ( s∈S X s )n , n ≥ 1. 5.4 The space Ce(IX ) is a Lie algebra ideal in Ce(AX ). P The element ( s∈S X s )n is in the center of the Lie algebra Ce(A S ). Let us choose an element x0 ∈ X . Then there is a canonical isomorphism of Lie algebras Ce(AX ) = κ −→ C (A S ) = ODer SS (AX ). (125) Ce(IX )

THEOREM

(a) (b) (c)

Proof P Take cyclic words A = C (( x∈X X x ) · A1 ) ∈ Ce(IX ) and B ∈ Ce(AX ). Then [A, B], by the very definition of the commutator in the Lie algebra of cyclic words, is a sum of the terms corresponding to pairs (a generator in A, a generator in B). The pairs (a generator in A1 , a generator in B) clearly produce an element of the subspace IX . It is easy to see that the sum of the terms corresponding to the remaining pairs P ( x∈X X x , a generator in B) is zero; this is very similar to the proof of the fact that P a cyclic word provides a derivation killing s∈S X s , and the same kind of argument proves (b). So Ce(IX ) is a Lie ideal in Ce(AX ), and C (A S ) is a Lie algebra. The isomorphism (124) shows that one obviously has an isomorphism of vector spaces C (AX ) = −→ C (A S ). C (IX ) Passing to the Ce-quotients on the left, we get an isomorphism of vector spaces Ce(AX ) = −→ C (A S ). Ce(IX )

Notice that the element X xn0 is zero in Ce(AX ) but it is not zero in Ce(A S ). So we have to define the quotient C (A S ). Moreover, since any element of AX can be written modulo I (X ) as an element of A S (i.e., we again use (124)), this isomorphism commutes with the Lie brackets. The second equality in (125) follows from the description of Ker px0 and Proposition 5.1. The theorem is proved.


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(l)

We define the group Out SS E π1 (X N ) of outer semispecial equivariant automor(l) (l) phisms of π1 (X N ) as the invariants of the action of µ N on Out SS π1 (X N ), and we proceed similarly in the other cases. If XG := {0} ∪ G ∪ {∞}, x0 = {∞}, Theorem 5.4 provides a description of the Lie algebra ODer SS E A(G) of outer semispecial equivariant derivations of A(G): Ce(A ) G = XG −→ ODer SS E (AXG ). Ce(IX )

(126)

G

Combining this with Theorem 5.2(b), we get a description of the Lie algebra ODer SS E (L XG ) ⊂ ODer SS E (AXG ). 5.7. The distribution relations Recall that X N = P1 \{0, µ N , ∞} and there is canonical isomorphism GrW L(X M , v∞ ) = L(µ N ). The maps i N : X N M ,→ X M ,

z 7−→ z,

m N : X N M −→ X M ,

z 7 −→ z N

induce the surjective Lie algebra homomorphisms i N ∗ : L(µ M N ) −→ L(µ M ),

m N ∗ : L(µ M N ) −→ L(µ M )

given on the generators by i N ∗ : Y −→ Y,

X ζ −→

m N ∗ : Y −→ N Y,

0, ζ 6∈ µ M , X ζ , ζ ∈ µM ,

X ζ −→ X ζ N .

LEMMA 5.5 Elements of ODer SS E L(µ M N ) preserve Ker(i N ∗ ) and Ker(m N ∗ ).

Proof Ker(i N ∗ ) is generated by the elements X ζ , where ζ 6 ∈ µ N . Since any semispecial derivation preserves conjugacy class of X ζ , it in particular preserves the ideal generated by such X ζ . The statement about m N follows from the fact that equivariant derivations, by their very definition, commute with the natural action of the group G on L(µG ). Namely, let D ∈ Der E L(µ M N ). Then ξ∗ (D (X ζ )) = D (X ξ ζ ), and so m N ∗ sends D (X ζ ) and D (X ξ ζ ) to the same element. The lemma is proved. It follows that there are well-defined homomorphisms e iN , m e N : ODer SS E L(µ M N ) −→ ODer SS E L(µ M )

(127)

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A. B. GONCHAROV

uniquely characterized by the property that for any element l ∈ L(µ M N ) and a derivation D ∈ ODer SS E L(µ M N ) one has f ∗ ◦ D(l) = e f (D) ◦ f (l),

f = iN , mN .

(128)

The same arguments provide us with homomorphisms i N , m N : Out SS E L (l) (µ M N ) −→ Out SS E L (l) (µ M ).

(129)

Recall the maps (l) ψ N : Gal Q/Q(ζl ∞ N ) −→ Out SS E L (l) (µ N ). They, together with homomorphisms (129), provide the following two commutative diagrams (similar to (128)), where j M is the natural inclusion: ψ M(l)N Gal Q/Q(ζl ∞ M N ) −→ Out SS E L (l) (µ M N ) ↓ jM ↓ iN (l) ψM Gal Q/Q(ζl ∞ M ) −→ Out SS E L (l) (µ M ) and

ψ M(l)N Gal Q/Q(ζl ∞ M N ) −→ Out SS E L (l) (µ M N ) ↓ jM ↓ mN (l) ψM Gal Q/Q(ζl ∞ M ) −→ Out SS E L (l) (µ M ) (l)

The composition ψ M ◦ j M does not depend on the choice of the maps i N , m N . Passing to the Lie algebras, we get (l)

GrW G N M ⊂ Ker(e iN − m e N ) ⊗ Ql . The maps i N ∗ , m N ∗ provide the linear maps µ µ i N0 ∗ , m 0N ∗ : C A(µ M N ) M N −→ C A(µ M ) N . We set i N00 := i N0 ∗ and m 00N := N −1 m N ∗ . 5.6 The maps i N00 , m 00N preserve the Lie brackets. LEMMA

Proof This is a direct check.

(130)


457

Warning The map m 00N is a Lie algebra morphism only when restricted to the subspace of Ginvariant cyclic words. Since i N00 − m 00N : Y k 7−→ (1 − N k−1 )Y k ,

X

X gk 7−→ 0,

(131)

g∈µ M N

the identification Der S E A(µG ) = Ce(A(µG ))G leads to the Lie algebra maps Der S E A(µ M N ) −→ Der S E A(µ M ). Then they restrict to the maps i N0 , m 0N : Der S E L(µ M N ) −→ Der S E L(µ M ).

(132)

PROPOSITION 5.7 The distribution relations are equivalent to the conditions (l)

GrW G N ⊂ Ker(i L0 − m 0L ) for each integer L|N . Proof An element of Der S E L(µ N ) is given by expression X a(g0 : g1 : · · · : gm )n 0 ,...,n m · κµ N C (X g0 Y n 0 −1 · · · X gm Y n m −1 ).

(133)

(134)

The sum is over cyclic words. The a’s are coefficients in Q. They satisfy the relation a(g0 : · · · : gm )∗ = a(hg0 : · · · : hgm )∗ expressing the fact that the cyclic word (134) is G-invariant. It is in Ker(i L0 − m 0L ) if and only if X a(g0 : g1 : · · · : gm )n 0 ,...,n m = L −1 L w−m a(h 0 : h 1 : · · · : h m )n 0 ,...,n m , (135) h iL =gi

except the relation a(g : g) = L −1

X

a(h 0 : h 1 )

(136)

h iL =g

for m = 1 and g0 = g1 . This is precisely the distribution relation. Here is the origin of the exception. An element X a(g0 : g1 : · · · : gm )n 0 ,...,n m · C X g0 Y n 0 −1 · · · X gm Y n m −1 is in Ker(i L00 − m 00L ) if and only if equations (135) are valid. Then to get the maps (132) we first restrict to the subspace of those cyclic words that provide derivations of the

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A. B. GONCHAROV

P Lie algebra L(µ N ) and then kill the elements by Y 2 and g∈G X g2 . Therefore, thanks to (131), the kernel is increased by a one-dimensional subspace: we add the element P projected by i L00 − m 00L onto g∈µ N /L X g2 . The equation we thus remove is exactly P (136). Indeed, this equation just means that the g∈µ N /L X g2 component of (i L00 − m 00L ) (see (134)) is zero, which we no longer require. The proposition is proved. One has i L0 − m 0L : (Y +

X g∈µ N

So killing the elements (Y +

P

X g )2 7 −→ (1 − L)(Y +

X

X g )2 .

g∈µ N /L

X g )2 , we get well-defined maps

e i L0 , m e0L : ODer SS E L(µ N ) −→ ODer SS E L(µ N /L ), and moreover, since 1 − L 6= 0, we have Ker(i L0 − m 0L ) = Ker(e i L0 − m e0L ). LEMMA 5.8 One has e iL =e i L0 and m eL = m e0L .

Proof One checks that each of the maps satisfies condition (128). Since this condition characterizes these maps, the lemma follows. Now the distribution relations follow from (130), Proposition 5.7, and Lemma 5.8. Proof of Theorem 2.2 It follows immediately from the inversion relation, which just has been proved as the N = −1 case of the distribution relations. The argument is identical with the one given in the proof of Corollary 4.2. Proof of Conjecture 1.1 (l) To complete the proof of Conjecture 1.1 it remains to prove that G•• (µ N ) lies in the S E L(µ ) defined by the power shuffle relations (62). So these subspace of Gr Der•• N relations provide the most nontrivial constraints on the image of the Galois group. This is rather amazing since they are the simplest relations on the level of functions. There should exist a natural proof of them which is as transparent as the relations are. There exists, however, a motivic proof: relations (62) hold for the corresponding framed mixed Hodge structures => for the corresponding framed mixed Tate motives over Q(ζ N ) => valid for their l-adic realization. The detailed exposition will appear in [G9]. In Sections 7.3–7.4 we prove certain special cases of Conjecture 1.1 relevant to our situation which can be obtained by some ad hoc methods.


459

6. Cohomology of some discrete subgroups of GL2 (Z) and GL3 (Z) 6.1. The general scheme Let G be a reductive group over Q. Denote by A f the ring of finite adels for Q. Choose a finite index subgroup K f ⊂ G(b Z). Let Z ⊂ G be the maximal split torus in the center of G, and let Z 0 (R) be the connected component of identity of its R-valued 0 the connected component of the maximal compact subgroup points. Denote by K ∞ of the derived group G 0 (R). Set min 0 K∞ := K ∞ × Z 0 (R), max := maximal compact subgroup of G(R) × Z 0 (R). K∞

Choose a subgroup K ∞ sitting in between min max . K∞ ⊂ K∞ ⊂ K∞

One defines the modular variety corresponding to a given choice of the subgroups K f and K ∞ as follows: S KG∞ ×K f = G(Q)\ G(R)/K ∞ × G(A f )/K f . (137) In general it is, of course, not a variety in the algebraic geometry sense and not even a manifold, only an orbifold. A rational representation ρ : G ⊗Q Q −→ GL(V ) provides a local system on the orbifold (137): LV := G(R)/K ∞ × G(A f )/K f ×G(Q) V. max = O · R∗ , K min = SO ·R∗ . The space Example. We have G = GLm , K ∞ m m + ∞ + max GLm (R)/K ∞ is identified with positive definite symmetric (m × m)-matrices. The group GLm (R) acts on it by X 7−→ AX At . The symmetric space is max Hm = GLm (R)/K ∞ · R∗>0 = > 0 definite symmetric (m × m)-matrices /R∗+ .

One defines 0 := GLm (Q) ∩ K f ⊂ GLm (Z). If 0 is torsion free, then H ∗ (0, V ) = H ∗ S KG∞ ×K f , LV . e If 0 contains a finite index torsion-free subgroup e 0 , then H ∗ (0, V ) = H ∗ (e 0 , V )0/0 . We use the shorthand S0 for the modular variety corresponding to the subgroup 0. The Borel-Serre compactification S 0 is a compact manifold with corners. The

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A. B. GONCHAROV

boundary ∂ S 0 is a topological manifold. It is a disjoint union of faces that correspond bijectively to the 0-conjugacy classes of proper parabolic subgroups of G defined over Q. The closure of the face corresponding to a parabolic subgroup P is called a strata and denoted ∂ P S. There is a natural restriction map Res

H ∗ (S0 , LV ) −→ H ∗ (∂ S 0 , LV ). ∗ (∂ S , L ) is, by definition, the image of this map. The cohomology at infinity Hinf 0 V The computation of these groups is an important step in understanding the cohomology of LV . To carry it out one should determine first the cohomology of the restriction of LV to the Borel-Serre boundary ∂ S 0 . Pick a rational parabolic subgroup P. Let U P be its unipotent radical, and let M P := P/U P be the Levi quotient. Denote by N P the Lie algebra of U P . Take the M ⊂ M(R). In our case, P(A ) · K = subgroup P(R) ∩ K ∞ , and project it to K ∞ f f ∗ G(A f ). Then the cohomology H (∂ P S, LV ) of the restriction of the local system LV to the boundary strata ∂ P S is calculated using the Leray spectral sequence whose E 2 -term looks as follows: H p S KMM , H q (N P (Z), V ) => H p+q (∂ P S, LV ). (138) ∞

One has H q N P (Z), V = H q (N P , V ). The first step of the calculation of the cohomology at infinity of a subgroup 0 is the calculation of these groups for all of the strata. The Kostant theorem Let P be a rational parabolic subgroup of a reductive group G. Fixing Cartan and Borel subgroups H (C) ⊂ B(C) with H (C) ⊂ M P (C), we get a system of positive roots and the Weyl group W . Denote by W P the Weyl group for M P with W P ⊂ W . It is known that in each coset class W P \W there is a unique element of minimal possible length. Denote by W P1 the set of the minimal length representatives for W P \W . For any dominant weight µ, denote by L µ the irreducible representation of the group M P (C) with the highest weight µ. Let ρ be the half of the sum of the positive roots for G. 6.1 ([K, Theorem 5.14]) The M P (C)-representation H ∗ (N P , V ) is algebraic and is given by H ∗ (N P , V ) = ⊕ω∈W 1 L ω(λ+ρ)−ρ − l(ω) , THEOREM

P

where l(ω) is the length of ω, and [−l(ω)] indicates that the module appears in degree l(ω). If V is defined over Q, then so is H ∗ (N P , V ).


461

Let H be the maximal Cartan subgroup of GLm given by the diagonal matrices. Denote by [n 1 , . . . , n m ] the character of H given by (t1 , . . . , tm ) −→ t1n 1 · . . . · tmn m . Let P be a parabolic subgroup P containing H , and let M P be the Levi quotient of P. Then H is a maximal torus for M P . Denote by V[nP1 ,...,n m ] or simply V[n 1 ,...,n m ] the representation with the highest weight [n 1 , . . . , n m ] of M P . Example. The weights ρ for GL2 and GL3 are given by ρ = [1/2, −1/2],

ρ = [1, 0, −1].

6.2. Cohomology of GL2 (Z) The cohomology H ∗ (GL2 (Z), S w−2 V2 ⊗ ε2 ) vanishes for odd w. Indeed, if w is odd, the central element diag(−1, −1) acts by −1 in S w−2 V2 . So we assume that w − 2 is even. Clearly, H 0 GL2 (Z), S w−2 V2 ⊗ ε2 = 0. For any GL2 -module V over Q, the Hochshild-Serre spectral sequence corresponding to the exact sequence det

0 −→ SL2 (Z) −→ GL2 (Z) −→ {±1} −→ 0 gives + H ∗ GL2 (Z), V = H ∗ SL2 (Z), V . Here + is the invariants under the involution provided by conjugation by the matrix −1 0 0 1 . There is an exact sequence (see [Sh]) 1 SL2 (Z), S w−2 V2 −→ H 1 SL2 (Z), S w−2 V2 0 −→ Hcusp Res 1 −→ Hinf SL2 (Z), S w−2 V2 −→ 0. It admits a natural splitting as a module over the Hecke algebra. Let us compute the cohomology of the restriction of the local system L S w−2 V2 to the boundary strata ∂ B S corresponding to the upper triangularBorel subgroup B ⊂ M = ξ1 0 , ξ = ±1. We GL2 . In this case M B = T is the diagonal torus and K ∞ i 0 ξ2 need to compute M H p K∞ , H q (N B , S w−2 V2 ⊗ ε2 ) . We have S w−2 V2 ⊗ ε2 = L [w−1,1] and H (N B , S q

w−2

V2 ⊗ ε2 ) =

L [w−1,1] , q = 0, L [0,w] , q = 1.

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A. B. GONCHAROV

Therefore H 0 = 0 (since w − 1 is odd), and − 1 1 H 1 GL2 (Z), S w−2 V2 ⊗ ε2 = Hcusp SL2 (Z), S w−2 V2 ⊕ Hinf SL2 (Z), S w−2 V2 . 1 (SL (Z), S w−2 V ) are of the same dimenIt is known that the + and − parts of Hcusp 2 2 sion that, thanks to the Eichler-Shimura isomorphism, coincides with the dimension of the space of the holomorphic weight w cusp forms on SL2 (Z).

6.3. Cohomology of GL3 (Z) with coefficients in S w−3 V3 The main result is the following. THEOREM

6.2

We have H i GL3 (Z), S w−3 V3

 i = 0, w = 3,  Q, 1 w−3 H GL2 (Z), S V2 ⊗ ε2 , i = 3, =  cusp 0, otherwise. (139)

Proof Since the central element − Id ∈ GL3 acts by −1 on V3 , and hence on S w−3 V3 if w − 3 is odd, we get H i (GL3 (Z), S w−3 V3 ) = 0 if w − 3 is odd. We assume from now on that w − 3 is even. ∗ (0, V ) which we use below, see [LS, For the definition of cuspidal cohomology Hcusp Section 1.3].

6.3 The following sequence is exact:

PROPOSITION

(a)

i 0 −→ Hcusp GL3 (Z), S w−3 V3 −→ H i GL3 (Z), S w−3 V3 i −→ Hinf GL3 (Z), S w−3 V3 −→ 0. (b)

∗ (GL (Z), S w−3 V ) = 0. We have Hcusp 3 3

Proof Let sl3 be the Lie algebra of SL3 (R). Let π be a unitary irreducible representation of SL3 (R). Denote by Hπ∞ the space of C ∞ -vectors in π. Since for w > 3, S w−3 V3 is not self dual (i.e., the Cartan involution θ : g 7 −→ (g −1 )t transforms it to a nonisomorphic representation), one has, according to [BW, Chapter II, Proposition 6.12], H ∗ sl3 , SO3 ; S w−3 V3 ⊗ Hπ∞ = 0.


463

The proposition follows from the argumentation given in [LS, Chapter 1]. The boundary of the Borel-Serre compactification of our modular variety is a disjoint union of three faces, e(P1 ), e(P2 ), and e(B), which correspond to the GL3 (Q)equivalence classes of the following proper parabolic Q-subgroups:   ∗ ∗ ∗ P1 := 0 ∗ ∗ , 0 ∗ ∗

  ∗ ∗ ∗ P2 := ∗ ∗ ∗ , 0 0 ∗



 ∗ ∗ ∗ B := 0 ∗ ∗ . 0 0 ∗

Their nilradicals are denoted by N1 , N2 , and N ; the corresponding Lie algebras are N1 , N2 , and N . We calculate the groups (138). Applying Kostant’s theorem, we get the following:  q = 0,  L [w−3,0,0] , H q (N1 , S w−3 V3 ) = L [−1,w−2,0] , q = 1,  L [−2,w−2,1] , q = 2,  q = 0,  L [w−3,0,0] , q w−3 H (N2 , S V3 ) = L , q = 1,  [w−3,−1,1] L [−1,−1,w−1] , q = 2.  L [w−3,0,0] , q = 0,    L ⊕ L , q = 1, [−1,w−2,0] [w−3,−1,1] H q (N , S w−3 V3 ) =  L ⊕ L [−2,w−2,1] , q = 2,   [−1,−1,w−1] L [−2,0,w−1] , q = 3. Let ±1 0 , S1 := 0 GL2 (Z)

GL2 (Z) 0 , S2 := 0 ±1

  ±1 0 0 S :=  0 ±1 0  . 0 0 ±1

If there is a ±1 subgroup in one of these groups, S? , which acts by −1 on a module V , then H i (S? , V ) = 0 for all i. This simple remark shows that we get a nonzero contribution only in the following cases: (i) for the group S1 : L [w−3,0,0] and L [−2,w−2,1] [−2]; (ii) for the group S2 : L [w−3,0,0] and L [−1,−1,w−1] [−2]; (iii) for the group S: L [w−3,0,0] and L [−2,0,w−1] [−3]. p,q We consider them now case by case, indicating the value ( p, q) for the E 2 -term of the spectral sequence where they appear. Recall that w − 3 is even. (i) For the strata ∂ P1 S we have the following: H 0 (S1 , L [w−3,0,0] ) = L [w−3,0,0] ,

( p, q) = (0, 0),

(140)

464

A. B. GONCHAROV

and H 1 (S1 , L [−2,w−2,1] ) = H 1 GL2 (Z), S w−3 V2 ⊗ ε2 − 1 1 = Hcusp SL2 (Z), S w−3 V2 ⊕ Hinf SL2 (Z), S w−3 V2 , ( p, q) = (1, 2).

(141)

The other cohomology groups are zero. (ii) Further, for the strata ∂ P2 S, Q, w = 3, 0 H (S2 , L [w−3,0,0] ) = 0, w > 3,

( p, q) = (0, 0),

H 1 (S2 , L [w−3,0,0] ) = H 1 GL2 (Z), S w−3 V2

1 = Hcusp SL2 (Z), S w−3 V2

+

,

( p, q) = (1, 0).

The other cohomology groups vanish. (iii) Finally, for the strata ∂ P2 S the only nonzero cohomology groups are   Q, q = 0, H 0 S, H q (N , S w−3 V3 ) = Q, q = 3,  0 otherwise. Since ∂ S = ∂ P1 S ∪ ∂ P2 S,

∂ B S = ∂ P1 S ∩ ∂ P2 S,

we have the Mayer-Vietoris long exact sequence · · · −→ H ∗ (∂ S, LV ) −→ H ∗ (∂ P1 S, LV ) ⊕ H ∗ (∂ P2 S, LV ) −→ H ∗ (∂ B S, LV ) −→ · · · . It is easy to see that the differential in this complex maps isomorphically the group (140) to the group from step (iii) with q = 0. Further, it follows from the definitions that the differential maps isomorphically 1 from (141) onto the group from step (iii) with q = 3. the group Hinf Therefore the boundary cohomologies are ( + 1 Hcusp SL2 (Z), S w−3 V2 , i = 1, i H ∂ S, L S w−3 V3 = − 1 Hcusp SL2 (Z), S w−3 V2 , i = 3. i (∂ S, L To determine the subgroup Hinf S w−3 V3 ) one usually uses the theory of Eisenstein cohomology classes originated by G. Harder (see [G7] for details in our case). However, just in our case there is another approach. Namely, we show in the next 1 (SL (Z), S w−3 V )− section that H 3 (GL3 (Z), L S w−3 V3 ) contains the subgroup Hcusp 2 2


465

(see (163)). The proof uses [G4, Theorem 1.3] and some simple arguments presented in Section 7.1 which are independent of the rest of the paper. Therefore − 3 1 Hinf ∂ S, L S w−3 V3 = Hcusp SL2 (Z), S w−3 V2 . (142) 1 is zero. The Poincar´ It remains to show that Hinf e duality provides a nondegenerate pairing H 1 ∂ S, L S w−3 V3 ⊗ H 3 ∂ S, L S w−3 V ∨ −→ Q. (143) 3

Employing the Cartan involution θ of the group GL3 which transforms V3 into V3∨ , we get − 3 1 Hinf ∂ S, L S w−3 V ∨ = Hcusp SL2 (Z), S w−3 V2 . (144) 3

If x ∈ H 1 (S, L S w−3 V3 ) and y ∈ H 3 (S, L S w−3 V ∨ ), then Res(x) ∪ Res(y) = 0. There3 fore (142) together with the Poincaré pairing (143) imply that Res(x) = 0. The theorem is proved. 6.4. Cohomology of 01 (3; p) Recall that if 0 is a torsion-free subgroup of GL3 (Z), then H ∗ (0, Q) = H ∗ (S0 , Q) = H ∗ (S 0 , Q). R. Lee and J. Schwermer derived in [LS, Section 1.8] an exact sequence ∗ ∗ (S 0 , C) −→ 0. 0 −→ Hcusp (0, C) −→ H ∗ (S 0 , C) −→ Hinf

One has H q (S0 , Q) = 0 for q > 3. Further, H 1 (S0 , Q) = 0 by [BMS, Chapter 16]. The cuspidal cohomology satisfies the Poincaré duality. Therefore 2 3 dimHcusp 01 (3; p), Q = dimHcusp 01 (3; p), Q . (145) THEOREM 6.4 Let p be a prime number. Then   Q,    0, i Hinf 01 (3; p), Q = 1 Hcusp 0(2; p), V2 ,     H 1 0(2; p), ε ⊗ Q2 ⊕ Q( p−3)/2 , 2 cusp

q q q q

= 0, = 1, q ≥ 4, = 2, = 3. (146)

Proof Let 0(m; p) be the full congruence subgroup of level p of GLm (Z). The cohomology of the subgroup 0(3; p) has been computed by Lee and Schwermer [LS]. Let us recall (and slightly correct) their result.

466

A. B. GONCHAROV

Set SL± m (F p ) := GLm (Z)/ 0(m; p). Here F p is ∗ Hinf (S 0(3m; p) , Q) is an SL± m (F p )-module. Consider the following subgroups in SL± 3 (F p ): e1 P

e2 P

  ∗ ∗ ∗ := ∗ ∗ ∗  , 0 0 ±1   ±1 ∗ ∗ :=  0 ∗ ∗ , 0 ∗ ∗

the finite field of order p. Then

 ∗ e  M1 := ∗ 0  ±1 e  B := 0 0

 ∗ 0 ∗ 0 , 0 ±1  ∗ ∗ ±1 ∗  . 0 ±1

Denote by P10 , M10 , and P20 the similar subgroups of SL3 (F p ). Let e 01 (m; p) be the 0 (resp., e projection of 01 (m; p) onto SL± (F ). Denote by ξ ξ ) the nontrivial onep i m i 0 e dimensional representation of Pi (resp., Pi ) pulled from the one of the GL2 -part of the Levi quotient given by the determinant. Denote by [V] the isomorphism class of a representation V . Recall that V2 is the standard two-dimensional GL2 -module. f3 be the generalized Steinberg representation of SL± (F p ); it sits in the Let St 3 exact sequence SL± (F p )

2 0 −→ Q −→ ⊕i=1 Ind Pe 3 i

SL± (F p )

3 Q −→ Ind e B

f3 −→ 0. Q −→ St

It was proved in [LS] (see Section 2.5 there) that, considered as an SL3 (F p )module,

q Hinf (S 0(3; p) , Q)

 Q,      0, SL (F ) SL (F ) 1 = (0(2; p), V2 ) , 2 Ind P 0 3 p ξ10 ⊕ Ind P 0 3 p Hcusp   1 1    ⊕2 IndSL0 3 (F p ) H 1 (0(2; p), ε2 ) ⊕ [St e 3 ], cusp i=1 P

q = 0, q = 1, q ≥ 4, q = 2,

(147)

q = 3.

i

In fact, for q = 3 in [LS] the module SL3 (F p ) 1 2 ⊕i=1 Ind P 0 Hcusp (0(2; p), Q) i

1 (0(2; p), Q)] should be changed to appears, but the Pi0 -module [Hcusp 1 [Hcusp (0(2; p), ε2 )]. As abelian groups they are isomorphic. See also an argument in Section 7.7 which shows that Q has to be changed to ε2 . One can check, following the arguments in [LS], that, as an SL± 3 (F p )-module, q Hinf (S 0(3; p) , Q) in the nontrivial cases q = 2, 3 looks as follows:


467

q Hinf (S 0(3; p) , Q)

 ± SL±3 (F p ) 1  2 IndSL3 (F p ) e ξ ⊕ Ind Pe Hcusp (0(2; p), V2 ) , q = 2, 1 e P1 1 = ±  ⊕2 IndSL3 (F p ) H 1 (0(2; p), ε ) ⊕ [St e 3 ], q = 3. 2 cusp e i=1 P

(148)

i

0 (We replaced SL3 by SL± 3 and put tilde instead of over Pi and ξ1 .) ∗ To compute Hinf (01 (3; p), Q)) we need to find the invariants of the action of e 01 (3; p) on these SL± 3 (F p )-modules. We use the following general statement.

