Problems in Analysis. A Symposium in Honor of Salomon Bochner

PROBLEMS

IN ANALYSIS A Symposium in Honor of Solomon Bochner

ROBERT C. GUNNING GENERAL EDITOR

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1970

Copyright © 1970, by Princeton University Press All Rights Reserved L. C. CARD 76-106392

I. S. B. N. 0-691-08076-3

A.M.S. 1968: 0004

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Foreword A symposium on problems in analysis in honor of Salomon Bochner was held in Fine Hall, Princeton University, April 1-3, 1969, to celebrate his seventieth birthday, which took place on

August 20, 1969. The symposium was sponsored by Princeton University and the United States Air Force Office of Scientific Research; the organizing com-

mittee consisted of W. Feller, R. C. Gunning, G. A. Hunt, D. Montgomery, R. G. Pohrer, and W. R. Trott.

This volume contains some of the papers delivered by the invited speakers

at the symposium, together with a number of papers contributed by

former students of Professor Bochner and dedicated to him on this occasion. The papers were received by June 1, 1969.

Contents PART I: LECTURES AT THE SYMPOSIUM

On the Group of Automorphisms of a Symplectic Manifold, by EUGENIO CALABI

I

On the Minimal Immersions of the Two-sphere in a Space of Constant Curvature, by SHIING-SHEN CHERN

27

Intersections of Cantor Sets and Transversality of Semigroups, by 41

HARRY FURSTENBERG

Kahlersche Mannigfaltigkeiten mit hyper-q-konvexem Rand, by HANS GRAUERT and OSWALD RIEMENSCHNEIDER

61

Iteration of Analytic Functions of Several Variables, by SAMUEL KARLIN and JAMES MCGREGOR

81

A Class of Positive-Definite Functions, by J. F. C. KINGMAN Local Noncommutative Analysis, by IRVING SEGAL

93 111

PART II: PAPERS ON PROBLEMS IN ANALYSIS

Linearization of the Product of Orthogonal Polynomials, by RICHARD ASKEY

131

Eisenstein Series on Tube Domains, by WALTER L. BAILY, Jr.

139

Laplace-Fourier Transformation, the Foundation for Quantum Information Theory and Linear Physics, by JOHN L. BARNES

157

An Integral Equation Related to the Schroedinger Equation with

an Application to Integration in Function Space, by R. H. CAMERON and D. A. STORVICK

175

A Lower Bound for the Smallest Eigenvalue of the Laplacian, by JEFF CHEEGER

195 ix

CONTENTS

X

The Integral Equation Method in Scattering Theory, by C. L. DOLPH

201

Group Algebra Bundles, by BERNARD R. GELBAUM

229

Quadratic Periods of Hyperelliptic Abelian Integrals, by R. C. GUNNING

239

The Existence of Complementary Series, by A. W. KNAPP and E. M. STEIN

249

Some Recent Developments in the Theory of Singular Perturbations, by P. A. LAGERSTROM

261

Sequential Convergence in Lattice Groups, by SOLOMON LEADER

273

A Group-theoretic Lattice-point Problem, by BURTON RANDOL

291

The Riemann Surface of Klein with 168 Automorphisms, by HARRY E. RAUCH and J. LEWITTES

297

Envelopes of Holomorphy of Domains in Complex Lie Groups, by 0. S. ROTHAUS

309

Automorphisms of Commutative Banach Algebras, by STEPHEN SCHEINBERG

319

Historical Notes on Analyticity as a Concept in Functional Analysis, by ANGUS E. TAYLOR

.Q/-Almost Automorphic Functions, by WILLIAM A. VEECH

325 345

Problems in Analysis A SYMPOSIUM IN HONOR OF SALOMON BOCHNER

On the Group of Automorphisms of a Symplectic Manifold EUGENIO CALABI1

1. Introduction Let X be a connected, differential manifold of 2n dimensions. A symplectic

structure on X is the geometrical structure induced by a differentiable exterior 2-form co defined on X, satisfying the following conditions:

(i) The form w is closed: dw = 0; (ii) It is everywhere of maximal rank; this means that the 2n-form con (nth exterior power of w) is everywhere different from zero, or equivalently, the skew-symmetric (2n) x (2n) matrix of coefficients of to, in terms of a basis for the cotangent space, is everywhere nonsingular.

A classical theorem, ordinarily attributed to Darboux, states that a 2n-dimensional symplectic manifold (i.e., a manifold with a symplectic structure) can be covered by a local, differentiable coordinate system {U; (x)} where (x) _ (x1, ..., x2n): U--* R2n, in terms of which the local representation of the structural form co becomes

(I.l)

u = dx' A dx2 + dx3 A dx4 +

+ dx2n -1 A dx2n

n

dx2j - 1 A dx2' ; 7=1

such a system of coordinates is called a canonical system. The purpose of this study is to describe the group G of automorphisms of a symplectic manifold, i.e., the group of all differentiable automorphisms of X which leave the structural 2-form co invariant, and the invariant subgroups of G. The group G can also be characterized as mapping canonical coordinate systems into canonical systems. ' The research reported here was supported in part by the National Science

Foundation.

1

EUGENIO CALABI

2

Two normal subgroups of G are distinguished immediately as follows: DEFINITION 1.1. Let (X, w) be a 2n-dimensional symplectic manifold and let G be the group of all symplectic transformations of X. If X is not compact, we denote by Go the subgroup of G consisting of all symplectic transformations of X that have compact support; that is to say, a sym-

plectic transformation g c G belongs to Go if and only if g equals the identity outside a compact region of X. DEFINITION 1.2.

Let (X, w) be a 2n-dimensional symplectic manifold and

let G be the group of all symplectic transformations of X. We denote by G0,0 the subgroup of G called the minigroup generated by the so-called locally supported transformations, defined as follows: a transformation g c G is called locally supported if there exists a canonical coordinate system {U; (x)} defined in a contractible domain U with compact closure, such that the support of g lies in U. The minigroup G0,0 and its corresponding Lie algebras are introduced here merely for expository convenience. In Section 3 it will be shown that

the commutator subgroup of the arc-component of the identity in Go coincides either with G0,0 or with a normal subgroup of codimension 1 in G0,0 (see Theorem 3.7, Section 3).

An elementary example of a locally supported transformation is the following: let {U; (x)} be a canonical coordinate system in X; let its range

V = (x)(U) - R2n contain the ball {Wli(ti)2 < Al for some A > 0; choose a real-valued differentiable function 0(r) of a real variable r >_ 0 with support contained in a closed segment [0, A] with A' < A. Then it is easily verifiable that the transformation f in R2n (with the 2-form w = n

=1

dt2i-1

A dt2'),

(t) -> (t') = f(t) with t12'-1 = t2'-1 cos 19(r) - t2' sin 19(r),

t'2' = t2j-1 sin i9(r) + t2j cos 79(r), 2n

(r=(t1)2;1 {=1

<J

n

is a symplectic transformation which equals the identity for r >_ A'. There-

fore its restriction to V defines via the coordinate map (x) a symplectic transformation in U that can be trivially extended by the identity map in X - U to a locally supported symplectic transformation in X. We shall state here the main results of this study in a preliminary form; more precise and stronger versions of these are repeated as theorems in the later sections.

AUTOMORPHISMS OF A SYMPLECTIC MANIFOLD STATEMENT 1.

3

The groups G, Go, and Go,0 are infinite dimensional Lie

groups in terms of the Whitney C°° topology (in the case of G) and the compactly supported C°° topology (in the case of Go and G0,0). STATEMENT 2.

The minigroup G0,0 is a closed, normal subgroup of Go;

the quotient group Go/G0.0 is locally isomorphic to the de Rham cohomology group Ho(X, R), the first cohomology group of X with real coefficients and compact support. STATEMENT 3. If X is not compact, the completion G1 0 of Go,o in the compact-open topology of presheaves (i.e., the group obtained by adjoining

to G0,o the infinite products of sequences of g. E Go,o, where for each compact K - X only finitely many of the g,, differ from the identity in K) is a closed, normal subgroup of G; the quotient group G/G1,o is locally isomorphic to the de Rham group H1(X, R), i.e., to the first cohomology group with closed support. STATEMENT 4. The group G1,0 has no connected, closed, normal subgroups other than the identity; in particular its commutator subgroup is an open subgroup. The same is true of G0,0, of course, if X is compact (in which case G0,0 = G1,0). On the other hand, if X is not compact, the commutator subgroup Go,o of G0,0 is normal in Go, has codimension equal

to 1, relative to G0,0, and has no connected, closed, normal subgroups other than the identity. The next two sections deal with the Lie group structure of the groups G and Go, emphasizing the relationship with the corresponding Lie algebras; the main tools used here are due to J. Moser [1]. In Section 3 we prove the four main statements just given at the Lie algebra level, and in Section 4 we expand the results at the group level, trying as far as we have succeeded to obtain results on the global structure of these groups and their closed, normal subgroups. 2. Infinite dimensional Lie groups of differentiable transformations

We shall summarize in this section some of the known facts about infinite dimensional Lie groups or pseudogroups of differentiable transformations, especially with regard to their relationships with the corresponding Lie algebras of tangent vector fields. We denote by G, H, and so forth, groups of differentiable transformations of a manifold; the corresponding pseudogroups of local transforma-

tions are denoted by G, H, and so forth; the associated Lie algebras of globally defined vector fields will be denoted by German capitals 03, S2, and so forth; the corresponding presheaves of local vector fields will be denoted by small German letters, such as g, Ij, etc.

4

EUGENIO CALABI

An infinite dimensional Lie group G of global, differentiable transformations in a manifold X (or alternately a pseudogroup G of local transformations) is an infinite dimensional, differential manifold (naturally we do not exclude from this notion finite dimensional manifolds), in the following sense: for any finite dimensional, differential manifold M a map 99: M-* G is defined to be differentiable, if and only if the corresponding evaluation map (p: M x X -. X, where q5(t, x) = (cp(t))(x), is differentiable (in the case of a pseudogroup, one requires also that the domain of definition of be open in M x X). For our present purposes, it seems irrelevant to fix any topology on G; we regard it rather as a compactoid, that is to say, we allow ourselves to consider the equivalence class of topologies that are compatible with the category of differentiable maps 9): M -. G just defined. The Lie algebra ($ associated to G is then the set of tangent vector fields (respectively pre-

sheaves of local vector fields) each defined as the equivalence class of differentiable paths in G originating at the identity with the obvious equivalence relation. The question that ordinarily arises here is how to recapture the arc component of the identity in G from the sheaf of germs g of Lie algebras determined by (53.

Any local cross section 0 in g (that is to say, each local vector field belonging to g) defines a one-parameter subpseudogroup of differentiable transformations by integrating the vector field to the corresponding autonomous flow. The composition of such one-parameter local flows defines a pseudogroup PG in the pseudogroup G associated to G. If G is finite dimensional, it is known classically that "G defines an open subpseudogroup of G. In the infinite dimensional case the same conclusion holds, if G acts real analytically, at least by considering first local cross sections of g and collating the resulting local transformations. In the C°° case for infinite dimensional Lie groups, several authors have shown examples to the effect that one-parameter subpseudogroups are not dense in a neighborhood of the identity, and the details of some of these examples give a strong indica-

tion that even the composition of the elements of such one-parameter systems may not fill out any neighborhood of the identity in G. Some authors have suggested a method based on affine connections; this method permits one to attach to each local vector field in (33 a differentiable path in

G originating at the identity but not constituting, in general, a oneparameter pseudogroup, so that the union of such paths fill out a neighborhood of the identity in G. This method is satisfactory in the case of differentiably acting pseudogroups definable in terms of a first-order, integrable differential system in the coordinate transformations but would require higher order connections for pseudogroups of a more complicated nature.

AUTOMORPHISMS OF A SYMPLECTIC MANIFOLD

5

The correspondence between presheaves of Lie algebras of local vector fields and pseudogroups of local differentiable maps can be established in a natural way only as a local one-to-one correspondence between differentiable paths. Thus the fundamental theorem on existence, uniqueness, and continuity with respect to initial data for ordinary differential equations establishes a one-to-one, locally biregular correspondence between differentiable, one-parameter families of local vector fields in g (i.e., paths in g) and local flows in X belonging to G (i.e., paths in G): any topological structure in the stalks of the sheaf of germs of one-parameter families of

vector fields in X belonging tog (provided that its definition includes minimal regularity conditions) yields a well defined topology on the stalks

of the corresponding sheaf of germs of G-flows. The corresponding topology on the sets of germs of elements of G is then obtained by passage to the quotient, assigning to each path in G (originating at the germ of the

identity) the germ of the terminal element c- G of the path. Thus one obtains from the sheaf of germs of Lie algebras g a sheaf G of germs of diffeomorphisms. The group G of global cross sections in G can be obtained without any difficulty if the manifold X is compact.

In the case where X is not compact, the corresponding sheaves g of germs of Lie algebras of vector fields and G of germs of pseudogroups of transformations lead to the corresponding global Lie algebras and groups in many ways, of which two are the most important: the ones with unrestricted support and the ones with compact support. They are obtained from the topologies of the corresponding stalks by "globalizing" them, in

the former case either by the compact-open extension of a uniform topology on the stalks (compact-open topology) or by a Whitney topology

and in the latter case by a uniform topology over uniformly compactly supported cross sections. It is worthwhile noting that the global groups obtained from the pathwise-connected sheaf of groups is not necessarily connected, as we know well in the case of the celebrated group of all differentiable, orientation-preserving automorphisms of spheres. 3. The Lie algebra of symplectic vector fields

We apply the concepts reviewed in the previous section to the case of the

Lie algebra associated with the group G or Go of automorphisms of a symplectic manifold (X, w). If q' is any exterior p-form and e is a vector field in a manifold, we denote

by e L p the interior product of e with p; this is the (p - 1)-form (identically zero if p = 0) whose value at a (p - 1)-vector 77, A A %_1 is given by (3.1) ( L q)( A ... A m,-1) = m( A ?,, A ... A

p-)

EUGENIO CALABI

6

The Lie derivative of c with respect to 6 is then obtained from the formula

[6,'] = d(e L p) + e L d9. It is well known that any differentiable path in the sheaf of germs of diffeomorphisms, originating at the germ of the identity at any x E X, leaves the form 9) invariant along the orbit of x, if and only if the path in the sheaf of germs of vector fields given by the differential of the given path yields a one-parameter family of germs of vector fields 6 satisfying [e, 9)] = 0. Thus the Lie algebra of the symplectic group G or Go is described locally by the vector fields 6 satisfying, since co is closed, d(6 L w) = 0.

(3.2)

We now introduce the Lie algebras corresponding to the groups G, Go, and G0,0 previously given in Definitions 1.1 and 1.2. PROPOSITION 3.1. The Lie algebras (l, (3o, and (3o,o corresponding, respectively, to the groups G, Go, and G0,0 are given by the global vector fields 6 on X satisfying (3.2) and, in addition,

(i) satisfying no further conditions in the case of 03; (ii) having compact support in the case of (BSc; (iii) generated, in the case of 030,,, by vector fields t=v, where each e, has compact support contained in a contractible domain U. admitting a canonical

coordinate system (x1,..., x2n), that is to say, such that there exists a function u,, with compact support in U. satisfying e,, L w = du,,,

or equivalently a

rrn

(3.3)

1

_

aua

ax2i ax21- 1 (au.

ax21- 1

ax21)

The proof of this proposition is clear. Since the form w is everywhere of maximal rank, the bundle map from the tangent to the cotangent vector bundle defined by assigning to r= the 1-form 6 L co is bijective. Thus, for instance, the Lie algebra (sS is isomorphic, as a vector space over the real numbers, to the set of all closed Pfaffian forms on X. This isomorphism induces a Lie algebra structure

on the set of all Pfaffian forms called the "Poisson bracket"; more precisely, for any 1-forma we denote by a# the uniquely defined tangent vector 6 such that 6 L w = a (at the bundle, sheaf, local, or global level); then the Poisson bracket of two Pfaffian forms a and P is defined to be {a, #} = [a#, P] L w.

AUTOMORPHISMS OF A SYMPLECTIC MANIFOLD

7

The Lie algebra of all C°° 1-forms (globally defined on X) with the Poisson bracket is isomorphic under the map {a -- a#} to the Lie algebra of all tangent vector fields; the vector subspace consisting of all closed 1-forms is a Lie subalgebra. It follows from (3.2) that this subalgebra is isomorphic to the algebra 03 of all vector fields 6 such that [e, co] = 0. The subalgebras No and G3o 0 of 03 are similarly characterized. We shall now define some vector subspaces of these Lie algebras that will, in fact, turn out to be ideals. DEFINITION 3.2. We denote by 03' the vector subspace of (53 consisting of all germs of vector fields e c 0 such that e L w is exact; that is to say,

0' consists of all the vector fields (du)# where u is an arbitrary differentiable

function on X. Similarly we denote by G%o the vector subspace of 030 consisting of all the vector fields (du)# where u is an arbitrary function on X with compact support. Finally, we denote by 05 the vector subspace of 03o consisting of the vector fields ;` = (du)#, where u is a function with compact support on X satisfying, in addition,

f uwn = 0.

(3.4)

X

REMARKS.

In the case of the unrestricted Lie algebra 0, one cannot

define the algebra that corresponds to (530 in the case of 030 i clearly, if X is

a compact manifold, then (530 = 03 and, in defining 03o = 0' the function u is determined by the vector field 6 = (du)# only up to an additive constant; therefore one can always choose u so as to satisfy (3.4); thus we have Oslo = 00 = (53'.

On the other hand, if X is not compact, the function u generating an element 6 E ', is uniquely determined by 6, since it is required by definition to have compact support; therefore the condition (3.4) becomes meaningfully restrictive, and indeed dim, (030/030) = 1.

Similarly, using the theorems of de Rham, for all symplectic manifolds

dimR (0/03') = dim, H1(X, R) = b1(X),

where b1(X) denotes the first Betti number of X with respect to real coefficients (and homology with compact support) while in the case of noncompact X

dim, (03/03o) = b°(X), where b? denotes the first Betti number for homology with closed support.

EUGENIO CALABI

8

We now show that the vector subspaces W', 0'0, and 030 are indeed Lie algebra ideals of 0 and (530i respectively. The commutator ideal of [CS, (s3] _ 32 while the commutators 00' = [(s30, 030] and (Sio =

PROPOSITION 3.3 (R. S. Palais).

is contained in [WO, moo]

($3',

are contained, respectively, in (Y0 and CsSo.

PROOF. Let 6 = a#, 1] _ fl# be arbitrary elements of (s3, that is to say, let a and P be closed 1-forms. Then {a, /} = [6, 77] L co is not only closed but indeed exact, for, with no assumption on and w, we have the following identities : [

(3.5)

,17] Lw = [6,,7 Lw] -,7 L [ ,w] = d(6 L (,7 L w)) + 6 L d(,, L w) - q L [;t, w]

d(6L("!Lw))+6L[,),w]-7L[6,w]-6L(,7Ldw)

where

w=6L(,7Lw); thus, under the assumptions dw = 0, [e, w] = [,7, w] = 0, we have [6,,?] L w = dw.

This shows that [($, 0] - (M' and, obviously, also [030, (530] 00', if 6 and (and hence w) have compact support. In order to show that 0533 c (530, we assume that = a#, ,7 = fl# belong to 03(,; if w is defined by (3.6), we have an elementary identity. (3.6)

ww" = (e L ("7 L w))w" = -n(6 L w) A (,7 L w) A w"-1

= -na A P A wn-1.

It follows that, if either of the two closed forms with compact support a or f3 is exact (say a = du, wherepu has compact support),

f ww" _ -n f d(up A wn 1) = 0. X

x

This completes the proof of this proposition. The next results show that the continuation of the commutator sequences yields no new ideals. For this purpose we recall the definition of the Lie algebra ($30,0 of the minigroup (Definition 1.2 and Proposition 3.1). Clearly, since 030,0 - (s3o, we have [Cs3o,o, 0o,o] - Wo,o 03", where (330,o is defined to be 030,0 n 03o; in other words (53'0,0 is generated by the vector

fields (du)#, where u is a function which has compact support in a contractible domain U admitting a canonical coordinate system and satisfies (3.4).

A UTOMORPHISMS OF A S YMPLECTIC MANIFOLD

9

LEMMA 3.4. The Lie algebras (530,0 and (330',0 coincide, respectively, with 033' and 03". The algebra (33' in the case of a noncompact manifold is generated by infinite sums with locally finite supports of elements of (330 = (` o,J.

Let 6 E (330. This means that 6 _ (du)#, where u is a differentiable function on X with compact support. Let {Uv}YE, be a locally finite, PROOF.

open cover of X by contractible open sets Uv, each of which admits a canonical coordinate system (Up, v(x)) and let be a differentiable partition of unity, where each (p. has compact support contained in U,; set uv = Pv u; then (du,)# E (350.0 and all but a finite number of them vanish, since u has compact support. This shows that (du)# E (330 can be expressed as a sum of elements in (330,0. Now suppose that 6 _ (du)# E (330:

since in this case X is assumed not to be compact and is connected, condi-

tion (3.4) is equivalent to the existence of a (2n - 1)-form 0 on X with compact support, such that d+o = uwn. Let 0v = cvcb be a decomposition of 0 by the partition of unity (ypv), and define uv by the condition

ui,wn = do, = d(p &)

Then clearly

(duv)# E (330 0i

showing that

(33"

is

generated additively

by (3300.

The final assertion about (33' now has to be proved only in the case where X is not compact. In this case an element 6 _ (du)# is defined by a function u on X whose support is unrestricted. Since X is connected and

not compact, the 2n-dimensional real cohomology group of X with unrestricted support is trivial; therefore the 2n-form uwn can be represented

by do, where 0 is a (2n - 1)-form. Decomposing 0 by means of the partition of unity p, as in the case of (3l0i we show that 6 can be expressed as an infinite sum of elements i=v E (330,0 with compact, locally finite supports; this completes the proof of the assertion. LEMMA 3.5.

The Lie algebra (330,0 coincides with its own commutator.

PROOF. Let 6 _ (du)# be a vector field in (530,0 defined from a function u with compact support K U where U is a contractible domain in which canonical coordinates (x) _ (xl, ..., x2n) can be defined. Expressing u in terms of these coordinates, the vector field 6 can be described by (3.3). In terms of these same coordinates, we have

CUn = n! dxl A ... A dx2n.

Since e E (330 0i the function u satisfies (3.4); in other words we have (3.7)

f

(x)(U)

u(x) dx' -

-

dx2n = 0.

EUGENIO CALABI

10

We shall show that there exists a set of 4k functions (vi, wi) (i = 1, 2, ..., 2n) each with compact support contained in U and satisfying, like the function u, equation (3.7) and satisfying 2n

du =

{dvi, dwi} i=1

or equivalently 2n

_ (du)# _

[(dvi)#, (dwi)#].

i=1

This means that u is determined from the other functions (using 3.3 and 3.5) by the formula 2n

(3.8)

u(x) _

n

i=1 i-1

avi awi avi aWi (8x21 8x21 - 1 - 8x21- 1 ax2j) .

Since the function u satisfies (3.4), there are, by virtue of Gauss' theorem, 2n functions z1, ..., z2n in X each with compact support in U such that u(x)

(3.9)

aai(x) i

As a matter of fact, the functions zi can be chosen so that their support is contained in an arbitrarily small, connected, open neighborhood of the

compact support K of the function u; in particular we let each of the functions zi have support contained in a compact K1 with K K1 - U. Now consider the functions vi = - (-1)iz(i - (-1)`) and wi = x; disregarding the bars, they satisfy (3.8). In fact we see that, for each i, avi 7=1

(ax2'

aw NO 8x21-1

-

avi 8x21-1

aw i ax2i)

az(i-1)h

- ax(i -1)i)

Consequently, summing both members of this equation, we have a solution of (3.8); however, the functions v, do not, in general, satisfy (3.4), nor do

wi = xi even have compact support. This can be remedied in a second stage, as we shall now proceed to do. Choose a non-negative valued, differentiable function h(x) which is identically equal to l in K1 and whose support is a compact domain K2 - U. We can then replace the functions wi = x' by h(x) xi; the new choice of wi will not affect equation (3.8) and the resulting functions h xi E (b,,o. Next, pick two more nonhave compact support, so that negative, nonzero, differentiable functions h' and h" with compact sup-

ports K3 and K, mutually disjunct and contained in U - K2. Since

AUTOMORPHISMS OF A SYMPLECTIC MANIFOLD

11

fu h'wn and fu h"wn are both positive, there exist real constants ci and c

(i = 1, 2, ..., 2n) such that

f (ui - cih')wn = f (h wi - c h")wn = 0 u

U

(1 < i < 2n).

Thus we can set

vi = vi - cih'; wi = h(x)x' - cl h"

(1 < i < 2n).

The replacement of the 4n functions vi and wi by vi and wi, respectively, does not affect the computation of the Poisson brackets {dvi, dwi}, so that (3.8) is still satisfied, while each of the functions vi and wi satisfies, as does the function u, equation (3.4). Hence (dvi)# and (dwo)# c 050 0 and 2.

(du)# _

i=1

[(dvi)#, (dwi)#]

This completes the proof of Lemma 3.5. Before stating the first main result on the Lie algebras (53 and W0, we need one more notion, that of the symplecting pairing. DEFINITION 3.6. In a symplectic manifold (X, w) we define the symplectic pairing to be the alternating, bilinear map of Ho(X, R) (first cohomology group with compact support and real coefficients) into the real

numbers, denoted by (u, v) as follows: let a, fl be closed 1-forms representing by means of de Rham's theorem the cohomology classes u, v, respectively. Then the symplectic pairing is given by (3.10)

(u, V)

=

fa A 9 A wn - 1

It is well known that the symplectic pairing is nonsingular in the case of the symplectic structure subordinate to a compact Kahler manifold, while it is identically zero in the case of the natural symplectic structure of the cotangent bundle of any differential manifold or, more generally, in

any (necessarily noncompact) symplectic manifold with n >- 2, where wn-1 is cohomologous to zero. THEOREM 3.7. Let (X, w) be a symplectic manifold, let 03 be the Lie algebra of all differentiable, symplectic, global vector fields, and let 03o be the Lie subalgebra of 00 consisting of vector fields with compact support.

Then the commutator algebra (352 = [(33, ($9] coincides with (SS' and (33/(332 is

isomorphic to H'(X, R); furthermore, if X is not compact, the commutator

EUGENIO

12

algebra O3 = [03o, (sIn] coincides with either W0 or (Si depending on whether

the symplectic pairing of Ho(X, R) is, respectively, nontrivial or identically zero; the third derived algebra (330 = [(s60, (§32] coincides in all cases with W',, The Lie algebras Wand (350 are each equal to their own commutator algebras. PROOF.

From Proposition 3.3 we have the inclusion relations

[(35, (Si] - (Si',

[(o, No] - (I3

[(In, (530] - 030-

In Lemma 3.5 it was established that [W0, (s5o] D (o. Since dimR ((So/(N") = 1, this means that (332 coincides with either (3 or (s30. It follows from equation (3.6) in the proof of Proposition 3.3 that W2 = 03o if and only if the symplectic pairing is nontrivial, and otherwise (s30 1 = (530 (this will be the case, in

particular, if Ho(X, R) = 0 or if w"-1 is cohomologous to zero). The corresponding conclusions in the case of (Si can be obtained by showing that (s3' is equal to its own commutator subalgebra. This can be done by the following argument. There exists an open cover { U,},,, of X by contractible canonical coordinate domains with the following additional property:

there exists a partition of the indexing system J into a finite collection N

of subsets, J = U J such that, for each µ, the open sets (Uy)VE,µ are pairwise

disjunct. Let (q9v) be a partition of unity subordinate to {Uv}vE, and let then (e,,)'=, is a finite partition of unity; any vector field E (Si'

e

VEJy

can be expressed as (du)# for some function u such that (after adding a suitable constant to u, if X is compact) uw" = do for some (2n - 1)-form

0, and let 0µ = e 0. Let u be defined by the equation

do and

consider the vector fields 6u = (du,,)#. By using the arguments of Lemma 3.5 in each Uv for v c J we see that, since the domains are pairwise disjunct, eu (µ = 1, 2, . ., N) can be expressed as a sum of commutators of 2n-pairs of vector fields with support in U Uv; the sum of these .

veJ,

N sums of commutators reproduces the given 6 E (35' represented as a finite sum of commutators of vector fields belonging to (Si'. This completes

the proof of the asserted commutator relations. This concludes the proof of the theorem. An immediate application of Theorem 3.7, to be used in the next section, is that we can obtain, using de Rham's theorem, a complete set of Abelian representations of the Lie algebras (Si and (s3o as follows. For any differentiable, compact, integral 1-cycle y and any 6 E (s3, we define (3.11)

JO(y) _ f 6 L w.

Similarly, if 6 E (s3o then, for any integral, locally finite, differentiable, integral 1-cycle y we define J(y)(ee) by the same formula. It is clear that the

AUTOMORPHISMS OF A SYMPLECTIC MANIFOLD

13

value of the integral depends only on the homology class of y: it establishes

the homomorphism of (ss (respectively Na) into hom (0H1(X, R), R) _ H'(X, R) [respectively into hom (H1(X, R), R) = Ho(X, R)]. We have seen that these two homomorphisms are surjective and are Lie algebra homomorphisms, if we regard the cohomology groups as Abelian Lie algebras. We can also construct geometrically a Lie algebra representation J' of 3o with kernel (83'0' onto the additive group R as follows. If X is

not compact, for each point x c X let y be an infinite, locally finite, differentiable 1-chain whose boundary 8y,, equals x. If 1; E

define the

function 0(6): X -> R by the equationfLw. (6)(x) =

(3.12)

YX

Since yx is defined modulo the group of 1-cycles and 6 E (sso, the function 0(6) is independent of the choice of the chain; in fact it coincides with the function u with compact support such that (du)# = 6. The representation of CSo into R with kernel N" then is obviously/,. by the integral

JIM

-

fX UCU,

y'(SW = fX WW.

We have at the moment no satisfactory, faithful geometrical representation of the non-Abelian nilpotent Lie algebra 03,,/Wo in the cases where [to, 3o] = Wo, i.e., where the symplectic pairing is not identically zero. The next result concerns the structure of Lie algebra ideals of 03 and Wo that give rise to normal subgroups. It is clear that any normal subgroup H of any (finite or infinite dimensional) Lie group G of differentiable transformations has as its Lie algebra an ideal in the Lie algebra of G. The converse, however, is false, unless the germs of the actions of G at each point are uniquely determined by their local power series expansion; this would happen, for instance, when G is finite dimensional or if its action is real analytic. In the case at hand, however, when G is the group of all automorphisms of a symplectic manifold (X, w), we know that there are nontrivial elements of G that act trivially in a nonempty open set, and nontrivially in another. In particular let U be an open subdomain of X

with U: 0, U $ X; then the Lie subalgebra Ou of (3 consisting of all vector fields 6 E 05 that vanish identically in U is locally transitive in X - U; clearly it is an ideal in 03; however, the group Gu of global transformations of X defined by (U is not a normal subgroup of X, since U is not invariant under G. This fact justifies the restrictive condition imposed below. DEFINITION 3.8. Let H be a Lie group of differentiable transformations of a manifold X and k) its Lie algebra of vector fields. A Lie algebra ideal

EUGENIO CALABI

14

5,)' - 52 is called stable, if it is invariant under the adjoint representation of

HinS. It is easy to verify that the local isomorphism classes of normal Lie subgroups of H are in one-to-one correspondence with the stable ideals of the Lie algebra of H. THEOREM 3.9. Let (X, w) be a 2n-dimensional symplectic manifold, and let G be the group of all global, symplectic automorphisms of (X, w); let Go be the subgroup of G consisting of the transformations with compact support and Go its third derived group. Then every nontrivial ideal of the Lie algebra W of G (or Wo of Go) that is stable under the adjoint action of G,, (indeed even of Go alone) contains the Lie algebra 050 of Go.

The proof of this theorem requires a lemma which is the analogue of the

lemma of Palais-Cerf in the case of symplectic transformations. It is formulated here in a form somewhat stronger than strictly necessary because it is of independent interest, and the proof given here is independent of that of the Palais-Cerf Lemma for the sake of completeness. LEMMA 3.10.

The identity component Go of the third derived group of Go

is strongly transitive on X in the following sense. Let g be a germ of a symplectic transformation at a point xo E X with target g(xo) = x,, let y be a simple, differentiable path from xa to xl parametrized by t (0 < t . R,

where d,, = S2(Go)/S2o(Go) is the universal covering group d,' of the identity component in Go. It follows trivially from deRham's theorem that the representations of d, G0, and do into H'(X, R), HI(X, R), and R, respectively, are surjective (except the third case, if applied to a compact manifold). This completes the proof of Proposition 4.3. It would be interesting to investigate how the representation X: Go - R must be modified to obtain a corresponding one of Go, i.e., what open subgroups of Go have representations Xl into quotient groups of R that are locally equivalent to X. This, however, has not been possible except in the case n = 1 of open surfaces X with an oriented area element w; in this case one knows from the theory of Teichmiiller-Ahlfors that the group Go is homotopically equivalent to the group of automorphisms of the fundamental group of X. In particular, the identity component of Go is simply connected, so that the representation Xl of this group onto R exists and coincides with X. It has been pointed out that, in the special case of the Cartesian plane R2 with co = dx A dy, the representation X of the group Go = Go of measure-preserving diffeomorphisms with compact support onto R had been found by Levi Civita. (I have been unable, however, to trace the reference.) This representation, intuitively, measures for each such diffeomorphism g, how much g "twists" the plane in the mean, in the clockwise sense.

We shall now give the partial modification of Proposition 4.3 that defines Abelian representations of the groups G and Go. DEFINITION 4.4. Let (X, co) be a symplectic manifold. We denote by A = A,, the additive subgroup of R made up of the numbers w[w] where w runs over all integral, compact, 2-dimensional homology

classes, w c 0H2(X, Z). If A is discrete, we say that (X, w) is a Hodgean manifold. THEOREM 4.5.

Let (X, co) be a Hodgean manifold; let oG denote the

open subgroup of the automorphism group G of (X, co) whose elements act as the identity on 0H1(X, Z); if X is not compact, let OG0 denote the open subgroup of Go whose elements act as the identity on H1(X, Z) (noncompact H1(X, R) [resp. Go -* H0 '(X, R)] homology). Then the representation X: G of Proposition 4.3 is locally equivalent to a representation OX: G -* hom (oH1(X, Z), R/A) [resp. oX: Go - hom (H1(X, Z), R/A)], where A is the additive subgroup of R consisting of the integral periods of to.

AUTOMORPHISMS OF A SYMPLECTIC MANIFOLD PROOF.

25

The two parallel statements (which are distinct if X is not

compact) concerning G and Go are entirely analogous in content and proof, so that we shall confine our statements mainly to the first case. Let g E OG and let y be any compact, 1-dimensional, singular differentiable cycle in X with integer coefficients, its homology class being denoted by [y] E OH1(X, Z). Then g(y) - y is a cycle which, by the definition of OG, bounds an integral 2-chain T, which is compact (since either y, in the first

case, or g, in the second case, has compact support). r is unique modulo the group OZ2(X, Z) of compact, integral 2-cycles, so that the integral J w is uniquely determined by g, w and y, modulo the discrete group A c R. In addition, if y is homologous to zero, i.e., y = aTO, then we can take T = g(TO) - TO and, since (o is invariant under g, J w = Jm%) _ io) w = 0, so that J ',.w depends only on the homology class of V. We thus obtain a map oX by passage to quotients, oX: oG - hom (OH1(X, Z), R/A), defined by (4.5)

oX(g)[Y] = f co;

0, = g(y) - Y.

Now suppose that g is (differentiably) homotopic to the identity under a homotopy h: I x X -->- X. This is a considerably weaker assumption than saying that g lies in the component of the identity in G (in the case of Go we assume the homotopy to be with compact support). Let h1 : X -->- X be the differentiable map defined by h1(x) = h(t, x), ho = identity, h1 = g; then he(y) - y is homotopic to the identity and the resulting chain homotopy h,k defines a 2-chain r = h*y for which we can define

a(g)([y]) = fY w

E R.

This map X has values in the additive group R and is a lift of the homo-

morphism x restricted to the open subgroup 1G - G of automorphisms that are homotopic to the identity ((identity component of G) c G c oG - G). In this case it is easy to see that by restricting X to the identity component of G we obtain the representation characterized in Proposition 4.3.

This completes the proof of the theorem. An analogous result in the case of open subgroups of Go has not been found, since the argument used in order to obtain the real-valued representation of its covering group Go requires in an essential way Lemma 4.2, i.e., that the homotopy h from the identity to g be within the group Go itself.

EUGENIO CALABI

26

5. Conclusions and problems

The most interesting conclusions of this paper in terms of further developments are probably related to Lemma 3.10, the analogue of the Palais-Cerf Lemma. It would be interesting to describe a set of sufficient conditions for a pseudogroup P of differentiable transformations to have the PalaisCerf property, i.e., that each germ of an element of r can be extended to a global transformation of X by an element of the group G of global crosssections of F over X (or, better, an element in the identity component of G). The corresponding question of the extension of a germ to global transformations belonging to a derived group G('`) of G appears to be much more complicated (see, for example, the case where r is the pseudogroup preserving a positive volume element con in an n-manifold). It is conjectured

that the existence of a syzygean resolution of the sheaf Or of germs of F-vector fields,

0 E 0j, E- (D-1 E_ (D-2 F ... ,

with each (D-' a fine sheaf, may be of significance here. In particular the length of such a resolution may describe the dimension of the primary

obstruction to the extension of a cross-section in 0r over a compact K c X to all of X. In the case of the symplectic group, Or is isomorphic by Proposition 3.1 to the sheaf of germs of closed 1-forms, so that the syzygean resolution can be described by

OF OrFd°F(R)F0 (,4P = sheaf of germs of p-forms), which has length 1. In the case where F consists of the germs of diffeomorphisms preserving a nowhere-vanishing n-form con, the corresponding resolution is

0 0 is a constant depending only on the system (35). With zo E D, zo 0, we integrate this inequality with respect to d6 d7j/I C - zoI over D. Remarking that 1

1

I(z - C)(zo - C)I (38)

f$

Iz - zoI z

A d < 2R, j

I

l

1

+

lzol'

MINIMAL IMMERSIONS OF THE TWO-SPHERE

33

the integration gives 27r

p(z) dx dy P(z) dzI + 4AR JJD Z'(Z - Z0)I JaD IZ'(Z - Zo)I

< 4R f

d6 d't

CC pp(1y)

JJD S'(S - Zo)I

or

p(z) dx dy < 4R ( P(z) I dz I .1 Dz(Z-z0)I JaDIz(z-zo)I We choose R so small that 27r - 4AR > 0. Then the integral at the

(2ir - 4A R) f f

I

left-hand side of the preceding equation is bounded as z0 -. 0. It follows I, and hence Iwa(S)Sis bounded as - 0. By (36) from (37) that exists. we see that lim z_o

LEMMA 2. Under the hypotheses of the theorem suppose wa = o(Iz' all r. Then wa = 0 in a neighborhood of z = 0. PROOF. Suppose, to the contrary, that p(zo) (37) by d6 d17 and integrate over D. This gives

2,, ff

'I d6 d'7 < 2R

(27T - 2AR) if

f

D

1),

0, Izol < R. We multiply

P(z)Iz - rI I dZI + 2AR

ff, P(z)I z

'I dx dy,

p(z)lz-rl dx dy < 2R f"D P(z)Iz-'I Idzl.

D

There exist two positive constants a and b independent of r such that the left-hand side of the above inequality is >alzol _r and the right-hand side is 0. Consideration of the terms in e in f)2 E, gives 4P A (D =0,

SHIINC-SHEN CHERN

36

so that I is of the form H(3)eµl .

It should be remarked that with this notation, Re

are the third

fundamental forms of the surface. The form of degree 6: (Hu3))291 6

(54) u

is independent of the choice of the frame field and, defined to be zero at a singular point, is well defined over the whole surface S2. With the local isothermal coordinate z it can be written as g(z) dz.s We shall show that g(z) is a holomorphic function. This follows immediately from the structure equations. In fact, consideration of the term in E2 in D2E1 gives

= 0.

{dk1 + ikj(2w12 - w34)} A

We can therefore put w34 = 2w12 + 01

(55)

where 01 is a real one-form. Substituting back, we obtain

(dk1 - ik101) A q, = 0 or (56)

01

id log k1i

mod q5.

Since D2E2 = 0, the terms involving e give, on using (55) and (56), (57)

Hv3)w,,,, - 3iw12Hµ3) = 0,

d)7,1, 3) +

mod p,

5 5 v 5 N,

v

from which it follows that (58)

d

(Hµ3')2 - 6i.12

(H,(, 3))2

0,

mod 99.

Using (32), we derive, as before, that g(z) is holomorphic. Thus the form (54) is an Abelian form of degree 6 and it must be zero. We can therefore define (59)

E3 = e5 + ie6

such that

DE2 = -k19pE1 - iw34E2 + k2gPE3.

MINIMAL IMMERSIONS OF THE TWO-SPHERE

37

Continuing in this way, we can define a local frame field eA at a regular point of S2, such that, if (60)

Es

=e2s-1 + ie2s,

s<m,

1

we have DEs = -ks_1-pEs_1 -iw2s-1,2sEs + ksg9Es+1,

(61)

1 <s<m, ko=km=0. The integer m is the smallest integer such that DEm is a linear combination of Em _ 1 and Em. Equations (61) will be called the Frenet-Boruvka Formulas for the minimal immersion of a two-sphere in X. Under the change (44), E, will be changed according to

Es -- E* = es`'Es

(62)

Generalizing (55) we set

2<s<m.

W2s-1,2s = S(X)12 + 9s-1,

(63)

Then the one-forms Bs_i, 2 < s < m, remain invariant under a change of the frame field in the tangent plane. Computing D2Es and using (41) and (64)

D2EE = 0,

s > 1,

we obtain

k2=-(c-K)0

(65) and

{dks + iks(Bs - Os-,)} A p = 0, (66)

dO,+i{ks -ks+1 -- (s+ 1)K}p A

=0,

00 = 0,

1 <s<m - 1.

These are the integrability conditions of (61), and they can be simplified. In fact, relative 'to the complex structure on S2 we use the operators a, a, and (67)

do = i (a - a).

From the fact that d log k, and Bs - O, from the first equation of (66),

are real one-forms, we obtain

0,- Bs_1 =dologks,

1 <s <m - 1,

6s = dc log (k1... ks),

I < s < m - 1.

which gives (68)

SHIING-SHEN CHERN

38

To a real-valued smooth function u let its Laplacian Au be defined by ddcu = 2 Aug) A q3.

(69)

Then, in view of (68), the second equation of (66) gives (70)

12A log (k1

ks) + k,

- ks+1 - z(s +

1)K = 0

or (71)

Z O log ks - k,'-, + 2ks - ks + 1 - iK =

0.

We summarize our results in the following theorem: THEOREM. Let S2 -* X be a minimal immersion of the two-sphere into a Riemannian manifold of constant curvature. There is an integer m such that the osculating spaces of order m (and dimension 2m) are parallel along the

surface. The singular points are isolated. At a regular point a complete system of local invariants is given by the quantities ks > 0, 1 < s < m - 1, which satisfy the conditions (71).

The simplest case is when the Gaussian curvature K is constant. If K = c, the tangent plane is parallel along the surface and the surface is totally geodesic. If K < c, it follows from (65) and then recursively from (71) that k3 are all constants. The same relations give (72)

K

2

m(m + 1)

c,

and all the ks can be expressed in terms of c and m. The surface is thus determined up to an isometry in the space. 6. The case when the ambient space is the N-sphere

Consider the special case, studied by Calabi, of the minimal immersion S2

(73)

SN'

where SN is the sphere of radius I in RN+1 (so that c = 1). The preceding theorem has the consequence that the surface must lie on an even-dimensional great sphere S2m. Without loss of generality we suppose N = 2m. In this case the tangent vectors of S2m can be realized in R2m+1, and the If x is a point vectors E., by vectors in the complex number space on the surface, we can write C2m+1.

(74)

dx =

E1 + -g1E1.

MINIMAL IMMERSIONS OF THE TWO-SPHERE

39

Moreover, the Levi-Civita connection is defined by orthogonal projection into the tangent spaces of S2m, and we have

DE1 = - px + dE1, (75)

DES = dE5,

1 < S.

We interpret ES as homogeneous coordinate vectors in the complex projective space P2m(C). Equation (61) with s = m shows that Em is an algebraic curve. It will be called the generating curve of the minimal surface. From the other equations of (61) we see that EmEm_1 is the line tangent to the generating curve at Em, etc., and EmEm _ 1 E1 is its osculating space of order m - 1. Since

(Er,Es)=0,

1 dim X dim A = dim A n A < 0,

if 2 dim A < dim X.

The first alternative leads to dim A = dim X whence A = X and the second alternative leads to dim A = 0. 2. Examples

For a first example we take X = the circle group K = R/Z. The integers act as endomorphisms on K, and if S is any multiplicative semigroup of integers, it may be thought of as acting on K. In [1] we obtained the following theorems: THEOREM 1. Let p and q be two integers > 1, with both not powers of the same integer, and denote by S, and S, the semigroups generated respectively by p and q. If A is a proper closed subset of K invariant under both S, and SQ then A is finite.

INTERSECTIONS OF CANTOR SETS

43

Theorem I may be regarded as a theorem in diophantine approximation. For if S is a semigroup containing both Sp and Sq and a is an irrational, then by Theorem 1, Sa = K. In particular we have THEOREM 2.

If S is a multiplicative semigroup of integers 0 not consisting

only of powers of a single integer and a is irrational, then for every e > 0 there exist s c S and r c Z with

l

a- s l< s

The condition that the semigroup not consist only of powers of a single integer is necessary for the conclusion. For if S - {an}, a > 0, and we take

a=

a-n2, then a is irrational and cannot be "well approxi-

a- In +

mated" by fractions with denominator in S. Theorem 1 is a sharpening of B. However it is also a consequence of B. For it can be shown without much difficulty (see [1]) that if A has the property in question, then the set of differences A - A is either all of K or it has 0 as an isolated point. The first alternative is impossible if dim A = 0, and the second implies that A is finite. Thus Theorem 1 would be a consequence of the following conjecture: Conjecture 1. The semigroups S, and Sq of Theorem 1 are transverse. Related to this is the following: Conjecture 2. For every irrational x e K,

dimS,x+dimSgx> 1. Of course when x is rational then Sx and Sqx are finite.

If we associate with a real number its expansion to the base p, then multiplying by p corresponds to shifting the expansion. A subset of K invariant under Sp corresponds to a family of one-sided infinite sequences with entries 0, . ., p - I and shift invariant. In particular, if we consider .

the set of all sequences whose entries belong to a proper subset of {0, . . ., p - 1}, these will determine a closed S,, invariant set. We will call such a set a Cantor p-set. Conjecture I implies that if p and q are not powers of the same integer, then each Cantorp-set is transverse to each Cantorq-set. We note that the Cantor p-set corresponding to the subset {al, . . ., a,} rz (0,. . ., p - 1} has Hausdorff dimension log r/log p (see [I ], p. 35). Moreover if A is any proper closed S, -invariant set, then A is contained in some

proper Cantor p"'-set for some m. It follows that dim A < dim K unless A = K. This verifies condition A(ii). An interesting consequence of Conjecture 2 is the following: Conjecture 2'. Suppose that p and q are not powers of the same integer. Then in the expansion of pn to the base pq every digit and every combination of digits occurs as soon as n is sufficiently large.

44

HARRY FURSTENBERG

For concreteness let us take p = 2 and q = 5 so that pq = 10. Suppose that some combination of digits b1, . . ., b, was missing from the expansions of infinitely many 21. We do not lose generality by supposing that b1 0, b, 54 0. Choose a subsequence of these n which increases very rapidly, say

{n,}. Form the number _

5-ni. If n{+1 - n; -* oo then S51~ is countable

and by Conjecture 2, dim Sloe = 1. Hence Sloe is dense in K. We consider the decimal expansion of C. First, to obtain the decimal expansion of 5-n, we take the expansion of 2n+ and move the decimal point over n; places: 5-n, = .00 ... Oai,a ... ai;.

a,1 is the expansion of 2n+ so that b1.. b, does not occur in the foregoing. We now assume the n, to increase so rapidly that none of the Here a11a

blocks of a k overlap. Then b1

b, does not occur in the decimal expansion

of e, and Sloe cannot be dense. Thus Conjecture 2' is a consequence of Conjecture 2.

To find other candidates for transversality we exhibit another pair of transformation semigroups for which hypothesis B is valid. Let r be an integer > 1. Denote by I. the ring of r-adic integers. I, is the completion of the integers Z in the non-Archimedean metric for which the distance between two numbers goes to 0 as their difference is divisible by high powers of r. Each element of I, has a unique expansion as an infinite series:

x=

W

w,,rn wn e {0, ..., r - 1}. If p divides some power of r then the

0

r

1

operation x - I p1 ([y] denoting the greatest integer in y) is uniformly continuous on the integers and extends to a transformation Dp on I,. One verifies that Dpq = D,DQ. We now take X = I,. We shall see presently that as a sequence space I, is endowed with a natural notion of dimension for which dim I. = log r. We now have THEOREM 3.

Let r = pq where p > 1 and q > I are not powers of the

same integer. Let S, and S9 denote, respectively, the semigroups of transformations of I. generated by Dp and D9. If A is a proper closed subset of I, invariant under both S' and S, then dim A = 0.

Theorem 3 is a non-Archimedean analogue of Theorem 1 but it is also a consequence of the latter. We shall sketch a proof. Let S2, denote the space

of all doubly infinite sequences w = (wa) with entries in {0, ..., r - 1}. T: 4, 4, denotes the shift operation. We have a map T + : 92, - I, with m -1 warn. Suppose (w) _ : 0, -> K with +(w) _ warn as well as 0 now that A is a closed subset of I, invariant under DP and D.. Then it is

-

INTERSECTIONS OF CANTOR SETS

45

invariant under Dpq = D, which is simply the shift operation on one-sided sequences. Let A be the set of sequences w in S2, satisfying 7+(Tmw) E A for each m. It can be shown that A is infinite if dim A > 0. The operations Dp and D, may be extended to 72, and one finds that they leave A invariant. For

_

Dp

warn/

0

Wnrn 0

with (3)

wn =

1

P

+ qwn + 1

(mod r),

and (3) defines an operation on O. Now take B the foregoing and the shift invariance of A that if

It follows from tnrn c B so is

6nr n,

where

6n = [L--iI + qen (mod r). But this means that if x E B, so is qx. Similarly B is invariant under x --> px.

