Causal structures in lie groups and applications to stability of differential equations

CAUSAL STRUCTURES IN LIE GROUPS AND APPLICATIONS

TO STABILITY OF DIFFERENTIAL EQUATIONS

BY

STEPHEN MARK PANEITZ

Bachelor of General Studies (BGS) University of Kansas,

1976

Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

May, 1980

@)

Stephen Mark Paneitz

Signature redacted

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1980

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-OF TFCHNCLOGY

JUM 11 1980

CAUSAL STRUCTURES IN LIE GROUPS AND APPLICATIONS TO STABILITY OF DIFFERENTIAL EQUATIONS

by

STEPHEN MARK PANEITZ

Submitted to the Department of Mathematics on May 5, 1980 in partial fulfillment of the requirements for the degree of

Doctor of Philosophy.

ABSTRACT

According to results of Kostant, a real simple Lie algebra &j, has an invariant convex cone if and only if al-is Hermitian symmetric. We classify and explicitly describe such cones in the classical algebras, specifically, all such cones in sp (n,JR) and all open or closed invariant convex cones in su(p,q), so*(2n), and so(2,n).

The universal covering group G receives a left- and

right-invariant causal structure from any such cone. In the tube-type cases G is shown to be globally causal under

all such structures, and likewise for certain of the structures in the other cases.

The open cones are found to consist of elliptic

elements, each within a unique maximal compact subalgebra. This mapping of positive elliptics -to maximal compacts

(K4hlerization) is shown to be real-analytic. The same results hold for infinite-dimensional analogues of sp(n,2) and so*(2n). In these two cases the Cayley transform is a causal mapping of an open subset of

y

into the group. Some of the mapping's other properties are collected.

Let 6 = (g E Sp(n,tR): g within a unique maximal compact subgroup3, an open set having ₂n connected components 0". . The inverse images of the S. in the universal covering Sp(n, ) correspond naturally to the well-known regions of strong stability of linear periodic canonical differential equations. Two stability criteria are found for certain such equations of positive type, applicable in finite or infinite dimensions. One involves the operator bounds of the coefficients, the other an

invariant Hilbert-Schmidt bound.

Thesis Supervisor: Irving E. Segal

To the memory of

my grandfather,

Ernest W. Engelkemier

(1895 1978)

Let us then provisionally adopt the unispace cosmos, and seek to analyze free propagation over very long times, and its effect on the measurement of frequency of light.

I. E. Segal

Let us go then, you and I,

When the evening is spread out against the sky... T. S. Eliot

A CKNOWLEDGEMENTS

It is a pleasure to thank Professor Irving Segal for his abundant instruction and encouragement during the last two years, and especially for many inspiring discussions of mathematics and physics. I hope the latter can continue and soon become a little less one-sided.

Formal thanks are due the National Science Foundation for the financial support of a fellowship during my first three years here, and the M.I.T. Mathematics Department for affording opportunities to gain some teaching experience, and secondly for an R.A. this term. I also owe a great deal to many professors at the University of Kansas; however, the list would be too long, and I would rather thank them in person anyway.

Marge Zabierek deserves praise for a beautiful typing

job and such a cheerful disposition. I'm glad she enjoyed doing this thesis so much, especially the tables and graphs.

Most of all I thank my parents, Joanne and Marvin Paneitz, for their love and constancy and everything else. I thank David Goering and Holly and Warren Schoming for the friendship and "spiritualizing" of nearly ten true years, and not only summers. My Watertown housemates, Daniel

"Jackson" Friedman and John "Johnson" Lutostanski, deserve public recognition for skillfully instructing an erstwhile raving liberal. May they always be free to choose.

The MIT math dept. has been a friendly and intriguing place to work and converse, and it will not be easy to

leave. And I have appreciated the Pleasant Folks of Cambridge.

Do I dare

Disturb the universe?... I have measured out my life

with coffee spoons. -Prufrock again

6

TABLE OF CONTENTS ABSTRACT. . . . .. DEDICATION. . . . .. ACKNOWLEDGEMENTS. . . . . INTRODUCTORY REMARKS. . . . .. Page . .

2

.9.

3

.9.

4

. .

8

CHAPTER 1. 1. 2.

3.

4.

5.

CHAPTER II.

6.

7.

8.

9.

CHAPTER CHAPTER III. 10. 11. 12. 13. 14. IV. 15. 16. 17.

18.

Preliminaries . . . .

Preliminaries on Convex Cones . . . . Existence of the Minimal Cone . . . . Structure of Hermitian Lie Algebras . Certain Classical Linear Groups . .

Classification of the Orbits. . . . .

Invariant Convex Cones in sp (n,JR) R .

Conventions for sp(n.,R) . . . - - -

--Conjugacy ofPositive Elliptics in sp(V). Uniqueness of Causal Cones in sp(n, R)

Classification of Cones in sp(n,JR) Invariant Convex Cones in su(p,q) .

-Conventions for su(p,q) - - - - . - .

Conjugacy of Positive Elliptics in u(p,q) Minimal and Maximal Causal Cones

in su(p,q) .*.0.0.0.0.*.*.a.0.*.*.*.0.0

Orbits in ~C of su(p,q) . . . ..

Classification of Open and Closed Cones

in su(p,q),. .0.0.0.0. . .. *.0. . . . .

Invariant Convex Cones in so*(2n) .

Conventions for so*(2n) . . . . Conjugacy ofPositive Elliptics in so*(>) Noncompact Convexity in so*(2n) . .

Minimal and Maximal Causal Cones

in so*(2n). . . . - . . .11 . 12 9 16 - 21 . 25 - 36

-. 36

39

42

.45

55

55

58 61 65 70 81 81

84

87 93 -. . . . -0

Page 19. Classification of Orbits and Cones

in so*(2n) . . . 97 CHAPTER CHAPTER CHAPTER V. 20. 21. 22. vi. 23. 24. VII. 25. 26. 27. 28. CHAPTER VIII. 29. 30. CHAPTER Ix. 31. 32.

33.

Invariant Convex Cones in so(2,n). . -. 102* Conventions and Preliminaries for so(2,n). 102 Noncompact Convexity in so(2,n) . - *- . 107 Classification of Orbits and Cones in so(2,n) . . . *. . . . - - -*112*

Global Causality of the Covering Groups. 119 Definitions and Isomorphisms . . .

119

Causal Actions on ghilov Boundaries. . . 123

KMhlerization in Causal Lie Algebras . . 132

Uniqueness of the Complex Structure. . . 132

Generalizations to Unbounded Operators . 141 Analyticity of Kghlerization in sp(r). . 145 An Example and Analytic Continuation: SU(1,1). . . . .0 . . 0 .0 .0 .9 . 147 The Cayley Transform . . . .

155

General Properties . . .

155

Cayley Transforms for Causal Groups. . . 158

Applications to Differential Equations . 162 Preliminaries. . . .