6.5 e Let ρ : P1 −→ Aut(W ) be a representation trivial on the unipotent radical and ±1 subgroup and such that H 0 (SL± 2 (F p ), W ) = 0. Then LEMMA

SL± (F p )

Ind Pe 3 1

W

e 01 (3; p)

= W 01 (2; p) . e

Proof The left-hand side is given by the space of W -valued functions f (g) on SL± 3 (F p ) satisfying the condition f (γ gy) = ρ(y −1 ) · f (g),

e1 . γ ∈e 01 (3; p), y ∈ P

The coset e 01 (3; p)\ SL± 3 (F p ) is identified with the set of rows X = (x 1 , x 2 , x 3 ) ∈ 3 e1 acts on it from the right. So we need to determine the space of F p − 0. The group P the W -valued functions f (X ) such that f (X y) = ρ(y −1 ) f (X ). Any nonzero vector (x1 , x2 , x3 ) is e 01 (3; p)-equivalent to (0, 1, 0) or (0, 0, x). The stabilizer of (0, 1, 0) is e e 01 (2; p) × {±1}. So f (0, 1, 0) ∈ W 01 (2; p) . The stabilizer of (0, 0, x) is SL± 2 (F p ) × ± {±1}. Since H 0 (SL2 (F p ), W ) = 0, we have f (0, 0, x) = 0. The lemma is proved.

Applying the lemma, we get the following spaces of e 01 (3; p)-invariants. SL± 3 (F p ) e (i) For Ind ξ1 it is zero. (ii)

For

(iii)

For

(iv)

e1 P SL± (F p ) Ind Pe 3 1 SL± (F p ) Ind Pe 3 i

1 (0(2; p), V ) it is H 1 (0 (2; p), V ). Hcusp 2 2 cusp 1

1 (0(2; p), ε ) it is H 1 (0 (2; p), ε ). Hcusp 2 2 cusp 1 f3 is ( p − 3)/2. The dimension of the space of e 01 (3; p)-invariants on St

Argumentation (i) Indeed, ξ(g) = −1 for g := diag(−1, 1, 1) ∈ e 01 (3; p). (ii) and (iii) These are clear.

468

(iv)

A. B. GONCHAROV

This is a corollary of the following elementary statements, which are checked similarly to the proof of the lemma above:

e p−1 0 (3; p) Q 1 =3· , 2 p−1 SL± (F p ) e 0 (3; p) dim Ind Pe 3 Q 1 = + 1, i 2 and the trivial module Q is, of course, e 01 (3; p)-invariant. So we get p−1 p−1 p−3 3· −2 +1 −1= . 2 2 2 The theorem is proved. SL± (F p )

3 dim Ind e B

7. Proofs of the theorems from Section 2 7.1. The Soulé elements (l) Since G•,−1 is abelian, we may identify it with the Ql -points of the corresponding (l)

algebraic group. So it makes sense to talk about projection of ϕ (l) (σ ) on G•,−1 . Let (l)

(l)

ϕ−w,−1 (σ ) be the component of this projection in Gr G−w,−1 . In [So] Soulé constructed for each integer m > 1 a Gal(Q(ζl ∞ )/Q)homomorphism ab χm : Gal Q/Q(ζl ∞ ) −→ Ql (m). He proved that it is zero if and only if m is even. Let I2m−1 (e : e)∨ be the generator of D−2m+1,−1 dual to I2m−1 (e : e). It follows from the key lemma [I1, Lemma B] that ϕ (l) (σ ) =

(1 − l m−1 )−1 χm (σ )ξ I2m−1 (e : e)∨ (m : odd ≥ 3). (m − 1)!

7.2. The depth 1 case The distribution relations, proved in Section 5.7, in the depth 1 case just mean that (l) G−w,−1 (µ N ) ⊂ ξµ N D−w,−1 (µ N ) ⊗ Ql . Easy classical arguments combined with the Borel theorem show that dim D−w,−1 (µ N ) = dim Hom K 2w−1 (S N ), Q . For w = 1 it is a reformulation of the Bass theorem, and the general case is completely similar. From the motivic theory of classical polylogarithms at N th roots of unity we get the following (see Theorem 2.1): (l) dim G−w,−1 (µ N ) = dim Hom K 2w−1 (S N ), Q .


469

Indeed, it has been proved in [BD] (see also [HW]) that motivic classical polylogarithms at N th roots of unity provide classes generating K 2w−1 (S N ) ⊗ Q; in particular, (−w) the l-adic regulators of these classes have been computed. Denote by H1 the part (−w,−m) of degree −w in H1 and similarly by H1 the part of degree −w, depth −m in H1 . We conclude that (l) (−w) (l) (l) (149) H1 G N /F−2 G N = G−w,−1 (µ N ) = Hom K 2w−1 (S N ), Ql = ξµ N D−w,−1 (µ N ) ⊗ Ql . (150) 7.3. Description of the image of the Galois group: The depth 2 case THEOREM 7.1 (a) Conjecture 1.1 is valid in the depth 2 case, that is, (l) G•,≥−2 (µ N ) ,→ ξµ N D•,≥−2 (µ N ) ⊗ Ql . (b)

Moreover, (l) G•,−2 (µ N )

= ξµ N

D•,−2 (µ N ) Ker δ : D•,−2 (µ N ) −→ 32 D•,−1 (µ N )

∨

.

Proof (l) Part (b) implies part (a). Since GrW G N is a quotient of the fundamental Lie algebra L T Ql (S N ) (see Section 2.7), one has (−w)

H1

(l)

(l)

G N /F−3 G N

Therefore

(−w,−2)

H1 (l)

(l)

⊂ Hom K 2w−1 (S N ), Ql .

(l)

(l)

G N /F−3 G N

= 0.

(151)

(152)

(l)

Indeed, [G N , G N ] ⊂ F−2 G N , so a nontrivial depth −2 part of H1 plus (151) makes the left-hand side of (132) bigger then the right-hand side. If G is a Lie algebra with a filtration F• indexed by integers n = −1, −2, . . . such that F−1 G = G , then the Lie algebra F F G /F−3 G is isomorphic to the Lie algebra GrF ≥−2 G := Gr−1 G ⊕ Gr−2 G . (l)

(l)

(l)

In particular, G N /F−3 G N is isomorphic to G•,≥−2 (µ N ). Therefore thanks to (152) we have (−w,−2) (l) H1 G•,≥−2 (µ N ) = 0. (153) (l)

(l)

This means that G•,≤−2 (µ N ) is generated by Gr G•,−1 (µ N ).

470

A. B. GONCHAROV

Therefore (l)

(153)

G•,−2 (µ N ) =

(l) (l) G•,−1 (µ N ), G•,−1 (µ N )

(150)

= ξµ N [D•,−1 (µ N ), D•,−1 (µ N )] ⊂ ξµ N D•,−2 (µ N ) .

This is equivalent to the statement of the theorem. The theorem is proved. Remark. If N = 1 or if N = p is a prime and • = −2, we prove that the space Ker δ : D•,−2 (µ N ) −→ 32 D•,−1 (µ N ) is zero. It may not be zero otherwise. The simplest example appears when • = −2 and N = 25. It would be interesting to construct explicitly the elements in the kernel. Using Theorem 7.1, one can show that if N is not prime, then ξµ N (D•1 (µ N )) could (l) be bigger than G• (µ N ). 7.4. The image of the Galois group for the m = 3, N = 1 case THEOREM 7.2 One has (l) G−w,−3 ,→ ξ(D−w,−3 ) ⊗ Ql . Proof When w is odd, this follows from Theorem 2.2. So we may assume that w is even. The same argumentation as in the proof of Theorem 7.1 gives (−w,−3) (l) (l) H1 G N /F−4 G N = 0. (154) (l)

(l)

(l)

However, a priori the Lie algebra G N /F−4 G N may be different from G•,≥−3 (µ N ). (l)

(l)

Recall that GrW G N is noncanonically isomorphic to G N . We show that if N = 1 and w is odd, the weight w, depth ≥ −3 parts of the standard cochain complexes (l) of GrW G (l) and G•• are isomorphic. For the depth ≥ −2 parts this has been proved above. So the discrepancy between them can appear only if there are elements x ∈ F3 GrW (G (l) )∨ such that δ(x) ∈ F1 GrW (G (l) )∨ ∧ F1 GrW (G (l) )∨ . (l) ∨ Since F1 GrW p (G ) can be nonzero only if p is odd, the weight of δ(x) must be even. This contradicts the assumption that w is odd. Therefore (−w,−3) (l) (−w,−3) (l) (l) H1 G•,≥−3 = H1 G N /F−4 G N = 0. (155)


471

Therefore this together with the previous theorem implies that when w is odd, (l) (l) G−w,≥−3 is generated by iterated commutators of triples of elements of G•,−1 . Since (l)

(l)

G•,−1 = ξ(D•,−1 ) ⊗ Ql , we conclude that G−w,−3 coincides with the subspace generated by iterated commutators of triples of elements of ξ(D•,−1 ) ⊗ Ql and so belongs to ξ(D•,−3 ) ⊗ Ql . The theorem is proved. (l)

(l)

Let L•,−k ⊂ G•,−k be the subspace generated by iterated commutators of k elements (l)

(l)

(l)

(l)

of G•,−1 . Then L•,≥−m := ⊕m k=1 L•,−k is a bigraded Lie subalgebra of G•,≥−m . COROLLARY 7.3 (l) (l) We have L•,−m = G•,−m for m = 2, 3.

Remark. We proved recently that, assuming Conjecture 1.1, one has (l) (l) 1 G−w,−4 /L−w,−4 = Hcusp GL2 (Z), S w−2 V2 ⊗ ε2 ⊗ Ql . b•• of depths 2, 3 7.5. The cohomology of the Lie algebras D•• and D Recall the isomorphism of bigraded Lie algebras b•• (G) = D•• (G) ⊕ Q(−1, −1), D where Q(−1, −1) is a one-dimensional Lie algebra of the bidegree (−1, −1). So b•• (G): there is the decomposition of the standard cochain complex of D b•• (G) = 3∗ D•• (G) ⊕ 3∗ D•• (G) ⊗ Q(1,1) . 3∗ D

(156)

It provides a canonical morphism of complexes b•• (G) −→ 3∗ ∂m : 3∗(m) D (m−1) D•• (G)[−1]. b•• (µ N ), alSurprisingly, it is easier to describe the structure of the Lie algebra D though the Galois-theoretic or motivic meaning of its Q(−1,−1) -component is unclear; it should correspond to ζ (1) or, better, Euler’s γ -constant, whatever it means. Remark. (i)

Notice the classical formula X d log 0(1 − z) = ζ (m)z m−1 dz,

where we set ζ (1) := γ .

m≥1

(ii)

One has γ =−

∞

Z

e−t log tdt = − 0

Z 0

∞

dt ◦ (e−t dt). t

472

A. B. GONCHAROV

(The iterated integral on the right has to be regularized.) So the Euler γ constant should be thought of as an “irregular” period of a motive. It was proved in [G4, Theorem 1.3] (see also Lemma 2.3) that i b•• ) = H i−1 GL2 (Z), S w−2 V2 ⊗ ε2 , i = 1, 2, (157) H(w,2) (D i b•,• ) = H i GL3 (Z), S w−3 V3 = 0, i = 1, 2, 3. H(w,3) (D (158) THEOREM 7.4 Denote by H(w,m) the weight w, depth m part of H . Then

i i−1 H(w,2) (D•• ) = Hcusp GL2 (Z), S w−2 V2 ⊗ ε2 , i H(w,3) (D•,• )

= 0,

i = 1, 2,

i = 1, 2, 3.

(159) (160)

Proof Consider the inclusion of complexes provided by the depth 2 part of (156): b ∂2 b•,1 −→ b•,1 . D•,2 −→ 32 D•,1 ,→ D•,2 −→ 32 D D

(161)

Here b ∂2 is the composition of the map ∂2 followed by the natural inclusion D•,1 ,→ b•,1 . The isomorphism µ from [G4, Theorem 1.2] is easily extended to an isomorD phism between the complex M∗(2) ⊗GL2 (Z) S •−2 V2 and the complex on the right of (161), which takes τ[1,2] M∗(2) ⊗GL2 (Z) S •−2 V2 just to the subcomplex on the left. This together with Lemma 2.3 proves (159). Notice that the second summand D•,1 [−1] in the decomposition (156) of the b•• provides the Eisenstein part of the cohomology. depth 2 part of 3∗ D The depth 3 part of the decomposition (156) is δ δ δ (162) D•,3 −→ D•,2 ⊗ D•,1 −→ 33 D•,1 ⊕ D•,2 −→ 32 D•,1 ⊗ Q(1,1) . Therefore we see that 3 1 b•• ) contains as a direct summand Hcusp H(w,3) (D GL2 (Z), S w−3 V2 ⊗ ε2 . (163) This is the last bit needed to prove Theorem 6.2. Now we can start using Theorem 6.2. By [G4, Theorem 1.3] and Theorem 6.2 we have i b•• ) 1.3 H(w,3) (D = H i GL3 (Z), S w−3 V3 ( 0, i = 1, 2, 6.2 = (164) 1 Hcusp GL2 (Z), S w−3 V2 ⊗ ε2 , i = 3. Therefore, using (159), we see that the complex on the left of (162) is acyclic. This proves formula (160). Theorem 7.4 is proved.


473

7.6. Proofs of Theorems 2.4, 2.6, and 2.10 (l) Set D•• := D•• ⊗ Ql . By Corollary 4.2 we get D−2w,−1 = 0. Therefore (l)

(l)

G•,−1 = D•,−1 . (l)

(l)

S E L(l) , this implies that Since D•• and L•• are Lie subalgebras of Gr Der•• (l) (l) L•• ,→ D•• .

(165)

Thus there are commutative diagrams where [, ] are the commutator maps: (l)

32 L•,−1 ↓ [, ] (l) L•,−2

(l)

=

(l)

−→ 32 D•,−1 ↓ [, ] (l) ,→ D•,−2

(l)

L•,−2 ⊗ L•,−1 ↓ [, ] (l) L•,−3

and

−→ ,→

(l)

(l)

D•,−2 ⊗ D•,−1 ↓ [, ] (l) D•,−3

LEMMA 7.5 1 We have H(•,m) (D•• ) = 0 for m = 2, 3.

Proof Since H 0 (GL2 (Z), S w−2 V2 ⊗ ε2 ) = 0, the lemma follows from formulas (159) and (160). The lemma means that the right vertical arrows in the diagrams above are surjective. Therefore the left diagram shows that the map (l)

(l)

L•,−2 ,→ D•,−2

is surjective and hence is an isomorphism. Thus the top arrow in the right diagram is an isomorphism. Therefore the second diagram shows that the map (l)

(l)

L•,−3 ,→ D•,−3

is also an isomorphism. Summarizing, we have (l)

(l)

L•,−2 = D•,−2

and

(l)

(l)

L•,−3 = D•,−3 .

(166)

Combining this with Corollary 7.3, we get (l)

(l)

G•,−2 = D•,−2

and

(l)

(l)

G−w,−3 = D−w,−3 ,

w is odd.

(167)

After this we reduced the study of the depth −2 and −3 quotients of the Lie (l) (l) algebras G•• and Gb•• to the study of the corresponding quotients of the Lie algebras b•• , which were investigated in [G4]. Therefore Theorems 2.4(a), 2.6(a) D•• and D and (b), and 2.10 follow from [G4, Theorem 1.2], Lemma 2.3, Proposition 6.3(b), and [G4, Theorem 6.2]. It remains to prove the formula from Theorem 2.6(c). Thanks to Theorem 2.2 we may assume that w is odd.

474

A. B. GONCHAROV

Computation of the dimensions (i) Consider the generating series am (x) :=

X w>0

w dim 3m (w) (D•,1 )x .

We know that a1 (x) = x 3 /(1 − x 2 ). Thus one has x 3+5+···+(2m+1) am (x) = Qm . 2i i=1 (1 − x ) In particular, a2 (x) =

x8 (1 −

x 2 )(1 −

x 4)

,

a3 (x) =

x 15 . x 4 )(1 − x 6 )

x 2 )(1 −

(1 −

(ii) The ring of modular forms for SL2 (Z) is generated by the Eisenstein series E 4 (z) and E 6 (z). By the Eichler-Shimura theorem the dimension of the space of modular forms for SL2 (Z) coincides with dim H 1 (GL2 (Z), S w−2 V2 ). Therefore X

dim H 1 GL2 (Z), S w−2 V2 x w =

w≥2

(iii) Let dm (x) :=

X

1 x 4 )(1 −

(1 −

x 6)

− 1.

dim Dw,m x w .

w>0

Computing the Euler characteristic of the depth 2 part of the standard cochain complex of the Lie coalgebra D•• and using Theorem 7.4, we get d2 (x) − a2 (x) = −

1 (1 −

x 4 )(1 −

x 6)

+ 1.

Therefore, using the formulas above, we get d2 (x) =

x8 (1 −

x 2 )(1 −

x 6)

.

This formula is equivalent to Theorem 2.4(b). (iv) Computing the Euler characteristic of the depth 3 part of the standard cochain complex of the Lie coalgebra D•• and using Theorem 7.4, we get d3 (x) − d2 (x) · a1 (x) + a3 (x) = 0. Therefore, using the formulas above, we get d3 (x) =

x 11 (1 + x 2 − x 4 ) . (1 − x 2 )(1 − x 4 )(1 − x 6 )


475

This formula is equivalent to Theorem 2.6(c). These results also provide the computation of the right-hand sides of the formulas appearing in [G4, Theorems 1.4 and 1.5]. These theorems in particular imply that the double shuffle relations provide a complete list of constraints on the Lie algebra of the image of the Galois group in (l) Aut π1 (P1 − {0, 1, ∞}, v∞ ) in the depths −2 and −3. 7.7. Proofs of Theorems 2.14 and 2.15 They are similar to the proofs in Section 7.3. Recall that p is a prime number. We use (l) shorthands like D• (µ p ) for the diagonal Lie algebra D•1 (µ p )(l) . By Theorem 2.12, (l)

(l)

Gr L−1 (µ p ) = D−1 (µ p ). This implies that

L•(l) (µ p ) ,→ D•(l) (µ p ).

(168)

Thus there are commutative diagrams (l)

(l)

=

32 L−1 (µ p ) −→ 32 D−1 (µ p ) ↓ [, ] ↓ [, ] (l) (l) L−2 (µ p ) ,→ D−2 (µ p ) and

(l)

(l)

L−2 (µ p ) ⊗ L−1 (µ p ) −→ ↓ [, ] (l) L−3 (µ p ) ,→

(l)

(l)

D−2 (µ p ) ⊗ D−1 (µ p ) ↓ [, ] (l) D−3 (µ p )

LEMMA 7.6 1 (D (µ )) = 0 for m = 2, 3. We have H(m) • p

Proof b•• (G)) = H 1 (D•• (G)). This implies Thanks to decomposition (156) one has H 1 ( D b• (G)) = H 1 (D• (G)). The following crucial result is provided by [G4, TheoH 1( D rems 1.2, 6.1, and 6.2] (see also [G9]): i b• (µ p ) = H i−1 01 (2; p), ε2 = H i−1 01 ( p), Q − , H(w,2) D (169) i i b• (µ p ) = H 01 (3; p), Q , i = 1, 2, 3. H(w,3) D (170) In particular, 1 b H(2) D• (µ p ) = H 0 01 (2; p), ε2 ,

1 b H(3) D• (µ p ) = H 1 01 (3; p), Q . (171) Both groups are zero: for the first one it is clear, and for the second, see [BMS, Chapter 16] or use Kazhdan’s theorem. The lemma is proved.

476

A. B. GONCHAROV

So the right vertical arrows in the diagrams above are surjective. Using the same arguments as in Section 7.6, we get Lie algebra isomorphisms (l)

L≥−m (µ p ) = D≥−m (µ p ) ⊗ Ql ,

m = 2, 3.

(172)

After this Theorem 2.14 follows from Theorem 7.1 and the results of [G4]. Similarly, Theorem 2.15 follows from (172) and the results of [G4] (see also [G9] for a more elaborate exposition of the results of [G4]). These results imply that the double shuffle relations provide a complete list of constraints on the weight = depth part of the Lie algebra of the image of the Galois (l) group in Aut π1 (X p , v∞ ) in the depths −2 and −3. 7.8. The Lie coalgebra D•un (µ p ) Recall that p is a prime number. Let us define a linear map v p : D• (µ p ) −→ Q by v p : Dm (µ p ) 7−→ 0

for m > 1,

v p : I1,1 (1 : ζ pa ) 7 −→ 1

(ζ pa 6 = 1).

Since there is no distribution relation when p is prime, the map v p is well defined. Let D•un (µ p ) ,→ D• (µ p ) be the codimension 1 subspace Ker(v p ). PROPOSITION 7.7 D•un (µ p ) is a sub-Lie

coalgebra of D• (µ p ).

Proof Let V be a vector space, and let f ∈ V ∗ . There is a map ∂ f : 3n V −→ 3n−1 V, v1 ∧ · · · ∧ vn 7−→

n X

∂ 2 = 0,

(−1)i−1 f (vi )v1 ∧ · · · ∧ b vi ∧ · · · ∧ vn .

i=1

In particular, the map v p provides a degree −1 map ∂v p : 3n D• (µ p ) −→ 3n−1 D• (µ p ). There is an exact sequence ∂v p

un 0 −→ 32(m) D•un (µ p ) ,→ 32(m) D• (µ p ) −→ Dm−1 (µ p ) −→ 0.

So to prove the proposition we need to check that the composition δ

∂v p

Dm (µ p ) −→ 32(m) D• (µ p ) −→ Dm−1 (µ p )


477

is zero. To simplify notation we set {g0 : · · · : gm } := I1,...,1 (g0 : · · · : gm ). Then ∂v p ◦ δ{g0 : · · · : gm } = ∂v p

m+1 X {gi+1 : gi+2 : · · · : gi−1 } ∧ {gi−1 : gi } i=0

+ {gi−1 : gi } ∧ {gi : gi+1 : · · · : gi−2 } .

(173)

Since {gi−1 : gi } = 0 if gi−1 = gi , we calculate (173) as follows. We locate g0 , . . . , gm+1 cyclically on the circle. Say that a string in this cyclic word is a sequence of letters gi , gi+1 , . . . , g j such that gi = gi+1 = · · · = g j , gi−1 6= gi , and g j 6 = g j+1 . The cyclic word splits into a union of strings. For instance, if all gi ’s are distinct, we get m + 1 one-element strings. The sum (173) is equal to X ∂v p {gi+1 : gi+2 : · · · : gi−1 } ∧ {gi−1 : gi } strings

X

+

∂v p {g j : g j+1 } ∧ {g j+1 : · · · : gi−1 : gi } .

strings

The sum of the two terms corresponding to a given string is obviously zero, so (173) is zero. The proposition is proved. It follows from Proposition 7.7 that the map ∂v p : 3∗(m) D• (µ p ) −→ 3∗(m−1) D• (µ p )[−1] is a homomorphism of complexes. 7.9. The depth 2 part of the cochain complex of D•un (µ p ) and the cuspidal cohomology of 01 (2; p) THEOREM 7.8 There is a canonical isomorphism of complexes D2 (µ p ) −→ 32 D1un (µ p ) = τ[1,2] M∗(2) ⊗01 (2; p) Q . Proof This theorem is a version of the results proved in [G2, Sections 3.5 and 3.6]. Let us b1 (µ p ) by making it zero on Q(1,1) . We define a map extend v p to D δ0

b1 (µ p ) −→ D1 (µ p ) ⊕ D1 (µ p ) 32 D by setting δ 0 := (−∂, ∂ + v p ), that is, ( β 0 ⊕ {1 : ζ p } − {1 : ζ pα }, α, β 6= 0, {1 : ζ pα } ∧ {1 : ζ pβ } 7−→ β β −{1 : ζ p } ⊕ {1 : ζ p }, α = 0, β 6= 0.

478

A. B. GONCHAROV

Thus we get the following complex, placed in degrees [1, 3]: δ

δ0

b1 (µ p ) −→ D1 (µ p ) ⊕ D1 (µ p ). D2 (µ p ) −→ 32 D

(174)

THEOREM 7.9 The complex (174) is canonically identified with the complex M∗(2) ⊗01 (2; p) Q.

Proof We start with a lemma describing more explicitly the rank 2 modular complex. LEMMA 7.10 The double shuffle relations for m = 2 are equivalent to the dihedral relations

hv0 , v1 , v2 i = −hv1 , v0 , v2 i = −hv0 , v2 , v1 i = h−v0 , −v1 , −v2 i. Proof The double shuffle relations in h·, ·, ·i generators look as follows: hv0 , v1 , v2 i + hv1 , v0 , v2 i = 0,

hv1 , v2 − v1 , −v2 i + hv2 , v1 − v2 , −v1 i = 0.

Changing variables u 0 = v1 , u 1 = v2 − v1 , and u 2 = −v2 , we write the second of them as hu 0 , u 1 , u 2 i + h−u 2 , −u 1 , −u 0 i = 0. The lemma follows. It follows from this lemma that the modular complex for GL2 (Z) is canonically isomorphic to the following complex of left GL2 (Z)-modules: ∂ ∂0 Z GL2 (Z) ⊗ D2 ξ2 −→ Z GL2 (Z) ⊗ D1,1 ξ1,1 −→ Z GL2 (Z)/ e B , where D2 ⊂ GL2 (Z) is the order 12 dihedral subgroup, ±1 ∗ 0 ±1 ±1 0 e , , B := D1,1 = , 0 ±1 ±1 0 0 ±1 ξ2 is the character of D2 given by the determinant, and ξ1,1 is a nontrivial character 0 of D1,1 killing ±1 0 ±1 . The differentials commute with the left action of the group GL2 (Z) and so are determined by their action on the unit in GL2 . They are given by −1 1 1 0 1 0 0 −1 ∂: 7−→ − − − , 0 1 0 1 1 −1 −1 0 1 0 0 1 1 0 0 ∂ : 7−→ − . 0 1 1 0 0 1


479

The coset 01 (2; p)\ GL2 (Z) is identified with the set of rows {(α, β) ∈ F p2 − 0}. One can identify it with the set of nonzero rows {(α, β, γ )} with α + β + γ = 0. Then Z (α, β, γ ) ∈ F p3 − 0|α + β + γ = 0 1 , M(2) = Z 01 (2; p)\ GL2 (Z) ⊗ D2 ξ2 = the dihedral relations 2−0 Z (α, β) ∈ F p 2 . M(2) = Z 01 (2; p)\ GL2 (Z) ⊗ D1,1 ξ1,1 = (α, β) = −(β, α) = (±α, ±β) 3 = Z[0 (2; p)\ GL (Z)/ e M(2) B] = the abelian group with the generators [β, 0] 1 2 and [0, β], where β 6= 0, and the only relation is symmetry under β 7−→ −β. The complement X 1 ( p) − Y1 ( p) consists of p − 1 cusps. The natural covering X 1 ( p) −→ X 0 ( p) is unramified of degree ( p − 1)/2. There are two cusps on X 0 ( p), the zero and ∞ cusps. So there are ( p − 1)/2 cusps over zero and over ∞. Under the identification with D1 (µ p ) ⊕ D1 (µ p ) they correspond to the summands D1 (µ p ). The desired isomorphism of complexes is defined by γ

(α, β, γ ) 7−→ {ζ pα , ζ pβ , ζ p }, (α, β) 7−→ {ζ pα , ζ p−α } ∧ {ζ pβ , ζ p−β }, [β, 0] ⊕ [0, β 0 ] 7−→ {ζ pβ , ζ p−β } ⊕ {ζ pβ , ζ p−β }. 0

0

To check that it is indeed an isomorphism of the vector spaces, notice that simγ β ilarly to Lemma 7.10, the only relations among the elements {ζ pα , ζ p , ζ p } are the dihedral symmetries. Notice also that {1, 1, 1} = 0 and the only distribution relation P β −α−β 0 = {1, 1, 1} = α,β {ζ pα , ζ p , ζ p } follows from the skew symmetry. Theorem 7.9 is proved. Theorem 7.8 follows immediately from Theorem 7.9. 7.10. The level p Galois Lie algebra is not free for p ≥ 5 LEMMA 7.11 For a prime p one has (l)

D1 (µ p ) ⊗ Ql = G1,1 (µ p ),

(l)

D2 (µ p ) ⊗ Ql = G2,2 (µ p ).