By Theorem 1, B is therefore either finite or all of K. One sees that this implies that A is either finite or all of 1 r. But the former implies that dim A = 0 and the latter implies that A = J. In analogy with the Archimedean case we formulate the following: Conjecture 3. The operations Dp and DQ are transverse on I,. For a final example we take X = Ip where p is a prime. Let it be a p-adic integer divisible by p but not by p2. In addition to the standard p-adic

representation, each x c I, has a representation x = wnlrn with W. E 0 {0, ..., p - 1}. The shift operation on sequences now induces a trans-

formation D. on I , with Dnx = 7r-1(x - h(x)) where h(x) E {0, ..., p - 1} and h(x) = x (mod p). Now let ir, and 7r2 be two such p-adic numbers. Conjecture 4. Dn, and D,2 are transverse on Ip provided vr,7ri is not a root of unity. Of course 7r27rl ' is, in any case, a p-adic unit. In case 7r1 = p and

7T2 = pq where q is relatively prime to p, then it may be shown that Conjecture 4 is a consequence of Conjecture 3. We omit the proof. 3. Strong transversality

The present investigation is devoted to a result which lends further plausibility to the conjectures of the preceding section. To formulate this we shall identify subsets of the circle K with subsets of the real line. A

closed subset A - [0, 1] will be called a p-set if pA - A u (A + 1) u

HARRY FURSTENBERG

46

(A + 2) v

v (A + p - 1). This corresponds to an S, -invariant subset of K. Now if A and B are subsets of the line, it is easy to see that

dimAnB=dimA x Bn10, where 10 represents the diagonal line x = y of the plane. Similarly if l is an

arbitrary line of the plane, 1: y = ux + t, then dim (uA + t) n B

dimA x Bnl. The remainder of this paper is devoted to a proof of the following property of p-sets. THEOREM 4.

Let A be a p-set and B a q-set where p and q are not powers

of the same integer, and let C = A x B. Let S be an arbitrary number. If there is some line I with positive, finite slope which intersects C in a set of dimension > S, then for almost every u > 0 (in the sense of Lebesgue measure) there is a line of slope u intersecting C in a set of dimension > S.

We can reformulate this using a variant of transversality. We say that two closed subsets A, B of the line are strongly transverse if every translate

A + t of A is transverse to B. We remark that for arbitrary closed A, B, almost every translate of A is transverse to B. From Theorem 4 we deduce the following theorem: THEOREM 5. Let A and B be as in Theorem 4. Assume that for a set of u of positive Lebesgue measure the dilation uA of A is strongly transverse to B. Then A and B are strongly transverse.

For, if not, there is a line with positive slope that intersects C in a set of dimension > dim A + dim B - I (resp. 0). Then for almost all directions this will be the case and dim (uA + t) n B > dim A + dim B - I (resp. 0), and since dim A = dim uA, uA and B are not strongly transverse. In this connection let us mention a further conjecture: Conjecture 5. For arbitrary compact subsets A, B of the line, almost all dilations of A are strongly transverse to B. The reason for expecting almost all dilations of a set to be well behaved with respect to another set is the following result (for which I am grateful to J.-P. Kahane): THEOREM 6. If A and B are arbitrary compact subsets of the line, for almost all dilations uA of A we have

dim A + dim B, if this is < 1, dim (B + uA) = 1' otherwise.

Here B + uA denotes the set of sums ux + y, x E A, y c B. The foregoing is a special case of a more general result asserting that if C is a compact set in the plane with dim C = y, y < 1, then in almost every

INTERSECTIONS OF CANTOR SETS

47

direction C projects onto a linear set of dimension y. This is easily proven using the characterisation of dimension in terms of capacities (see [2]). Clearly, Conjecture 5, together with the theorem we are about to prove, imply the validity of Conjecture 1. In the non-Archimedean case a similar result is true. Specifically consider subsets A, B of 4. Ip is contained in the field of all p-adic numbers and we

may speak of lines in the p-adic plane intersecting A x B. We shall be interested in lines of the form y = ux + t where the "slope" u is a p-adic unit. THEOREM 7.

Let A be a closed subset of I, invariant under D,1, B a

closed subset invariant under D,,2, and assume 7r27r1 1 is not a root of unity. There is a subgroup U of finite index in the group of p-adic units such that

if a single line y = u°x + to intersects A x B in a set of dimension > S, where u° E U, then for almost every u e U (with respect to Haar measure on U) there is a line with slope u intersecting A x B in a set of dimension > S. The proofs of Theorems 4 and 7 are very similar and we shall confine our

attention to the former. 4. Trees and dimension

Let A be a finite set with r elements. Denote by Q, the product A x A x

Ax

endowed with the usual topology that renders it a compact

Hausdorff space. We shall denote by A* the free semigroup generated by W

A: A* _ u An where A° consists of the empty word which we denote 0

by 1, and where multiplication is by juxtaposition. If a E An we shall write 1(a) = n. We shall use the term factor to denote left factor: a is a factor of T if T = ap for some p. DEFINITION 1.

(I)

A subset A - A* is called a tree if

I E0.

(ii) If a E A then every factor of a c A. (iii) If a E A then some as c A for a E A. The following example will justify the terminology:

as a

1

/ \

aaa

--- aab

ab - aba

b

ba

baa bab

HARRY FURSTENBERG

48

One can set up a correspondence between closed subsets of S2A and trees in A*. If A c 4A let A* denote the set of all initial segments (including 1) occurring in sequences of A. Clearly A* is a tree. Conversely if A is a tree, we associate to it the set of all infinite sequences all of whose initial seg-

ments are words in A. It is easily seen that this is a one-to-one correspondence. (If, however, A is not required to be closed, then the set associated to A* is the closure of A.)

A section of a tree A is a finite subset II - A satisfying the following conditions: DEFINITION 2.

(i) With finitely many exceptions, each element of A has a factor in II. (ii) If p is a factor of a and both p, a c H then p = a. If a,, ala2i ..., a,a2, ..., an is an increasing sequence of elements of A, then one of these must eventually possess a factor in II by (i), and by (ii) it follows that the foregoing sequence intersects II in exactly one element. We see from this that a section of the tree A* corresponds to an irredundant open covering of the compact set A. If II is a section we denote by 1(II) the minimum of 1(a), a c H. DEFINITION 3. The dimension of a tree A is defined as the g.l.b. of the set of A with the property: 3 sections II - A with 1(II) arbitrarily large and

e-Aua> < 1. QEII

If A is a closed subset of S2 we set dim A = dim A*. The connection with Hausdorff dimension is given in the following: LEMMA 1.

... , r -

If A = {O,

A=

1l 111

1

Wnr-n:

1) and A is a closed subset of 52,,, set

w= (wn) e A r C [0, 1]. )

Then

dim A = log r x Hausdorff dim (A).

The proof is straightforward. One notices as a consequence that the dimension of a set depends only on the "geometry" of the associated tree and not upon how one labels its vertices. Thus if in A, each a c A is followed by exactly m successors aa;, then dim A = log m. We introduce into the space of trees a compact Hausdorff topology by setting

D(A'A')

m+ 1 ifAnAm=A'nAm but AnAin+1# A'nAm

If A is a tree and if p e A, then {al pa c- A} is a tree which we denote All.

INTERSECTIONS OF CANTOR SETS

49

DEFINITION 4. If A is a tree we denote by -9(0) the closure of the set {0°, p c 0} in the space of trees. The trees of -9(0) are called derived trees

of A.

Just as trees correspond to sets in S2A we can define objects corresponding to measures on SZA as follows:

A real-valued function 0 on A* is a T-function if

DEFINITION 5.

(i) 0 < 0(a) < 1, (ii) O(1) = 1,

(iii) 0(a) _

O((Ta). aeA

The set of T-functions will be denoted by TA.

The support 101 of a T-function is always a tree. Conversely every tree is the support of some T-function. If µ is a probability measure on 1 A and we denote by pc& the set of sequences that begin with p, then the function O(p) = P(PQA) clearly represents a T-function. The space TA is a compact metric space where we set

li(O, 0') =

2-")JO(a) - 0'(a)I. aeA'

The convergence of this series follows from the fact that I O(a) = 1. 1(a) = n

REMARK. Quite generally, if II is a section of 101 we have

aerl

0(a) = 1.

This is a special case of a still more general assertion: if p is a factor of an element of II then O(a) = O(p), the sum being extended over those a E II of which p is a factor. To see this, note that it is obviously true if p c n. If p 0 rl then all its successors pa, a E A are factors of elements of II and it suffices to establish the assertion for these. But then our assertion follows

by induction on max l(a) - l(p). aer!

If 0 is a T-function, then for each p e 101 we may set __

O(Pa)

B°(a)

B(P)

O° is again a T-function and 1O°1 = DEFINITION 6.

10 1 0.

If 0 is a T-function we denote by _q(O) the closure in

TA of the set {Oa: O(a) > 0}. The functions of p1(0) are derived T-functions of O.

HARRY FURSTENBERG

50

We can define a notion of dimension for T-functions: DEFINITION 7.

If 0 is a T-function we set 0(a) log

dim 0 =

lim. inf. LT a section of 101

1

0(a)

0011

aerI

Bat/(a ()l )

(i) dim 0 < dim 101. (ii) If A is a free, dim A = sup {dim 0: 101 - A}. LEMMA 2.

PROOF.

If e`t(a) < 1 0011

then

0(,)e-Al(a)+

log 1/0(a) < 1

ae11

whence by Jensen's inequality (using

0(a) = 1):

0(a) log (e-Tt(0)+ log 1(0(0)) < 0

aen or

- A : 0(0)1(0) +

0(a) log 0(a) < 0, a

a

so that A > dim 0, and hence dim 101 >_ dim 0. This proves (i). To prove (ii) we need only prove that for A < dim A, 3 0 with 101 c A and dim 0 >_ A. We use [2, Theoreme II, p. 27], which asserts that if E is

a compact subset of the line with Hausdorff dimension >P, then there exists a probability measure it on E with µ([a, b] < C lb - all'). If A = A* then A - QA corresponds to a subset of [0, 11 obtained by viewing QA as expansions of real numbers to the base r (= card. A). This set has Haus-

dorff dimension > A/log r. Choose g = A/log r and let µ be a measure satisfying the foregoing inequality. We lift µ to a measure µ on A and we find that µ(pL2A) = µ(I) where I is an interval of length r-1(0). Thus if 0(P) = µ(Pie), B(P) < Cr-au°) or log

0p) > -log C + /1(p) log r = -log C + Al(p)

so that I

lim. inf.

ocn

0(P) log

M

°011

0(p) >

0(01(0

Here we use the fact that 0(01(0 > 1(II). °en

This completes the proof of Lemma 2.

A.

INTERSECTIONS OF CANTOR SETS

51

5. Markov processes on TA

The space of T-functions, TA, is endowed with a natural set of transition probabilities which induce a family of Markov processes on TA. If 0 E TA, the numbers 0(a), a c A constitute a probability distribution on A, and we may define transition probabilities on TA by assigning probability 0(a) to the transition 0 - 0a. We call these the canonical transition probabilities on TA. Fixing an initial distribution for a TA-valued variable z0, there is determined a unique process {zn} in accordance with these transition probabilities. Inasmuch as 0(a) is a continuous function of 0, it follows

that the induced Markov operator transforms continuous functions to continuous functions. There will exist stationary measures; these constitute a compact convex set, and the extremals of this set determine ergodic stationary processes {zn}. DEFINITION 8. A C-process is a stationary Markov process on TA with the canonical transition probabilities.

The main step in the proof of Theorem 4 is provided by the following theorem. THEOREM 8.

If A is a tree with dim A > 8, then there exists an ergodic

C-process {zn} such that with probability 1, 1 znI is contained in trees of-9(0), and such that with probability 1, dim Iz,1 > S.

The purpose of this theorem is to exhibit an abundance of trees of dimension > 8 once we have a single tree with dimension > S. The following shows how a single T-function 0o may generate a C-process: DEFINITION 9.

A measure µ on TA is compatible with a T-function 0o if

there exists a sequence Nk - oo such that for each continuous function fon TA, (4)

lim f .f(0) dµ(0) = k-1 Nk + 1 TA

0o(T)f(0o) n=0 zcAnnIBol

LEMMA 3. If µ is compatible with 0 then µ has its support on -9(0). In addition µ is a stationary measure for the canonical transition probabilities. PROOF.

The first statement is obvious and the second involves a

straightforward verification. If p(O, 0') denotes the canonical transition probabilities, then p(0,, 00) = 0°(Ta). 00(-)

HARRY FURSTENBERG

52

To show that µ is stationary we must show that J

(:, P(0, 0')f(0')) dµ(0) = f f(6) dµ(6)

But lim

Nk + 1

IG n=o

0o(T) > P(0o, 00f(0) s

aeA Nk

= lim

Nk +

1

+ Nk

1

6o(Ta)f(00 ) a

Nk+1

= lim

0o(r)f(0o) =

ff(0) dµ(6)

T

If µ is a stationary measure on TA we set

DEFINITION 10.

p(0, 0') log

E(µ) = fTA LEMMA 4.

n=1

p(0 0') dµ(0)

If µ is compatible with 00 then E(µ) >_ dim 00.

PROOF.

I

E(µ) = lim

k-w Nk 1+ 1

n

= lim k-. Nk + 1

i

00(') 60(la) logg 0o(r)

a

au(ra){log Oo(T) - log 00(ra)}

1

n

a

u=)

+1

m

kl ao Nk + 1

6o(ra)

60(T) log

0 (T).

Now let IIk denote the section j00j n ANk+1. Then the last expression in the proof just given is the same as 0o(r) log lim

2EIIk

1

60(r) >

00(T)1(r)

dim 00,

SEIIk

which proves the lemma. LEMMA 5. For any T -Junction 00 there exists an ergodic stationary measure µ for the canonical transition probabilities with support onJi(00) and with E(µ) > dim 00. PROOF.

To begin with, there exist measures µ' compatible with 00 since

we can choose {Nk} so that (4) converges for every function in ''(TA). µ' is supported on the derived set of 00. The set of stationary measures

53

INTERSECTIONS OF CANTOR SETS

with support in (Oo) forms a compact convex set, and it follows that µ' can be expressed as an integral of ergodic measures with support in 1(Oo): µ' = f µw dv(w). It follows that

E(FL') = fE(w) dv(w);

therefore some E(µ,,) > E(µ') > dim Oo. If µ is an ergodic stationary measure, then for almost all 0 with respect to µ, dim 0 > E(µ). LEMMA 6.

Let {zn} denote the TA-valued variables of the stationary

PROOF.

Markov process determined by p. Define the function H on TA:

O(a) log (

H(O) = aEA

a)

H(O) is a bounded continuous function. By the ergodic theorem we have, with probability 1, H(zo) + H(z1) + ... + H(zn_ 1)

fH(o) dµ(O) =

Now let A be any number < E(µ), and denote by S(n, A) the subset of the underlying probability space for which some average

H(zo) + H(zl) +

+ H(zn_ 1)
_ n. We have P(S(n, A)) -* 0 as n -* oo. Hence the conditional oo for almost all zo. Now given probability P(S(n, A)Izo) -* 0 as n zo = 0, the conditional probability that z1 = Oa', z2 = Oala2, ..., zn = eala2. a is O(a1a2 an). Let II be a section of 101 with 1(fl) > n. Given zo = 0, we can assert that there will be a value of m for which zm = Oal .. am and a = a1 am E lI. We will then have

G(a) = G(al...am) =

H(O) + H(Oal) + ... + H(Oal..am_ 1) > m

Oal...0al...am corresponds to a point of S(n, A). Since unless the sequence the conditional probability of obtaining this sequence is 0 ( a ,---am), we conclude that the sum of the 0(a) for which a c 11, and G(a) < A, is less than P(S(n, A)Izo = 0). Hence

O(a)G(a)l(a) > A{l - P(S(n, A)Izo = 0)}

.(5) It

O(a)1(a).

HARRY FURSTENBERG

54

We proceed to evaluate the left side of (5). First a preliminary observation. Let pla denote that p is a proper factor of a; pI II denotes that p is a proper factor of an element in II. By our remark following Definition 5 we have (6)

°la,aen

0(a) = 0(P)

Now by (6),

0(a)G(a)1(a) _ H(0°)0(a) n

oen ala

H(01)0(p). °1>z

Also aln

H(0°)0(P) _

O(pa) log °Irl aeA

pln

aeA

O(pa)

0(Pa) log

0(pa) - B(P) log

0(P)

0(a) log 0Ja). UCII

Hence

0(a) log 0(0') > n

0(a)1( a)

A{ I - P (S(n, A) zo = 0)} -> A

as n

oo a.e.

This proves the lemma. Taking stock of what has occurred in the foregoing lemma we see that we have arrived at a proof of Theorem 8.

6. Proof of Theorem 4 Let p and q be two positive integers not both powers of the same integer and let A - [0, 1 ], B - [0, 1 ] be respectively a p-set and a q-set. We shall

assume q > p. If we set g(x) = [px] and h(y) = [qx], then A is closed under x -* px - g(x), and B is closed under y --* qy - h(y). Let us define two transformations of the unit square:

1(x, Y) = (Px - g(x), Y) 12(x, Y) = (px - g(x), qy - h(y)). Then A x B is invariant under both transformations t1 and (D2. Each (D; transforms a line into a finite number of line segments, and if I is a line

(7)

with slope u, then each line of X1(1) has slope u/p and each line of 4 2(1) has slope qu/p. Now suppose that I is a line that intersects A x B in a set of dimension > S. The same will be true of at least one of the lines of X1(1) and of one

INTERSECTIONS OF CANTOR SETS

55

of the lines of X2(1). Proceeding in this manner we find infinitely many lines with the property that they intersect A x B in a set of dimension > S.

If the first of these lines had slope u, we will find all the slopes u' = qmu/p", n > m, represented. Note that the set of u' of this form is dense in (0, co) precisely when p and q are not powers of the same integer. For that is equivalent to log q/log p being irrational, which implies that log U, =

logp(mlogq _ n + logu) log p

log p

is dense in the reals. Thus under the hypotheses of the theorem, we will obtain a dense set of slopes with the desired property. The machinery of the preceding section will be invoked to extend the conclusion to almost all u'. Note, however, that the present argument shows that it suffices to establish the assertion of the theorem for slopes in some finite interval. We shall do so for 1 5 u < q. We now introduce a number of spaces that will play a role in proving Theorem 4. We denote the subset A x B of the unit square in the plane by C. L will denote the set of lines Y = uX + t with 1 dim D(10), and let fi be a section of D(10) with e-01(o) < 1. If oer

(x, y) E 10 n C then y(x, y, 10) begins with some sequence of II. If (x', y'), (x", y") are two points corresponding to the same element a of 11 then, by

HARRY FURSTENBERG

58

Lemma 9, Ix' - x"I < p-ua). Hence II determines a covering of 10 n C by intervals of respective length p-`(°). But we have {p-uQ)}$Ilog P < 1, 0

so that

log p x dim (10 n C). It follows that dim D(10) > log p x

dim (10 n C).

Conversely suppose a > dim (10 n C) and that {J;} is a covering of jJ I < H -1p-". Here JJJ denotes the length of the interval J. Define n, by p-nt-1 < jJjI < p-nt. Next take the set of all segments in 10 n C with

D(10) of length n which occur in y(x, y, 10) where (x, y) is a point of J;. Denote this set H.. By Lemma 10, there are at most H elements in II;. Now U Fl, = H' is a finite set which clearly contains a section of D(10). We have He-a logPn, =

e-alogP1(a) _
dim D(10). This proves the lemma.

Hp"

1J11a < 1,

One can also introduce the notion of T-functions in the context of L-trees. DEFINITION 12.

A TL-function is a function 0 on U L" satisfying 1

(i) 0 has its support in some D(10),

(ii)0 8 (with probability 1).

INTERSECTIONS OF CANTOR SETS

59

We are now in a position to prove Theorem 4. Suppose dim (1o n C) > S. By Lemma 11, dim D(10) > 8/log p. Now apply Theorem 9 and let {zn} denote the stationary process provided by the theorem. For any T-function, denote by 1(0) the line determined by 10 c D(1(0)). Consider now the stationary, line-valued process 1(zn). Since with probability 1, dim Iznl > 8/log p, a fortiori, dim D(1(zn)) > 8/log p. By Lemma 11, we have, with probability 1, dim (1(zn) n C) > S. We now determine the distribution of the slopes of 1(z,,). From the form of the canonical transition probabilities we see that if un is the slope of 1(zn), the slope of 1(zn+ 1) is given by

gun if un < p,

un+1 =

P

IUn ifun> P-

P

Let v denote the distribution of log q Then v is a measure on the unit interval invariant under the transformation t -- t -

log

q (modulo 1). But

this is a version of the irrational rotation of the circle and Lebesgue measure is the unique invariant measure. It follows that the distribution

of u is a measure equivalent to Lebesgue measure. The assertion of Theorem 4 is now a consequence of the Fubini theorem. HEBREW UNIVERSITY JERUSALEM

REFERENCES 1. FURSTENBERG, H., "Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation," Mathematical Systems Theory, 1 (1967), 1-49. 2. KAHANE, J: P., and R. SALEM, Ensembles Parfaits et Series Trigonometriques. Paris: Hermann & Cie., 1963.

Kahlersche Mannigfaltigkeiten mit hyper-q-konvexem Rand HANS GRAUERT UND OSWALD RIEMENSCHNEIDER Einleitung

1953 zeigte K. Kodaira [3], daB die Kohomologiegruppen Hs(X, F) fur s < n verschwinden, wenn X eine n-dimensionale kompakte kahlersche Mannigfaltigkeit and Fein negatives komplex-analytisches Geradenbundel auf Xist. Hierbei bezeichnet Fwie ublich die Garbe der Keime von lokalen holomorphen Schnitten in F. Kodaira verwandte beim Beweis dieses Ergebnisses neben der harmonischen Analysis wesentlich eine Ungleichung,

die zuerst von S. Bochner angegeben wurde (man vgl. [2]). Bis heute ist es nicht gelungen, diesen Satz fur vollstandige projektiv-algebraische Mannigfaltigkeiten auf algebraischem Wege herzuleiten. Man gelangte jedoch zu mehreren Verscharfungen and Verallgemeinerungen. Von Y. Akizuki and S. Nakano wurde bewiesen [1], daB in der obigen Situation sogar Hs(X, F ®S2') = 0 ist fur r + s < n, wenn S2' die Garbe der Keime von lokalen holomorphen r-Formen bezeichnet. Nakano [6] gewann danach eine Aussage fiber negative Vektorraumbundel V auf kompakten kahlerschen Mannigfaltigkeiten X: Es gilt stets Hs(X, V) = 0 fur s < n. SchlieBlich iibertrug E. Vesentini [8] 1959 einige Resultate auf semi-negative Geradenbundel. Da kompakte komplexe Mannigfaltigkeiten mit solchen Geradenbiindeln nicht mehr notwendig kahlersch sind, stellte sich die Frage, ob die "Vanishing"-Aussagen wesentlich an der KahlerStruktur hangen. Diese Frage wurde, ebenfalls von Vesentini, im positiven Sinne beantwortet. Ein Geradenbundel F heiBt negativ, wenn es eine offene, relativ-kompakt

in F gelegene Umgebung der Nullschnittflache gibt, die in jedem Rand-

punkt streng pseudokonvex ist. Ein Vektorraumbundel mit der entsprechenden Eigenschaft nennen wir schwach negativ. Es hat sich gezeigt, daB negative Vektorraumbundel (im Sinne der Definition von Nakano) 61

GRAUERT AND RIEMENSCHNEIDER

62

auch schwach negativ sind. Das Umgekehrte gilt jedoch nicht. Es gibt sogar, wie wir zeigen werden, schwach negative Vektorraumbundel V fiber dem p2 mit H1(P2, V) 0 0. Wir zeigen dies auch fur beliebige projektivalgebraische Mannigfaltigkeiten. In der vorliegenden Arbeit sollen die Satze von Kodaira, Akizuki and Nakano weiter verallgemeinert werden. Zu diesem Zwecke mussen Ergeb-

nisse von J. Kohn [4] wesentlich herangezogen werden. Wir betrachten eine n-dimensionale komplexe Mannigfaltigkeit X, die relativ-kompakter offener Teilbereich einer komplexen Mannigfaltigkeit X ist. Auf X sei eine kahlersche Metrik gegeben. Ferner sei der Rand OX von X glatt and beliebig oft differenzierbar. Ist 8X streng pseudokonvex (wir sagen auch 1-konvex), so zeigen wir, daB Hs(X, 0) fur s < n and HS(X, S2") far s > 0 verschwindet. Dabei ist 0 = 00 die Strukturgarbe von X and Hs(X, 0) die s-te Kohomologiegruppe mit kompaktem Trager. Ersetzt man die

1-Konvexitat durch die schwachere q-Konvexitat, q > 1, so sind im allgemeinen Hs(X, 0) fur s < n - q and Hs(X, Stn) fur s > q von Null verschieden, obgleich man zunachst das Gegenteil vermuten mochte. Ein Beispiel hierzu kann aus einem schwach negativen Vektorraumbundel uber dem P2 gewonnen werden, fur das nicht das Vanishing-Theorem

von Nakano gilt. Wir mussen daher die q-Konvexitat fur q > 1 durch eine starkere Forderung ersetzen, die wir als Hyper-q-Konvexitat bezeichnen.

In Teil

1

der Arbeit werden bekannte Ergebnisse zusammengestellt,

die spater benotigt werden. Der Teil 2 enthalt den Beweis der Hauptaussage,

and in Teil 3 wird gezeigt, wie dieser Beweis in Beziehung steht zu den Vanishing-Theoremen von Kodaira, Akizuki and Nakano. Ferner werden dort Gegenbeispiele gebracht. 1. Ergebnisse von J. Kohn

Wir stellen zunachst einige bekannte Tatsachen aus der Theorie der kahlerschen Mannigfaltigkeiten zusammen. Es sei X eine dberall n-dimensionale komplexe Mannigfaltigkeit; es bezeichne A' den C-Vektorraum der (beliebig oft) differenzierbaren komplexen Differentialformen vom Grade I auf X and den Vektorraum der Formen vom Typ (r, s). Es A.

gilt also A' = 0 A". Die totale Ableitung d: A'--* A'+' 1513t sich in '+s=I

der Form d = d' + d" schreiben, wobei d':

die totale

Ableitung nach den Variablen z1, . . ., z,, and d": A- -->- A'.s+1 die totale

Ableitung nach den 2k, ..., z,, bezeichnet. Es gilt: d2 = d'2 = d"2 = 0, d'd" + d"d' = 0, do = dpi, d'(p = d"p and d(9 A 0) = dp A 1' + (-1)'c A do, wenn p c A' ist (die letzte Formel gilt analog auch fur d' and d").

KAHLERSCHE MANNIGFALTIGKEITEN B.

63

Es sei auf X eine (positiv-definite) hermitesche Metrik ds2 =

gy;, dzy df, gegeben. Dann ist der Metrik ds2 ein Anti-Isomorphismus Ar,s-* An-r,n-s, d.h. eine R-lineare bijektive Abbildung mit *cq) = c*q1, c e C, zugeordnet, so daB T2q' = (-1)'qq fur p c- At gilt. Eine positivdefinite hermitesche Metrik auf X heiBt kahlersch, wenn die zugeordnete

(reelle) Form w: = 2 > gy;, dzy A df, geschlossen ist, d.h. wenn dco = 0 gilt.

gy;, dzy dz eine hermitesche Metrik in X DEFINITION 1. Es sei dS2 = and x0 E X ein Punkt. Unter einem geodatischen Koordinatensystem zu xo and ds2 versteht man ein System von lokalen komplexen Koordinaten z1, ... , zn in einer Umgebung U(xo), so daB xo die Koordinaten zy(xo) = 0, v = 1, ..., n, besitzt and fur alle v, µ = 1, . . ., n gilt: gf,,(xo) = 8y,,, dgya(xo) = 0. Geodatische Koordinaten heiBen in der angelsachsischen Literatur auch

"normal coordinates". Es gilt: Eine hermitesche Metrik ist genau dann kahlersch, wenn es zujedem Punkt x0 E X ein geodatisches Aoordinatensystem gibt.

Hat man in xo ein geodatisches Koordinatensystem, so gilt fur ='ayj "'

A

dzV,

A

d& E A',' im Punkte x0: T9, = b(r, s, n)

(1)

b(r, s, n) =

sign (v, *v) sign (µ, *p)ayµ

dA*y A dS,k,,,

(-l)n(n-1)/2+s(n-r)2r+s-nin >

wobei *v ein (n - r)-tupel bezeichnet, in dem genau alle natiirlichen Zahlen von 1 bis n stehen, die nicht in v = (v1,

..., v,) vorkommen.

Wir setzen wie 0blich 8 = - *d *, 8' _ - *d'*, 8" _ --id"*. Man hat 8=8'+8", 82=8'2=8"2=0, 8'8"+8"8'=0, 6' , =8"9. S' and 8" sind homogene Operatoren vom Typ (- 1, 0) bzw. (0, - 1), d.h. 8': A'.s--> A'-1,s' 8": A',5- A',s-1 Wir definieren nun den reellen LaplaceOperator A and den komplexen Laplace-Beltrami-Operator durch = d6 + 8d and : = d"6" + 8"d".

Ist w = 2

g,,-, dzy A dz die der hermiteschen Metrik ds2 zugeordnete

(1, 1)-Form, so definiert man schlieBlich noch Operatoren L: Al - A'+2 and A: A'-* A`- 2 durch LT : = p A to and A : = (-1)'*L*. Eine Form p heiBt primitiv, wenn Aq, = 0 ist. Es gilt fur primitive (r, s)-Formen die Gleichung (2)

*Le _ !- 1)1(1 +1)/2

e!

(n-1- e)!

is-rLn-l-e(p

GRA UERT AND RIEMENSCHNEIDER

64

fur e = 0, 1, ..., n - 1 and 1 = r + s. Ist p E A' mit 1_< n, and gilt Ln-1

(P = 0, so folgt 99 = 0.

Falls die gegebene Metrik kahlersch ist, gilt insbesondere ist dann ein reeller Operator: (p man in diesem Fall die folgenden Relationen:

d'L = Ld',

d"L = Ld",

S'A = A8',

S"A = AS",

S'L - LS' = -id", d'A - Ad' = - iS",

d'S"+8"d'=0, and

A = d'8' + S'd'; g1. AuBerdem hat

S"L - LS" = id',

d"A - Ad" = i8',

d"S'+8'd"=0,

ist vertauschbar mit den Operatoren L, A, d', d", S', and S".

Im folgenden sei X stets mit einer kahlerschen Metrik ds2 = gvµ dzv dz versehen. Es sei weiter V - X ein komplex-analytisches C.

Vektorraumbundel vom Range m uber X. Dann gibt es eine Uberdeckung U = {U,: t E I} and Trivialisierungen T,: U, X Cm --> V I U,, so daB die durch r,-' o T,(z, w) = (z, e,K o w) definierten Abbildungen e,K: U, n U, GL(m, C) einen holomorphen Kozyklus bilden (hierbei bezeichnet w, _ (w1i, ..., w'm') das m-tupel der Faserkoordinaten uber U,). Das zu V duale Biindel V* wird bezUglich derselben Uberdeckung U durch den Kozyklus (e K) -1 definiert (dabei bezeichnet e,K das Transponierte von e,K).

Wir bezeichnen den C-Vektorraum der Formen mit Koeffizienten-wir

sagen auch: mit Werten-in V durch Ar,s(V) and setzen A'(V) := A',s(V). Lokal wird 91 E Ar,s(V) auf U, gegeben durch einen Vektor

r+s_1 9,, = 9, 1 U, _

q1(m')), dessen Komponenten gewohnliche Differentialformen vom Typ (r, s) auf U, sind, so daB auf U, n UK gilt: q1, = e,K o TK Da {e,K} ein holomorpher Kozyklus ist, wird durch (d" q1), (d" q1), ... , d"qq(ml)) eindeutig ein Operator d": Ar.s(V)-->. Ar.s+1(V)

definiert. Ebenso ist L: A'(V) --*Al+2(V) durch (LT), = (Lg)1), ..., Lg1'm)) erklart. Wir setzen weiter voraus, daB auf den Fasern von V eine hermitesche Form gegeben sei; d.h. auf jedem U, ist eine (von z c U, differenzierbar abhangende) positiv-definite hermitesche Matrix h, = (h;',) gegeben, so daB die Form von der Gestalt h;kwkw; ist. Auf U, n UK gilt dann:

h, = e,,,hKeK,. Man kann eine solche Metrik stets einfiihren, sofern X parakompakt ist. Wir wollen zunachst zeigen, daB man dann die Trivialisierungen in spezieller Weise wahlen kann. DEFINITION 2. Eine Trivialisierung T,: U, X C n -± V I U, heiBt normal im Punkte xo E U,, wenn (h,(xo)) die Einheitsmatrix and dh,(xo) = 0 ist.

KAHLERSCHE MANNIGFALTIGKEITEN

65

SATZ 1. Zu jedem x0 E X gibt es eine Umgebung U = U(xo) and eine in x0 normale Trivialisierung T*: U x Cm(wi , ... , w,*n) -- V I U. Ist r : U x Cm(w1, ... , wm) -* V I U eine weitere normale Trivialisierung, so ist T- 1 o T* in x0 eine unitdre Transformation, and es ist dort dfik = 0, wenn fiik = fik(x) die Komponentenfunktionen der Abbildungsmatrix Bind.

Es sei hik(xo) = Sik and T-1 o T* von der Form: wi = JikWk = aikvwk zv + Glieder hoherer Ordnung. Wir setzen weiter voraus,

BEWEIS.

w* + k,v

daB die Koordinaten z1i ..., zn von xo Null sind. Dann folgt: i,k

hikwk W* = 7, hikwkwi i,k

hik(Wk

i,k

+

(hik +

ak7vw*zv +

...)(w*

(aikvzv + akivzv)

+''.

duloi`zv + .. .

+ WkW*l

v

d.h.

laikvzv + akivzv) + ...

hik(z) = hik(z) + V

Es gilt somit dh*k(xo) = 0 genau dann, wenn hik,V(xo):= hik,zv(xo) = - aikv ist. Ist T schon eine normale Trivialisierung, so muB also df k = 0 gelten.

Das ist auch richtig, wenn f k(xo) nicht gleich Sik ist; denn die Transformation (f k(xo))-1 o T-1 o r* fuhrt ebenfalls eine normale Trivialisierung in eine normale fiber and hat die vorausgesetzte Form. Um die Existenz zu zeigen, diirfen wir annehmen, daB schon hik(xo) _

Sik gilt; denn das kann man ja durch eine lineare Transformation in Cm(w1i

..., wm) erreichen. Es ist dann y: wi = w* - k,v hik.,(xo)wk Z. fur

hinreichend kleines U = U(xo) eine fasertreue biholomorphe Abbildung von U x Cm(wi , ... , w,*n) nach U x Cm(w1i... , w,,), so daB T* = T O 'y eine Trivialisierung mit den gewunschten Eigenschaften ist. Auf Grund der Eindeutigkeitsaussage von Satz 1 sind Differentialoperatoren von hochstens erster Ordnung auf AT's(V) schon dann eindeutig definiert, wenn sie in jedem Punkte xo E X in Bezug auf eine normale Trivialisierung invariant gegenuber unitaren Transformationen im Cm angegeben sind. Sei also T: U x Cm -- > V I U eine in xo E U normale and rj U = (r1i ..., rpm) in xo: Trivialisierung, so setzen wir fur p c *4p: _ (*411, ... , *p.), d'q, : _ (d'rp1, ... , d'rm) and erhalten somit globale *:Ar,s(V)-.An-r,n-s(V*) and d':Ar-s(V)-±Ar+1,s(V). Wir Operatoren definieren weiter analog zu den C-wertigen Formen: Aqp _ (-1)i*L*p, = d"S" + S"d". 91 E A1(V), S" _ -*d"* and

GRAUERT AND RIEMENSCFINEIDER

66

Ist r: U, x Cm --- VI U, eine beliebige Trivialisierung, q' E and q', = -pI U, = (ml, ..., mm), so sei d'q7, : = (d'q'(,'), ..., d'p(m'). Ist @, =

hl- 'doh,, so wird durch (ep), : = dop, + 0, A 99, ein globaler Operator a: A''s(V) -Ar+1.s (V) definiert (vgl. Nakano [6]), der in normalen Trivialisierungen mit d' ubereinstimmt. Da weiter in normaler Trivialisierung 8"L - LS" = id' gilt, so folgt allgemein

d' = a = i(LS" - S"L). Im Falle des trivialen Bundels V = X x C hat man als hermitesche Metrik die euklidische Metrik auf den Fasern and die Gleichheit A'-s. Es stimmen alle in diesem Abschnitt definierten Operatoren mit den entsprechenden Abbildungen aus Abschnitt B uberein. D. Wir bezeichnen mit 0 die Strukturgarbe von X and mit Or die Garbe der Keime von lokalen holomorphen r-Formen 9) = av1 ...v, dzvl A A dzv,. Ferner bezeichne, falls V--> X ein komplex-analytisches Vektorraumbundel fiber X ist, V die Garbe der Keime von lokalen holo-

morphen Schnitten in V and AI(V) bzw. Ar's(V) die (feine) Garbe der Keime von lokalen differenzierbaren Formen mit Werten in V vom Grade 1

bzw. vom Typ (r, s). Wegen des Lemmas von Dolbeault ist die Sequenz

0 - V ®Or-->- A','(V)

Ar.1(V)

...

fur alle r = 0, . ., n exakt. Setzt man

A*.n(V) {qq c

.

0 0}

so sind die de Rhamschen Gruppen definiert and durch Zr,s(V)/B',s(V), and wegen der obigen exakten Sequenz von feinen Garben gilt: H',s(V) -- HS(X, V ®S2').

Setzt man A''s(V) = {p c- Ar,s(V):Tr q' ist kompakt in X}, ZI,s(V) _ {q E A's(V):d"p = 0}, BT.S(V) = d"A's-1(V) and Hcr.s(V) = Zc"'s(V)/ so gilt entsprechend Hc.s(V) = Hc(X, V ®O'r),

wobei Hs (X, V die s-te Kohomologiegruppe von X mit kompaktem Trager and Werten in V ® Or bezeichnet. Ist V = X x C das triviale Bi ndel, so schreiben wir and x C) and Hc''s(X x Q. Man hat in diesem Fall anstelle von

Hr.s -- HS(X, Or), E.

Hc's = Hc(X, Or).

Im folgenden sei I eine n-dimensionale komplexe kahlersche

Mannigfaltigkeit and V - X ein komplex-analytisches Vektorraumbundel

uber X. Ferner sei X - X ein relativ-kompakter Teilbereich; der Rand

KAHLERSCHE MANNIGFALTIGKEITEN

67

OX von X sei glatt and beliebig oft differenzierbar. Daruber hinaus gebe es zu jedem Punkt xO E aX eine Umgebung U = U(x0) and eine in U beliebig oft differenzierbare streng q-konvexe Funktion p(x) mit dp(x) = 0 fur alle

x c U, so dal3 X n U = {x c X:p(x) < 0}. Wir setzen X also als streng q-konvex voraus. Jede andere Funktion p1(x) mit diesen Eigenschaften unterscheidet sich von p(x) in einer Umgebung von xo nur durch einen positiven, beliebig oft differenzierbaren Faktor.

Ein komplexer Tangentialvektor im Punkte xO E X an aX ist ein

aav

a

a

a

+ a,, ai so daB fur alle ae a aav 8i im Punkte xO gilt: (a)(p) = 0. Diese Definition ist

kontravarianter Vektor e _

av

a zv

,

v

az + unabhangig von der speziellen Auswahl von p. Die Menge TXO(aX, C) der v

v

komplexen Tangentialvektoren im Punkte xO E X an aX ist ein komplex(n - 1)-dimensionaler Vektorraum, der (als reeller Untervektorraum) in dem reell-(2n - 1)-dimensionalen Vektorraum Tx,(aX) der reellen Tangentialvektoren im Punkte xO E X an OX enthalten ist. Es sei schliel3lich T,,. = TXO(X) der reelle Vektorraum der Tangentialvektoren in xO an X.

Dann ist eine Differentialform p vom Grade I auf k im Punkte xO eine alternierende /-Form auf TXO.

Wic bezeichnen nun mit 41-s(V) den Untervektorraum derjenigen (r, s)-Formen auf X mit Werten in V IT, die noch auf dem Rand aX von X X: eine (r, s)beliebig oft differenzierbar sind, d.h. es ist so ist-da die Einbettung Form auf X mit Werten in V}. Ist p c id: aX y X beliebig oft differenzierbar ist-id*cp eine C°°-Form vom Typ (r, s) mit Werten in VIaX. Wir setzen p X := id*p. Im Falle des trivialen x C). fur Biindels schreiben wir wieder Wir sagen, daB die Beschrankung einer Form T _ dz,, A ... A E 40,s(V) auf dem komplexen Rand dzv, A pv, .. v, E mit von X verschwindet, wenn fur alle xo E aX and alle e,, ..., e, E TXO(aX, C) der Wert pv1...,,(x(,)(e1, ... , is) = 0 ist, and schreiben dafur auch Tl aaX = 0. Es gilt fur 0-Formen plaaX = 0 genau dann, wenn 991 aX = 0 ist. Die eben definierte Eigenschaft stimmt mit dem Begriff "complex normal" in Kohn-Rossi [5] iiberein. Es gilt namlich SATZ 2.

p E A'"s(V) verschwindet genau dann auf dem analytischen Rand

von X, wenn es zu jedem Punkt x0 E aX eine Umgebung U = U(xo) - X,

eine reelle C' -Funktion p(x) auf U mit dp(x)

0 fair alle x E U and

X n U = {x c U:p(x) < 0} and Formen 01 and 02 auf U vom Typ (r, s - 1) bzw. (r, s) gibt, so daf3 (pI U = d"p A 01 + P02 ist. BEWEIs. Es sei xO E 3X. Nach Voraussetzung gibt es eine Umgebung U = U(x0) and eine reelle C'-Funktion p(x) in Umit X n U = {p(x) < 0}.

GRA UERT AND RIEMENSCHNEIDER

68

Wir konnen U so klein wahlen, daB man in U komplexe Koordinaten zl, ..., zn mit den folgenden Eigenschaften einfuhren kann: xo hat die Koordinaten z1 = .. = zn = 0, and p(z) ist (eventuell nach Multiplikation mit einer positiven Konstanten) von der Form p(z) = z1 + zl + Glieder hoherer Ordnung. In xo gilt 'pI aaX = 0 fur p c genau dann, wenn alle Komponenten von pp(xo) den Faktor d21 enthalten, da die Ebene {zl = 0} die komplexe Tangente an aX in xo bildet. Da in xo auf3erdem d"p = dz1 gilt, kann man in xo aus T den Faktor d"p eindeutig ausklammern. Auf these Weise bestimmt man ein 0 auf U vom Typ (r, s), so daB

auf U n aX die Gleichheit p, = d"p A 0 besteht. Mit 02 = (1/p) x (p - d"p A 01) folgt die Behauptung. Gilt umgekehrt (p = d"p A 01 + pb2, so ist in xo: 99 = d51 A 01, d.h. q,jaax = 0. Man beweist weiter leicht: Fur T E gilt pI aaX = 0 genau dann, wenn fur alle a E

An-r.n-s-1(V*) das Integral

cpAa fax

verschwindet. Hierbei bedeutet A das Produkt A',s(V) x Ar+r*.s+s*, das wegen der Paarung V x V* --> C wie bei C-wertigen Formen definiert werden kann. F.

Aus dem letzten Kriterium folgt unmittelbar, daB die Raume 0,,s(V) {p) E Ar,s(V):T = d"o, Y' E Ar,s-1(V)} Cr r.s(V)

{97 E Ar.s(V):p7 = 8"0, 0 E Ar.s+1(V), ,k,l,IaaX = 0} 8"p

{p) E

= 0, *TIaax = 0}

in Bezug auf das hermitesche Skalarprodukt (p, Y') : = f, T A *Y', x

pp, 0 E Ar.s(V)

paarweise orthogonal sind. Wir setzen weiter ,41,s(V) : = {p7 E Ar.s(V): *pI aaX = *d"pIaaX = 0}.

Da X streng q-konvex ist, gibt es nach Kohn [4] fur s > q einen beschrank-

ten Operator N:

so daB fur 91 E Ar,s(V) gilt: h : = 99

-

Setzen wir 991 := 8"Nq1 and 992:= d"Nqp, so folgt 91 = and 8"9)2 d"p1 + d"p2 + h mit d"(p1 Man hat also das Zerlegungstheorem Ar.s(V) = 58r.s(V) O ¶ s(V) O 'D'-s(V) fur s 1 q. Ist nun 99 E Ar,,(V) mit 9' = 990 + h mit 99o c

dl"qp

= 0 and s > q, so gestattet q1 die Zerlegung and h E Sar.s(V). Die harmonische Form h

KAHLERSCHE MANNIGFALTIGKEITEN

69

hangt nur von der de Rhamschen Klasse von T ab. Man erhalt so einen Isomorphismus

H'.s(V) '-' ''s(V),

s 1 q.

Zur Untersuchung der Kohomologie mit kompaktem Trager setzen wir ,Dr.s(V)

{q, c-

0, pIaa,X = 0}.

A',s(V):d"9) =

Der *-Operator gibt einen Isomorphismus 5Dr' (V) -->n_r.n-s(V*) Aul3erdem hat man nach Serre [7] fur q-konvexe Raume einen Isomorphismus H"-s(X, y* 0 S2n-') - H' (X, V ® S2') fur s < n - q. Also gilt auch Hcl.s(V)

s < n - q.

Im nachsten Teil werden wir zeigen, daB S2°.s(V) = 0

fur s < n - q gilt, sofern X hyper-q-konvex and V semi-negativ ist. Insbesondere ist dann gezeigt:

H'(X,U)=0, Hs(X, f2n) = 0,

s < n - q, s ? q.

2. Beweis des Hauptresultates

Es sei f im folgenden eine n-dimensionale komplexe kahlersche Mannigfaltigkeit, V4. X ein komplex-analytisches Vektorraumbundel uber f vom Range m, and auf den Fasern von V sei eine hermitesche Form gegeben. Ferner sei X - Xein offener relativ-kompakter Teilbereich von X mit glattem, beliebig oft differenzierbarem Rand X. Wir wollen untersuchen, unter welchen Voraussetzungen an V and 8X die Raume i'.s(V) verschwinden. Dazu leiten wir zunachst eine wichtige Gleichung A.

her.

Es sei q, E dann folgt d'q, = i(LS" - S"L)q, = -iS"Lp, and wegen *S"L , = (-1)'d"*Lp,, I = r + s, d(d'q, A *Lp) = d"(d'q) A *L9,) = d"d'q, A *Lqp + (- I)'+ 'd'q n d"*Lq, ergibt sich mit Hilfe des Stokes'schen Satzes

(d'c, d'q,) = i(d'q,, S"Lq,) = i

-i f

Jx

d'cp A *S"Lq

d'q, A *L9, +

if

A *L-,.

GRAUERT AND RIEMENSCHNEIDER

70

d" O, so ergibt sich d"d'p

Definiert man noch wie iiblich X

d"(d,'99 + 0 A q') = d"O A 9' = 2,riX A p and somit unsere Hauptgleichung

- (d'q,, d'q,) = 2-(x A p, Lq)) + i fox d'rp A *Lq).

(3)

Es gilt (x A 99, Lqv) > 0 fur 99 E A°,s(V), s < n, wie wir im nachsten Abschnitt zeigen werden, wenn V I X ein semi-negatives Vektorraumbundel im Sinne von Nakano ist. Im Abschnitt C untersuchen wir die Bedingungen an 8X, unter denen auch das Randintegral i f?,x d'T A *Lcp >_ 0 wird. Da

stets (d',p, d'p) 0 gilt, mul3 dann bei semi-negativen Bundeln sogar fax d'q' A *L9 = 0 gelten. Daraus wird dann in Abschnitt D unter den gegebenen Voraussetzungen

0 folgen.

B. Es sei p E s < n, and 0 : _ (x A q)) A *L9) E An,n. Ist x° E X ein Punkt, so konnen wir in einer Umgebung U von x° eine

Trivialisierung von V finden, die in x° normal ist. Gilt bezuglich dieser Trivialisierung 99 _ (99i, ... , Tm), X = (Xii), so folgt in x°:

_

(x A q')t A *Lggt =

i,i

t

(Xii A (pi) A *LTi, s

and daraus wegen *Lpt = iYs +1)/2.(n

l

Y

s

-

1)i

Ln

S

- s - 1)I i,i

Xii A 97i A (Pi A wn-s-1.

Wir konnen weiter in U komplexe Koordinaten einfuhren, die in x° geodatisch sind. Es ist dann

w=t Setzt man rpi =

dz.Adz,,.

tav,...v. dzvl A

A dzvs, i = 1, ..., m, and

< vs_1 < n:

fur 1 0in xo? Es sei p eine reelle differenzierbare Funktion in einer Umgebung U von

xo mit dp Vz 0 and U n X = {x c U:p(x) < 0}. Dann gilt U n aX = {x e U:p(x) = 0}, and es ist dplBX = d'plaX + d plBX = 0. Da p auf dem analytischen Rand von X verschwindet, gibt es nach Satz 2 in einer

Umbegung von x° eine Darstellung p = d"p A 0 + p+bo. Damit folgt + das heiBt: d'p A q)JBX = d'd"p A 0 A d'p A q=1J8X = (-1)1+1d'p A d'p A 9' _ (d'd "p A 0 - d "p A d'o + d'P A 0o + pd'Oo) A (d'P A d'd"p A 0 A q=rJBX, and wegen Formel (2) hat man in x0: (is-r+1 i y=id'pA*Lqp_(-1)ul+12.(n

(5)

_ _(_l)u!-1)/2.(n

l-1)!d9'A9'A

con-1-1

i s-r+1

1- 1)! d'p A d'd"p A 0 A 1 A wn-1-1,

wobei auf der rechten Seite stets die Beschrankung auf aX zu nehmen ist. Wir fiihren nun in x0 geodatische Koordinaten ein, so daB dort p von der Gestalt ist: p(z) = x1 + Glieder hoherer Ordnung. Es gilt in x0: 4' = I d21 A 0; wir konnen deshalb annehmen, daB qb kein dz1 enthalt. Setzen wir 9) =

I

1 0.

Da fur eine positive differenzierbare Funktion a die Gleichheit d'd"(ap)I aaX = ad'd"pIBaX besteht, 1st die obige Definition unabhangig von der speziellen Wahl von p. Aus (7) folgt unmittelbar SATZ 4. Hat X die Eigenschaft B(r, s) (d.h. gilt B(r, s) in jedem Punkt xo E 15X), so ist ft r jede primitive Form qp E 4',s(V) mit r + s < n and

opI aaX = 0, deren KoeffizientJ en nicht sdmtlich auf aX verschwinden:

id'p A *Lq>>0. x

Wir wollen im folgenden die Eigenschaft (iii) aus der Definition 4 naher untersuchen. Es sei

0_

av, ...v,,,,l ...µs dzv, A ... A dzvr A dza, A ... A dzps 2 0 sei V = X x C das triviale Biindel. Dann folgt aus unserer Hauptgleichung and den Abschnitten B and C: 0 < (d'g), d'99) = - 27T (x A w, Lq)) - i fox d'99 A *Lp < 0, ax

d.h. (d'q), d'p) = 0 and i

lax

d'qp A *Lq = 0. Wir erhalten also d'97 = 0,

(12)

and wegen Satz 4 gilt mit q'; _ ! ;av,µdjv A d,,: jav,...

(13)

vr.li:...µs l bX = 0

tt, :5 n und 1 1 and der Satz fur Formen vom Grade < I - I schon hergeleitet, so ist wegen Ap E SDT - 1.s -1 die Form 9' primitiv, and die Behauptung folgt aus Satz 6. Es ist den Verfassern nicht bekannt, ob Satz 8 anwendbar ist.

3. Die Satze von Akizuki and Nakano. Beispiele

Es sei X = X eine kompakte kahlersche Mannigfaltigkeit and Vein negatives komplex-analytisches Vektorraumbundel uber X. Aus unserer Hauptgleichung folgt dann A.