162

Regions of Stability . . . .

164

Two Stability Criteria . . .

168

. . 0 0 0. . . . 179

REFERENCES.. BIOGRAPHICAL NOTE - AFTERWORD . . . 182

INTRODUCTORY REMARKS

The main ideas of chapters one to five on the classi-fication of invariant convex cones in the classical

Hermitian symmetric Lie algebras, seem to be the following.

1) We use the orbit classification of Burgoyne and

Cushman [ 2] from time to time, particularly to deal with the boundaries of the cones and so(2,n) . These are the only really technical results we quote, and the above

authors claim they "only need quote one result, Sylvester's theorem on the signature."I Also, the form of the orbits in the open maximal cones on su(p,q) could have been determined by the idea of Theorem 14.2, just as was done

for the corresponding open maximal cones in so*(2n) (Theorem 17.3), independently of classification. 2) The quadratic mapping

v E V -+ 1(.v,v) E r*

where V is a fundamental representation space and $ an invariant symplectic form (and a rather different

expression, T(-.e,f) , for so(2,n) ; see section 21) is shown to single out orbits with particular positivity properties (Theorems 6.1, 10.1, and 15.1), connecting the geometries of the Lie algebra and the space V, and is

explicitly described cones (Theorems 8.2, 12.3, 18.2).

3) The argument of sections 7, 11, and 16 seems well-known but appears in other context, such as the Klein-Gordon

equation in Minkowski space

Z cp + my2 = 0 m > 0

In section 7 the form &(-,.) is the energy norm

> 3 12 + _m21c2 + ftp d

and gl(-,-) is the Lorentz-invariant norm

L

f43c

+23

where C = (m2_)1/4 and say ep,/p E C0(I3

4) The "noncompact convexity" theorems 14.2, 17.3, 21.3, and possibly 9.2 for sp(n,JR) (a "discrete" version), are in the direction of generalizing to noncompact groups Horn's theorem for Hermitian matrices (see section 14) and

Kostant's generalizations to maximal compact subgroups of semisimple Lie groups, within [16]. The classification of the open and closed invariant convex cones and all

invariant convex cones for sp(nR) (Theorems 8.1, 14.4,

19.3, and 22.2) is a fairly immediate consequence, just as

Horn's theorem easily classifies the invariant convex cones in u(n)

Chapter six is a fairly direct application of the preceeding chapters. The uniqueness and analyticity of

Kdhlerization (for sp(V) , at least), shown in chapter

seven, 25.2, 25.4, and 27.2, are valid in infinite

dimensions and, like chapter nine, suggested by problems in quantum field theory. Chapter eight is a rather

non-systematic collection of properties of the mapping

@: X MI+X I - x

of the Lie algebra to the Lie group. One remark that should have been made there, is that all group elements in the

range of the Cayley transform restricted to the cones 01 , described in Proposition 30.3, commute with a unique

complex structure, by Theorem 25.2. The proof of the

latter applies equally to SO*(V) , defined in section 16. One comment on notation: A C B means A is contained

CHAPTER I: Preliminaries.

1. Preliminaries on Convex Cones.

We need some standard facts about convex cones as, for example, in [6 ], to which we refer for proofs. In this section all vector spaces are finite-dimensional.

Definition. Let E be a real vector space. C;E is convex if Xx+ (1-X) EC whenever x,yEC and XE [0,l . C Q E is a cone if xEC implies XxEC for all X>0 . A non-trivial cone is a nonempty cone C satisfying

.01 / C / E .

Elementary properties of convex cones are, for

example: the closure or interior of a convex cone is also a convex cone. Less intuitive is the following

Lemma 1.1. A convex cone which is dense in E is equal to E

Let be the vector space dual to E . There is a

tseparating hyperplane _{theorem" for convex cones.}

Lemma 1.2. If C is a closed convex cone in E and x ( C,

then there exists f E E such that f(x) > 0 and f(y) ! 0

for all y E C .

Corollary 1.3. A convex cone, not all of E , is in some

It is convenient to introduce a real positive-definite scalar product <.,-> in E to state the duality theorem for cones, but this is not strictly necessary as E E canonically.

Definition. Given any cone C in E, let C* = [y: <y,x> s 0 for all x EC). C* is a closed convex cone, called the dual cone of C.

Theorem 1.4. For any cone C in E, (C*)* is the closed convex hull of C. In particular, if C is closed and convex, (C*)* = C .

Proof. It follows from the definitions and Lemma 1.2.

2. Existence of the Minimal Cone.

In this section we recall the results of Kostant on the existence of invariant convex cones in semisimple Lie

algebras.

Definition. A causal cone in a real Lie algebra is a non-zero closed convex cone C in o invariant under the adjoint group of and satisfying, C f - C = (0).

Proposition 2.1. Any non-trivial invariant convex cone C in a simple real Lie algebra satisfies C An- C C o.

Theorem 2.2 (Kostant). Let G be a semisimple Lie group acting on a real finite-dimensional space V. Assume that K is a maximal compact subgroup of G. Then there exists a non-trivial G-invariant convex cone C in V satisfying C n - C;UJ() iff V has a non-zero K-invariant vector.

Proof. The proof is given in [19], but the ideas are useful later so we repeat them here. Let C be a cone with the given properties. By Lemma 1.2, _fEEV such that

f(x) ,0 x E C and f(z)>0 for some z E C . Then

w =

U

k(z)dk

is K-invariant and f(w)> 0 so w/0

Conversely, let w/0 be K-invariant. Let k be the Lie algebra of K and o = k +p the Cartan decomposition. As G is a matrix group, its complexification Gc acts on

V = V+iV, and any subgroup Gu corresponding to k+ip is compact. We can assume K cGu , and Gu leaves invariant some complex Hilbert structure <-,-> . All XEk+ip are

skew-Hermitian on Vc , so all X E p are Hermitian and

g E exp p positive-definite Hermitian.

Any gE G can be written uniquely as (expX)k for

XEp, kEK. Thus <gw,w>=<(expX)w,w>>0. Now <guv> =

<u,G(g~ )v) VgEG , u,vEV , where 8: G->G: (expX)k-+

(exp -X)k is the Cartan involution corresponding to

e: X+Y->X-Y for XEk , YEp . Letting C denote the

Corollary 2.3. Let o be a semisimple Lie algebra and G the adjoint group of o. Let K be any maximal compact subgroup of G, with Lie algebra k. Then there is a non-trivial G-invariant convex cone C in satisfying

C f

-

C= (O} if k has a non-trivial center.

Proof. Take V= in Theorem 2.2. As K acts irreducibly on the p's of the simple components of o , any non-zero K-fixed vector must be in k.