Proof The first equality follows immediately from Theorem 2.12. The second follows from Theorem 7.1 (b) and Lemma 7.6. 7.12 2 (D (µ )) 6 = 0 for a prime p ≥ 5. One has dim H(2) • p LEMMA

480

A. B. GONCHAROV

Proof There is the following complex (compare with (174)): D2 (µ p )

where

δ

−→

32 D1 (µ p )

δ 00

D1un (µ p ),

−→

(175)

δ 00 : {1 : ζ pα } ∧ {1 : ζ pβ } 7 −→ {1 : ζ pβ } − {1 : ζ pα }.

The map δ 00 is obviously surjective. Thus p−3 2 D• (µ p ) ≥ dim D1un (µ p ) = . dim H(2) 2 The lemma is proved. COROLLARY 7.13 (l) The Galois Lie algebra G•• (µ p ) is not free for a prime p ≥ 5.

Proof (l) (l) (l) (l) The w = 2 part of G•• (µ p ) is G2,1 (µ p ) ⊕ G2,2 (µ p ). Observe that G2,1 (µ p )∨ is in the kernel of the coproduct δ since the Galois Lie coalgebra is, by its very definition, graded by the depth. Thus δ (l) (l) 2 (l) (µ p ) = Coker G2,2 (µ p )∨ −→ 32 G1,1 (µ p )∨ . H(2,2) G•• It follows from Lemma 7.11 that the complex in the parenthesis is isomorphic to the weight two part of the standard cochain complex of the Lie algebra D• (µ p ). So Lemma 7.12 implies the corollary. 7.11. The depth 3 part of 3∗ D•un (µ p ) THEOREM 7.14 One has   0, i = 1,  2 un i 1 Hcusp 01 (3; p), Q ⊕ Hcusp 01 (2; p), V2 , i = 2, H(3) D• (µ p ) =   H 3 0 (3; p), Q, i = 3. cusp 1 Proof There is an exact sequence of complexes: δ

D3 (µ p ) −→ ↓= δ

D3 (µ p ) −→

δ

D2 (µ p ) ⊗ D1un (µ p ) −→ ↓ D2 (µ p ) ⊗ D1 (µ p ) ↓ ∂v p D2 (µ p )

δ

−→ δ

33 D1un (µ p ) ↓ 33 D1 (µ p ) ↓ ∂v p

−→ 32 D1un (µ p )

(176)


481

A choice of splitting Q −→ D1 (µ p ) of the map v p provides a splitting of the middle complex into a direct sum of the top and bottom complexes. Taking into account canonical decomposition (156), we end up with b• (µ p ) = 3∗ D•un (µ p ) ⊕ 3∗ D•un (µ p ) [−1] ⊕ 3∗ D• (µ p ) [−1]. 3∗(3) D (3) (2) (2) Notice also a noncanonical splitting 3∗(2) D• (µ p ) = 3∗(2) D•un (µ p ) ⊕ D1un (µ p )[−1]. Therefore we have b• (µ p ) = 3∗ D•un (µ p ) ⊕ 3∗ D•un (µ p ) ⊗ Q2 [−1] ⊕ D•un (µ p )[−2]. 3∗(3) D (2) (3) This implies that 1 Hcusp 01 (2; p), ε2 ⊗ Q2 ⊕ Q( p−3)/2 3 (0 (3; p), Q). This has been known to us through Theis a direct summand of Hinf 1 orem 6.4. On the other hand, this gives an additional confirmation that the result of 3 (0(3; p), Q) has to be corrected, as was explained in Section [LS] concerning Hinf 6.3. Using the relation (169)–(170) between the depth 2 and 3 pieces of the cohob• (µ p ) and the cohomology of groups 01 (2; p) and mology of the Lie coalgebra D 01 (3; p), and combining this with the description of the cohomology of the group 01 (3; p) given in the previous chapter, in particular Theorem 6.4, we arrive at the proof of Theorem 7.14.

b• (µ p )): Remark. Here are some cycles in H 2 (D γ

γ

{ζ pα , ζ p−α } ⊗ {ζ pα , ζ pβ , ζ p } + {ζ pβ , ζ p−β } ⊗ {ζ pβ , ζ pα , ζ p }. They probably generate H 2 , and one should be able to prove directly using these 1 (0 (2; p), V ). b• (µ p )) = Hcusp cycles that H 2 (D 1 2 7.12. Proof of Corollary 2.16 Let us introduce the following notation. Let D• be a Z+ -graded Lie algebra. Denote by χ(m) (D• ) the Euler characteristic of the degree m part of the standard cochain complex of D• . 7.15 For m = 2, 3 one has COROLLARY

( χ(m) D•un (µ p )

=

1 dim Hcusp 01 (2; p), ε2 , m = 2, 1 dim Hcusp 01 (2; p), V2 , m = 3.

(177)

482

A. B. GONCHAROV

Proof For m = 2 this was done before. For m = 3 this follows from the above theorem and the Poincaré duality for cuspidal cohomology for 01 (3; p). Corollary 2.16: The m = 2 case. Thanks to Lemma 2.3 and Theorem 7.8, 1 dim D2 (µ p ) − dim 32 D1un (µ p ) = 0 − dim Hcusp 01 (2; p), ε(2) . One has dim D1un (µ p ) =

p−3 ( p − 3)( p − 5) => dim 32 D1un (µ p ) = . 2 8

Using [Sh, Proposition 1.40], we have for p ≥ 5, − p2 − 1 p−1 1 dim H 1 X 1 ( p), Q = dim Hcusp 01 (2; p), ε(2) = 1 + − . 24 2 (In our case, using the notations of [Sh], µ = ( p 2 − 1)/2, ν2 = ν3 = 0, ν∞ = p − 1 and 0 1 ( p) := 01 ( p)/ ± Id.) Therefore (l)

(172)

dim L−2 (µ p ) = dim D2 (µ p ) =

( p − 5)( p − 1) . 12

Corollary 2.16: The m = 3 case. We use Corollary 7.15 and the following result (see [Sh, Theorem 2.25]): p2 − 1 p − 1 p − 1 1 dim Hcusp 01 (2; p), V2 ⊗ ε2 = 2 − + . 24 2 2 Using this and (172), computation of the Euler characteristic gives (l)

( p − 1)( p − 5)( p − 3) 24 ( p − 3)( p − 5)( p − 7) 1 − − dim Hcusp 01 (2; p), V2 ⊗ ε2 48 ( p − 5)( p 2 − 2 p − 11) = . 48

(172)

dim L−3 (µ p ) = dim D3 (µ p ) =

We believe that there exists an explicit degree m polynomial in p giving dimDm (µ p ) b• (µ p )) similar to (177). Notice that for all m. There should be a formula for χm (D 2 there is no closed formula for dim Hcusp (01 (3; p)). b• (µ p )) seems to be a feasible problem for the following The computation of χ4 (D reasons. First, according to [G9], the relation between the depth m part of the cohob• (µ p ) and cohomology of 01 (m; p) is still the case for mology of the Lie coalgebra D m = 4: i b• (µ p ) = H i+2 01 (4; p), ε4 , 1 ≤ i ≤ 4. H(4) D


483

Here ε4 is the character of 01 (4; p) given by the determinant. Therefore 6 X b χ4 D• (µ p ) = (−1)i dim H i 01 (4; p), ε4 . i=3

Further, the cuspidal cohomology of 01 (4; p) is in the degrees 4 and 5, and the rest ∗ ; that is, of the cohomology is given by the cohomology at infinity Hinf ∗ ∗ 01 (4; p), ε4 −→ 0 0 −→ Hcusp 01 (4; p), ε4 −→ H ∗ 01 (4; p), ε4 −→ Hinf is an exact sequence. This can be proved using the following facts: besides the trivial representation there is only one unitary cohomological representation of SL4 (R) (the so-called special representation), and this representation is tempered (see [VZ, Theorems 5.6 and 6.16] for the general results). Then we employ arguments similar to the one given in [LS, Chapter 1]. Since the cuspidal cohomology satisfies the Poincaré duality, we get 6 X i b χ4 D• (µ p ) = (−1)i dim Hinf 01 (4; p), ε4 . i=3

Problem. Compute the right-hand side of this formula. We expect that the cuspidal cohomology of 01 (3; p) will not appear in the answer, so we should have an explicit formula for it. 7.13. The Galois groups unramified at p (l) (l) Let us assume that p 6= l. Let G−m (µ p )un be the maximal quotient of G−m (µ p ) unramified at the place 1 − ζ p . It is easy to see that (l) G−1 (µ p )un = Hom the group of the cyclotomic units in Z[ζ p ], Ql . It follows from Theorem 7.8 that (l)

(l)

G−2 (µ p )un = G−2 (µ p ),

(l)

(l)

L−3 (µ p )un = L−3 (µ p ).

7.16 (l) The Lie algebra G• (µ p )un is not free for a prime p ≥ 11. Assuming Conjecture 1.1, the above statement is valid for p = 7.

THEOREM

(a) (b)

(l)

Remark. We do not know whether the Lie algebra G• (µ5 )un is free or not.

484

A. B. GONCHAROV

Proof By Theorem 7.8 and Lemma 2.3 one has 2 1 dim H(2) G•(l) (µ p )un = dim Hcusp 01 (2; p), ε2 . The dimension on the right is known to be greater than zero for p ≥ 11. Part (a) is proved. For p = 7 we need to go to depth three. By Theorem 7.11 one has 2 1 D•(l) (µ p )un ≥ dim Hcusp 01 (2; p), V2 . dim H(3) The dimension on the right is known to be greater than zero if p ≥ 7. It follows (l) (l) from Conjecture 1.1 that D3 (µ p ) = G3 (µ p ). This, along with the already known (l) (l) result Dm (µ p )un = Gm (µ p )un for m = 1, 2, implies that the left group above is 2 (G (l) (µ )un ). Part (b) of the theorem is proved. isomorphic to dim H(3) p • Acknowledgments. I am grateful to M. Kontsevich and G. Harder for very useful discussions. The initial draft of this paper was submitted in 1998 to Mathematical Research Letters. I am very much indebted to the referee from this journal for correcting some gaps and errors and making many suggestions, incorporated into the text, that greatly improved and clarified the exposition. The initial draft of this paper was written during the summer and fall 1998 at the Max Planck Institute (MPI, Bonn) and reported at the MPI Conference on Polylogarithms at Schloss Ringberg (August 1998). I am grateful to MPI for its hospitality. Corrections to paper [G4] (1) In formulas (10), (35), (36), and the formula one line before (35), one needs to replace S w−2 V2 with S w−2 V2 ⊗ ε2 , where ε2 is as above. (2) In Theorem 6.1, 0 is a finite index subgroup of GL2 (Z); replace H i−1 (0, V ) with H i−1 (0, V ⊗ ε2 ). (3) Section 4.1, first two lines: use Z[X ] instead of Z[[X ]]; line four: use Z[Pm ] instead of Z[[Pm ]]. (4) In Theorem 6.2, 0 is a finite index subgroup of GL3 (Z). References [BMS]

[Be] [BD]

H. BASS, J. MILNOR, and J.-P. SERRE, Solution of the congruence subgroup problem for

´ SLn (n ≥ 3) and Sp2n (n ≥ 2), Inst. Hautes Etudes Sci. Publ. Math. 33 (1967), 59–137. MR 39:5574 465, 475 A. A. BEILINSON, Polylogarithms and cyclotomic elements, MIT preprint, 1989. 400 A. A. BEILINSON and P. DELIGNE, Motivic polylogarithms and Zagier’s conjecture, manuscript (version of 1992). 400, 404, 423, 424, 469


485

[BK]

S. BLOCH and I. KRIZ, Mixed Tate motives, Ann. of Math. (2) 140 (1994), 557–605.

[B]

´ A. BOREL, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole Norm. Sup.

[BS]

A. BOREL and J.-P. SERRE, Corners and arithmetic groups, Comment. Math. Helv. 48

[BW]

A. BOREL and N. WALLACH, Continuous Cohomology, Discrete Subgroups, and

MR 96c:11062 (4) 7 (1974), 235–272. MR 52:8338 405 (1973), 436–491. MR 52:8337

[D1]

[D2] [D3] [Dr]

[G1]

[G2] [G3]

[G4]

[G5] [G6]

[G7]

[G8] [G9] [Gr]

[HM]

Representations of Reductive Groups, 2d ed., Math. Surveys Monogr. 67, Amer. Math Soc., Providence, 2000. MR 2000j:22015 462 P. DELIGNE, “Le groupe fondamental de la droite projective moins trois points” in Galois groups over Q (Berkeley, 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989, 79–297. MR 90m:14016 397, 398, 400, 404, 405, 417, 419 , letter to D. Broadhurst, June 1997. 418, 426 , letter to the author, July 2000. 426 V. G. DRINFELD, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q), Leningrad Math. J. 2 (1991), 829–860. MR 92f:16047 398, 403, 443 A. B. GONCHAROV, “Polylogarithms in arithmetic and geometry” in Proceedings of the International Congress of Mathematicians (Zürich, 1994), Vol. 1, 2, Birkhäuser, Basel, 1995, 374–387. MR 97h:19010 401 , The double logarithm and Manin’s complex for modular curves, Math. Res. Lett. 4 (1997), 617–636. MR 99e:11086 397, 398, 401, 402, 477 , “Mixed elliptic motives” in Galois Representations in Arithmetic Algebraic Geometry (Durham, England, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 147–221. MR 2000e:11084 426 , Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998), 497–516. MR 2000c:11108 397, 398, 401, 402, 410, 413, 414, 416, 426, 429, 432, 433, 442, 464, 472, 473, 475, 476, 484 , Multiple ζ -numbers, hyperlogarithms and mixed Tate motives, preprint MSRI 058-93, 1993. 397, 408, 433 , Galois groups, geometry of modular varieties and graphs, Proceedings of the Arbeitstagung 1999 (Bonn), preprint MPI-50f (1999), http://mpim-bonn.mpg.de/ 398, 401, 402, 414, 416 , Multiple ζ -values, Galois groups and geometry of modular varieties, Proceedings of the Third European Congress of Mathematics (Barcelona, 2000), arXiv:math.AG/0005069 464 , Multiple polylogarithms and mixed Tate motives, arXiv:math.AG/0103059 426, 432, 433 , Multiple zeta values, modular complexes, Feynman diagrams and geometry of symmetric spaces, to appear. 401, 402, 414, 416, 458, 475, 476, 482 A. GROTHENDIECK, “Esquisse d’un programme” in Geometric Galois Actions, I, London Math. Soc. Lecture Note Ser. 242, Cambridge Univ. Press, Cambridge, 1997, 5–48. MR 99c:14034 398 R. HAIN and M. MATSUMOTO, Weighted completion of Galois groups and some

486

A. B. GONCHAROV

conjectures of Deligne, preprint, 2000. 423 [Ha] [HW]

[I1] [I2]

[I3]

[I4]

[K]

[LS]

[Sh]

[So]

[VZ] [Z1]

G. HARDER, letter to the author, 1999. A. HUBER and J. WILDESHAUS, Classical motivic polylogarithms according to

Beilinson and Deligne, Doc. Math. 3 (1998), 27–133. MR 99k:19001a 400, 404, 424, 469 Y. IHARA, Profinite braid groups, Galois representations and complex multiplication, Ann. of Math. (2) 123 (1986), 43–106. MR 87c:11055 398, 405, 468 , “Galois representation arising from P1 − {0, 1, ∞} and Tate twists of even degree” in Galois Groups over Q (Berkeley, 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989, 299–313. MR 90k:11067 , “Braids, Galois groups, and some arithmetic functions” in Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 99–120. MR 95c:11073 398 , Some arithmetical aspects of Galois action on the pro- p fundamental group of b π1 (P1 − {0, 1, ∞}), preprint 1229, 1999, http://kurims.kyoto-u.ac.jp/˜kenkyubu/paper/all.html 398, 408 M. KONTSEVICH, “Formal (non)commutative symplectic geometry” in The Gelfand Mathematical Seminars, 1990–1992, Birkhäuser, Boston, 1993, 173–187. MR 94i:58212 443, 444, 460 R. LEE and J. SCHWERMER, Cohomology of arithmetic subgroups of SL3 at infinity, J. Reine Angew. Math. 330 (1982), 100–131. MR 83k:22029 462, 465, 466, 481, 483 G. SHIMURA, Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures 1, Iwanami Shoten, Tokyo; Publ. Math. Soc. Japan 11, Princeton Univ. Press, Princeton, 1971. MR 47:3318 461, 482 C. SOULE´ , “On higher p-adic regulators” in Algebraic K -Theory (Evanston, Ill., 1980), Lecture Notes in Math. 854, Springer, Berlin, 1981, 372–401. MR 83b:12013 405, 424, 468 D. VOGAN and G. ZUCKERMAN, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), 51–90. MR 86k:22040 483 D. ZAGIER, “Values of zeta functions and their applications” in First European Congress of Mathematics (Paris, 1992), II, Progr. Math 120, Birkhäuser, Basel, 1994, 497–512. MR 96k:11110

Department of Mathematics, Brown University, 151 Thayer, Box 1917, Room 314, Providence, Rhode Island 02912, USA; [email protected]


ON INJECTIVITY AND NUCLEARITY FOR OPERATOR SPACES EDWARD G. EFFROS, NARUTAKA OZAWA, AND ZHONG-JIN RUAN

Abstract An injective operator space V which is dual as a Banach space has the form e R(1−e), where R is an injective von Neumann algebra and where e is a projection in R. This is used to show that an operator space V is nuclear if and only if it is locally reflexive and V ∗∗ is injective. It is also shown that any exact operator space is locally reflexive. 1. Introduction A Banach space V is said to be injective if for any Banach spaces W1 ⊆ W2 , each contraction ϕ : W1 → V has a contractive extension ϕ˜ : W2 → V . In particular, V = C is an injective Banach space by the Hahn-Banach theorem. (We consider only complex Banach spaces in this paper.) The injective Banach spaces, dual spaces, and second dual spaces were characterized in the 1950s by L. Nachbin [36], D. Goodner [15], J. Kelley [24], M. Hasumi [20], and A. Grothendieck [16] (see [32] for a comprehensive survey, including the theory of second duals). This work may be summarized as follows. THEOREM 1.1 Let V be a Banach space. (i) V is injective if and only if it is isometric to C(X ) for a Stonean space X . (ii) The dual space V ∗ is injective if and only if it is isometric and weak∗ homeomorphic to an L ∞ (X, µ)-space. In this case, V is unique to within isometry, and it is isometric to L 1 (X, µ). (iii) The second dual space V ∗∗ is injective if and only if one has diagrams

`n∞ σ

V

%

&τ id

−−−−→

V

DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 3, Received 16 July 1999. Revision received 27 November 2000. 2000 Mathematics Subject Classification. Primary 46L07; Secondary 46L08. Effros and Ruan’s work partially supported by National Science Foundation grant numbers DMS-9801324 and DMS-9877157. Ozawa’s work supported by the Japanese Society for Promotion of Science. 489

490

EFFROS, OZAWA, AND RUAN

of contractions which approximately commute in the point-norm topology. It has been known for more than twenty years that analogous results hold for C ∗ algebras (see, e.g., [4]). This theory has the drawback that it is concerned with algebras of operators rather than linear spaces of operators. In this paper we show that with one unavoidable change involving local reflexivity, the corresponding results hold for arbitrary operator spaces. Furthermore, we prove that any exact operator space is locally reflexive, another result that was first proved for C ∗ -algebras (see [28]). This essentially completes a cycle of ideas which had been explored in a variety of papers, including [44], [28], and [41]. An operator space V is a linear space of bounded linear operators on a Hilbert space H. Each matrix space Mn (V ) has a canonical norm determined by the operator norm on Mn B (H ) = B (H n ). The central role of these matrix norms is the distinguishing feature of operator space theory, and in particular, they are used in the abstract (Hilbert space-free) description of the operator spaces (see [43]). Owing to the parallel with quantum mechanics, in which one replaces functions with operators, operator spaces are often thought of as “quantized Banach spaces.” The morphisms of operator space theory are the linear mappings that behave well with respect to all of the matrix norms. A linear mapping of operator spaces ϕ : V → W determines linear mappings ϕn : Mn (V ) → Mn (W ) : [vi j ] 7 → ϕ(vi j ) , and ϕ is said to be completely bounded if kϕkcb = sup kϕn k : n ∈ N < ∞. Similarly, ϕ is said to be completely contractive (resp., completely isometric) if kϕkcb ≤ 1 (resp., ϕn is isometric for all n ∈ N). The space CB(V, W ) of all completely bounded operators has a natural operator space structure (see [11]). An operator space V is injective (in the operator space sense) if for any operator spaces W1 ⊆ W2 , each complete contraction ϕ : W1 → V has a completely contractive extension ϕ˜ : W2 → V . From the Arveson-Wittstock-Hahn-Banach theorem (see [39, Th. 7.2]), B (H ) is an injective operator space. This notion had been studied for C ∗ -algebras and von Neumann algebras some years before operator spaces were introduced. Although little is known about the injective C ∗ -algebras, the injective von Neumann algebras were characterized by A. Connes in terms of the hyperfinite von Neumann algebras (see [5]). This remains one of the central theorems of modern abstract analysis.

ON INJECTIVITY AND NUCLEARITY FOR OPERATOR SPACES

491

In contrast to the situation for Banach spaces, an injective operator space need not be a C ∗ -algebra. Nevertheless, there is a natural analogue for Theorem 1.1(i). We define an off-diagonal corner of a unital C ∗ -algebra A to be an operator subspace of the form e A (1−e), where e is a projection in A. The following analogue of Theorem 1.1(i) was proved in [44, Ths. 4.3 and 4.5]. THEOREM 1.2 Suppose that V is an operator space. Then the following are equivalent: (i) V is injective; (ii) there is a linear completely isometric mapping of V onto an off-diagonal corner of an injective C ∗ -algebra.

In this paper we begin by proving the following analogue of Theorem 1.1(ii). THEOREM 1.3 Suppose that V is an operator space that is dual as a Banach space. Then the following are equivalent: (i) V is injective; (ii) there is a linear completely isometric weak∗ homeomorphism of V onto the off-diagonal corner of an injective von Neumann algebra. Moreover, there is a canonical operator space matrix norm on the (unique) Banach space predual V∗ of V for which the given isometry (V∗ )∗ ∼ = V is a complete isometry.

The proof of this result depends upon a more general result for off-diagonal corners of operator algebras. For this purpose it is useful to take a more algebraic approach to these spaces. An off-diagonal corner of a C ∗ -algebra is closed under the ternary product (x, y, z) 7→ x y ∗ z. In fact, the off-diagonal corners of C ∗ -algebras essentially coincide with the norm-closed ternary rings of operators (or TRO’s; see §2). Similarly, the off-diagonal corners of von Neumann algebras are just the weak∗ -closed ternary algebras (W ∗ -TRO’s). Ternary algebras were introduced by M. Hestenes [21]. Many of the elementary theorems of operator algebra theory carry over without change to ternary algebras (see [19], [52], and [27]). We need the more difficult fact that a TRO that is dual as a Banach space is a W ∗ -TRO (see Theorem 2.6). This was proved by H. Zettl [52], who used an intricate argument involving Hilbert modules. We have included a simplified proof of his theorem. Our approach is based upon a TRO generalization of J. Tomiyama’s well-known characterization of the contractive conditional expectations in [50] (see Theorem 2.5).

492


We use Theorem 2.6 to extend several results known for operator algebras to general operator spaces. It is well known that a von Neumann algebra is injective if and only if it is semidiscrete. In Proposition 3.1 we show that with the natural definitions, this is the case for arbitrary dual operator spaces. We also characterize the preduals of injective dual operator spaces in Proposition 3.2. A C ∗ -algebra A is nuclear if and only if its second dual A∗∗ is an injective von Neumann algebra (see [4]). In Theorem 4.5 we prove the appropriate operator space analogue of Theorem 1.1(iii). We show that an operator space V is nuclear if and only if V is locally reflexive and V ∗∗ is injective. Local reflexivity is an essential condition in this result since E. Kirchberg [28] has constructed a separable nonnuclear operator Q space V for which V ∗∗ = ∞ n=1 Mn . ∗ Turning to additional C -algebraic results, we prove that an operator space is nuclear if and only if it is locally reflexive and has the weak extension property (see [8, Prop. 5.4] for the C ∗ -algebraic version). In Corollary 4.8 we generalize a C ∗ -algebraic theorem of Kirchberg [28] by showing that any exact operator space is locally reflexive (i.e., C 0 ⇒ C 00 in the terminology of R. Archbold and C. Batty [1] and [8]). Our arguments rest, in part, upon the fact that an operator space is locally reflexive if and only if that is the case for its separable subspaces (see Theorem 4.3), and that any separable exact operator space can be embedded in a separable nuclear operator space (see Theorem 4.7). The last part of Theorem 1.3 has the following interesting consequence. If an injective operator space has a Banach space predual, then it has an operator space predual. C. Le Merdy has shown that this is false in general (see [35]). He proved that the dual Banach space V = B (H )∗ (where H = `2 (N)) can be given an alternative operator space structure for which V is not the dual of an operator space. In §5 we give some additional examples, using von Neumann algebras rather than their duals. As pointed out by Zettl, one may associate a natural Hilbert module with a TRO, and in fact, these notions are essentially equivalent. We do not use this approach in this paper, but we have included a few remarks in the appendix. 2. Dual ternary ring of operators A ternary ring of operators (TRO) between Hilbert spaces K and H is a norm-closed subspace V of B (K , H ) which is algebraically closed under the ternary product (x, y, z) ∈ V × V × V 7→ x y ∗ z. A TRO V ⊆ B (K , H ) is called a W ∗ -TRO if it is weak∗ closed in B (K , H ). If V is a TRO (resp., a W ∗ -TRO) contained in B (K , H ), then there is a natural operator space matrix norm on V obtained by identifying Mn (V ) with a subspace of Mn (B (K , H )) = B (K n , H n ) for all n ∈ N. In this case, each Mn (V ) is a TRO


493

(resp., W ∗ -TRO) contained in B (K n , H n ). Given TRO’s V and W , a linear map π : V → W is called a TRO-homomorphism if it preserves the ternary product π(x y ∗ z) = π(x)π(y)∗ π(z) for all x, y, z ∈ V . We call π a TRO-isomorphism if it is also a linear isomorphism from V onto W . It is easy to see that the weak∗ closure V of a TRO is a W ∗ -TRO. If V is an off-diagonal corner of a unital C ∗ -algebra A, that is, there is a projection e ∈ A such that V = e A(1 − e), then V is obviously a TRO. On the other hand, let us suppose that V is a TRO contained in B (K , H ). We fix the notation V ] = {x ∗ : x ∈ V }, and V W = linear span{x y : x ∈ V, y ∈ W }. Then V V ] and V ] V are ∗-subalgebras of B (H ) and B (K ), and we let C and D denote their norm closures, that is, the C ∗ -algebras generated by V V ] and V ] V , respectively. Since V is norm closed, CV ⊆ V

and

V D ⊆ V,

(1)

and thus we may regard V as a (C, D)-bimodule. If we let A be the unital C ∗ subalgebra of B (H ⊕ K ) defined by C1 + C V (2) A= V] C1 + D and if we let e ∈ A be the projection e=

1 0 0 0

,

(3)

then we have the completely isometric TRO-isomorphism V ∼ = e A(1 − e). This shows that the off-diagonal corners of unital C ∗ -algebras are, up to TRO-isomorphism, the TRO’s. In much the same manner, we may identify the off-diagonal corners of von Neumann algebras with the W ∗ -TRO’s. If V is an off-diagonal corner of a von Neumann algebra R, then it is immediate that V is a W ∗ -TRO. On the other hand, if V is a W ∗ -TRO contained in B (K , H ), we let M and N denote the weak∗ closures of

494


CI H + V V ] and CI K + V ] V in B (H ) and B (K ), respectively. Then M and N are von Neumann algebras on the Hilbert space H and K . Since V is weak∗ closed, MV ⊆ V

and

V N ⊆ V,

and thus R=

M V]

V N

(4)

is a von Neumann subalgebra of B (H ⊕ K ). Letting e be the projection given in (3), we have the completely isometric and weak∗ continuous TRO-isomorphism V ∼ = e R(1 − e). Our goal in this section is to show that any TRO V that is the Banach space dual of some Banach space must be a W ∗ -TRO. To understand the significance of this result, let us suppose that V is an arbitrary weak∗ -closed subspace of B (H ). From standard Banach space theory, V may be identified with the Banach space dual of B (H )∗ /V⊥ , where V⊥ = ω ∈ B (H )∗ : ω|V = 0 . But from elementary operator space theory, B (H ) is also the operator space dual of B (H )∗ , and as a result, V is the operator space dual of V∗ (see, e.g., [13, Prop. 4.2.2]). In particular, for each n ∈ N, we may identify Mn (V ) with the Banach space b V∗ , and therefore the unit ball of Mn (V ) is weak∗ closed. By contrast, if dual of Tn ⊗ an operator space V is a dual Banach space, that is, it is the dual of a Banach space V∗ , we cannot conclude that the unit ball of Mn (V ) is weak∗ closed for n > 1. A TRO V ⊆ B (K , H ) is said to be nondegenerately represented on K and H if V K is norm dense in H and if V ] H is norm dense in K . In this case it is easy to see that the C ∗ -algebras C and D are nondegenerately represented on H and K , respectively. The weak∗ closure V is a W ∗ -TRO with M = C 00 , N = D 00 , and R = A00 . If V is a TRO, then V V ] V is norm dense in V

(5)

since if we let C be the norm closure of V V ] and if we let {cα } be a positive and contractive approximate identity in the C ∗ -algebra C, then kx − cα xk = kx x ∗ − cα x x ∗ − x x ∗ cα + cα x x ∗ cα k1/2 → 0.