-(XAp,Lp)>0 fur alle 9) E f','(V) = {qp E A',s(V):d"(p = 8"p = 0}. Andererseits wurde in Abschnitt B von Ted 2 gezeigt, daB (X A T, Lcp) > 0 ist fur 99 E

s < n. Da auf Grund der Gleichung (4) (X A p, Lp) = 0 nur dann gilt, wenn p = 0 ist, erhalten wir das Vanishing-Theorem von Nakano: Hs(X, V) = S5°.s(V) = 0,

s < n.

B. Ist Fein negatives Geradenbiindel uber der kompakten kahlerschen Mannigfaltigkeit X = X, so ist X eine positiv-definite globale (1, 1)-Form auf X, die geschlossen ist. Man kann daher X als die einer kahlerschen Metrik auf X zugeordnete Form betrachten and erhalt mit Lp : = X A p stets (X A p, LT) > 0. Andererseits ist wegen der Hauptgleichung -(X A p, Lq)) > 0 fur Formen 9) E ,Dr, '(F), d.h. Lp = 0. Daraus folgt im Falle r + s < n schon p = 0. Damit ist auch das Vanishing-Theorem von Akizuki and Nakano bewiesen:

r+s 2. F bezeichne das zu einem Hyperebenenschnitt gehorende komplex-analytische Geradenbundel; F ist somit positiv. Es gibt dann in der 1-ten Tensorpotenz F1 von F, I > 1°, schon n + 1 Schnittflachen

.s°, s1, ..., sn, die keine gemeinsame Nullstelle in Y haben. Man erhalt deshalb einen Epimorphismus (n + 1)(9 --> F1-* 0, wobei 0 die Strukturgarbe von Y bezeichnet. Wir tensorieren mit F-1 and bekommen so einen Epimorphismus e: (n + 1)F-1 - F7-1. Es sei Vder Kern von E. Die Garbe V ist lokal frei and wird von einem n-rangigen Untervektorraumbundel V von (n + 1)F-1 geliefert. Da F-1 and mithin auch (n + 1)F-1 negativ ist, muB V schwach negativ sein.

Mit H°(Y, (n + 1)F-1) = H1(Y, (n + 1)F-1) = 0 folgt aus der exakten Garbensequenz 0 - V - (n + 1)F-1 - F1-1 - 0: H1(Y, Y) - H°(Y, F1-1),

1 > lo.

Da schlief3lich H°(Y, F1-1) von Null verschieden ist fur hinreichend groBes, 1° erhalt man

H1(Y,Y)

0.

Nach deco Satz von Nakano muBte jedoch H1(Y, V) = 0 gelten, wenn V negativ ware. D. Zum SchluB entwickeln wir aus dem vorigen Abschnitt ein Beispiel dafur, daB unser Vanishing-Theorem nicht mehr richtig bleibt, wenn man die Hyper-q-Konvexitat im Falle q > 1 durch die schwachere q-Konvexitat ersetzt. Es seien Y and V wie in Abschnitt C gewahlt. Es gibt eine hermitesche Form hlkwkwl auf den Fasern von V, so daB durch {: hikwkwi < 1} eine relativ-kompakte streng pseudokonvexe Umgebung der Nullschnittflache definiert wird. Auf den Fasern des dualen Bundels V* ist dann ebenfalls eine hermitesche Form k"w*w* gegeben, wobei (h'k) = (h1) h = (hik), ist. Obwohl in normaler Trivialisierung beide Formen ubereinstimmen, ist die Nullumgebung Q hlkw*w* < 1} nicht mehr 1-konvex. SchlieBt man

jedoch die Fasern von V* zu komplex-projektiven Raumen ab and bezeichnet man mit V* das auf these Weise erhaltene projektive BUndel, so

ist das Komplement X der oben definierten Umgebung der Nullschnittflache bezuglich V* eine n-konvexe Umgebung der unendlich fernen,

Punkte V. = V* - V*. Wir zeigen, daB HAX, 0) 0 ist. Es ist dann das gewunschte Beispiel in X gefunden. Es sei U = {UU:1 E I} eine endliche Steinsche 0berdeckung von Y. Nach Konstruktion gibt es einen Kozyklus 6 c Z1(U, V), der nicht kohomolog

KAHLERSCHE MANNIGFALTIGKEITEN

79

zu Null ist. Da die Elemente von V y E Y, Linearformen auf Vy sind, erhalt man durch 6 einen Kozyklus E Z'(!1, 0), wobei ft das Urbild der Uberdeckung 11 bezuglich der Projektion V* -* Y bezeichnet. 1st = so sind die Funktionen e 1 2 auf den Durchschnitten Cl,,,, holomorph and entlang der Fasern von V* linear; sie besitzen deshalb im Unendlichfernen eine Polstelle erster Ordnung.

Es sei WR = {j h'kw*w* > R} c - X eine Umgebung von V.. Die Kohomologieklasse von fl V* - WR IaBt sich dann nicht nach V* fortsetzen. Ware dies namlich doch der Fall, so gibt es wegen U, = U, x pn and H'(UU, (9) = 0 ein y e Z'(U, 0) and ein 7 E C°(UI I7* - WR, (9), so daB y = e - 8,7 in V* - WR ist. Entwickelt man dann e, y and ,7 in eine Potenzreihe nach den Koordinaten auf den Fasern and bezeichnet man

mit e('), y() and Y'I den linearen Anted, so gilt e(i) = 6, yI" = 0 and mithin 6 = g,7(') im Gegensatz zu der Voraussetzung. Aus den Vorhergehenden folgt weiter, da3 auch ;'jX - WR nicht nach X hinein fortgesetzt werden kann. Wir stellen nun die Kohomologieklasse von elX - V. bezuglich der Dolbeault-Isomorphie durch eine (0, 1)-Form p dar. Es sei E eine beliebig oft differenzierbare positive Funktion auf X, die in X - WR identisch 1 ist and in einer Umgebung von V. identisch verschwindet. Offenbar hat 0 = d"(E(p) kompakten Trager. Gabe es eine

(0, 1)-Form a in X mit kompaktem Trager and d"a = 0, so hatte man d"(ET - a) = 0. 1st R > I hinreichend klein, so bestehtjedoch in X - WR die Gleichheit T = ET - a. p ware also nach X hinein fortsetzbar. Da dies nicht der Fall ist, wie eben gezeigt wurde, reprasentiert d"(Ep) nicht die Nullklasse von He (X, 0). UNIVERSITAT GOTTINGEN

LITERATUR 1. AKIZUKI, Y., and S. NAKANO, "Note on Kodaira-Spencer's proof of Lefschetz theorems," Proc. Japan. Acad., 30 (1954), 266-272. 2. BOCHNER, S., "Curvature and Betti numbers I, II," Ann. of Math., 49 (1948), 379-390; 50 (1949), 77-93.

3. KODAIRA, K., "On a differential geometric method in the theory of analytic stacks," Proc. Nat. Acad. Sci., 39 (1953), 1268-1273. 4. KOHN, J. J., "Harmonic integrals on strongly pseudo-convex manifolds, I and II," Ann. of Math., 78 (1963), 112-148; 79 (1964), 450-472. 5. KOHN, J. J., and H. Rossi, "On the extension of holomorphic functions from the boundary of a complex manifold," Ann. of Math., 81 (1965), 451-472. 6. NAKANO, S., "On complex analytic vector bundles," J. Math. Soc. Japan, 7 (1955), 1-12. 7. SERRE, J.-P., "Un theoreme de dualite," Comment. Math. Helv., 29 (1955), 9-26. 8. VESENTINI, E., "Osservazioni sulle strutture fibrate analitiche sopra una variety kahleriana compatta, I, 1I," Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur. (8), 23 (1957), 231-241; 24 (1958), 505-512.

Iteration of Analytic Functions of Several Variables SAMUEL KARLIN AND JAMES McGREGOR' Professor Salomon Bochner has contributed notably in many areas of mathematics including the theory of functions of several complex variables and the theory of stochastic processes.

In line with these interests, this paper reports results on iteration of holomorphic functions of several complex variables motivated by investigations pertaining to multi-type branching Markoff processes. Apart from its intrinsic importance and independent interest, iteration of holomorphic

mappings plays a fundamental role in celestial mechanics, population genetics, numerical analysis, and other areas. Consider a vector-valued mapping (1)

{

f(z) = (J1(z),J2(z),...fJP(Z))' Z = (Z1, Z2, ..., ZP) E ZP,

where Z,, denotes the space of p-tuples of complex numbers. Suppose the mapping has a fixed point in its domain of definition and that at the fixed

point the Jacobian matrix is nonsingular and all its eigenvalues are of modulus less than 1. One of the main objectives is to determine a canonical representation for the iterates (2)

f(z; n) =.f(.f(z; n - 1), 1),

n = 1, 2, .. .

f(z; 0) = z from which the complete structure and the asymptotic behavior of iterates of high order is easily ascertained. The case p = 1 has a long history and an extensive literature and has been the subject of considerable recent research, e.g., see Jabotinsky [6], Baker [1], Karlin and McGregor [8], [9] and Szekeres [17]; the last named 1 Research supported in part at Stanford University, Stanford, California, under contract N0014-67-A-0112-0015. 81

KARLIN AND McGREGOR

82

work contains a substantial bibliography of earlier writings on this topic. Consider a complex-valued function of one complex variablef(z) analytic

in the neighborhood of the origin and satisfying f(O) = 0 so that the origin is a fixed point. It was already known in the nineteenth century that if c = f'(0) satisfies 0 < I c I < 1, then the limit function (3)

A(z) = li m f (cnn)

exists, is holomorphic at z = 0 and satisfies (4)

A(f(z)) = cA(z).

The function A(z) may be characterized as the unique solution of the functional equation (4) holomorphic at 0 with prescribed initial conditions A(O) = 0, A'(0) = 1. If the mapping inverse to w = A(z) is z = B(w), then, from

A(f(z, n)) = cnA(z),

n = 0, ± 1, ± 2,...,

we obtain the representation of the iterates (5)

f(z, n) = B(cnA(z))

The formula (6)

f(z; t) = B(etb0 A(z)),

-00 < t < 00

defines a continuous extension of f(z; n) where each f(z; t) is analytic at 0 and keeps this point fixed. It is not difficult to show that (6), for the various

determinations of log c, gives the only continuous embeddings of the iterates off in a one-parameter group (see [8]). The expression (6) provides a canonical representation of the group of

mappings f(z; t) with the nature of the dependence on z and t clearly displayed. DEFINITION 1.1. If z O(z) is a holomorphic mapping defined in a neighborhood of the fixed point of f(z) with the same fixed point, then the mapping z -.g(z) = 0-1(f(ys(z))) is called conjugate to f(z).

It is clear that for g conjugate to f then g(z; n) = 0-1(f(o(z); n)),

and the structure of the iterates of g is readily discernible from those off and conversely. The representation (5) asserts that if f(z) is holomorphic at 0, f(0) = 0,

and f'(0) = c (0 < Icl < 1), then f(z) is conjugate to the linear function g(z) = cz.

ITERATION OF ANALYTIC FUNCTIONS

83

In this article we outline the analogues of (5) and (6) in the case p >= 2 and indicate some applications of the representation formula. In order to state the principal result we require some additional notation. There is given a mapping z f(z) = (fl(z), ..., f(z)) in accordance with (I) with each component function holomorphic in a neighborhood of the origin. Suppose for definiteness that 0 is a fixed point of f(z). Thus each ,(z) admits a power series expansion in the p variables z1, z2, . . ., z, without constant term, of the form .fv(z) =

J=A

av(j)z>

where A denotes the set of all j = (ji, j2, . , jp), iv non-negative integers not all zero and z' = zil z22 . zjp. The length of the vector z = (z1, ... , z,) is denoted by IzI = max (Izll, Iz2l, ..., Izpj). The gradient matrix of the mapping at the fixed point is .

C =P0) = 11 C.a 11,

(7)

where a C"°

az13

a,/3=1,2....,p. Z=0

The gradient plays the role, for general p, of the multiplying factor f'(0) = c occurring in the case p = 1. Our main assumption is that every eigenvalue A of C satisfies (8)

0 IA.I>

lA2I>...> !API>0.

(Repeated A's are to appear consecutively.)

Algebraic complications may develop which do not arise in the case p = 1. These are of two kinds: (i) The canonical form must necessarily be recast whenever C possesses nonsimple elementary divisors. This situation

can occur only when p > 2. (ii) A more substantial modification is required whenever algebraic relations among the eigenvalues {Al, A2i ..., A2}

exist. An algebraic relation is an identity of the form (9)

ArlA22... AP', = A

KARLIN AND McGREGOR

84

where r1, r2, ..., rp are positive integers, and

P

u=1

rµ > 2. When algebraic

relations are not present and C admits no elementary divisors, a simple diagonal canonical form prevails. Even under these circumstances a further technical obstacle pointed out by Bellman [2] and other authors is that the most straightforward generalization of the limit formula (3) is no longer forthcoming. It is correct that the first component function A1(z) can, in fact, be constructed in the spirit of (3), but the other component functions A,(z), v = 2, ..., p, cannot be so simply determined. The form of the general canonical representation is the content of the following basic theorem we write 1i I = THEOREM 1.2

Let the assumption (8) hold. There exists a mapping

A(z) = (A1(z), A2(z), (10)

I.iYI for j c 0 V=1

..., Ap(z)) holomorphic at the origin such that A(0) = 0,

Suv

dAu I==o

and Av(z) satisfies a functional relation (11)

Av(f(z; n))=Avj A,,(z)+

l

8 where a(v) is the smallest integer such that the linear parts of A1, A2, . . ., Av are linearly independent. Since each A,(z) is holomorphic in jzj < 1 + E, the functions A'(z) = Ail(z)A22(z) ... App(z),

jcA

are all in .V These functions are evidently linearly independent and span LEMMA.

If a, b are holomorphic in jzj < 1 + E and there are complex

numbers a, P and positive integers m, n such that (T - a)ma = 0, (T - P)nb = 0, then

(T -

a,)m+n-lab

= 0.

By repeated application of this lemma one shows that for sufficiently large n = n(j) j c- A. (T - A')nA' = 0,

ITERATION OF ANALYTIC FUNCTIONS

87

Thus the basis AJ, j e A, provides a complete decomposition of the space JXP'

into eigenmanifolds of T. The eigenvalues of T are the numbers A' = A?I ... App,

j E A,

each listed with proper multiplicity (trivially, zero is not an eigenvalue of T).

In particular, each of the functions wa is a linear combination of the finitely many A' such that A' = ya. Because of the way the Av have been ordered, in any nontrivial relation Av = A', j E A, Ij I >_ 2, only ja with < v can be different from zero. Consequently, when the right member of TAv = Twa(v) = Av(Av + ya(v)wa(v)-1)

is expressed as a linear combination of the function A', the form (11) for n = 1 is obtained. The result for general n follows readily. We also see that the dimension of S is

N=p+R where R is the number of relations.

We pointed out, following the statement of Theorem 1, that when algebraic relations existf(z) is not generally conjugate to a linear mapping. However, it is possible to represent the mappingf(z) in a higher dimensional space such that the conjugacy expression corresponds to that of a linear mapping. We illustrate the embedding first with an example in the case p = 2. Here, the eigenvalues A1, A2 of C fulfill exactly one relation, an

identity A2 = Ai where n is an integer, n

2. For the example f(z) _

(Az1, A2z2), 0 < A < 1, we have a relation Al = A, A2 = Al but nevertheless the map is linear; in fact we can take Av(z) = z,,, v = 1, 2, and then

h(Q = (14)

On the other hand, for the example f(z) = (Az1, A2z2 + zi), (0 < A < 1),

we again have a relation Ai = A2 = A2, but a simple computation reveals thatf(z) is not conjugate to Cz. With the two dimensional mapping in (14) we shall associate a linear mapping g(s) in the space of three complex variables s = (s1, s2i s3), viz. (15)

g(S) = (AS1, A2S2, A2S3 + S2).

It is easy to see that the algebraic surface

{s:s2=sfl is mapped into itself by g(s). The surface 4' is parametrized by the coordinates S1, S3. In fact, the map U defined by

UI I=

zl z2

KARLIN AND McGREGOR

88

provides a holomorphic and globally univalent map of the space of two complex variables onto tel. The inverse mapping is simply the restriction to .4' of the projection V, Si

V

S2

S3

= (S3)

It is apparent that

.f(z) = V(g(U(z))) Thus the mapping (15), although not conjugate to a linear transformation,

is conjugate to the restriction, to an invariant algebraic manifold, of a linear transformation in a space of higher dimension. This situation is general. Suppose there are exactly R relations. Define

N = p + R. Then we have

THEOREM 2. There exists a univalent polynomial mapping U of Z, onto an algebraic surface .4' in ZN, and a nonsingular linear mapping g of ZN

onto ZN such that

(i) If 1 is a linear subspace of ZN and Y - ff then 1 = ZN. (ii) Uh = gU. Two remarks are worth appending: (i) The inverse mapping U -1 of mil onto Zp is implemented by a linear map of ZN onto Zp. (ii) The relation between h and g extends at once to the iterates hn and g". That is,

Uh"=g"U,

n=0,±1,±2,.... Applications

MAPPINGS WHICH COMMUTE WITHf(z). Let e(z) be a mapping of Zp into Zp which is holomorphic in a neighborhood of the origin, leaves the origin fixed, and commutes with f(z); that is,

e(f(z)) = f(e(z))

for all z in a neighborhood of the origin. The following Theorem serves to characterize such mappings.

ITERATION OF ANALYTIC FUNCTIONS THEOREM 3.

89

Using the notation of Theorems 1 and 2, the formula

E = U(A(e(A-1(U-1))))

sets up a 1-1 correspondence between the set of all mappings e(z) of Z holomorphic in a neighborhood of the origin, leaving the origin fixed and commuting with f, and the set of all linear mappings E of ZN which map Ll into itselfand commute with g.

The important special case where no algebraic relations are present leads to the corollary. Under the conditions of Theorem 3, if no relations exist, COROLLARY. then e(z) commutes with f(z) if and only if e has the form e(z) = A -1(E(A(z))) where E(z) is a linear mapping of Zp onto itself commuting with the mapping h(z). SUPERCRITICAL MULTI-TYPE BRANCHING PROCESSES. Consider a multitype branching Markov process of p-types with probability generating

function (p.g.f) ff(z1,

..., zp)

a,(j)z' for a single individual of the with

type, v = 1, 2, . . ., p; i.e., the coefficient av(j) is interpreted as the probability that a single parent of the with type produces I j I offspring consisting ofjl individuals of type I J2 of type 2, .. . ,. j, oftypep where j = (j1,.j2,

Obviously av(j) > 0 and

,j,)

a,(j) = I for each v. Let X(n) = (X1(n),

X2(n), ... , Xp(n)), n = 0, 1, 2, ... be the vector random variable depicting

the population structure at the nth generation; i.e., X,(n) denotes the number of individuals of the with type in the nth generation. The temporally homogeneous transition probability matrix governing the fluctuations of population size over successive generations is defined implicitly by the generating function relation (16)

Pr{X(n + 1) = jI X(n) = i}z' = [f(z)]t

(we employ the previous notation here). The identity (16) characterizes

Markov branching processes in that each individual bears offspring independently of the other existing individuals. Thus, if the current population makeup consists of i, individuals of type r (r = 1, 2, . ., p), .

then the p.g.f. describing the next generation is given by [J1(Z)]'1LJ2(Z)]i2...

and thus (16) obtains.

UW]in

KARLIN AND McGREGOR

90

A standard result in the theory of Markov branching processes is that

Pr{X(n) = jIX(0) = i}zi = [f(z; n)]' is valid for jzj < 1, where f(z; n) denotes the nth iterate of f(z; 1) = f(z). By the nature of the stochastic process at hand, each of the functions f(z) is defined in the polydisc IzI < I and the vector 1 = (1, 1, . . ., 1) is 0 for all v, µ. In this circumstance the existence .

of another invariant point 7T = (7f1, 7f2, ..., irp) with 0 < it < I (i = 1, 2, ..., p) is assured; i.e., f(ir) _ iT. Moreover, the iteratesf(z; n) converge to 7T as n - oo for all jzj < 1, z 0 1 (see Harris [4], or Karlin [7], Chap. 11). If we assume f(0) # 0, then it follows that 7i > 0 for all i. We can now prove Under the conditions stated above, the eigenvalues of the

PROPOSITION 1.

matrix C = lj (af,,/8zµ)(Ir) 11 satisfy 0 < 1 Ai I < 1 (i = 1, 2, ... , p).

Introduce the vector 1; = (V7,, 1l;,-, ... , 1/). An applica-

PROOF.

tion of the Schwartz inequality yields Jvl v ")

_

/V

av(i)(

1/2

1/2

")' < (>z av(i)it(17)

Strict inequality prevails because of the stipulations 0 < f,(0), (af/8zµ)(1) > 0 (v, ,u = 1, 2, ..., p) and since p(D) > 1. Because each ff(z) is convex on

the domain _9 = {z: 0 _ C C. (z - r) for z c L',

where the notation signifies that the inequality holds componentwise. Substituting z = 1/ and referring to (17) we find for the positive vector

u= V; - -,T that u > Cu

with strict inequality in each component. This inequality implies, by virtue of the well known Frobenius theorem pertaining to matrices with positive entries, that all eigenvalues of C are in magnitude less than 1. The proof of Proposition 1 is complete.

ITERATION OF ANALYTIC FUNCTIONS

91

If C is nonsingular, which is the case except if { ff(z)}p=1 are "essentially"

functionally dependent, then Theorems I and 2 are relevant for the study of the iterates f(z; n) in the neighborhood of the fixed point 7T. In particular,

utilizing the canonical representation implicit in (11), it is not difficult to establish the strong ratio theorem. Specifically, it can be proved for any prescribed nonzero integer vectors i, j, k, 1

Pr{X(n) = jIX(0) = i}

-. Pr{X(n) = kIX(0) = 1} exists. It is also possible to develop a spectral representation of the process

(see Karlin and McGregor [8], [9]) and with its aid to uncover several right and left invariant functions of probabilistic importance. The elabora-

tion of these applications as well as the proofs of Theorems 1 to 3 and related matters will be exposed in further publications. We learned at the symposium whose proceedings are summarized in this volume that Sternberg [13] gave a proof of Theorem I using an elegant

iterative scheme. He focused attention on the case where C admits no nonsimple elementary divisors. Undoubtedly his method, which is sketchy in places, can be adapted to handle this case as well. See also Sternberg

[14], [15], [16], Hartman [5, Chapter 9], Chen [3] and Nelson [11] for discussion of the corresponding problem with mappings of class C'` for suitable k.

Numerous other authors have recently worked on the problem of Theorem I and succeeded in establishing partial cases of Theorem 1; e.g., see Schubert [12] and references given there. They are apparently unaware of Sternberg's important contribution on this subject. Our method of proof of Theorem 1 differs from Sternberg's and may be of some interest. STANFORD UNIVERSITY

REFERENCES BAKER, I. N., "Fractional iteration near a fixed point of multiplier, 1," J. Austral. Math. Soc., 4 (1964), 143-148. BELLMAN, R., "The iteration of power series in two variables," Duke Math. J., 19 (1952), 339-347. CHEN, K. T., "Normal Forms of Local Diffeomorphisms on the Real Line," Duke Math J. 35 (1968), 549-555. HARRIS, T. E., Theol.r of Branching Processes. Berlin: Springer-Verlag, 1967. HARTMAN, P., Ordinary Differential Equations. New York: John Wiley & Sons, Inc., 1964. JABOTINSKY, E.,

457-477.

"Analytic iteration," Trans. Amer. Math. Soc., 108 (1963)

KARLIN AND McGREGOR 7. KARLIN, S., A First Course in Stochastic Processes. New York: Academic Press,

92

1966.

8. KARLIN, S., and J. MCGREGOR, "Embeddability of discrete time simple branching processes into continuous time branching processes," Trans. Amer. Math. Soc., 132, No. 1 (1968), 115-136. 9.

"Embedding iterates of analytic functions with two fixed points into

continuous groups," Trans. Amer. Math. Soc., 132, No. 1 (1968), 137-145. 10. KARLIN, S., and J. MCGREGOR, "Spectral theory of branching processes, I and II," Z. Wahrscheinlichkeitstheorie, 5 (1966), 6-33 and 34-54. 11. NELSON, E., The Local Structure of Vector Fields, part of lecture notes at Princeton University, Section 3, 1969. 12. SCHUBERT, C. F., "Solution of a generalized Schroeder equation in two variables," J. Aust. Math. Soc., 4 (1964), 410-417. 13. STERNBERG, S., "Local contractions and a theorem of Poincare," Amer. J. Math., 79 (1957), 809-824. 14. , "On the structure of local homeomorphisms," Amer. J. Math., 80 (1958), 623-631. 15.

"The structure of local homeomorphisms, III," Amer. J. Math., 81

(1959), 578-604. 16.

"Infinite Lie groups and the formal aspects of dynamical systems,"

J. Math. Mech., 10 (1961), 451-474. 17. SZEKERES, G., "Regular iteration of real and complex functions," Acta Math., 100 (1958), 203-258. 18.

, "Fractional iteration of entire and rational functions," J. Aust. Math. Soc., 4 (1964), 129-142.

A Class of Positive-Definite Functions J. F. C. KINGMAN Among the many distinguished results associated with the name of Salomon Bochner, one of the more widely known is his characterization of the positive-definite functions, those continuous functions f of a real variable for which any choice of the values tl, t2i ..., t,, makes

(f(t - ta); a, 9 = 1, 2, .

(1)

.

., n)

a non-negative-definite Hermitian matrix. These are exactly the functions expressible in the form (2)

f(t) = 5eitc(dw),

for totally finite (positive) measuresq on the Borel subsets of the real line. In the case of a real-valued function f (necessarily even), (2) takes the form (3)

f(t) = Jcos wt (dw),

and may then be regarded as concentrated on the interval [0, oo). In the theory of probability these functions arise most commonly as the

autocovariances of stationary stochastic processes. Thus let Z = (Zt; - oo < t < oo) be a stationary process, with E(Ztl) finite, and let f(t) be the covariance of ZS and Zs+t (which is necessarily independent of s). Then

f is a positive-definite function, and 0 is the "spectral measure" of the process.

A particular case arises in the theory of Markov chains (for which we

adopt the notation and terminology of Chung [I]). Suppose that X = (XI; -co < t < oo) is a stationary Markov chain on a countable state space S, with (4)

it=P(Xs=i), Pti(t) = P(Xs+t = .I I XS = 0, 93

J. F. C. KINGMAN

94

for t > 0, i, j c S. That is to say, X is a stochastic process such that, for il, i2, ., i,, e S and tl < t2 < < t., . .

7Ti, (5)

P{Xtn = la (a = 1, 2, ... , n)} = J

n

l

a=2

Pin - i.ia(ta

- to - l)'

As is usual, the chain is assumed to be standard in the sense that, for each i c S, (6)

lim pi;(t) = 1. t-.o

Let a be a particular state in S (with, to avoid triviality, define a stationary process Z by

7ra

> 0), and

Zt = 1 if Xt = a, and (7)

=0 if XX:a.

The autocovariance function f of Z is clearly given by (8)

f(t) _ -a{Pas(t) - "a}.

Substituting this into (3), we obtain (9)

Paa(t) = ra + f Cos u,Ndw),

with A = ra * Hence the function Paa, which is a typical diagonal element of the transition matrix (10)

Pt = (Pii(t); i,j c- S),

is positive-definite.

Not every Markov transition matrix Pt can arise from a stationary Markov chain; this will be possible if and only if (for some a), lira Paa(t) > 0. t -.

However, Kendall [6] has shown that conclusion (9) holds quite generally

even without this condition and, furthermore, that the measure A is absolutely continuous. Kendall's argument uses sophisticated Hilbert space techniques, but simpler arguments exist (cf. [15], [10]). The diagonal elements pi, of Markov chain transition matrices therefore

form a subclass (denoted by ") of the class of real, positive-definite functions, and the problem arises of characterizing YM. In a sense the answer is given by Markov chain theory, for a function p belongs to e,#

A CLASS OF POSITIVE-DEFINITE FUNCTIONS

95

if and only if there exists a family (pt;; i, j = 0, 1, 2, ...) of functions on (0, oo), satisfying W

Po(t) > 0, (12)

i=o

P1,(t) < 1,

W

Pt,(s + t) =

k=0

Ptk(S)Pki(t),

and

imPtt(t) = 1,

and such that p = Poo. But this, although it (apparently) removes the problem from the province of the theory of probability, cannot be said to provide more than a "solution in principle." To proceed further, observe that the function paa satisfies functional inequalities similar in form, but in addition to those defining the positivedefinite functions, which can be written as det (f(I to - to1)) > 0. (13) For example, if s, t > 0,

0 0, (26)

un = P(XX = aI Xo = a).

The corresponding assertion in continuous time is false.

Comparing (23) with (25), we see that any function p c 1 has the property that the sequence (p(nh)) belongs to S9 for all h > 0. Conversely, it is possible to show that a function continuous in [0, co] with (p(nh)) e ' for arbitrarily small h > 0 belongs to .)'. This means that some properties of functions in notably those referring to behavior for large t, can be

J. F. C. KINGMAN

98

deduced from the extensive theory of renewal sequences. For example, a celebrated result of Erdos, Feller, and Pollard [3] states that, if (un) is any renewal sequence for which the set {n > 1; u,, > 0} has 1 as its greatest common divisor, then lim u,,

n-.

exists. Hence, if p c 9,, lim p(nh) n-m

exists for all h > 0. Together with the uniform continuity of p, this implies that

p(co) = lim p(t)

(27)

t-.W

exists. This is a straightforward deduction, but a systematic technique for

deriving asymptotic properties by the "method of skeletons" has been elaborated in [9]. If in (24) we set z = e B1t, multiply by h, and let h --* 0 (keeping 0 > 0 fixed), the left-hand side converges to the Laplace transform (28)

r(9) = I

p(t)e-°t dt

0

of p. The limiting behavior of the right-hand side of (24) may be examined with the aid of Helly's compactness principle, and the result is the following fundamental characterization of JI: THEOREM 2. If p belongs to 9,, there exists a unique (positive) measure µ on (0, co] such that (29)

fro and such that, for all 0 > 0, (30)

(1 - e-X)µ(dx) < co

fo, p(t)e-0t dt = {B +

(1 - e-B")µ(dx)}

Conversely, if µ is any measure satisfying (29), there exists exactly one continuous function p satisfying (30), and that function belongs to

In other words, (30) sets up a one-to-one correspondence between the class 9 and the set of measures satisfying (29). It is important to note that (29) does not imply that µ is totally finite, although it must have µ(e, co) < co for all E > 0. Indeed, an easy Abelian argument from (30) identifies the total mass of µ with the limit q in (16): (31)

q = t-(0, cc].

A CLASS OF POSITIVE-DEFINITE FUNCTIONS

99

Letting 0 -->. 0 in (30) gives

f

(32)

p(t) dt =

0

1,

so that µ{oo} > 0 implies p(oo) = 0. On the other hand, if µ{oo} = 0, it may be shown that 1+

p(oo)

(33)

fx(dx)} 1

In case q is finite, (30) may be written

r(0) = B+ q - Je0(dx)) n=0

(0 +

i

q)-n-1{ f e exµ(dx)ln JI

q)-n-1 fe_0n(dx),

(0 + n=0

where µn is the n-fold Stieltjes convolution of µ with itself. This inverts to give (34)

p(t) _

n=0

f ?rn{q(t - x)}q-nµn(dx), 0

where 7rn denotes the Poisson probability (35)

Trn(a) = e - "an/n !.

This formula may be given a probabilistic interpretation as follows. Let

0 = To < Tl < T2 < .

. be random variables whose differences Tn = Tn+l - T. are independent, with distributions given by

P(-r, < x) = I - e-qx = q-1 µ(0, xl

(n even), (n odd),

for x > 0. Define a process (Z,; t > 0) to be equal to I on the intervals IT2m, T2m+jl and 0 elsewhere. Then routine calculations show that Z (which is a special type of alternating renewal process) has finite-dimensional distributions given by (19), where (36)

p(t) =

M=0

P(T2m < t < T2m+ )

Moreover, t

P(T2m < t < T2.+1) = f Tm{q(t - x)}q-mµm(dx), so that (34) and (36) coincide.

100

J. F. C. KINGMAN

The question then arises of giving a similar construction for the case q = µ(0, co] = oo. This is a more complex problem, but the key to its solution lies in the observation that expressions similar to (30) arise in the

theory of additive processes (processes with stationary independent increments). Indeed, for any µ satisfying (29), there exists an additive process (71t; t > 0) (nondecreasing, and increasing only in jumps) such that E(e-° t) = e-ewce),

(37)

where (38)

0(9) _ J(1 - e-ex)µ(dx).

Adding a deterministic drift, we obtain a process

et = t + 7t, with (39)

E(e-"t) = e-t(e+m(e)) = e-tl.(B)

It has been observed by Kendall (the proof being easiest if a strong Markov version of 77 is used) that the process defined by

Zt = 1 if 6s = t for some s, (40)

= 0 otherwise,

satisfies (19), where p is the function corresponding to µ in (30). Our starting point was the positive-definiteness of functions in Y,#, and it is therefore pertinent to remark that, as implied by the title of this article, the functions in . are also positive-definite. Indeed, it is proved in [10] that, if p E Y, there exists a non-negative integrable function.f such that (41)

ff(w)costdw.

P(t) = P(oo) + 0

No very simple proof of this result appears to be known. Although " is therefore a subset of the class of real positive-definite functions, it differs from that class in failing to be convex. This fact is important as the source of many of the difficulties encountered by the explorer of 9

It will perhaps help to consider a few examples of functions in 9. Suppose for instance that the measure µ is concentrated in an atom of mass q at a single point a. Then (34) shows that

= (42)

P(t)

[t/a7

7n{q(t - na)}. n=O

A CLASS OF POSITIVE-DEFINITE FUNCTIONS

101

This is an oscillating function, converging to the limit p(oo) = (I + qa)-1.

It is differentiable except at the point a, where it has left- and rightderivatives :

D_p(a) =

-qe-aa,

D+p(a) = q -

qe-11.

According to a famous theorem of Austin and Ornstein (see [1], Section 11.12) the functions pt, in a Markov chain are all continuously differentiable in (0, oo). It therefore follows that every function in A is so differentiable,

and therefore that (42) cannot define a member of .ill. Thus " is a proper subset of Y The local behavior of the particular p-function (42) is quite typical. For any µ satisfying (29) the function (43)

m(t) = µ(t, co]

is nonincreasing, right-continuous, and integrable over (0, T) for any finite T. Equation (30) may be thrown into the form (44)

r(O) = B-1{ 1 + ' m(t)e-0t dt } Jo l J

Expanding this formally, we obtain (45)

p(t) = I - f t b(s) ds,

where (46)

b(s) =

(_ 1)n lmn(s ), n=1

and Mn is the n-fold convolution of m with itself. It is shown in [12] that this formal expression is always valid, that the series (46) is uniformly absolutely convergent in every compact subinterval of (0, oo), and that Mn (n > 2) is continuous. Since m is continuous except for jump discontinuities at the atoms of µ, it follows that every function in 3 has leftand right-derivatives in (0, oo), and that (47)

D+p(t) - D_p(t) = µ{t}

for all t > 0. In particular, p is continuously differentiable in any interval free from atoms of p. Reversing the logic, the Austin-Ornstein theorem on the continuous differentiability of members of Y .I# is seen to be equivalent to the assertion

that, if p c 9,W, the corresponding measure µ is nonatomic in (0, oo). In

J. F. C. KINGMAN fact more is true, for a result of Levy (see [13], I) implies that it is absolutely continuous (except perhaps for an atom at oo). We shall return later to the problem of describing the measures p which correspond to members of Y For another kind of example, consider any renewal sequence-(un). By the 102

theorem of Chung quoted previously, there exists a discrete-time Markov chain satisfying (26). Denote its transition matrix by P. If c is any positive number, and I denotes the identity matrix, then (48)

Q = c(P - I)

defines the infinitesimal generator of a q-bounded Markov chain, with transition matrix

Pt = exp {c(P - I)t} = e - `t exp (c1P) ct) n P.

= e-ct n=0

rn(Ct)P',

n= =0

where 7Tn is given by (35). Hence, from (26) Trn(Ct)Un.

Paa(t)

n=0

Thus, for any (un) E .4 and any c > 0, the function (49)

P(t) =

Unin(Ct)

n=0

belongs to 9', and so also to 9. Not every p c L A' is expressible in the form of (49), since (49) implies that, for instance,

p'(0) = -c(1 - u1) > -00. If, therefore, 2 denotes the class of functions (49) with (un) E M and c > 0, we have the strict inclusions (50)

.2 -9,# -9.

For any p c . and any integer k, the sequence (p(nk -1) ; n = 0, 1, 2.... belongs to .?, and hence the function (51)

Pk(t) _

P(nk -1)irn(kt ) n=0

A CLASS OF POSITIVE-DEFINITE FUNCTIONS

103

belongs to 9. If S11 21 ... are independent Poisson variables with mean t, then Pk(t) = E{P[k-1151 + b2 + ... + WD,

and the weak law of large numbers therefore implies that

lira pk(t) = p(t)

(52)

k-w

for all t > 0. Thus P2 is dense in 9 (in the product topology), and so afortiori is Y. It is the fact that 9./ff is not closed in 9 that makes its identification such a delicate problem. It is of course possible that interesting topologies for Y exist in which 9.A1 is closed, but such topologies would need to be strong enough to prevent Pk converging to p, for all p E . - Y. If pl and P2 are two p-functions, construct as in Theorem I processes Z1, Z2 on distinct probability spaces Q, 522 to satisfy (19) with p = pl and p = P2, respectively. On the product space 4 = 521 x 522 define a new process Z by (53)

Zt(W1, W2) = Z1t((01)Z2t(0J2)

Using the product measure on 0, it is immediately apparent that Z satisfies (19), with (54)

P(t) = Pl(t)P2(t)

Thus the product of two p-functions is itself a p-function, and consequently the product of two members of 9 is in 9 It follows that, under the operation of pointwise multiplication, Y is a commutative Hausdorff topological

semigroup with identity. The arithmetical properties of this semigroup have been studied by Kendall [8] and Davidson [2], who have shown that it exhibits many of the properties classically associated with the convolu-

tion semigroup of probability distributions on the line. The latter is of course isomorphic to the semigroup of positive-definite functions (under pointwise multiplication) of which Y is a subsemigroup.

The fact that 9 is closed under pointwise multiplication permits the construction of further examples of p-functions. A particularly important use of this device is due to Kendall [7]. Let A, a be positive numbers, and construct a Poisson process 11 of rate A on (0, oo). Define Zt to be equal to 1

if no points of 11 lie in the interval (t - a, t), and 0 otherwise. Then, for

0 = tp < t1 < t2 < ... < tn, n l P{Zt. = 1 (a = 1 , 2, ..., n)} = P no points of H in U (ta - a, ta) }

a=1

e-AL

,

JJJ

J. F. C. KINGMAN

104

where L is the measure of the set n

U (ta - a, ta) n (0, co). a=1

It is not difficult to check that n

L=

min (ta - to _ 1, a), a=1

so that Z satisfies (19) with (55)

p(t) = exp {- A min (t, a)}.

Thus the function (55) belongs to Y for all A, a > 0 (though not to 9.,t because it is not differentiable at t = a). Since 9 is closed under multiplication, it contains every function of the form m

p(t) = exp

l

Ai min (t, ai) } i=1

with A j, ai > 0. The compactness of the space of p-functions then implies that, if A is any measure on (0, oo) with

Jmin(1,x)A(dx) < oo, then (56)

p(t) = exp {-f min (t, x)A(dx)}

defines an element of Y But the functions of the form

¢(t) =

J

min (t, x)A(dx)

are exactly the continuous, non-negative, concave functions on [0, oo) with 0(0) = 0. We have therefore proved that, for all such 0, the function (57)

p(t) = e-&(O

belongs to ''. Notice that the set of functions of this form is a convex subset

of the nonconvex set 9 The p-functions (57) play a central role in Kendall's theory, for they are the infinitely divisible elements of 9,, those possessing nth roots for all n. That they have this property is obvious; to see that they are the only ones

A CLASS OF POSITIVE-DEFINITE FUNCTIONS

105

let f be any function with the property that fa E 91 for arbitrarily small values of a. Then f > 0 and we may define 0 = -logf. From (18) with

n=3,

{

0 < F(tl, t2, tt'3; J a) {'

{'

=.fa(((t3) -J a(tl)fa(t3 - t1) _ a(t2)J a(t3 - t2) + {'

('

J a(t2- tl)J a(t3- t2) a(t1)('

= at -0t3) + c(t3 - t1) + 0(t2) - c(t2 - t1)) + O(a2) Since this holds for arbitrarily small a, we must have

qS(t3) - ON - t1) < 0(t2) - q(t2 - t1), which implies that 0 is concave.

Among the functions of the form (57) are the completely monotonic functions, which can be written in the form (58)

p(t) =

Je_tYv(dy)

for probability measures v on [0, oo). It is shown in [13, 11] that these

belong not merely to 3 but also to 9,#; they are indeed exactly the functions paa which can arise from reversible Markov chains. Thus far the only p-functions discussed have been standard, but there

are interesting examples of p-functions which fail to satisfy (15). For example, if the variables Zt are independent, with P(Zt = 1) = b, then (19) is satisfied with (59)

p(t) = b

(t > 0).

Hence (59) defines a p-function whenever 0 < b < 1, and this is not standard unless b = 1. Since the product of p-functions is a p-function, the function (60)

p(t) = bp(t)

is a nonstandard p-function for 0 b < 1 and p E These nonstandard p-functions are continuous except at the origin, but there are more irregular ones. Suppose for instance that G is any proper additive subgroup of the line, and define Zt to be I with probability one if t c- G, and 0 otherwise. Then (19) is satisfied with p the indicator function of G n [0, oo). If G is measurable, it necessarily has measure zero, so that p = 0 almost everywhere. In a sense these are the only two possibilities for measurable p-fu nctions. More precisely, the following result is proved in [14]. THEOREM 3. Let p be any p-function. Then exactly one of the following statements is true: (i) p is standard,

J. F. C. KINGMAN

106

(ii) there is a standard p -function p and a constant 0 < b < 1 such that

p(t) = bp(t), (iii) p is almost everywhere zero, (iv) p is not Lebesgue measurable.

It is also worth remarking that, in case (ii), as, more obviously, in case (iv), the process Z in Theorem 1 cannot be chosen to be measurable. It has already been noted that 9 is not closed as a subset of the compact space of p-functions, and it becomes relevant to identify the closure : of 9.. Since

b = lim pn(t), n-w

where

pn(t) = exp {-n min (t, -n-1 log b)} is of the form (55), the constant p-function (59) belongs to general p-function

as does the

bp(t) = lim pn(t)p(t) n-w

of type (ii). The important question is therefore which p-functions of type

(iii) belong to.. If in (42) we set a = 1 - q-'a and let q -* oc, we obtain the p-function (61)

p(t) = 1T(at)

=0

(t integral) (otherwise),

which is thus a p-function of type (iii) belonging to .. On the other hand, not every p-function belongs to Y. To see this, it suffices to note the inequality, discovered independently by Freedman and Davidson, which states that for every p c 9 and s < t, (62)

P(s) ? I + {p(t) -

}liz,

so long as p(t) >. This must therefore hold also for all p e

In par-

ticular, any p E9 of type (iii) must satisfy (63)

p(t) < 1,

for all t > 0. This is certainly not true of all p-functions of type (iii). It seems very likely that the upper bound j in (63) can be improved considerably. The maximum value of (61) is e-1 (when a = t = 1), and

all the evidence suggests that this is the sharp upper bound to p(t) for p c 9 of type (iii). It will be seen, therefore, that the identification of 9 is closely involved with establishment of inequalities between the values of p, true for all p e 9 These problems are extremely difficult, mainly because

A CLASS OF POSITIVE-DEFINITE FUNCTIONS

107

of the lack of convexity of 9; the exact form of the set (64)

{(p(l ), p(2)); p n Y} g [0, 1 ] x [0, 11

is not even known. The most important unsolved problem of the theory remains, however,

the determination of the class 9,# of those standard p-functions which can arise from Markov chains. In the one-to-one correspondence (30), 9./1

(as a subset of 0') corresponds to a subset' of the class of measures satisfying (29). The elements of #' are measures on (0, oo], but it is easy to show that the value of the atom at infinity is irrelevant to the question of membership of t'. Thus attention may be restricted to those it with µ{oo) = 0 (that is, to recurrent states); the corresponding subset of . ' is denoted by .JY.

There are of course many ways of combining Markov chains to form new ones, and these may be used to establish structural properties of the class M. Thus it is shown in [13, 1] that -0 is a convex cone, closed under countable sums, where these are consistent with (29). There are of course measures in M having infinite total mass, but these cause no real difficulty, since it can be shown (see [13, IV]) that every measure in # is the sum of a countable number of totally finite measures in .i i The convolution of a finite (or even infinite, under suitable convergence conditions) number of finite members of ./,ff is again in M. At the same time it must be stressed that ill is not closed in any obvious topology, since 9.0 is dense in r

Using these properties of M, we can at once exhibit large classes of elements of It is, for instance, immediately apparent that .h' contains any measure with density of the form a)

amn t me - nt

(65) m=0n=1

with amn > 0, so long as (29) is satisfied. On the other hand, we have already seen that measures in .ill must be absolutely continuous (with respect to Lebesgue measure). Moreover, it

is a consequence of another result of Austin and Ornstein ([1], p. 149) that the density of any nonzero measure in .idl must be everywhere strictly positive, and this density may also be shown to be lower-semicontinuous. These facts go a long way toward determining the class . K, and thus 9. , but a complete solution remains elusive.

The class i' N consists, of course, of all diagonal Markov transition functions p;i, and it may be asked whether corresponding results are available for the nondiagonal functions p;, (i .j). To describe the theory which has been developed to answer this and related questions, which depends on the concept of a quasi-Markov chain (see [11], [13]), would

J. F. C. KINGMAN take us too far afield, but the answer may be stated thus. The nondiagonal Markov transition functions p;; are exactly those which may be expressed as convolutions 108

P1 * dA * P2,

(66)

where pl and p2 belong to ..ill,

a = J pi(t) dt < oo, and A is a measure on [0, oo) with aA[0, oo) < 1,

which, apart from a possible atom at the origin, belongs to . The whole characterization problem for Markov transition probabilities therefore depends on the cone .ill. NOTE ADDED IN PROOF (May 1970): The problem of characterizing J'.111

is now solved, and simple necessary and sufficient conditions are known for a measure to belong to M. An account will appear as number V in the series [13]. UNIVERSITY OF SUSSEX

REFERENCES 1. CHUNG, K. L., Markov Chains with Stationary Transition Probabilities. Berlin: Springer-Verlag, 1967. 2. DAVIDSON, R., "Arithmetic and other properties of certain Delphic semigroups," Z. fur Wahrscheinlichkeitstheorie and verwandte Gebiete, 10 (1968), 120-172. 3. ERDOS, P., W. FEELER, and H. POLLARD, "A theorem on power series," Bull. Amer. Math. Soc., 55 (1949), 201-204. 4. FEELER, W., "Fluctuation theory of recurrent events," Trans. Amer. Math. Soc., 67 (1949), 98-119. 5. , An Introduction to Probability Theory and its Applications. New York: John Wiley & Sons, Inc., 1957. 6. KENDALL, D. G., "Unitary dilations of one-parameter semigroups of Markov

transition operators, and the corresponding integral representations for Markov processes with a countable infinity of states," Proc. London Math. Soc., 9 (1959), 417-431.

"Renewal sequences and their arithmetic," Symposium on Probability Methods in Analysis. Berlin: Springer-Verlag, 1967, pp. 147-175. , "Delphic semigroups, infinitely divisible regenerative phenomena, and 8. the arithmetic of p-functions," Z. fur Wahrscheinlichkeitstheorie and verwandte Gebiete, 9 (1968), 163-195. 9. KINGMAN, J. F. C., "Ergodic properties of continuous-time Markov processes and their discrete skeletons," Proc. London Math. Soc., 13 (1963), 593-604. "The stochastic theory of regenerative events," Z. fur Wahrscheinlichkeits10. 7.

theorie and verwandte Gebiete, 2 (1964), 180-224. 11.

"Linked systems of regenerative events," Proc. London Math. Soc., 15 (1965), 125-150.

109 A CLASS OF POSITIVE-DEFINITE FUNCTIONS 12. KINGMAN, J. F. C. "Some further analytical results in the theory of regenerative 13.

events," J. Math. Anal. Appl., 11 (1965), 422-433. , "Markov transition probabilities," Z. fur Wahrscheinlichkeitstheorie and

verwandte Gebiete, I: 7 (1967), 248-270; II: 9 (1967), 1-9; III: 10 (1968), 87-101; IV: 11 (1968), 9-17. 14.

, "On measurable p-functions," Z. fur Wahrscheinlichkeitstheorie and verwandte Gebiete, 11 (1968), 1-8.

15. LoYNES, R. M., "On certain applications of the spectral representation of stationary processes," Z. fur Wahrscheinlichkeitstheorie and verwandte Gebiete, 5 (1966), 180-186.

Local Noncommutative Analysis IRVING SEGAL 1. Introduction

Despite the inroads of linear functional analysis, much of analysis still deals ultimately not with vectors in a linear space, but with functions defined on a suitably structured point set, generally endowed with properties reminiscent of physical space. It seems as if some notion of physical space is quite possibly a primordial concept in the human mind and inevitably colors all our perceptions of nature and formulations of natural law. The mathematical emphasis on equations which are local in geometrical

space (or transforms of such equations) and the complementary physical idea that the primary forces in nature are exerted through local interactions (i.e., by a kind of physical contact in the underlying space) may similarly reflect the apparent lack of firm roots in human perception of any other equally broadly applicable type of model. In any event, the fact is that mathematicians have always concerned themselves largely with local processes, structures, and equations, a notable instance being the theory of partial differential equations; and that physically, locality of the interaction is probably the most general nontrivial guiding principle which has as yet been found effective. A classical situation in which ideas of locality have been quite successfully merged with ideas of causality, relativistic invariance, and other such foundational theoretical physical desiderata, is the theory of hyperbolic partial differential equations. In this connection, as well as quite generally in connection with partial differential equations, one deals with an entirely explicit and well developed notion of a local function of a given function. Specifically, a "local" function of a given class of functions f is a mapping from this class to another class, which has the form:.f -± g, where g(x) = rp(f(x)) for all x, and T is a given function defined on the range of values of f. This- notion of locality may be characterized intrinsically in various ways, depending on the mathematical contexts of the functions involved. For example, within the context of abstract measure theory, g is a local 111

112

IRVING SEGAL

function off if and only if it is measurable with respect to every sigma-ring with respect to which f is measurable. Within a linear C °° framework, a sheaf-theoretical characterization has been given by Peetre.