It is clear that any invariant convex cone in a semi-simple & is contained in the direct sum of invariant cones in the simple factors, and we restrict to the simple case from now on. It is well known that if cg is a real simple Lie algebra the dimension of the center Z(K) is either 0

or 1. Thus by Proposition 2.1, a simple admits a non-trivial invariant convex cone iff dimZ(k) =1. The inte-gration argument above shows that in this case there are unique minimal causal cones tC . By the simplicity, the positive-definite K-invariant form used in the proof of

Theorem 2.2 was on oBe(-.,)=-B(-,.) up to a scalar, where B is the killing form on o. We use later the fact

that B,(X,Y) _{0 V X,Y E 7 0}

We see that no compact or complex simple Lie algebra admits a causal cone. By the classification the only

algebras that do are sp (n,BI) (n 1) , su(p,q) (pq2 l) so*(2n) (n z 3) , so(2,n) (n -3) , and two exceptional

algebras. _{We list information about the dimensions of these} algebras below, using the notation of [ 9]. Recall that char o= dimp - dim k where o=_k +p is any Cartan decom-position, and rank is the dimension of any maximal abelian subalgebra of p.

2 2

sp (n.,EI) dim k =n2, dim p =n 2+ n; ranko n ;charoflO V n

su(p,q) dimk = p2 +q 2 - 1 , dimp = 2pq;rankyq;charT>0 when p = q-, charm=OC when p=qa +1 , and char < 0 otherwise.

so*(2n) dimk=n , dimp=n 2 - n ; rankj,=[n/2] ; char < 0 V'n

so(2,n) dimk=jn(n-1) +1ldimp=2n ; rank%.= 2;

char o>0 for n=3,4 and charog<0 otherwise. e III (a real form of e6 ) dim k = 46 , dimp = 32 ; rank = 2 .

e VI (a real form of e ₇ ) dimk=79, dimp=54 _{; rank1= 3 .}

A glance at the list on p. 346 of [9 proves the following

Corollary. A complex simple Lie algebra oy admits a non-compact real form having a causal cone iff the adjoint group corresponding to a compact real form of ct is not simply connected.

Let c be one of the Lie algebras in the list in Section 2, and = k + p a fixed Cartan decomposition. Let

03

be its complexification, and setk = k + ik pC = P +ip . Let c be the one-dimensional center of k

h a maximal abelian subalgebra of k , and set hc = h +ih Then cC hO chc * We have k = k 9c , e where

k = [k,k] is a compact semisimple Lie algebra.

'-s5

We recount the notation and analysis in [ 9],

p. 312-16. It is shown there that hC is a Cartan

C

subalgebra of q . Let denote the s-et of roots of

C with respect to hC . (A root is always nonzero.) If

cc

a.Et,

is the root space decomposition, then each d satisfies either Ca kc , in which case the root is called compact, or C pC , in which case a. is called noncompact. We have the decompositions

kc =hc + Ec

; running over the compact roots and $ over the noncompact roots.

of the killing form to ih , a real positive-definite inner product. (For our purposes, any positive multiple of B would serve equally as well.)

Choose compatible orderings in the real duals of ic and ih . V aE6 take HM Eih such that a(H) = 2 and B(H, -) proportional to a . Also take E E

such that i(EB+E_) , E -E_ E _k + ip , and

[E ,E_] = H V a E 6 . Let G+ be the set of positive and noncompact roots. As in [15] V a E Q+ set

X = E + E

y =-i(E-E )

These vectors are a basis for p . We have

[H ,X

3

= 2iYM

(1) [H .,Y = -2iX

[X ,Y I = 2iH

for all a EQ+

Let 6b+ C be the positive roots with respect to the ordering, and let L+= a 16...,a) be the simple positive roots. The z. are a basis for the dual of ih and any a. EA+ is a nonnegative integral linear

noncompact and positive.

The following is well known.

Proposition 3.1. a(iZ) = 0(iZ) VaS E Q+ , and all but exactly one of the a E & are compact.

i 0

Proof. As [p,p] c , if a,$ are noncompact roots,

0= [Z,[E,E0]] = (a +$)(Z)[BLEE $ .

Thus a.+ C

E implies (a. +s)(Z) = 0 . We recall the

lemma on root systems ([13, p. 45 1): B(a.,$) < 0 implies a +s is a root (so B(a,$) > 0 implies a.- is a

root). Therefore

Ia(Z)I

/7$(Z)f

implies a J$ .

Thus one can put the noncompact roots into mutually perpendicular subspaces, each singled out by a particular value of j.(Z)f . As shown above, root spaces corre-sponding to different values must commute (and recall

[E ,E$] = 0 for a., E Q+)*

We wish to show that each compact root a is in one of these subspaces. Now there must be some noncompact

$ such that is a root; otherwise [E ,p] = 0

and [X E k: [X,p] = 0) would be a non-trivial ideal in q , contradicting simplicity. Thus a + $1= $2 for

noncompact 51,42 'M = $2 - l ,is a root, and 5 and

$2 must be in the same subspace.

The root system is then decomposed into non-trivial mutually orthogonal systems, contradicting simplicity,

unless all a.(Z)| =

IS(Z)I

for a,S noncompact. Assume then that a(iZ) = 1 VaM EQ+ . Let Sl'' n be the noncompact roots in +. Any noncompact root involves one but no others, as ( +)(iZ) = 2 . If y,8 are noncompact and involve different $ 's , then

[E ,E5] = 0 , and the noncompact roots are again

partitioned non-trivially unless n = 1

If Q(iZ) = I V z E Q , we have [Z,Ej = -iE

and [Z,E ] = iE , so [Z,X ] = Y and [Z,Y ] = -X

V a E Q+ . Thus ad Z: p -+*p is a complex structure on

p.

In [9 ] it is shown that there exists a set SC Q+ such that

t = RX

5EE ₀

is maximal abelian in p , and Y t 5 are not roots

V Y,6 E S . Let h- be the real span of iH , z E 0

Then h C h and these iH are orthogonal, so

dim h- = dim ax . Let h+ be the orthocomplement of h in h with respect to the killing form. In [15] it is

shown that h + is the centralizer of or in h , and that the complexification of h + e So is also a Cartan

subalgebra of'

2.

It may be shown that the centralizer of orc in k , usually denoted ' , is spanned by h+ and all i(EaM+E).9 , Ea- E as above, where a is

compact and iH. E h+

We introduce one more bit of notation; as is standard, let Z0 =-(i/2)3 H E h . Then

[Z - Z0,Xa] = Y - [Zo,X] = Y -Y =0 ,

so Z- Z0 E h+ , and we have decomposed

Z = (Z-ZQ + Z0 E h+ 9 h~

Only in the symplectic case is h+ = 0 , but often Z =

even if' h+ /0 0. Whether Z = or not depends only

on and not on the choices we have made, as is proven in [15]:

Proposition 3.8. Z = Z iff Z is contained in a three-dimensional simple subalgebra of'o .