(6)

It follows that V is a nondegenerate (C, D)-bimodule, that is, that C V and V D are dense in V. In fact, from Cohen’s factorization argument C V = V and V D = V (see [22, (32.22)], [51, Prop. 2.4]), but we do not need this stronger result.


495

PROPOSITION 2.1 Every TRO-homomorphism is completely contractive, and every injective TRO-homomorphism is completely isometric.

Proof Using a simple spectral argument, L. Harris proved in [19, Prop. 3.4] that every TROhomomorphism (resp., injective TRO-homomorphism) π : V → W is contractive (resp., isometric). Given TRO’s V and W and a TRO-homomorphism, π : V → W , we have that for each n ∈ N, πn : Mn (V ) → Mn (W ) is also a TRO-homomorphism (resp., injective TRO-homomorphism), and the result follows. It follows that, as in the case of a C ∗ -algebra, the operator space structure on a TRO does not depend upon the given TRO representation. M. Hamana has proved a refinement of this result in [17] and [18]. If ϕ : V → W is a linear surjection of TRO’s, then the following are equivalent: (i) ϕ is a 2-isometry; (ii) ϕ is a complete isometry; (iii) ϕ is a TRO-isomorphism. A norm-closed subspace J in a TRO V is called a TRO ideal in V if J V ] V ⊆ J and if V V ] J ⊆ J . It follows that V J ] V ⊆ J since from (5), J J ] J is norm dense in J , and therefore V J ] V is contained in the norm closure of (V J ] J )J ] V . Since J is an ideal, the latter is contained in J . The following result may be found in [18, Prop. 2.1]. We have included a proof for the convenience of the reader. PROPOSITION 2.2 Let V be a TRO (resp., W ∗ -TRO), and let J be a TRO ideal (resp., weak∗ -closed TRO ideal). Then V /J is a TRO (resp., W ∗ -TRO) with the induced ternary product and operator space structure.

Proof We only consider TRO’s since the argument for W ∗ -TRO’s is similar. Let J be a TRO ideal in V . Then the quotient operator space V /J = {x J = x + J : x ∈ V } has a ternary product given by x J y ∗J z J = (x y ∗ z) J .

496


We let A= and

" I =

C1 + C V]

JV] + V J] J]

V C1 + D

J ] J V + V]J

# .

Then I is a closed ideal in A, and we have the completely isometric TROisomorphisms V ∼ = e A(1 − e) and J∼ = eI (1 − e), where e ∈ A is the projection given in (3). There is a natural complete contraction π : V /J → A/I given by π(x J ) = x I . It is clear that π preserves the ternary product π(x J y ∗J z J ) = π(x)π(y)∗ π(z) for all x J , y J , z J ∈ V /J , and it maps V /J onto the off-diagonal corner e I A/I (1−e I ) of the C ∗ -algebra A/I , where we let e I = e + I denote the corresponding projection in A/I . To complete the proof, we only need to show that π is a complete isometry. Given x J ∈ V /J , we have

π(x J ) = inf kx + yk : y ∈ I

≥ inf e(x + y)(1 − e) : y ∈ I = inf kx + yk : y ∈ J = kx J k. This shows that π is an isometry. A similar matrix norm calculation shows that π is a complete isometry. Therefore, V /J is completely isometrically TRO-isomorphic to e I (A/I )(1 − e I ). If ϕ : V → W is a TRO-homomorphism, then J = ker ϕ is a TRO ideal in V . The mapping ϕ induces an injective (completely isometric) TRO-homomorphism from V /J into W . It follows that ϕ(V ) is closed in W and ϕ(V ) is a sub-TRO of W . If the TRO-homomorphism ϕ is surjective, it must be an exact complete quotient mapping in the sense that for each n ∈ N, ϕn maps the closed unit ball of Mn (V ) onto the closed unit ball of Mn (W ). This is an elementary result for quotients of C ∗ -algebras. From above, we may identify V /J with e I (A/I )(1 − e I ) where I is a closed ideal in


497

a C ∗ -algebra A and where e is a projection in A. Given w ∈ V /J with kwk ≤ 1, we may choose a ∈ A with kak ≤ 1 and with ϕ(a) = w. The element v = ea(1−e) ∈ V satisfies kvk ≤ 1 and ϕ(v) = w. The same argument can be used for matrices. If V is a TRO and if r ∈ V is a partial isometry, that is, rr ∗r = r , we let er = rr ∗ ∈ C and fr = r ∗r ∈ D denote the range and support projections of r , respectively. It is easy to prove (see [19] and [52]) that the norm-closed subspace V(r ) = {x ∈ V : er x = x = x fr } = {er x fr : x ∈ V } is a unital C ∗ -algebra with multiplication and involution given by x ◦ y = xr ∗ y and

x ] = r x ∗r.

The unital element is given by r . If V is a W ∗ -TRO contained in B (K , H ), it is clear that V(r ) is a weak∗ -closed subspace of er B (K , H ) fr = B ( fr K , er H ), and thus it is a von Neumann algebra. The following important result of Zettl [52, Prop. 4.3] shows that if V is a TRO and if it is a Banach space dual, then V has an abundance of such partial isometries. Using this result, we are able to “locally” apply von Neumann algebra results to such dual TRO’s. We have included a somewhat modified proof. PROPOSITION 2.3 Let V be a TRO that is also a dual Banach space; that is, V = (V∗ )∗ for some Banach space V∗ . If ϕ ∈ V∗ and kϕk = 1, then there exists a partial isometry r ∈ V and states p ∈ C ∗ and q ∈ D ∗ such that

ϕ(x) = p(xr ∗ ) = q(r ∗ x) and p(c) = ϕ(cr ),

q(d) = ϕ(r d)

for all c ∈ C, d ∈ D, and x ∈ V. In particular, ϕ(x) = ϕ(er x fr ). Proof If ϕ ∈ V∗ and kϕk = 1, we may select an element x in the unit ball V1 of V with ϕ(x) = 1. It follows that Re ϕ assumes the maximum value 1 on V1 . Since V1 is V∗ -compact and convex and since K = x ∈ V1 : Re ϕ(x) = 1

498


is a face of V1 , K contains an extreme point r of V1 . We have that Re ϕ(r ) = 1 ≥ ϕ(r ) implies that ϕ(r ) = 1. From [52, Lem. 1.3], r must be a partial isometry in V . We define p(c) = ϕ(cr ) and q(d) = ϕ(r d) for all c ∈ C and d ∈ D. We have that kqk ≤ 1 and q( fr ) = ϕ(rr ∗r ) = ϕ(r ) = 1. Thus q is a state on D. Similarly, p is a state on C. We have a contractive linear functional fr ϕ on V defined by ( fr ϕ)(x) = ϕ(xr ∗r ) = p(xr ∗ ). Since ( fr ϕ)(r ) = ϕ(rr ∗r ) = ϕ(r ) = 1, it follows that k fr ϕk = kϕk = 1. From the proof of [7, Lem. 3.1], it follows that fr ϕ = ϕ, and thus ϕ(x) = ϕ(x fr ) = ϕ(xr ∗r ) = p(xr ∗ ). A similar argument shows that ϕ(x) = ϕ(er x) = ϕ(rr ∗ x) = q(r ∗ x). From these relations, we have ϕ(x) = ϕ(er x) = ϕ(er x fr ). PROPOSITION 2.4 A TRO-isomorphism between dual TRO’s is automatically weak∗ continuous, and thus the Banach space predual of a dual TRO is unique.

Proof Let V and W be TRO’s with Banach space preduals V∗ and W∗ , and let π : V → W be a TRO-isomorphism from V onto W . It suffices to show that for each ϕ ∈ W∗ , ϕ ◦ π is contained in V∗ . Given ϕ ∈ W∗ , it follows from Proposition 2.3 that there is a partial isometry r ∈ W such that ϕ(w) = ϕ(er w fr )


499

for all w ∈ W . Since π is a TRO-isomorphism, r˜ = π −1 (r ) ∈ V is a partial isometry in V . It is easy to see that π restricted to V(˜r ) induces a ∗-isomorphism θr˜ from the von Neumann algebra V(˜r ) onto the von Neumann algebra W(r ) . If xα → x ∈ V in the weak∗ topology, then er˜ xα fr˜ → er˜ xα fr˜ ∈ V(˜r ) in the weak∗ topology. Since θr˜ is a ∗ isomorphism between von Neumann algebras, it is automatically weak∗ continuous, and θr˜ (er˜ xα fr˜ ) converges to θr˜ (er˜ x fr˜ ) with respect to the weak∗ topology in W . Applying ϕ ∈ W∗ , we conclude ϕ π(xα ) = ϕ er π(xα ) fr = ϕ θr˜ (er˜ xα fr˜ ) converges to ϕ π(x) = ϕ er π(x) fr = ϕ θr˜ (er˜ x fr˜ ) . This shows that ϕ ◦ π is weak∗ continuous on V . Let V be a sub-TRO of W , and let P be a projection from W onto V . Then P is called a conditional expectation if the following hold: C1 P(x y ∗ w) = x y ∗ P(w) for all x, y ∈ V and w ∈ W ; C2 P(wx ∗ y) = P(w)x ∗ y for all x, y ∈ V and w ∈ W ; C3 P(xw∗ y) = x P(w)∗ y for all x, y ∈ V and w ∈ W . The following is the TRO analogue of Tomiyama’s conditional expectation theorem in [50] (see also [37, Th. 5]). 2.5 Let V be a sub-TRO of W , and let P be a projection from W onto V . Then the following are equivalent: (i) P is completely contractive; (ii) P is contractive; (iii) P is a contractive conditional expectation. THEOREM

Proof The implication (i) ⇒ (ii) is obvious. Let us prove (ii) ⇒ (iii). Passing to the second duals, and replacing V and W by their weak∗ closures, we may assume that V and W are W ∗ -TRO’s and that P is weak∗ continuous. We can choose Hilbert spaces H and K and an identification of V ⊆ W with an inclusion of weak∗ -closed sub-TRO’s of B (K , H ). If we let M denote the weak∗ closure of V V ] in B (H ), it is evident that M V ⊆ V and M W ⊆ W. Given a projection e ∈ M and w ∈ W , we claim that P(ew) = e P(w).

500


To see this, we let f = 1 − e ∈ M. Since f P(ew) ∈ V , we have for any λ ∈ R,

2 2 (1 + λ)2 f P(ew) = f P ew + λ f P(ew)

2 ≤ ew + λ f P(ew)

2 = kewk2 + λ2 f P(ew) . In the last equality we used the fact that if b and c are operators with orthogonal ranges, then b∗ c = c∗ b = 0 and

kb + ck2 = (b + c)∗ (b + c) = kb∗ bk + kc∗ ck = kbk2 + kck2 . Since λ ∈ R is arbitrary, we obtain f P(ew) = 0 or, equivalently, P(ew) = e P(ew). Similarly, we obtain e P( f w) = 0 and hence e P(w) = e P(ew). It follows that P(ew) = e P(ew) = e P(w). Since the linear span of projections in M is weak∗ dense in M, given any a ∈ V V ] ⊆ M, there exists a net {aα } of the linear combination of projections in M such that aα converges to a in the weak∗ topology. It is clear that aα w converges to aw in the weak∗ topology in W and, by the weak∗ continuity of P, that P(aα w) converges to P(aw) in the weak∗ topology in V . Thus we have proved condition C1, P(aw) = a P(w). Condition C2 follows from a similar argument; that is, we have P(wb) = P(w)b for all b ∈ V ] V and w ∈ W . It remains to prove C3, that is, P(xw∗ y) = x P(w)∗ y for x, y ∈ V and w ∈ W . Fix x, y ∈ V and w ∈ W . It suffices to show that ϕ P(xw∗ y) = ϕ x P(w)∗ y for all ϕ ∈ V∗ . Given ϕ ∈ V∗ with kϕk = 1, we may select an element x0 in the closed unit ball V1 of V such that ϕ(x0 ) = 1. Since P is a weak∗ continuous contractive projection from W onto V , ϕ ◦ P ∈ W∗ and ϕ ◦ P(x0 ) = kϕ ◦ Pk = 1. It is evident that K = x ∈ V : Re ϕ(x) = 1 and Re ϕ ◦ P(x) = 1


501

is a face of V1 , and in particular, the extreme points of K are extreme in V1 . We may repeat the argument in the proof of Proposition 2.3 for both ϕ and ϕ ◦ P. Briefly, we may find a partial isometry r in K , and we may obtain a state pV on the ∗ -algebra V V ] and a state pW on the ∗ -algebra W W ] such that ϕ(x) = pV (xr ∗ )

and

ϕ ◦ P(w) = pW (wr ∗ )

for all x ∈ V and w ∈ W , respectively. It follows that ϕ(x) = pV (xr ∗ ) = pV (r x ∗ ) = ϕ(r x ∗r ) for all x ∈ V , and similarly, ϕ ◦ P(w) = ϕ ◦ P(r w∗r ) for all w ∈ W . Thus we have ϕ x P(w)∗ y = ϕ r y ∗ P(w)x ∗r = ϕ ◦ P(r y ∗ wx ∗r ) = ϕ P(xw∗ y) . This proves (iii). To prove (iii) ⇒ (i), we show that if w = [wi j ] ∈ Mn (W ) and kPn (w)k Mn (V ) > 1, then kwk Mn (W ) > 1. We suppose that W is a sub-TRO of B (H ) for a Hilbert space H . The C ∗ -algebra D = V ] V acts nondegenerately on K = V ] V (H ); that is, for each ξ ∈ K \{0} there is a d ∈ D with dξ 6 = 0 (see [40, §2.2.4]) and, in turn, an x ∈ V with xξ 6 = 0. It follows that x ∗ xξ 6= 0 and K (x, ξ ) = V ∗ V x ∗ xξ 6 = {0}. Applying Zorn’s lemma, we let K α = K (xα , ξα ) (α ∈ I ) be a maximal family of mutually orthogonal subspaces of K of that form. The K α span K since if ξ ∈ K \{0} and ξ ⊥ K α for all α ∈ I , then given x ∈ V with xξ 6 = 0, K (x, ξ ) ⊥ K α for all α, a contradiction. Taking dot products, it is evident that the subspaces V K α are mutually orthogonal in H . Thus the subspaces (K α )n are mutually orthogonal and span K n , and the spaces Mn (V )(K α )n = (V K α )n are mutually orthogonal in H n . Since Pn (w) ∈ Mn (V ) maps (K α )n into (V K α )n , there is an α ∈ I with kPn (w)|(K α )n k > 1. We have K α ⊆ V ] V ξα , and since V V ] V xα∗ xα ξα ⊆ V V ] xα ξα , V K α ⊆ V V ] xα ξα . Let ξ = ξα and η = xα ξα . Then there exist b j ∈ V ] V and ai ∈ V V ] for i, j = 1, . . . , n such that X X kb j ξ k2 < 1, kai ηk2 < 1 j

and we have

i

X P(wi j )b j ξ, ai η > 1. i, j

502


Take ε > 0 so that X kb j ξ k2 < 1 − εkξ k2

X

and

j

kai ηk2 < 1 − εkηk2 .

i

Let b = ε I + j b∗j b j and a = ε I + i ai∗ ai . Let ξ˜ = b1/2 ξ and η˜ = a 1/2 η. Moreover, let b˜j = b j b−1/2 ∈ D and a˜i = ai a −1/2 ∈ C. Then we have X kξ˜ k2 = (bξ, ξ ) = εkξ k2 + kb j ξ k2 < 1 P

P

j

and thus kξ˜ k < 1. Similarly, we have kηk ˜ < 1. Also, it is easy to check that

X

X

∗

b˜j b˜j ≤ 1 and a˜i ∗ a˜i ≤ 1.

j

i

Finally, we have X X 1< P(wi j )b j ξ, ai η = P(wi j )b˜j ξ˜ , a˜i η˜ i, j

i, j

X X = a˜i ∗ P(wi j )b˜j ξ˜ , η˜ = P a˜i ∗ wi j b˜j ξ˜ , η˜ i, j

i, j

X

X

1/2

X

∗ 1/2

≤ a˜i ∗ wi j b˜j kξ˜ kkηk ˜ < a˜i ∗ a˜i k[wi j ]k b˜j b˜j i, j

i

j

≤ [wi j ] . This completes the proof. Our proof of (ii) ⇒ (iii) was based on E. Lance’s approach to Tomiyama’s result (see [49, §9]) and arguments from [52]. The proof of (iii) ⇒ (i) is essentially the same as that given in [46, §1.6]. 2.6 Suppose that V is an off-diagonal corner of a unital C ∗ -algebra. If V is dual as a Banach space, then it is weak∗ homeomorphic and completely isometric to an offdiagonal corner of a von Neumann algebra R. Moreover, V has a unique Banach space predual V∗ (up to isometry), and there exists a canonical operator space matrix norm on V∗ such that we have the complete isometry V ∼ = (V∗ )∗ . THEOREM

Proof The uniqueness of the Banach space predual was proved in Proposition 2.4.


503

Let us suppose that V is a TRO and that V is the Banach space dual of a Banach space V∗ . We wish to show that one can define operator space norms on V∗ for which V is the operator dual space of V∗ . The second dual TRO W = V ∗∗ is a W ∗ -TRO. If we let ι be the canonical inclusion of V∗ into V ∗ , then P = ι∗ is a weak∗ continuous norm one projection from W onto V . By Theorem 2.5, P is a conditional expectation from W onto V . It follows from C1 that the kernel J = ker P = ι(V∗ )⊥ ⊆ W is a weak∗ -closed subspace of W such that V V ] J ⊆ J. Since V is weak∗ dense in its second dual W and since the ternary multiplication on W is weak∗ continuous in each component, V V ] J is weak∗ dense in V W ] J and thus weak∗ dense in W W ] J . Since J is weak∗ closed, we have W W ] J ⊆ J. A similar argument using condition C2 shows that J W ] W ⊆ J. Therefore, J is a weak∗ -closed TRO ideal in W . From Proposition 2.2, V = W/J is a W ∗ -TRO, and therefore it is the dual of an operator space V˜∗ . Owing to the unicity of the predual, we may identify the underlying Banach space of V˜∗ with V∗ . The matrix norms on V˜∗ determine matrix norms on V∗ for which V is the operator dual of V˜∗ . 3. Dual injective operator spaces We can now prove our characterization of the dual injective operator spaces. Proof of Theorem 1.3 (ii) ⇒ (i). If R is an injective von Neumann algebra and if e ∈ R is a projection, then e R(1 − e) is closed in the weak∗ topology, and thus it is a dual Banach space. From Theorem 1.2, we also have that e R(1 − e) is injective. (i) ⇒ (ii). From Theorems 1.2 and 2.6, we may assume that V = e R(1 − e), where R is a von Neumann algebra and where e is a projection in R. If we let c(e) and c(1 − e) be the central covers of e and (1 − e) in R, and p = c(e)c(1 − e), then V = e( p R)(1 − e). Thus, replacing R by p R, we may initially assume that e and (1 − e) have central covers equal to 1. Using standard comparison theory, we can find another projection f ∈ R with central cover 1 for which f e and f (1 − e). Thus we have partial isometries

504


v, w ∈ R with v ∗ v = f , vv ∗ = f 2 ≤ 1 − e, and w∗ w = f, ww∗ = f 1 ≤ e. It follows that the space f R f is completely isometric to f 1 R f 2 ⊆ e R(1 − e) since there is a complete isometry from f R f onto f 1 R f 2 given by f x f ∈ f R f 7 → w( f x f )v ∗ = f 1 (wxv ∗ ) f 2 ∈ f 1 R f 2 . It is evident that x 7→ f 1 x f 2 is a completely contractive projection of the injective operator space V = e R(1−e) onto f 1 R f 2 , and thus both f 1 R f 2 and f R f are injective operator spaces. Since f has central cover 1, it follows from multiplicity theory that there is a Hilbert space K for which we have a von Neumann algebraic isomorphism B (K )⊗R ∼ = B (K )⊗ f R f.

(This follows, e.g., if one considers the representation commutants in [6, Prop. 5.3.1].) Since f R f is injective, we conclude that B (K )⊗R and thus R∼ = (e11 ⊗ 1) B (K )⊗R (e11 ⊗ 1) are injective. We say that a dual operator space V is semidiscrete if there exist diagrams of weak∗ continuous complete contractions Mn(α) ϕα

V

%

&ψα id

−−−−→

V

which approximately commute in the point-weak∗ topology (see [10] for the equivalent von Neumann algebra notion). PROPOSITION 3.1 Suppose that V is a dual operator space. Then the following are equivalent: (i) V is injective; (ii) V is semidiscrete.

Proof (i) ⇒ (ii). It follows from Theorem 1.3 that there is an injective von Neumann algebra R and a projection e ∈ R such that V ∼ = e R (1 − e) is a completely isometric weak∗ homeomorphism. Then the result is an immediate consequence of the fact that all injective von Neumann algebras are semidiscrete. (ii) ⇒ (i). Given operator spaces W0 ⊆ W and a complete contraction s : W0 → V , the compositions ϕα ◦ s : W0 → Mn(α) are complete contractions, and thus they


505

have completely contractive extensions sα : W → Mn(α) . Then {ψα ◦ sα } is a net ˆ ∗ )∗ , and it must have a subnet conof complete contractions in CB(W, V ) = (W ⊗V verging to a complete contraction s˜ ∈ CB(W, V ) in the weak∗ topology. It is easy to check that s˜ is a completely contractive extension of s. This shows that V is an injective operator space. The preduals of injective operator spaces were considered in [31]. An operator space Z has the operator space local lifting property if given a diagram

Z

ϕ

Y  yq

−→ Y/W

where W is a closed subspace of an operator space Y, q is the quotient mapping, and ϕ is a complete contraction, then for each finite-dimensional subspace E of Z and ε > 0, there is a mapping ϕ˜ : E → Y with kϕk ˜ cb < 1 + ε and q ◦ ϕ˜ = ϕ|E . This is not the same as Kirchberg’s notion of the “local lifting property” in [26], in which only C ∗ -algebraic quotients are considered. The following result settles a question posed in [31]. PROPOSITION 3.2 Suppose that V is a dual operator space with the operator predual V∗ . Then the following are equivalent: (i) V is injective; (ii) there exists an injective von Neumann algebra R and a projection e ∈ R such that V∗ is completely isometric to (1 − e)R∗ e; (iii) V∗ is completely isometric to a completely contractively complemented operator subspace of the operator predual R∗ of some injective von Neumann algebra R; (iv) there exist diagrams of complete contractions

Tn(α) ϕα

V∗

(v)

%

& ψα id

−−−−→

V∗

which approximately commute in the point-norm topology; V∗ has the operator space local lifting property.

Proof (i) ⇒ (ii). This is an immediate consequence of Theorem 1.3.

506


(ii) ⇒ (iii). This is trivial. (iii) ⇒ (iv). Let us assume that R is injective and that Q : R∗ → V∗ is a completely contractive projection. Since R is semidiscrete, there exist weak∗ continuous complete contractions sα : R → Mn(α) and tα : Mn(α) → R such that tα ◦ sα → id R in the point-weak∗ topology. It follows that the preadjoints (tα )∗ : R∗ → Tn(α) and (sα )∗ : Tn(α) → R∗ are complete contractions and that (sα )∗ ◦ (tα )∗ : R∗ → R∗ converges to id R∗ in the point-weak topology. By a standard convexity argument, we may have complete contractions ϕ˜α : R∗ → Tn(α) and ψ˜ α : Tn(α) → R∗ such that ψ˜ α ◦ ϕ˜α → id R∗ in the point-norm topology. It follows that ϕα = ϕ˜α |V∗ : V∗ → Tn(α) and ψα = Q ◦ ψ˜ α : Tn(α) → V∗ are complete contractions such that ψα ◦ ϕα → idV∗ in the point-norm topology. (iv) ⇒ (v). This is proved in [31, Th. 3.2]. (i) ⇔ (v). This is proved in [31, Th. 5.1] (this is due to M. Junge) and [31, Th. 5.5]. 4. Nuclear operator spaces An operator space V is nuclear if there exist diagrams of complete contractions Mn(α) ϕα

V

%

& ψα id

−−−−→

V

which approximately commute in the point-norm topology. The usual convexity argument shows that it is equivalent to assume that these diagrams commute in the point-weak topology. R. Smith showed that for C ∗ -algebras it is also equivalent to the usual definition of nuclearity introduced by Lance (who used completely positive contractions; see [33] and [47]). 4.1 Let V be a nuclear operator space. Then for every separable subspace V0 of V , there exists a separable nuclear operator subspace V1 of V such that V0 ⊆ V1 . PROPOSITION

Proof Let V0 be a separable operator subspace of V , and let E 1 ⊆ E 2 ⊆ · · · ⊆ E k ⊆ · · · ⊆ V0 be an increasing sequence of finite-dimensional operator subspaces of V0 such that ∪E k is norm dense in V0 . In the following we use an induction procedure to construct a separable nuclear operator subspace V1 of V such that V0 ⊆ V1 . 1 } be a finite subset in the For k = 1, we let F1 = E 1 , and we let {v11 , . . . , vm 1 closed unit ball of F1 , such that for every v ∈ F1 with kvk ≤ 1, there exists some


507

v 1j such that kv − v 1j k < 1/8. Since V is nuclear, there exists a diagram of complete contractions Mn 1 σ1

%

&τ1

id

V

V

−−−−→

such that kτ1 σ1 (v 1j ) − v 1j k < 1/8 for j = 1, . . . , m 1 . By the triangle inequality, this implies that

τ1 σ1 (v) − v < 1/2 for all v in the closed unit ball of F1 . For k = 2, we let F2 = E 2 + τ1 σ1 (F1 ). Then F2 is a finite-dimensional subspace 2 } be a finite subset of the closed of V containing E 2 as a subspace. We let {v12 , . . . , vm 2 unit ball of F2 such that for every v ∈ F2 with kvk ≤ 1, there exists some v 2j such that kv − v 2j k < 1/16. Since V is nuclear, there exists a diagram of complete contractions σ2

V

Mn 2

%

&τ2 V

−−−−→

such that kτ2 σ2 (v 2j ) − v 2j k < 1/16 for j = 1, . . . , m 2 . By the triangle inequality, this implies that kτ2 σ2 (v) − vk < 1/4 for all v in the closed unit ball of F2 . Continuing in this manner, we obtain an increasing sequence of finite-dimensional operator spaces F1 ⊆ F2 ⊆ · · · ⊆ Fk ⊆ · · · ⊆ V and a sequence of completely contractive diagrams σk

%

V

Mn k

&τk Fk+1 ⊆ V

−−−−→

such that kτk σk (v) − vk < 1/(2k ) for all v in the closed unit ball of Fk and k ∈ N. Let V1 be the closed union ∪Fk . Then V1 is a separable operator subspace of V containing V0 as a subspace. From the above construction, it is easy to see that the diagrams of complete contractions σk

V1

%

Mn k −−−−→

&τk Fk+1 ⊆ V1

converge to the identity mapping on V1 in the point-norm topology. Therefore, V1 is nuclear.