It seems logically natural and is scientifically relevant to extend the treatment of local functions, and the differential equations involving them, to the case of functions whose values are not necessarily numerical but may lie in a noncommutative algebra. This extension must be distinguished from

the familiar and important extensions to the cases of functions having values in either a given linear space or in a given vector bundle over the space. A natural type of algebra in the present connection, by virtue of its simplicity, technical effectiveness, and physical relevance in the quantum theory of radiation, is an algebra of operators in a Hilbert space. When the algebra is Abelian, one is more or less back in the familiar vector-valued

function case. When the algebra is not commutative, and there is no a priori information concerning the commutators of the basic operators, one is in a very general situation. With relatively strong regularity assump-

tions, such as compactness of the underlying space, boundedness of the operators involved, and uniform continuity of the mapping from the space to the operators, one has a framework explored by Atiyah and shown by him to clarify greatly the Grothendieck theory of vector bundles, from an analytical point of view. Within this context, it would be straightforward to extend the usual treatment of local functions, indeed in such a way that the idea in the theory of hyperbolic equations that the intervention of local lower-order terms does not essentially enlarge the region of influence of the basic equation carries over. However, even considerably weaker regularity assumptions are too strong to appear compatible with either of

two additional desiderata which are mathematically interesting and physically relevant: (i) invariance under an open simple Lie group, such as the Lorentz group; (ii) simple nontrivial assumptions on the form of commutators, such as the assumption that [f(x),f(x')] should lie in the center of the algebra, and not be identically zero, for any two points x and x'. I shall treat here aspects of a theory which is compatible with, and indeed makes essential use of such desiderata, for two reasons: (a) mathematically interesting, indeed quite unexpected, results emerge; (b) three centuries of the development of the mathematical theory of light can be summarized in the statement (now relatively very well established) that light is described in a categorical way by Maxwell's equations for operatorvalued functions satisfying these desiderata. More specifically, I shall first treat the question of the meaning and existence of nonlinear local functions of the generalized and relatively

singular functions which are the only ones satisfying the indicated desiderata. So-called "white noise" is an example of the type of singular

LOCAL NONCOMMUTATIVE ANALYSIS

113

function whose powers have no a priori meaning, but arise naturally in a nonlinear local theory compatible with (i) and (ii); it is more singular than heretofore treated functions, but its suitably renormalized powers will be shown to exist in a definite mathematical sense, and indeed to exist as a

strict function at individual points of the space in question; and these powers remain local functions, despite the formally infinite renormalizations involved. Second, an initial step in the "solution" (again in a sense which is radically generalized, but basically conceptually simple) of differential equations involving such renormalized local products will be treated in a somewhat more regular case than that of white noise. This case is nevertheless interesting in that it involves singularities which are comparable to apparently crucial ones of "quantum electrodynamics," as suitably mathematically formulated. There is little that is distinctive here about white noise, and logically the

present considerations should be extended to generalized stochastic processes from the standpoint pioneered by Bochner [1] and Wiener, and

developed by their students, among whom the work of Cameron and Martin has been especially relevant. Historically, the cited work was the chief mathematical source for analysis in function space, but the conception of quantum field theory initiated by Dirac, Heisenberg, Pauli, and others was scientifically stimulating if not mathematically entirely cogent. Local noncommutative analysis should help to round out and connect these rather distinct developments.

2. Nonlinear functions of white noise A. The intuitive concept of "white noise" may be formulated in a mathematically general way as follows. Let M be an abstract Lebesgue measure space: M = (R, R, r), where R is a given set, R is a given sigmaring of subsets of M, and r is a countably additive measure on R. The white noise on M is the isonormal process over the space H = RL2(M) of all real square-integrable functions on M, where this process is defined as follows. If H is a given real Hilbert space, a concrete isonormal process over H is a linear mapping, say 0, from H into the random variables (i.e., measurable functions modulo null functions) on a probability measure

space (i.e., one of total measure one), having the property that if x1,

.

.

., x,,

is an arbitrary finite set of mutually orthogonal vectors in H, then the random variables ¢(x,), ..., j(x,,) are mutually independent. This is equivalent to the property that for every finite set of vectors yl, ..., ym in H, the random variables q(yl).... , ¢(ym) have a joint normal distribution, and that E(c(x)) = 0, E(q(x)q(y)) = c<x, y> for arbitrary x and y in H,

114

IRVING SEGAL

where c is a constant. It will be no essential loss of generality for present purposes to assume that c = 1. It may be suggestive, although entirely unnecessary mathematically, to employ the symbolism

q(x) - f9(a)x(a)da,

x c L2(M),

where the purely symbolic function p corresponds to the intuitive white noise, i.e., the p(a) are for different a, mutually independent, identically distributed, normal random variables, of mean 0, and of variance of such "infinity" that the L2-norm of O(x) in L2 over the probability space is the same as the L2-norm of x in L2(M). It should perhaps be emphasized that, a priori, neither p(.) nor p(x) exists in a strict mathematical sense; qp( ) is a "generalized random function" or "stochastic distribution." Nevertheless, as is well known, formal linear operations on the p(.), including differential and integral operators, can be given mathematically effective formulation and treatment by their transformation into suitable mathematical operations on the unexceptionable object gs( ). These methods of treatment of linear aspects of "weak" functions break down when it is attempted to apply them to nonlinear considerations. Indeed, one might be tempted to believe that no effective notion of nonlinear function of a white noise can be given, since the apparently simpler concept of a nonlinear function of a Schwartzian distribution has proved incapable of effective nontrivial development. The fact is, however, that there is an additional structure which is frequently present in the case of weak stochastic or operational functions, which is lacking in the familiar case of a numerical function, on which such a concept may be based, and indeed, the function and its powers may be represented by mathematical, i.e., nonsymbolic, strict (i.e., nongeneralized) functions on M, in these cases. B. The linear theory of weak processes can to a large extent be pursued independently of the precise space M on which the process is a generalized

function. Thus the white noise, as defined earlier, is primarily a pure Hilbert space construct, and its basic theory can proceed quite independently of whether this space is a function space, as in Wiener's theory of the "homogeneous chaos," and if so, of how it is represented. The nonlinear theory, however, depends in an essential way on the multiplicative structure in the algebra of numerical functions on M. One could deal with this in part by introducing suitable topological linear algebras, but it is simpler, particularly in connection with the determination of an actual function on M representing the processes in question, to give a concrete treatment for

LOCAL NONCOMMUTATIVE ANALYSIS

115

functions on the given space M, endowed with appropriate additional structure to be indicated later. The problem is that of legitimizing in a mathematically effective and formally valid way expressions of the form F(p(a)), where qp(a) is the previously indicated symbolic white noise, and F is a given function of a real variable. It is too well known that this cannot be done in line with the usual mathematical ideas to require elaboration here; even the simplest nontrivial case F(A) _ A2 is totally elusive, for the square of the white noise on a measure space without atoms appears as an infinite constant on the space. I shall show that if one gives up for the moment the literal interpretation of the application of the nonlinear function F to a weak function as the limit of its applications to approximating strict functions, and insists only that the resulting generalized function i(a) - F(P(a)) should be a "local" function of p(a), in an intrinsically characterized sense, whose transformation properties under vector displacements in function space are in formal agreement with those of the intuitive function F(gp(a)), then one obtains essentially unique results in the crucial cases of the powers, F(A) = As'. These powers behave in many ways as do conventional powers, and are the limits of renormalized powers of strict functions, in a way which

clarifies their nature and illustrates the natural occurrence of additive "divergences," or apparent infinities. My method is a development from one used earlier in connection with quantum fields and Brownian motion. In terms of the conventional mathematical representation x(t) of the latter, the situation may be described as

follows. It was shown by Wiener that a suitable version of x(t) has the property of possessing fractional derivatives x(e)(t) of all orders e < ; for orders e > 2, the fractional derivative exists only in a generalized sense, as a stochastic distribution for example; for e = 1, the derivative is simply white noise on R1. Thus there is no problem in the definition of a power of x(e)(t) if e < z ; when e = 12, a literal interpretation leads to the result that x(112)(t)2 = oo with probability one; but with a suitable "renormalization," the powers of x(112'(t) are well-defined, finite, nontrivial stochastic distribu-

tions. For z < e < 1, the powers cannot exist in the same sense as in the case e = -1- (see [4]). However, they exist essentially as generalized functions ("distributions") on the probability space. This conclusion does not appear

valid in the still more singular case e = 1. It is shown here, by a method applicable to a variety of processes similar to white noise, that with a suitable increase in the regularity of the test functions employed, a similar conclusion holds. Due, however, to the failure of the Soboleff inequalities in an infinite-dimensional space, it is convenient to emphasize the use of Hilbert space methods, and to treat the generalized functions as sesquilinear forms on a dense domain of regular vectors in L2(S2), where Q

116

IRVING SEGAL

denotes the probability space in question, rather than as linear functionals. The formal correspondence is this: if f is a function, the associated sesquilinear form F is given by the equation: F(g, h) = f fgh. The case e = z has essentially the same singularity with respect to the formation of renormalized powers as a so-called scalar relativistic quantum process in a two-dimensional space time, as a process in space at a fixed time. The cases I < e < I similarly involve singularities quite comparable to those of n-dimensional such processes for n > 2. White noise is thus more singular than any such quantum process, as a process in space at a fixed time.

C. A natural mathematical category in which to treat renormalized powers of generalized stochastic processes is that of the ergodically quasiinvariant (EQI) processes. A linear map ¢ from a real (topological) linear vector space L to the random variables on a probability measure space, say S2, will here be called a process (it is also known as a "weak distribution," or as a "generalized random process," in the dual space to L); more precisely, it is a version of a process, the process itself being the class of all equivalent versions, relative to the usual equivalence relation (see [5]) of

agreement of probabilities defined by finite sets of conditions. It is no essential loss of generality to suppose that the random variables /(L) form a separating collection for the space Q, i.e., the minimal sigma-ring with respect to which they are all measurable is, modulo the ideal of all null sets, the full sigma-ring of Q; and this supposition will be made throughout. Such a process is called quasi-invariant if any of the following equivalent conditions holds: (i) For every y E L* (the dual space), there exists an invertible measurable transformation on S2, carrying null sets into null sets, and such that

q(x) - ¢(x) + <x, y> (for suitably chosen S2; otherwise the measurable transformation need not exist as a point transformation on 4, but only as an automorphism of the Boolean ring of measurable sets modulo the ideal of null sets). (ii) For every y E L*, there exists a unitary operator U on the Hilbert space K = L2(S2), such that UT(x) U -1 = W(x) + <x, y>I, where W(x) denotes the operator in K consisting of multiplication by q(x) (in general, an unbounded but always normal operator), I being the identity operator on K. (iii) There exists an automorphism of the algebra of measurable functions on S2, modulo the ideal of null functions, which carries (the equivalence class of) q(x) into (that of) O(x) + <x, y>.

These are various ways of asserting the "absolute continuity" of the transformations z -* z + y in the dual space L* relative to the weak

LOCAL NONCOMMUTATIVE ANALYSIS

117

probability measure in L* corresponding to ¢; when the process arises from a conventional countably additive probability measure in L* in the natural fashion they are equivalent to absolute continuity in the conventional sense; it was in this form that absolute continuity was first shown in function space, in the work of Cameron and Martin [2]. More specifically,

this work shows that the transformation g -*g +f is absolutely continuous in C [0, 11 with respect to Wiener measure, provided f satisfies a certain regularity condition. This result is closely related to (and follows from) the absolute continuity of arbitrary vector displacements in Hilbert space, relative to the isonormal process. When L is finite-dimensional, every process corresponds to a probability measure on L*, which is quasiinvariant if and only if it is absolutely continuous with nonvanishing density function. Ergodicity is similar generalization of the ergodic property of vector displacements in R", i.e., the nonexistence of invariant measurable sets

other than null sets or their complements. More specifically, an EQI process is a quasi-invariant one such that only the constants are invariant under all the automorphisms of the algebra of random variables induced from vector displacements in V. Again, the ergodicity of Wiener space relative to vector translations was shown by Cameron and Martin; and this is closely related to (and deducible from) the ergodicity of the isonormal process in a Hilbert space relative to vector displacements in the space. Suppose then that ¢ is an ergodic quasi-invariant process on a space L of functions x(.), so that symbolically ¢(x) ' f P(a)x(a) da, and consider in an intuitive preliminary way the problem of treating the "square" 0(2) such that 0(2)(x) ,.

f (9'(a))2x(a) da.

Mathematically, the indicated integral is in general entirely nebulous, but in a formal way it is immediately visible that it has the following properties : (i) Multiplication by q(2)(x) commutes with multiplication by q(x'), for

all x and x' (this follows from the commutativity of the operations of multiplication by given numerical functions). (ii) Under the automorphism of the algebra of random variables on S2 which carries O(x) into O(x) + F(x), where F(x) = J x(a)f(a) da (so that, symbolically, ft-)), 0(2'(x) is carried into 012)(x) + 2¢(fx) + f f 2x.

Now ergodic quasi-invariance implies formally that if any such function 0(2) exists, it is unique within an additive constant; it is totally unique if it

is integrable, and the normalization is made that its expectation value vanishes. It develops that such a notion of renormalized power can be

IRVING SEGAL

118

applied to generalized as well as strict functions on 4, and that in this way one is led to an effective definition of the renormalized powers of weak EQI processes through their intrinsic characterization by properties similar to (i) and (ii) and their normalization by vanishing expectation value.

Before this can be done, however, a suitable dense space of regular "test" functions on Q must be specified. A natural space to consider in this connection, invariantly attached to the underlying Hilbert space H = L2(M), and one studied intensively by Kristensen, Mejlbo, and Poulsen (see [3]), is a direct analogue to the space of all infinitely differentiable functions with infinitely differentiable Fourier transforms, all in L2i

over Rn (the so-called "Schwartz" space); however, the vectors of this space are insufficiently regular to serve as test functions, either in connection with a treatment of generalized functions as linear functionals or a treatment as sesquilinear forms. A suitable space of test functions may be derived from the imposition of suitable additional structure to the space M. There are two natural generalizations of R" from the standpoint of white noise theory: (i) the replacement of R" by an arbitrary complete Rieman-

nian manifold, and (ii) its replacement by an arbitrary locally compact Abelian group. In generalization (i) there is a natural distinguished operator, the Laplacian, which induces in a natural way a test function domain in L2(S2) which is effective at least in the case of R. It is probably

effective quite generally, but the present methods, designed to treat generalization (ii), leave this question open. In place of the Laplacian, an

arbitrary translation invariant operator B in L2(G), G being the given group, having the property that its spectral function on the dual group G* has the property that -1 E Lp(G*) for all sufficiently large p, will be used. The corresponding test functions are not necessarily bounded on S2, and for this and other reasons it is convenient to deal with sesquilinear forms, rather than linear functionals, on this domain of test functions. A mathematical statement of the existence of renormalized powers of white noise is conveniently made with the aid of the concept of the isonormal symmetric (quantum) process over a given complex Hilbert space H. This is the system (W, K, v, P) composed of a complex Hilbert space K, a mapping `F from H to the self-adjoint operators in K having the property that e`7(x)e+"(y) = e(ii2) Im <x.y>eiY(x+y) for arbitrary vectors x and y in H, a

unit vector v in K, and a continuous unitary representation r of the full unitary group on H by unitary operators on K such that

r(U)v = v,

r(u)'(x)r(U)-1 ='F(Ux)

for all unitary operators U on H and vectors x in H, such that v is a cyclic vector for the operators e`F(x', x c H; and uniquely characterized among such systems, within unitary equivalence, by the property that if for any

LOCAL NONCOMMUTATI VE ANALYSIS

119

self-adjoint operator X in H, df(X) denotes the self-adjoint generator of the one-parameter unitary group 1'(e`tx), t e R1, then d F(X) > 0 whenever X > 0. The connection with white noise comes about from the circumstance that if H, is any real subspace of H such that H is the direct sum, over the

real field, of H, and iHr, then the restriction map TIHr, relative to the expectation values defined by the functional E(A) = for any operator A in the ring of operators generated by the e"(x', x c Hr, is, within a certain isomorphism, the isonormal (stochastic) process over H apart from a factor of 212. For the nature of this isomorphism, see [6]; it will suffice here to point out that the 'F(x) for x e Hr are mutually commutative and determine a maximal Abelian ring of operators; this then implies by general theory that they may be identified with the multiplication operators associated with certain real measurable functions acting on

L2(S2) for a suitable measure space 0; from the circumstance that E(e`'F`x') = e-(1/4)IIx'12 = E(e`6W121'2), 0 may be chosen as the probability

space associated with white noise in such a way that respective expecta-

tion values, on the one hand for operators in K and on the other for random variables on S2, are equal. NOTATION. For any positive operator A in a Hilbert space H, DM(A) denotes the common parts of the domains of the e' (n = 1, 2, ...); while [D,(A)] denotes this set as a linear topological space in the topology in which a generic neighborhood of 0 consists of all x such that IIAkxjl < E for k < k°, for some a and k°.

THEOREM 2.1. Let G be a given locally compact Abelian group and B a given real, positive, translation-invariant operator in L2(G) such that E Lp(G*) for all sufficiently large p, where G* denotes the dual (character) group of G. Let ('F, K, v, U) denote the standard normal symmetric process over the Hilbert space H = L2(G). Let H denote dF(B). Then for arbitrary a c G and j = 0, 1, 2.... there exists a sesquilinear form 011(a) on [D,(H)] such that:

(i) The map (a, u, u') --> g1'l(a)(u, u') is continuousfrom G x

x

[D.(H)] into C1. (ii) pl"(a)(u, u') = 1, so that the indicated convolution exists and is again in this space, by the Hausdorff-Young Theorem. On the other hand, I gI2 is also in all L,(G*), under the indicated assumption on g, so that the inner product is indeed finite. It remains to consider the integral over region II. In this region, e < 2d; the integration over kn - s+ le

kr therefore contributes at most const. dcO t

,

if f, I g1, and the B(k,) -1 for j > n - s + I are replaced by constants which bound them. The resulting integral over the k1i ..., k,-, is then bounded by one of the form: const. f d - 2(a/r) + const. dk1 dk,,-s, which is finite if a is chosen sufficiently large. Taking the greater of the two a's involved, the required inequality has been established. REMARK 3.1.

The exponent r in the required decay for f, while possibly

best possible in the generality of Theorem 2, is not best possible in the relativistic case. Indeed, an appropriate specialization of the foregoing argument shows that in the case r = 2, the conclusion of Theorem 3.1 is valid if only I f(l) I = 0(111 -1). More specifically, the argument is the same

except that because of the slower decay for f, N(k) must be taken as B(k)-2. In this relativistic case, this is in Lp for p > 2; the two-fold convolution of such a function with itself is in Lq for q > 3 and hence has finite inner product with Ig12. This shows that f t)2:g(x) dx is, as a function of t, and as a Hilbert-space operator, a distribution of first order;

and it is known that it is not a distribution of zero order, i.e., a strict function. Similar considerations are applicable to higher powers in the relativistic case. If, for example, r = 3 and I f (l) I = 0(111 -2), the method given shows that the conclusion of Theorem 3.1 is implied by the convergence of the integral (*) with N(k) taken as B(k)-113. In the relativistic case, this func-

tion is in Lp for p > ; (in four space-time dimensions); the three-fold convolution with itself of a function in this case is in Lq for q > 3; it therefore has a finite inner product with I g12. COROLLARY 3.1. If, in addition, f is a real even function (or a translate of an even function), then the operator indicated in Theorem 3.1 has a selfadjoint extension. PROOF. The case of a translate of an even function reduces to that of an even function via transformation with I'(t'), t' being the translation in question. It is known (and easily seen) that there exists a unique conjugation J on K such that Jv = v and Jj(x, t)J-1 = O(x, - t); when f is even and g is real, the cited operator is real relative to J, and, being densely defined, has a self-adjoint extension.

IRVING SEGAL

130

REMARK 3.2. In colloquial language, Corollary 3.1 and Remark 3.1 imply that if 0 and 0 are independent scalar quantum fields (in four-

dimensional space-time), with periodic boundary conditions in space, and

if f(t) = 1 - It/el for It! < e and is 0 otherwise, then f :0(x, t)¢(x, t)2: f(t) dx dt has a self-adjoint extension as a conventional operator on the domain D.(H). The same is perhaps true for "quantum electrodynamics," which has a somewhat similar "interaction Hamiltonian," but this has not yet been verified. The cited functions f(t) are particularly convenient for generalized Riemann product integration of the type indicated earlier, for a simple sum of translates can represent a function identically one in an arbitrarily large interval; similar functions are available for higher values of r. MASSACHUSETTS INSTITUTE OF TECHNOLOGY

REFERENCES 1. BOCHNER, S., Harmonic Analysis and the Theory of Probability. Berkeley, Cal.: University of California Press, 1955. 2. CAMERON, R. H., and W. T. MARTIN, "Transformations of Wiener integrals under translations," Annals of Mathematics (2), 45 (1944), 386-396. 3. KRISTENSEN, P., Tempered Distributions in Functional Space, Proceedings of the Conference on Analysis in Function Space, W. T. Martin and I. Segal, eds. Cambridge, Mass.: M.I.T. Press, 1964, Chap. 5, pp. 69-86.

4. SEGAL, I., "Transformations in Wiener space and squares of quantum fields," Advances in Mathematics 4 (1970), 91-108. 5.

"Tensor algebras over Hilbert spaces, II," Annals of Mathematics, 63 (1956), 160-175.

6. 7.

, "Tensor algebras, I," Trans. Amer. Math. Soc., 81 (1956), 106-134. , "Nonlinear functions of weak processes, I and II," Journal of Functional Analysis 4 (1969), 404-456, and in press. (These articles are referred to as "I"

and "II".) 8.

, Local Nonlinear Functions of Quantum Fields, Proceedings of a Conference in Honor of M. H. Stone. Berlin: Springer (in press).

Linearization of the Product of Orthogonal Polynomials RICHARD ASKEYI

In [3] Bochner showed that there is a convolution structure associated with ultraspherical polynomials which generalizes the classical L1 convolu-

tion algebra of even functions on the circle. Bochner uses the addition formula to obtain the essential positivity result. In [18] Weinberger shows that this positivity property follows from a maximum principle for a class

of hyperbolic equations. Hirschman [8] has dualized this convolution structure and has proven the required positivity result by means of a formula of Dougall which linearizes the product of two ultraspherical polynomials. We will prove a theorem which gives most of Hirschman's results as well as other positivity results which were not previously known. In particular we obtain a positivity result for most Jacobi polynomials. In view of the known duality for the classical polynomials as functions of n

and x, this suggests strongly that the Bochner-Weinberger result can be extended to Jacobi polynomials. This is the only missing step in proving the positivity of some Cesaro mean for most Jacobi series, and this result can then be used to solve some LP convergence problems for Lagrange interpolation.

The theorem we prove is concerned with the problem of when an orthogonal polynomial sequence pn(x) satisfies n+m

(1)

Pn(x)Pm(x) = >

k=ln-ml

akPk(x),

ak _ 0.

Any sequence of orthogonal polynomials satisfies a recurrence formula xPn(x) = anpn+l(x) + PnPn(x) + Ynpn-1(x), Nn is real, anYn+1 > 0. We normalize our orthogonal polynomials pn(x) by pn(x) = xn + . Then (2) takes the form (2)

p-1(x) = 0,

(3) P1(x)pn(x) = Pn+1(x) + anpa(x) + bnpn-1(x) 1 Supported in part by NSF grant GP-6764. 131

RICHARD ASKEY

132

In order to have (1) we must have an > 0. bn > 0 is a general property of orthogonal polynomials. Our main theorem is THEOREM 1 .

I f (3) holds f o r n = 1, 2,

... , an > 0, b,, > 0 and an+ 1 > an,

bi+1 > b,, then (1) holds for n, m = 0, 1,

....

By symmetry we may assume m < n and we will prove the theorem by induction on m. Since it holds for m = 1, n = 1, 2.... we may assume it holds for m = 1, 2, ..., l and prove it for m = l + 1, 1 < n. We have Pi+1(x)Pn(x) = P1(x)p,(x)Pn(x) - a,p,(x)Pn(x) - b,p1_1(x)Pn(x) = p1(x)P.+ 1(x) + anp,(x)Pn(x) + bnp1(x)Pn -1(x) - a,p1(x)Pn(x) - b1p1-1(x)Pn(x) = p1(x)Pn+1(x) + (an. - a1)p,(x)Pn(x) + (b, - b,)P1(x)Pn-1(x) + bi [p1(x)Pn-1(x) - Pi -1(x)Pn(x)]

Since an - a, > 0 and bn - b, > 0 and b, > 0 by the induction assumption we are finished if we can take care of the last term. Using (3) again, we see that Pdx)Pn-1(x) - P'-1(x)Pn(x)

= Pn-1(x)[P1(x)P,-1(x) - a1-1p1_1(x) - b1_1P1-2(x)]

- p1_1(x)[Pl(x)Pn-1(x) = (an-1 - al-1)Pn-1(x)Pt-1(x) + b,-1

an-1Pn-1(x) -

bn-1Pn-2(x)]

X [P1-1(x)Pn-2(x) - Pn-1(x)P1-2(x)]

+ (bn_1 -

b`-1)Pj-1(x)Pn-2(x)

Continuing in this fashion we have terms that have positive coefficients except possibly for the last one p1(x)Pn-I(x) - Pn-t+1(x). But we use (3) again to see that this term is an _,pn _,(x) + bn _, pn _, _ 1(x) and an -1 > 0, bn_, > 0. This completes the proof of Theorem 1. The Charlier polynomials (see [6]) normalized in this way satisfy (4)

d1(x; a)dn(x; a) = do+1(x; a) + ndn(x; a) + andn_1(x; a),

a > 0,

and Theorem 1 is immediately applicable. Similarly, Theorem I is applicable to Hermite polynomials since (5)

H1(x)Hn(x) = Hn+1(x) + 2nHn_1(x).

Here HH(x) = 2nxn + (6)

; thus the normalized polynomials K,,(x) satisfy

Kl(x)Kn(x)

= Kn+1(x) + 3 Kn-1(x)

ORTHOGONAL POLYNOMIALS

133

For Laguerre polynomials L,",(x) we let Qn(x) _ (- 1)nLn(x)/n! Then the recurrence formula (7)

(n + 1)Ln+ 1(x) - (2n + a + 1 - x)Ln(x) + (n + a)L,", _ 1(x) = 0

becomes

Q1(x)Qn(x) = Qn+1(x) + 2nQn(x) + n(n + a)Qn-1(x),

(8)

and Theorem 1 applies if a > - 1.

The Meixner polynomials pn(x) = F(-n, -x, g; 1 - 1/y), g > 0, 0 < y < 1 satisfy

(9) - x(y - y

- (n + n + N)/ mn(x) + y 91n - 1(x)-

mn(x) = (n + 9)9'n+1(X)

See [12]. If we normalize these /polynomials accordingly Kn(x) = [1

(N 1/Y)]n

pn(x) = xn + ... ,

then (9) becomes (10)

K1(x)Kn(x) = Kn+1(x) +

n0(1

-+ y

y))

Kn(x) +

- y)2- 1) Kn 1(x)

ny(n + S I

and the assumptions of Theorems I are satisfied for 0 < y < 1, p > 0. We now consider the most important special case, that of the Jacobi polynomials Pn""3 (x). They satisfy

Pna.9)(x)= l

(11)

r(2n+a+9+ 1) xn+.. 2F(n+a+9+1)n!

and

2(n + 1)(n+a+g+ 1)(2n+a+p)Pn+i)(x)

=(2n+a+9+

(12)

1)[(2n+a+fl)(2n+a+P+2)x+a2_P2]

2(n + a)(n + fl)(2n + a + fl +

x

If we define R(na.a)(x) = 2nr(n

+a+

+ 1)n! pna,a)(x) = xn + .. .

F(2n+a+/3+1)

we have Ria.0 )(x)R(a.9)(x)

= R(n+Bi(x) + (13)

+

a-

(a + fl + 2)(a + 9) Rn./t)(x) a+p+2 [ 1 - (2n+a+fl+2)(2n+a+g)J 4(n + a + /3)(n + a)(n + g)n

(2n+a+fl+ 1)(2n+CZ +p)2(2n+cc +f- 1)

= Rca.9)(x) + anR(a.9)(x) + g nR(a,a)(x) n n-1 n+1

Rna.a)(x)

RICHARD ASKEY

134

It is clear that an ? 0, f > 0 if a > P and that an+1 ? an if either a + P > 0 or a = fi. We take first the case a = which is much easier. Then

_

_

(n + 2a)n

4n2 + Ban

1

4 [4n2 + 8an + 4a2 - 1]

(2n + 2a + 1)(2n + 2a - 1)

l

(4a2 + 1)

r1

41

4n2 + 8an + 4a2 + 1

and clearly Nn+l > fn if a > i. (1) fails for -1 < a = fi < -1 since the 5 coefficient of p3(x) in p2(X)p3(x) _ akpk(X) is negative. This still leaves k=0

us with an open problem because the theorem is known to hold for

a = fl > - (see [9]), and it would be desirable to have this result follow

z theorem. It is not surprising that the proof fails for these from a general values of a, since the author has long suspected that the theorem should be easier to prove for large a. This is because is much larger than

-1 < x < 1, x fixed, for a large and large n. There are two other special cases which we can handle easily, a + fl = 0

anda+fl= 1. If a + P = 0 then

_ 4n2(n2 - a2) fln

- n2 - a2

(4n2 - 1)4n2

1

4n2 - 1 = 4

[1

Pnif a2-4 >_0ora>=i. If a+fi= 1then

and fl+ P"

_ 4(n + 1)n(n + a)(n + 1 - a)

- (n + a)(n + 1 - a)

(2n + 2)(2n + 1)2(2n) _

4(n + 1)2

(n + '1)2 - ( - a)2

4(n +

l2)2

and i}1 >_ fn if a>_ 1, a = 1 -/3. A natural conjecture is that Nn+1 > Nn if

However, to have Pn+1 > P. and Nn - 4 we must have fin < 4 and it is easy to check that ,8n > 4 for large n if a2 + N2 < 1. In spite of the seeming complexity of the problem of finding out when Pn+ 1 fln, it is relatively easy. We let k = (a + P)/2. Then _

(n2 + 2kn)(n2 + 2kn + as)

(n + k)2[(n + k)2 + 4]

If k2 > 4 + ap, or a

- r _ L

k2

(n + k)2] [

_

(k2 - ' - aP)l

(n + k)2 - 4 J

1, we are finished since each of the factors on the right-hand side is an increasing function of n. A long routine calcula-

ORTHOGONAL POLYNOMIALS

tion allows us to conclude that Pn+1 >

a-/3 /3 and a + /3 > - 1; therefore we shall spare the reader the calculations which led to these results. The last result adds very little to la ± 91 > 1, a > 191 in any case. As was pointed out in [2] there is a symmetry between (a, /3) and (/3, a). We have Pna,a)(-X)

= (-l)"P//'a)(X);

thus the results we obtained for a > 0 by normalizing so that could be obtained for a < /3 if we normalize so that 1) > 0.

0

The methods used to prove Theorem 1 can be used to prove other theorems as well. Theorem 2 is equivalent to Theorem 1, but it is worth stating because it has the same form as Weinberger's theorem. THEOREM 2.

LetLnk(n) = k(n + 1) + ank(n) + fnk(n - 1), n = 0, 1, 2,

... , k(- 1) = 0. If a(n), n = 0, 1, ... , is given we consider the difference equation (14)

Ona(n, m) = Oma(n, m),

n, m = 0, 1, ..

where a(n, 0) = a(0, n) = a(n), a(n, -1) = a(-1, n) _ 0. Then if a(n) 0, n = 0, 1, . . .,and an > 0, 9n+1 > O, an+1 > an, 9"+2 > N n+i, n = 0, 1, . ., we have a(n, m) > 0, n, m = 0, 1, .... .

The proof is exactly the same as the proof of Theorem 1. (14) can be considered as a strange type of hyperbolic difference equation and this raises the question of which other hyperbolic difference equations have a maximum principle. We give one other application of this type of argument. About twenty years ago P. Turan established a very interesting inequality for Legendre polynomials (15)

P"-1(x)Pn+1(x) - [Pn(x)]2 < 0,

-1 < x < 1.

Inequalities of this sort were exhaustively studied by Karlin and Szego for most of the classical polynomials in [13]. It has been thought that these inequalities were in some sense restricted to the classical polynomials because they are the only polynomials which satisfy second-order equations in both n and x. We shall obtain a Turan-type inequality for a wide

RICHARD ASKEY

136

class of orthogonal polynomials which will contain the known Turan inequality for Hermite polynomials. THEOREM 3.

Let pn(x) satisfy pn(x) = xn +- and P1(x)Pn(x) = Pn+1(x) + bnPn-1(x)

(16)

where p1(x) = x + a and bn > 0, bn+1 >= bn, n = 1, 2, .... Then

-00 < x < 00, n = 1, 2, .... For n = 1, (17) is obvious since po(x)p2(x) - pi(x) _ -b1 < 0. We Pn-1(x)Pn+1(x) - pn(x) < 0,

(17)

complete the proof by induction. Pn+1(x)Pn-1(x) - Pn(x) = PiPnPn-1 - bnPn-1 - Pn

= Pn[Pn + bn-1Pn-2] - bnPn_1 - Pn

(18)

2

_ (bn-1 - bn)pn-1 + bn-1[PnPn-2 -

P2n-11

It is interesting to observe that Theorem 3 is in one sense a best possible theorem. In [13, p. 131] it was pointed out that if pn(x) satisfies (17) and rn(x) = knpn(x), then rn(x) satisfies (17) if knkn+2 > 0, knkn+2 - kn+1 < 0, n = 0, 1, .... In [1] it was observed that these conditions are necessary if Pn(X) = anxn + . Thus the polynomials pn(x) with the normalization are a natural class to consider. For this class, bn > 0 and pn(x) = xn + bn+1 > bn are necessary conditions for (17) to hold. In (18) if we take x to be a zero Ofpn-1(x) we see that bn-1 > 0 is necessary. Also in (18), if we look at the highest power of x which appears on the right-hand side it is (bn_1 bn)x2n-2; thus bn_1 < bn is necessary. Whenever non-negative numbers occur in a problem it is natural to see if any probability is lurking in the background. We have no probabilistic meaning for the conclusion, but the hypothesis does have a probabilistic interpretation. Karlin and McGregor [11] have shown that to each birth

and death process on the non-negative integers with zero a reflecting barrier there corresponds a sequence of orthogonal polynomials whose spectral measure is concentrated on [0, co). These polynomials Qn(X) satisfy

Q0(x) = 1, (19)

-xQ,,(x) = N'nQn-1(x) - (An + µn)Qn(x) + AnQn+1(x), I1o = 0, A. > 0, Ian+1 > 0, n = 0, 1, ....

From (19) it is easy to see that Qn(x) =

-1 A0A1

n

. A.-,

xn + ...

thus

Rn(x) = A0A1... An-1(

1)nQn(x)

ORTHOGONAL POLYNOMIALS

137

is normalized in the same way we normalized our polynomials. (19) then becomes (20)

RI(x)Rf(x) = Rn+,(x) + (A. + Fin + A0)Rn(x) +

µnAn_1Rn_1(x)

A sufficient condition that the coefficients in (20) satisfy the hypothesis of Theorem 1 is that An+I > A. > 0, N'n+1 > Fin > 0. In terms of birth and death processes this condition says that the rate of absorption from state n to state n + 1 which is given by A,, does not decrease with n nor does the rate of absorption from state n to state n - 1 which is given by µn. We conclude with some references to earlier work on Theorem 1. For Legendre polynomials (1) was stated by Ferrers [7] and proofs were given shortly thereafter by a number of people. Dougall [4] stated (1) for ultraspherical polynomials and a proof was first given by Hsu [9]. For Hermite polynomials (1) was given by Nielson [15] and for Laguerre polynomials the ak were first computed by Watson [17] and given in a different form so that it was obvious that (1) holds by Erdelyi [5]. Hylleraas [10] proved

(1) for Jacobi polynomials for a = fl + 1 and in unpublished work Gangolli has remarked that (1) holds for a = k, fi = 0; a = 2k + 1, P = 1; a = 7, P = 3, k = 1, 2, .... The case P _ --, a >_ P follows from the case a = fl in a standard way. Thus we have new proofs of (1) in all

the cases that have previously been established except a = fl + 1, -1 < fi < -I and -I < a = fl < -1. However, in all the cases except the ones considered by Gangolli, ak was explicitly found, and it is sometimes necessary to have it exactly to use (1). ak can be computed explicitly in the case of Jacobi polynomials, but it seems impossible to use it in the

form that it has been found [14, (3.7)]. In a preprint just received, G. Gasper has proved (1) for Jacobi polynomials,

a>

a+p+1

>_ 0,

UNIVERSITY OF WISCONSIN MADISON

REFERENCES 1. ASKEY, R., "On some problems posed by Karlin and Szego concerning orthogonal polynomials," Boll. U.M.I., (3) XX (1965), 125-127.

2. ASKEY, R., and S. WAINGER, "A dual convolution structure for Jacobi polynomials," in Orthogonal Expansions and Their Continuous Analogues. Carbondale, Ill.: Southern Illinois Press, 1968, pp. 25-26. 3. BOCHNER, S., "Positive zonal functions on spheres," Proc. Nat. Acad. Sci., 40 (1954), 1141-1147.

4. DOUGALL, J. "A theorem of Sonine in Bessel functions, with two extensions to spherical harmonics," Proc. Edinburgh Math. Soc., 37 (1919), 33-47. 5. ERDELYI, A., "On some expansions in Laguerre polynomials," J. London Math. Soc., 13 (1938), 154-156.

RICHARD ASKEY

138

6. ERDELYI, A., Higher Transcendental Functions, Vol. 2. New York: McGraw-Hill, Inc., 1953. 7. FERRERS, N. M., An Elementary Treatise on Spherical Harmonics and Subjects Connected With Them. London, 1877. 8. HIRSCHMAN, I. I., Jr., "Harmonic Analysis and Ultraspherical Polynomials," Symposium on Harmonic Analysis and Related Integral Transforms. Cornell, 1956.

9. Hsi), H. Y., "Certain integrals and infinite series involving ultraspherical polynomials and Bessel functions," Duke Math. J., 4 (1938), 374-383. 10. HYLLERAAS, EGIL A., "Linearization of products of Jacobi polynomials," Math. Scand., 10 (1962), 189-200. 11. KARLIN, S., and J. L. McGREGOR, "The differential equations of birth and death

processes and the Stieltjes moment problem," Trans. Amer. Math. Soc., 86 (1957), 489-546. 12.

, "Linear growth, birth and death processes," J. Math. Mech., 7 (1958),

643-662. 13. KARLIN, S., and G. SzEGO, "On certain determinants whose elements are orthog-

onal polynomials," J. d'Anal. Math., 8 (1961), 1-157. 14. MILLER, W., "Special functions and the complex Euclidean group in 3-space, II," J. Math. Phys., 9 (1968), 1175-1187. 15. NIELSEN, N. "Recherches sur les polynomes d'Hermite," Det. Kgl. Danske Viden. Selskab. Math. fys. Medd. I, 6 (1918). 16. SzEGO,G. Orthogonal Polynomials. Amer. Math. Soc. Colloq. Pub. 23. Providence, R.I.: American Mathematical Society, 1959. 17. WATSON, G. N., "A note on the polynomials of Hermite and Laguerre," J. London Math. Soc., 13 (1938), 29-32. 18. WEINBERGER, H., "A maximum property of Cauchy's problem," Ann. of Math., (2) 64 (1956), 505-513.

Eisenstein Series on Tube Domains WALTER L. BAILY, JR.1

Introduction

We wish to prove here that under certain conditions the Eisenstein series for an arithmetic group acting on a tube domain [8], [9] in Cm generate the field of automorphic functions for that group. The conditions are that the domain be equivalent to a symmetric bounded domain having a 0-dimensional rational boundary component (with respect to the arith-

metic group) [3, Section 3] and that the arithmetic group be maximal discrete in the (possibly not connected) Lie group of all holomorphic automorphisms of the domain. More precisely, under the same conditions, we prove that certain types of polynomials in certain linear combinations of the Eisenstein series generate a graded ring R such that the graded ring

of all automorphic forms of weights which are multiples of a certain integer with respect to the given arithmetic group, satisfying a certain "growth condition at infinity," is the integral closure of 91 in its quotient field. The objective in proving this result is that by using it one can prove that the field generated by the Fourier coefficients of certain linear combin-

ations of Eisenstein series is also a field of definition for the Satake compactification of the quotient of the domain by the arithmetic group [3]; when that field is an algebraic number field, and especially when it is the rational number field, one may hope for arithmetic results to follow. Our methods here have been adapted largely from the proof of this result in a special case (see [12]).

We now describe our results in more detail. Let G be a connected, semisimple, linear algebraic group defined over the rational number field Q. Let R be the real number field. We assume at the outset that G is Q-simple, that G°n (the identity component of G,,,) is

centerless (i.e., has center reduced to {e}) and has no compact simple

factors, and that if K is a maximal compact subgroup of GR, then = KM, is a Hermitian symmetric space. (Later it will be easy to relax 'The author wishes to acknowledge support for research on the subject matter of this paper from NSF grant GP 6654. 139

140

WALTER L. BAIL Y, JR.

these assumptions somewhat.) Then Gis centerless (see [3, Section 11.5% the absolutely simple factors of G (all centerless) are defined over the algebraic

closure of Q in R, and we may write (see [3, Section 3.7]) G = .gk,QG', where G' is absolutely simple and k is a totally real number field. Moreover,

X is isomorphic to a symmetric bounded domain D in C". We make the further assumption that X is isomorphic to a tube domain (1)

Z={X+iYECM1YES},

where S is a homogeneous, self-adjoint cone in R"'; according to [8, Sections 4.5, 4.9, and 6.8 (Remark 1)], this can also be expressed by saying

that the relative R-root system RE of G does not contain the double of any element of E, i.e., that RE is a sum of simple root systems of type C. In [3, Section 3], we have defined the concept of "rational boundary component of D" and have proved that a boundary component F of D is rational if and only if the complexification P of N(F) = {g c- GR1 Fg = F}

is defined over Q. We now add the assumption that there exists a rational

boundary component FO of D such that dim FO = 0. Then 3; may be identified with the tube domain Z in such a way that every element of N(F0) acts by a linear affine transformation of the ambient vector space C' of Z and such that every element of the unipotent radical U = U(F0) of N(F0) acts by real translations. If F is any boundary component of D, let

Nh(F) be the normalizer of N(F) in the group Gh of all holomorphic automorphisms of X; in particular, let Nh(Fo) = Nh. Let r be an arithmetic subgroup of Gh; i.e., I' n GZ is of finite index in

P and in Gz. Put F' = GR n F and let FO = F n Nh. Clearly F' is a normal subgroup of F. If g c Gh and Z E Z, letj(Z, g) denote the functional determinant of g at Z. We shall see that for y c- F0, j(z, y) is a root of unity.

Let GQ = GQ n GR and if a E GQ, let FO, = F n aNha -1 and let la be the least common multiple of the orders of all the roots of unity which occur in the form j(*, y) for y c a-1FO aa. Then for any sufficiently large positive integer 1, divisible by Ia, the series (2)

E,.a(Z) _

ver/ro.a

j(Z, ya)1

converges absolutely and uniformly on compact subsets of T and represents

there an automorphic form with respect to F. Let G*Q be the normalizer in Gh of G. We shall see that F c G. Let NQ = Nh n G. Then we shall also see that G*Q is the disjoint union of a finite number of double cosets FaNQ, where a runs over a finite subset A

EISENSTEIN SERIES ON TUBE DOMAINS

141

of G. For each a c A, let ca be a complex number and denote the assignment a i- ca by c. We put aEA

Our main interest will be for the case when ca 0 for all a c A. Let Z = `;/F and let 23* be the Satake compactification (see [3]) of 8. The main result of this paper is the following theorem : THEOREM. Let I' be a maximal discrete arithmetic subgroup of Gh, let other notation be as above, and fix the mapping c such that ca 0 for all a c A. Then every meromorphic function on `B* can be expressed as the I > 0, quotient of two isobaric polynomials in the Eisenstein series where 1° is the l.c.m. of all la for a c A. If B is a basis for the module of isobaric polynomials of sufficiently high weight in these, then the elements

of B may be used as the coordinates of a well defined mapping W of Q * into complex projective space. The variety 52,E = W( *) is birationally equivalent

to 3*, %* is the normal model ofand if k is a field of definition for V_ (e.g., the field generated by all Fourier coefficients of all elements of B), then there exists a projective variety defined over k which is biregularly equivalent

to i*. 1. Notational conventions

In what follows, if H is a group, g, an element of H, and X, a subset of H, let 9X = gXg-1. If P is a subgroup of H, the phrase "P is self-normalizing" will be used to mean that P is its own normalizer in H. If H is topological, H° will denote the component of H containing the identity. Use C and Z to denote the complex numbers and rational integers, respectively. 2. Tube domains

In this section only, G is a centerless, connected, simple linear algebraic group defined over R. With notation as in the introduction, the space X = K\G° is isomorphic to a tube domain given by (1). The cone ft may be described as the interior of the set of squares of a real Jordan algebra J with R'" as underlying vector space. We denote by N the Jordan algebra norm in J. We have [Gh: GO] = I or 2, and the domains included under each case are described in [3, Section 11.4]. In every case, GR contains an element o which acts on Z by a(Z) = -Z -1 (Jordan algebra inverse), and j(Z, a) N(Z) for a certain positive integer P. In each case where [Gh: G°R] = 2, Gh contains an element r of order two, not in GR, such that

142

WALTER L. BAIL Y, JR.

r operates on Z by a linear transformation of C°`. Hence, j(Z, r) (a constant function of Z). Let RT be a maximal R-trivial torus of G with relative R-root system RE and simple R-root system RA. Then RE is of type C and we may speak of the compact and noncompact roots in RE (see [3, Section 1]). Precisely one simple root a is noncompact. Let S be the 1-dimensional subtorus of RT on which all simple R-roots vanish except a. The centralizer(S) of S 1

and the positive R-root groups in G generate a maximal R-parabolic subgroup P of G. Let Pl = P n GR. Then Pl = N(F0) for some 0-dimensional boundary component Fo of (the bounded realization of) X, and X may be identified with T in such a way that every element of Pl acts by a linear affine transformation on the ambient affine space C'4 of Z and the unipotent radical U of P1, by real translations. Using the Bruhat decomposition relative to R, one may prove that if g E GO., then g e Pl is also necessary in order for g to act by a linear transformation of Cm. Direct calculation shows that T normalizes Pl in each case where [G,: GI] = 2, and since Pl is self-normalizing in GO (see [3, Section 1.5(1)]), it follows that r and Pl generate the normalizer Nh of Pl in Gh and that Nh = Pl U TPl. Hence, j(Z, g) is a constant function of Z if and only if g E Nh; and N(F0) = Nh n GO. Moreover, a E V'(RT), and viewed as an element of the relative R-Weyl group of G, sends every R-root into its negative. Then if g E.(RT)a, we have j(Z, g) = c is a constant, since

ff(RT) - s(S) acts linearly on Z and in particular each element of it multiplies the norm function N by a constant. 3. Relative root systems

We now lift the assumption that G be absolutely simple and return to the general assumptions of the introduction. Then G = -qk,QG', where k is a totally real algebraic number field and G' is simple and defined over k. According to [3, Section 2.5], we may choose a maximal Q-trivial torus QT, a maximal R-trivial torus RT, and a maximal torus Tin G such that T is defined over Q and QT c RT - T. Then for suitable maximal k-trivial

torus kT', R-trivial torus RT', and maximal k-torus T' in G' we have kT' - RT' - T', QT - . k,Q(kT'), T = ,,,QT'. We take compatible orderings (see [3, Section 2.4]) on all the root systems. If P is a Q-parabolic subgroup of G and U is the unipotent radical of P, then P = Mk,QP' and

U = Mk,QU', where P' is a k-parabolic subgroup of G' and U' is its unipotent radical; if P is maximal Q-parabolic, then P' is maximal k-parabolic, and even maximal R-parabolic in G'. If a c RE, then the restriction of a to QT is either zero or a simple Q-root; in the latter case, a is called critical; each simple Q-root is the restriction of precisely one

EISENSTEIN SERIES ON TUBE DOMAINS

143

critical R-root for each irreducible factor (see [3, Sections 2.9, 2.10]). If F

is a rational boundary component, then N(F)c = P is a maximal Qparabolic subgroup of G (see [3, Section 3.7]). Until further notice, let F be

the 0-dimensional rational boundary component Fo of the introduction, let P = N(F)c, and let U be the unipotent radical of P. We may assume the relationship between P, QT, and QA to be that P contains the minimal Q-parabolic subgroup of G generated by .(QT) and the unipotent subgroup of G generated by the positive Q-root spaces. By [5, Section 4.3], this determines P within its GQ-conjugacy class. Since dim F = 0, it follows

that the noncompact simple root ai in each irreducible factor Gi of G is critical, that QE is of type C, and, using the latter fact to define the compact

and noncompact Q-roots, that each a, restricts onto the noncompact, simple Q-root P. It follows from this that the noncompact positive (respectively noncompact negative, respectively compact) R-roots are those in whose restriction to QT, expressed as a sum of simple Q-roots, [3 appears with coefficient + I (respectively - 1, respectively 0). If S denotes the I-dimensional subtorus of QT annihilated by all simple

Q-roots except [i, then P is generated by £(S) and U. We have RT Y(QT) - .'(S), and all elements of f(S) n GR act by linear transformations on Z. Let g e .K(QT)R represent the element of the relative Weyl group QW which sends every Q-root into its negative. Then g normalizes Y(QT) and sends RT into another maximal R-trivial torus of f(QT). Hence, we may find h E (QT), such that gh E .N'(RT), and we may assume gh c GR because RTR meets every component of GR (see [5, Section 14.4]). Replace

g by gh. By our preceding considerations, it is clear that Ad g transforms every positive noncompact R-root into a negative one.

On the other hand, for each irreducible factor Gi of G, let ai be the element of G?R sending Z; into -Z, 1, and let or = ]1 ai. Then Ad or sends i

every R-root into its negative. Hence, ag normalizes and so belongs to P1. It follows thatj(Z, g) = const rl Ni(Z2)- vi, where Ni is the Jordan algebra i

norm for the ith irreducible factor, Zi is the component of Z in the ith irreducible factor of Z, and vi are certain strictly positive integers.

4. The Bruhat decomposition

Retaining the notation of Section 3, let N+ denote the unipotent subgroup of G generated by the subgroups of G associated to the positive Q-roots. Let N- be the similarly defined group associated to the set of negative

Q-roots. Of course, NR is connected and therefore NQ - G. Let GO

144

WALTER L. BAIL Y, JR.

denote the subgroup of GQ generated by all the groups 9NQ for g E GQ (see [13]). Obviously GO - GA. The main facts we need are the following (see [13, Section 3.2 (18) et al.]): (i) The group GQ can be written as the disjoint union of the sets (4)

NQ w_T(QT)QNQ

(Bruhat decomposition),

where w runs over a complete set of representatives of the cosets of K(QT)Q in .A"(QT)Q, and (ii) GQ = -T(QT)Q GQ. It follows from (ii) that

each of the elements w appearing in (i) can be taken in GQ - G. Since Po = -T(QT) N+ is a minimal Q-parabolic subgroup of G, it follows from the remark just made and from [5, Section 4.13] that if two Q-parabolic subgroups of G are conjugate, then they are conjugate by an element of

GQ - G. Let P be an arithmetic subgroup of G. Since GO has no compact simple factors, IF is Zariski-dense in G. Let d(P) be the algebra of all finite linear

combinations with rational coefficients of elements of r (viewed as matrices from the matrix representation of G). In the Bruhat decomposition (4), if w c . V(QT) is such that Ad w(N+ ) n

N- has dimension strictly smaller than that of N+, then the Zariski closure of NQ wY(QT)QNQ in G is a proper algebraic subset of G. Since

QE is of type C, there is one element wo of QWQ which sends every positive Q-root into its negative; thus, Ad wo(N+) n N- has the same dimension as N+ ; and wo is the only element of QWQ for which this is true. Therefore, since r' is Zariski-dense in G (see [4, Theorem I]), we have that r' n NQ wO

(QT)QN+

is nonempty. Every element of NQ can be written in the form u n', where u c UQ and n' E NQ n .(S). Since wo normalizes s(S) (by taking every compact Q-root into another), it follows that if yo E F' n NQ wo.'(QT)QNQ ,

then we may write yo = uwop, where p e PQ, and we may assume wo, p E GQ, adjusting by an element of ff(QT)Q if necessary.