Finally, we compute the action of A = exp or% on Z . (The action of A on h is used a great deal in

the next several sections.) Let X = t aX Eatr. The

relations (1) imply ME ₀

Ad(exp X)(Z) = Z - Z0 + Ad(exp X)(-(i/2) I H)

M ES ₀

= Z - Zo - (i/2) (cosh 2tZ)H

MES

- (1/2)

3

(sinh 2t)Y.

Corollary 3.2. Any invariant convex cone in containing Z contains all

c₀(Z - Zo) +3 c(-iHM) E h

for c₀ ,c satisfying 0 < c₀ < 2 min c

Proof. Average over positive and negative t in the above.

4. Certain Classical Linear Groucs.

To classify some of the invariant convex cones in the

classical Hermitian Lie algebras we need information about

their orbits, and the work [ 2] of Burgoyne and Cushman, from the point of view of the lowest-dimensional repre-sentation, is ideal for our purposes. The exceptional algebras will probably require a more abstract approach,

which is currently being suggested by the similarities and analogies seen in the following treatments of the four classical series.

In this section we review their description of the classical linear groups, and in the next describe their

classification of the orbits. It would not simplify matters at this stage to restrict to the Hermitian case, and we

refrain from doing so until the end of section five.

Let V be a finite-dimensional complex vector space, always non-zero. If A: V -+ V is linear, det A denotes the usual complex determiant. Let G = GL(V) . Let a be any conjugate-linear operator a: V -+ V such that a = I. a induces an automorphism of G of order two.

We consider the real groups H aT= (g E H: ga = ag)

for H = G and certain special proper subgroups H of G

Such a and a' are called equivalent with respect to H if a= ga'g for some g E H. If this happens H aTand Hat are isomorphic.

The two cases a = I induces quite different structures on V .

(a) a2 = I . V = [v E V: av = v) is a real form of V ,

and any g E H leaves V invariant.

a7 a

det g E B for any g E H . 2

(b) a = -I . In this case V must have even dimension,

and V receives the structure of a

= (a+bi+cj+dk: a,b,c,d Ew1) denote the quaternions as usual. If a +Oj E 0 for

a,S E T and v E V , define (a+Oj)v =

Mv+Sa(v) . (Note aj = ji for a E .

We use the anti-automorphism

a+bi+cj+dk E a

- a+bi-c j+dk E

%

denoted X -+ X ; it satisfies (p)q = ₄qpq

V Xp E Q . NoteXq is not necessarily real, and usually Xq / x .

Again, det g EB for any g E H Real forms of GL(V) .

The first case involves no a ; we assume only a

nondegenerate Hermitian form T*(-,-) on V , and define

G = (g E G: *(g-,g.) = *(-,-)} . fdet g = 1Vg E G

and G is connected. The subgroup of det g = 1 is connected and isomorphic to an SU(p,q) .

(a) a2 =I. All a are equivalent with respect to G G e GL(n,2R) where n = dim V . G has two components, and SL(n,]R) is connected. (b) a2 = -I . All a are equivalent with respect to G

G U*(2n) , where 2n = dim V . G has

two components, and is isomorphic to the

general linear group on a quaternionic vector

Real Forms of O(V,T)

Let T be a nondegenerate symmetric complex-bilinear form on V . We consider only those a as above such that

)= T(7,) . Let H = (g E G: T(g.,g-) = t(,*)l

(a) a2 =I. There are several equivalence classes of the a's with respect to H , depending on the (real) signature of T restricted to V , denoted T+ . H has either two or four components, and det g = -l for each g E H

Each H aTis isomorphic to some 0(p,q) .

(b) a2= -I . Let dimV = 2n. V becomes a quaternionic vector space as above, and defining

T_(u,v) = T(U,V) + T(UaV)j for uv E V

T is a nondegenerate Z-valued form on V

It safisfies

(2) T_( Xu,4v) =XT_(UV)u

and T_(u,v) = T_ (v,u)q for all u,v E V , X,p E Q . There is one equivalence class for a , and '-bases eA exist for V such that T-(e ,e1 ) = [ 5 ]. Each

g E H automatically has determinant

1,

just as for the symplectic case, and

H a- SO*(2n) is connected.

Real Forms of Sp(V, )

bilinear form consider only Let H=(g E

(a) a2 = I

(b) a2= -I .

on V - dim V is necessarily even. We those a such that T(c ,a-) =

-G: (ge,ge)

There is a unique equivalence class for a with respect to H , and H C Sp(n,JR) is

connected. Each g E H a has determinant 1. Again V becomes a quaternionic vector

space, and T is defined on V by the same expression as above. (2) still holds, but now T(u,v) = -T(v,u) , and there are

various equivalence classes for a with respect to H , classified by the signature

of T . That is, there are k-bases (e) such that T(er'es) = t rs Each g E H

has determinant 1, and H r Sp(p,q) is connected.

5.

Classification of the Orbits. (Burgoyne and cushman)

One may designate any one of the groups (real or complex) of the previous section by G(V,a,T) , using the notation there. The Lie algebra of G(V,a,T) , written

L(Va,T) , is just those linear endomorphisms of V

commuting with a and skew with respect to T . However, as in [ 2 ], either T or a may not actually occur in the definition of the group or algebra. Moreover, r may

There is the notion of a type 6, an equivalence class of pairs (A,V) where A E L(V,a,T) , under the obvious notation of equivalence. dim 6 (= dim V for all

(AV) E t) is well defined. The connection between types and orbits is the following

Proposition 5.1. Let A,B E L(VaT) . Then there exists g E G(V,a,T) such that g Ag = B iff (AV) and (B,V) belong to the same type.

The sum of two types is defined to be just the obvious T-orthogonal, a-invariant direct sum. A type 6 is

indecomposable if it cannot be written as the sum of two other types.

Given any A E L(Va,T) , one can write it uniquely as A = S+N , where S,N E L(V,a,T) , S is semisimple,

N is nilpotent, and SN = NS .

Definitions. 1) Let m ' 0 be the unique integer such that Nm /'d0 and Nm+1 = 0 in the above. m is called the height of (A,V) , and the notation ht 6 for any type & is well defined.

2) Let K = ker Nm ; then K2WNV. If K = NV , we say (A,V) is uniform, and clearly one can speak of uniform types.

3) If ht L = 0 call A a semisimple type. A semisimple type is uniform.

There is a natural mapping of uniform types to semi-simple types, defined as follows. Let t be uniform and m=ht&. If (AV) E& put V=V/NV, andfor vEV

put v + NV. Define Acand T on V by

r = Tv , = av , and (5,r) = T(u,NmV) . Since T is nondegenerate on V and (A,V) is uniform, ~ is non-degenerate on V . G(V,a,~ ) and L(Va,~) are well defined, and A E L(TJ,&,Th) . Let

3

denote the type containing (A,v) ; 3 is semisimple. Note i has nothing to do with complex conjugation in T

Remark. G(V,Th) may be a group in a different class than G(V,aT) : if T is complex bilinear and T(u,v) = X' (V'u) X = l , then

Th

satisfies 'r(uv) = X(-l)m-h(xr,5) . If T denotes T* one can assume that ~r is again an Hermitian form by replacing

%

by i if necessary.