508


An operator space V is said to be locally reflexive if for any finite-dimensional operator space E, each complete contraction ϕ : E → V ∗∗ can be approximated in the point-weak∗ topology by complete contractions ψ : E → V . Equivalently, for any finite-dimensional subspace F ⊆ V ∗ and ε > 0, we can find a complete contraction ψε : E → V such that

kψε kcb < 1 + ε and ψε (x), f = ϕ(x), f (7) for all x ∈ E and f ∈ F (see [9, Lem. 6.4]). The following result is due to L. Ge and D. Hadwin [14]. (The averaging argument used in their proof may also be found in [42].) LEMMA 4.2 Let V be an operator space, and let E be any finite-dimensional subspace of V ∗∗ . For any finite-dimensional subspace F ⊆ V ∗ , ε > 0, and n ∈ N, there exists a linear isomorphism ψ from E onto a subspace E ψ ⊆ V such that (i) kψn k ≤ 1, kψn−1 k ≤ 1 + ε, hψ(x), f i = hx, f i for all x ∈ E and f ∈ F, (ii) (iii) ψ|E∩V = id E∩V . THEOREM 4.3 An operator space V is locally reflexive if and only if that is the case for each separable subspace.

Proof Since any subspace of V is locally reflexive, it is evident that the first condition implies the second. Conversely, let us suppose that each separable subspace of V is locally reflexive. It suffices to show that if ϕ : E → V ∗∗ is a complete contraction and if F ⊆ V ∗ is finite-dimensional, then for each ε > 0, there exists a mapping ψε : E → V satisfying (7) for all x ∈ E and f ∈ F. It follows from Lemma 4.2 that we may find a mapping ψ (n) :E → V such that (n) kψn k < 1 + 1/n, and

(n)

ψ (x), f = ϕ(x), f for all x ∈ E and f ∈ F. The norm-closed linear span V0 of the union of subspaces ψ (n) (E) with n ∈ N is separable in the norm topology, and we can regard ψ (n) as a sequence in B(E, V0∗∗ ). Since the closed ball of radius 2 is compact in the pointweak∗ topology on the latter space, we may choose a limit point ψ : E → V0∗∗ of the sequence ψ (n) . We have that if r ≤ n, then

(n) (n)

(ψ )r ≤ (ψ )n ≤ 1 + 1 , n


509

and thus kψr k ≤ 1. It follows that kψkcb ≤ 1. Furthermore, we have that

ψ(x), f = x, f for all x ∈ E and f ∈ F. Since, by assumption, V0 is locally reflexive as an operator space, if we are given ε > 0, we may find a mapping ψε : E → V0 satisfying (7) with φ = ψ for all x ∈ E and f ∈ F. THEOREM 4.4 If V is a nuclear operator space, then V is locally reflexive.

Proof Let us first suppose that V is separable. Kirchberg has shown that the hypotheses imply that there exist a left ideal L and a right ideal R of the C A R algebra B = M2 ⊗ M2 . . . such that V is completely isometric to B/(L + R) (see [28]). If we let l and r be the supporting projections of L and R, then e = 1 − r and f = 1 − l are closed projections in B ∗∗ such that V ∗∗ is completely isometrically and weak∗ continuously isomorphic to e B ∗∗ f . Since B is nuclear, B ∗∗ is injective, and thus the same is true for V ∗∗ = e B ∗∗ f. In order to see that V is locally reflexive, we use the commutative diagram B  πy

,−→

V = B/(L + R) ,−→

∗∗ B S y |

V ∗∗ = e B ∗∗ f

where π is the quotient mapping, each row is the usual inclusion of a space into its second dual, and the right mapping is given by x 7→ ex f. Given a finite-dimensional operator space F and a complete contraction ϕ : F → V ∗∗ ⊆ B ∗∗ , we can use the fact that nuclear C ∗ -algebras are locally reflexive to approximate ϕ in the point-weak∗ topology by a net of mappings ϕγ : F → B. It follows that the mappings π ◦ ϕγ : F → V converge to ϕ in the point-weak∗ topology, and thus V is locally reflexive. For the general case, we have from Theorem 4.3 that it suffices to prove that if V0 ⊆ V is separable, then V0 is locally reflexive. From Proposition 4.1, we have V0 ⊆ V1 , where V1 is norm separable and nuclear, and thus V1 is locally reflexive. It follows that V0 is locally reflexive. An operator space V is said to have the weak expectation property if, given a Hilbert space H and a completely isometric injection π : V ,→ B (H ), there exists a completely contractive projection P : B (H ) → π(V ) (the σ -weak operator closure)

510


such that P ◦ π(v) = π(v) for all v ∈ V. It suffices to prove this for the “universal representation” πu : V → B (Hu ) determined by taking the direct sum of all complete contractions ϕ : V → Mn (n ∈ N); πu determines a weak∗ homeomorphic completely isometric mapping π˜ u of V ∗∗ onto π(V ). It is a simple matter to verify that if V = A is a C ∗ -algebra, this coincides with the notion introduced by Lance [33]. The equivalence (i) ⇔ (iv) in the following proposition was proved for C ∗ -algebras in [8, Prop. 5.4]. 4.5 Suppose that V is an operator space. Then the following are equivalent: (i) V is nuclear; (ii) V is locally reflexive, and V ∗∗ is semidiscrete (i.e., injective); (iii) V is locally reflexive, and V ∗ has the operator space local lifting property; (iv) V is locally reflexive, and V has the weak expectation property. THEOREM

Proof (i) ⇒ (ii). If V is nuclear, then from Theorem 4.4, V is locally reflexive. To see that V ∗∗ is semidiscrete, consider the diagrams E

V ∗∗ ↓σ˜

⊆

 

θy

−→

V ∗∗ x  

Mn σ%

&τ

V

−−−−→

V

where the bottom triangle is approximately commutative in the point-norm topology and where E is a finite-dimensional subspace of V ∗∗ . Given a finite-dimensional subspace F ⊆ V ∗ , we may choose θ : E → V such that kθ kcb < 1 + ε and hθ(x), f i = hx, f i for x ∈ E, f ∈ F. The composition σ ◦ θ : E → Mn has a weak∗ continuous extension σ˜ : V ∗∗ → Mn satisfying kσ˜ kcb < 1+ε. To see this, we note that the inclusion mapping ι : E ,→ V ∗∗ is the adjoint of a mapping ρ : V ∗ → E ∗ , and since ι is completely isometric, ρ must be a complete quotient mapping. In particular, if we indicate weak∗ continuous mappings with the subscript w∗ , we have the commutative diagram Mn (V ∗ ) CB

w∗

|| (V ∗∗ ,

→

Mn (E ∗ ) ||

Mn ) → CB(E, Mn )


511

in which the top and bottom rows are quotient mappings. It follows that σ ◦ θ ∈ C B(E, Mn ) is the restriction of a weak∗ continuous mapping σ˜ ∈ CB(V ∗∗ , Mn ) with the desired norm property. We conclude that the diagrams Mn σ˜

%

V ∗∗

&τ V ∗∗

−−−−→

with σ˜ weak∗ continuous approximately commute in the point-weak∗ topology. (ii) ⇒ (i). This is immediate from the diagrams V ∗∗ σ S

V ∗∗

−−−−→

&

%τ Mn

| σ0

%

V

S

|

&τ0 −−−−→

V

in which we let σ0 be the restriction of σ, and we use local reflexivity to approximate τ in the point-weak∗ topology by complete contractions τ0 . (ii) ⇔ (iii). This is a consequence of Proposition 3.2. (ii) ⇒ (iv). This is trivial if one uses the universal representation of V. (iv) ⇒ (ii). This was proved in [31, Prop. 6.2]. The equivalences (i) ⇔ (ii) and (ii) ⇔ (iv) above were proved for C ∗ -algebras in [4] and [8], respectively. An intriguing aspect of the C ∗ -algebraic situation is that one does not need the condition of local reflexivity for (i) ⇔ (ii). As we remarked in the introduction, (ii) ⇒ (i) is false for operator spaces if one does not assume V is locally reflexive. The situation is clarified by the following result (see [3, p. 72] and the discussion of [28, Lem. 2.8(i)], where a more general result is proved for operator systems). LEMMA 4.6 If A is a C ∗ -algebra, then any completely positive contraction ϕ : Mn → A∗∗ can be approximated in the point-weak∗ topology by completely positive contractions θ : Mn → A.

If A is a C ∗ -algebra such that A∗∗ is injective, then we have approximately commuting diagrams Mn(α) ϕα

A∗∗

%

&ψα id

−−−−→

A∗∗

512


with ϕα and ψα completely positive contractions. We may then use Lemma 4.6 to approximate ψα by completely positive contractions ψα0 : Mn(α) → A, and continue as before. Turning to another property that was first considered for C ∗ -algebras, an operator space V is said to be exact if for each finite-dimensional subspace E ⊆ V and ε > 0, there exists an n ∈ N, a subspace S ⊆ Mn , and a linear isomorphism ϕ : E → S for which kϕkcb ≤ 1 and kϕ −1 kcb ≤ 1 + ε. An equivalent property for C ∗ -algebras was first introduced by Kirchberg [25]. Kirchberg and N. Phillips [29] have shown that any separable exact C ∗ -algebra is ∗ -isomorphic to a C ∗ -subalgebra of the Cuntz algebra O2 (which is nuclear). However, there is an example of a separable nuclear operator space (in fact, an operator system) which is not unital completely isometric to an operator subspace of any nuclear C ∗ -algebra (see [30, Cor. 18]). For operator spaces, we have the following interesting alternative result. THEOREM 4.7 If V is a separable exact operator space, then it is completely isometric to a subspace of a separable nuclear operator space.

Proof We let E 1 ⊆ E 2 ⊆ · · · be an increasing sequence of finite-dimensional operator subspaces of V such that V = ∪E k . We claim that there is a sequence {n(k)} of natural numbers and a diagram of mappings E1 ↓

i1

Mn(1)

⊆

···

⊆ ϕk−1

ϕ1

,−→ · · ·

Ek ↓

⊆

Mn(k)

,−→

ik

,−→

ϕk

E k+1 ↓

i k+1

Mn(k+1)

⊆

···

ϕk+1

,−→ · · ·

such that (i) ki k kcb ≤ 1 and ki k−1 kcb ≤ 1 + 1/2k , (ii) ϕ

k is a complete isometry, (iii) i k+1|E k − ϕk ◦ i k cb < 1/2k . We proceed by induction. Since E 1 is exact, we may find an integer n(1), an operator space S1 ⊆ Mn(1) , and a linear isomorphism i 1 : E 1 → S1 such that ki 1 kcb

≤ 1 and i 1−1

cb

< 1 + 1/2. Suppose that n(k) and

i k : E k → Sk ⊆ Mn(k) are given. We construct n(k +1), i k+1 , and ϕk so that they satisfy the above conditions.


513

Since E k+1 is exact, there exist an integer m, an operator space Sk+1 ⊆ Mm , and a linear isomorphism j : E k+1 → Sk+1 such that k jkcb ≤ 1 and k j −1 kcb < 1 + 1/2k+1 . Since Mn(k) is injective, there is an α : E k+1 → Mn(k) such that α|E k = i k and kαkcb ≤ ki k kcb ≤ 1. Since Mm is injective, there is a β : Mn(k) → Mm such that β|ik (E k ) = j ◦ i k−1 and kβkcb ≤ ki k−1 kcb ≤ 1 +

1 . 2k

We let n(k + 1) = n(k) + m, and we define i k+1 : E k+1 → Mn(k+1) by i k+1 (x) =

α(x) 0

0 j (x)

.

If r ∈ N and x ∈ Mr (E k+1 ),

−1

(i )r (i k+1 )r (x) = kxk k+1

= jr−1 jr (x)

≤ k j −1 kcb jr (x)

1 ≤ 1 + k+1 jr (x) 2

1 ≤ 1 + k+1 (i k+1 )r (x) 2 and thus we have

−1

i k+1

cb

≤ 1+

1 .

2k+1

If we define ϕk : Mn(k) → Mn(k+1) by  ϕk (x) = 

x 0

 0 −1 , 1 1+ k β(x) 2

it is a simple matter to verify that we also have (ii) and (iii).

514


We next construct the inductive limit of the spaces Mn(k) . For k < l, we define a complete isometry ϕk,l : Mn(k) → Mn(l) by ϕk,l = ϕl−1 ◦ · · · ◦ ϕk . If k ≥ l, we let ϕk,l = 0. We let `∞ (N, Mn(k) ) denote the von Neumann algebra of bounded sequences (xl ) with xl ∈ Mn(l) , and we let c0 (N, Mn(k) ) be the ideal of sequences converging to zero. We have that A = `∞ (N, Mn(k) )/c0 (N, Mn(k) ) is a C ∗ -algebra, and for each k ∈ N we may define a complete isometry ϕk,∞ : Mn(k) → A : x 7 → ϕk,l (x) l + c0 (N, Mn(k) ). It is clear that ϕl,∞ ◦ ϕk,l = ϕk,∞ . Furthermore, we have that Y = ∪ϕk,∞ (Mn(k) ) is a separable nuclear operator space. For each x ∈ ∪E k , we define i(x) = lim ϕk,∞ ◦ i k (x). k→∞

This limit exists since if k < l,

ϕl,∞ ◦ il (x) − ϕk,∞ ◦ i k (x)
0 such that ||| · |||n ≤ Ck · kn for all n ∈ N. PROPOSITION 5.1 The operator space `∞ (B (H )), {||| · |||n } is a dual Banach space, but it is not the dual of an operator space.

516


Proof Since ||| · |||1 = k · k1 , the first assertion is obvious. Let us suppose that `∞ (B (H )), {||| · |||n } is completely isometric to the operator space dual of an operator space (X, {||| · |||n }). Owing to the uniqueness of the Banach space predual of a von Neumann algebra, we may identify the Banach spaces (X, ||| · |||1 ) and `1 (K (H )∗ ), k · k1 . We then have that `1 (K (H )∗ ) is provided with two matrix norms {k · kn } and {||| · |||n }, for which the dual operator space matrix norms are the matrix norms {k · kn } and {||| · |||n } on `∞ (B (H )). Given an operator space V and its operator dual V ∗ , we have for each [vi j ] ∈ Mn (V ), k[vi j ]k = sup k[hvi j , flt i]k : [ flt ] ∈ Mn (V ∗ ), k[ flt ]k ≤ 1 ; similarly, if [ flt ] ∈ Mn (V ∗ ), then k[ flt ]k = sup k[hvi j , flt i]k : [vi j ] ∈ Mn (V ), k[vi j ]k ≤ 1 (see [13, §3.2]). Since |||·|||n ≥ k·kn on Mn (`∞ (B (H ))), it follows that |||·|||n ≤ k·kn on Mn (`1 (K (H )∗ )). On the other hand, for each n ∈ N and [(vikj )] ∈ Mn (c0 (K (H ))), we have limU k[vikj ]op kn = 0 and thus |||[(vikj )]|||n = k[(vikj )]kn . It follows that for any [(ωikj )] ∈ Mn (`1 (K (H )∗ )), n hX i o

|||[(ωikj )]|||n = sup hωikj , vltk i : v = [(vltk )] ∈ Mn (`∞ (B (H ))), |||v|||n ≤ 1 k

i o n hX

≥ sup hωikj , vltk i : v = [(vltk )] ∈ Mn (c0 (K (H ))), |||v|||n ≤ 1 k

i o n hX

= sup hωikj , vltk i : v = [(vltk )] ∈ Mn (c0 (K (H ))), kvkn ≤ 1 k

= k[(ωikj )]k Mn (`1 (K (H )∗ )) . We conclude that ||| · |||n = k · kn on Mn (`1 (K (H )∗ )) for all n ∈ N, and by duality, ||| · |||n = k · kn on `∞ (B (H )), which is a contradiction. Appendix As we have briefly discussed in §2, if V is a TRO, then there is a natural nondegenerate (C, D)-bimodule structure on V . V is also a faithful (C, D)-bimodule; that is, for any c ∈ C and d ∈ D, cx = 0 for all x ∈ V implies that c = 0, and xd = 0 for all x ∈ V implies that d = 0. To see the second implication, let us assume that


517

d ∈ D satisfies xd = 0 for all x ∈ V . Then y ∗ xd = 0 for all x, y ∈ V . This shows Pn Pn that ( i=1 yi∗ xi )d = 0 for all i=1 yi∗ xi ∈ V ] V , and thus d 0 d = 0 for all d 0 ∈ D. 0 ∗ If we let d = d , it follows that d = 0. A similar argument can be applied to the left action of C. We have a nondegenerate and faithful Hilbert right D-module structure HV (resp., Hilbert left C-module structure V H ) on V defined by (x, y)V = x ∗ y resp., V(x, y) = x y ∗ for x, y ∈ V . Let L (HV ) denote the space of all adjointable right D-module mappings, and let K (HV ) denote the space of all compact right D-module mappings on the Hilbert right D-module HV . Since the module structure is nondegenerate and faithful, we have the canonical C ∗ -algebraic identifications C∼ and M(C) = M K (HV ) = L (HV ), = K (HV ) where M(C) is the multiplier algebra of C given by M(C) = {x ∈ C 00 = M, xc and cx ∈ C for all c ∈ C} (see [40, p. 78], [34, Chap. 2]). Similarly, if we let L (V H ) and K (V H ) denote the spaces of adjointable left Cmodule mappings and compact left C-module mappings, respectively, on the Hilbert left C-module V H , we have the canonical C ∗ -algebra isomorphisms D = K (V H ) and M(D) = M K (V H ) = L (V H ). If V is a nondegenerate W ∗ -TRO contained in B (K , H ), then M(C) ⊆ M = C 00 ⊆ B (K ). On the other hand, since M V ⊆ V and V M ⊆ V , we have MC ⊆ C and C M ⊆ C. This shows that M = M(C), that is, M∼ = L (HV ). Similarly, we have N∼ = L (V H ). It is thus apparent that the von Neumann algebras M and N considered in (4) do not depend on the choice of a nondegenerate representation of V . The following result of Smith [48] is an elementary consequence of the above discussion. (It also follows from a purely algebraic argument in [21].)

518


PROPOSITION A.1 Every finite-dimensional TRO is injective.

Proof If V is a finite-dimensional TRO, then the C ∗ -algebra C = K (HV ) and D = K (V H ) constructed above are clearly finite-dimensional. Therefore, the C ∗ -algebra A in (2) is finite-dimensional and thus injective. Therefore, V = e A(1 − e) is an injective TRO. We also have a useful refinement of Theorem 2.6. PROPOSITION A.2 If a W∗ -TRO V has a separable predual, then we may assume that R in (4) has a faithful representation on a separable Hilbert space.

Proof Using the notation introduced above, Zettl proved that there is a natural isometry ∗ γ M(C) ∼ = L M(D) (V ) = B M(D) (V ) ∼ = V ⊗ M(D) V∗ . In these identifications, we regard V as a right M(D)-module, and the expressions L M(D) (V ) and B M(D) (V ) denote the adjointable and the bounded M(D)-module γ mappings. The space V ⊗ M(D) V∗ is a suitable quotient of the Banach space projective tensor product (see [52, Cor. 4.10]). The duality is determined by the pairing hc, x ⊗ f i = f (cx) for c ∈ M(C), x ∈ V , and f ∈ V∗ . Let us suppose that V∗ is separable, and let us fix a norm dense sequence f k ∈ V∗ . Then the closed unit ball V1 is a compact metrizable space in the σ (V, V∗ )-topology, and in particular, it has a sequence xn that is σ (V, V∗ )-dense in V1 . We let 3 be the set of elementary tensors xn ⊗ f k , and we let 3 D ⊆ M(C)∗ be the image of 3 under the quotient mapping. 3 D is countable, and we claim that the linear span of 3 D is γ norm dense in V ⊗ M(D) V∗ . If this is not the case, then there exists a c ∈ M(C) such that f n (cx j ) = 0 for all j and n. But that in turn implies that cx j = 0 for all j. Since x 7→ cx is continuous in the weak∗ topology, it follows that cx = 0 for all x ∈ V1 , and thus c = 0. Thus M∗ = M(C)∗ is norm separable. A similar proof shows that N∗ = M(D)∗ is separable, and thus M V R= V] N


519

also has a separable predual. A dense sequence of normal states determines a faithful normal representation of R. Acknowledgments. We are indebted to Bernie Russo for drawing our attention to Zettl’s important paper [52]. Ruan wishes to thank S.-H. Kye for stimulating discussions related to Propositions 3.2 and Theorem 4.5. References [1]

R. J. ARCHBOLD and C. J. K. BATTY, C ∗ -tensor norms and slice maps, J. London Math.

[2]

D. P. BLECHER, The Shilov boundary of an operator space and the characterization

[3]

M. D. CHOI and E. G. EFFROS, Nuclear C ∗ -algebras and the approximation property,

Soc. (2) 22 (1980), 127–138. MR 81j:46090 492 theorems, J. Funct. Anal. 182 (2001), 280–343. MR CMP 1 828 796

[4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17]

Amer. J. Math. 100 (1978), 61–79. MR 58:2317 511 , Nuclear C ∗ -algebras and injectivity: The general case, Indiana Univ. Math. J. 26 (1977), 443–446. MR 55:3799 490, 492, 511 A. CONNES, Classification of injective factors: Cases I I1 , I I∞ , I I Iλ , λ 6 = 1, Ann. of Math. (2) 104 (1976), 73–115. MR 56:12908 490 J. DIXMIER, Les C ∗ -algèbres et leurs représentations, Cahiers Sci. 29, GauthierVillars, Paris, 1964. MR 30:1404 504 E. G. EFFROS, Order ideals in a C ∗ -algebra and its dual, Duke Math. J. 30 (1963), 391–411. MR 27:1847 498 E. G. EFFROS and U. HAAGERUP, Lifting problems and local reflexivity for C ∗ -algebras, Duke Math. J. 52 (1985), 103–128. MR 86k:46084 492, 510, 511 E. G. EFFROS, M. JUNGE, and Z.-J. RUAN, Integral mappings and the principle of local reflexivity for noncommutative L 1 -spaces, Ann. of Math. (2) 151 (2000), 59–92. MR 2000m:46120 508 E. G. EFFROS and E. C. LANCE, Tensor products of operator algebras, Adv. Math. 25 (1977), 1–34. MR 56:6402 504 E. G. EFFROS and Z.-J. RUAN, Representations of operator bimodules and their applications, J. Operator Theory 19 (1988), 137–158. MR 91e:46077 490 , Mapping spaces and liftings for operator spaces, Proc. London Math. Soc. (3) 69 (1994), 171–197. MR 96c:46074a , Operator Spaces, London Math. Soc. Monogr. (N.S.) 23, Oxford Univ. Press, New York, 2000. MR CMP 1 793 753 494, 516 L. GE and D. HADWIN, Ultraproduct for C ∗ -algebras, preprint. 508 D. B. GOODNER, Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89–108. MR 12:266c 489 A. GROTHENDIECK, Une caractérisation vectorielle-métrique des espaces L 1 , Canad. J. Math. 7 (1955), 552–561. MR 17:877d 489 M. HAMANA, “Injective envelopes of dynamical systems” in Operator Algebras and Operator Theory (Craiova, Romania, 1989), Pitman Res. Notes Math. Ser. 271,

520

[18] [19] [20] [21]

[22]

[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

[35] [36] [37]


Longman Sci. Tech., Harlow, England, 1992, 69–77. MR 94a:46100a 495 ˇ , Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama Univ. 22 (1999), 77–93. MR 2001a:46057 495 L. A. HARRIS, A generalization of C ∗ -algebras, Proc. London Math. Soc. (3) 42 (1981), 331–361. MR 82e:46089 491, 495, 497 M. HASUMI, The extension property of complex Banach spaces, Tôhoku Math. J. (2) 10 (1958), 135–142. MR 20:7209 489 M. R. HESTENES, A ternary algebra with applications to matrices and linear transformations, Arch. Rational Mech. Anal. 11 (1962), 138–194. MR 27:169 491, 517 E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis, Vol. 1: Structure of Topological Groups, Integration Theory, Group Representations, Grundlehren Math. Wiss. 115, Springer, Berlin, 1963. MR 28:158 494 K. K. JENSEN and K. THOMSEN, Elements of K K -Theory, Math. Theory Appl., Birkhäuser, Boston, 1991. MR 94b:19008 J. L. KELLEY, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323–326. MR 13:659e 489 E. KIRCHBERG, The Fubini theorem for exact C ∗ -algebras, J. Operator Theory 10 (1983), 3–8. MR 85d:46081 512 , On nonsemisplit extensions, tensor products and exactness of group C ∗ -algebras, Invent. Math. 112 (1993), 449–489. MR 94d:46058 505 , On restricted perturbations in inverse images and a description of normalizer algebras in C ∗ -algebras, J. Funct. Anal. 129 (1995), 1–34. MR 95m:46094a 491 , On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), 35–63. MR 95m:46094b 490, 492, 509, 511 E. KIRCHBERG and N. C. PHILLIPS, Embedding of exact C ∗ -algebras in the Cuntz algebra O2 , J. Reine Angew. Math. 525 (2000), 17–53. MR 2001d:46086a 512 E. KIRCHBERG and S. WASSERMANN, C ∗ -algebras generated by operator systems, J. Funct. Anal. 155 (1998), 324–351. MR 99f:46085 512 S.-H. KYE and Z.-J. RUAN, On the local lifting property for operator spaces, J. Funct. Anal. 168 (1999), 355–379. MR 2000k:47109 505, 506, 511 H. E. LACEY, The Isometric Theory of Classical Banach Spaces, Grundlehren Math. Wiss. 208, Springer, New York, 1974. MR 58:12308 489 [E.] C. LANCE, On nuclear C ∗ -algebras, J. Funct. Anal. 12 (1973), 157–176. MR 49:9640 506, 510 , Hilbert C ∗ -Modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, Cambridge, 1995. MR 96k:46100 517 C. LE MERDY, On the duality of operator spaces, Canad. Math. Bull. 38 (1995), 334–346. MR 97a:46022 492, 514 L. NACHBIN, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28–46. MR 11:369a 489 M. NAKAMURA, M. TAKESAKI, and H. UMEGAKI, A remark on the expectations of operator algebras, K¯odai Math. Sem. Rep. 12 (1960), 82–90. MR 40:3325 499


521

[38]

W. L. PASCHKE, Inner product modules over B ∗ -algebras, Trans. Amer. Math. Soc.

[39]

V. I. PAULSEN, Completely Bounded Maps and Dilations, Pitman Res. Notes Math.

[40]

G. K. PEDERSEN, C ∗ -Algebras and Their Automorphism Groups, London Math. Soc.

[41]

G. PISIER, “Exact operator spaces” in Recent Advances in Operator Algebras (Orléans,

182 (1973), 443–468. MR 50:8087 Ser. 146, Longman Sci. Tech., Harlow, England, 1986. MR 88h:46111 490 Monogr. 14, Academic Press, London, 1979. MR 81e:46037 517

[42] [43] [44] [45] [46]

[47] [48] [49] [50] [51] [52]