In the notation of the introduction, we see that P - GQ, since G is centerless (see [4, Theorem 2]). Also F normalizes F' and hence normalizes the group algebra W(P) (over Q) of P. That group algebra contains GQ. In fact, consider the left regular representation of G, on its group algebra over C, let V be the vector space over C spanned by all the matrices in P', and let W be the vector space over C spanned by all the matrices in GQ; clearly, VQ = and V - W; since P' GQ, F. VQ = VQ, and since P' is Zariski-dense in G, it follows that G- V = V; since the identity matrix

e c V, we have G, - VV, hence W = V and GQ - WQ = VQ. Moreover, since the adjoint representation of G on V is faithful (G is centerless) and defined over Q, it is an isomorphism of algebraic groups, defined over Q,

EISENSTEIN SERIES ON TUBE DOMAINS

145

between G and its image. Hence, GQ may be identified with the set of Q-rational points in the image of G. Therefore, GQ is its own normalizer in Gc. From the foregoing, we also see easily that P normalizes GQ, i.e., F - G*Q, and that G*Q n G' = G. Hence, GQ is of finite index in G. If P* is any Q-parabolic subgroup of G and if H is any arithmetic subgroup of G, then G. is the union of finitely many disjoint double cosets HaP*Q. Thus in particular we may write

GQ = U r'aP0,

(5)

from which we have GQ = U F'aN(FO)Q. Since P is self-normalizing, we acA,

see from property (ii) of GQ that GQ* = G°Q NQ = G4. NQ. Then it obvious that we have GQ* = U FaNQ (disjoint union) (6)

is

aeA

for some finite subset A of G. 5. Eisenstein series

We know that G,/G0 is a finite Abelian group of type (2, 2, ... , 2), if not

trivial. With notation as before and a c GQ*, define Fo,a = P n aNh, Po a = P' n aNh ro = roof ro = Po e. We know that aP' GQ and we also know (see [3, Section 3.14]) that there exists a positive integer da such

that if y c a -1 Pa n Nh, then j (Z, y)da = 1, and by Section 2, if p E Nh,

then j(Z, p) is constant as a function of Z. Hence, since Fo,a/Po.a is Abelian of type (2, 2,..., 2), if not trivial, we have j(Z, y)2da = 1 if

'p0.a. Let the set A be as in (6) and let d be the l.c.m. of all da, a E A. Let I be a positive integer divisible by d. For each a that we have chosen, select a complex number ca, and then form the series E,,a given in (3). By our choice of 1, it is clear that the values of the individual termsj(Z, ya)` do not depend on the choice of representatives y of the cosets of Po,a in F. If yEa

this series converges, it clearly defines a holomorphic automorphic form on Z with respect to P. If b E P, write b = gp, with g = gb c GO, p = Pb E NQ. Write P as a disjoint union of double cosets P'bFo,a, b c B. Since P is normal in F, we have P'bFo,a = bF'Po,a and \ I( j(Z, by'a)' I. Ei.a(Z) = (7) bEB \ y'Er'Ir'o,a

/

Using the cocycle identity (see [3, Section 1.8 (1)]) for j, the convergence of E,,a for sufficiently large I then follows from known convergence results on Eisenstein series [3, Section 7.2]. Writing b = gp as above, we have

j(Z, bya) = j(Z, g(pyp-')(pap-')p) =

gy,al),

WALTER L. BAIL Y, JR.

146

where c is the constant value ofj(Z, p), a1, g E GQ (since NQ normalizes the

latter), and y1 E Pb = pT'p-1 -_ GQ. Let Pb,al = a1Nh n Pb. Then the inner sum in (7) becomes .1(Z, gy1a1)1 = Eal.b.l(Z ),

(8) Y1EPb/rb.a1

which is an automorphic form of weight 1 with respect to the arithmetic group 9pb. Therefore, (DFE,,a may be calculated for any rational boundary component F by applying information already available (see [3, Sections 3.12, 7.7]) on the limits of Poincare-Eisenstein series for arithmetic groups contained in G. If g is a holomorphic automorphism of a domain D - Cm, then j(Z, g) is a nowhere vanishing holomorphic function on D. Using this

fact and the information just referred to, one sees easily that for any rational boundary component F, (DFE,,a, if not -0, is the sum of a normally convergent (in general, infinite) series of lth powers of holomorphic functions of which not all are identically zero, such that each term not identically zero is a nowhere vanishing holomorphic function on F.

If F =

aE A, we have for a' (DFa(E,,a.) = lim j(Z, a-1)`E1,a.(Za-1) Z-FO

= lim > (j(Z, a-1)j(Za-1, ya'))` Z-.FC Yerlro.a (9)

= lim

7, j(Z, a-1ya')`,

Z-.F0 YEr/ro.a.

and all terms on the right go to zero except those for which a-1ya' E Nh; the existence of such terms implies a = a' and y c PO,a, which leaves, in case a = a', only one term with nonzero limit; it follows that we have (10)

(Fa(El,a-) =

(Kronecker delta symbol).

Therefore, (10')

(DFa(El.c) = Ca.

Since each orbit of zero-dimensional rational boundary components under r contains Fa for precisely one a E A, we see that if the mapping c: A ->. C 0 for every 0-dimensional, rational is nonzero on all of A, then boundary component F1. By the transitivity of the (D-operator (see [3, Section 8]), it follows that for any rational boundary component F, q)FE,,c is not identically zero. Consequently, by the previous remarks, (DFE,,c is the sum of a series of lth powers of nowhere-vanishing holomorphic functions on F. The following is easily proved (see [1, Lemmas V, VI]):

EISENSTEIN SERIES ON TUBE DOMAINS

147

LEMMA 1. Let ai be an absolutely convergent series of complex numbers, not all zero. Then there exists a positive integer m such that

dam#0.

If f is an integral automorphic form (see [3, Section 8.5]) with respect to F, then f may be viewed as a cross-section of an analytic coherent sheaf on Z = Z/F, such that that sheaf has a unique prolongation to an analytic coherent sheaf on the Satake compactification Z* of Z, and such that f has a unique extension, which we denote by f *, to a cross-section of the extended sheaf (see [3, Section 10.14]). We know from [3, Section 8.6] that each of the series E,,0 is an integral automorphic form with respect to F. From this, from Lemma 1, and from the discussion preceding the lemma, we obtain at once: PROPOSITION 1. There exists a finite set ' of positive integers I such that the members of the family {E,'a},E..aEA have no common zero's on 3*. If c: a -->. ca is given such that ca 0 0 for all a c A, then, by increasing the

size of 2 if necessary, we may assert that the members of the family {E,*,,},E' (fixed c) have no common zero's on Z*.

The proof is obvious.

Referring to the first paragraph of this section, let to be a positive integer divisible by d such that for all positive integers 1 divisible by to the series E,,0 converge. Let n be a positive integer divisible by l and let An be the linear space over C spanned by all expressions of the form (11)

Eii.a, ... E,µ

where ai E A, such that 1 , > 0, lo1li, i= and 11 + - + 1, = n. An element of A. will be called an isobaric polynomial (in the Eisenstein series) of degree n. Denote by A* the linear space of the extensions to l* of the elements of A. We make the same set of definitions with the functions E,,a replaced by E,,, for a fixed, nowhere zero c and denote the spaces corresponding in this way to An and to A*n by An,, and A*,, n, respectively. Clearly A*.a - A. Then we have from the proposition immediately: COROLLARY. If l n and if n is sufficiently large, then the elements of A* (respectively of A* a) have no common zero's on 28*. Clearly, if p, q c A.*, q 0, then p/q is a meromorphic function on Z*; or it may be identified with a meroinorphic function on Z invariant under F.

7. Projective imbeddings

Let A* be any subspace of A* such that the elements of A* have no common zero's on Z* and let A0, ..., A be a basis of A*. Then the assignment

ZH [A0(Z):...:AM(Z)]

148

WALTER L. BAIL Y, JR.

determines well defined holomorphic mappings of Z and of Z3* into CPM, then0 is which we denote by ) and 0i,,,, respectively. Let al = a (complete) projective variety. The functions A, ..., A induced on T by

..., AM are automorphic forms of weight n. If k is a suitably large positive multiple of n, then a basis of automorphic forms of weight k affords an injective holomorphic imbedding of 3* in some projective A0,

space such that 8 ' is mapped biregularly onto its image. It follows (see [2,

pp. 353-354]) that since the ,1i are nonconstant and without common zero's, the mapping /'(A) has the property that for every x E2Z3, b(-i (x) is a finite set. Hence, we may speak of the degree of the mapping 0(r): F8* -->U,

which is equal to the degree of the algebraic extension of the field of rational functions on i3 by the field of those on W. Moreover, if loll, 1 > 0, then the image of E,*a or of E1*C under restriction to the image of any

rational boundary component F in Z3* is a nontrivial cross-section (if not identically zero) of an ample sheaf, and is hence nonconstant.

Furthermore, we see at the same time that cri can be extended to a mapping of the union of all rational boundary components which is continuous in the Satake topology (see [3, Section 4.8]).

We now let A* = An c for fixed, nowhere-vanishing c. Let On be the mapping associated to a basis of A*, let Jan = n( 3*), and let do be the degree of On. It is clear that d > do+,o. Therefore, there exists an n' such that do = dn. for all n > n'. Also it is easily seen (by the Jacobian criterion) that if p is a regular point of 3 - 3* and if On, induces a biregular mapping of a neighborhood of p onto a neighborhood of a regular point of'2 n,,

then the same is true for all n > n1. We wish to prove the following proposition : PROPOSITION 2.

The degree dn. is equal to one.

We proceed to the proof of this by stages. If g c G,, for any real number r, let

L9,, = {Z E ZI Ij(Z, g)I = r}; this is a real analytic subset of Z and is different from Z, and hence is of measure zero if j(Z, g) is not constant as a function of Z, i.e., if g 0 Nh. Since the series (2) converges uniformly on compact sets, it follows that the number of the surfaces Lya , passing through any compact subset of Z for fixed r is finite. If I j(Z, ya)I = d I j(Z, y'a')I, where d is a positive real constant, y, y' E P, a, a' E A, then we see easily from the cocycle relation that a = a' and y = y' ya, Ya E I'p a From the termwise majoration of the series (2) in a truncated Siegel domain, afforded by [3, Section 7.7(i)], we conclude that on an Fo-adapted

EISENSTEIN SERIES ON TUBE DOMAINS

149

truncated Siegel domain S in Z, there exists a series of positive constants which majorizes the series j (Z, b _ 1 ya),

(lol1, 1 > 0)

yer/ro,a

on Cam, for any a, b c A (to reduce this to the form considered in [3, Section

7.7(i)], write b-lya = b-la(a-1ya)). Moreover, from this we conclude a or if

that, if S is suitably chosen, then we have I j(Z, b - lya)I < I if b b = a and y 0 Define the subset tea, of Z by

{Z E ZI I j(Z, b-lya)l < l if b e A - {a} or if b = a, y *Po.a}.

Then from this discussion and from properties of the Satake topology (see [3, Section 4]) we have at once the following lemma: LEMMA 2. There exists a dense subset of X such that T - e is of measure zero and such that for Z E ', the terms in the series for E,,, are all distinct. For each a c A, the set Y,, is open, and nonempty, and contains an Fo-adapted Siegel domain. The closure of `9a in the Satake topology contains a neighborhood of Fo in that topology.

Consider only n > 0 divisible by lo. Let An be the inverse image under

On Y On of the diagonal of CP"n x CP"n, where µn + 1 = dim An,a. Clearly An contains the diagonal of 8* x 3*, and An - 4n+/o Without loss of generality, we may assume n' > 0 chosen such that An = An. for all n > n' (because {An} is a decreasing sequence of algebraic sets). Consider now only n > n'. We have seen that 0n 1(x) is finite for all x E U. We now prove the following lemma: LEMMA 3. If x0 = n(F0), then x0 = &(F1) for every 0-dimensional, rational boundary component F1 and n 1(x0) is precisely the set of all

0-dimensional, rational boundary components. PROOF.

It is sufficient to consider the case F1 = F. for some a c A.

Since the result is evidently independent of the choice of basis for An* 0, we may assume the basis to consist of monomials in the series E,,,. Then by (10'), the image under n of Fa is the point in projective space rep-

resented by [1:

.:1]; thus tiin(F0) = t1in(Fa) for all a. The converse

requires further considerations. Let F be any rational boundary component of Z (we may have F = Z)

with dim F > 0, and let Z E F. The function FFE,,a on F is an automorphic form of positive weight on F with respect to the homomorphic image I'F of N,(F) n P in the full group of holomorphic automorphisms of F, and PF is an arithmetically defined discontinuous group acting on F. Without loss of generality, we may assume that F0 is a rational boundary

WALTER L. BAIL Y, JR.

150

component of F. We may view F (see [9, Section 4.11 ]) as a tube domain on which NhF = Nh(F) n Nh acts by linear transformations; with a certain abuse of notation, let FF,a = FF n aNhF. Then by [3, Section 7] it is easily seen that (DFE,,a is the Eisenstein series (12)

jF(Z, ya)°F',

ca aEAp

Z E F,

verp/rp,a

where qF is a positive rational number (not a multi-index!), jF(Z, g) is a nonzero constant e9 times the functional determinant of g as a transformation of F, and a runs over the set AF of elements of A such that Fa is F-equivalent to a rational boundary component of F. By assuming ab initio suitable divisibility properties for 1, we may assume that qFl has the necessary divisibility properties for the tube domain F. This having been

said, we now prove that n 1(xo) n Z is empty and note that the same argument will prove that /n 1(x0) n F is empty by merely supplying the subscript F where needed.

If Z E Z and

a(Z) = xo, then we have E,. (Z) # 0 and fk(Z) _

Ek,,c(Z)/E,,c(Z)kwill have the same value for all k (because xo = [1: . . . :1 ]). Following [12, p. 126], we introduce the function

(A -

MM(A) = acA

Yer/ro,a

ya))-`)-1.

This series converges if A is not one of the discrete set of points E,,0(Z) ya))-' of C, because of the convergence of the series for E,,c itself, and uniformly so on any bounded subset of C not meeting that discrete

set. Hence, MZ is a meromorphic function on C with infinitely many distinct poles. On the other hand, MZ is holomorphic at the origin and its

power series expansion is -

k=1

fk(Z)Ak-1. By hypothesis, all fk(Z) are

equal, hence MZ(A) _ -fk(Z)/(1 - A) has at most one simple pole-which is absurd. This completes the proof of the assertion that n 1(xo) is simply the set of 0-dimensional rational boundary components. It follows from Lemmas 2 and 3 that there exists a neighborhood 9Z of

xo such that T n n 1(92) c (U Ya) F. Let Y* = (U Ya). a

a

Suppose now that the degree d,,, is greater than 1. We assume 1o chosen such that E,o,, is not identically zero. Henceforth, fix c and let E,,c = E,.

Since d,,. > 1, one sees easily that there exist points Z1, Z2 E 9'*, say, Zt E .tea,, i = 1, 2, in distinct orbits of r such that (i) Eta(Z,)

0, i = 1, 2;

(ii) the canonical images of Z1 and Z. are regular points of Z - 'Ii*; (iii) (Z1) = (Z2) and their common value w,, is a regular point of

EISENSTEIN SERIES ON TUBE DOMAINS

151

din, and Y'n is a biregular mapping of a suitable neighborhood of Z, onto a neighborhood of w,,,, i = 1, 2; (iv) W. 0,n(3* - F8).

Defining S = Y'n(Z* - i), we may find a small neighborhood 91i of Zi contained in i = 1, 2, such that T1 n 912'r is empty, and such that 0n is a biregular analytic mapping of 9i onto a neighborhood 91 of wn in

4n - Sn, i = 1, 2. Let T denote the composed mapping (nI

2)-1

°

(Y'nl X11) of 91 onto 5912 i because of the properties of Y'n and 1,, which are

stable for large n (q.v. supra), 9p is independent of the choice of n for sufficiently large n, say n > n'. Then 9' is biregular and we may assume E,0 # 0 on SJ21 U 912. Let 1 = 1o. With fk(Z) = E,(Z) _ k E,k(Z) as before, our conditions imply that fk(Z) = fk((p(Z)), k = 1, 2, ... , Z E 9l1. 8. Extension of the mapping cp

We have the following lemma: LEMMA 4.

The mapping 9' may be extended to a holomorphic automor-

phism of Z. PROOF.

The main ideas here are those of [12, pp. 126-130]. Define

Mz(A) as before, when E,(Z) 0. By hypothesis, E, 0 in 911 U 912. If ya))-'} is a discrete set. Z E 91 U 912, then the set of points Moreover, in any region of X x C avoiding poles of Mz(A), Mz(A) is an analytic function, and its poles for any fixed Z E Z such that E,(Z) 0

are the points of that discrete set, each of which is a simple pole with residue equal to the number of times a term of given value occurs in the Eisenstein series. As before, -Mz(A) _

fk(Z) = fk(cp(Z)), k = 1, 2,

fk(Z),1k 1. Then the equations k=1

..., imply that the sets of poles of Mz and of

M,,izj are the same. If Z E Yap, then by the cocycle identity j (Z, yb) = j (Zi, at:-')-' j(Zi, a% 1Yb),

Zi = Z ai,

and jj(Zi, a, 1yb)I < 1, unless b = ai and y E Po,a,, in which case J j (Z,, ai 1ya,) I = 1; therefore, (cat j (Z, a,)) - 'E,(Z) is the pole of Mz(A) having the smallest absolute value, i = 1, 2. Hence, we have (13)

(calj(Z, a1))-`Ei(Z) = (ca2j(p(Z), a2))-`E1(9,(Z))

The equality of the other poles of Mz and Mm(z) gives the system of equations (14)

Ya2))-`E&p(Z)) = (cbzj(Z, Yzbz))-`E1(Z)

WALTER L. BAIL Y, JR.

152

Dividing (13) by (14) and using the cocycle identity gives (15)

j (rp(Z) - a2, a2 'ya2)` = (cbzj (Z, yzbz)(calj (Z, a1)) -1)1,

where bzi yz c G** depend, at first, on Z. However, the number of possibilities for each Z is at most countable, and using the fact that an analytic function not identically zero vanishes on at most a set of measure zero, we conclude that there exists a function f from the set of cosets P/Po,a, for each a, into G** such that for suitable constants d, we have (16)

j(p(Z)a2, az lya2)` = d,(j(Z,f(y))j(Z, a,)-1)`.

Since the set of lth roots of unity is finite and the function j(Z, a), for fixed a, is a nonvanishing holomorphic function on all of Z and in particular on the neighborhood W,, we may conclude that (17)

j(9p(Z)a2, az

1ya2)-1

= h,,(Z),

where h, is a nonvanishing holomorphic function on all of Z. We now proceed to solve the system (17). The group a2 l['a2 is also arithmetic. We let X be the one-to-one analytic mapping of 9?1 into Z obtained by first applying 9) and then translating by a2. Our main problem becomes that of solving the equations (18)

.

j (a(Z ), y) -1 = h,(Z ),

for an analytic mapping a of Z into itself such that a = X in a neighborhood of Z1, where y runs over an arithmetic group IF, and h7 are nonvanishing holomorphic functions on Z. We now choose yo = uwop c F, as in Section

4 and let y run over the elements Ay,,, where A runs over the lattice A = U n Pl in UR. Identifying UR with the group of real translations in Z, the system (18) for these y becomes (19)

j(a(Z) + u + A, wop)-1 = g,°(Z),

where g°° is holomorphic in Z. By Section 3, this system becomes (20)

J %#a(Z) + u + A),) = g,\(Z),

where g = const. g°, is holomorphic in Z. Since we know that a solution X = a of the system (18) exists in a neighborhood of Z,, it will follow that if a subsystem of the system (18) has a unique analytic solution for all Z E Z, then this solution must be an analytic continuation of X, and hence, must be a solution of the full system (18) for all Z E Z. + nk. If z; E C"', Let n,, ... , nk be positive integers and m = n1 + write z, = (z;;)7=1....,11,, and if z E Cm = rj Cn+, write z = (z1)i=1,...,k. If also z' E Cm, define z + z' by the usual component-by-component addition.

EISENSTEIN SERIES ON TUBE DOMAINS

153

In what follows, let pi be a homogeneous polynomial of degree di > 0 in n, indeterminates with complex coefficients, viewed as a function on C' i in

,

an obvious way. We assume that the first partial derivatives api , j = 1, ni are linearly independent polynomials for each i. Let v1, ... , vk be positive integers and put p =PI'' pkk. Let A be a Zariski-dense subset of C76. LEMMA 5. There exists a finite number of points ra E A - Cm, where a runs over a finite indexing set B', such that if D is a connected open subset

of Cm and if { fa}aEB. are analytic functions on D for which the system

p(a(z) + ra) = fa(Z)

(21)

has an analytic solution a mapping some open subset of D into Cm, then the system (21) has a unique solution a defined on all of D, a: D -. C°1. PROOF.

View the polynomials p1,. .., Pk as polynomials in independent

sets of indeterminates {x1}11 .....n;, i =1.....k. It is an elementary exercise to prove that if II denotes the linear span, in the vector space of polynomials with complex coefficients, of the set of higher (i.e., of order higher than 1) partial derivatives of p and if 7f1, ... , iN is a basis of n, then iT1,... , 'TNi 8p

8xi;' j = 1, ..., ni, i = 1, ..., k are linearly independent polynomials. It follows from Taylor's expansion that if y c Cm, then (22)

+ Y) = P(Y) + a p, (Y)zii +

P(z

i.i

#cB

PAYWAZ),

where B is some finite indexing set, Pfl and Q, are homogeneous polynomials such that deg Pa + deg Qs = deg p and PQ, P E B, are linearly independent elements of H. Therefore we may find ra E A, a E B', card B' _

in + card B, such that the determinant of the matrix

I

a

Vp aXknk

(ra), {Pa(ra)}6EB)aX11

aaB'

is nonzero (because the product of Zariski-dense sets is Zariski-dense in the product variety). Then it is possible to solve uniquely the system of equations (23)

a pf

(ra)ai,(z) +

BeB

PB(ra)aD(z) = fa(Z) - p(ra)

154

WALTER L. BAIL Y, JR.

for the "unknowns" ai;(z) and a,&), and it is clear that ai,(z) is analytic in all of D. Since a solution does exist in an open set and since D is connected, it follows that a = (as,) is analytic in all of D and is the unique solution there to the system (21). This proves the lemma.

To demonstrate the applicability of Lemma 5 to our situation, one notes the systems of equations (18) and (21) and observes (e.g., by an easy case-by-case verification) that the Jordan algebra norm Wi always satisfies the requirements (i.e., that pi be homogeneous and that the polynomials 69P' be linearly independent) imposed upon each of the polynomials pi in

axi;

Lemma 5. This computation is left to the reader. Finally we note that any translate A + u of the lattice A of UR is Zariski-dense in C", (the complexification of UR).

Thus we have obtained an analytic mapping a of Z into the ambient space C'' of Z such that a coincides with X in a small neighborhood of Z1. Now view the noncompact Hermitian symmetric space Z as imbedded in its compact dual Z` (see [14, Section 8.7.9]). Then a2 E GR extends to a holomorphic automorphism of the compact, complex manifold Z°, and

then a2-1 o a = a' is an analytic mapping of T into Z` which coincides with p in a small neighborhood X21 of Z1. Our next step is to show that a' maps `3` into the closure Z* of Z in Z`. It is easy to see that for n n' there

exists a proper Zariski-closed subset A' of Un such that Sn

such that if X; = on 1(.Nn) and .N,, _ ,n

then X -

.N and ,,

is a

covering manifold of Q3* - .N',' and the latter is a covering manifold of Un - Nn. (It is sufficient that .Nn include the singular points of Un, the set Sn, the images of the singular points of Z*, and the image of the set of regular points x of Q* at which On is not locally biregular-the intersection of the last set with Z is, modulo singular points of Z, a proper Zariskiclosed subset of 3 by the Jacobian criterion.) Let AV = {`n. The set Z - .K is connected and open and contains Z1. Let Zo be any point of Z - .iV and join Z1 to Zo by a path ¢. Let 0a, be the image of ¢

under a'. We have n a' = //n in a neighborhood of Z1. It is clear that On(qi) is a path in Jn - .Nn and by the homotopy covering theorem (see [14, Sections 1.8.3-4]) can be raised to a unique path in Z - .A' beginning at Z2, since 'iin(Z1) = n(Z2). By analytic continuation of the relation on o a' = cn it is clear that this covering path beginning at Z2 can be no other than ¢$.. Hence, a'(Z0) must be a point of Z. Therefore, a'(Z - .N') -

Z - N - T. Viewing Z as the bounded domain D, we have a'(D) - D, since a' is continuous in D and Z - 'V is dense in Z. By a known property (see [9, Section 4.8]) of the boundary components of D, since a'(D) n D

is not empty, a'(D) n (D - D) must be empty; thus a'(D) - D. Let a" be the similarly defined mapping associated to q-1. Since, also, a"(D) c D

EISENSTEIN SERIES ON TUBE DOMAINS

155

and a'a" = a"a' = identity (because this is true in a small open set, 9"tl or W2i to begin with), we see that both a' and a" are bijective maps of D onto itself. This completes the proof of Lemma 4. 9. Conclusions

Thus we see that there exists a biholomorphic map a: D -* D, a E Gh - P such that Y'n o a = On. Let r* be the subgroup of Gh generated by a and P.

Clearly r

r*, so that r* cannot be a discrete subgroup of Gh. By

construction, Y'n is constant on any orbit of r and also on any orbit of the group generated by a and hence is constant on any orbit of P*. Since P* is not discrete, it is not property discontinuous on Z, and thus there is an orbit w of P* in T with a limit point Zo. Hence, in every neighborhood of Z0, there exist infinitely many points mapped by 0. onto the same point of On. But this contradicts what we already know about On. The contradiction having come from the assumption that dn. > 1, we conclude that do = 1 for n > n'. This proves Proposition 2. Therefore, On is a proper birational

mapping of the normal variety 8* onto On, n ? n', and for w EOn, On 1(w) is finite. Since do = 1, we conclude from Zariski's main theorem (see [10, p. 124]) that 8* is simply the normal model of On, and hence there exists a projective variety biregularly equivalent to 8* and defined over any field of definition for On. Since by Chow's theorem (see [6]), any

meromorphic function on On is a rational function expressible as the quotient of two isobaric polynomials of like weight (see [3, Section 10.5]), the proof of the theorem stated in the introduction is complete. COROLLARY 1.

Let G be a connected, semisimple algebraic group defined

over Q such that GR has no compact simple factors. Let K be a maximal compact subgroup of GR and let r be a maximal discrete arithmetic subgroup of G'R°. Assume that 3r = K\G°R is Hermitian symmetric, and assume that Gh is connected. Then the same conclusions as those in the theorem hold. PROOF.

Let Z be the center of G (which is not assumed to be centerless),

and let a: G - G' = G/Z. Then 7r(IF) is maximal, discrete, arithmetic in GR = Gh, and we may apply the theorem. The cases where Gh is not connected are described in [3, Section 11.4]. COROLLARY 2.

Let G, X, and r be either as in the statement of the

theorem or as in the statement of Corollary 1. Then the conclusions of the theorem, or of Corollary 1, respectively, are true if the module of isobaric polynomials in {E,,,:} of sufficiently high weight is replaced by the module of isobaric polynomials in {E,,a}jai,, PROOF.

1>0, aEA

of sufficiently high weight.

The proof is trivial because the first module mentioned is a

submodule of the second.

WALTER L. BAILY, JR.

156

The group F is called unicuspidal if all minimal Q-parabolic subgroups of G are IF-conjugate. Then, if P is as before and F is unicuspidal, we have GQ = P - PQ = PQ P. I', and thus the set A may be chosen to consist of e alone. Let EI = EI,1 (i.e., ce = 1). We then have the following corollary: COROLLARY 3. If P is unicuspidal, the cross-sections E* have no common zero's on X3*, and the isobaric polynomials of sufficiently high weight in these induce a mapping 0 of 93* onto a complete projective variety of which 9* is the projective normal model.

THE UNIVERSITY OF CHICAGO

REFERENCES 1. BAILY, W. L., Jr., "On Satake's compactification of V,,," Amer. J. Math., 80 (1958), 348-364.

, "On the theory of 9-functions, the moduli of Abelian varieties, and the moduli of curves," Ann. of Math., 75 (1962), 342-381. 3. BAILY, W. L., Jr., and A. BOREL, "Compactification of arithmetic quotients of bounded symmetric domains," Ann. of Math., 84 (1966), 442-528. 4. BOREL, A., "Density and maximality of arithmetic groups," J. f. reine u. ang. Mathematik, 224 (1966), 78-89. 5. BOREL, A., and J. TITS, "Groupes reductifs," Publ. Math. I.H.E.S., 27 (1965), 55-150. 6. CHOW, W. L., "On compact complex analytic varieties," Am. J. Math., 71 (1949), 893-914. 2.

7. HELGASON, S., Differential Geometry and Symmetric Spaces. New York: Academic Press, Inc., 1962.

8. KORANYI, A., and J. WOLF, "Realization of Hermitian symmetric spaces as generalized half-planes," Ann. of Math., 81 (1965), 265-288. , "Generalized Cayley transformations of bounded symmetric domains," Am. J. Math., 87 (1965), 899-939. 10. LANG, S., Introduction to Algebraic Geometry. New York: Interscience Publishers, Inc., 1958. 11. RAMANATHAN, K. G., "Discontinuous groups, II," Nachr. Akad. Wiss. Gottingen Math.-Phys. Klasse, 22 (1964), 154-164. 12. SIEGEL, C. L., "Einfuhrung in die Theorie der Modulfunktionen n-ten Grades," Mathematische Annalen, 116 (1938-1939), 617-657. 9.

13. TITS, J., "Algebraic and abstract simple groups," Ann. of Math., 80 (1964), 313-329. 14. WOLF, J., Spaces of Constant Curvature. New York: McGraw-Hill, Inc., 1967.

Laplace-Fourier Transformation, the Foundation for Quantum Information Theory and Linear Physics JOHN L. BARNES 1. Two-dimensional choice information in a discrete function

For clarity this paper begins with the very simple examples shown in Figs. la, lb, and 2. These illustrate the idea that a simple function can contain choice information (see [2]) in two ways. In these figures the elements chosen are shaded. To construct messages they are placed in positions located in intervals on the scale of abscissas called cells and on

the scale of ordinates called states. It is assumed that they are chosen independently and that the binary locations are chosen with equal statistical

frequency. In Fig. la the information is contained in the height, i.e., ordinate of the function. In Fig. lb the information is contained in the position, i.e., the first or second place, in the cell. For the transmission of information the function illustrated in Fig. la would be said to be heightmodulated. Sometimes "amplitude" or envelope modulation are the terms if the carrier is a sinusoidal wave. Fig. lb would be said to be position- or time-modulated. Instead of position, time rate of change of position is often used. The terms are then pulse position or pulse rate modulation. Pulse

radar, sonar, and neural axon spike transmission, as well as phase or frequency modulation in radio are in this class. For height modulation over a 2-space-dimensional field one can select "black and white" still photographs. Here shades of gray give the height and are levels of energy. Both height and position modulation may be used jointly as illustrated in Fig. 2. Here the message function has independent choice information carried in the height and horizontal position of the selected elements. If

a coherent source such as a laser beam is used to form a 2-D space interference pattern on a photographic film, then called a hologram, by 157

JOHN L. BARNES

158

Height States

nSH 2

0 I

0

tZ -Icy 0

4

3

2

I

nCH Cells

mH -nSHnCH=24=2dH mH = 16 messages

dH = 4 bits of information Fig. la Height modulation

Position States nSP to

OF

i

P

2x0

1

t 2210 I

1

0

0 1

2.0

I

2

I

40-

0 M '0

2

I

nCp

nSP

=

24

=

3

2dP

mp = 16 messages

dp = 4 bits of information Fig. lb Position modulation

4 nCp Cells

LAPLACE-FOURIER TRANSFORMATION

Height

Position States

States

2

AFp

n

nSH

I

S-W

0

1

159

2,0

I

0

22 0

0 I

2,0

I

2

0

0 Z -I

2

I

mH tP = nSH CH

n,,,,

4

3

dH. 2 dp = 2 4 24 = 2

= 2 dH iP

mH,P = 256 messages

dHtp = 8 bits of information Fig. 2 Joint height and position modulation

summing exposure from a direct beam and from one reflected from a 3-D space object; and if then the direct beam is subtracted out by passing it through the hologram, there will result a presentation which retrieves the

3-D space picture from the 2-D space hologram. In this holographic process the retention of position, as well as height, modulation provides the depth information missing from an ordinary photograph. In line with this terminology, ordinary photography could be called "halfography" since it uses only half (approximately) the information available in the 3-D space physical world. It will be seen later that ordinary probability and statistics correspond to "halfography" since they often present approximately half the information available. By the use of (usually uniquederivative) complex analysis (see [9]) in place of real analysis, probability and statistics can be made whole. 2. Laplace-Fourier transformation as a path to the analytic complex domain

While a single set of symbols would suffice to show the mathematical relations to be discussed in what follows, particular symbols carry associated physical meaning to engineers and physicists beyond the form of the mathematical relations (see [10]). Hence the same mathematics will frequently be stated in several sets of symbols.

nCP, CH

JOHN L. BARNES

160

Professor Bochner's deep understanding of functional transform theory first influenced my thinking through his 1932 book [7] and his supervision of my graduate research at Princeton University.

3. Uncertainty theorems from the 2'-3 transformation

The fundamental uncertainty principle was known intuitively by the German composer Johann Sebastian Bach (see [2]) and used by him about 1730. Through the intervening years it has become known in successively more precise forms. A modern version given by Norbert Wiener at a 1924 Gottingen seminar could be expressed in our form as follows (see [3]): z

If f(t), ddt ), dd t) exist and EL2

WIENER'S UNCERTAINTY THEOREM.

on R, and if the bilateral Fourier-integral transform .F,, f(t)] °= F(iw), then (U1)

(El)

(a) F(iw) E L2 on I, At AIiw1, and (b) I (c) 1 = At Al iwl for f(t) a Gaussian pulse,

in which At

tai, Aw °= 2ou, - aii,,i, where energy c

e(t) °

fj=-W

.f2(tl) dti,

fury= -iao

e(t) e(oo)

F(iw2 72 En(iw) °=

,

E(Iw) E(ico)

1/2

[J_ .

t2 den(t)]

i.

Qia,

fw=-iw

Ilwl

2 dEE(iw) lie i2Tr

Note that by the Rayleigh-Parseval Theorem IE(ioo)i = e(co). Two other physical representations of this theorem arise. See [2] for the first. Here time tk is replaced by distance-space coordinate Xk, k = 1, 2, 3; time-angular frequency, iwk, is replaced by distance-space angular frequency, i/3k; and energy ek(tk) is replaced by linear momentum pk(tk). Then the conclusion is: (a') Fk(iflk) E L2 on I, (Ui) (b') 1 Axk DliPkk = 1, 2, 3, (E') (c') 1 = Oxk 0l iIk if fk(xk) is a Gaussian pulse.

Y1[f(tk)] A F(Sk), tk is time, Sk A- Qk + iwk, Wk 4 21Tfk,

f(Tk)e-{kk dmk

k = 1, 2, 3,

* "°_" means "equals by definition."

{kF(Sk)-

{k°'1U(gk)]

skF(Sk>

f(pk)e - Ikok dqk

7

sk-" Af(tk)]

°T1[f(Tk)1

2' [f(Xk)]

YKy1[f(Xk)] A YkF(YA

f(tk)e-'ktk dtk

A

Sk

Sk

Yrk

Sk

-multiplied

Stieltjes (Type 1)

Stieltjes (Type 1) and yyk -multiplied

Sk

k=1,2,3.

k=1,2,3,

Y1[f(gk)I A F(Sk, Tk is magnetic-flux coordinate, Sk - Xk + t k

f(Xk)e-YkXk dXk

"tk=0-

f(Tk) d(I - e-{k0k)

m

Transform Domain

f(Xk)e-YkXkdXk 45fI[J(Xk)1 A F(Yk), Xk is distance-space coordinate, yk A ak + isk,

f(Xk) d(I - e - YkXk)

xk k _0

f

,

(tk)e-'ktk dtk

Ordinary unilateral

f(tk) d(I - e-'ktk) A -FI[f(tk)] 4

0

k=0+ J

Sk fmk = 0 _

Yk

Sk

xk-0-

m

tk= 0 -

E

m

,uk=

k

f

Original Domain

INTRODUCTION OF SYMBOLS THROUGH SEVERAL VERSIONS OF THE LAPLACE-FOURIER TRANSFORMATION

action

k = 1,2,3.

gk(c'k)

9'k,

in direction xk,

Ik,

electric charge,

d'pk

dam o

k = 1, 2, 3.

linear momentum,

k

dxpk(xk)

= ek(tk) energy,

da, a

dtk

dat o

Then by definition:

9'k

in the xk direction,

tk

ao(mk)JJJJJJ

at(tk)

In rectangular coordinates let

Original Domain = skA(Sk),

tkA(bk, A

Yk' °I[ax(xk)] A YkA(yYyk),

Isk'el[at(tk)]

Ck-r' [gk('Pk)]

Yk'e![Pl (xk)] S

yyk

A- Xk + i#k,

Yk A ak +

Sk A ak + iCUk,

skA(Sk) (Ek(Sk) Pk(yk) = Sk skA(Sk) - Skat(0), YkA(Sk) exist, then = Yk YkA(yYk) - YkaX(0), Qk(Sk) = Sk CAW - Ckam(0).

Qk(Sk,

Pk(Yyyk),

Ek(Sk),

A(ioo) = nh,

tk

Dirac (1931)

de Broglie (1923)

Planck (1900)

0,1,2.... :

k = 1, 2, 3.

(Th )

for n = 0, 1, 2, ... :

a`a(0)J)

AQk(i'Iik) = nh A(iy k),

AEk(iak) = nh A(iwk), APk(isk) = nh D(isk),

Planck's Law de Broglie's Law Dirac's Law k = 1, 2, 3.

(L)

here. Then by Theorem (Thl) under restrictions (I) and assumptions (A), and in terms of increments,

9'k

Choose the origins of Xk so far to the left that ax(0) . = 0, since transient response is not of interest

at(0)l

A(iik) ° A(ioo) = nh,

A(isk)

A(iwk) °= A(ioo) = nh,

in which Planck's quantum of action h °A 27Th.

tyk-tm

lim

Ok-tm

lim

to,k-tw

lim

Introduce the asymptotic physical assumptions for n

Next restrict consideration to the imaginary line in the complex domain:

If

The Derivative Theorem [13] here becomes:

Let

Transform Domain

LAPLACE-FOURIER TRANSFORMATION AS THE BASIS OF DERIVATIVE-TRANSFORM THEOREMS

o

gkl 4k 0

¢,

Primal, Covariant

dz dz = gkl dzk dz'

bb:

dP. dP = g dPk dPl ;

dC dC = gkl dzk d)

Primal, Covariant dy dy = gkl dy' dY'

__

k, l = 1, 2, 3.

dP dP = g" dPk dPl

Dual, Contravariant

Transform Domain

= dPdPt. -t ° IdPI2(e`Ldp)2 = IdPI2et2Lp,

Dual, Contravariant

(dfi)2

(I)

Q

Itotal flux)

Itotal distancel

2 (total flux magnitude)2 9M W dot`

o'

= gkl dzk dzt'

(total distance magnitude)'

= gkl dt;k dot'

Itotal flux spectruml2

Rayleigh-Parseval Theorem Cross-Domain Invariance Itotal distance spectrum) = _ Itotal momentum) Itotal flux spectrum) Itotal chargel =

= gk` d9k d9l

(total charge magnitude)2

= gkl dPk dfi

(total momentum magnitude)2

Itotal distance spectrum 12 = gkl dyk dyt'

Itotal charge spectrum)

Itotal momentum spectrum)

gk` dQk d67

Itotal charge spectruml2

total momentum spectruml2 = gk' dPk dpi

d4. d4 = gk` dqk ddl gk` dQk dQl ----------------------------------------------------------- ----------------------------- ----------------------------

0

dpt o IdPI e`Ldp,

Original Domain

Example: dP ° IddIe'Ldp,

COMPLEX METRICS

JOHN L. BARNES

164

FIG. 3 ACTION FUNCTIONS FROM ENERGY, LINEAR MOMENTUM, LAPLACE-FOURIER

Original Domain

0 Xk 41 k

LAPLACE-FOURIER TRANSFORMATION

165

AND CHARGE FUNCTIONS, AND THEIR IMAGES IN THE TRANSFORM DOMAIN

Transform Domain

IS A( i wk)I k

A(i(3k)I k

kA(i*k)I

L/

10,

i-

i0

i0

Iwk 'Qk i *

k

JOHN L. BARNES

166

FIG. 3 ACTION FUNCTIONS FROM ENERGY, LINEAR MOMENTUM, LAPLACE-FOURIER

Original Domain

In the second representation the replacements are: magnetic flux coordinate 9'k for xk, k = 1, 2, 3, magnetic flux-angular velocity i,k for ifik, and electric charge gk(Pk) for pk(xk). Then the conclusion is: (a") Fk(icbk) c L2

(Ul) (Ei)

on I,

k = 1, 2, 3,

(b") 1 < (c") 1 = ATk A j i0kI

if fk(9 k) is a Gaussian pulse.

The partial ordering relation (U1) provides a foundation for quantum choice-information theory (see [2]). Together with (Ui) and (Ui) it leads to three quantum uncertainty theorems as follows. QUANTUM UNCERTAINTY THEOREMS. Multiply (U1), (U,), (U') by nh,

n = 0, 1, 2,..., to obtain nh co.

Because the {Pk} forms a complete orthonormal sequence and the {06V} is bounded, we have (a,, (p) - (a, (p) for all p c .4' This completes the proof of the lemma. LEMMA 2.

The cluster set `,(H, zo) is nonempty and weakly closed

whenever H(z) is a bounded function. PROOF.

The fact that ',(H, zo) is nonempty follows from Lemma I.

To show that the cluster set W ,,(H, zo) is a closed set, let {ak} be any sequence

contained in `' _ 'U(H, zo) which converges weakly to an element a E X, and we shall prove that a e W.

THE SCHROEDINGER EQUATION

179

We shall assume without loss of generality that H(z) is bounded by one and consider {p)n} a complete orthonormal sequence in 3l°. We form the

metric 0(a, /) _

n=m 1 n=1 2n

I(a, qqn) - (8, pn)I which gives a topology equiva-

lent to the weak topology inside the unit ball. Since a' ---->-a weakly, given any e > 0, there exists K such that for k > K > 0, 0(a', a) < E/2. Because each a' is an element of ((, for each a' ak weakly. there exists a sequence {z; }, z; E Q, lim z; = z,, and H(z;) ;

For each index k, we choose an index jk such that I zo - z kI < 1/k and A(H(z k), ak) < 1/k. The sequence {z,*} defined by z,* = z k has the property

that zk k zo and because 0(H(z,*), a) 0(H(zk ), ak) + 0(ak, a) we have 0(H(z,*), a) < e whenever k K + 2/E and so H(z,*) --*a weakly and consequently `' is closed. LEMMA 3. If the cluster set consists of only one element a, the function possesses a as a weak limit, i.e., if zn --* zo and `t (H(z), zo) = {a} then lim (H(zn) - a, q,) = 0 for every 99 E X n-.

PROOF.

This follows from Lemma I by a reductio ad absurdum

argument.

4. Operator-valued function space integrals

We now introduce the operator-valued function space integral that was defined in [1]. Let A be a parameter which eventually will assume complex

values but for the present is real and positive. Let C[a, b] = c C[a, b], real and continuous on [a, b]}, and let Co[a, b] = x(a) = 0}. Let F be a complex-valued functional defined on C[a, b] such that the following Wiener integral exists: (4.1)

(IA(F)0)(f) = f

F(A-1'2x + 6)0(A

"2x(b) + 6) dx

C0[a.b]

for 0 E L2(-oo, oo) and real 6. Let IA(F)b denote the function which maps e into (IA(F)/)(e) for almost all values of e in (-co, oo). Let IA(F) denote the operator that maps 0 into IA(F)/. We extend the definition of I;4(F) to complex values of the parameter A in two different ways. Using analytic extension, we define I>an(F) to be the operator-valued analytic function of A which agrees with IA(F) for real A and is analytic through Re A > 0. It is clear that if I,, (F) exists it is unique.

We also make a sequential definition which in a large class of cases

CAMERON AND STORVICK

180

coincides with Ian(F). Let us consider functionals defined on the larger space B[a, b]:

defined and continuous on [a, b] except for B[a, b] = a finite number of finite jump discontinuities}.

Let a be any subdivision of [a, b], a: a = to < tl < t2 < let norm a = max (t, - t,-,). For any x c B[a, b], let (4.2)

X'(0 =

< to = b, and

if tj_1 < t < t, for j = 1, ..., n if t = b.

rx(t7_1) x(b) )

Let

Fa(x) = F(xa)

(4.3)

for x c B[a, b]. Let us define the function fa(v(,, ... , vn) by the equation fa[x(to),..., x(tn)] = F(xa) which holds for all x c B[a, b]. For Re (A) > 0 let (I;,(F)0)(6) _ (4.4)

a)112 .. (tn - to 0(vn)

exp -

1)1(2 J

A(v; - v; _ 1)2

n

7=1

2(t,-t, _,)

w .(n)

f

M6, v,, ... , vn)

dv1... dvn,

where 0 E L2(-cc, oo) and vo - 6. We now define (4.5)

hen (F) = w lim I;(F), a-0

where w lim denotes weak limit. In Theorem I of [ 1 ] we have shown that if F(x) is bounded and con-

tinuous in the uniform topology on B[a, b], I;seq (F) exists and equals IA(F) for real positive A. We have shown in [1] that for a large class of functionals F, I and are equal for Re (A) > 0. When I;°(F) and Ire°(F) exist and are equal for Re A > 0, we shall call their common value IT(F).

Let F be a functional for which Ixeq(F) and I;°(F) exist and are equal for Re A > 0. Then if the following weak limit exists, we define

Jq(F) =

w lim

A-iq, Re A> 0

I,,(F).

In order to obtain the existence of the weak limit Jq(F), we shall first prove a theorem showing that certain functions defined by cluster sets satisfy our integral equation.

THE SCHROEDINGER EQUATION

181

5. The integral equation for cluster values THEOREM 2.

Let B(t, u) be continuous almost everywhere in the strip

R: 0 < t 5 to, -oo < u < oo, let I B(t, u) I S M for (t, u) E R, and let q be any real number, q = 0. Let 0 E L2(-oo, oo) and let Ft(x) = exp

(5.1)

f t B(t

- s, x(s)) ds }

o

JJJ

Let G(C, A) - G(t, e, A) _ (I (Ft)cb)(e), where = (t, e). Then every element F(C) F(t, e) of `'(G( , A), - iq) satisfies the integral equation:

cf

q

F(t,

= (-) q

(5.2)

1/2

iq( 2t - u)z ) du

,1(u) exp (

tM

+ (2qi)

J

fo

(t - s)

uz 0(s, u)P(s, u) exp (iq(e

- u)2

2(t - s) ) duds

for almost all (t, 6) E R.

The proof of this theorem parallels the proof of Theorem 9 of [I], but it differs at significant points. For the convenience of the reader the entire proof is presented. PROOF. Let t and q be given real numbers, t c (0, to) and q 0 0. By applying Lemmas 4 and 5 of [1, p. 527] to B*(s, x(s)) = B(t - s, x(s)) we observe that B(t - s, x(s)) is measurable for almost every x E Co[0, t], and is measurable in the product space. Thus FF(x) is Wiener integrable on

Co[0, t] and hence also on Co[0, to]. By Theorem 4 of [I], IA(Ft) exists and

is bounded for Re A > 0,

e1to = M'. Hence we obtain II I?,(Ft)0II < M'11011

and

J(IjFt)0)(6)J2 d6 < M'2110II2.

Upon integration with respect to t we obtain m

f t° Ja

F. I(IA(Ft)o)(e)I2 d6 dt

If we set (IT(Ft)1)(6) = G(t, e, A) = Fubini theorem J JG(C, A)12 dd

A) where M'2toJJblI2

R

and

JIG(-, A) II
2/(t-3)] dude ds.

We now consider the double integral:

F. F-

9,(C)h(u)e-cni2xZ-u)2 du d;~

where h(u) = O(s, u)G(s, u, A)

and A = t

AS

By Lemma I of [1], since I1hII < oo, we may observe that

(A 112 -.

Ih(u)Ie-cx8 ni2uC-u)2 du

11h1j;

CAMERON AND STORVICK

184

hence by the Schwarz inequality

RA1/2 (-a--) F.

f°°

Jh(u)Ie-cxen/2)(4-u)2 dude
,/2)[(4-U)2/(t

6(s, u)G(s, u,

2)i duds de

9(s, u)G(s, u, A) . e-(A/2)[cC-u)2/(t -s)] dC du ds.

(5.12)

We now consider the Hermite polynomials Hk(C) and write

f

ft

W Hk(6)e-1212

o

(t

-s)

1/2

O(s, u)G(s, u,

ds)1/2

ft

=

f

(t

d)e-cr/2x«-U)2/t-s?] duds de

f - O(s, u)G(s, u, A)Dk(u, t

A

where

tk(u, A) =

Hk(6)e-Z212e-nc

-u)2/z de

and A =

Now we consider the Fourier transform of ck, 1

.y ('Dk) = (201/2 f

Ok(u, A)etu" du,

and observe that F(Hk(t)e -X2/2) = (jp)ke - 02/2_

A

t - s

S) du,

THE SCHROEDINGER EQUATION

185

Because the Fourier transform of a convolution is the product of Fourier transforms we have fTw

Ok(U, A)etu0 du = .F(tk) =

(27r )1/2

Al/2/2

(277)1/2(ip)ke-n2/2

27T(t - S)

a-Da/2n

(lP)ke (T+t-s)0212A

Because the Fourier transform is an isometric transform, we have II(DkII =II`((Dk)II = [2

-

1/2

s)

J0

P2ke-(1A12+(t-s)ReA)P2/IAI2.dp

]

If we let _ (1A12

+ (t - s) Re a IAI2

P2,

)

then

IAI(IAI2

1/2

ao

27T(t - S)IAI2k+1 II(DkII

+ (t - s) Re A)k+1/2 f ak-1/2e-0

do]

and using the gamma function we have - S)IAI2k (IAI2 +(t-s)ReA)k + 12

E

r(k +

)]1/2

We shall later perform a similar analysis for differences of (Dk's. We shall undertake to establish both the pointwise limit and the limit in the mean for 1 k. We begin with the pointwise limit. Set A = s). For a fixed value of u, lim

dC = f J

Hk(6)e-42/2exp

(t (S - u))

S

2(t

s)

This follows from Lebesgue's convergence theorem because Ie-(n/2)(Z-u)2I = e-Ren(Z-u)2/2 < 1

if Re A > 0,

and we have

Hk(6)e

-,2,12 e

-(A12)(1-U)21 < I Hk(e) I e - Z212

Thus we have the pointwise limit for Re A > 0 and for each fixed u, lim 40k(u, Au) = 4Dk (u,

AW

-i9)

-s

If we now apply Lemma 1 of [1] to the function (Dk,

(')k(u, A) = fT Hk(

)e-f2/2e-(A/2)(,-'u)2 dS,

de.