They prove the following results.

Theorem 5.2. The decomposition of a type L = 61 +... +L

into indecomposable types (which clearly exists) is unique.

Proposition 5.3. If N is uniform it is uniquely deter-mined by ht & and 3

Proposition 5.4. If 6 is indecomposable then 6 is

uniform and I is indecomposable.

uniform type 6e from ht 6 and the semisimple type , as in Proposition 5.3. Let (S,E) E E and S E L(E,a,r). Let V be the direct sum of m+l copies of E , written conveniently as E +NE +... +NmE . Extend a and S to

V , acting componentwise on this decomposition, and let N act on V in the obvious way. Then Nm0#O ,Nm+ = 0 , and ker Nm = NV . To define T on V , let T(NrE,NsE) = 0

unless r+s = m _{and in that case let T(Nre,Ne} 2) = (-1) 2 (e,e2) V el,e2 E E . Then clearly G(V,a, T) is

one of the above groups., A = S+N E L(Va,T) , and

(A,V) E &

By the above, the problem of the classification of

the orbits under the adjoint group of any of these algebras reduces to the determination of the semisimple indecomposable types, at least in the case where G is connected. This description is straightforward for the complex groups and the U(p,q) . The results are as follows. Let A be a semisimple indecomposable type for either a complex G or

U(p,q) and (S,W) E .

G = GL(W) . W is one-dimensional and 6 is determined by a single eigenvalue C T . Denote this

type by L(C) .

G = O(WT) . If S = 0 W is one-dimensional. There

is a basis element e such that T(e,e) = I

G=TU(p,q) _.

If S / 0 the set of eigenvalues for S is C,-CI for some 03/ E C. W is

two-dimensional and there is a basis (e,f} such that Se = Ce , Sf = -Cf , T(e,e) = T(f,f) = 0 and t(e,f) = 1 . Denote this type by t(C,- C).

Whether S

/

0 or not, dim W = 2 , and there is a basis (e,f) and C E C such that

Se = Ce , Sf = -Cf , and T(e,f) = 1. Denote this type also by 6(C,-C) .

dim W is either 2 or 1. When dim W = 2

there is a basis (e,f) and C E C such that C / -C where T(e,e) = (ff) = 0 ,

*(e,f) =

I

, and Se = Ce , Sf = - f . Denote this type by t(C,-)

When dim W = 1 there is C E C such that

= -C and a basis element e such that Se = Ce and T(e,e) = l. The two signs

give different types. Denote these by 6t(C)

For the real groups G(V,cT) one proceeds as follows. An indecomposable semisimple type

N

for

G(V,a,T) clearly gives rise to a type Ac for the corre-sponding complex group G(V,T) just omit a . Let

(SV) E . c is a sum of semisimple indecomposable types for G(V,

t)

, so let 6 be an indecomposable component of 6 . Let (S,W) E where W Q V . Clearly (S,aW)

is also a type for G(VT) , _{and we have either} aW = W or or aW l W = (0) . Since a = I and & is indecomposab-le we have three possible decompositions for c.

(a) c=(+a& and /c

(b) & =(&+ aA and =1

(c) &C= &0and tc=aC

Let eig c denote the set of eigenvalues of S on W. They prove the following

Lemma A.l. (a) occurs iff eig ( eig . Lemma A.2. Suppose S

/

0 . Then (b) occurs iff a=2 and all elements of eig 6t are real.

We now consider each of the six classes of

fixed-point groups and indicate the classification of and notation for the semisimple indecomposable types, based mostly on Lemmas A.1 and A.2.

G = GL(V)

a

=

I

. Let I) 2)

3)

2 a2=-I . Let I) S=

t(Q).

(a) by A.l; type (

= / 0 ; _{(c) by A.l and A.2; type A()} C = 0 ; (c) because if (b) and W = e ,

D(e+ae) is a-invariant, contrary to indecomposability; type (0)

A = (Q) .

C

= {

/

0 ; (b) by A.2; type A(C,C)

= 0 ; (b) as dim V must be even; type 6(O,0).

G = (V

a2= -IT.

Let 6 = A(0)

1) Clearly (c) ; sig T+ = (1,0) or (0,1) ;

two types A(0)

Let Al=

A(Cr-C)

, C/0

2)

C

/

t ; (a) by A.l; sig + = (2,2) ; type

3)

C

= ; (c) by A.l, A.2; sig T = (1,1) ;

type

A(C,-C)

4)

C =-Z

; again (c); sig T+ = (2,0) or (0,2);

two types A

(C,-C)

Let 6 = (0)

1) (b) as V must be even-dimensional; type

A(0,0)

Let A = 6(C,-C) , C /0

2)

C /

t ; (a) by A.1; type

A(C,-CC,-C)

3)

C

= ; (b) by A.2; type

A(C,-C,C,-()

4)

C

= - ; (c) by A.l, A.2; two types L*(C,-C) depending on the signature of T(',S-)

(V is one-dimensional over Q.

G = Sp(VLT)

a = I1.Let _S =

Y(C,-C).

2)

3)

1) G

/

tt ; (a) by A.1; type (

2) =

/

0 ; (c) by A.1, A.2; type (C,-C) = -i / 0 ; (c) by A.1, A.2; T(S-,-) either positive or negative definite on V; two

types

(

4)

= 0 ; clearly (c); type 6(0,0) 2 _{=- I . Let 6c = t(,-0}

1) c

/

te ; (a) by A.l; type t(,-CC,-)

2)

;

=

/

0 ; (b) by A.2; type A(-',2-')

3) C= -

/

0 ; (c) by A.1, A.2; two types

A6(,-) depending on sig (-,')

4) e = 0 ; clearly (c); again two types At(0,0) This information is summarized in the following table taken from

[

2 ]

-The Semisimple Indecomposable Types Type Conditions A(c) -I

&(C,

-)

A

(C)

sym

A(o)

+I sym A (C,-C) A -(0)

A(C-C = 3 /0 C = -C /0 C = -Z / 0 A(0,0) alt +I alt

A

(C,-C)

A

(C,-C)

C =

-C /

0

C

=

-C

Using the construction outlined before, it is now a simple matter to list the indecomposable types. We do so

2 __

*

-I sym

for the Hermitian symmetric Lie algebras, and in fact for all the o(p,q) . Write a = a. according to whether

= tI , and set 5s = (-l)m/2 for m even where e = t1

Indecomposable Types for Hermitian Lie Algebras

Conditions Signature GL(V, *) [>SU(pq) ] O(V,a+' [= o(p,q)j 0 (Va,-)

[=

So*(2n) Sp(V,a+' [=Sp(n,IR)I Am Am ( C , - ) C-Am () '-6m(c,- ,z,-:) Am(c,-c)

t4(C,-0)

am (0,0) t1m(0,0) 6m (c I - c I, I-0 Am(C 1 -0 '~ 6e(C,-c) Am(oo)

L (0,90)

E: (c,.-C)

t (0.0)

C / - (m+l,m+l) m even(i(m+1+6) ,2(m+l-s)) m odd (mlMl) C / tC (2(m+l),2(m+l)) 0 = (m+l,m+l) m even(m+l+5,m+1-) m odd (m+l,m+l) m even ({(m+l+5) , (m+l- )) m odd (M+lM+l) C C m m C C C m 3-0 even odd 3/ C even odd Group Type

The indecomposable types which can contribute to a type for o(2,n) are quite limited, and listed below.