France, 1992), Astérisque 232, Soc. Math. France, Montrouge, 1995, 159–186. MR 97a:46023 490, 514 A. G. ROBERTSON and R. R. SMITH, Liftings and extensions of maps on C ∗ -algebras, J. Operator Theory 21 (1989), 117–131. MR 90g:46082 508 Z.-J. RUAN, Subspaces of C ∗ -algebras, J. Funct. Anal. 76 (1988), 217–230. MR 89h:46082 490 , Injectivity of operator spaces, Trans. Amer. Math. Soc. 315 (1989), 89–104. MR 91d:46078 490, 491 S. SAKAI, A characterization of W ∗ -algebras, Pacific J. Math. 6 (1956), 763–773. MR 18:811f A. M. SINCLAIR and R. R. SMITH, Hochschild Cohomology of von Neumann Algebras, London Math. Soc. Lecture Notes Ser. 203, Cambridge Univ. Press, Cambridge, 1995. MR 96d:46094 502 R. R. SMITH, Completely bounded maps between C ∗ -algebras, J. London Math. Soc. (2) 27 (1983), 157–166. MR 84g:46086 506 , Finite dimensional injective operator spaces, Proc. Amer. Math. Soc. 128 (2000), 3461–3462. MR 2001d:46087 517 ˘ ˘ , Modular Theory in Operator Algebras, Abacus Press, Tunbridge Wells, S¸. STRATIL A England, 1981. MR 85g:46072 502 J. TOMIYAMA, Tensor products and projections of norm one in von Neumann algebras, lecture notes, University of Copenhagen, 1970. 491, 499 H. H. ZETTL, Ideals in Hilbert modules and invariants under strong Morita equivalence of C ∗ -algebras, Arch. Math. (Basel) 39 (1982), 69–77. MR 84i:46060 494 , A characterization of ternary rings of operators, Adv. Math. 48 (1983), 117–143. MR 84h:46093 491, 497, 498, 502, 518, 519

Effros Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555, USA; [email protected] Ozawa Texas A&M University, College Station, Texas 77843, USA; [email protected]; University of Tokyo, Komaba, 153-8914, Japan Ruan Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA; [email protected]


QUANTUM COHOMOLOGY RINGS OF GRASSMANNIANS AND TOTAL POSITIVITY KONSTANZE RIETSCH

Abstract We give a proof of a result of D. Peterson identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of GLn . The totally positive part of this subvariety is then constructed, and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of C. Vafa, K. Intriligator, and A. Bertram for the structure constants (Gromov-Witten invariants). Finally, we use the positivity of these Gromov-Witten invariants to prove certain inequalities for Schur polynomials at roots of unity. 1. Introduction The group U + of unipotent upper-triangular matrices in GLn have on their coordinate ring a nice basis with positive structure constants. Namely, one has the dual of the classical limit of G. Lusztig’s geometrically constructed canonical basis of the quantized enveloping algebra. The existence of this basis has been closely tied (see [10]) to the “totally positive part” of U + (the matrices with only nonnegative minors). In this paper we study certain remarkable subvarieties Vd,n of U + which come up in the stabilizer of a particular standard principal nilpotent element e as closures of the 1-dimensional components under Bruhat decomposition. By a theorem of Peterson [16], the quantum cohomology rings of Grassmannians may be identified with the coordinate rings of these varieties. Therefore, like U + itself, these varieties have coordinate rings with “canonical” bases on them (this time coming from Schubert bases) and with positive structure constants. Most of Peterson’s results, in particular this one, are unpublished. But [8] is at least a reference for the “Peterson variety,” which is Peterson’s approach to encoding all the quantum cohomology rings of partial flag varieties in one go, and into which DUKE MATHEMATICAL JOURNAL c 2001 Vol. 110, No. 3, Received 26 July 2000. Revision received 4 December 2000. 2000 Mathematics Subject Classification. Primary 20G20, 15A48, 14N35, 14N15, 05E05. Author’s work begun at the Institute for Advanced Study, Princeton, New Jersey, with the support of National Science Foundation grant number DMS 97-29992. Author’s work currently supported by Engineering and Physical Sciences Research Council grant number GR/M09506/01. 523

524

KONSTANZE RIETSCH

the varieties Vd,n may be embedded naturally. We study the varieties Vd,n here from the point of view of explicitly describing their totally positive parts. We show that the totally positive elements come in a single one-parameter family, and we give closed formulas for the values of the Schubert basis on it. Furthermore, we obtain an upper bound (coming from this totally positive part) for certain Schur polynomials evaluated at roots of unity. Since our main approach is to study the coordinate rings directly and by elementary means, we also find some new and very simple proofs for results about them implied by the isomorphism with quantum cohomology. This includes properties of their Schubert bases, their explicit presentation (and thus Peterson’s theorem), and finally a formula of Vafa, Intriligator, and Bertram for the structure constants. The paper is organized as follows. After an initial section on the centralizer of the principal nilpotent element e and its Bruhat decomposition, we focus in on the special subvarieties Vd,n mentioned above. In Section 3 we state Theorem 3.4 on the coordinate ring of Vd,n and prove it partially. What makes the variety Vd,n so accessible to elementary methods is that its points are very easy to construct explicitly. This is explained next. In Section 4 we prove some orthogonality formulas for values of Schur polynomials at roots of unity. These facilitate the final step in the proof of Theorem 3.4 and are also the essential ingredient for the proof of the Bertram-VafaIntriligator formula in Section 6. Finally, after a little review of total positivity in Section 7, we determine the totally positive part of Vd,n . The totally nonnegative matrices in Vd,n have a very beautiful explicit description, as do the values the Schubert basis elements take on them. In Section 9 we write down how these totally positive matrices factor into products of elements of simple root subgroups. In Section 10 we explain the interpretation of the coordinate ring of Vd,n in terms of quantum cohomology, and we put into context some of the results encountered in the earlier sections. We also note that one property of this ring that we have not been able to derive in an elementary way, and which is not at present known other than following from the geometric definitions, is the positivity of the structure constants. (The Vafa-Intriligator formula that computes these is worse than alternating.) We use this positivity property in Section 11 to prove an inequality for values of Schur polynomials at roots of unity.

2. The stabilizer of a principal nilpotent 2.1. Preliminaries We recall some standard facts and notation for GLn (C). Let B + , B − be the subgroups of upper-triangular, respectively, lower-triangular matrices in GLn (C), and let U + and

QUANTUM COHOMOLOGY AND TOTAL POSITIVITY

525

U − be their unipotent radicals. Their intersection is the maximal torus T of all diagonal matrices in GLn (C). Let e1 , . . . , en−1 and f 1 , . . . , f n−1 be the standard Chevalley generators in the Lie algebras of U + and U − , respectively. So ei is the (n × n)-matrix with unique nonzero entry 1 found in position (i, i + 1), and f i is its transpose. Let X ∗ (T ) denote the character group of T , written additively, and let 4 ⊂ X ∗ (T ) be the set of roots. Denote by 5 = {αi ∈ 4| i = 1, . . . , n − 1} the usual system of simple roots defined by tei t −1 = αi (t)ei for all t ∈ T . The resulting sets of positive and negative roots are denoted by 4+ , respectively, 4− . We identify the Weyl group of GLn , the symmetric group, with the group W ⊂ GLn (C) of permutation matrices. And we let s1 , . . . , sn−1 denote the usual Coxeter generators of W . So s j corresponds to the adjacent transposition ( j, j + 1). W acts in the usual way on X ∗ (T ) preserving 4, and for any w ∈ W , the length `(w) is the number of positive roots sent to negative by w. Let w0 be the order-reversing permutation and so the longest element in W . 2.2. (U + )e and its Bruhat decomposition Definition 2.1 Let us fix the principal nilpotent element e=

n−1 X

ei ∈ gln (C).

i=1

Let (U + )e := {u ∈ U + | ueu −1 = e}, the stabilizer of e in U + . This is an abelian subgroup of U + of dimension n − 1. The elements of (U + )e are precisely those elements of U + of the form   1 x1 x2 . . . xn−2 xn−1  1 x1 xn−2     ..  .. ..   . . .   (2.1) u= . ..   . x1 x2      1 x1  1 We can thereby explicitly identify the coordinate ring C[(U + )e ] with the polynomial ring C[x1 , . . . , xn−1 ]. 2.3 To decompose (U + )e by the Bruhat decomposition, we need to look a bit more closely at the Weyl group. Let K be a subset of {1, . . . , n − 1}. Then to K we associate the

526

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parabolic subgroup W K = hsk ik∈K of W and its unique longest element w K . We also consider the element w K := w0 w K , which is the minimal length coset representative ˆ be the set {1, . . . , n − 1} \ {k}. in w0 W K . Let {k} We begin with two lemmas that we learned from Dale Peterson. 2.2 [15] The elements w K ∈ W defined above are characterized by w K ∈ W | K ⊂ {1, . . . , n − 1} = w ∈ W |w · 5 ⊂ (−5) ∪ 4+ . LEMMA

Proof Suppose that w = w K is the longest element in W K = hsk | k ∈ K i. Then w K has the property that it sends 5 K = {αk |k ∈ K } to −5 K , while all other simple roots remain positive. This implies the inclusion ⊆. Now let w ∈ W lie in the right-hand side. So w · 5 ⊂ (−5) ∪ 4+ . Then let K + be defined by K = { j | − α j ∈ w · 5}. By this definition w−1 K w · 5 ⊂ 4 . Therefore −1 w = w K . Note that w K w · αi is positive also when w · αi ∈ 1+ since any positive P root sent to −1+ by w K must lie in k∈K N αk . This cannot be true of w · αi since w−1 · αk ∈ −5 for any k ∈ K . LEMMA 2.3 [15] Bruhat decomposition induces

(U + )e =

G

(U + )e ∩ B − w K B − .

(2.2)

K ⊂{1,...,n−1}

Proof By Bruhat decomposition we can write u = b1 w0 wb2 for some b1 ∈ B − , b2 ∈ U − , and w ∈ W . Since u · e := ueu −1 = e, we have wb2 · e = w0 b1−1 · e. P The right-hand side of this equation is of the form n−1 j=1 m j f j + x for an uppertriangular matrix x and some m j ∈ C. The left-hand side is w · (e + y) for some lower-triangular matrix y. It follows from their equality that w · 5 ⊂ (−5) ∪ 4+ . By Lemma 2.2 we have w = w K for some K ⊂ {1, . . . , n − 1}. Definition 2.4 Let 1 j ∈ C[(U + )e ] be the top right-hand corner ((n − j) × (n − j))-minor of u in


527

(2.1), 

xj

  x j−1   1 j = det    

x j+1 xj .. .

x2 j−n+1

.. .. ..

xn−1 . . .

..

.

xj x j−1



    ,   x j+1 

(2.3)

xj

where we set x0 = 1 and xk = 0 for k < 0. We can now give an explicit description of the parts in the decomposition (2.2) of (U + )e . LEMMA 2.5 For u ∈ (U + )e , define K u := { j ∈ {1, . . . , n−1} | 1 j (u) = 0}. Then u ∈ B − w K u B − . In particular, (U + )e ∩ B − w K B − = u ∈ (U + )e | 1 j (u) = 0 for all j ∈ K . (2.4)

Proof By Lemma 2.3 we have u ∈ B − w K B − for some K ⊂ {1, . . . , n − 1}. We need only show that K = K u . Let v1 , . . . , vn be the standard basis of Cn , and let {vi1 ∧ vi2 ∧ V · · · ∧ vid | 1 ≤ i 1 < · · · < i d ≤ n} be the corresponding basis of d Cn . Then 1 j (u) is the matrix coefficient 1 j (u) = u · (v j+1 ∧ · · · ∧ vn ), v1 ∧ · · · ∧ vn− j of u expressed in this standard basis. Write u = b1 w0 w K b2 for some b1 ∈ B − and b2 ∈ U − . Then we have u · (v j+1 ∧ · · · ∧ vn ) = ±b1 w0 w K · (v j+1 ∧ · · · ∧ vn ). Therefore 1 j (u) 6= 0 precisely if w0 w K · (v j+1 ∧ · · · ∧ vn ) = ±v1 ∧ · · · ∧ vn− j or, equivalently, if w K · (v j+1 ∧ · · · ∧ vn ) = ±v j+1 ∧ · · · ∧ vn . Now consider the maximal parabolic subgroup W{ j} ˆ . It can as the group of permutation matrices preserving the subsets v j+1 , . . . , vn in Cn . Therefore condition (2.5) is equivalent to is, to j ∈ / K.

(2.5) be characterized v1 , . . . , v j and w K ∈ W{ j} ˆ , that

Remark 2.6 The preceding lemmas can be generalized to arbitrary reductive linear algebraic groups (where the 1 j ’s are replaced with the corresponding matrix coefficients in the fundamental representations).

528

KONSTANZE RIETSCH

3. The variety Vd,n and its coordinate ring We now go over to studying in detail the 1-dimensional components of (U + )e , correˆ These are very special in that their elements—and especially sponding to K = {d}. their totally positive elements—can be constructed explicitly and in an elementary way. Studying the higher dimensional components involves much deeper results, and this more general case will be treated in the forthcoming paper [17]. Definition 3.1 (The variety Vd,n ) Define Vd,n to be the closed subvariety of GLn (C) defined as ˆ − Vd,n := (U + )e ∩ B − w {d} B , ˆ

where w{d} is the Weyl group element defined in Section 2.3. Let C[Vd,n ] denote the coordinate ring of Vd,n as a reduced affine algebraic variety. Definition 3.2 (The ring 3d,n ) For k = 1, . . . , n, set Yk ∈ Z[X 1 , . . . , X d ] to be the the (k × k)-determinant X1 X2 · · · .. 1 X1 . . . . . .. .. . .. .. .. . X X 1 2 1 X

(3.1)

1

In other words, Yk = det(X i− j+1 )i, j=1,...,k , where X 0 := 1 and X k := 0 if k ∈ / {0, . . . , d}. We define 3d,n to be the ring given in terms of generators and relations by 3d,n := C[X 1 , . . . , X d ]/ (Yc+1 , . . . , Yn−1 ) , where c = n − d. Remark 3.3 (The element q and quantum cohomology) Define q := (−1)d+1 Yn in 3d,n . Then we may identify 3d,n ∼ = C[X 1 , . . . , X d , q]/ Yc+1 , . . . , Yn−1 , Yn + (−1)d q . Therefore the ring 3d,n coincides with q H ∗ (Grd (n), C), the quantum cohomology ring of the Grassmannian of d-planes in Cn , by the presentation found in [19] and [20] (see Sec. 10 for more on this).


529

By a theorem of Peterson [16], the quantum cohomology rings of Grassmannians are identified with coordinate rings of the varieties Vd,n . (In fact, his statement is a generalization of this to arbitrary type.) Using the known presentation of these quantum cohomology rings in type A, Peterson’s result for that case may be stated as follows. THEOREM

3.4 (D. Peterson)

φ

There is an isomorphism 3d,n → C[Vd,n ] which takes X j to x j for j = 1, . . . , d. Our first aim is to give a direct proof of this theorem. We do not make use of any of the properties of quantum cohomology until Section 11, keeping our treatment completely elementary. So, in particular, any facts about 3d,n which are required are proved by hand, even if they can be deduced from the isomorphism with quantum cohomology. 3.5 The algebra homomorphism C[X 1 , . . . , X d ] → C[Vd,n ] defined by X j 7→ x j is surjective and has as kernel the radical of the ideal generated by the elements Yc+1 , . . . , Yn−1 defined in Definition 3.2. LEMMA

Proof Let y1 , . . . , yn ∈ C[(U + )e ] be defined by yk := det xi− j+1 i, j=1,...,k , where x0 := 1 and x j := 0 if j ∈ / {0, . . . , n}. It suffices to show that Vd,n is the vanishing set of xd+1 , . . . , xn−1 , yc+1 , . . . , yn−1 . By Lemma 2.5 the variety Vd,n consists of the matrices in (U + )e for which all the minors 1 j with j 6= d vanish (see (2.3)). Consider the minors 1n−1 , 1n−2 , . . . , 1d+1 . Note that 1n−1 = xn−1 . The vanishing of the minors 1n−1 , . . . , 1d+1 implies inductively that the coordinates xd+1 , . . . , xn−1 vanish. The converse implication is immediate. Let u be the matrix from (2.1). It is clear that the inverse matrix to u is given by   1 −y1 y2 · · · (−1)d yd · · · (−1)n−1 yn−1   .. ..   . 1 −y1 .     . . . .   . .     d −1 (−1) y .  d u =      ..     .     1 −y1 1

530

KONSTANZE RIETSCH

So 11 (u) = yn−1 = (−1)n−1 1n−1 (u −1 ). Moreover, 1 j (u) = (−1)n−1 1n− j (u −1 ) for all j = 1, . . . , n − 1 (see [12, I.(2.9)]). Therefore the conditions 11 (u) = · · · = 1d−1 (u) = 0 are equivalent to 1c+1 (u −1 ) = 1c+2 (u −1 ) = · · · = 1n−1 (u −1 ) = 0, and these are equivalent to the conditions yc+1 = · · · = yn−1 = 0. Remark 3.6 It follows from Lemma 3.5 that any element in 3d,n gives rise to a function on Vd,n since we now have a well-defined map φ : 3d,n → C[Vd,n ]. We make free use of this by letting X (u) := φ(X )(u) for any X ∈ 3d,n and u ∈ Vd,n . To fully prove Theorem 3.4, it remains to show that 3C d,n is reduced. This is done in Section 5.2. Constructing elements in Vd,n We now explicitly construct elements inside Vd,n . For m 1 , . . . , m d ∈ C, let   1 m1 m2 . . . md 0 0   .. ..   . . 1 m1     .. ..  . . 0 1     ..  . md  u¯ n (m 1 , . . . , m d ) :=   ∈ GLn (C). (3.2)   . . . .  .. .. .. ..       1 m1 m2     1 m1  1 Consider the roots z 1 , . . . , z d ∈ C of the polynomial z d − m 1 z d−1 + · · · + (−1)d m d = 0, counted with multiplicities. So m j = E j (z 1 , . . . , z d ), where E j is the jth elementary symmetric polynomial in z 1 , . . . , z d . We define u n (z 1 , . . . , z d ) = u¯ n E 1 (z 1 , . . . , z d ), . . . , E d (z 1 , . . . , z d ) . (3.3) There is now a nice description for when a matrix u¯ n (m 1 , . . . , m d ) lies in Vd,n in terms of the roots z 1 , . . . , z d . 3.7 The matrix u n (z 1 , . . . , z d ) for complex numbers z 1 , . . . , z d lies in Vd,n if and only LEMMA


531

if either z 1 = · · · = z d = 0 or z 1 , . . . , z d are all distinct and z 1n = · · · = z dn . All elements of Vd,n are of this form. Furthermore, the element q ∈ 3d,n defined in Remark 3.3 evaluates on u n (z 1 , . . . , z d ) ∈ Vd,n to q(u n (z 1 , . . . , z d )) = (−1)d+1 z 1n . The main argument in Lemma 3.7 could be proved using the presentation of the quantum cohomology ring via the Landau-Ginzburg potential (see [13, Sec. 8.4] or, more originally, the work of D. Gepner [6] for q = 1). The following is another proof. Proof Let Hk be the kth complete elementary symmetric polynomial. Then by the JacobiTrudi identity, yk (u n (z 1 , . . . , z d )) = Hk (z 1 , . . . , z d ). So by Lemma 3.5, Vd,n consists of all u n (z 1 , . . . , z d ) for which Hc+1 (z 1 , . . . , z d ) = · · · = Hn−1 (z 1 , . . . , z d ) = 0. Consider the recursion Hm (x1 , . . . xk ) = Hm (x1 , . . . , xk−1 ) + xk Hm−1 (x1 , . . . , xk ). Therefore Hm−1 (x1 , . . . , xk ) = 0 =⇒

Hm (x1 , . . . , xk ) = Hm (x1 , . . . , xk−1 ),

(3.4)

and if the xi are all nonzero, the opposite implication holds as well. Now suppose Hc+1 (z 1 , . . . , z d ) = · · · = Hn−1 (z 1 , . . . , z d ) = 0 for z 1 , . . . , z d ∈ C. Then the vanishing of Hc+1 implies Hc+2 (z 1 , . . . , z d ) = Hc+2 (z 1 , . . . , z d−1 ). If d > 2, this again equals zero and it follows that Hc+3 (z 1 , . . . , z d ) = Hc+3 (z 1 , . . . , z c−1 ) = Hc+3 (z 1 , . . . , z d−2 ). We repeat this process until we finally get that Hn (z 1 , . . . , z d ) = Hn (z 1 , . . . , z d−1 ) = · · · = Hn (z 1 ) = z 1n . This implies by symmetry that z 1n = z 2n = · · · = z dn = Hn (z 1 , . . . , z d ). Suppose one of the z i 6 = 0. Then the same holds for the others. To prove that the z i must be distinct, look one step back. We have 0 = Hn−1 (z 1 , . . . , z d ) = Hn−1 (z k , zl ) for any k 6= l ∈ {1, . . . , d}. But Hn−1 (z, z) = nz n−1 is nonzero for z 6= 0. Hence z k 6= zl . Now suppose we are given d distinct z 1 , . . . , z d with z 1n = · · · = z dn . It remains to check that Hc+1 , . . . , Hn−1 vanish or, equivalently (since the z i are nonzero), that Hn (z 1 , . . . , z d ) = · · · = Hn (z 2 , z 1 ) = Hn (z 1 ) = z 1n . We claim that Hm+n (z 1 , . . . , z d ) = z 1n Hm (z 1 , . . . , z d ).

532

KONSTANZE RIETSCH

This is obvious if d = 1. For d > 1 it follows by induction on d using z 1n = z 2n and the divided difference identity Hn+m (z 1 , . . . , z d ) =

Hn+m+1 (z 1 , z 3 , . . . , z d ) − Hn+m+1 (z 2 , z 3 , . . . , z d ) . z1 − z2

Now for any k = 1, . . . , d we have Hn+1 (z 1 , z 3 , . . . , z k ) − Hn+1 (z 2 , . . . , z k ) z1 − z2 H (z , z , . . . , z k ) − H1 (z 2 , . . . , z k ) 1 1 3 = z 1n = z 1n . z1 − z2

Hn (z 1 , . . . , z k ) =

This concludes the proof. 4. Orthogonality formulas for Schur polynomials at roots of unity We have already encountered the elementary symmetric polynomials in d variables E 1 , . . . , E d and the complete homogeneous symmetric polynomials H1 , H2 , . . . . At this point the other Schur polynomials also enter the picture. Denote by Sλ the Schur polynomial in d variables corresponding to the partition λ of at most d parts. We write such a partition λ as a d-tuple of integers λ = (λ1 , . . . , λd ) with λ1 ≥ λ2 ≥ · · · ≥ λd ≥ 0. As a visual aid, partitions are often represented by their Young diagrams (see [12]). There are many definitions of the Schur polynomials. For instance, the Schur polynomials are the characters of the polynomial representations of GLd (symmetric polynomials in the eigenvalues). We recall also that the Schur polynomials may be obtained from the elementary and complete homogeneous symmetric polynomials by the Jacobi-Trudi and dual Jacobi-Trudi identities. Let λt = (λt1 , . . . , λtc ) be the conjugate partition to λ obtained by exchanging rows and columns in the Young diagram; then E λt +1 · · · E λt +c−1 Hλ1 E λt H · · · H λ +1 λ +d−1 1 1 1 1 1 Hλ −1 E λt · · · E λt +c−2 Hλ2 · · · Hλ2 +d−2 E λt −1 2 2 2 2 , Sλ = = .. .. . . H ··· ··· Hλd E λt +c−1 ··· ··· E λt λd −d+1 c

c

where E k = Hk = 0 for k < 0. The Schur polynomial Sλ is homogeneous of degree |λ| := λ1 + · · · + λd , the size of the partition. Also, the Schur polynomials form a basis of the ring of symmetric polynomials in d variables (for further background on Schur polynomials, we refer to [12]). Definition 4.1 (Partitions inside a box and PD) Let Sh(d, c) denote the set of partitions whose Young diagram or shape fits into a


533

d × c box. In other words, Sh(d, c) is the set of partitions λ = (λ1 , . . . , λd ) of length at most d such that λ1 ≤ c. We use the shorthand notation (m k ) for the partition (m, . . . , m, 0, . . . , 0) with m occurring k times. So, for example, (cd ) is the longest partition in Sh(d, c). For a partition λ ∈ Sh(d, c), define PD(λ) := (c − λd , c − λd−1 , . . . , c − λ1 ). The notation PD stands for Poincaré duality (see Section 10). Definition 4.2 (The set Id,n ) We fix the primitive nth root of unity ζ = exp(2πı/n). Let ζ I := (ζ i1 , . . . , ζ id ) be an unordered d-tuple of distinct nth roots of (−1)d+1 . Then I = (i 1 , . . . , i d ) may be chosen uniquely such that −(d − 1)/2 ≤ i 1 < i 2 < · · · < i d ≤ n − (d + 1)/2 and the i k ’s are all integers (resp., half-integers) if d is odd (even). Denote the set of all such d-tuples I by Id,n . Let c = n − d. If I = (i 1 , . . . , i d ) ∈ Id,n , then denote by Iˆ = (iˆ1 , . . . , iˆc ) the unique c-tuple, −(d − 1)/2 ≤ iˆ1 < iˆ2 < · · · < iˆc ≤ n − (d + 1)/2, such that ˆ ˆ ζ i1 , . . . , ζ id , ζ i1 , . . . ζ ic enumerates all nth roots of (−1)d+1 . There is a bijection ( )t : Id,n → Ic,n (4.1) which takes I = (i 1 , . . . , i d ) to I t := (n/2 − iˆc , . . . , n/2 − iˆ1 ). We note that this bijection corresponds exactly to transposition of shapes, ( )t : Sh(d, c) → Sh(c, d), after the identification Sh(d, c) ↔ Id,n , λ = (λ1 , . . . , λd ) 7→ Iλ =

d + 1 2

+ λd − d, . . . ,

We have Iλt = (Iλ )t . Furthermore, if kI k :=

Pd

k=1 i k ,

d +1 + λ1 − 1 . 2

(4.2)

then kIλ k = |λ|.

We show the following identities for Schur polynomials. 4.3 (Orthogonality formulas) I, J ∈ Id,n , and let z 1 , . . . , z d , t ∈ C. Then P Qd Qc J J − jˆl ), λ∈Sh(d,c) Sλ (z 1 , . . . , z d )SPD(λ) (ζ ) = S(cd ) (ζ ) k=1 l=1 (1 − z k ζ P I J d I I 2 λ∈Sh(d,c) Sλ (tζ )SPD(λ) (tζ ) = δ I,J (n S(cd ) (tζ ))/| Vand(ζ )| , P I d I 2 J λ∈Sh(d,c) Sλ (ζ )Sλ (ζ ) = δ I,J (n /| Vand(ζ )| ),

PROPOSITION

Let (1) (2) (3)

where | Vand(ζ I )| stands for the absolute value of the Vandermonde determinant Q ik ij k< j (ζ − ζ ) and where the bar in formula (3) stands for complex conjugation.

534

KONSTANZE RIETSCH

LEMMA 4.4 If λ ∈ Sh(d, c) and if I ∈ Id,n , then we have the following equality of values of Schur polynomials: SPD(λ) (ζ I ) t (4.3) Sλt (ζ I ) = = Sλ (ζ I ). S(cd ) (ζ I )

Proof To begin, recall that S(cd ) is just the character of the cth power of the determinant representation of GLd . The right-hand side equality follows from Sλ (ζ I ) = Sλ (ζ −I ) and the general formula Sλ (z 1−1 , . . . , z d−1 ) =

SPD(λ) (z 1 , . . . , z d ) S(cd ) (z 1 , . . . , z d )

for the character of the dual representation of GLd . ˆ We now prove that Sλt (ζ (n/2)− I ) = Sλ (ζ −I ). Let 1 ≤ k ≤ d, and suppose λ ∈ Sh(d, c) is the partition (1k ) := (1, . . . , 1, 0, . . . , 0) with 1 appearing k times. So Sλ = E k , the kth elementary symmetric polynomial, and Sλt = Hk , the kth complete homogeneous symmetric polynomial. Then we have p(t) :=

d Y

(1 + tζ −i j ) = 1 + E 1 (ζ −I )t + E 2 (ζ −I )t 2 + · · · + E d (ζ −I )t d ,

j=1

r (t) :=

d Y

1

j=1

ˆ (1 − tζ (n/2)−i j )

= 1 + H1 (ζ (n/2)− I )t + H2 (ζ (n/2)− I )t 2 ˆ

ˆ

+ · · · + Hd (ζ (n/2)− I )t d + 0 ˆ

+ 0 + Hn (ζ (n/2)− I )t n + higher order terms. ˆ

And since r (t)−1 p(t) =

n Y

(1 + tζ −(d+1)/2+k ) = 1 + (−1)d+1 t n ,

k=1

we get r (t)(1 + (−1)d+1 t n ) = p(t). Comparing the first d coefficients on either side, ˆ we see that E k (ζ −I ) = Hk (ζ (n/2)− I ). So the required identity is proved for λ = (1k ), where k = 1, . . . , d. For all other partitions in Sh(d, c), the formula follows from this special case using the Jacobi-Trudi identity.