CAMERON AND STORVICK

186

we obtain the inequality 27T

(j)

0k(', A) 11

1/2

2

In order to establish the limit in the mean for the Gk's, we observe the general property; if a sequence of functions fn possesses the property that lim f .(x) = g(x) and if lim fn(x) = h(x) almost everywhere, then g(x) n-m

n- w

h(x) almost everywhere. Now if µ, v are distinct positive integers, t-

- -

AU

T

t A9-

2

A-

S)

S)

S) - (A

= 21r(t - s) J

-

r

-

-v

t s)/(2Av)P2

(IP)ke--t

]

(A)"2

[(ip)ke (A )1/2

2

S)

lp)ke-(A +t-s)/(2A )P2

r(-jP)ke-G

(AU) 1/2

dP

1/2

We have established that the integral tends to 0 as µ, v - oo, since the integrand tends to zero as µ, v - oo and is dominated by a function

integrable with respect to p and independent of µ an4 v, namely Thus by dominated con(8e-P212p2k)/Iql for IA > 1Igk, IAvl > +Iq!.

vergence, we have established that

t-

S)

-

t

All

S)

--0

as p, v -* oo, and by the general property indicated above, we have a.e. lim (Dk(u,

n-

t SS an )

- k(u

-iq)

Let us consider the functions Ik(A)

f0

(t aS)1/2 J

9(S, u)G(s, u,

A)tk(u,

S) du

t

and

Jk(q) =

s

Jo (t

as )1/2 J -.

O(s,

u)I'(s,

u)4Dk(u

Then Ik(on) - Jk(q) = A + .2,

t

du. L

S

THE SCHROEDINGER EQUATION

187

where t

1 = f (t -S)1/2

O(s, u)G(s, u, An)

1

0

(u, t

an

iq )J du ds

t

s) - -Dk(u,

and

2=

t I

(t -

l s )1/2

u)q)k(u,

f

-lq)

O(s,

[G(s, u,

A,,) - P(s, u)] du A

By equations (5.5) and (5.12), we have FM 94C)G(t, 6, An) dC

=

f

W

(2;t)112

f

Pk(e) 1/2

+

Mm

u )2) du dC

0(u) exp (-'\n(e t

fo (t -1s)1/2 f

O(s, u)G(s, u, An)

W

u)2

f- - Tk() eXp ( 2(t-s) =

fF.. Tk(C)

1/2

A

f

m

An(S - u)

2t

J _ . 0(u) exp (

)

d6 du ds

(A.) 1/2 )dud +Ik(on)

We next note that if Rt - (0, t) x (-oo, oo) s)1/2 O(s, u)(Dk(U, t

(t for, if we consider

i9) E L2(Rt);

JJT

2

-iq

-16)1/2

(t

e('s,

u)(Dk(u, t

du ds

)

and apply Lemma 1 of [1] to the function 'Dk, we obtain I.Pkll2

f Iok(u, A) I2 du I

r

and

ffl

-Iq)I2du

(Dk I

(

u't-s

< 21T(t - s) IqI

II

k112.

Thus we have established that

f ot f

IC-1S)1/2 0(s,

t

1S) I2 du dS -

M

IgI2 II

kll2

CAMERON AND STORVICK

188

and have proved that (t

_ls)1/2 O(s, u)(Dk(u, t -7 J e L2(R1)

Thus by equation (5.3), we obtain .f2 - 0 as n -- co. The integral 1

ft

1

B(s, u)G(s, u, An)

Jo (t - S)1/2

L'Dk(u, t

an

S) -'Dk(u, t -iq

du ds, 1

and by an application of the Schwarz inequality, we have

ft Ii'II

- k(''S) - ( ,-tq) t- S (t - S)1/2

o(t-S)1/"MIIG(s, ,A)II )ktS)

Me"tof


IqI/2, by bounded convergence we have .f1 --> 0. Consequently Ik(an) - Jk(q) -* 0 as n ---> oo, i.e.,

f

Hk(S)e-1212

J-s) 1o t

(t

1

112

f

O(S, u)G(s, u, A)

e- (Al2)[ -u)21(t -S)] du ds d6

(5.13)

- Jk(q)

asn -*oo. By applying Lemma 10 of [1] to the inner two integrals of e

1 Jk(q) = So (t - s)1/2

f

0(s, u)I'(s, u)

TkMe

(t4/2)M-U)2/(t -S>]

de duds

and using f(u) = O(s, u)F(s, u) we have t

Jk(q) = So

Q)

(t - s)1/2f- k() f

= (2T

fo f °°

O(s, u)r(s, u)e+j9cC

pk(e)S(s, 6) de ds

U)2/2(t

S) du d ds

THE SCHROEDINGER EQUATION

189

where

)=

AS'

q

1I2 (.)

O(s u)F(s

(2i(t - s)

du,

and by Lemma I of [1] and (5.3) g(s, ) II < M11r(S, )M < lim inf Al 11 G(., -, An) 11 < MM l(to)112 11011.

M11F(.,

n-.

Thus by the Fubini theorem 2-i) 1/2

Jk(q) _

9

f-

f

9'k(e)

e) ds de

f

J

t

f-

1

o (t -

S) 1/2

O(s, u)r(s, u)

(5.14)

e+tg(1-u)2/2(t-s) du ds di;.

Thus we have proved by equations (5.13) and (5.14) that li

/2

(An

m

)1

(5.15)

_

f

91k() fo

(fq 1/2 ri)

f

f

°°

ao

'k() f

_

(- s)2(-s)

O(s, u)G(s, u2

eXp

2(t

- u)2)

iq(6 - u)2

O(s, u)r(s, u)

(t - s)1/2 exp ( 2(t - s)

du ds d(t

du ds d

.

We now turn our attention to the first integral on the right-hand side of the equation. By dominated convergence, since 9'k c L1, lim

TkMe-(,1n/2)[cf-u)z/tJ de = f

9)k(

)ei9(Z-u)2/2t d5

n -+ m

for all fixed u. By Lemma I of [1] we see that

J.

-u)'/2t1 d$ < C9'd

S

for some constant c.

Hence the pointwise limit can be replaced by the weak limit (27r)1/2

f

&(u) 91k(e)e-(An/2t)[(

lim

FW

(_I_) 112 27Th

O(u) J

J

-u)21 de du

,

d6 du.

We interchange the order of integration on both sides, using the Fubini theorem for the limitand on the left and Lemma 10 of [1, p. 542] on the right. We can do the latter because ,/9pk e L1 n L2. We obtain (27rt )1/2 1 [m n

(5.16)

f-

0

91k(S)

W(u)e-(A/2)[(f- u)2)t1 du d$

J (_I_) 112 27rit

fT qk(6) f '

0(u)e`a(e-u)2/2t du d6.

CAMERON AND STORVICK

190

Let us multiply the following integral equation of Theorem 8 of [1], valid for Re A > 0, (_)ltz f0(u)e-1(4 - U)2/2t du

G(t, 6, A) _

)112

(5.17)

+

ds

f o(t t - S)

"2 f

O(s, u)G(s, u, A)e-(A/2)[(4-u)2(t-s)7 du

2/2 and integrate both sides with respect to i; from to co and pass to the limit as n -* co with A replaced by An. by

Hk(e)e-

-cc

Since we know that the limit on the right-hand side exists, it follows that the limit on the left-hand side exists also, and we have by (5.15), (5.16), and (5.17) lim

n-m

f

f

9'k()G(t, 6, An) d6 = m

qiiz c>

(Tt)

f

(u )

l

q

()1/2

+

exp ( lq(e 2t

ft (fl o

w

u)z

Tk(6)r*(t, e) d;±

) du

iq(- u)z

B(s, u)l7(s, u)

f

-

eXp

(t - S)1/2

( 2(t - s)

duds

for each t c (0, to] and almost all 6 E (-co, co). Thus for each fixed t, 0 < t < to, the weak limit of G(t, , an) is r*(t, e), w lim G(t,

(5.20)

n-m

,

An) = r*(t, 6).

Letting /3(t) E Lz(0, to) we write Pk(6)G(t, 6, An) d6 dt

f 0 p(t) lim

=

(5.21)

f

to

o

/3(t) f

9Pk(6)r*(t, 6) d;~ dt.

By dominated convergence we have lim f oo /3(t) f

(Pk(')G(t, 6, An) d6 dt to

(5.22)

=

f

o

/3(t) f

k( )r*(t, 6) de dt.

By the definition of r(t, 6) we have

n_ lim J to /3(t) f 0

9,k(6)G(t, 6, An) d'; dt =

to /3(t) 0

(' I

'pk(e)r(t, e) de dt,

THE SCHROEDINGER EQUATION

191

and therefore r(t, e) = P*(t, e) almost everywhere in R, and the proof of Theorem 2 is complete. COROLLARY 1. Under the hypotheses of Theorem 2, for each fixed t E (0, to], the right-hand side of equation (5.2) as a function of e is an

)ll.

element of L2(-co, co) and its norm is bounded by 11011 + MAo(to)112, where

Ao = jr(.,

PROOF. Let t be fixed, 0 < t < to. By Lemma 1 of [1], the first term of the right-hand side as a function of 6 is an element of L2(-00,00)-

Indeed,

2cry rW (q)"2 7rit

(u) exp

X

(iq(6 - u)2) du < II0II

Using the notation of Theorem 2 we have

r(., ) EL2(R), and thus there exists a null set So, such that if s 0 So,

r(s, ) E L2(-co, 00)-

Now ao

to

A

o

f

=

F(s, u) 2 ds du.

J

0

Let A(s) F(s, then A2(s) = f and A(s) E L2(0, to). We set

O(s, u)r(s, u)

1l1/2

f

Q(s, t, 6) =

r(s, u)12 du; thus A10 = f o`0 [A(s)]2 ds,

(t - s)1/2

exp

(

iq(e - u)2 2(t - s) ) du.

By Lemma 1 of [1], for any fixed t c (0, to] and for s 0 So and 0 < s < t, Q(s, t, e) is of L2(-co, oo) in 6, and 11 Q(s, t, ) < A(s)M. Then we have by the Schwarz inequality that

f.

I fot Q(s, t, 6) dsl2

t

d6 = f

f0 Q(s, t, 6) ds f0 Q(s', t, 6) ds' d6

f.. = =

fo fo t

t

o

o

t

t

f f f-

Q(s, t, 6)Q(s', t, 6) ds ds' d6

Q(s,

t, e)Q(s', t, 6) de ds ds'

< f of A(s) A(s') M2 ds ds' S M2Aot. 0

0

Thus we have proved that for each fixed t c (0, to], f .t Q(s, t, 6) ds e L2(-oo, co) as a function of 6.

CAMERON AND STORVICK

192

Let G(t, e, A) satisfy the hypotheses of Theorem 2.

COROLLARY 2.

Let Al, A2, ... be a sequence such that Re An > 0, lim A. = -iq, and w lim G(t, 6, An) = F(t, $) over R. Then for each t e (0, to], it follows that n-m

w lim G(t, e, An) = P*(t, ;r) exists over (-co, oo). Moreover r* = I' almost

n--

everywhere on R, and r* is given by (5.19) for each t c (0, to] and almost all real 6. PROOF.

This follows from equation (5.18).

6. Existence theorem THEOREM 3.

Let O(t, u) be continuous almost everywhere in the strip

R: 0 < t < to, -oo < u < oo, let

I O(t, u) I

< M for (t, u) E R, and let q

be any real number, q 0 0. Let 0 E L2(-oo, oo) and let

Ft(x) = exp {

(6.1)

t O(t

l Jo

- s, x(s)) ds}

Then for each fixed t E (0, to], JQ(Ft)o exists and is an element of L2(-oo, co).

Moreover, if F(t, i;) = (Jq(FF)+b)(6) for (t, 6) E R, then for each t E (0, to] we have F(t, (6.2)

) _ ()

112 0

G(t, 6, A) = F(t, 6).

(t,t)

An

By Corollary 2 of Theorem 2, for any sequence {An}, Re An > 0, - iq, we have for each t c (0, to], w lim G(t, e, An) = F*(t, e) over n- m

(-oo, co). Thus we have for each fixed t E (0, to]

w lim (IAn(Ft)b)() = r*(t, 0

(6.5)

n--

for almost all 6 E (-oo, oo). Since the sequential limit exists for all sequences {an} and r* is unique by (5.19), we have (6.6)

(J0(Ft)cb)(e) =

r*(t, e).

w lim

A -iq, Re A> 0

Thus IF as defined in the hypotheses of the theorem exists, equals F* and satisfies the integral equation (6.2). 7. Deterministic theorem for JJ(F)

We now show that with a slightly stronger hypothesis, Theorem 5 of [1] can be made deterministic, i.e., its conclusion can be shown to hold for all nonzero q instead of merely almost all q. THEOREM 4.

Let B(s, u) be bounded and almost everywhere continuous in

R. Let F(x) = exp { f to B(s, x(s)) ds}. Then if q is real and q :A 0, J0(F) exists as a bounded linear operator taking L2(-co, 00) into itself. Let B*(s, u) = B(to - s, u) and Ft*(x) = exp { f o B*(t - s, x(s)) ds} = exp { f o B(to - t + s, X(s)) ds}, for 0 < t < to. Then by Theorem 3, Jq(F*) PROOF.

exists as a bounded linear operator taking L2(-c0, 00) into itself. In particular, this is true when t = to. However, Fo(x) = exp {

f

to

B(s, x(s)) ds } = F(x).

o

JJ

Hence Jq(F) exists and is a bounded linear operator taking L2(-c0, 00) into itself. REMARK.

Clearly the interval [0, to] can be changed into the interval

[a, b] used in the definition of IA(F) and J0(F).

REFERENCE 1. CAMERON, R. H., and D. A. STORVICK, "An operator valued function space integral and a related integral equation," J. Math. Mech., 18 (1968), 517-552.

A Lower Bound for the Smallest Eigenvalue of the Laplacian JEFF CHEEGER Various authors have studied the geometrical and topological significance of the spectrum of the Laplacian O2, on a Riemannian manifold. (The excellent survey article of Berger [2] contains background, references, and open problems.) The purpose of this note is to give a lower bound for the smallest eigenvalue A > 0 of O2 applied to functions. The bound is in terms of a certain global geometric invariant, essentially the constant in the

isoperimetric inequality. The technique works for compact manifolds of arbitrary dimension with or without boundary. The author wishes to thank J. Simons for helpful conversations and in particular for suggesting the importance of understanding the following example of E. Calabi. Consider the "dumbbell" manifold homeomorphic to S2, shown in Fig. 1. The pipe connecting the two halves is to be thought of as having fixed length I and variable radius r.

Fig. 1

One sees that A -> 0 as r --- 0. Calabi's original argument involved consideration of the heat equation,

atT = 02T.

A somewhat more direct argument is as follows: Let f be a function which is equal to c on the right-hand bulb, -c on the left-hand bulb and 195

JEFF CHEEGER

196

changes linearly from c to - c across the pipe. (c chosen so that fM f 2 = 1.) Then fm f = 0 and bulbs, 10 Ilgrad f II : tLi One has by Stokes' theorem

f

f Ilgradll2 ,:

Clearly A -± 0 as r -> 0. The Calabi example makes it evident that in bounding A from below, it is not enough to consider just the diameter or volume of M. It also suggests DEFINITION 1.

(a) Let M be a compact n-dimensional Riemannian manifold, 8M A(S) Set h = inf ' where A( ) denotes (n - 1)-dimensional area, V( min V(M;)

denotes volume, and the inf is taken over all compact (n - 1)-dimensional

submanifolds S, dividing M into submanifolds with boundary M,, M, with M = M, U M2, and 8M, = S. (b) If 8M

0, set

A(S)

h = inf V(M,)' 3

where we stipulate S n 8M and there is a submanifold with boundary M, such that S = BM,. M, is necessarily unique.

In the preceding definition 8M, M1i M2, S are not assumed to be connected. THEOREM.

In the situation just described A >_ *h2. (If 8M = ¢ we

assume f * df I M = 0.) PROOF.

If M is not orientable, it will suffice to look at its 2-fold

orientable cover. Let f be the eigenfunction corresponding to A. We make the assumption that f has nondegenerate (and therefore isolated) critical points. If this is not the case we use an obvious approximation argument based on Sard's theorem, which will be left to the reader. First note that for any region R, such that f * df I M = 0, (2)

A

Ilgrad f II2 fR f2

.IR

fRf2 (fR 1/Igrad fll2) (3)

(fR

(fRf2)2 (4)

f ell )2

I (fR 4

Ill/grad

2IfI - lgradf f

(fRf2)2

2)2

11 )2

THE SMALLEST EIGENVALUE OF THE LAPLACIAN

where the inequality in

(3) is

197

obtained by squaring the Schwarz

inequality. We now assume that zero is not a critical value of f (Again if this is not the case the argument undergoes a trivial modification.) Now the submani-

fold Z = {xl f(x) = 0) divides M into n-dimensional submanifolds with boundary Ml = {xlf(x) > 01 and M2 = {xlf(x) < 0}. It is in asserting that Z, Ml, and M2 exist that we are using the information that f is a nonconstant eigenfunction (A 0 0) and hence must take on positive and negative values. Let h, hl, and h2 be the constants corresponding to M, Ml, and M2. Clearly if, say, V(M1) < V(M2), then hl > h. It will then suffice to prove the estimate for the submanifold with boundary Ml and, moreover, the same argument will work for any manifold with boundary. Now the regions of Ml lying between the critical levels of f2 have a

natural product structure L x I given by the level surfaces and their orthogonal trajectories. We introduce product coordinates (x, t) by choosing local coordinates {xi} on some L and setting t = f2. Since dt is orthogonal to dxi, the volume element dv may be written in coordinates as

dV = vl(t, x) dt x v2(t, x) dx.

(5)

Since f 2 = dt, v,(t, x) = a I we have a

grad t, at/

v1(t, x) =

II grad f 2

(6)

a

a

at

at

1. dt (a) t = Let V(t) denote the volume of the set {x e Ml1 f2(x) > t}. V(t) is

(7)

continuous and differentiable.

f

(8)

II grad f 2 II

M1

(f . dv = IL (J

II grad f 2 II

vl v2 dt) dx.

By (7) this is equal to (9) (10)

IL

v2 (fo

dt) dx

=f

o

v2

dx) dt

(f$

= f A(Li) dt > h, f V(t) dt 0

0

_ -hi f t

d c

V(t) = V(M1) - f {f vl(x, t) v2(x, t) dx} dt, 0

L

JEFF CHEEGER

198

and t = f2. Thus (11) becomes h,

: t{ f

J0

JL

vl(t, x) v2(t, x) dx} dt

= h, J

(13)

m

{fL

0

= h, f

(14)

t . V1.0, x) . v2(t, x)

dx} dt

f 2. dV.

Squaring the inequality (8)-(14) and dividing through by (fM, f 2)2 yields (fM1fgrad (JM,f j

(15)

II)2

> h2 > h2

2

Combining (15) with (2)-(4) completes the proof.' In dimension 2, it is relatively easy to see that h is always strictly greater

than zero. In fact, let V(M) = V, and let c be such that a metric ball of radius r < c is always convex. Then, if A(S)

(16)

c

min V(M,) = V

it follows that each component of S must lie in a convex ball. On such balls

the metric g satisfies k E > g > k E where E is the Euclidean metric in normal coordinates. Hence h > 0 is implied by the usual isoperimetric inequality in the plane. Now, according to a theorem of the author (see [3]), c may be estimated from below by knowing a bound on the absolute value of the sectional curvature SM, an upper bound for the diameter d(M), and a lower bound for the volume. Once this is done, it is elementary that k may be estimated from Is, 1. This yields COROLLARY.

that if

If dim M = 2, 8M = ¢, then given 8 there exists a such

+ d(M) + IsMI < 8, then A > V

In case dim M > 2 the situation is not so elementary but 6.1 and 6.2 of [4], or [5], will still imply that h > 0.2 Actually the results of [4] and [5] show the existence of an integral current T whose boundary in S, such that A(S) divided by the mass of the current is always bounded away from zero independent of S. However, since T is of top dimension, it is known that T 1 The equality f,,,11 grad f211 dV = f 0 A(L,) dt is actually a special case of the "coarea formula" (see [6]). 2 Thanks are due to F. Almgren for supplying these references.

THE SMALLEST EIGENVALUE OF THE LAPLACIAN

199

may be taken to be either Ml or M2. If 9M 0, the fact that h > 0 may also be deduced from Theorem 1 of [1] without too much difficulty. It would be of interest to generalize the argument given here to A2

acting on k-forms. Singer has pointed out that this would give a new proof of the fact that the dimension of the space of harmonic forms is independent of the metric and that the techniques might be applicable to other situations. To date, we have not been able to accomplish this except in the case dim M = 2. The essential point here is that for any eigenvalue of 02 on 1-forms one can find an eigenform of the form df, where f is an eigenfunction corresponding to A. This observation is probably of little help if n > 2. UNIVERSITY OF MICHIGAN

and SUNY, Stony Brook

REFERENCES 1. ALMGREN, F., "Three Theorems on Manifolds with Bounded Mean Curvature," B.A.M.S., 71, No. 5 (1965), 755-756. 2. BERGER, M., Lecture notes, Berkeley Conference on Global Analysis, 1968. 3. CHEEGER, J., "Finiteness Theorems for Riemannian Manifolds," Am. J. Math. (in press). 4. FEDERER, H., and W. FLEMING, "Normal Integral Currents," Ann. of Math, 72 (1960).

5. FEDERER, H., "Approximating Integral Currents by Cycles," A MS Proceedings, 12 (1961). 6.

, "Curvature Measures," Trans. A.M.S., 93 (1959), 418-491.

The Integral Equation Method in Scattering Theory C. L. DOLPH' 1. Introduction

The subject of this essay seems singularly appropriate to this occasion for several reasons. Much of the material on which it depends stems from the Berlin and Munich schools where Salomon Bochner spent many of his early mathematical years. The foundations of the theory are perhaps best expressed via the Bochner integral (see Wilcox [1] and Dolph [2]). Finally, the subject has reached some sort of culmination for the direct problems with the as yet unpublished work of Shenk and Thoe [3], while the inverse problem has achieved a significant new impetus from the recently published work of Lax and Phillips [4] and that of Ludwig and Morawetz [5]. In this review I hope to give some indication of the history and status of this subject in the hope that perhaps a new generation of mathematicians

can be interested. For this reason very few theorems will be proved in detail, and the discussion will be limited to the appropriate operators with independent variables restricted mainly to two- and three-dimensional spaces, although almost all that will be said will be valid, with modifications, for n > 2. At the outset it is necessary to make a clear distinction between the basic physical problems which will be treated and the method of the title, since, 1 The author wishes to express his great gratitude to P. Lax and R. S. Phillips for

making their new unpublished work [4] available to him as soon as possible; to S. Sternberg [42] for the same reason; to P. Werner for making the dissertations of all of his recent students including those of Kussmaul [52] and Haf [63] available; to C. H. Wilcox, who furnished me a Xerox copy of his Edinburgh lecture notes; and in particular to N. A. Shenk II, and D. Thoe for their unremitting efforts both by mail and personal contacts to keep me informed of their progress. I would also like to express my thanks to H. Kramer, R. Lane, and R. R. Rutherford for the stimulating summer they provided at G. E. Tempo 1968. I have taken the liberty to reproduce some things from those in Section 7; see also [82]. Sponsored in part by NSF grant GP-6600. 201

C. L. DOLPH

202

in many cases, the physical problems can be treated by other methods, the most extensive such treatment of which is to be found in the work of Lax and Phillips [6]. The mathematical problem is as follows: Let Q in R" with n = 2 or 3 be an unbounded domain with a compact C2 hypersurface F as boundary, where r may have several components or be empty. In the former case P is considered as the disjoint union Pl U P2 with each F; either empty or consisting of some of the components of P. Let v be the unit normal to r

which points into 4, P' the interior bounded by F, and a a real-valued Holder continuous function on IF,, and let q(x) be a real-valued Holder continuous function on Q u P which satisfies q(x) = 0[exp (-2aixl)]

(1.0)

as lxi -* o0

for some a > 0. It should be noted that this latter assumption is more general than that used by Lax and Phillips [6] who require a q of compact support and a domain which is star-shaped. Further let K denote the subset {k; Im k > - a) of the complex plane if n is odd and the portion

{k001Imki 0 was taken by Miller's student Werner [47], who replaced the simple double-layer Ansatz by one

C. L. DOLPH

206

of the form O(x, k) =

avy Fk[(x - s)µ(s) dSs]

+ f Fk[(x - y),-(y) dV ]}, r

(2.2)

where, in n-dimensional Euclidean space,

F} [kl xl ] = ±4 (1)H[kIx];

P= n +

(2.3)

2

Hp1),

2;

- H Hpz)

and in which T is an unknown volume potential. Under this assumption the difficulties associated with the interior Neumann problem disappear and one has merely to deal with the first part of the Fredholm alternative to get existence and uniqueness since one can set up a vector integral equation involving a compact operator in a suitable Banach space. One obtains, in

addition, analyticity in k for the solution for all of k, Im k >_ 0 and this method was used subsequently by Shenk [17] and is part of the technique in the current version of Shenk and Thoe [3] which will be discussed in Section 6. From a practical point of view the presence of an additional volume integral makes numerical calculations immensely more difficult. As a result several people-Leis, another student of Mi.iller [48], Werner and Birkhage [49], and Pannic [50]-replaced the double-layer assumption by one which reads, for both the Dirichlet and Neumann problems, as follows:

qi(x, k) = Jr (avs - i"))Fk(x - s)µ(s) dSs, where

=7j(k)=1

forRek>0

= -1 for Re k < 0. Under this assumption one needs only the first part of the Fredholm alternative. Pannic also went on to treat the corresponding exterior electromagnetic problems although here I find the treatment somewhat obscure. As Werner and Greenspen [51] have shown for the Dirichlet problem and Kussmaul [52] for the Neumann problem, it is possible to obtain excellent results numerically for at least the cylinder problem and certainly the absence of the volume integral is of tremendous help. I fullyanticipate that a subsequent version of Shenk and Thoe [3] will employ it also.

As sometimes happens in mathematics, good mathematics results, ironically enough, come from a method failure rather than from the true

SCATTERING THEORY

207

physical situation. To return to the Maue integral equation (2.1), I conjectured that what was true there was true in general. This led Wilcox and me to suspect that a technique of Tamarkin's [53], combined with a theorem of Rellich [27], could be used to prove that this was always the case. This we were able to establish. Subsequently, for his Edinburgh address of 1967,

Wilcox developed an extensive Fredholm-type treatment valid for the entire complex plane. Since it seems unlikely that these results will ever be published I shall use this occasion to state them in some detail.

Before doing this, a few comments on the radiation condition are in order. 3. The radiation condition

There are many possible forms for a radiation condition and even today new ones are being devised for special purposes (compare Ludwig and Morawetz [54]). In 1956, Wilcox [56] gave a particularly elegant reformula-

tion of that of Sommerfeld, in which for any solution ¢(x, k) in C2 he showed that it was sufficient to require that

ii

lm

Rao r=R

(a

8r-

ik) 0I Z dS _

0.

Earlier, in 1950, Kupradze [56] gave the following conditions (compare the 1956 German translation [57]):

r(8 - ik)¢ = efskro(1) ro(r) = efskro(1),

where the plus sign holds for Im k < 0 and the minus sign for Im k > 0. Subsequently, most workers, including Shenk and Thoe [3], have used a 1960 reformulation due to Reichardt [58] for the plane and a 1965 generalization of Schwartz [411. If Fk (x - y) is any fundamental solution of (1. 1),

these conditions are merely the requirement that the surface integral arising from Green's formula, namely:

L. [s-Fx - s) - Fk(x - s) 8v- 1(s)] dS5 converge to zero as r -* oo. For k 0 and 0 < arg k < Tr this condition is equivalent to the Sommerfeld radiation condition if one takes Fk(x - y) to be that given in (2.3). The elegant formulation of a radiation condition due to Wilcox [59] in 1959 seems somehow to have been overlooked, probably since he did not

C. L. DOLPH

208

explicitly state its validity for the entire complex plane at the time of its introduction. To state it, let k c [K - [0]] and let v be a solution of

Ov+k2v=0

(3.1)

for

1xI > C.

Then he observes that the mean, Mx ,[v] =

1 f v(x + rw) dw, 47r Ji(0i=1

is in fact equal to Mx,T[v] = v+(x) err + v _ (x) e

ikr

r = r(x),

where eikr

v(x) = ± 9;ik

f,

8 (0i=1

ar + (--

ik) rv(x + rw) dw

is defined for all x c R3 (independently of r >_ ro(x)) and v,,(x) are entire solutions of (k). v is a k-radiation function for, if v E C2[1 ], v is a solution of (3.1) for x such that

J

-I.1 (ar

- ik)rv(x + rw) dw = 0.

For Im k > 0 these conditions are equivalent to those of Sommerfeld. With this result we can return to the Kupradze-Maue-Weyl integral equation. 4. The Fredholm integral for the exterior Dirichlet problem

Following Wilcox [60], we assume that F is a closed bounded surface in R3 of class C2, that S2 is the connected exterior of r, and that F' is the interior of F. The problem to be solved is that of finding a solution of L

+ k2¢ = p(x) for x c S2

and

q(x) = 0

for x E F.

Here p(x) is a prescribed function-the source-and k is the wave number in the set {K - (0)} but is otherwise an arbitrary complex number. Let Oo(x, k)

1

fCiklx - vl

Ix-

Y

I p(Y)dY

SCATTERING THEORY

209

be the solution for the case in which S2 = R3 and F = 0. In the general case, let

¢(x, k) = ¢o(x, k) + v(x, k) for x E 4, where v(x, k) is a k-radiation function for 4 in the sense of Wilcox such that v(x, k) E C'(S2) and

v(s, k) = - co(s, k) for x e P. As in the Kupradze-Maue-Weyl equations, the double-layer assumption implies a representation for v(x, k) in the form (4.1)

etlkl Ix-sl

a

v(x, k) = fr avs 2lTI x - s I

µ(s) dSs,

where v is a unit exterior-directed normal on P. This leads to the following integral equation (in view of the well-known jump relation): (4.2)

µ(s) = Jr K(s, s'; k)(s') dS= ¢o(s, k)

for s, s' E P,

when K(s' s

k)

1

a el kJ Is-s'I

27r avs,

I s - s' I

Then in terms of iterates of the kernels, one can represent the resolvent kernel of (4.2) in the form

R(s, s'; k) = N(D(k) k)'

D(k) 0 0,

where

R(s, s'; k) = K(s, s'; k') + Jr K (s, s"; k)R(s", s'; k) dSs

R(s, s'; k) = K(s, s'; k) + f R(s, s"; k)K(s ", s'; k) dSs..

r

for all k such that D(k)

0. This in turn implies that the double-layer density admits the representation

µ(s, k) = v(s, k) +

Jr

R(s, s' ; k)v(s', k) dS

C. L. DOLPH

210

and that the total field admits the representation ¢(x, k) = f G(x, y; k)p(y) dy. Here the resolvent Green's function has the following explicit expression:

G(x, y; k) _

I

etlkl Ix-yI

-4w x - Y + Jr

K(x, s; k)

eilklls-yl

eilkl Is' yl

f R(s, s , k) Is, - yl is - Y1 + Jr

dSs.

As such it is defined for all k except for those values of D(k) = 0. Let us examine the nature of these zeros. First of all D(k0) = 0 if and only if the homogeneous integral equation

µ(s, k) -

(4.3) If

Jr

K(s, s; k)µ(s') dSs = 0.

µ(s, k0) 0 0 is a solution of (4.3), consider v(x) =

Jr K

(x, s'; ko)(s', ko) dSs,

so that v is a k-radiation function, v c C2[S2] n C1[U] and v vanishes on F.

Thus if Im ko > 0, then v(x) - 0 in Q. Hence av

av

avx+

r

aVx-

r

Ov+kov=0;xcF'

=0,

and

v(s, ko) = µ(s, ko) + = 2µ(s, ko)

Jr

K(s, s'; k)µ(s', ko) dSs

0,

so that v is an eigenfunction of the interior Neumann problem for k2. It is not difficult to show that there are no zeros in Im k > 0 and that the only

zeros in Im k > 0 are of the form k0 = [A ]1"2, where 0 < A0 < Al < A2 , the eigenvalues of the interior Neumann problem. The case solved by Maue shows that there can be zeros in Im k < 0. Consideration of the integral equation adjoint to (4.3) in the usual Fredholm sense shows that the lattei occur also if and only if D(k) = 0. Returning to the representation (4.1), one has the following Dolph-Wilcox theorem referred to in the Introduction :

SCATTERING THEORY

211

THEOREM 4.1. The poles of the total field cb(x, k) at k = [A ]112 are all removable. Hence the only poles of the scattered field come from the region

Im k < 0 at those values for which D(k) = 0.

To prove this, suppose for simplicity that ko is a simple zero of D(k) with Im k = 0. Then one has

D(k) = (k - ko)D1(k);

D1(ko)

0.

Then (4.4)

O(x, k) = 00(x, k) +

Jr

K(x, s'; k)p(s', k) dSs-,

where

µ(s, k) _ 0o(s, k) + f R(s, s'; k)¢o(s', k) dSs, r _-1(s) ko

(4.5)

+ 0o(s) + 01(s)(k - ko) +...

Moreover, (4.6)

µ(s, k) -J K(s, s'; k)µ(s; k) dSs. _ ¢o(s, k).

The substitution of (4.5) into (4.6) and the use of a series expansion in powers of (k - ko) implies, since 00(x, k) is analytic at k = ko, that (4.7)

0-1(s) -

Jr.

K (s, s';

ko)o_1(s') dSs, = 0.

On the other hand, the substitution of (4.5) into (4.4) followed by expansion yields O(x, k) _

-1(x) + fo(x) + O1(x)(k - ko), + ... k - ko

where

0-1(x) = Jr K(x, s', ko)o -1(s') dS' is obviously a radiation function for ko, 0_, E C2 (Q) n C'(4) and , s', ko)o-1(s') dSs = 0 0-.'(s) = -0-1(s) + JKES r

by (4.7). Thus -I(x) = 0 in 0, and hence, by Rellich's uniqueness theorem, or that of ordinary potential theory if ko = (0), the pole is

C. L. DOLPH

212

removable and does not affect the scattered field. A similar proof can be given for higher-order poles. If one considers the spaces LZ(O-) = {o,

E L2(S2), 0

lal

m},

one can show that L2(O) is the closure of Co(Q) in L21(4) and one can introduce a self-adjoint operator A in L2(4) whose domain is L2(S2) n {q, A0 e L2(S2)}

and whose spectrum is the positive real line, as was first shown by Rellich [27]. Moreover, the associated resolvent operator

RT = (A - AI)-1 is defined on the entire complex plane deleted by the positive-real axis. As such, for this cut plane it is a bounded operator which maps L2()) into itself and is also a strongly analytic function of A there. If Im k > 0, then the resolvent is an integral operator and then G(x, y, k) as given earlier is its kernel with A, = k2. Moreover, the relation [G(k)p](x) =

J

G(x, y; k)p(y') dy'

defines d mapping of C(4) into C1(S2) which is meromorphic in the strong

sense. The associated resolvent having G(x, y, k) as its kernel defines a bounded operator of C0(S2) into C1(1) for Im k > 0 which is strongly analytic in the latter Frechet space. For Im k < 0, the resolvent operator possesses an analytic continuation which is meromorphic with all its poles in Im k < 0. A different method was first used by Dolph, McLeod, and Thoe [61 ] to obtain the meromorphic character of the resolvent kernel for the situation to be discussed in the next section, and general theorems have been obtained subsequently by Steinberg [62] and by Haf [63]. We shall

return to this subject in Section 5. Finally, to conclude this section, we should note that McLeod [64] treated the two-dimensional problem and obtained results similar to those of Wilcox [60] by a method closer to the spirit of Dolph et al. [611. 5. The Schrodinger equation and the meromorphic "spin off "

Physicists over a long period of years have studied the analytic continua-

tions of the spherically symmetric Schrodinger equation to the region Im k < 0. This history can be found either in the work of Wu and Ohmura

[6] or in the review paper by Newton [7]. It was from the latter that McLeod, Thoe, and I initiated an attempt to achieve similar results without

SCATTERING THEORY

213

the restriction of spherical symmetry on the potential. In this we succeeded and our results are published in Dolph et al. [61 ] but not without the help

of MacCamby and Werner who directed our attention to the possibility of adapting a result of J. D. Tamarkin's [66] to the case at hand. The widely known Dunford-Schwarz result [67] did not imply that the continuation was meromorphic.3 In brief, we first introduced the enlarged Hilbert space H, = L2[dµ(x)] where dµ(x) = exp [-alxl] dx with an inner product

0> = f 0(y)i(y)

e-2aiy1 dy

and a norm 11011H1 =

«,'>.

The introduction of this space avoids the so-called "exponential catastrophe" which results if one works in a time-independent way in the usual Hilbert space appropriate for the region Im k > 0. When, in this enlarged

Hilbert space, under the assumption (1.0), k is restricted to the region Im k > -a, the integral operator (5.1)

Tk0 =

1

eiikl Ix-yl

-4I f x - y

p(y)0(y) dµ(y),

where p(y) = e2alxlq(x) is an analytic compact family of operators of the Hilbert-Schmidt class with the property that 11 T(k) I

H, = 0[(Im k)"2] as Im k - oo.

Moreover, [I - T(k)]-1 is a meromorphic operator-valued function for Im k >-- -a and has a pole at ko if and only if there exists a 0 such that 0 = T(k)¢ in H,. Explicitly, it was the resolvent kernel of the scattering integral equation (see Dolph et al. [61] or Ikebe [11]) which could be continued since it is not apparently possible to continue the resolvent relations of Hilbert. We also showed that the poles were symmetrically placed with respect to the imaginary k-axis and that those in the region Im k > 0 all occurred on Re k = 0. We also discussed the scattering operator in the form given by Ikebe [70] and showed (i) that under these same conditions it could be continued analytically through the region IIm kJ < a; (ii) that it was meromorphic in this region with poles confined

to the subregion -a < Im k < 0 and to the non-negative imaginary axis; and (iii) that the poles in the next-to-last region were also symmetrically placed with respect to the imaginary axis. We did not, however, relate the singularities of the integral operator and/or resolvent Green's kernel to a Similar results were announced by Ramm [68]. I have just received a reprint of his paper [69] containing the full details.

214

C. L. DOLPH

those of the scattering operator. However we did give an example which indicated that the region of continuation to Im k > -a was in agreement with the known results of radially symmetric potentials and was the best possible. This example led us to suspect, but this has not been verified as yet, that under the hypothesis (1.0), in contrast to that of compact support for q(x), there will always be a contribution from infinity in the lower halfplane due to the occurrence of a branch cut along Im k = -a. Subsequent to our joint work, Thoe [22] gave a proof under a differentiability condition on q(x) that there were only a finite number of poles in any strip in the region and made a beginning on the associated asymptotic series for the wave equation Im k > 0. (There is a known simple relation between the reduced wave equation with potential when derived from the wave equation and when derived from Schrodinger's equation.) This relation is, for example, explicitly spelled out in Shenk [20]. Independently, McLeod [71 ] introduced ellipsoidal coordinates and in a manner reminiscent of the way physicists prove the convergence of the Born series (see Newton [72], Zemach and Klein [73], and Khuri [74]), he was able to obtain Thoe's result without the additional differentiability condition. One

essentially integrates by parts in this new coordinate system and then makes appropriate estimates. The results on the meromorphic nature of a family of compact operators have subsequently been considerably generalized by students of Phillips and Werner, respectively, namely Steinberg [62] and Haf [63]. Steinberg obtains his results by working with the projection operator defined by the contour integral involving the resolvent (see, for example, Riesz-Nagy

[75]) while Haf bases his on the Schmidt procedure of approximation by degenerate kernels. Since Steinberg's results are the most general and will be used in the next section, we shall state them explicitly. From the viewpoint of one who teaches applied analysis, however, the method used by Haf is probably more easily understood today by graduate engineers and physicists. Haf's thesis also contains several detailed applications of his results to various scattering problems. Following Steinberg, we let B be a Banach space and 0(B) be the bounded

operators on B. Let Kl be a subset of a complex plane which is open and connected. A family of operators T(k) is said to be meromorphic in Kl if it is defined and analytic in Kl except for a discrete set of points where it is assumed that in the neighborhood of one such point ko (5.2)

T(k) =

j= -N

T,(k - k0)',

where the operator T, E 0(B), N > 0, and where the series converges in the

uniform operator topology in some deleted neighborhood of k0. The

SCATTERING THEORY

215

family T(k) is said to be analytic at ko if N = 0 in (5.2). Then one has the following theorems: THEOREM 5.1. If T(k) is an analytic family of compact operators for k E K1, then either [I - T(k)] is nowhere invertible in K1 or [I - T(k)] 1

is meromorphic in K1. THEOREM 5.2. Suppose T(k, x) is a family of compact operators analytic in k and. jointly continuous in (k, x) for each (k, x) E S2 x R (R is the set of real numbers) then if [I - T(k, x)] is somewhere invertible for each x, then its inverse is meromorphic in k for each x. Moreover, if ko is not a pole

of [I - T(k, xo)]-1, then the inverse is jointly continuous in (k, x) at (ko, x0) and the poles of the inverse depend continuously on x and can appear and disappear only at the boundary of S2, including the point at infinity.

In addition to these results, Steinberg also gives sufficient conditions for poles to be of order one.

Again these results illustrate the usefulness of the interplay between mathematics and physics. 6. Intimations concerning the unified Shenk-Thoe theory

All of what precedes has been unified for the problem given by the relations (1.0)-(1.3) under the hypotheses stated below them. To state some of these results we rewrite the Werner relation in the form

O(x, k) = S,7(x) + Dµ(x) + Vr(x), where the first term stands for a single layer, explicitly

S-q(x) = -2

rl

Fk(x - s),7(s) dSs,

the second term stands for the double layer of (4. 1), and the third term is a

slightly modified volume potential of (2.2). One next defines a matrix operator

1AS AD AV

M = Mt S

D

V

S

D

V

In this matrix, the third row is obtained by applying the operator

8v-a(x)

216

C. L. DOLPH

under the integral signs in the definitions of S, D, V and A = A(k) = q(x)e2aixi

= k2 +

i2aIxI

for x E S2

for x e P.

Let B;, i = 1, 2, 3, denote three Banach spaces where B1 = C(P1), B2 = Ca(P2) for a fixed a, 0 < a < 1, and B3 = [T E C(S2 u P)], where T has continuous extension to f u r and to r, u r, and where under the hypothesis (1.0) the norm in B3 is TIIB3 = sup exp [-ajxI][jr(x)I]. The following theorem is now established : THEOREM 6.1. The operator M maps B1 x B2 x B3 into itself' and this mapping is completely continuous for each k and analytic in k for

Im k > - a (with a logarithmic singularity at k = 0 if n is even). It is next shown that every outgoing solution q(x, k) can be obtained by solving (6.1)

[I + M(k)][Te-2alxj, p., 77] = [f, e2alxl, f2,f3]

in the product Banach spaces. In connection with Steinberg's results discussed in Section 5, it also follows that [I + M(k)]-1 is a meromorphic function of k in all Im k > -a and this in turn yields meromorphic con-

tinuation of the outgoing solutions from Im k > 0 to all of Im k > -a, the only poles occurring at the poles of the last inverse. Except for these poles, these analytically continued solutions are unique and satisfy the outgoing radiation condition. At the poles the "complex eigenvalues" occur and the outgoing solutions are not unique. However, as in the usual Riesz-Fredholm theory, a solution will exist if and only if the [f,] involved are in the null space of the "adjoint" of [I + M(k)]. As in Werner's work [47] and that discussed in Section 4, this approach eliminates the (removable) singularities of the corresponding interior problems. The resolvent Green's operator can be constructed and used to build a spectral theory for real k completely analogous to that of Ikebe [I1] and Shenk [17] for this general problem, but its description is too difficult and lengthy to

present here. Perhaps more importantly, the poles of the operator [I + M(k)]- I can be related to those of the scattering operator in a very. precise sense, thus answering the question left open by Dolph et al. [61] about the relationship of singularities of the scattering operator and the nontrivial solution of the extended integral operator discussed there. If one lets A denote the self-adjoint extension of the operator defined by

SCATTERING THEORY

217

equations (1.1)-(1.3) in L2[QJ, then the scattering operator S(k) for the "Schrodinger wave equation" applicable to this problem, taken in the form

AU(t), d dtt) = is a unitary operator on L2[Si-1] for each positive k, where Si-1 is the unit hypersphere in n dimensions. As such, S(k) has a meromorphic extension (with a branch point at k = 0) if n is even to the entire complex i

k-plane and has no nonzero real poles. The poles of S(k) in the upper halfplane are simple, lie on the imaginary axis, and have as their square the nonzero eigenvalues of A. Shenk and Thoe now define a resonant state as a nontrivial solution of u

Tk u

u(s) a -

F,+ (x - s) - Fk (x - s) avS u(s) dS.

- f Fk (x - y)q(y)u(y) dVy in H2 °[S2] satisfying the boundary condition (1.2) and (1.3). Such a function will automatically satisfy

[-A+q-k2]u=0 as a distribution in n, since for H2' °[S2] one has

[-A-k2]Tkv=0 as distributions in Q. Suppose now for simplicity that one considers the poles of S(k) with Im k 0 for all i, j, and since each guts, is an isometry, m

m

n

G

j=1 i=1

where Iy

6j(x)0i(x)gU,v,(x)tv,(y(x))

n

j=1i=1

ej(x)Y'i(x) II y II =

YII,

is the norm of y in Po(i) (see [3]): Ilyll = sup ly(x)I, where x

IY(x)I = tu(y(x)) II A, x c U c k We now write, since supp y = K, fK i=1

tvgj(x)Y(x)) dx +

f

G j=1

fK\K j=1 i=1

6j(x)tv,(Y(x)) A +

j(x)0i(x)gUtv,(x)tv,(y(x)) dx

J K\K j=1

ej(x)tv,(Y(x)) A

f

2 i=1 Sj(x)Y'i(x)gU,v,(x)tv,(y(x)) dx. K\K 2 j=1 m

Because of the obvious inequality:

6,(x)tv,(Y(x))

< Ilyll, we find

III(Y; {Ui}i=1, {&i};=1) - I(y; {Vj}in 1, {ej}'jn-=l)IIA < 21,,(KIK)IIYII

Consider the set of all triples r = (M, { Ui}i= 1, {;}i= 1) where

(i) .4 is a coordinate bundle in the equivalence class of.9; -4 is engendered by a refinement GW of GW ; the coordinate maps of -4 are found by restricting those of -4.

(ii) {Ui}i=1 - ill is a covering of K. (iii) {;}i=1 is a partition of unity subordinate to {Ui}i=1. We partially order {T} by the inclusion order among the associated {ill}. We now assume that the measure µ on G is regular and that for all N in

some neighborhood base .A' = {N} for G, µ(8N) = 0, where 8N = N n (G\N) = boundary of N. If G is a locally compact group, these conditions

BERNARD R. GELBAUM

232

obtain (see [3], [6]). According to the main result in [3], for e > 0 there is a refinement W0 of QI, a coordinate bundle 2° in the equivalence class of 6'

such that Vo is the associated covering and such that the associated coordinate maps are derived from those of . by restriction and there is a compact K c K such that (a) µ(K) > µ(K) - e1211YJ1,

(b) for Uo, Uo E Wo, x c Uo n Uo n K there obtains gu0u (x) = e. Furthermore, if . is any coordinate bundle whose associated covering I& is a refinement of Gllo and whose associated coordinate maps are derived from those of -40 by restriction, then (b) holds with "Uo, Uo E G&o," etc., replaced by "U', (U" E Gll,",, etc. In consequence, for any such .4, 1& we find II,,

II AY; {Ui}i=1,

1) - AY; {Vj}m=1, {tSj})T 111 A < ---

We shall write I,(y) for I(y; {Ui}i=1{Y'i}i=1) These remarks are the basis for the following theorem: THEOREM 1.

Let y c Po(") and let

be a coordinate bundle (in the

.

equivalence class oft) with associated covering QI. Then for e > 0 there is a T(e) then

T(e) such that if T1i T2

IIIT1(Y) - "(Y)l A < e. PROOF.

Let T(e) = To - (-4°, {Uio}n=1, {Y'io}; = 1} where -40 is chosen as

in the argument above, Uio E Q/ , etc.

If T1i T2 >_ T(e), let T3 arise from any common refinement I3 of the coverings V, and Pte associated to T1 and 7-2. In other words, T3 = p3i {Wk}k=1, {'qk}k=1), where, in particular, the open covering associated to -43 is some common refinement Pt3 of Gut and 1&2. We let T1 = {Ui}1!1=1, {0X% 1), T2 = 042i {Vj};'_1, {6j};' ) and we calculate

,"(Y) - 4h) = f

0j(x)t U,(Y(x)) dx - f n

=

17k(x)twk(Y(x)) dx

G k=1

G t_ 1

r

q

k 0i(x)"]k(x)t U,(Y(x)) dx -

J G i=1 k=1

q

?,(x)twk(Y(x)) dx.

J G k=1

We show now that gvtwk(x) = e on Ui n Wk n R. Indeed we recall that implicit in the definition of each T is the fact that the associated coordinate maps are restrictions of those given for .4. This means that for To, TI, T2, T3 P such that o,(U) - U and such that by definition there are maps a,: °Il, on p- 1(U), r = 0, 1, 2, 3. Let us consider tvt and t,,,. By Tu1 = definition tUt - ta1(Ut),

tWk = ty3(Wk)

GROUP ALGEBRA BUNDLES

233 ''//,,

/

If a1(Ui) = a3(Wk), then '/i(x)'7k(x)t U,(Y(x)) = Y' i(x)'Ik(x)twk(Y(X)). If a,(U4) n a3(Wk) _ 0 then ci(x),Ik(x) = 0 and Oi(x)'k(x)tu,(Y(x)) = Oi(x)9gk(x)tWk(y(x)).

If x c- a1(Ui) n a3(Wk) n K,

where R - K and

gu0u,0(x) = e on Uio n U,, n R, then tut = ta,(u,) = gol(Ut)a3(Wk)ta3(Wk) ga,(U,)a3(Wk)tWk. However, gal(U,)a3(Wk) is generated by -Ta,(U09?',(Wk) and, when

restricted to R, the latter map is the identity. Thus on K, tu,(y(x)) _ twk(Y(x)) and again Y'i(x)']k(x)t U,(Y(x)) = Oi(x)?7k(x)twk(Y(x))

In conclusion, we may write I,1(y) - I,(y) as

G =

For xEK,

fi

+

fG\K

=

L

n

q

+

JK\R

q

inn

i=1 k=1

Y'i(x)'7k(x)tU,(Y(x)) =

i=1 k=1

0 (x)17k(x)tWk(Y(x))

q

Tk(x)tWk(Y(x))

k=1

Thus fK = 0. Since µ(K\K) < e/2IIyII we see that II ls,(Y) - It3(Y) I A < 2.