( ,-C, ,-C) am( C, -0) S(e,-0) S(0)

m

=0 m = 0,1 m = 0, e = -l; m = 2 with c = 1 m = , c =;h in = 0, e = t1; in = 2, e = tl; in =

4

with S= -l (0,0) =l1

Am (0,90)

CHAPTER II. Invariant Convex Cones in sp(n,]R).

6. Conventions for sp(n,).

It will be convenient to work with fixed models for the classical Hermitian Lie algebras, fixed Cartan

decompositions, etc. The notation is as in section 3. We take

A B A,B,C real nxn matrices,

1'

= C -At B,C symmetric, A arbitrary

A B kB = - A E : A skew, B symmetric A B B E A,B symmetric r: B -A 0 D h = E D diagonal -D0 W 0 -c =

Let d1. ( = l,...,n) be linear functionals spanning the dual of h

0 D

d. - D..

-D 0 d

Then

noncompact roots} = '-i(d1+d k): 1 j<krn; 1 : 1i2d1j4n compact roots} = ti(da 1 -y: 1j<ks}

and the positive roots we take to be the above with +

only. Also,

simple positive roots} = t+= i(d₁-d2),...,i(dn -dn) ,i2d

We give the H., Xa, Y.for mEQ . For a= i(d.+dk

+0 kk (I < k 19n) =

tIjt

k

A=0 0 B B I where D = -iE. -For = i2_d (0ID .--D 0 E A= B = Ej + Ekj (1

j

n) XA 01 XO =AY

where H =-iE , A = B=E

We have then Z=

3=Z

, h=h

1 0) _{A 0}

= Pi2d ,...i2dn} _{, r =}

3-

_E

The cone in h obtained from Corollary 3.2 is

, and take

A diagonal}

10 -D1

E _h: D has positive diagonal entries}

=D

0Y

Define B(XY) = Tr(XY) for X,Y E g.If

x A B X = C -At ' D G H -D t E , we have B(XY) = 2

31A

D1 + I B1 1H +

i,j i,j iuJ

C. G..

The Cartan involution is 9: X and B9(XY) = -B (Xt9y),

0 B)

so

A B D G

BGK

;j

:ii)=

2 Ai 4 Di4 + + C 1 H .

B C -A t I H -D i Ci Hi

2n

Def ine the symplec tic form g(.-,.-) on E by

y=

-x-for x,y,u,v E In.

Theorem 6.1. Let u EBR2n and X Ecy. Then a(Xu,u) =

Be(XY) where c(Yv,v) 0 for all v EE2n

v. A B

Proof. Let u = and X = . Then 2(Xu,u) =

w C -A

(Cv)-v - 2(Av).w - (Bw)-w =

3C

vv - B wwj

AB D G

- 2 A 1w v4 =

JBA

(-D

= B6(XY) , where

D = -w vi , G. . = -w w. , and H = V v . Clearly

Y E c.

QJ

~ ~

2n

Te

Secondly, let = v E In Then

a(Yv,v) = (Hx)-x - 2(Dx)-y - (Gy)-y

S7x H x. - 2 x D y - y G

i ,j i,j ins

(xv) (x.v) + 2(x.v) (w.y) + (y.w) (y.w)

7. Conjugacy of Positive Elliptics in sp(*)

Let V. be a real Hilbert space, possibly infinite dimensional, with inner product (-,-) . We assume that

z

has a complex structure J , i.e., an orthogonal with

square -I . Define the symplectic form a(x,y) = S90(J1xY)

for x,y E %V c is skew and nondegenerate. (V, < ,>)

is then a complex Hilbert space, where <-,

-90(-,-) + i(-,-) ; note g(-,-) = 6(J-,a) . is really

the primary bilinear form here, but we need some 90 to define the topology.

Write

11-112

= c.,-> or sometimes 1112 for

0

clarity. In this section all operators are bounded. Given a (real-linear) operator B: V(->4V , 1B1 denotes the

norm of B . Let Sp(V) be the group of invertible linear operators V -+ V preserving 6 . Call B: V -> V

infinitesimally symplectic if B is skew with respect to

C . The set of infinitesimally symplectic operators we

denote q , or sp(W) .

Theorem 7.1. Let B E satisfy l(Bv,v) - k(v,v)V v E',

for some k > 0 . Then there exists S E Sp(Q) such that

SBS- JH 'where H is a positive self-adjoint operator

on (%, < , >) (i.e., H commutes with J) . Furthermore,

the spectrum of H is bounded as follows:

for all v E VW.

Proof. Clearly

IBlI

2 sup c(Bv,v) , and in fact by

IIv IIl

polarization one can prove equality, but we do not need this. It follows from the identity %0(Bv,w) =

c(Bu_,u_) - c(Bu+,u+) where u. = (v t Jw)/2 and the positivity of 67(B-,*) .

Define S(v,w) = a(Bv,w) for all vw E V . S is symmetric as B E m and 6 is skew. We have

(3)

JIBI

J

vt2~Svv)

kIl v1

2 'V

v E

'V

so 3S and S are real Hilbert structures defining equivalent norms on V . We have S(*,I,.) =

;

(J~1Bl,-) ,

and J is orthogonal for 9 iff B commutes with J

J 1B is a symmetric operator with respect to S , bounded below by k and above by

IBII

, so B is bijective.

Hence it has a bounded inverse, and

|IB

1| 1/k

Now B is skew with respect to ., so if B=K B B=

KJ-B 2 is the polar decomposition in (W,S) , all three operators B,K, and -B2 commute. We have K2 = -I , and

7

is invertible and positive-definite symmetric in (H,). Furthermore, K E Sp(W) as

B22

6(Kv,Kw) = C((B/A-B)v, (B/-B )w)

-Sj((/-1Bl)v, (BJB)w) - (v,-B Iv)=1vw

(4) ) = g(QB)_( <.

and & defines yet another equivalent norm on V . Its advantage over 9 is that

<.> =+ i(,.)

defines a complex Hilbert structure on V (with complex structure K) with the same imaginary part as that of <

It is not difficult to see that the topological equivalence of (V, < , >) and (V, <, >1) implies there exists a unitary equivalence

S:# (V,< , > 1) - x < J, >)

We have JS = SK , S E Sp(g) , and SBS = J(SJ-B2S-I) So, we set H = Sf- Sl, and HJ = JH because K commutes with /B.