535

Proof of Proposition 4.3 Recall the classical identity due to D. Littlewood, X

Sλ (z 1 , . . . , z d )Sλt (w1 , . . . , wc ) =

λ∈Sh(d,c)

c d Y Y

(1 + z i w j ),

(4.4)

i=1 j=1

which is a consequence of the usual orthogonality of Schur functions (see I. Macdont ald [12, I.(4.30 )] or W. Fulton [4, App. A]). Now replace (w1 , . . . , wc ) by ζ J in (4.4) and apply Lemma 4.4 to get orthogonality formula (1). Formulas (2) and (3) follow by furthermore replacing (z 1 , . . . , z d ) by ζ I and checking that   if I 6 = J , d c 0 YY i k − jˆl d n (1 − ζ ζ ) =  2 if I = J .  k=1 l=1 Vand(ζ I ) This is immediate in the case I 6= J . To verify the identity for I = J , compute c d Y Y ˆ Vand(ζ I ) 2 (1−ζ ik ζ −il ) = k=1 l=1

Y

(1−ζ ζ

i m −ir

m,r ∈{1,...,d}, m6=r

d Y c Y ˆ ) (1−ζ ik ζ −il ), k=1 l=1

which is the same as d Y

"Q

n r =1 (1 −

k=1

zζ ik ζ r −(d+1)/2 ) 1−z

# .

(4.5)

z=1

Since this polynomial in z simplifies to Qn (1 − zζ ik ζ −(d+1)/2+l ) 1 − zn p(z) = l=1 = = 1 + z + · · · + z n−1 , 1−z 1−z we have p(1) = n and (4.5) equals n d . Remark 4.5 Passing from “column orthogonality” to “row orthogonality” in Proposition 4.3, we also get the formulas 2 1 X Sλ (ζ I )Sµ (ζ I ) Vand(ζ I ) = δλ,µ (4.6) d n I ∈Id,n

and

I 2 1 X I I Vand(ζ ) Sλ (ζ )SPD(µ) (ζ ) = δλ,µ , nd S(cd ) (ζ I )

(4.7)

I ∈Id,n

for any λ, µ ∈ Sh(d, c). Formula (4.6) looks as though it should have an explanation in terms of representation theory of GLd . Formula (4.7) is related to the analogue of Poincaré duality in the quantum cohomology ring (see Sec. 10.5).

536

KONSTANZE RIETSCH

5. Applications to the ring 3d,n In this section we finish the proof of Theorem 3.4, that 3d,n is the coordinate ring of Vd,n . For a partition λ ∈ Sh(d, c), let sλ ∈ 3d,n be defined by X t X λt +1 · · · X λt +c−1 λ1 1 1 X λt · · · X λt +c−2 X λt2 −1 2 2 . (5.1) sλ = .. . X λt +c−1 ··· ··· X λtc c Note that this looks like the dual Jacobi-Trudi formula. Therefore these elements evaluate on a point u = u n (z 1 , . . . , z d ) in Vd,n to sλ (u) = Sλ (z 1 , . . . , z d ). 5.1. The quantum Pieri rule We need to know how these sλ behave under multiplication by the generators X k of 3d,n . This is explained by the (dual) quantum Pieri rule stated in Theorem 5.1. The quantum Pieri rule was first found by Bertram [2], who proved it for the ring q H ∗ (Grd (n)) using some geometry. But later Bertram, I. Ciocan-Fontanine, and Fulton [3] also gave an algebraic and combinatorial proof inside the ring 3d,n . THEOREM 5.1 (Dual quantum Pieri rule; see [3, Prop. 4.2]) Let λ ∈ Sh(d, c) and k ∈ {1, . . . , d}; then the following formula holds in 3d,n : X X X k sλ = sν + q sµ , ν

µ

where the first sum is over all ν ∈ Sh(d, c) such that |ν| = |λ| + k and ν1t ≥ λt1 ≥ ν2t ≥ · · · ≥ νct ≥ λtc ≥ 0. The second sum is over all µ such that |µ| = |λ| + k − n and λt1 − 1 ≥ µt1 ≥ λt2 − 1 ≥ µt2 ≥ · · · ≥ λtd − 1 ≥ νlt ≥ 0. Such a formula for multiplication at q = 1 also shows up in the earlier work [6] of Gepner coming from physics. And Gepner notes that this specialized ring is semisimple. The final step of our proof of Theorem 3.4 uses an insight similar to Gepner’s. 5.2. Conclusion of the proof of Theorem 3.4 By Lemma 3.5 it remains only to prove that the homomorphism φ : 3C d,n → C[Vd,n ] C is injective. In other words, let p ∈ 3d,n , and suppose the image of p in C[Vd,n ], again denoted by p, is zero. Then we must show that p = 0 in 3C d,n . At the outset, p is some polynomial in the X k ’s. By repeated application of the P dual quantum Pieri rule, we can write p as a linear combination p = λ∈Sh(d,c) pλ sλ , where pλ ∈ C[q]. Then because p vanishes as a function on Vd,n , we get that X 0 = p u n (tζ I ) = pλ (t n )Sλ (tζ I ) (5.2) λ


537

for any element u n (tζ I ) ∈ Vd,n . Now for any fixed t ∈ C, consider the vector ( pλ (t n ))λ ∈ CSh(d,c) . By (5.2) this vector lies in the orthogonal complement to the span h(Sλ (tζ I ))λ i I ∈Id,n (with respect to the standard Euclidean inner product on CSh(d,c) ). But the vectors (Sλ (tζ I ))λ for varying I ∈ Id,n are all linearly independent by the orthogonality formula in Proposition 4.3(2), and therefore span CSh(d,c) . So it follows that the vector ( pλ (t n ))λ = 0. Since this holds for all t ∈ C, we have that all of the pλ = 0 in C[q]. Hence p = 0, as required. 5.2 (Schubert basis) The elements {q k sλ | k ∈ Z≥0 , λ ∈ Sh(d, c)} form a Z-basis of 3Z d,n = Z[X 1 , . . . , X d ]/(Yc+1 , . . . , Yn−1 ). COROLLARY

Proof By repeated application of the quantum Pieri rule, any element in 3Z d,n is expressible k as a Z-linear combination of elements q sλ . These are all linearly independent by exactly the same argument as in the proof of Theorem 3.4. 6. Structure constants of 3d,n In this section we give an elementary proof of a formula for the structure constants of 3d,n . Formula (6.1) was first noted by Bertram [2] to hold in the isomorphic ring q H ∗ (Grd (n)) as a consequence of his quantum Giambelli formula (see Theorem 10.2) and the Vafa-Intriligator formula for Gromov-Witten invariants. The mathematical proof of the Vafa-Intriligator formula by B. Siebert and G. Tian [19] depends on higher-dimensional residues and the geometric definition of the structure constants. We give a simple proof for 3d,n below which instead uses Theorem 3.4 and the orthogonality formula from Proposition 4.3. 6.1 Let λ1 , . . . , λ N ∈ Sh(d, c), I ∈ Id,n , and t ∈ C∗ . Let P be an arbitrary homogeneous symmetric polynomial in d variables: (1) We have the following formula for P(tζ I ) in terms of Schur polynomials: X P(tζ I ) = m νP (t)Sν (tζ I ), PROPOSITION

ν∈Sh(d,c)

where m νP (t) = (2)

2 1 X P(tζ J )Sν (t −1 ζ −J ) Vand(ζ J ) . d n J ∈Id,n

For any homogeneous symmetric polynomial m, X m(ζ J ) = 0 unless deg(m) ≡ 0 mod n. J ∈Id,n

538

(3)

KONSTANZE RIETSCH

In particular, m νP = 0 unless |ν| ≡ deg(P) mod n. Let p be a (nonzero) homogeneous element in 3d,n . Then the coefficients pλ,k of X p= pλ,k q k sλ λ∈Sh(d,c), k∈Z≥0

written out in the basis {q k sλ | k ∈ Z≥0 , λ ∈ Sh(d, c)} of 3d,n are given by the formula ( P 1 J J 2 if kn + |λ| = deg( p), J d J ∈Id,n p u(ζ ) Sλ (ζ )| Vand(ζ )| n pλ,k = 0 otherwise. COROLLARY 6.2 (Bertram-Vafa-Intriligator formula) The structure constants for multiplication in the basis {q k sλ | λ ∈ Sh(d, c), k ∈ N} of 3d,n , X sλ sµ = hsλ , sµ , sν ik q k sPD(ν) , k∈N,λ0 ∈Sh(d,c)

are computed by J 2 1 X J J J Vand(ζ ) hsλ , sµ , sν ik = d Sλ (ζ )Sµ (ζ )Sν (ζ ) n S(cd ) (ζ J )

(6.1)

J ∈Id,n

whenever cd + kn = |λ| + |µ| + |ν|, and otherwise by hsλ , sµ , sν ik = 0. Proof Formula (6.1) is an immediate consequence of Proposition 6.1(3) and Lemma 4.4. Proof of Proposition 6.1 P For J ∈ Id,n , define the symmetric polynomial S J := ν∈Sh(d,c) Sν (ζ −J )Sν . By Theorem 4.3(3) we have S J (ζ I ) = δ I,J (n d /| Vand(ζ J )|2 ). Therefore P(tζ I ) =

2 1 X P(tζ J ) Vand(ζ J ) S J (ζ I ) d n J ∈Id,n

=

1 nd

X

X

2 P(tζ J ) Vand(ζ J ) Sν (t −1 ζ −J )Sν (tζ I ).

ν∈Sh(d,c) J ∈Id,n

So (1) is proved. P For (2), notice that M(t) := J ∈Id,n m(tζ J ) satisfies M(t) = M(ζ t). Therefore M is a polynomial in t n , and unless n divides deg(m), we must have M = 0. Since (deg(P) − |ν|) is the degree of m νP , the rest of (2) also follows.


539

To prove any identity in 3d,n , it suffices (by Theorem 3.4) to check it “on points,” that is, evaluated at the elements of Vd,n . Therefore we can deduce (3) directly from (1) and (2). Remark 6.3 (q=0) The ring 3d,n specializes to 3d,n = C[x1 , . . . , xd ]/(yc+1 , . . . , yn ), the usual cohomology ring of the Grassmannian, at q = 0. The (images of the) sλ with λ ∈ Sh(d, c) now form a basis of this ring, and the structure constants for this basis are the wellknown Littlewood-Richardson coefficients. Therefore formula (6.1) for structure constants in 3d,n gives when k = 0 a formula for Littlewood-Richardson coefficients. 7. Total positivity (A quick review) Before proceeding in Section 8 to determining the totally positive part of Vd,n , we give some background and motivation for doing this. Total positivity, introduced by I. Schoenberg in the 1930s, is the study of invertible matrices with exclusively nonnegative real minors (and nonnegative entries in particular). We, strictly speaking, call such elements of GLn totally nonnegative and, if all minors are in fact strictly positive, then totally positive. The definition of total positivity is basis-dependent (or more precisely, dependent on a “pinning” of GLn ; see [10]), but totally positive matrices also have nice intrinsic properties. For example, F. Gantmacher and M. Krein showed that all totally positive matrices are regular semisimple with distinct positive real eigenvalues. The totally positive matrices form an open semialgebraic subset of the real points GLn (R) which we denote by GLn (R>0 ) in analogy with the standard notation GLn (K) for the K-valued points of GLn when K is a field. We also write GLn (R≥0 ) for the totally nonnegative part of GLn . In the early 1990s Lusztig [10] took the theory of total positivity a big step further and generalized it to all reductive algebraic groups, at the same time relating it (in the simply laced case) to canonical bases and their deep positivity properties. Let us look in more detail at the special case of the subgroup U + , to give a taste of what the totally positive part looks like and how it is related to the canonical basis. We use the notation introduced in Section 2.1. Let U + (R) = U + ∩ GLn (R). We consider GLn (R) and any subsets as endowed with the usual Hausdorff topology. Inside U + (R) we define U + (R≥0 ) to be the semigroup of all totally nonnegative matrices in U + . This is a semigroup since the totally nonnegative matrices can be characterized as acting by nonnegative matrices in all exterior powers of the standard representation of GLn (with their standard bases). In the 1950s A. Whitney proved that U + (R≥0 ) is the semigroup generated by the elements exp(tei ) for t ∈ R≥0 . The following more precise description of U + (R≥0 )

540

KONSTANZE RIETSCH

was given by Lusztig in [10]. For w ∈ W , let Uw+ = U + ∩ B − w B − . Then we define Uw+ (R>0 ) to be the set of totally nonnegative matrices in Uw+ . For w equal to the simple reflection si , it is easy to see that ∼

R>0 −→ Us+i (R>0 ), t 7→ xi (t) := exp(tei ) and that Us+i (R>0 ) is a connected component of Us+i (R) ∼ = R∗ ; hence the notation + + Usi (R>0 ) rather than Usi (R≥0 ). Moreover, for general w, taking a reduced expression w = si1 · · · sik and multiplying together the corresponding maps give a (semialgebraic) isomorphism ∼

(R>0 )k −→ Uw+ (R>0 ),

(7.1)

(t1 , . . . , tk ) 7 → xi1 (t1 ) · . . . · xik (tk ) (see [10, Prop. 2.7]). The largest of these real semialgebraic cells, Uw+0 (R>0 ), which corresponds to the longest element in W , is also called the totally positive part of U + and denoted by U + (R>0 ). It is open in U + (R), and its closure is the set of all totally nonnegative matrices in U + , denoted by U + (R≥0 ). Now let us describe the connection with canonical bases. Consider the coordinate ring C[U + ] as the graded dual to the enveloping algebra U + of the Lie algebra Lie(U + ). Then C[U + ] has a basis B given by the dual canonical basis (obtained from the canonical basis of the dual quantum enveloping algebra after specializing the quantum parameter to 1). In fact, it also has a nice Z-form that is spanned by this canonical basis. And the products of basis vectors are nonnegative linear combinations of basis vectors: the structure constants are nonnegative integers. This last point is very deep as it follows from Lusztig’s geometric construction of the coproduct in the quantized universal enveloping algebra (see [9]). And this positivity property enters crucially into the proof of the following theorem. THEOREM 7.1 (see Lusztig [10]; see also [11, Sec. 3.13]) Let u ∈ U + . Then

u ∈ U + (R>0 ) ⇐⇒ b(u) ∈ R>0

for all b ∈ B,

u ∈ U + (R≥0 ) ⇐⇒ b(u) ∈ R≥0

for all b ∈ B.

and

We see in the next section exactly similar results for the variety Vd,n , where the dual canonical basis is replaced by the Schubert basis.


541

8. The totally positive part of Vd,n We begin with some general observations about Vd,n . First Vd,n is a 1-dimensional affine variety with a C∗ -action, given explicitly by t · u¯ n (m 1 , . . . , m d ) = u¯ n (tm 1 , t 2 m 2 , . . . , t d m d ) (see (3.2)). We identify 3d,n with C[Vd,n ] by the isomorphism of Theorem 3.4. Consider the morphism therefore defined by q ∈ 3d,n , q : Vd,n → C. This map is C∗ -equivariant for the C∗ -action on C given by t ·z = t n z for t ∈ C∗ , z ∈ C. And the fiber over 1 is precisely {u n (ζ I ) | I ∈ Id,n }. So q is an dn -fold covering over C∗ , ramified at zero. Also, two elements u n (ζ I ) and u n (ζ J ) in the fiber of q over 1 lie in the same C∗ -orbit if and only if ζ J = ζ k ζ I for some k = 1, . . . , n. Therefore Vd,n consists of precisely n1 dn regular C∗ -orbits and one trivial one (given by the identity matrix). Consider now the real form 3R d,n = R[X 1 , . . . , X d ]/(Yc+1 , . . . , Yn−1 ) = Z 3d,n ⊗Z R, and consider the corresponding set of real points Vd,n (R) = Vd,n ∩GLn (R) of Vd,n . If d 0 := d2 is the greatest integer part of d2 , then there are precisely dn0 real points in the fiber of q over 1. So Vd,n (R) is made up of dn0 regular R∗ -orbits and the one trivial one, and q : Vd,n (R) → R simply looks as indicated in Figure 1.

.. .. .. .

.. .. .. .

q

n d 2

q

Figure 1

We identify exactly one of the branches over R>0 as the totally positive part of Vd,n . Let us give the precise definitions first. Definition 8.1 Let Vd,n (R≥0 ) := Vd,n (R)∩U + (R≥0 ), the set of totally nonnegative matrices in Vd,n . Recall that, by definition, ˆ − Vd,n = (U + )e ∩ (B − w {d} B ).

542

KONSTANZE RIETSCH

Therefore in analogy with the case of U + described in Section 7, we define ˆ Vd,n (R>0 ) := Vd,n (R≥0 ) ∩ (B − w {d} B − ). The remaining Bruhat decomposition of Vd,n (R≥0 ) is very simple since, by Lemma 2.3 or Figure 1, we have Vd,n (R≥0 ) = Vd,n (R>0 ) t {id}.

Definition 8.2 Let I0 = (−(d −1)/2, −(d −1)/2+1, . . . , (d −1)/2) ∈ Id,n . So ζ I0 is in a sense the “positive-most” d-tuple of roots of (−1)d+1 . Then we define a one-parameter family of matrices in Vd,n (R) by u >0 : R → Vd,n (R),

where u >0 (t) := u n (tζ I0 ).

(8.1)

Note that u n (tζ I0 ) does indeed lie in Vd,n (R) since the d-tuple ζ I0 is (up to permutation) invariant under complex conjugation. Let us determine the matrix entries E 1 (tζ I0 ), . . . , E d (tζ I0 ) of u n (tζ I0 ) explicitly. We use a variant of the classical q-binomial theorem of Gauss, with ζ playing the role of q (cf. [12, Exam. I.2.3]). For m, k ∈ N, let [m]ζ := and

ζ m/2 − ζ −m/2 ζ 1/2 − ζ −1/2

h m i [m]ζ [m − 1]ζ · . . . · [m − k + 1]ζ := . k ζ [1]ζ [2]ζ · . . . · [k]ζ

Then we have the following version of the binomial formula: (1 + ζ −(d−1)/2 t)(1 + ζ −(d−1)/2+1 t) · . . . · (1 + ζ (d−1)/2 t) h d i h d i h d i =1+ t+ t2 + · · · + t d−1 + t d . 1 ζ 2 ζ d −1 ζ In other words, the elementary symmetric polynomials evaluated on tζ I0 give h d i E j (tζ −(d−1)/2 , tζ −(d−1)/2+1 , . . . , tζ (d−1)/2 ) = t j . j ζ Since ζ = exp(2πı/n), we have that (ζ m/2 − ζ −m/2 )/(ζ 1/2 − ζ −1/2 ) = sin(m(π/n))/ sin(π/n). Therefore 1)(π/n) I0 j sin(d(π/n)) sin (d − 1)(π/n) · · · sin (d − j + E j tζ =t . sin(π/n) sin 2(π/n) · · · sin j (π/n) This gives the x j entry of u >0 (t). To determine also the minors of u >0 (t) and prove total positivity, we need a bit more notation.


543

Definition 8.3 Associate to a partition λ its Young diagram, and let the boxes in λ be labeled by their (x, y) coordinates in the usual way, as if they were entries of a matrix. We write (i, j) ∈ λ if (i, j) are the coordinates of a box in λ, so j ≤ λi and i ≤ λtj . Let hl(i, j) denote the hook length of the box labeled by (i, j); that is, hl(i, j) = λi +λtj −i − j +1. THEOREM 8.4 Recall the definitions of sλ and q (from (5.1) and Remark 3.3). (1) For any partition λ, the value of sλ on u >0 (t) is given by the following hook length formula: Q sin (d − i + j)(π/n) |λ| (i, j)∈λ . sλ u >0 (t) = t Q (i, j)∈λ sin hl(i, j)(π/n)

(2)

The value of q is given by q(u >0 (t)) = t n . u >0 The map R −→ Vd,n (R) restricts to give homeomorphisms (for the standard Hausdorff topology) between ∼

R≥0 −→ Vd,n (R≥0 )

(3)

and

∼

R>0 −→ Vd,n (R>0 ).

So the u >0 (t) for t ∈ R≥0 are precisely all the totally nonnegative elements in Vd,n . An element u ∈ Vd,n lies in Vd,n (R≥0 ) if and only if sλ (u) ∈ R≥0 for all λ ∈ Sh(d, c). It lies in Vd,n (R>0 ) precisely if sλ (u) ∈ R>0 for all λ ∈ Sh(d, c).

Proof (1). From the definitions it follows that sλ (u >0 (t)) = Sλ (tζ I0 ) = t |λ| Sλ (ζ I0 ). Now the identity in (1) is equivalent to Q (i, j)∈λ sin (d − i + j)(π/n) I0 . Sλ (ζ ) = Q (i, j)∈λ sin hl(i, j)(π/n) This equality is again the consequence of a classical formula. It is a variant of Littlewood’s identity for Sλ (1, q, q 2 , . . . , q k ) (see [12, Exam. I.3.1]). The proof goes by writing the Schur polynomial (by Weyl’s character formula) as quotient of two determinants. So Sλ (x −(d−1)/2 , x −(d−1)/2+1 , . . . , x (d−1)/2 ) = 1λd (x)/1∅d (x), where 1λd (x) is the generalized d × d Vandermonde determinant Y 1λd (x) = det(x (−(d−1)/2+i−1)(λ j +d− j) ) = (x (λi −λ j +i− j)/2 − x (λi −λ j + j−i)/2 ). 1≤i< j≤d

Replacing x by ζ and (x (λi −λ j +i− j)/2 − x (λi −λ j + j−i)/2 ) by 2i sin((λi − λ j + i − j)(π/n)) and cancelling the numerator against the denominator in Sλ , the desired formula follows. The identity for q is immediate from Lemma 3.7.

544

KONSTANZE RIETSCH

(2) and (3). To prove that any u ∈ U + is totally nonnegative, it suffices to check that all the minors with connected column sets are nonnegative, by a classical result. For u = u >0 (t) = u n (tζ I0 ) these minors are precisely Schur polynomial values Sλ (tζ −(d−1)/2 , tζ −(d−1)/2+1 , . . . , tζ (d−1)/2 ), where λ is a partition that fits inside a j × (n − j) box for some j = 1, . . . , n − 1. In other words, λ must be a partition with all hook lengths less than n. Therefore in the formula in (3), the denominator is always 6= 0 (if t 6= 0). From the numerator it follows that sλ (u >0 (t)) 6= 0 precisely if λ ∈ Sh(d, c). Suppose now that t > 0 and λ ∈ Sh(d, c). Then for any box (i, j) in λ, both d − i + j as well as the hook length are positive integers less than n. Therefore the values sin((d − i + j)(π/n)) and sin(hl(i, j)(π/n)) in the formula are all positive, and thus these minors are greater than zero. For t = 0 all Sλ except for S(0,...,0) ≡ 1 vanish. This implies (3) and one direction in (2). It remains to prove that there are no other matrices in Vd,n (R≥0 ). Suppose, therefore, u = u n (tζ I ) is totally nonnegative, where t ∈ C∗ and I ∈ Id,n . We have observed that all of the sλ for λ ∈ Sh(d, c) must take nonnegative values on u. Since u n (tζ I ) ∈ Vd,n (R), we know that q(u n (tζ I )) = t n must be real. In fact, q can be expressed as q = X d Yc in 3d,n , which follows by expanding the determinant (3.1) for Yn to Yn =

n−1 X

(−1)k+1 X k Yn−k = (−1)d+1 X d Yc .

k=1

Therefore q(u(tζ I )) = t n must be positive real. So t = ζ k r for some k ∈ {1, . . . , n} and r ∈ R>0 . We can replace t by r and ζ I by ζ J := ζ I +(k,...,k) to get u n (tζ I ) = u n (r ζ J ). If ζ J = ζ I0 , then u n (r ζ J ) = u >0 (r ), so we may assume that this is not the case. By orthogonality formula (2) from Theorem 4.3, we then have X Sλ (r ζ J )SPD(λ) (r ζ I0 ) = 0. λ∈Sh(d,c)

Since all the SPD(λ) (r ζ I0 ) are strictly positive, the Sλ (r ζ J ) cannot all be nonnegative. This finishes the proof. 9. Factorization of elements in Vd,n (R>0 ) The totally positive part Vd,n (R>0 ) lies, in particular, in U +{d}ˆ (R>0 ). Hence we can w factorize its elements into products of elements xi (t) = exp(tei ) (with t ∈ R>0 ) as described in (7.1).


545

Definition 9.1 For k ≤ d and m ≤ c, let B(k, m) := (i, j) | d − k + 1 ≤ i ≤ d, 1 ≤ j ≤ m . Then B(d, c) labels the set of boxes in the Young diagram of (cd ) and we may think of B(k, m) as parameterizing a subset. We introduce a partial ordering on B(k, m) by (i, j) (i 0 , j 0 ) ⇐⇒ i ≥ i 0 and j ≥ j 0 . There is a unique smallest element (d, m) and largest element (d − k + 1, 1). When taking products over B(k, m), we consider the factors ordered compatibly with this Q partial order (in increasing order). So, for example, the product (i, j)∈B(d,c) sd−i+ j ˆ

of simple reflections in the Weyl group gives w{d} . The different allowed orderings of the factors correspond to different reduced expressions. For each pair (k, m) we also have a well-defined monomial Y ed−i+ j e(k,m) = (i, j)∈B(k,m)

in the universal enveloping algebra U + of Lie(U + ). Next we define ˆ

x : (C∗ ) B(d,c) → U + ∩ B − w{d} B − , Y xd−i+ j (a(i, j) ). a = (a(i, j) ) 7 → x(a) := (i, j)∈B(d,c)

So the restriction of x to (R>0 ) B(d,c) is the parameterization map from (7.1). Finally, let 1[i, j] ∈ C[U + ] be the minor with row set 1, . . . , 1+ j −i and column set i, . . . , j. 9.2 Let u = x(a) for a = (a(i, j) ) ∈ (C∗ ) B(d,c) . (1) The (m × m)-minor 1[k+1,k+m] of u is given by Y 1[k+1,k+m] (u) = LEMMA

a(i, j) .

(i, j)∈B(k,m)

(2)

The entries of a are computed in terms of the minors u by a(d−k+1,m) =

1[k+1,k+m] (u)1[k,k+m−2] (u) , 1[k+1,k+m−1] (u)1[k,k+m−1] (u)

where k ∈ {1, . . . , d} and m ∈ {1, . . . , c}.

546

KONSTANZE RIETSCH

The entries a(i, j) can also be worked out using A. Berenstein and A. Zelevinsky’s generalized Chamber Ansatz (see [1]). This method for factorizing elements of Uw+ (R>0 ) is more general in that it works for any w. But it gives the entries in the factorization of an element x ∈ Uw+ (R>0 ) in terms of minors not of x itself but of another matrix −1 (x), which in turn can be expressed using minors of x. In the case at hand, it z = ηw would also be possible to show that their map ηw is essentially the identity on these particular elements. This explains why formula (2) looks like the one given by a single application of the Chamber Ansatz. Proof (1) Let v1 , . . . , vn be the standard basis of Cn . Then the minor in question is just the matrix coefficient hu · vk+1 ∧ · · · ∧ vn , v1 ∧ · · · ∧ vm ∧ vk+m+1 ∧ · · · ∧ vn i

(9.1)

V in the fundamental representation n−k Cn of GLn (C). Now consider the monomial Q e(k, m) = (i, j)∈B(k,m) ed−i+ j in U + . It has the property that e(k, m) · vk+1 ∧ · · · ∧ vn = v1 ∧ · · · ∧ vm ∧ vk+m+1 ∧ · · · ∧ vn . In fact, e(k, m) is the unique (up to scalar) monomial in the ei ’s satisfying

e(k, m) · vk+1 ∧ · · · ∧ vn , v1 ∧ · · · ∧ vm ∧ vk+m+1 ∧ · · · ∧ vn 6= 0. Q Then embed u = (i, j)∈(cd ) xd−i+ j (a(i, j) ) into the (graded) completion of the enveloping algebra and expand and multiply the exponentials to get an infinite series Q of monomials in the ei ’s. The coefficient of (i, j)∈B(k,m) ed−i+ j in this series is preQ cisely (i, j)∈B(k,m) a(i, j) . Since this is the only monomial that contributes to matrix coefficient (9.1), the statement follows. Part (2) follows directly from (1) by cancellation. 9.3 If u = u >0 (t), then

PROPOSITION

(1)

sin (i + j − 1)(π/n) , u= xd−i+ j t sin (d − i + j)(π/n) (i, j)∈B(d,c) Y

(2)

where the order of multiplication is such that the indices of the xd−i+ j spell a ˆ reduced expression of w{d} . An element u ∈ Vd,n lies in Vd,n (R>0 ) precisely if s(m k ) (u) > 0 for all rectangular partitions (m k ) with m ≤ c and k ≤ d.