Using a similar argument for I, and I, and then combining inequalities we conclude that Ii,(Y) - Is2(Y)IIA < e. Since A is complete the Cauchy net {I,(y)} has a limit, which we denote by I(Y)

Before proceeding to discuss the properties of I(y) we note that the value of I(y) is, a priori, dependent upon -4. We shall show that despite this situation, the space L'(G, 4) based on the "integral" I is isometrically isomorphic to L'(G, A). Furthermore, if G is a locally compact group then

L'(G, 4) as an algebra is isometrically isomorphic to L'(G, A). The informal conclusion we reach is that the bundle structure of 0, is irrelevant to the structure of L1(G, -ff). We now prove that (1) I(-1Y1 + %Y2) = a11(Y1) + a2I(Y2), al, a2 E C. IIYII,(supp y).

(1i) III(Y)II

(i) Let Ki = (supp Yi); i = 1, 2, Y3 = -1Y1 + a2y2, then supp Y3 K3 - K1 U K2. For -r = {Ui}i=1i {ci};=1) the expressions Iy(Y3), II(Y1), n and I,(y2) are meaningful, if U Ui - K3. For such r it is clear that i=1

IX-1Y1 + a2Y2) = a1II(Yl) + a210(Y2)

BERNARD R. GELBAUM

234

On the other hand if Ti are associated to y;, let the underlying coverings be °11;, i = 1, 2. We can then construct in an obvious fashion a -r whose underlying covering is a refinement of °1l1 and W2 and whose {U;};=1 covers

K1 U K2. For such a T the equation just given holds. The existence of I(y) for all y c P0(e) implies that ultimately I,(y;) and I,.(y,) are close. Thus for T associated to K we find lira It(«lYl + a2Y2) = I(alyl + a2Y2)

For T arising from T1 and T2 as described, we note that they are cofinal in the set of T associated to K and that IT(alYl + (X2y2) = alk(yl) + a21,(Y2) = allt,(Yl) + a21'2(y2)-

Passage to the limit now yields the required linearity of I.

(ii) For all T, JI,(y)JJ < IIYIIFt(supp y). Hence the conclusion follows.

Since .2/ consists of isometrics, for y c Po(i) and x c G, ly(x)I = tU(y(x))II A is independent of U. Furthermore ly(x)I is continuous on G, supp IY(x)I = supp Y(x) We define

IYI = IG Iy(x)I dx,

and thereby define a norm on P0('). The completion of F0(i) in this norm will be denoted by L1(G, (fl. We observe that L'(G, 6) is a Banach space insensitive to the bundle structure of 6. For y c Po(i) we show III(y)IIA < YI.

Indeed III,(y)IIA < Iyi by virtue of the definition of I,. The result follows since I(y) = lim Ic(y).

2. L'(G, 4) amd L'(G, A) By L'(G, A) we mean the set of Bochner-integrable A-valued functions on G. We shall establish an isometric isomorphism between L'(G, off) and

L'(G, A). For this purpose we denote by C0(A) the set of continuous compactly supported A-valued functions on G. Let . be an arbitrary coordinate bundle in the equivalence class of'. Let 41 be the underlying open covering of G, {99u} the set of coordinate maps, and {go.,) the set of coordinate transformations (transition functions) defined by {qpc}. For y c- Fo(e), let K be the support of y and let

GROUP ALGEBRA BUNDLES

235

41 be a covering of K with an associated subordinate partition of unity 1. As we did earlier, we denote by T a triple (.V, { Ui}; = 1, {0i}i= 1) consisting of a coordinate bundle .4 in the equivalence class of ff, an open covering {Ui};1 of K, where {Ui}i - °ll, G& is a refinement of 0& and {99u} for -4 arise from {po} for . by restriction. For each T let {Ui}i=1

TT(Y)(x) _

i=1

0i(x)t u,(Y(x)) E Co(A)

Clearly Ti is linear.

For e > 0 there is a

K such that µ(K\K) < e/211yll and there is a

To such that for T > Tp X E K.

Ti(Y)(x) = T10(Y)(xo),

This follows from the argument used in the establishment of I. Thus, we see that for 7-1i T2 > To l.. T 1(Y) - Ti2(Y) II A dx =

5K +

< e. K\K

K\K

Thus {TT(y)} is a Cauchy net in L'(G, A) and we denote its limit by T(y).

From the construction of T(y) we see that T(y)(x) = 0 if x 0 K and for each e > 0 and corresponding K, To, TT(y)(x) = T(y)(x), x e If en - 0 we may choose corresponding Kn, Tn so that R. - Kn+1, and µ(K\Kn) < en. Thus k c N, x c K.

T(y)(x) = TTn(y)(x) = Tin+k(Y)(x),

Hence T(y)(x) is continuous on 1J Kn and thus a.e. on K. Since all T, are n=1

linear, so is T. Next we note that Jc

11T(Y)(x)IIA dx = SG IY(x)I dx = IYI

Hence T is an isometry on Fo(e) (in particular T is one-to-one). Furthermore, if H(x) E C0(A), we may write n

H(x) =

0i(x)H(x) i=1

and then find yi(x) (see [2]) such that tu,(y;(x)) = bi(x)H(x). Thereupon, n if y(x) = yi(x) then i=1

n

Ts(Y)(x) _

i=1

n

TT(Yi)(x) _

i=1

n

7=1

n

0itu;(Yt(x)))

_

n

1=1J=1

0icg-'(xu,)tu;(Yi(x))-

BERNARD R. GELBAUM

236

If supp H = K, then supp y - K and by arguments used earlier, for e > 0, there is a compact k -- K, .(K\K) < e so that for all r > r, and XEK n

TT(Y)(x) _

n

i=1 J=1

n

O1tui(Yi(x)) _

i=1

tu,(Yi(x)) = H(x).

Hence T(y)(x) = H(x) a.e. and thus, viewed as a subspace of L'(G, A), CO(A) is the complete image by T of Po(i), viewed as a subspace of L1(G, S). Thus we have

L1(G, ) -

FO(6) T C0(A) c L'(G, A),

where T is an isometric isomorphism between dense sets. Clearly T is extendable to L'(G, (ff) and thus this extension, again denoted by T, implements an isometric isomorphism between L1(G, 6) and L'(G, A).

The map I: Po(i) - . A may be extended uniquely to a map denoted again by I: L1(G, &) -. A. In terms of this map I we may define a multiplication in L'(G, d) when G is a locally compact group and µ is Haar measure. The natural definition extends classical convolution. We begin by fixing a coordinate bundle .4 and all the associated apparatus used in the definitions of T, and I,, T and I. Since i(x)t u,(Y(x)),

TT(Y)(x) _ i=1

we see that IC(Y) =

fG

T,(Y)(x) dx,

and, in consequence of earlier inequalities, we conclude that

I(y) =

T(y)(x) dx. fe

We show that, if y, 8 E F(e), then T(yS) = T(y)T(8). Indeed, if (M, { Ui}i =1i {iii};= ) is given we observe that {Oil;}i ; is also a partition of unity subordinate to {Ui}i=1. We let -r' = (M, {Ui}i=1, {Y'iY'1}i;=1).

The set of all r' is cofinal in the set of all r. Then n

Tr(Y8)

n

_ i=1)=1 Oi(x)Oj(x)tUi(Y(x))tu,(8(x)) n

n

_

Oi(x)oj(x)guiu,(x)tu,(Y(x))tu,(8(x)) i=17=1 n

_

i=1

n

Oi(x)gU'U,(x)tU,(Y(x)) I &i(x)tU,(8(x)) f=1

GROUP ALGEBRA BUNDLES

237

As in earlier arguments, for suitable k in the union of the supports of y and 8 and for suitable -r we find T.,.(y8)(x) = T,(y)(x)Tt(8)(x)

for x c K.

Using the cofinality of {r-'} in {T} we conclude that

T(y8)(x) = T(y)(x)T(8)(x) a.e. in x.

Thus we see that if for any

E Fo(b) we write 77x(y) - ij(y-'x), then

I(YSx) =

T(Y8x)(y) dy fG

= f T(Y)(y)T(Sx)(y) dy a

=

fc

T(Y)(y)T(S)(y-'x) dy

= T(y)*T(8)(x) E Co(A).

Thus we define

(Y*S)(x) = T -'[I(y8 )] and conclude that T(Y*S)(x) = T(Y)*T(S)(x)

It is now clear that T is an algebraic isometric isomorphism between L'(G, e) and L'(G, A). UNIVERSITY OF CALIFORNIA IRVINE

REFERENCES 1. GELBAUM, B. R., "Tensor products and related questions," Trans. Amer. Math. Soc., 103 (1962), 525-548. 2. "Banach algebra bundles," Pacific Journal of Mathematics, 28 (1969), 337,

3.

4.

-349. , "Fibre bundles and measure," Proceedings of the American Mathematical Society, 21 (1969), 603-607. , "Q-uniform Banach algebras" Proc. Amer. Math. Soc., 24 (1970), 344-353.

5. GROTHENDIECK, A., "Produits tensoriels topologiques et espaces nucleaires," Mem. Amer. Math. Soc., No. 16 (1955). 6. HERZ, C. S., "The spectral theory of bounded functions," Trans. Amer. Math. Soc., 94 (1960), 181-232.

7. JOHNSON, G. P., "Spaces of functions with values in a Banach algebra," Trans. Amer. Math. Soc., 92 (1959), 411-429. 8. STEENROD, N., The Topology of Fibre Bundles. Princeton, N.J.: Princeton University Press, 1960.

Quadratic Periods of Hyperelliptic Abelian Integrals R. C. GUNNING'

1.

Consider a compact Riemann surface M of genus g > 0, represented in the familiar manner as the quotient space of its universal covering surface M by the covering translation group F. For the present purposes the only thing one needs to know about the surface M is that it is a simply con-

nected noncompact Riemann surface. The group P is properly discontinuous and has no fixed points; and the quotient space M/P is analytically equivalent to the Riemann surface M. The image of a point z E M under an automorphism T E P will be denoted by Tz. Some fixed but arbitrary point zo c M will be selected as the base point of the covering space M.

The Abelian differentials on the surface M can be viewed as the P-invariant complex analytic differential forms of type (1, 0) on the surface M. Since M is simply connected, any such differential form 0 can be written as 0 = dh, for some complex analytic function h on the surface

M; the function h is then uniquely determined by the normalization condition h(zo) = 0. Since 0 is F-invariant, it is apparent that h(Tz) _ h(z) - 0(T) for some complex constant 0(T), for any element T c P; and clearly B(ST) = 0(S) + 0(T) for any two elements S, T c P. The mapping T--> 0(T) is thus a homomorphism from the group F into the additive group C of complex numbers; this homomorphism will be called the period class of the Abelian differential 0. As is well known, the period classes of the Abelian differentials on the Riemann surface M form a g-dimensional subspace of the 2g-dimensional complex vector space Horn (P, C) consisting of all group homomorphisms from F into C. Now suppose that 01 = dhl and 02 = dh2 are two Abelian differentials on M, where the functions hl, h2 on M are both normalized by requiring 1 This work was partially supported by NSF Grant 3453 GP 6962. 239

240

R. C. GUNNING

that h1(zo) = h2(zo) = 0. The product a = h102 = h1 dh2 is an analytic differential form of type (1, 0) on the surface A? and clearly satisfies the functional equation (1)

a(Tz) = a(z) - 01(T)02(z)

for any element T e r. Again, since M is simply connected, this differential form can be written as a = ds for some complex analytic function on the

surface M; the function s is uniquely determined by the normalization condition s(zo) = 0. It follows from (1) that the function s satisfies the functional equation (2) s(Tz) = s(z) - 91(T)h2(z) - a(T) for some complex constant a(T), for any element T e F. The mapping T-* a(T) will be called the period class of the quadratic expression a = h1 dh2 in the Abelian integrals on the surface M; the set of all such mappings will be called, for short, the quadratic period classes of the Abelian differentials on M. The aim of this paper is the explicit determination of these quadratic period classes for the special case of a hyperelliptic Riemann surface; in this case, the quadratic periods are quadratic expressions involving the ordinary period classes of the Abelian differentials on M. In general, the quadratic period classes may be transcendental functions

of the Abelian period classes. Section 2 of this paper is devoted to a general discussion of the properties of these quadratic period classes, for an arbitrary Riemann surface; Section 3 contains the explicit calculations

for a hyperelliptic surface; and Section 4 provides the motivation for studying these quadratic period classes, with a discussion of their role in the general theory of Riemann surfaces. 2.

A quadratic period class is not a homomorphism from the group IF into the complex numbers; but it follows readily from (2) that it does satisfy the relation (3) a(ST) = a(S) + a(T) - 01(S)02(T) for any two elements S, T e F. This sort of relation is a familiar one in the cohomology theory of abstract groups (see [2]). In general, for any two group homomorphisms 01i 02 a Hom (I', C), the function 01 u 02 defined on P x F by (01 u 02)(S, T) = 01(S)02(T) satisfies the relation (01 u 02)(S, T) - (01 u 02)(RS, T) + (01 u 02)(R, ST) - (01 U 02)(R, S) = 0

for any three elements R, S, Te I'; that is to say, in the cohomological language, 01 U 02 is a 2-cocycle of the group F with coefficients in the

HYPERELLIPTIC ABELIAN INTEGRALS

241

trivial P-module C. Equation (3) is simply the assertion that this cocycle is cohomologous to zero, indeed, that it is the coboundary of the 1-cochain a. Perhaps it should be noted in passing that requiring the cocycle 0 U 02 to be cohomologous to zero does impose nontrivial restrictions on the homomorphisms 0 and 02; indeed, these restrictions are precisely the Riemann bilinear equalities on the periods of the Abelian differentials 0 and 02. This can be verified by a straightforward calculation, choosing the canonical presentation for the group P and examining the restrictions imposed by (3) and the group relations in P. However, it is more instructive to note that the correspondence 0 x 02 --± 0 U 02 can be viewed as a bilinear mapping

H1(r, C) ® H1(r, C) - H2(r, C), since Hom (P, C) - H1(P, Q. Since M is contractible, there are further canonical isomorphisms HQ(r, C) H°(iQ/P, C) -= H°(M, C), for q = 1, 2, and with these isomorphisms, the bilinear mapping just given can be

identified essentially with the usual cup product operation in the cohomology of the space M. The Riemann bilinear equalities amount precisely to the conditions that the cup products of any two analytic cohomology classes in H1(M, C) are zero (see [I]), which yields the desired assertion.

The collection of all the 1-cochains a c C'(F, C) satisfying merely the coboundary relation (3) form a complex linear set of dimension 2g; for this set is nonempty, and any two elements of the set differ by a 1-cocycle, that is, by an element of Horn (r, Q. The problem is to select the unique quadratic period class in this set; this appears to be a nontrivial problem in general. A much simpler problem that one might imagine tackling first is the problem of determining the analytic 1-cochains in this set, where a 1-cochain a satisfying (3) is called analytic if there is some complex analytic function s on the Riemann surface M for which the functional equation (2)

holds. Note that the analytic 1-cochains form a complex linear set of dimension g; for it is clear from (2) that two analytic 1-cochains differ by an analytic class in Hom (F, C), that is, by the period class of an Abelian differential on the surface M. It is actually quite easy to determine the

analytic 1-cochains, using the Serre duality theorem and the familiar technique for calculating that duality (see [I]). The details will be given elsewhere.

In at least one special case it is very easy to calculate the quadratic periods for an arbitrary Riemann surface; for when 0 = 02 it is clear that s = hi/2, and hence that a(T) 01(T)2/2 for any element T e P. This

242

R. C. GUNNING

observation can be extended to yield a symmetry principle for the quadratic periods in general. Let B; = dh;, i = 1, . . ., g, be a basis for the Abelian differentials on M; and let T oi,(T) be the period class of the quadratic expression (7i, = hi01, i, j = 1, . . ., g. Then since, clearly, s;, + s,; = hih,, it follows immediately that (4)

,t,(T) + a,(T) _ -B1(T)0,(T)

for any element T E P. Finally, it should be noted in the general discussion that the quadratic periods, unlike the ordinary Abelian periods, depend on the choice of the base point zo e M. To examine this dependence, select another base point zo ; and indicate the various expressions just considered, renormalized to vanish at zo*, with an asterisk. Thus clearly h* (z) = ht(z) - h;(zo*) and s*(z) = si,(z) - hi(zo)h,(z) - s1,(zo**) + hi(zo*)h,(zo); and it follows immediately for the quadratic periods that (5)

*(T) = -ti(T) + ei(T)hi(z) - ei(T)hi(z )

for any element T E P. Note that the change is skew-symmetric in the indices i, j, as of course it must be in consequence of the symmetry principle (4).

3.

Suppose now that the Riemann surface M is hyperelliptic, so that it can

be represented as a two-sheeted branched covering of the Riemann sphere P, branched over points to, t1i . . . , t2g+1 in P; and for simplicity, suppose further that the base point zo of the universal covering surface M is chosen to lie over the point to c P. Select disjoint smooth oriented paths

Tk in P from the point to to the various points tk, k = 1, ... , 2g + I. Each pathTk in P lifts to two separate paths Tk, Tk in M; these two paths are disjoint except for their common initial point, the single point po c M lying over to, and their common terminal point, the single point pk E M lying over tk. In turn, each path Tk lifts to a unique path Tk in A? with initial point the base point zo c M and terminal point denoted by zk. Since the points zk and zk lie over the same point of M, they are necessarily

congruent under the covering translation group r; therefore, there is a uniquely determined element Tk E F such that zk = Tkzk. Note that the paths T; and TkTk have the same terminal point; hence the path (T.)' (TkTk)-1, obtained by traversing first the path Tk with its given orientation

and then the path TkTk with the opposite of its given orientation, is a connected piecewise-smooth oriented path from zo to Tkzo in M.

HYPERELLIPTIC ABELIAN INTEGRALS

243

THEOREM 1. For a hyperelliptic Riemann surface represented as above, the quadratic periods for a basis 0i = dhi of the Abelian differentials on

the surface are given by (6)

aij(Tk)

0i(Tk)Oj(Tk)/2

fori,j = 1,...,gandk = 1,...,2g+ 1. PROOF.

Introduce the hyperelliptic involution P: M -- M, the analytic

automorphism of the Riemann surface M corresponding to the interchange of the sheets in the covering M -* P; this automorphism has period

2, and the quotient space M/P under the cyclic group generated by P is analytically equivalent to the Riemann sphere. The key to the proof is the observation that Bi(z) + 0i(Pz) = 0 for any Abelian differential 0i on the surface M. To see this, note that 0i(z) + 0i(Pz) is an analytic differential form on M which is invariant under the automorphism P, hence corresponds

to an Abelian differential on the quotient space M/P - P, and thus must be identically zero. Now let zk(t) E Tk be the point lying over t E Tk, so that

the functions zk parametrize the paths Tk by the coordinates along the paths Tk. Then 0i(zk(t)) + 0i(zk(t)) = 0 for all t E Tk, as above, and consequently hi(zk(t)) + hi(zk(t)) = constant, for all t E Tk; since z, (to) _ zk(to) = z0, it follows from the normalizations adopted that actually

hi(zk(t)) + hi(zk(t)) = 0 for all t E Tk. For the quadratic expressions aij = hi0j it then follows also that aij(zk(t)) = oij(zk(t)) for all t c T, Applying these and the earlier observations, note that fTkC

_-

a(Z) = J

I('

TkTk - Tk o

52 Laij(Z)

- 0i(Tk)Bj(Z)] -

Tk

f

fl

aij(Z)

Tk

[aij(Zk(t )) - aij(Zk(t ))J - OZ(Tk) 77

k

f

2

0j(Z)

k

Bi(Tk)hj(Zk)

Similarly note that

= 0,(Tk)

-

JTkZ0

I(`

0j(Z) = ZO

2

1

J TkTk - Tk(ZI

0j(Z)

[0,(Z(t)) - 0 (Z(t))] 2 f12 0(z) Ilk k = 2hj(zk) The desired result follows immediately from these two formulas.

To see that the preceding theorem gives a complete description of the quadratic periods of the Abelian differentials on a hyperelliptic Riemann

R. C. GUNNING

244

surface, it is necessary to show that the elements Tk generate the entire covering translation group P. It is quite easy not only to do this, but also to determine the relations between these generators; the results are well known, but since it appears hard to find a sufficiently explicit reference for them, the details will be included here. Matters are somewhat simplified if a bit more care is taken with the notation. In particular, suppose that the

branch points tk E P are so numbered that the paths , occur in counter2g+1

clockwise order around the point to. The domain P - U , is simply k=1

connected; thus the portion of the Riemann surface lying over it consists of two disjoint sets each homeomorphic to that domain. These two sets will be labeled sheets I and 2 of the covering, in some fixed but arbitrary order. This having been done, let Tk be that path in M lying over Tk which has sheet v along its right-hand side, for v = 1, 2. Having fixed a base point zo e R lying over the point po c M, there is a canonical isomorphism between the covering translation group r and the fundamental group Ir1(M, po). Specifically, for an element T E F, select any path in A? from zo to Tzo; this path covers a closed path at po in M, representing the element in 7r1(M, po) associated to T under this isomorphism. It is clear that the element Tk E F is associated to the element of vr1(M, po) represented by the path (Tk.)(Tk)-1 obtained by traversing first

the path Tk in the direction of its orientation and then the path Tk in the opposite direction to its orientation; this element of ar1(M, po) will also be denoted by Tk. The problem is now to show that these elements generate the entire fundamental group 7r1(M, po). Note that any closed path based

at po in M can be deformed continuously into a path contained in the 29+1

29+1

subset K = U (Tk V T2); for clearly the set Ko k=1

= k=1 U

Tk

is a deformation

retraction of P - t for any point t E P - KO, and the lifting of this retraction to M accomplishes the desired result. Since K is a wedge of circles, v1(K, po) is the free group generated by the paths tracing out these component circles; but these are precisely the paths representing the elements Tk, hence these elements Tk generate the group ar1(M, po), as asserted. Moreover, since the Tk are free generators for the group 7rl(K, po), the relations between these elements in -jT1(M, po) all follow from the exact homotopy sequence of the pair (M, K), which clearly has the form ir2(M, K, Po)

vra(K, Po) --> ir1(M, po) -* 0.

The pair (M, K) is evidently 1-connected; thus it follows from the Hurewicz isomorphism and excision that, with the notation of [4], r2(M, K, po)

H2(M, K)

H2(M/K, K/K)

Z

Z,

HYPERELLIPTIC ABELIAN INTEGRALS

245

noting that M/K is a wedge of 2-spheres. Thus the full group 7r2(M, K, po)

is generated by two elements A, A2, together with the natural action of the fundamental group; and since this action of the fundamental group commutes with the exact homotopy sequence, all relations among the generators of Tk of ir1(M, po) are consequences of the relations derived from the elements 8L1, 802 of 7r1(K, po). It is clear geometrically that Al, 02 correspond to the obvious mappings from the closed unit disk to sheets I and 2 of the covering respectively, with their common boundary K; and that 801 = T1T2.. ' T2g+1 and 802 = Tl 1T2 1 ..' T29+1. Therefore the relations in 7r1(M, po) are consequences of the following two relations: (7)

7'1T2... T29+ 1 =T29+1 ... T2T1 = I.

It must be pointed out that the quadratic periods are not quite so simple as they might first appear to be from the symmetry formula (4) and formula (6) of Theorem 1. These periods are essentially non-Abelian in nature, and for a general element of the group F they have a rather more complicated form than they have for the special generators Tk. For instance, one may

wish to calculate the quadratic periods for a canonical set of generators of the fundamental group ir1(M, po). Introducing the elements Ak, Bk E 7r1(M, po) defined by Ak = T2kT2k+1' ' T29+1T2k1, Bk = T2k+1T2k1,

fork = 1, ..., g,

a simple calculation shows that these are also generators for 7ri(M, p,), and that the relations between them are consequences of the single relation A1B1A11B1 1A2B2A2 1B2 1... AgBgA9 'B91 = 1.

Thus these elements can be viewed as a canonical set of generators of the fundamental group of the surface M. It then follows by a straightforward calculation from formulas (3) and (6) that the quadratic periods for these generators have the form -2aij(Ak) = Oi(Ak)Oj(Ak) +

BZ(Ak)OJ(Bk... B9) -

Oi(Bk... B9)Oj(Ak)

9

+ 1 [01(A1)0j(B1) - Bi(B)Bj(A1)] l=k

-2aij(Bk) = Oi(Bk)O(Bk) + Ol(Bk)O,(AkBk+ 1 ... Bg) - Oi(AkBk+l ... B9)O.(Bk)-

In each case the first term, which is symmetric in i and.j, can be determined

directly from the symmetry formula (4); the interest actually lies in the remaining terms, which are skew-symmetric in i and.j, and which reflect the non-Abelian nature of the quadratic periods.

246

R. C. GUNNING 4.

That all the quadratic Abelian periods with base point a Weierstrass point

can be expressed so simply in terms of the usual period matrix of the Riemann surface reflects the particular simplicity of hyperelliptic surfaces. Even allowing an arbitrary base point, the expressions remain quite simple, in view of (5). For a general Riemann surface the situation is rather more

involved. Even sidestepping the matter of the choice of base point, by considering only those linear combinations of the quadratic Abelian periods which are independent of the base point, there remains the question

whether the resulting expressions are actually functions of the period matrix of the surface; that is to say, any expressions in the quadratic Abelian periods which are independent of the base point can be viewed as

functions on the Teichmuller space, but need not be functions on the Torelli space (see [3]). The detailed investigation of this matter is of some interest, involving as it does the relatively lightly explored area of nonAbelian problems on Riemann surfaces, but will be reported elsewhere. The mere fact that these quadratic periods are of a non-Abelian character is, of course, an insufficient justification for considering them extensively.

However, on the one hand, the quadratic periods do arise naturally in studying some problems on compact Riemann surfaces. For example, any finitely sheeted topological covering space of a compact Riemann surface has a natural Riemann surface structure; fixing the topological type of the covering, the period matrix of the covering space is a well defined function of the period matrix of the base space, and it is of some interest to know

this function more explicitly. Equivalently, given a compact Riemann surface M of genus g, the problem is to determine explicitly the period matrix of the unique compact Riemann surface M' of the same genus g

such that a fixed subset G c M x M', the graph of the topological covering, is homeomorphic to an analytic subvariety of the manifold M x M'. The Abelian part of the problem can be handled quite easily, for Lefschetz' theorem determines the period matrices of the surfaces M' such that the graph G is homologous to an analytic cycle on the manifold M'. There are numerous surfaces M' admitting correspondences onto M having the topological type of the covering mapping, however. A detailed examination of the analogue of Riemann's bilinear relations picks out the period matrix of the unique covering space among all such surfaces M' in terms of the quadratic periods. In order for this approach to be very useful, more detailed knowledge of the quadratic periods on arbitrary Riemann surfaces seems necessary. On the other hand, the quadratic periods do have some relevance to the

old problem of determining which Riemann matrices are the period

HYPERELLIPTIC ABELIAN INTEGRALS

247

matrices of Riemann surfaces. The Riemann bilinear equalities correspond to the vanishing of the cup product of any two analytic period classes, or

equivalently, to the fact that the product B1 A 02 of any two Abelian differentials is cohomologous to zero. Actually, of course, the product B1 A 02 vanishes identically; this is equivalent to the fact that the differen-

tial form a = h102 is closed, that is, that the quadratic periods are well defined. Note that the Abelian differentials 0 = dh have simple critical points in the sense that h is a finite branched covering map at any zero of 0; and conversely, if 0 = dh is any closed complex-valued differential form with simple critical points on a two-dimensional differentiable manifold,

the manifold has a unique complex structure for which 0 is an Abelian differential form. Now if 01, ... , 0, are any closed complex-valued differen-

tial forms with simple critical points on a two-dimensional differentiable manifold, and if Bi A 0, = 0 for all i, j, it follows immediately that all the forms determine the same complex structure; that is, that these are precisely the Abelian differential forms on some Riemann surface. Thus one is led to suspect that the period conditions beyond the Riemann bilinear

relations may involve the quadratic periods; but again, some more detailed knowledge of the quadratic periods on arbitrary Riemann surfaces seems necessary before pursuing this line of investigation further. PRINCETON UNIVERSITY

REFERENCES 1. GUNNING, R. C., Lectures on Riemann Surfaces. Princeton, N.J.: Princeton University Press, 1966. 2. HALL, MARSHALL, The Theory of Groups. New York: The Macmillan Company, 1959.

3. RAUCH, H. E., "A transcendental view of the space of algebraic Riemann surfaces," Bull. Amer. Math. Soc., 71 (1969), 1-39. 4. SPANIER, EDWIN H., Algebraic Topology. New York: McGraw-Hill Book Co., Inc., 1966.

The Existence of Complementary Series A. W. KNAPP' AND E. M. STEIN2 1. Introduction

Let G be a semisimple Lie group. The principal series for G consists of unitary representations induced from finite-dimensional unitary represen-

tations of a certain subgroup of G. These representations are not all mutually inequivalent, and their study begins with a study of the operators that give the various equivalences-the so-called intertwining operators.

For G = SL(2, R), these operators are classical transformations. The principal series can be viewed conveniently as representations on L2 of the line or L2 of the circle. In the first case, the operators are given formally by scalar multiples of

(l.la)

f(x)- f

f(x

- y)IyI"dy

and

(l.lb)

f(X) - fT f(x - y)(sign y)IyI - +it dy.

The operator (1.1a) is fractional integration of the imaginary order it and is also known as a Riesz potential operator of imaginary order; for t = 0, the operator (l.lb) is the Hilbert transform. If the principal series instead is viewed on the circle, the operators are less familiar analogs of these, given formally in the case of (l.la) by (1.2)

.f(B)

_> f 2n ff(B - 9))(1 - cos

9,)-'-it)/2 dp.

0

In [3] the authors investigated the operators that generalize (1.1) to an arbitrary group G of real-rank one in order to determine which representa-

tions of the principal series are irreducible. The idea was roughly that 1 Supported by National Science Foundation Grant GP 7952X. 2 Supported by AFOSR Grant AF68-1467. 249

250

KNAPP AND STEIN

reducibility occurs exactly when the operator generalizing (1.1) can be interpreted as a bounded operator given as a principal-value integral. Here we shall study these operators for the same groups G for a different purpose. We wish to determine what unitary representations of G can be obtained by inducing from nonunitary finite-dimensional representations of the special subgroup. In other words, we ask what the representations are of the complementary series of G. We shall treat the problem of existence of complementary series by considering analytic continuations of the operators generalizing (1.1) and (1.2). The essential question will be to determine which of the continued operators are positive-definite in a suitable sense. [In (1.1) the operators (1.1a) are positive-definite when it is replaced by a real number between 0 and 1, and the operators (I.1 b), with it replaced by a complex parameter, are never positive-definite.] The ideas used in answering this question will be given in Section 2, and a more precise exposition will follow in the later sections. Most of the arguments will involve operators A(z) generalizing (1.2), rather than (1.1), but at one point indicated in Section 3 we shall pass to the operators generalizing (1.1). This passage back and forth between integration on a compact group and integration on a noncompact group appears to play an important role in our work. Our results are special, in that we work only with semisimple groups of real rank 1. Among other results concerning existence of complementary series in special situations are those of Kostant [4] (for general G but only for "class 1" induced representations) and Kunze [5] (for complex semisimple groups G).

The sections of the paper are arranged as follows. The notation and motivation are in Section 2, the precise definition of complementary series and the main theorem (Theorem 3.3) are in Section 3, and a discussion of the applicability of the main theorem is in Section 4. Since it is our intention to present here only the main ideas, we defer most proofs until another time. The authors wish to thank S. Rallis for stimulating conversations about this work.

2. Notation and heuristics

In what follows, G will denote a connected semisimple Lie group with finite center. Let G = ANK be an Iwasawa decomposition of G, let 0 be the Cartan involution of G corresponding to K, let M be the centralizer of A in K, let M' be the normalizer of A in K, let p be half the sum of the positive restricted roots, and let N = ON. Then MAN is a closed subgroup whose finite-dimensional irreducible unitary representations are all of the

THE EXISTENCE OF COMPLEMENTARY SERIES

251

form man -k A(a)a(m), where A is a unitary character of A and a is an irreducible representation of M. The principal series of unitary representa-

tions of G is parametrized by (Q, A) and is obtained by inducing these representations of MAN to G. These representations may be viewed as operating on a space of functions on K by restriction. That is, let a operate on the finite-dimensional space Va and let H° be the subspace of members f of L2(K) ® Va such that, for each m in M,

f(mk) = a(m)f(k) for almost all k in K. Define operators on H° by eaxck"'A(exp H(kx))f(K(kx))

(2.1)

xcG

where the notation on the right refers to the Iwasawa decomposition kx = exp H(kx) n K(kx). The representation Ua,' is unitarily equivalent with the member of the principal series corresponding to the pair (a, k).

The definition (2.1) of a representation in the Hilbert space H° also makes sense when A is a nonunitary character of A. In this case, Ua,"(x) is a bounded operator with norm < sup JA(exp H(kx)) 1, but it is not unitary. kcK

We call these representations the nonunitary principal series. Somewhat imprecisely, the complementary series consists of those representations of the nonunitary principal series that can be made unitary by redefining the inner product. (A precise definition will be given in Section 3.) Temporarily we shall proceed only formally and see what has to happen for a representation to be in the complementary series. Suppose < , > is an inner product for which Ua-' is unitary. This inner product will be given by an operator, possibly unbounded, say = (L.f, g)

is the usual inner product on L2(K) given by integration. The Here means that condition that Ua."(x-1)*L,

where the adjoint is defined relative to ( , ). On the other hand, we have the lemma below, which follows from a change of variables. LEMMA 2.1.

Ua."(x-1)* = U0.x-1(x)

We conclude that (2.2)

Ua."-1(x)L

KNAPP AND STEIN

252

with L 0. There is a theorem of F. Bruhat [1], in the case that A is unitary, that most U6'1' are irreducible and that U°'T and U'-" are equivalent if and only if there is some member m' of M' such that am", defined by

am'(m) = a(m'mm- 1), is equivalent with r and such that Am' = µ. If we assume (slightly inaccurately) that these facts persist for nonunitary A, then we expect that, for most A, if (2.2) holds, then L is unique up to a am' equivalent with a and Am' scalar and there exists some m' in M' with equal to A-'. From now on, assume that dim A = 1 (the real-rank one case). Then M'/M has order 2. Fix an m' in M' but not in M, and introduce a complex parameter z by the definition A(a) = ezoH(a)

If A corresponds to z, then A-l corresponds to -2 and Am' corresponds to - z. From what we have just said, there are only two possibilities:

(i) z = -z. That is, z is imaginary and A is unitary; hence principal series.

is in the

(ii) am' is equivalent with a, and -z = -z. That is, it is possible to define a(m'), and z is real.

Thus we are looking for an operator L such that LU°'z = U°' -2L, and we expect it to be unique up to a scalar for most z. Such an operator was obtained by Kunze and Stein [6] for Re z > 0. It is (2.3)

A(z).f(ko) = f e(1-8)ploga(km')a(m')a-1(m(km'))f(kko) dk, x

where the notation on the right refers to the decomposition of G into MANN, namely

km' = m(km') a(km') . n n; this decomposition exists uniquely for all k not in M. In short, for (a, z) to give a representation of the complementary series, am' and that z must be real. In we expect that a must be equivalent with

this case if z > 0, the inner product should be a multiple of (A(z)f, g). That is, a multiple of A(z) must be a positive Hermitian operator. A(z) is always Hermitian for z real, and it is positive if and only if its kernel is a positive-definite function.

For z > 1, we can settle the question of positivity immediately. For such a value of z, the kernel vanishes at the identity and is continuous and bounded on K; since it is not identically 0, it is not positive-definite. Thus there should be no complementary series for z > 1.

THE EXISTENCE OF COMPLEMENTARY SERIES

253

Our approach to the question of positivity when 0 < z < 1 involves complex methods. To begin with, z --> A(z)f is analytic for Re z > 0, and, if f is smooth, we show that this function of z extends to be meromorphic in the whole plane. Denoting the new operators, defined for Re z < 0, by A(z) also, we shall see that z ---> A(-z)A(z)f is meromorphic in the whole

plane. For z purely imaginary and not 0, A(-z)A(z) is an intertwining which is irreducible by operator for the unitary representation Bruhat's theorem. Thus A(-z)A(z)f = c(z)f with c(z) scalar for z imaginary. If we introduce a suitable normalization B(z) = y(z)-'A(z), we shall obtain B(-z)B(z)f =f for imaginary z, hence for all z by analytic continuation. Suppose B(O) is the identity. Then for B(zo) to fail to be positive-

definite for some positive z0, the equality B(-z)B(z)f = f says that either B(z) or B(-z) must have a singularity for some z with 0 < z < z0. Thus an investigation of the singularities of B(z) will be the key to the whole problem of the existence of complementary series associated with a. 3. The existence theorem

We continue to assume that G has real-rank one. The representations U°,z(x) of the nonunitary principal series, which was defined in Section 2, are parametrized by the finite-dimensional irreducible unitary representa-

tions a of M and by the complex number z, which corresponds to the character a -* ezO'(a' of A. The space H° on which U°'z operates is a subspace of L2(K) ® V° that

depends on a but not on z, and the action of K, by right translation, is independent of z. The space of C `° vectors for is the subspace of C ' functions in H°; thus, it too is independent of z. We denote this subspace

by C'(a). We shall say that U-z is a member of the complementary series if there

exists a positive-definite continuous inner product

on C°°(a) x

C°°(a) such that (3.1)

=

for all x in G and all f and gin C°`(a). If there is a nontrivial positivesemidefinite continuous inner product on C°°(a) x C0°(a) such that (3.1) holds, we shall say that U°,z is a member of the quasi-complementary series.

In either case, the continuity of the inner product, equation (3.1) for x in K, and the Schwartz Kernel Theorem together imply the existence of a continuous operator L mapping C°°(a) into itself such that = (Lf, g)

for f and g in C°°(a). Here L2(K) ® VV.

denotes the usual inner product on

254

KNAPP AND STEIN

As in Section 2, equation (3.1) translates into the fact that L intertwines and its contragredient. Applying Lemma 2.1, we see that (3.2)

LUa'z(x) = Ua' - t(x)L

for all x in G. Recall that m' is a fixed member of M' that is not in M. If a is equivalent with om then it is possible to extend a to a representation of all of M' on Va; that is, we can regard a(m') as defined. In this case, we define the operator A(z) for Re z > 0 by equation (2.3). The operator A(z) (actually a slight variant of it) was considered in [6]. It was shown that the kernel (3.3)

e(I -z)O log a(km')a(m')a- I(m(km'))

is an integrable function of k for Re z > 0 and hence that A(z) is a bounded operator on Ha. Moreover, A(z) satisfies (3.4)

A(z)Ua'z(x) = Ua'-z(x)A(z).

(For Re z < 0, the expression (3.3) is not an integrable function, and we consequently shall not deal directly with this case. In any event, one expects that Ua'z is in the complementary series if and only if Ua, -z is and that, in this case, Ua.z and Ua - z lead to the same unitary representation.) LEMMA 3.1. Fix a and z, and suppose Re z > 0. Unless a is equivalent with am' and z is real, the only continuous linear operator L on C'(a) satisfying (3.2) is 0. If a is equivalent with am" and z is real, then the continuous operators on Cm(a) satisfying (3.2) are exactly the scalar multiples of A(z). A(z) is bounded and Hermitian.

Before we pass to a study of the analyticity of A(z), let us observe that the A(z) have a common finite-dimensional resolution. Specifically, let HD be the subspace of Ha of functions that transform under K according to the equivalence class D of irreducible representations of K. HD is finitedimensional since Ha c_ L2(K) ® Va, and it is independent of z. Then each A(z) maps each HD into itself, by equation (3.4) for x in K.

If f is in Cm(a), the mapping z -. A(z)f is an analytic mapping of {Re z > 0} into C CO(a). We shall be concerned with extending this mapping

to a meromorphic function defined in the whole complex plane. It will be

enough to consider the simpler function z - . A(z)f(l), where

1

is the

identity of K, provided we prove joint continuity of this function in z and f.

Since the singularities of the kernel (3.3) occur only for k in M, we can suppose that f is supported near M, particularly away from M' - M.

THE EXISTENCE OF COMPLEMENTARY SERIES

255

This turns out to mean that we can transform the whole problem to a problem about the simply connected nilpotent group N. In fact, using the change-of-variables formula of [2, p. 287], we find that (3.5)

A(z)f(1) =

fNe(1-z)Ploga(Ym')a(m')u-1(m(ym')){e(1+z)PH(vf(K(y))} dy.

The notation here is the same as in formulas (2.1) and (2.3). The ingredients of this formula are technically much simpler than those of formula (2.3), and we consider them one at a time. First we make some comments about N. The restricted roots of the Lie

algebra g of G are either 2a, a, 0, -a, -2a or a, 0, -a. In this notation, N = exp (9 g-a l3+ g_2a). Let p = dim g-a and q = dim 9 -2a. The group A acts on N by conjugation; geometrically this action looks like dilations, except that the g-a directions are dilated twice as fast as the g - a directions. Now consider the first factor in the integrand of (3.5). Although it is not

necessary to do so for the present problem, one can compute this factor explicitly. If y = exp (X + Y) with X E g - a and Y E g then3 (3.6)

e(1- z)p log a(vm') = (+c2II XII4 + 2c11 Y

112)-(p+29)(1-z)/4,

where the norm is that induced by the Killing form of g and where c = (2p + 8q)-'. Put I y I = e- P log a(Ym') Then the function (3.6) has an important property of homogeneity relative to A: if b is in A, then Ibyb-1I = e-2PH(b)lA

Next we consider a(m')a-1(m(ym')), which we shall denote a(y). This is a matrix-valued function defined everywhere but at the identity and satisfying the homogeneity property a(byb-1) = o(y) for all b in A. Finally we consider the factor e(1 +2)PH(Y f(K(y)). This function is a

smooth function of compact support in N because f is assumed to be supported away from M' - M. The function depends on the complex parameter z but is entire in the variable z since pH(y) has no singularities. To see that (3.5) extends to be meromorphic in the whole z-plane, we choose a continuous function p(r) of compact support on [0, co) so that 4'(I yI)f(K(y)) = ft-(y)), we expand e(1+z)PH(Y f(K(y)) about y = identity in a finite Taylor series with remainder term, we collect the polynomial terms of

the same homogeneity relative to A, we multiply both sides of the expansion by p(IyI), and we substitute into (3.5). The terms of the expansion can be computed well enough to conclude the following: each term but the remainder has a meromorphic extension with at most one pole, that one simple and occurring at an integral multiple of z = - (p + 2q) -1, and the 3 That this explicit formula holds might be guessed from an earlier formula that S. Helgason had derived for exp { - 2pH(y)}.

KNAPP AND STEIN

256

remainder term gives a contribution analytic in a large right-half plane. Collecting these results, we have the following theorem: THEOREM 3.2.

Let f be in C °°(a). As a mapping into C a0(a), the function

z - . A(z)f has a meromorphic extension to the whole complex plane with singularities only at the non-negative integral multiples of - (p + 2q)-1. The singularities at these points are at most simple poles. The poles can occur only

at integral multiples of -2(p + 2q)-1 if a(y) = a(m')a-1(m(ym')) satisfies (3.7)

o(exp (-X + Y)) = a(exp (X + Y))

for X E g a and Y E g _ Za. Moreover, the mapping (z, f) A(z)f for z in the regular set and f in C-(a) is a continuous mapping to C°°(a). REMARKS.

Condition (3.7) holds for all a for the Lorentz groups

SOQ(n, 1), the Hermitian Lorentz groups SU(n, 1), and the symplectic Lorentz groups Sp(n, 1), but it fails for the spin groups Spin(n, 1). In any case, the parameter z is normalized so that z = 1 corresponds to p; therefore, z = 2(p + 2q)-1 corresponds to the restricted root a. The result for SOQ(n, 1) that the only poles of A(z)f are simple and are at multiples of -a was obtained by Schiffmann [7]. Using Theorem 3.2, we can now define A(z) for all z. To proceed further, however, we need more information about a. It is possible to show, under the additional assumptions on G that G is simple and has a faithful matrix

representation, that some representation D of K, when restricted to M, contains a exactly once.4 By the reciprocity theorem, this means that K acts irreducibly on some HD 0 0. Fix such a D = D°, and let v(k) be a nonzero member of H 0. Since A(z) commutes with K, we obtain (3.8)

A(z)v = y(z)v

for a complex-valued meromorphic function y(z). Define

B(z) = y(z) -'A(z). As we shall see in Section 4, there is no complementary series associated

with a near z = 0 unless the unitary representation U°'0 is irreducible. [And for G = Spin(n, 1) or SU(n, 1) there is no complementary series for any z unless this condition is satisfied.] We therefore assume now that U°'0 is irreducible. A necessary and sufficient condition for this irreducibility is given as Theorem 3 of [3]. The condition implies that y(z) does have

a pole at z = 0, which implies that the operators B(z) are uniformly bounded on compact subsets of 0 < Re z < c if c is sufficiently small. The irreducibility and equation (3.4) then imply that B(0) = I. 4 Independently J. Lepowsky has obtained this result and a generalization in his thesis at the Massachusetts Institute of Technology.

THE EXISTENCE OF COMPLEMENTARY SERIES

257

The definition of y(z) is arranged so that B(-z)B(z)f = f for all f in C°°(a) and for all z. In fact, Theorem 3.3 implies that B(-z)B(z)f is meromorphic in the whole plane. But B(-z)B(z) for purely imaginary O intertwines U"with itself. By Bruhat's irreducibility theorem in [1], B(-z)B(z) = c(z)I with c(z) scalar for z imaginary. Applying both sides to z

v, we see that c(z) = 1 for z imaginary. That is, B(-z)B(z)f = f for z imaginary. By analytic continuation, B(-z)B(z)f = f for all z. B(z) preserves each HD, and B(0) = I. Fix D, and suppose B(z°)ID is not positive-definite for some z° > 0. Then either B(z) has a pole nearer 0, or some B(z)f has a 0, in which case B(-z) has a pole, because B(-z)B(z) = I. The poles of B(z) (and similarly for B(-z)) arise when A(z)f has, for some f, a pole of higher order (possibly negative) than does y(z). These are the ideas behind the main theorem: THEOREM 3.3.

Suppose that G is simple, that G has a faithful matrix

representation, and that dim A = 1. Let a be an irreducible finite-dimensional

unitary representation of M satisfying the necessary conditions above, namely, that

(i) a is equivalent with am', where m' is a member of M that is not in M, and (ii) the unitary representation U6,0 is irreducible. Define A(z) by (2.3) and y(z) by (3.8). Let z° be the least number > 0 such that, for some f c C °°(a), z -. A(z)f has a pole at - z° and y(z) does not or

such that y has a zero at z° or at - z°. Then z° > 0 and the parameters (a, z) give rise to representations of the complementary series for 0 < z < z° with inner product = y(z) -1

JK

(A(z)f, g)v, dk

for f and g in C -(a). It is a simple matter to see also that the parameters (a, z°) give rise to a representation of the quasi-complementary series. It can happen that this representation is the trivial representation of G.

We should emphasize why the number z° in the theorem is strictly positive. The set whose least member is z° consists at most of the positive

nonzero integral multiples of (p + 2q)-1 and the non-negative values z such that one of the meromorphic functions y(z) and y(-z) vanishes. This set is discrete, and it does not contain z = 0 because y(z) has a pole at

z=0.

KNAPP AND STEIN

258

4. Further investigation of the singularities of B(z)

For the case that u is the trivial representation of M, we can choose the eigenvector v of the A(z) to be a constant function. The associated function y(z) is closely related to Harish-Chandra's c function (see [2]), and we can obtain very explicit results as a consequence. For G of general real rank, the c function can be defined as the analytic continuation in )u to the whole complexified dual of the Lie algebra of A of the function

c(µ) =

e° +o)H(x) dx.

IN

Let us return to the real-rank one case. PROPOSITION 4.1. Let u be the trivial representation, and choose v to be a constant function. Then the function y(z) is given by y(z) = c(- izp). There-

fore

F(J(p + q + 1))I'(j(p + 2q)z) z = 2 (P+2q)(1-z)/2 Y() I'(+ (p + 2q)(1 + z))]P(l (p + 2 + (p + 2q-)z))' where p = dim g_a and q = dim g-2a PROOF. e(1 - z)o log a(k) dk

.e(1 - z)p log a(km') A

y(z)

K

K

=I

e(1 -z)p log a(k) dk = f'V e(1 -z)p log a(x(x))e2DH(x) dx,

K/M

t he last equality following from [2] (see p. 287). If x = ank c ANK, then k = a-1(an-1a-1)x c MANN. Hence a(K(x)) = exp (-H(x)), and we obtain y(z) = IN e(1+z)oH(x) dx, as required. The formula for y(z) then follows from [2] (see p. 303). In [4], B. Kostant obtained the existence of complementary series for a

trivial and G of any real rank. For a first application of Proposition 4.1, we shall compare his results in the rank-one case with what we can prove from Theorem 3.3 and Proposition 4.1 when a is trivial. Using the explicit value of y(z) in the proposition, we see that y(z) is nonvanishing for -1 < z < I and that the only poles of y(z) for -1 < z 5 0 occur at the non-negative integral multiples of -2(p + 2q)-1. If q = 0, there is a pole at every multiple of -2p-1 less than 1. If q > 0, then p is even and a pole occurs at every multiple satisfying - (p + 2)(p + 2q)-1
0, limf(x, E) = 1. As a lowest-order approximation we then take go(x) = 1,

and expect Ord 1 to be a domain of validity, i.e., if - god tends to zero uniformly in the semi-infinite interval x >= c where c is any constant >0. On the other hand, because of (4.6b), go cannot be valid near x = E. We therefore introduce a stretched coordinate, using the scale E. This leads us

back to the original variable x.8 We now see that go = 1 is the outer solution, analogous to (2.6). It is not uniformly valid near the origin and we need a correction ho, i.e., a solution satisfying the condition at x = E but which need not be valid at large x. The solution ho as given by (4.4) is

an inner solution analogous to (2.11), and there is no a priori reason to expect it to satisfy the boundary condition at x = oo. Thus the Stokes paradox is shown to be due to a faulty interpretation of the Stokes equation (4.3). Following a central idea of singular perturbation theory we instead

expect to determine A in (4.4) by matching ho with go. The matching condition (2.12) obviously makes no sense in the present case and we have to use the more general formulation of Section 3. To do this we first have to determine the domain of validity of ho. We note that if we introduce the variable x,,, defined by ,+)x = x, into (4.6a) and then take the formal limit of the resulting equation, keeping x,

fixed, we obtain the "Stokes" equation (4.3) whenever q = o(1). In an obvious sense the formal domain of validity of (4.3), relative to (4.6a), is the set of 0-classes (4.7)

'Y,- = {XI X < Ord 1}.

We now invoke a heuristic principle (also due to Kaplun). There exists a solution ho(x, c) of (4.3) whose maxi(4.8)

mal domain of validity, as an approximation to f(x, E), includes the formal domain of validity of (4.3).