By (4) we have orthogonal equivalences

(VS) 2 1/4

(9) ()-B

)0

of real Hilbert spaces. As (-B2)1/4 commutes with

tU? , is positive-definite symmetric in

also. By the equivalence, H has the same property in

and is self-adjoint in (/,< , >) as HJ = JH It remains to prove the bounds on H . They are

( B //k/2) 2 (k ,.)2 3/2/ B J1/2) 1.12

or

(11(|BI3 2 2. 1 2

For the upper bound of (5), (-B2w,w) = S(Bw,Bw)

J |B J 0go(Bw, Bw)

S |B113 wI2 e, (I BI3 /k) lwl2

o 0 0

the first and third inequalities following from (3). For the lower bound,

&(Bw,Bw) k%(BwBw)

0 0

2 k(1/1B 1 ) |w l2

by (3) and

IB

1

1I

I/k shown earlier.

8. Uniqueness of Causal Cones in sr(n,J)R.

The goal of this section is to prove Theorem 8.1. We use the notation of section 6, and in order to apply

Theorem 7.1 we indicate the connection between the

notations of sections 6 and 7. Our real Hilbert space V

0 -I 0 -I v -w

structure J is

;

note that

corresponds to i(v+iw) = -w + iv for v,w E BR . Also <v+iw,x+iy> = V-w + w-y + i(w.x - v.y , so the two

definitions of the symplectic form a agree.

M N"

A short computation shows that g = E Sp(n,r)

t t t t t t

equivalently MN = NM ,

QP

= PQt ,and MQt -NPt = I.

Thus g = -. Also X E gl(2n,B) commutes with

J iff X has the form

3]

for AB E gl(n,R) . We find g E Sp(niI) in the subgroup K corresponding to k iff g=

K J

where AA + BBt = Iand AB = BA ,

,B A

or AtA + BtB =I and A B= BtA. As shown on p. 343 of [9

1,

K is algebraically isomorphic to U(n) acting o Cn , i.e. I = (A+iB)(A+iB)* gives the same equations.

Let S be the set of all non-trivial invariant convex cones in , and let C 0= X E : 1(Xv,v) > 0OV v 3/'O

in I 2n

Theorem 8.1. C0 is nonempty and in S . Any C E S contains either C or -C but not both. tC are the

0 0

unique open cones in S . C0= (X E : ca(Xv, v) > Ow*v E n

and tC are the unique closed cones in S

Proof. Clearly J E SO and C0 ES. Let C E C. By

Lemma 1.1 C E S also, and by the proof of Theorem 2.2

2n

c(X,-)

is a symmetric form on FR for all X Et. Since all norms are equivalent in a finite-dimensional space C is open. Thus J E nC implies aW E C ACQ0

Let W be the B in Theorem 7.1. The invariance of

C implies JH E C as in the theorem, and the diagonal-ization of JH by a unitary implies that C also contains

0

C

-f7]I = 0 -f

J D _{-I 0 ,}

0 _D

D _{0 ,}

for some diagonal matrix D with positive entries.

Now conjugation by a,a~ E A and averaging (C is

0 -D'

convex) shows that C also contains any

[

Eh where D'i D. for i = 1,...,n. As C is a cone it must

contain all ,] D positive diagonal. The argument can now be repeated to show C₀ Q C . But C cannot contain

C₀ 0 and -C ₀ by Proposition 2.1.

Of the remaining three statements we show how the first follows from the last two. Let C be an open cone in C- , and assume C = C0 . As C fl -7 = (0 we must

have C CC. If XEC-C C C - C 0 v E F2n such that

1(Xv,v) = 0 and v

/

0 . C being open implies 'Lc > 0

such that Y = X - J E C C but 1(Yv,v) < 0 , a contradiction.

To prove 70 =

(X

Ec.: (Xv,v) 'a 0 V v E R

note first that clearly C C S . Conversely, if X E S

then X + eJ E C 0 e > 0 , so X E7C0.

C CC₁ so that C0 QC1 =C . if COaC then

C* C C = -70 by Theorem 8.2 below. But we have proven

that then -C0 C C{ must occur, hence -C C{, a

contradiction.

Theorem 8.2. C = -CO , i.e., the closed cone C is

self-dual with respect to any B form.

Proof. We saw in section 2 that B8(X,Y) 0 VX,y E -CO

which is just - ₀ CC₅ . If now X

9'

-t SvEE 2n such that aI(Xv,v) > 0 . By Theorem 6.1 BO(X,Y) > 0 for some

Y E O , so X g C .

00

Corollary 8-.3. Co _{is the topological interior of C0}

Proof. The interior of C0 contains C₀ and is an

invariant convex cone, and so equals C0 by the uniqueness.

9. Classification of Cones in sp n,E)

We have seen that any invariant convex cone C (or its negative) in sp(nY) must sit between C₀ and 00 0

Let C+= (C E C: C₀ Q C QC0} . It turns out that C+

is a finite collection, and we will be able to classify its elements. For example, in sp(2,JR) there are 12 such

cones in C+

Any C E C+ can be considered a collection of types

, each of which is a sum of indecomposable types

is in 0 iff each L is in the Cj s for the (possibly) lower-dimensional 1-orthogonal subspaces. Thus to determine the types on the boundary of C0 it suffices to consider the indecomposable types.

Using the notation and analysis in Section 5, it is clear that X E C₀ determines a type whose indecomposable components are all of the form 6 (,-) for 0 /

C

E iR. tC are the eigenvalues of this type, so the eigenvalues of any X E 0 must also lie on the imaginary axis. Thus

it suffices to consider only Am(O) (m odd), Lm(0,0) (m even) , and 6(C,-C) (C = -3 / 0)

Theorem 9.1. Only the types A+(0) , 60(o0) , and 1 +(C,-C) are in Co

Proof. Let A = S + N be a representative of one of the

above types of height m . The representation space is a direct sum E+NE +... +NmE , where l(.,Nm-) is non-degenerate on E

Assume first m 1 , take v,w E E , and consider

a(A (Nm-1v+w) ,N M-1v+w)

= a(Aw,w) +1(-I)m-'a(AN -2v,v) + 21(Nmv,w)

2m-2 nm+l iff m 2

3

, in which case the first and second terms vanish, and the expression is negative for

the proper choice of v and w .

and this is negative if w is sufficiently large and of the proper sign.