547

Proof (1) The minor 1[k+1,k+m] of u n (tζ I ) is computed by S(m k ) (ζ I ), by the dual JacobiTrudi formula. For u >0 (t) this gives Y sin (d − i + j)(π/n) I0 km 1[k+1,k+m] u >0 (t) = S(m k ) (tζ ) = t . sin (hl(i, j))(π/n) k (i, j)∈(m )

Therefore u >0 (t) =

(i, j)∈B(d,c) x d−i+ j (a(i, j) ),

Q

a(i, j)

where a(i, j) is computed by

sin (i + j − 1)(π/n) =t sin (d − i + j)(π/n)

after applying Lemma 9.2(2) and cancelling in the numerator and denominator. (2) It is clear since by (7.1) and Lemma 9.2, the totally positive part of U + ∩ ˆ − B w{d} B − itself is determined by the inequalities 1[m+1,...,m+k] > 0. In fact, this criterion is the earlier result of Berenstein and Zelevinsky [1], which we have just reproved in our special case. Now because 1[m+1,...,m+k] V = s(m k ) , we are done. d,n

Note that the reduced set of cd inequalities given in Proposition 9.3(2) to describe Vd,n (R>0 ) inside variety Vd,n (R) is still surprisingly large, for a 1-dimensional variety. It is the same as the set needed to describe the positive part Uw{d}ˆ (R>0 ) in the whole cd-dimensional variety Uw{d}ˆ . 10. Quantum cohomology of Grassmannians This section is a brief collecting together of some basic facts about the (small) quantum cohomology ring of a Grassmannian. Quantum cohomology originally comes from the work of Vafa and E. Witten and ideas from string theory. Since then, it has had a big impact on mathematics, with much work being done to make the theory rigorous and to expand on it. Our main reference is Bertram’s self-contained mathematical treatment for the Grassmannian (see [2]; see also [5] and [13] and the references therein for a more general introduction to the mathematical theory of quantum cohomology (in the algebraic geometric or symplectic setting, resp.)). Let Grd (n) denote the variety of all d-dimensional subspaces of Cn . As before, we fix c = n − d. So c is the codimension while d is the dimension. 10.1. Schubert basis The quantum cohomology of Grd (n) is in the first instance a module over the polynomial ring in one variable Z[q] defined by q H ∗ Grd (n) = Z[q] ⊗Z H ∗ Grd (n), Z . (10.1)

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KONSTANZE RIETSCH

Therefore it has a Z[q]-basis given by the classical Schubert basis of H ∗ (Grd (n), Z). We briefly recall its definition here. Consider a flag V• of subspaces V• = (V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn ) and a partition λ ∈ Sh(d, c). Then the Schubert variety X λ (V• ) ⊂ Grd (n) is defined concretely as X λ (V• ) = W ∈ Grd (n) | dim(W ∩ Vc+i−λi ) ≥ i . X λ (V• ) is a closed subvariety of Grd (n) of (complex) codimension |λ|. Since any flag can be transformed into any other by the action of GLn (C), the homology class of X λ (V• ) does not depend on the choice of flag V• . By the classical result of C. Ehresmann, the Schubert varieties give rise to a basis of the homology of Grd (n). Let us denote the Poincaré dual basis of cohomology by {σλ | λ ∈ Sh(d, c)}. So we have σλ ∈ H 2|λ| (Grd (n)). 10.2. Poincaré duality Consider the Poincaré duality pairing ( , ) : H ∗ Grd (n) × H ∗ Grd (n) → Z defined by (σ, τ ) = (σ ∪ τ )[Grd (n)], where [Grd (n)] ∈ H2cd (Grd (n)) is the fundamental class. We recall that in terms of the Schubert basis elements, this pairing takes the form (σλ , σµ ) = δλ,PD(µ) , (10.2) where as before PD(µ) = (c − µd , c − µd−1 , . . . , c − µ1 ). 10.3. Ring structure As a ring, q H ∗ (Grd (n)) does not agree with H ∗ (Grd (n)) ⊗Z Z[q]. The new ring structure is a q-deformation of the old with structure constants for the Schubert basis given by 3-point Gromov-Witten invariants (see below). Let us view our Schubert basis elements {σλ | λ ∈ Sh(d, c)} as lying inside q H ∗ (Grd (n)) by identifying σλ with σλ ⊗ 1. Let λ, µ, ν ∈ Sh(d, c) such that |λ| + |µ| + |ν| = dc + kn (the dimension of the moduli space of degree k holomorphic maps CP 1 → Grd (n)), and fix three flags V• , V•0 , and V•00 in general position. Then hσλ , σµ , σν ik is defined as the number of degree k holomorphic curves CP 1 → Grd (n) taking 0, 1, and ∞ ∈ CP 1 to points in X λ (V• ), X µ (V•0 ), and X ν (V•00 ), respectively. If |λ| + |µ| + |ν| 6= dc + kn, then hσλ , σµ , σν ik := 0. Multiplication in q H ∗ (Grd (n)) is defined by X hσλ , σµ , σPD(ν) ik q k σν . σλ · σµ = ν,k


549

It is a remarkable fact proved now in more general contexts by Y. Ruan and Tian [18] and also M. Kontsevich and Yu. Manin [7] that the quantum cohomology product is associative. For the Grassmannian the associativity of the above-defined multiplication is also derived directly in [2]. With this product, q H ∗ (Grd (n)) becomes a graded ring with grading defined by deg(σλ ) = |λ| and deg(q) = n (or everything doubled to agree with the grading of the ordinary cohomology). Note that the structure constants of this ring are nonnegative integers. Also, when q is set to zero, only the k = 0 terms in the product formula remain, and we recover the definition of the usual cup product. 10.4. Presentation and structure constants Recall the definition of the ring 3Z d,n from Definition 3.2 (and Cor. 5.2). THEOREM 10.1 [19], [20] The ring q H ∗ (Grd (n)) is generated by the d Schubert classes σ(1) , σ(1,1) , . . . , σ(1d ) . The assignment ∗ d+1 3Z Yn 7 → q (10.3) d,n → q H Grd (n) : X k 7 → σ(1k ) , (−1)

defines an isomorphism of rings. It is also known that, as in the classical cohomology, the determinants Y1 , . . . , Yc map to the Schubert classes σ(1,0,...,0) , . . . , σ(c,0,...,0) . In fact, Bertram proved that all the classical Giambelli formulas hold also in the quantum cohomology ring. THEOREM 10.2 (Quantum Giambelli formula; see [2]) For λ ∈ Sh(d, c) the element sλ ∈ 3Z d,n defined in (5.1) is mapped to the (quantum) Schubert class σλ under the isomorphism (10.3).

With Theorem 10.2 we recover the Bertram-Vafa-Intriligator formula for the GromovWitten invariants from our algebraic result Corollary 6.2. THEOREM 10.3 [2, Sec. 5] The Gromov-Witten invariant hσλ , σµ , σν ik , where |λ| + |µ| + |ν| = nk + cd, is computed by

J 2 1 X J J J Vand(ζ ) Sλ (ζ )Sµ (ζ )Sν (ζ ) hσλ , σµ , σν ik = d n S(cd ) (ζ J ) J ∈Id,n

(see also [19] for a more general version of the Vafa-Intriligator formula).

550

KONSTANZE RIETSCH

10.5. Poincaré duality for q H ∗ (Grd (n)) We end this section with the geometric interpretation of “row-orthogonality” formula (4.7). One can define an analogue of the Poincaré duality pairing on q H ∗ (Grd (n)). This is a Z[q]-bilinear map ( , )q : q H ∗ Grd (n) × q H ∗ Grd (n) → Z[q] which takes (σ, σ 0 ) to the coefficient of σ(cd ) in σ · σ 0 . The pairing ( , )q specializes to the usual Poincaré duality when q goes to zero. For two Schubert basis elements σλ and σµ , this pairing is simply computed by (σλ , σµ )q = hσλ , σµ , σPD(cd ) ik q k

(10.4)

if n divides |λ| + |µ| − cd with quotient k. And it is zero if |λ| + |µ| − cd is not divisible by n. Now by Theorem 10.3, equation (10.4) becomes J 2 1 X J J Vand(ζ ) (σλ , σµ )q = d Sλ (ζ )Sµ (ζ ) . n S(cd ) (ζ J ) J ∈Id,n

Thus (4.7) corresponds to the following geometric statement. 10.4 [20, Sec. 3.1], [2, Lem. 2.5] Let λ, µ ∈ Sh(d, c). Then the pairing ( , )q defined above takes the same form PROPOSITION

(σλ , σµ )q = δλ,PD(µ) as the classical Poincaré duality pairing. 11. An inequality for Schur polynomial values In this section we give a little application of the positivity of structure constants in 3d,n . The positivity of structure constants is one property of 3d,n coming from its isomorphism with q H ∗ (Grd,n , C) for which we know no elementary explanation. The Vafa-Intriligator formula does not obviously give something positive. And there is a generalization of the Littlewood-Richardson rule to quantum cohomology in [3], but it is an alternating formula. PROPOSITION 11.1 Let I0 = (−(d − 1)/2, . . . , (d − 1)/2), let I ∈ Id,n be any other element, and let ζ = exp(2πi/n), as before. Then for any partition λ ∈ Sh(d, c), Sλ (ζ I ) ≤ Sλ (ζ I0 ). (11.1)


551

Remark 11.2 An equivalent formulation of the above result is to say that Sλ (ζ i1 , . . . , ζ id ) ≤ Sλ (ζ −(d−1)/2 , ζ −(d−1)/2+1 , . . . , ζ (d−1)/2 ) for any d-tuple of distinct nth roots of unity ζ i1 , . . . , ζ id and partition λ ∈ Sh(d, c). Proof Consider q H ∗ (Grd (n), C)q=1 := q H ∗ (Grd (n), C) ⊗C[q] C[q]/(q − 1), the specialization of the quantum cohomology ring at q = 1. We may identify this ring with the ring of functions on the fiber of Vd,n over q = 1, that is, with functions on the finite set {u n (ζ I ) | I ∈ Id,n }. We have that q H ∗ (Grd (n), C)q=1 is a finite-dimensional vector space with basis given by (restriction of) the Schubert basis σλ . P Another basis of this space is the set consisting of all σ I := ν∈Sh(d,c) Sν (ζ I )σν for I ∈ Id,n . By Proposition 4.3(3) we have nd σ I u n (ζ J ) = δ I,J , Vand(ζ I ) 2 so these are up to scalar the characteristic functions of the points in the set {u n (ζ I ) | I ∈ Id,n }. Now consider the multiplication operator [σλ ] : σ 7 → σλ σ on q H ∗ (Grd (n), C)q=1 . Then clearly, σλ · σ I = Sλ (ζ I )σ I . So σ I is an eigenvector of the multiplication operator [σλ ] with eigenvalue Sλ (ζ I ). Moreover, the set {σ I | I ∈ Id,n } is a complete eigenbasis, in fact, a simultaneous eigenbasis for all the multiplication operators [σ ], where σ ∈ q H ∗ (Grd (n), C)q=1 . Consider the set Y of all σ ∈ q H ∗ (Grd (n), R)(q=1) such that all the eigenvalues of [σ ] are distinct. Then Y is obtained from the real vector space q H ∗ (Grd (n), R)(q=1) by removing certain lower-dimensional linear subspaces. Therefore, in particular, Y is open dense in q H ∗ (Grd (n), R)(q=1) . Now recall that the multiplication operators [σλ ] on q H ∗ (Grd (n), C)q=1 are given in terms of the Schubert basis by an dn × dn -matrix Aλ with nonnegative integer entries (Gromov-Witten invariants). We can approximate the matrix Aλ to arbitrary precision > 0 by some other nonnegative matrix X Aλ := Aλ + µ Aµ , where 0 < µ < , µ∈Sh(d,c)

which has only simple eigenvalues, by the previous paragraph. We now apply the following version of the Perron-Frobenius eigenvalue theorem to Aλ .

552

KONSTANZE RIETSCH

THEOREM (Perron-Frobenius; see [14, Th. I.4.2]) Suppose A is an (m × m)-matrix with all entries in R≥0 . Then A has an eigenvalue r in R≥0 with a nonnegative eigenvector and such that all other eigenvalues of A have absolute value less than or equal to r .

Let A = Aλ , chosen as above. Since all of its eigenvalues are simple, the eigenvectors σ I are unique up to scalar. Therefore using Theorem 8.4, it follows that (up to positive scalar) σ I0 is the only nonnegative eigenvector. So by the Perron-Frobenius theorem, P its eigenvalue Sλ (ζ I0 ) + µ µ Sµ (ζ I0 ) is the maximal eigenvalue. Now choosing sufficiently small, we obtain that Sλ (ζ I0 ) must be a maximal eigenvalue of Aλ . This proves inequality (11.1). Acknowledgments. We were very fortunate to hear many inspiring lectures by Dale Peterson and would like to thank him here. We are also grateful to Shahn Majid for some useful discussions during the writing of this paper. References [1]

A. BERENSTEIN and A. ZELEVINSKY, Total positivity in Schubert varieties, Comment.

[2]

A. BERTRAM, Quantum Schubert calculus, Adv. Math. 128 (1997), 289–305.

[3]

A. BERTRAM, I. CIOCAN-FONTANINE, and W. FULTON, Quantum multiplication of

[4]

W. FULTON, Young Tableaux, London Math. Soc. Stud. Texts 35, Cambridge Univ.

[5]

W. FULTON and R. PANDHARIPANDE, “Notes on stable maps and quantum

Math. Helv. 72 (1997), 128–166. MR 99g:14064 545, 547 MR 98j:14067 536, 537, 547, 548, 549, 550 Schur polynomials, J. Algebra 219 (1999), 728–746. MR 2000k:14042 536, 550 Press, Cambridge, 1997. MR 99f:05119 535

[6] [7]

[8]

[9] [10]

cohomology” in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, 1997, 45–96. MR 98m:14025 547 D. GEPNER, Fusion rings and geometry, Comm. Math. Phys. 141 (1991), 381–411. MR 92j:81271 531, 536 M. KONTSEVICH and YU. MANIN, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562. MR 95i:14049 548 B. KOSTANT, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ρ, Selecta Math. (N.S.) 2 (1996), 43–91. MR 97e:17029 523 G. LUSZTIG, Introduction to Quantum Groups, Progr. Math. 110, Birkhäuser, Boston, 1993. MR 94m:17016 540 , “Total positivity in reductive groups” in Lie Theory and Geometry, Progr. Math. 123, Birkhäuser, Boston, 1994, 531–568. MR 96m:20071 523, 539, 540


[11]

[12]

[13]

[14] [15] [16]

[17] [18] [19]

[20]

553

, Total positivity in partial flag manifolds, Represent. Theory 2 (1998), 70–78, http://www.ams.org/journal-getitem?pii=S1088-4165-98-00046-6 MR 2000b:20060 540 I. G. MACDONALD, Symmetric Functions and Hall Polynomials, 2d ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995. MR 96h:05207 530, 532, 535, 542, 543 D. MCDUFF and D. SALAMON, J-Holomorphic Curves and Quantum Cohomology, Univ. Lecture Ser. 6, Amer. Math. Soc., Providence, 1994. MR 95g:58026 531, 547 H. MINC, Nonnegative Matrices, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1988. MR 89i:15001 551 D. PETERSON, Quantum cohomology of G/P, lecture course, Massachusetts Institute of Technology, Cambridge, 1997. 526 , Quantum cohomology of G/P, Seminaire de Mathematiques Superieures: Representation Theories and Algebraic Geometry, Universite de Montréal, Canada, 1997, unpublished lecture notes. 523, 529 K. Rietsch, Quantum cohomology of partial flag varieties and totally positive Toeplitz matrices, preprint, 2001. 528 Y. RUAN and G. TIAN, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259–367. MR 96m:58033 548 B. SIEBERT and G. TIAN, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), 679–695. MR 99d:14060 528, 537, 549 E. WITTEN, “The Verlinde algebra and the cohomology of the Grassmannian” in Geometry, Topology, & Physics (Cambridge, Mass., 1993), Conf. Proc. Lecture Notes Geom. Topology 4, Internat. Press, Cambridge, Mass., 1995, 357–422. MR 98c:58016 528, 549, 550

Department of Pure Mathematics and Mathematical Statistics, Cambridge University and Newnham College, Wilberforce Road, Cambridge CB3 0WB, United Kingdom; current: Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford, OX1 3LB, United Kingdom; [email protected]


ON THE ZEROS OF ζ 0 (s) NEAR THE CRITICAL LINE YITANG ZHANG

Abstract Let ρ 0 = β 0 + iγ 0 denote the zeros of ζ 0 (s), s = σ + it. It is shown that there is a positive proportion of the zeros of ζ 0 (s) in 0 < t < T satisfying β 0 − 1/2 (log T )−1 . Further results relying on the Riemann hypothesis and conjectures on the gaps between the zeros of ζ (s) are also obtained. 1. Introduction Throughout this paper s = σ + it is a complex variable; T is a large parameter, and L = log T ; ρ 0 = β 0 +iγ 0 denotes the zeros of ζ 0 (s), the first derivative of the Riemann zeta function; a sum over ρ 0 is repeated according to multiplicity. The distribution of the zeros of ζ 0 (s), as well as its relationship with that of the zeros of ζ (s), has been investigated by many authors (cf. B. Berndt [1], J. Conrey and A. Ghosh [3], C. Guo [8], [9], N. Levinson and H. Montgomery [10], K. Soundararajan [13], A. Speiser [14]). In [1] Berndt proved N1 (T ) =

T L − log(4π) − 1 + O(L), 2π

(1.1)

where N1 (T ) denotes the number of the zeros of ζ 0 (s) in 0 < t ≤ T , counted according to multiplicity; in [14] Speiser proved that the Riemann hypothesis (RH) is equivalent to ζ 0 (s) having no zeros in 0 < σ < 1/2. Recently, Soundararajan [13] introduced the following functions (for ν ∈ R): X m − (ν) = lim inf N1 (T )−1 1, (1.2) T →∞

m + (ν) = lim sup N1 (T )−1

X

T →∞

β 0 ≤1/2+ν/L 0 2.6. He conjectured that m − (ν) ≡ m + (ν)(= m(ν)), m(ν) is continuous, m(ν) > 0 for all ν > 0, and m(ν) → 1 as ν → ∞. In the present paper we first establish an unconditional result on m − (ν). Namely, we prove the following. THEOREM 1 If ν is sufficiently large, then

m − (ν) > 0. From Theorem 1 and [10, Theorem 4], we can easily derive the following. COROLLARY

There exist positive constants ν and c such that X 1 > cT L |β 0 −1/2| 0. To state this result precisely, we adopt a standard notation in the theory of the Riemann zeta function (cf. E. Titchmarsh [15, Chapter 9]), which is also used elsewhere, as follows. The zeros of ζ (s) on the upper half-plane are arranged as ρ1 , ρ2 , . . . with ρn = βn + iγn and 0 < γ1 ≤ γ2 ≤ · · · . If a zero is multiple with multiplicity m, then it appears precisely m times consecutively in the above sequence. For α > 0, define X D − (α, T ) = 1 (1.4) γn ≤T γn+1 −γn 0 we have D − (α) > 0. Although the statement of SGZ is independent of RH, any significant progress that is known toward this conjecture relies on RH (cf. Conrey, Ghosh, D. Goldston, Gonek, and D. Heath-Brown [4], Soundararajan [12]). In fact, on assuming RH, SGZ is a consequence of Montgomery’s pair correlation conjecture. The second theorem of the present paper is the following.

558

YITANG ZHANG

THEOREM 2 (Assume RH and SGZ) For any ν > 0 we have m − (ν) > 0.

In [13] Soundararajan also conjectured that, on assuming RH, the following two statements are equivalent: (i) lim inf(β 0 − 1/2) log γ 0 = 0; (ii) lim inf(γ + − γ ) log γ = 0, where γ denotes the positive ordinate of the zeros of ζ (s), and γ + is the least ordinate of a zero of ζ (s) with γ + > γ . On using the same argument as in the proof of Theorem 1, it can be shown that (ii) implies (i). Actually, we prove a slightly stronger result as follows. THEOREM 3 (Assume RH) Let α1 and α2 be positive constants satisfying α1 < 2π and r α1 −1 . α2 > α1 1 − 2π

(1.6)

If ρ = 1/2 + iγ is a zero of ζ (s) such that γ is sufficiently large and γ + − γ < α1 (log γ )−1 , then there exists a zero ρ 0 of ζ 0 (s) such that |ρ 0 − ρ| < α2 (log γ )−1 . The proof of Theorem 3 is brief and similar to that of Theorem 2. In fact, we can also derive Theorem 2 from Theorem 3 on using another result of Fujii [7, Theorem 1]. It should be remarked that our argument is based on relating the following two different subjects in the theory of the Riemann zeta function to each other: the vertical distribution of the zeros of ζ (s) and the horizontal distribution of the zeros of ζ 0 (s). In fact, Theorems 2 and 3 exhibit examples of such relationships. We believe that further investigations in this field are possible. 2. Preliminary lemmas In this section we establish several preliminary results that are equally used in the proofs of the three theorems. Let s , h(s) = π −s/2 0 2 ξ(s) = h(s)ζ (s), η(s) = h(s)ζ 0 (s). For σ > 0, ξ(s) and η(s) have the same zeros as ζ (s) and ζ 0 (s), respectively. The following simple relations are repeatedly used:

ON THE ZEROS OF ζ 0 (s) NEAR THE CRITICAL LINE

i m ξ (m)

1 2

+ it ∈ R

(m ≥ 0),

η0 h0 ζ 00 (s) = (s) + 0 (s). η h ζ

559

(2.1) (2.2)

LEMMA 1 Suppose t > 5. Then ζ (1/2 + it) = 0 if and only if Re{η(1/2 + it)} = 0.

Proof By (2.1) and the relation ξ 0 (s) =

h0 (s)ξ(s) + η(s) h

we get

1 h0 1 + it = − Re + it ξ + it . 2 h 2 2 0 Since Re{h / h(1/2 + it)} > 0 for t > 5, the assertion follows. Re η

1

Applying Hadamard’s theorem to the entire function (s − 1)2 ζ 0 (s) and noting that ζ 0 (0) 6= 0, we get Y s 0 (s − 1)2 ζ 0 (s) = e As+B 1 − 0 es/ρ , (2.3) ρ 0 ρ

where A and B are certain constants and where ρ 0 runs through all the zeros of ζ 0 (s). By [10, Theorem 9], all the zeros of ζ 0 (s) in σ ≤ 0 are real and can be arranged as ρ10 , ρ20 , . . . , with −(2n + 2) < ρn0 < −2n. (2.4) It follows from (2.3) that X 1 ζ 00 2 1 (s) = − + A + + + 61 , ζ0 s−1 s − ρ0 ρ0 0

(2.5)

β >0

where 61 =

∞ X n=1

1 1 + . s − ρn0 ρn0

(2.6)

Now, assume σ = 1/2. By a well-known formula for the 0-function, ∞

X 1 h0 1 (s) = − + O(1). h 2n s + 2n n=1

(2.7)

560

YITANG ZHANG

Since

∞ X 1 1 + 1, ρn0 2n

∞ X

n=1

n=1

1 1 − 1 s − ρn0 s + 2n

by (2.4), it follows from (2.6) and (2.7) that h0 (s) + 61 1. h

(2.8)

Combining (2.2), (2.5), (2.8), and the simple result X

1 < +∞, ρ0

Re

β 0 >0

we conclude that, in case σ = 1/2 and ζ 0 (s) 6= 0, Re

X η0 1 (s) = Re + O(1). η s − ρ0 0

(2.9)

β >0

It is easy to see that, for s = 1/2 + it, the error term in (2.9) is continuous in t. If η(1/2 + it) 6 = 0, then −

X β 0 >0

Re

1 = F1 (t) − F2 (t), 1/2 + it − ρ 0

(2.10)

where X

F1 (t) = −

Re

1 , 1/2 + it − ρ 0

(2.11)

Re

1 . 1/2 + it − ρ 0

(2.12)

β 0 >1/2

F2 (t) =

X 0 γn . Since γ1 > 14, by Lemma 1 we have η(1/2+it) 6= 0 for t ∈ (γn , γn+1 ). Thus, we can define a function arg η(1/2+it) which is continuous on (γn , γn+1 ). For γn < t1 < t2 < γn+1 , by (2.13) we get Z

t2

Z

1/2+it2

F(t) dt = − Im t1

1/2+it1

1 1 η0 (s) ds = arg η + it1 − arg η + it2 . (2.16) η 2 2

562

YITANG ZHANG

Note that the left-hand side of (2.16) is continuous in both t1 and t2 . If it is greater than or equal to π for some t1 , t2 ∈ (γn , γn+1 ), t1 < t2 , then F(t3 ) ≡ π/2 (mod π ) for some t3 ∈ (γn , γn+1 ), contradicting Lemma 1. This proves the first assertion. Now, assume that both ρn and ρn+1 are simple zeros and βn = βn+1 = 1/2. In this case the continuous function arg η(1/2 + it) is well defined on [γn , γn+1 ], and (2.16) holds with t1 = γn , t2 = γn+1 . By Lemma 1 we now have 1 1 π arg η + iγn ≡ arg η + iγn+1 ≡ (mod π ). 2 2 2 This proves the second assertion. 3. Proof of Theorem 1 In this and the next sections we repeatedly use the following simple result: 1 T L + O(T ). (3.1) 2π Some unconditional results on the gaps between the zeros of ζ (s) were obtained by Fujii [7]. (In his paper the proofs are not given.) For the purpose of proving Theorem 1, only a special case of the corollary to [7, Theorem 2] is needed, which can be stated, in a slightly different form, as follows. N (T ) =

LEMMA 5 There exist constants c1 > 0 and λ > 1 such that the following holds: Let S1 = n : n < N (T ), γn+1 − γn > 2πλL −1 .

Then for large T , X

1 > c1 T L .

n∈S1

Proof See Heath-Brown’s notes for [15, Chapter 9]. Let λ be as in Lemma 5. Since λ − 1 > 0 and (1 + λ)(2λ)−1 < 1, we can choose positive constants c2 and ν such that c2 < λ − 1 and

2 ν λ+1 > . ν + 2π λ 2λ

(3.2)

(3.3)

Let δ = ν L −1 ,

θ = 2π λL −1 ,

(3.4)

ON THE ZEROS OF ζ 0 (s) NEAR THE CRITICAL LINE

and let

563

n 1o S2 = n : n < N (T ), βn = . 2

LEMMA 6 Suppose that n ∈ S1 ∩ S2 and γn > T L −1 . Then Z γn+1 X 2 F2 (t) dt ≥ 1, 1+ πc2 γn 0 |ρ −ρn | . ν + 2π λ 2 By (3.8) and (3.10)–(3.12) we get Z γn+1 log L π(λ − 1) F2 (t) dt ≥ +O + O(γn+1 − γn ). 2 L γn λ

(3.12)

(3.13)

Since both of the error terms in (3.13) are o(1) (see [15, Theorem 9.11]), it follows from (3.2) and (3.13) that, for large T , (3.6) holds. This completes the proof of Lemma 6. Proof of Theorem 1 Let

n 1o S3 = n : n < N (T ), βn < . 2 By the symmetry of the nontrivial zeros of ζ (s) with respect to the critical line, we have X X 1+2 1 = N (T ) − 1. (3.14) n∈S2

n∈S3

Let ν satisfy (3.3), and let c be chosen such that 4c ν −1 c1 − 4c − . 0 cT L . β 0

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