This solution must be of the form (4.4) since validity on Ord E necessitates that the boundary condition (4.2b) be satisfied. We now have to find A, possibly as a function of c, by matching. Note that matching should be possible according to our heuristic principles. If Ord 1 is in the maxi8 This circular argument serves to illustrate an important point. Historically, the Stokes equation was derived in the manner in which (4.3) was derived from (4.2). However, from the point of view of theory of singular perturbations, the formulation (4.6) is the principal formulation. It is the solution go(x) which is being perturbed, not the solution ho(x); the latter one is a correction. This point of precedence is an important feature of the theory (see [2], p. 3).

P. A. LAGERSTROM

270

mum domain of validity ill of go, then by the extension theorem QI intersects Y-, which is a domain of validity of ho according to (4.8). To actually carry out the matching we use the technique of the intermediate limit. We

note that go - ho = I - log [1 + A(log xn + log q - log E)] and hence that Jim 1go - hoj = 0 if n

A (4.9)

e-1 -log E'

and if 17 is in the domain (4.10)

-9 = {Ord ill log 17 = o(log E)}.

We shall now show that -9 is actually the overlap domain. We first note that A is unique, within o(A). We must require that: (1) as Ord 71 varies over the overlap domain, application of Jim to I go - hot does not give

contradictory results for A, and that: (2) the overlap domain contain order-classes arbitrarily close to Ord 1. By this is meant that it must have a nonzero intersection with any interval (Ord , Ord 1) for any which is o(l). The requirement follows from the description of 'W and Y" above. It is seen that -9, as determined by (4.10), is the maximal domain satisfying both requirements. Note that, for instance, Ord E112 could not be in the overlap domain since one may find functions 17 satisfying Ord E112 < Ord 77 < Ord I which give contradictory values of A.

The domain -9 contains Ord 1 and even classes greater than Ord 1. Thus the maximal domain of validity of ho is slightly larger than the formal domain of validity of (4.3). In the present case the overlap domain is quite restricted. One finds, for instance, that Ord Ea < -9 < Ord E-" for any a > 0. Still the fact that ho and go have a nonempty overlap domain and satisfy the inner (4.6b) and outer (4.6c) boundary conditions, respectively, means that in principle we have obtained an approximation which is uniformly valid to order unity and also that the determination of A by matching is legitimate. Note that the manner in which c occurred in A was not determined a priori; instead it followed from the form of the equation and the matching. In the example discussed in Section 2 the functions go and ho could be obtained by applying a suitable lim to the exact solution: go is obtained

by choosing 77 = I (outer limit) and ho is obtained by choosing 77 = E (inner limit). The example discussed in the present section shows that this

viewpoint is too narrow. If we apply lim with q = Ea, 0 < a < 1, to the n function f defined by (4.6), we obtain log [(I - a)e + a]. (Note that in such a limit process f may be replaced by ho.) Thus, the idea of applying

THEORY OF SINGULAR PERTURBATIONS

271

limits to the equation and then finding solutions of the resulting approximate equations is more basic than looking for limits of the solution. In fact, the example in Section 2 could be studied from the point of view adopted in the present section. Also, the function go = 1 in this section was regarded as a limit and was determined by physical reasoning. However, if lim with 1 = o(-q) is applied to (4.6a) one obtains (4.10)

go

do=0.

Solving this equation using the boundary condition at infinity yields go - 1. Applying lim with 77 = 1 leaves the equation (4.6a) invariant. n

Using the extension theorem and the heuristic principle (4.8) we see that we need a solution of (4.6a) which matches both with go (valid at least for 77 >> 1) and with ho (valid at least for 77 0. A Riesz convergence ' in a lattice group L is a set of sequences in L satisfying:

(1.1) If [.fn] and [gn] c l then [f, + gn] E V.

(1.2) If f = 0 ultimately then [

E W.

(1.3) If [f,,] E l and [p(n)] is any sequence of positive integers tending to E W. oo then [ (1.4) If fnl < I g, for all n and [gn] E le then E V.

Define f -. f to mean [J - f] e W. In particular f

Let f, - f and g (I) f. A gn-.f A g,

PROPOSITION 1.

(II) .fn V gn -'-'> f V g,

('ii)

Ifnl if I. ' Work done under NSF-GP 7539.

273

g. Then

0 means

e W.

SOLOMON LEADER

274

PROOF.

{ Ifn A gn -J A gI : IJn n gn -f. A gI + IJn A 9-f/A gI

{

Ig,, - gI + Ifn -f1. Hence (i) follows from (1.1) and (1.3), (ii) follows similarly with cap replaced by cup, and (iii) follows from (1.4) since I.ffI - Ifl I < Ifn - fI I

E '' and fo e L then [fii-1] E le. Thus, adjoining PROPOSITION 2. If finitely many terms to a sequence belonging to le always yields a sequence belonging to W. PROOF. Let p(1) = 1 and p(n) = n - 1 for n > 1. Then [fp(,,,] E W by (1.3). By (1.2) [fn-1 - f,(,,)] E'. Hence [f. _J E' by (1.1).

PROPOSITION 3.

If 0 < u,t+1 < u and some subsequence [u,,,k] E W then

E W.

PROOF. By Proposition 2 we may assume that n1 = 1. Let Vk = u,,,. Define p(k) to be the last i for which ni < k. p(k) is well defined since n1 = 1 and n; < n;+1. Moreover p(k) - oo. Hence by (1.3) [vpck>] E V.

Now 0 -- 0. To show that c = 0 we shall find for any given E > 0 a subsequence [J,,(,)] such that for gk = Ifwk)I

C = lim a(gk) < E.

(7.4)

k-w

We choose our subsequence so that

a(gk - gk+D) < fk for all p.

(7.5)

This is done by choosing n(k) > n(k - 1) so that a(f, - fm) < fk for all

m, n > n(k). Since gk -

k

t=1

Igk - gtI -5 gk - Igk - gil 5 gt for i = 1,

k we conclude that the left side is a lower bound of uk = g1 A

Thus 0 5 gk < uk +

k

t=1

Igk - gt1 Hence by (7.5) we have k

0 < a(gk) < a(uk) +

t=1

a(gk - gt)
0 choose k large enough so that

a(f - f) < 3

(8.2)

for all i, j > k.

Then choose M large enough so that (8.3)

a(f - f, A mu) M and j < k.

Now 0 < (f,, - mu) + = fn - fn A mU < If, - fkl + I fk -./k A mu! + I fk A mu - f A muI < 21f, - fkl + (fk - mu)+. Hence (8.4)

a(f - mu) + < 2a(f, - fk) + a(fk - mu) +.

From (8.2), (8.3), and (8.4) we conclude that (8.5)

a(fi - mu) + < E for all m > M and all n.

PROPOSITION 12.

If f converges a-truncatedly to 0 and is uniformly

a-absorbed by some u, then a(f,) -* 0.

SEQUENTIAL CONVERGENCE IN LATTICE GROUPS

289

PROOF. From (8.5) and the triangle inequality a(fn) 0.

n-.w

PROPOSITION 13. If L has an a-limit unit then every a-Cauchy sequence which converges a-truncatedly to 0 must a-converge to 0. PROOF.

Apply Propositions I 1 and 12.

PROPOSITION 14.

A Banach lattice L has a norm-limit unit if and only

if there exists a countable subset E of L whose solid hull H is norm-dense in L. PROOF. Let a be the norm in L. If u is an a-limit unit let E consist of all integral multiples of u. That E has the stated properties is then trivial. Conversely, given E with the stated properties we may assume that E is contained in L. Enumerate the members of E to obtain a sequence [un].

Then choose coefficients an > 0 so that M

n=1

n=1

ana(un) < oo. Define u =

anun which exists because L is complete. Consider any f in L and any

h in H. Then IhI < un for some n. Hence IhI < mu for m > 1/an. Thus by

(8.4) we have a(f - f A mu) < 2a(f - h) + a(h - h A mu) = 2a(f - h) for m sufficiently large. Therefore

lim a(f - f A mu) < 2a(f - h) for all h in H.

(8.6)

m-.w

Since H is a-dense in L the left side of (8.6) must be 0. Thus u is an a-limit unit for L. PROPOSITION 15. PROOF.

Every separable Banach lattice has a norm-limit unit.

Apply Proposition 14.

PROPOSITION 16. In a Banach lattice L with norm a every a-convergent sequence is uniformly a-absorbed by some u in L+. PROOF. Apply Proposition 14 to the Banach lattice generated in L by the convergent sequence. Then apply Proposition 11.

RUTGERS UNIVERSITY THE STATE UNIVERSITY OF NEW JERSEY

REFERENCES 1. BIRKHOFF, G., Lattice Theory. New York: A.M.S. Coll. Pub. 25, 1948.

2. KAKUTANI, S., "Concrete representations of abstract L-spaces and the mean ergodic theorem," Ann. of Math., 42 (1941), 523-537.

SOLOMON LEADER

290

3. KAKUTANI, S. "Concrete representations of abstract M-spaces," Ann. of Math., 42 (1941),994-1024. 4. KIST, J., "Locally o-convex spaces," Duke Math. J., 25 (1958), 569-582. 5. LEADER, S., "Separation and approximation in topological vector lattices," Can. J. Math., 11 (1959), 286-296. 6. LUXEMBURG, W. A. J., and A. C. ZAANEN, "Notes on Banach function spaces," Proc. Acad. Sci. Amsterdam, A66 (1963) A67 (1964) A68 (1965) II

135-147 148-153

III

239-250

I

IV

251-263 V 496-504 VI 655-668 VII 669-681

VIII 104-119 IX 360-376 X 493-506 XI 507-518

XIVa 229-239 XIVb 240-248

XII 519-529

XVIa 646-657 XVIb 658-667

XIII 530-543

XVa 415-429 XVb 430-446

7. NAMIOKA, I., Partially Ordered Linear Topological Spaces. Providence: A.M.S. Memoir 24, 1957. 8. PERESSINI, A. L., Ordered Topological Vector Spaces. New York: Harper and Row, (1967).

A Group-theoretic Lattice point Problem BURTON RANDOL

1.

Let G = SL(2, R), F = SL(2, Z), and let P be the set of primitive integral lattice-points in R2. For g c G and r > 0, let N,(g) be the number of points in the set g(P) which intersect the disk Ixj < r. Then, inasmuch as y(P) = P for any y c F, it is evident that N,(g) can be regarded as a function on the quotient space G/l', which has a normalized Haar measure dg. Moreover, as a result of Siegel, the mean, or integral, of N,(g) over G/l' is simply -17rr2 (see [4], Formula 25).

Consider now the following question. What can we say about the variance of N,(g) over G/P? That is, what is the asymptotic behavior, as

r-* oo, of

v(r) = f

clr(N.(g) -

It will be shown here that, for any integer k, v(r) = (C(2))-1Trr2 + 0(r2 log-' r). Before passing to the proof, I would like to thank Robert Langlands for several informative conversations concerning L2(G/I'). 2.

Suppose f(x) is C- with compact support in R2 - {0}. Suppose, for the sake of simplicity, and since this is the only case with which we will be concerned, that f(x) is radial, i.e., f(Tx) =.f(x) for T e SO(2). Define

O(g) = I f(g(p)). Then PEP

(1)

JCIr IO(g)12 dg =

f

Res=c

+ (2iC(2))_1

f

Lf(2 - 2s)Lf(2s) ds

JRes=c

i(2 - 2s)Lt(2 - 2s)Lf(2s) ds, (2s)

where.'(x) = fRz f(y)e-2nl(x.y) dy, c is any real number greater than 1, and L9(z) = 2 fo g(p)pz dp for a radial function g(p). 291

BURTON RANDOL

292

This result comes from combining Formulas 9 and 22 in [1]. Godement

makes the additional hypothesis that fR2 f(x) dx = 0, but only to avoid the presence of a pole of LI(2 - 2s) at s = 1, and the correctness of (1) is not affected by removing this hypothesis (see also [3, 10a]). The constant which precedes the integrals in (1) differs from the constant which precedes the integral in Formula 9 of [1]. The reason for this is two-

fold. The latter constant should be (27ri)-1 instead of a-2, assuming that the measure of the circle-group is normalized, and we have, in addition, divided by 7r

in order to normalize the measure of G/F (see [2], p. 145).

Suppose now that {c6 (x)} is a sequence of C°° radial functions having compact support in R2 - {0}, and converging, except at the origin, to the indicator function of the unit disk, in the manner indicated cross-sectionally in Fig. 1. X

and let ¢,(x) be the indicator function of the r disk IxI < r. Then ¢,,,(x) -->. c,(x) monotonically, except at the origin. Define Nn,,(g) _ fin,,{g(p)}. Then it follows from Beppo Levi's Define 0.,,(x) = din

PEP

theorem that

LF INn.r(g)I2 dg -*

fclr

INr(g)II dg.

But by (1), LP INn.r(g)I2 dg = (2)

f

Res=c

+ (2i4(2)) -1

Lmn.,(2 - 2s)L,,n ,(2s) ds

f

Res=c

(2y - 2s) Y(2s)

Len ,(2 - 2s )L,5n,,(2s) ds.

Now Lmn ,(z) is an entire function of z, inasmuch as qn r(x) is C- with

compact support in R2 - {0}. Moreover, repeated integration by parts

Fig. I

A GROUP-THEORETIC LATTICE-POINT PROBLEM

293

shows that Lmn ,(z) is of rapid decrease in any vertical strip. The Fourier transform of 0,,,,(x) is again a radial function, C`°, of rapid decrease as oo, and satisfying ,,.,(O) -- Trr2 as n - - oo. From this it follows by jxj repeated integration by parts that L,$,(2 - 2s) is of rapid decrease near infinity in any vertical strip and is meromorphic with a simple pole at s = 1, having residue c,,, with c --> -2irr2, and possible additional poles

at s = 2, 3, 4, .... (There are no poles at s = 3/2, 5/2, 7/2, ...

since

c6,,,,(x) is radial, and hence its odd derivatives in the p-coordinate vanish

at p = 0.) Now (2 - 2s) has simple zeros at s = 2, 3, 4, ..., and is holomorphic in a neighborhood of Re s >_ i and vanishes only at s = 1;

(2 - 2s

thus we conclude that the expression

(2s)

Len ,(2 - 2s)Lm ,(2s) is

meromorphic in a neighborhood of Re s > 1, with a single simple pole at s = 1. Furthermore, since Len ,(2 - 2s) is rapidly decreasing in any strip, and since

(2 - 2s (2s)

hood of any strip

is of no worse than polynomial growth in a neighbor-

Res 2 is the genus, for the order of the automorphism group is attained (vide infra). Renewed interest attaches to it in light of the role

played by surfaces with automorphisms in the structure of spaces of Riemann surfaces (see, for instance, [17], the references therein to both authors' work, and [18]). The motivation for this paper is the desire to determine the action of Klein's group on the theta functions and constants attached to his surface.

We plan a subsequent publication on this and other aspects of theta functions on Klein's surface. It would be of interest to contemplate our process for a surface of higher genus. Macbeath, in [14] and [15] (see also [13]), has recently exhibited new examples of surfaces with automorphisms realizing Hurwitz's upper bound, in particular, one of genus seven with a simple group of 504 automorphisms.2 The reader will see clearly that our methods can be used in

principle to treat Macbeath's example, but he will also see that our geometrical method in that case would be quite tedious in application. After seeing our results Macbeath called our attention to Leech's work, in particular to papers [11] and [12], with the remark that answers to (i) and (ii) for Klein's and Macbeath's surfaces could be deduced from them. The first named author has since carried out the calculations with the aid of H. Lee Michelson. Our method here is more powerful in that we also obtain

the homological action of the extended groups obtained by adjoining reflections. Leech's method, which is purely group-theoretical, on the other

hand, utilizes a computer to do calculations that would otherwise be impractical. We call attention to an incorrect answer to (iii) by Hurwitz ([7], p. 159, criticized in [I]) and an abortive attempt on (i), (ii), and (iii) by Poincare in [16], p. 130, all noticed after the completion of our work.'

2. Klein's surface Klein originally obtained his surface S in the form of the upper half-plane identified under the principal congruence subgroup of level seven, r(7),

of the modular group F. In this form it is necessary to compactify the fundamental domain at its cusps. Klein's group then appears as 17/17(7), which is simple and of order 168. We need, however, another representation given by Klein, one which we recognize today as the unit circle uniformization of S. In the unit circle draw the vertical diameter Ll and another diameter L3 making an angle of IT/7 with Ll and going down to the right. In the lower semicircle draw 2 See note added in proof at end of paper.

THE RIEMANN SURFACE OF KLEIN

299

the arc L2 of the circle which is orthogonal to the unit circle and to L1 and

which meets L3 at the angle 7r/3. Let t be the non-Euclidean triangle enclosed by L1, L2, L3, and let R1, R2, R3 be the non-Euclidean reflections in L1, L2, L3, respectively. R1, R2, R3 generate a non-Euclidean crystallo-

graphic group, which we denote by (2, 3, 7)', with t as a fundamental domain. The images of t under (2, 3, 7)' are a set of non-Euclidean triangles each of which is congruent or symmetric to t according as the group element which maps t on it has an even or odd number of letters as a word in R1, R2, R3. These triangles form a non-Euclidean plastering or tesselation of the interior of the unit circle. The union of t and its image under R2 is a fundamental domain (with suitable conventions about edges) for the triangle group (2, 3, 7), which is the group generated by T = R3R1i

V = R3R2, U = R2R1 with the relations T7 = V3 = U2 = T-'VU = I. T, V, U are, respectively, non-Euclidean rotations of angles 2a/7, 27r/3, 7r counterclockwise about the vertices of t belonging to the angles 7r/7, 7/3,

7r/2. Now the uniformizing Fuchsian group of S is a normal subgroup

N - (2, 3, 7) of index 168. The quotient group F166 = (2, 3, 7)/N is Klein's simple group of automorphisms of S and is obtained from (2, 3, 7) by imposing the relation (UT-4)4 = I. A convenient fundamental domain for N is the circular arc (non-Euclidean) 14-gon A shown in Fig. 1. It will be noticed that t appears as the unshaded triangle immediately below and to the right of P°. There are 168 unshaded triangles, which are the images of t under F168, in A and 168 shaded triangles, which are the images of t

under anticonformal elements of (2, 3, 7)'. We let F166, the extended Klein group, be the group of 336 proper and improper (sense reversing) non-Euclidean motions obtained by adjoining, say, R1 to r166. Then 0 can be viewed either as the union of 336 shaded and unshaded triangles consisting of t and its transforms by F'168 or as the union of 168 double triangles (one unshaded, one shaded) consisting of the union of t and its image under R1 [thus a fundamental domain for (2, 3, 7)] and the transforms of this by F166. If the sides of 0 are labeled as in Fig. 1, then, Klein has given the side identifications induced by N in Table 1. A with those identifications is Klein's model of his surface S. TABLE 1

1+6 3 P5X3 + XBP1 + P1P8 + P3X2 + X11P10 + Plc X12 + X71PI + p2 Q7 + Q5P53

-i-4-f-e-c-d+c+i-i+ 11 -m+m -j

=-2-3-4-5=-81 82

P1o X11 + X2P3 + P8P1 + P1P7 + P13X13 + X4P5

-i-c+c+d-I-m+f+e-k-h+h+i

=3-7=-82

83 -> Q7P1 + P1 P8 + P3X2 + X11P10 + PS Q5

--f-e-d-c+c+i+j=-3+5=-83

From Tables 2 and 6 we see that R2 takes yl = 1 - 4 - 7 - 9 into P5X4 + X13P12 + P7Pl + P1 X; + X12P10 + P10P1 + P?P5i where we

have reversed the order of some pairs to have all positive signs and

THE RIEMANN SURFACE OF KLEIN

305

rearranged terms so that, with identifications, the terms fall in natural order, end-to-end. We next observe from Fig. and Table 6 that 1

PSX4-'P5P+PX4=-i-h,X13P12-X13Piz^ X13P+PP2=h+k, P7P1 +P1X. ^ P7 X7=1,X2l'11- X12Q- 11 +PP10=m+2+i, PoPi +P1Ps _p 310

Q-9-8-7-6-P5Q=j-9+3-7+1 -j.

Hence R2y1--i-h+h+k+1+m+2+i+j-9+3-7+ 1 -j=

k+l+m+2-9+3-7+1=7+2-9+3-7+1=2-9+3+ 1 = y1 + 81 + 82 + 83. In Table 7 - means "homologous to." Using Tables 4 and 7 to construct matrix representations f1i d2, M3 of the actions of R1, R2, R3 on the homology basis of Table 2 we now write down by composition the matrix representations MT 1, .14, M u of the

generators T-1 = R1R3i V = R3R2, U = R2R1 of F168 (we prefer for technical reasons to use T-1 instead of T as a generator). Observe that, consistent with the practice of [17], Lemma 8 and Proposition 9 (note the corrections in the reference in the bibliography), R1R3i for example, will be represented by .ill3Mli i.e., the reverse order of the geometric transformations.

(1 )

T

-1

-1

0

0

-1

-1

-1

0

0

-1

-1

0

0

0

0

1

1

1

-1

1

0

0

-1

-1

1

-1

0

0

1

0

I-1

0

0

-1

-1

-1

0

0 0 1

1=

dl

(2)

0

0

-1 -1

0

-1

0

0

0

0

0

0

0

0

1

0

0

0

0

1

-1

0

1

1

0

0

0

0

-1

/

0

0

-1

-1 '

1

1

0

1

(3)

0

0

-

1

0

0

0

0

0 -

1

0

0

0

0

0

0

-1

-1

0

0

0

0

0

1

0

0

0

0

0

-1

1

U

-1

RAUCH AND LEWI7TES

306

5. Computation of the normal period matrix

We can now use the results of Section 4 to compute the period matrix 7r over 81, 82, S3 of the normal Abelian integrals of the first kind on S with respect to the canonical homology basis y1, Y2, y3; 81, S2, 83 of Section 3 (see [17], p. 12 ff.). The technique is to observe that Tr is fixed under the transformations of the inhomogeneous Siegel modular group #3 corresponding to the matrices (1), (2), (3) in Sp(3, Z) of Section 4 (see [19] and [17], Section 2, C, p. 16). Thus we have

7r = (Air + B)(CTr + D) -1

or (4)

T((Car + D) = A7r + B,

where C`

A)

is any of the matrices in (1), (2), and (3). If we set .7r=

a

b

c

bde c

ef

((r is symmetric), and f = J4 (here B = C = 0) we find from (4)

b=-b - d

-b - c=-c - e -d - e=e or

(5)

2b=-d b=e -d 2e.

Similarly, choosing .& = . (, one finds

(i) a 2 + ab + ac + a + bc + b + (6)

C

+1=0

(ii) ab+ad+ae+be-c=0 (iii) ab+b2+bc+b+cd+d+ e =0 (iv) b2+bd+be+de-e- 1 = 0 (v) f=eb+cd+.ce+e2+d+e+ 1.

THE RIEMANN SURFACE OF KLEIN

307

Setting (5) in (6, iv) one obtains

2e2+e+1=0

(7)

or

e=

(8)

-I±V/i 4

We shall decide the choice of sign later. Substituting (5) in (6, ii), (6, iii), and (6, v), we obtain

c = e2 (9)

ae-e3+e2=0

a=e2-e

or

f=e2-e+1.

But from (7) one finds e2 = - - e/2 and hence e2 - e = - - 3e/2 and e2 - e + 1 = z - 3e/2, so that

- -1

3e

-2 e

-12

e

-2e

e

Now 7r as a normal period matrix must have a positive-definite imaginary part. Reference to (8) shows that we must pick the minus sign in e so that we have finally I

+ (10)

,r=

8

_1 _ 1/7i 4

4 3 8

+

_4 - 4 1 +1/7i

3i +

1

2

2

4

1/7 i

1

1/7 i

7

8

4

4

8

8

_ 1/7i 4

3\/7 i +

8

We observe that the entries of ,r all lie in the field k(1/-7), which is the field generated over the rational field k by the character of the representation (irreducible) induced on the differentials of first kind of S by x168 (see [18]). NOTE ADDED IN PROOF (May 1970): Macbeath's surface of genus 7 in [14]

was anticipated by Fricke in [22]. In pp. 265-270 of [1] Baker discusses Klein's surface without, as far as we can see, arriving at anything like our results. As he says, he does not construct any explicit homology basis, and

RAUCH AND LEWITTES

308

the period matrix he obtains is not proved to correspond, and probably does not correspond, to a canonical homology basis. THE CITY COLLEGE OF THE CITY UNIVERSITY OF NEW YORK THE HERBERT H. LEHMAN COLLEGE THE GRADUATE CENTER OF THE CITY UNIVERSITY OF NEW YORK

REFERENCES 1. BAKER, H. F., Multiply Periodic Functions. Cambridge, 1907. 2.

, "Note introductory to the study of Klein's group of order 168," Proc.

Cambridge Philos. Soc., 31 (1935), 468-481. 3. BURNSIDE, W., Theory of Groups of Finite Order. Cambridge, 1911.

4. EDGE, W. L., "The Klein group in three dimensions," Acta Math., 79 (1947), 153-223.

5. GORDAN, P., "Uber Gleichungen siebenten Grades mit einer Gruppe von 168 Substitutionen," Math. Ann., 20 (1882), 515-530; 11, ibid., 25 (1885), 459-521. 6. HURWITZ, A., "Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Mathematische Werke, Bd. I. Basel: Birkhauser, 1932, pp. 392-430. 7.

"Uber einige besondere homogene lineare Differentialgleichungen,"

ibid., pp. 153-162. 8. KLEIN, F., and R. FRICKE, Vorlesungen fiber die Theorie der elliptischen Modulfunctionen, Bd. I. Leipzig, 1890. 9. KLEIN, F., "Uber die Auflosung gewisser Gleichungen vom siebenten and achten Grade," Math. Ann., 15 (1879), 251-282. 10.

"Uber die Transformation siebenter Ordnung der elliptischen Funk-

tionen," Math. Ann., 14 (1879), 428-471. 11. LEECH, J., "Generators for certain normal subgroups of (2, 3, 7)," Proc. Cambridge Philos. Soc., 61 (1965), 321-332. 12. LEECH, J., and J. MENNICKE, "Note on a conjecture of Coxeter," Proc. Glasgow Math. Assoc. (1961), 25-29. 13. MACBEATH, A. M., Fuchsian Groups, mimeographed notes, Queen's College, Dundee, University of St. Andrews. 14. , "On a curve of genus 7," Proc. London Math. Soc., 15 (1965), 527-542. 15.

"On a theorem of Hurwitz," Proc. Glasgow Math. Assoc., 5 (1961),

90-96. 16. POINCARE, H., Sur ['integration algebrique des equations lineaires et les periodes

des integrales abeliennes, Oeuvres, t. 111. Paris: Gauthier-Villars, 1934, pp. 106-166. 17. RAUCH, H(ARRY) E., "A transcendental view of the space of algebraic Riemann surfaces," Bull. Amer. Math. Soc., 71 (1965), 1-39, Errata, ibid., 74 (1968), 767. 18. , "The local ring of the genus three modulus space at Klein's 168 surface," Bull. Amer. Math. Soc., 73 (1967), 343-346. 19.

"Variational methods in the problem of the moduli of Riemann surfaces," in Contributions to Function Theory. Bombay: Tata Institute of

Fundamental Research, 1960, pp. 17-40. 20. SPRINGER, G., Introduction to Riemann Surfaces. Reading, Mass.: AddisonWesley, 1957. 21. WEBER, H., Lehrbuch der Algebra, Bd. If. Braunschweig, 1908.

22. Fricke, R. "Ueber eine einfache Gruppe von 504 Operationen," Math. Ann., 52 (1899), 319-339.

Envelopes of Hol omorphy

of Domains in Complex Lie Groups O. S. ROTHAUS1

There are two well known results in the literature concerning envelopes of holomorphy that we want to single out here, namely the ones describing

the completion of Reinhardt domains and tube domains. The statement for the latter type we owe to Salomon Bochner. There is a feature common to both the results which is worth noting. Roughly speaking it may be described as follows: a domain in a complex Lie group which is stable under a real form of the group may be completed by forming certain "averages" in the complex group. In this note we shall give one reasonably general form of this phenomenon, which more or less includes the case of Reinhardt domains.

We shall follow the usual convention of letting German alphabetic characters denote the Lie algebra or its elements corresponding to the group denoted by the associated ordinary alphabetic character. Let G be a connected complex Lie group, ( 3 its Lie algebra. Let j, j2 = -identity, be the endomorphism of 03 giving the complex structure. Let ff.) be a real form of (3, so that 03 = S) (+ jS), and H the corresponding analytic subgroup of G.

We want to investigate the envelope of holomorphy of domains D contained in G which are H stable; i.e., Dh = D for all h c H. We make the following assumption governing the situation: H is a maximal compact

subgroup of G. This assumption has a number of useful consequences, mostly well known, which we now present. For these purposes, let G be the universal covering group of G, r the fundamental group of G, so that G G/P. Let 7r be the natural map from G to G, let R be 7r - I(H) and R, be the connected component of the identity in H. 1 This research was partially supported by NSF GP 8129. 309

O. S. ROTHA US

310

There are natural homeomorphisms of the identification spaces as indicated : G

GOHO H= H H

o

Ho

By a theorem of E. Cartan and others, G/H is Euclidean; on the other hand H/Ho is discrete. Since Euclidean space is simply connected, we have LEMMA 1.

Ha = H.

Clearly II is then a covering space for H. But G, viewed as a fibre bundle over G/H, fibre fl, is equivalent to a trivial bundle, since G/H is Euclidean. G being simply connected, it follows that LEMMA 2. H is the universal covering space of H.

There is an involutory automorphism or of W given by a(Tj1 +A2) _ ht - jI 2 for TI1, b2 E S.). This automorphism induces an involutory automorphism of G, whose subgroup of fixed points has connected component

of identity equal to H. Since H - P, the automorphism, still denoted a, may be viewed as an automorphism of G/P - G, fixing H/P ~- H. Since H also is compact, we have LEMMA 3.

G/H is a Riemannian symmetric space (see [1]).

The transvections of G/H are given, of course, by exp jT , tj c S2, and every element in G may be written uniquely as a transvection times an element of H. H being compact, it is well known that H decomposes into the direct sum of a semisimple algebra S, and (E, the center of S. (Y is the Lie algebra

of C, the connected center of H, which must be a torus, and S is the Lie algebra of S, a connected semisimple subgroup of H. Every element in H may be written, though not uniquely, in the form sc, s c S, c e C. For future purposes, we shall suppose that a basis c1, c2i ... ,c1 of S has been chosen so that exp A1c; = identity implies and is implied by

a;EZVi. Now let h -* r(h) be a finite d-dimensional representation of H; for our

purposes we suppose the representation is already unitary, so that r(h) E U(d).

There is induced a representation of the algebra S,), which we still denote by l The representation of can be extended to give a representation R of 03 by setting R(b1 +J72) = r(1l1) + ir( 2)

ENVELOPES OF HOLOMORPHY IN LIE GROUPS

311

for b1, b2 E S. The representation R of 03 now induces a representation R of G. R, restricted to H, is simply the lift of r to ff. Since a representation of a connected group is completely determined by the corresponding representation of the Lie algebra, it follows immediately that R is trivial on F. Hence R may be viewed as a representation of G/F G, and thus we have: LEMMA 4.

Every representation of H extends to a representation of G.

It is worth noting that the entries in the representation R are holomorphic functions on G. In fact R is easily seen to be the unique holomorphic extension of r. Also note that in the extension, transvections are represented by Hermitian definite matrices. Now let r be a representation of H in U(d) and R the representation of G in GL(d, C) extending r. GL(d, C) has an analytic structure J arising

from the fact that U(d) is a real form. The associated involutory automorphism -r of GL(d, C) is the familiar one mapping a matrix to its conjugate transpose inverse. We have by definition the equations R(j9) = JR(S) and

R(ag) = -rR(g).

The first of these equations asserts that R is a holomorphic map of G to GL(d, Q. From the second equation we can prove LEMMA 5.

Let k =

R(G) is closed in GL(d, Q.

be a sequence of elements in GL(d, C) converging to k

in GL(d, Q. Write g = ph with op,, = pv 1, ah = h,,; i.e., p, is a transwith -ra = av 1, -rb = b,,. Then vection and h E H. Write b = b,,. And it follows that the sequences a and b a and converge separately, to a and b, respectively. Since H is compact, R(H) is

compact, and it follows that b c R(H) - R(G). Let p = exp p,,, so a = exp R(p ). It follows that converges to an element in the Lie algebra, say u. Since the map of Lie algebras is a surjection, there is a p E j5? such that R(p) = u; then R(exp p) = exp u = a, completing the proof. Now let rbe a faithful representation of H in U(d). We claim that LEMMA 6.

R is a faithful representation of G.

Suppose R(g) = identity. Write g = pk with up = p ah = h. Then each of R(p) and R(h) are the identity. Since R(h) = r(h) we have h = identity. Now let p = exp j11; R(p) = exp ir(b1). ir(r)1) is Hermitian, and

O. S. ROTHAUS

312

since exponentiation is one-one on Hermitian matrices, r(1j1) = 0, which implies that tjl = 0, implying p is the identity. By assembling the last few results we can show THEOREM 1.

G is a Stein manifold.

Let R be the extension to G of a faithful representation of H. By Lemma 6, the entries of R give sufficient holomorphic functions to separate points. Let tjl, 1)2, ..., Tj,, be a basis of SD and put z = x + iYv, z =

..., zn) E C. For z in a sufficiently small neighborhood N of the origin in Cn, the map of N to G given by (z1i z2,

z--g = V

gives a complex analytic chart at the identity in G. We have R(exp

(x + (z))a,9

(J

belongs to M(d, C), the ring of d-dimensional matrices over the complex numbers, which may be identified as before with the Lie algebra of GL(d, Q. Since the exponential map of M(d, C) into GL(d, C) has a

holomorphic inverse in a neighborhood of the identity, and since the representation r is faithful, it follows readily that for z sufficiently close to the origin, z is a holomorphic function of the entries faQ(z). Thus a subset

of the faf(z) give holomorphic coordinates near the identity in G. An analogous argument takes care of arbitrary points in G.

Finally, the topology of GL(d, C) may be given as the topology it inherits as a subspace of M(d, Q. Since R(G) is closed in GL(d, C), it follows that R(G) topologized as a subspace of GL(d, C) is homeomorphic to G. Let K be a compactum in G. Let a be an upper bound for the moduli

of all of the entries of R(g) and

det R(g)

for g c K. Then the set A of

u = (ui;) E GL(d, C) such that I ui; 5 a and

I < a is compact in

det u GL(d, C). A n R(G) is also compact. Hence the holomorphically convex I

hull of K is compact, proving that G is holomorphically convex, and completing the proof that G is Stein. REMARK.

It may be shown that if G is a closed subgroup of GL(d, C),

G stable under -r, then H = G n U(d) is maximal compact in G, and H is a real form of G. Let A be the Casimir operator of S, and put

0 = -4,2 :E c?.

ENVELOPES OF HOLOMORPHY IN LIE GROUPS

313

Both 0 and V are in the center of the universal enveloping algebra of H or G. We shall regard A and V as differential operators for the manifold H by viewing the elements of the Lie algebra as left-invariant vector fields. Both operators are then formally self-adjoint.

Let h -- r(h) now be an irreducible representation of H, inducing representation t) -± r(b) of ). Restricted to Cam, the representation remains irreducible, since the elements of the center of SS are all represented by

scalar matrices. While restricted to (E, the representation is simply the representation of (E induced by a representation of the torus C. The representation r of S2 is completely determined by its restriction to S and Q, but not every representation of S? gives rise, of course, to one of H.

A representation r of C is completely determined by a - tuple K = (al, a2,

..., a,,) of integers, namely: r (exp

exp (27ri V

On the other hand, a representation of

if S is of rank 1, is determined

by an 1 tuple L = (bl, b2i ..., b) of non-negative integers as follows. Every irreducible representation is determined by its highest weight w, and there exist l fundamental weights wl, w2, ... , w, such that every highest weight w is of form

w = I biwi. Thus to every irreducible representation of H we may assign uniquely a pair (L, K) consisting of an l-tuple of non-negative integers, and a -tuple of integers. We shall parametrize the representations of H by all such pairs, and remind the reader to suppress in ensuing computations the pairs which do not correspond to any representation. Now let h --* r(h) correspond to pair (L, K). First of all we see that Vr(h)

av)r(h) = X(K)r(h)

and as a straightforward computation shows Or(h) = r(A)r(h), where r(A) is the matrix representing A in the extension of the representation to the universal enveloping algebra. r(A) is clearly a scalar matrix, r(A) = 77 identity, where 77 = q(w) is a constant depending on the highest weight. 77 has been computed by Freudenthal (see [2]) as follows. Let p = i sum of the positive roots of H. Then A(w) = (co, w) + 2(w, p),

where the symmetric bilinear form ( , ) is positive-definite. If w = biwi, L = (b1, b2, ... , b,), then we denote A(w) by A(L) and note that ,1(L) is an

314

O. S. ROTHA US

inhomogeneous quadratic form in the b, whose part of weight 2 is positive-

definite. Moreover, it is known that A(L) >= 0, and A(L) = 0 only for L = 0. The degree d = dL of the representation corresponding to the weight w fl( + W)' where the products are taken has been given by Weyl: d = fl( over the positive roots a. d is a polynomial in L.

Let D be an H-stable domain in G. Then E = 7r(D) is a domain in G/H, and D = 7r-1(E). We call E convex if it contains the geodesic segment connecting any two of its points. D is called convex if or(D) is convex. If E

is an arbitrary domain in G/H, the convex hull E of E is the intersection of all convex domains containing E. The convex hull of D is defined to be ar -1(-(D)).

The convex hull of E may be constructed as follows. Let E,i}1 be the union of all geodesic segments connecting points of E,, and put E0 = E. Then the E,, are an increasing sequence of open sets whose union is E. There is an analogous construction for the hull of an open, H-stable, set in G. Now let f be a real function defined on G/H. f is called convex if for any geodesic segment x(t), 0 < t 1, we have

f(x) < (1 - t)f(x(0)) + tf(x(1)). If f is defined on G, we say f is convex if f(gh) = f(g) for h c H, and viewed as a function on G/H it is convex as above. Clearly, if f is a continuous convex function on G, the set {g If(g) < l} is a convex domain. Our first basic result on convexity is as follows. Let r be a representation

of H in U(d) and R its extension to G. Let A be an arbitrary complex matrix of dimension d. By A* we denote the conjugate transpose of A. Then LEMMA 7.

f(g) = tr AR(g)[AR(g)]* is a convex function on G.

It is clear that f(gh) = f(g) for h e H.

Now let A > 0 and 0 < p < 1, q = I - p. By the inequality of the arithmetic and geometric mean A

q +pA.

If B is a Hermitian definite matrix, then B = exp C, for unique Hermitian C, and we denote by Bt the matrix exp tC, for any real t.

The inequality just given implies that qI + pB - B1' is Hermitian definite. Thus if P is also Hermitian positive-semidefinite, we have

trPB1 1} - R are semigroups under multiplication. Finally suppose R contains an annulus adjacent to the unit circle. THEOREM 1. There is an automorphism T of a semisimple commutative Banach algebra for which a(T) = R. PROOF. It is easy to see that R = U Rn, where each R. satisfies the requirement just given and, further, OR,, is rectifiable. If T,, can be constructed for Rn, then T = + Tn on the direct sum of the algebras will have a(T) = R. Thus we may assume OR is rectifiable. The annulus that R contains separates the plane into two components L10 and Q., containing 0 and oo, respectively. Let Fo = OR n S2o and r = OR n om. 319

STEPHEN SCHEINBERG

320

Let A be the family of all bounded analytic functions on R. The multiplication on A will be the Hadamard product; each member of A has a Laurent series on the annulus: (, anzn) * (> bnzn)

=

anbnZn.

To see that A * A si: A, write

f=fo+fW+ -M o

for each f in A. Clearly fo is analytic on Ro = 0-0 v R and sup If, 1 < Rp

const sup If 1. The analogous statement applies to f. R

Now f * g = fo * go + f. * g. and a standard (easy) computation shows that fo * go(z) = (27ri)1 fry fo(z/w)go(w)w-1 dw and f. * g.(z) _ (2lri)-1 rroff(z/w)gc(w)w-1 dw, with suitable orientation on F0, F. In these integrals go(w) and g.(w) are defined a.e.; the semigroup condition is just what is needed to ensure z/w c Ro when z c Ro and w c P., and similarly with 0 and oo interchanged. It is now evident that A becomes a Banach algebra when If 11 is defined

to be (a suitably large constant) sup If 1. It is also clear that A is semiR

simple, since the homomorphisms 0k(:E a,Zn) = a,, separate the members

of A [i.e., MD =

(27ri)-1

fizi_1f(z)z(-"" dz]. Let T be the shift f(z)-±

zf(z). T is obviously an automorphism of A with a(T) = R.

Examples with disconnected interior may be obtained by a slight modification of this construction. Let G be an open subset of {2 < IzI < 3}

with rectifiable boundary such that 8G meets {Izl = 2} in a totally disconnected set of positive length. Let A be the family of all bounded analytic functions on R = {1 < IzI < 2} V G such that the boundary values on the two parts agree a.e. on the common boundary. A is complete with respect to uniform convergence and is an algebra under Hadamard

product, since every member of A * A is analytic on {l < I z I < 4}. As before, If 11 = const lif 11 makes A into a Banach algebra. The condition on the boundary means that if f vanishes on {l < Izl < 2} then it must vanish on all of R; thus A is semisimple. As before Tf(z) = zf(z) defines an automorphism with spectrum R. This procedure can be repeated indefinitely and, together with the previous construction, will produce fairly complicated sets. If a(T) properly contains the unit circle, must it contain an annulus, must it be the closure of its interior, and must it have interior?

.

AUTOMORPHISMS OF COMMUTATIVE ALGEBRAS

If T is an automorphism of a semisimple commutative

THEOREM 2.

Banach algebra and Tn PROOF.

321

I (all n), then the spectrum of T is connected.

Via the Gelfand map we may assume that the algebra A in

question is a (not necessarily closed) subalgebra of C(X), that lif II > If 11

and that T acts as a homeomorphism of X: Tf(x) = f(Tx). Since o(T) contains the unit circle (see [2]), if it were disconnected it could be separated by a rectifiable path y lying in the interior or in the exterior of the unit disc. By taking T in place of T if necessary, we may assume that y lies in {Izl > 1}. Put S = (2iri)-1 fy A(A - T) dA. Then a(S) = a(T) n (inside of y)

and a(T - S) = a(T) n (outside of y). For any f c A, x c X, Sf(x) _ (27,i)-I fY (1 - A-1T)-1f(x) dA =

(2lri)-1 fr

[(2,i)-1 fy A-" dA]. This is justified since

A-nf(Tnx) dA = J.f(TnX). a

is uniformly > 1 for A E y. If y surrounds 0, this string of equalities gives Sf(x) = f(Tx), implying I r1I

S -- T. If y does not surround 0, we have Sf(x) = 0, implying S = 0. Either case contradicts the assumption that a(T) meets both the inside and outside of y. 2.

Given a sequence an > 0, let Aa = 1 X = 2 xnz L: Ilxll =

l

IXnIan < 001.

1

Aa is a Banach space isomorphic to 11. It becomes a Banach algebra under formal power-series multiplication if an+m f, and n Jim

so from the continuity at yo, continuity we obtain

n

x o yn o y-1 o yo = x o yo. Then by left

limxoyn = Jim x0ynoy-loYo oyoloy n

n

= x0'0 0Yo 10J = x0J and therefore (2) is continuous at y. We have now that (2) is continuous in each of the variables separately, and therefore by the Ellis separate con-

tinuity theorem (or in this case a fairly easy category argument) (2) is continuous on Xf x S. As stated previously, (i) and (ii) are established.

.W-ALMOST A UTOMORPHIC FUNCTIONS

349

Statement (iii) is obvious and (iv) is true because the induced action of S on Bf has norm compact orbits. Each orbit is Sh for some h c Bf. Our reformulation of Bochner-von Neumann almost periodicity is THEOREM 1.

A necessary and sufficient condition for f e ' to belong to

.sad is that f e BT.f whenever Ta f exists. PROOF. If f c .Q1, and if Taf exists, then it exists uniformly. Moreover, as is noted in (1) lim II Tj(t - an) - f(t) 11 = 0. That is, f c BTaf. For the n

converse we set S = Xf in Proposition 1 and note that properties (a) to (e) are trivial.

We will introduce a slightly more general terminology. Let F(t) _

[fi(t), f2(t), ... ] be a vector-valued function on T whose coordinate functions belong to V. There may be a finite or an infinite number of coordinates. BF will be the T-algebra generated by the coordinates and symbols like Ft, TaF, and XF have obvious definitions. If F and G are two such functions, F E B. will mean BF BG. XF is metrizable, and BF is isometrically isomorphic with C(XF) by the usual correspondence. DEFINITION 1.

A function f c ' is ,21-almost automorphic if the totality

Hf = {T _ aTa f I Ta f and T_ aTa f exist} is norm precompact, and if Hf generates a minimal T-algebra. REMARK.

If f is almost automorphic, then f is obviously at-almost

automorphic. Let (T, X) be a minimal flow where X is compact metrizable and T, as

usual, is Abelian. For each x c X define E(x) to be the set of y c X for which there exists a sequence a = {an} such that lim anx = y' exists and n lim - any' = y. We recall the following n

THEOREM A. (See [7], Theorems 1.1-1.2.) With notations as given previously the relation y ' x if y c E(x) is a closed, invariant equivalence relation. If Y = X/' , the natural flow (T, Y) is minimal and equicontinuous. Moreover, - is contained in any relation with the latter property. Now let f be si-almost automorphic, and let B be the T-algebra generated by Hf. Since Hf is norm precompact, it has a dense subset fl, f2, ... , and we set F = (fl, f2, ... ). Clearly, B = BF. By assumption B is minimal, and therefore (T, XF) is a minimal flow. In what follows a bitransformation group shall be a triple (T, X, S) which satisfies properties (i) to (iii) of Proposition I [with (T, X) minimal]. This terminology is due to Ellis. THEOREM 2.

Given f e ' the following statements are equivalent:

(i) f is d-almost automorphic.

WILLIAM A. VEECH

3 50

(ii) There exists F = (fl, f2i ...) such that f c BF and F E BT _aTaF whenever the limits exist. (iii) There exists a bitransformation group (T, X, S), a point x0 E X such

that Sx0 = E(xo), and p c C(X) such thatf(t) = p(txo). PROOF.

(i) - (ii). We associate F with the sad-almost automorphic

function f just given. To see that FE BT_T,F when the limits exist we shall apply Theorem A to (T, Xf) which, since f is minimal, satisfies the hypotheses of that theorem. Since the coordinates of Fare dense in Hf (= E(f)), the existence of T_ aTaF implies the existence of T_ aTah, h c Hf. Secondly,

because - is transitive, we have T_,Tah E Hf, h c Hf. Bf, is a minimal

T-algebra by assumption, and therefore T_,Ta: Hf -> Hf is a norm isometry. Now Hf is closed in Xf because ' is closed, and being norm precompact, it follows that Hf is norm closed (and so compact). An isometry of a compact metric space into itself must be onto, and therefore T_,T, is onto. If fi is any sequence such that T_$TBF exists, then clearly T_8TaFE BF, and afortiori T_,TT_sTsFE BT_,T,F. Iff,, is the nth coordinate function, we may choose /3 so that T_,T,T_aTQf, = f, [since E(,,) _ E(f )]. Therefore f,, c BT _aTaF, and since n is arbitrary, F E BT _ TaF (ii) (iii). We take X = XF, S = E(F), x0 = F, and p(tF) = f(t). The latter function extends to be continuous on X because f E BF. We verify properties (a) to (e) for F and S. (b) and (c) are trivial, and therefore, by Lemma I and Theorem A, (a) is true. (e) is a consequence of the transitivity of - and (d) is a consequence of symmetry. Now (iii) is a consequence of Proposition 1. (The change from f to F is purely formal.)

(iii)

(i).

Because Sx0 = E(x0), we have Sf = Hf. Moreover, if

B = {g c 'Ig(t) = G(tx0), G E C(X)}, then B is a minimal T-algebra, and SB si: B. Thus, in particular Hf si B, and Hf generates a minimal T-algebra. The theorem is proved. We conclude with some further remarks.

Let f be d-almost automorphic, and suppose E(f) = Xf. Then f c -27, and f is almost automorphic. That is, E(f) = {f}, which, coupled with the

preceding, implies f is constant. It follows that if f is a nonconstant d-almost automorphic function, Xf/- is not trivial, and Bf contains nonconstant almost periodic functions. If f is sad-almost automorphic, and if (T, X, S) is as in Theorem 2(iii), there is a natural quotient flow (T, XIS). If p: X -* XIS is canonical, and

if yo = px0, then E(x0) = Sxo implies E(y0) _ {yo}. That is, yo is an "almost automorphic" point, and (T, X) is a group extension (Furstenberg's terminology) of an almost automorphic flow. Conversely, if (T, X) is a group extension of an almost automorphic flow (T, Y), and if yo E Y is almost automorphic, then E(x0) = Sx0 if px0 = yo. The sat-almost auto-

d-ALMOST A UTOMORPHIC FUNCTIONS

351

morphic functions are precisely the orbit functions, f(t) _ g(txo), coming from group extensions of almost automorphic flows, subject to the restriction that pxo be almost automorphic.

If f is Q-almost automorphic, and if Taf is d-almost automorphic whenever the limit exists, it is possible to choose (T, X, S) so that (T, XIS) = (T, X/-). Then, in the terminology of [5], f is almost periodic with respect to the almost periodic functions. This result is similar in spirit to the state-

ment that f is almost periodic if and only if Taf is almost automorphic whenever the limit exists (see [6], Theorem 3.1). Using Theorem 2 it is easy to see that a-almost automorphic functions on T form a T-algebra. For example, if f and g are d-almost automorphic, and if Fand G are associated as in Theorem 2(ii), set H = (f1, 911f21921 . . . ) (with an obvious convention if one or both of the sets is finite). Then f + g, c BH, and H E BT_aTaf whenever the limits exist. A similar argument

shows the class to be closed under uniform limits. Self-adjointness and translation invariance are trivial. When T is the group of real numbers in its usual topology, the BebutovKakutani theorem (see [4]) implies that a separable minimal T-algebra has the form B1 for some f c W. Thus, if g is d-almost automorphic, there exists f E' with g c B1 and f e BT _aTar whenever the limits exist. We do not know if this is true in general. The functions f E'' for which f c BT _aTaI whenever the limits exist are precisely those functions for which (T, X1) is a (minimal) group extension of an almost automorphic flow. One can carry out a "Fourier analysis" similar in spirit to that of [5] for the .sad-almost automorphic functions. UNIVERSITY OF CALIFORNIA BERKELEY

INSTITUTE FOR ADVANCED STUDY

REFERENCES 1. BOCHNER, S., "A new approach to almost periodicity," Proceedings of the National Academy of Sciences of the U.S.A., 48 (1962), 2039-2043. 2. , "Beitrage zur Theorie der fastperiodischen Funktionen, I," Math. Ann., 96 (1926), 119-147.

3. ELLIS, R., "Locally compact transformation groups," Duke Math. J., 24 (1957), 119-126. 4. KAKUTANI, S., "A proof of Bebutov's theorem," Journal of Differential Equations, 4 (1968), 194-201. 5. KNAPP, A., "Distal functions on groups," Trans. Am. Math. Soc., 128 (1967), 1-40. 6. VEECH, W., "Almost automorphic functions on groups," Am. J. Math., 87 (1965), 719-751.

7. -, "The equicontinuous structure relation for minimal Abelian transformation groups," Am. J. Math., 90 (1968), 723-732.