Finally, if m = 1 take v,w E E and consider a(A(Nv+w),Nv+w) = -21(NSw,v) +I(Nw,w)

If S / 0 it is injective, and again this is not positive-definite.

The remaining cases are just those stated in the theorem. All are orbits in sp(IYR) sl(2,JR) . 60 is just

(

0).*(a _) is in the orbit ofa )iff

2 00 c aa b0

iff -a 2 -bc > 0 , b < 0 , and c > 0 . (c-a)/O is

in %0(0) iff -a-bc =0 ,b O, and c0. This

completes the proof.

To classify the cones we need some further machinery and a convexity lemma. Let En = l' 1 '''''n' n

2n

all x ,y -0 . We consider it a cone in E .Define

a semigroup 9 acting on En generated by

1) [(x1,y1),...] -*-+ ,X1)...] ,

2) [(xl.PyJ),... gxiy ),-- J~n' n

S(x) 1y1) ,'''' . , (xn yn)] for i=2,...,n,

3) ₁ [( . [ ,(l/X)y,...1 for X > 0 ,

and

4) [(xl1,y9, (x2,2,...+-*[(x 1,y1 y2 ),(x1+x 2 ,y92...

Let be the collection of all Schrbdinger bases

I = t(a,,b.) : @(a a5 ) = a(b ,b ) = 0, (a ,b1) =51}

and let C+ _{be the collection of all convex cones in En} which are invariant under g and contain 0n =

I[

(xl,71)I,-- n' n) ] : all x ,y > 0) . We will put

C

in 1-1 correspondence with C+ by means of the following maps. Given X E -0 and cp = (a ,b) 1 E I , define

M (X) = [((Xa₁,a),1(Xb,bj),...,(c(Xan,an),4(Xbnbn))] _{E En}

For any C E C-+ let EC be the convex cone generated by all M (X) for XEC , cpE .

Lemma 9.2. Let X E C and let the decomposition of the type of X be P summands of the form A+(C-Q

= - /0), N summands of the form (0) , and Z

of the form &(0,0) . Then n = P +N+Z . Let cp =

(ai,bi) E I exhibit this decomposition, so that M (X) has P pairs of the form (e,f) , e,f > 0 , N of the

form (X,0) or (0,X) , X > 0 , and Z of the form (0,0) Let # = (c ,d ) E I be arbitrary. We can assume t is ordered so that a(Xc ,c ).,(Xd ,d ) > 0 for i=l,...,P',

a(Xc ,c ) > 0 , 1(Xd ,di) = 0 for i= P' +1,...,P' +N', and $(Xc ,ci) = (Xd ,d ) = 0 for i = PI+Nl+1,...,n

Then Z S Z and N'+Z ' S N+Z

Proof. Assume that the diagonalizing basis p has been ordered so that the sentence above is true with (a.,b.)

respectively. Now each c. and d. is a linear combin-ation of the a. and b . If j = P'+N'+l,...,n , c. and d can involve no a or b with i = 1,...,P or a. with i = p+l,...,P+N . The 2Z'x2Z' matrix

r(ckcZ)

a(ckd

:(dk9cL) 7(dk-dL)J k,L,=P'+N'+l,...,n

(which depends only on the combinations of the a1,b for i = P+N+1,...,n) is of rank 2Z' as $ E E , but is equal

to At( 5IA where A is of order 2Zx2Z' , hence Z'1!9Z. Express likewise d. for j = P'+l,...,n . These

d can involve no a ,b for i = 1,...,P . Now the maximal dimension of an isottopic subspace in a 2(N+Z)-dimensional symplectic space is N+Z , and N'+Z' N+Z

follows.

Definition. If e E En has exactly N pairs of the form

(OX) or (%,0) , X > 0 , and Z pairs of the form (0,0)

say e gives (N,Z)

Theorem 9.3. EC E a+ for each C E C+ ,and the mapping

is a 1-1 correspondence.

Proof. E₀ clearly contains n since each C contains

the trivial basis changes a -* b1 ,b1 -+ -a and

1~ ~ 1*Xa

a -+%a, b, -+ (1/X)b .

Invariance under 4) follows from an Sp(2,R)

symmetry: given (a₁,b1,a2,b2 ) , make the change of basis

a1 -c = a1 a2 -+c2 = a₂ + ka

(6)

b -+d = b 1 - kb2 b₂ -+*d₂ = b2

for k E BR, which is symplectic. If (a₁,b₁,a₂ ,b₂) is part of a yqE i and X E C so M (X) E E , do (6) for

k = 1 to obtain p+',- , and (M (X) +% M(X)) E E

2 P+ CC

exhibits the invariance under 4). Thus EC E C+ *

Before proving the 1-1 correspondence, we first make a simplifying remark about the D E C+ . If e E D , then D also contains any other e' obtained from e by

changing any positive x or y. in e to any other positive x1 or yj This is trivial for the (x,0)

and (0,x) (X

/

0) pairs by (3), and accomplished for the (x,y) (x,y > 0) pairs by initially multiplying by a small constant, and then applying (3) twice so

(x,y) -+ (Xx,(1/X)y) and (x,y) -+ ((l1/X)x,y)

and averaging, to "raise" (x,y) by any desired amount.

Now let DE C+ . By the above D is characterized

by the set of pairs (N,Z) which the e E D give. (Of

Each (N,Z) determines many types 6 , as in the statement of Lemma 9.2; let C be the convex cone in C generated

by the representatives of these types determined by D

That E = D follows from the fact that D is convex,

Lemma 9.2, and the observation that if D contains e

giving (N,Z) where Z < n (e ' 0) , then it also contains all points giving (N',Z') where Z' S Z and N'+Z' S N+Z

by invariance under 1) to 4). Thus C -* E is surjective.

Finally, suppose EC = E for ClC2 E C+ . We must show that each type in Cl is in C2 also. Let & be a type in C and X E representing t . If

p E is a diagonalizing basis as in Lemma 9.2,

e = M (X) E E and gives a pair (N,Z) . By hypothesis e E BC , so e is a convex combination of M

(Y)

for

2

Yi E C2' cEi E

Observe that if x,y E Bn , and x,y , and x+y give (N1,Z1), (N2'Z2), and (N3,Z3) respectively, then

Z3 : min(Z₁, Z2) , and N3+Z3 min(N1+Z1, N2+Z2) . Thus for some Y E C2 and cp C , M9(Y) gives (N₁,Z1) where

Z S z and N+Z : N1+Z1 . Now Y may be diagonalized by

some E , and M(Y) gives (N2,Z2) . By the lemma,

N2+Z2 2 N +Z and Z2,so N2+Z2 2>N+Z and Z 2 Z

It remains to see that these inequalities and Y E 02 imply _{X E C2}

Via the basis $ , Y acts on n copies of ]R