Signature Redacted Signature of Author

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

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 Signature of Author. • D;p;rtm·;'·~t-'6f'• M; th~-;a t;;s-;, M;J 5'. 1980

- - ./i Signature redacted Certified by .•• . . . . , ...... ,. . . . Thesis Supervisor Signature redacted

.Accepted by. • ARBHl'VES . • • • : • ·• • • • • . . • • • • • • .n,,:::::;Ar.·,.,T:r.n,;. •ric- - .Jrc: r.-1.,v ... , .,; Uvt: t; l'ivi 11l L Cha:.rman., Departmental- Com.mi ttee OF TFCHNCLOGY JUM 11 1980

LIBRARIES 2

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 Title: Professor of Mathematics 3

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 4

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. 5

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 .

TABLE OF CONTENTS Page

ABSTRACT...... 2 DEDICATION...... 9. 3 ACKNOWLEDGEMENTS...... 9. 4 INTRODUCTORY REMARKS...... 8

CHAPTER 1. Preliminaries ...... 11 1. Preliminaries on Convex Cones ... . 2. Existence of the Minimal Cone ... . . 12 3. Structure of Hermitian Lie Algebras . 9 16

4. Certain Classical Linear Groups . . - 21 5. Classification of the Orbits...... 25

CHAPTER II. Invariant Convex Cones in sp (n,JR)R . - 36

6. Conventions for sp(n.,R) .. . ------. 36 7. Conjugacy ofPositive Elliptics in sp(V). 39 8. Uniqueness of Causal Cones in sp(n, R) 42 9. Classification of Cones in sp(n,JR) .45

CHAPTER III. Invariant Convex Cones in su(p,q) . - - 55 10. Conventions for su(p,q) --- - . - . 55 11. Conjugacy of Positive Elliptics in u(p,q) 58 12. Minimal and Maximal Causal Cones in su(p,q) .*.0.0.0.0.*.*.a.0.*.*.*.0.0 61 13. Orbits in ~C of su(p,q) ...... 65 14. Classification of Open and Closed Cones in su(p,q),. .0.0.0.0. . .. *.0. ... . 70

CHAPTER IV. Invariant Convex Cones in so*(2n) . 81 15. Conventions for so*(2n) ...... 81 16. Conjugacy ofPositive Elliptics in so*(>) 84 17. Noncompact Convexity in so*(2n) . . 87 18. Minimal and Maximal Causal Cones in so*(2n)...... - . . -0 93

. 7

Page

19. Classification of Orbits and Cones in so*(2n) ...... 97

CHAPTER V. Invariant Convex Cones in so(2,n). . -.* 102 20. Conventions and Preliminaries for so(2,n). 102 21. Noncompact Convexity in so(2,n) . - *- . 107 22. Classification of Orbits and Cones

in so(2,n) . . . .. *.. . . - - -*112*

CHAPTER vi. Global Causality of the Covering Groups. 119

23. Definitions and Isomorphisms ...... 119

24. Causal Actions on ghilov Boundaries. . . 123

CHAPTER VII. KMhlerization in Causal Lie Algebras . . 132 25. Uniqueness of the Complex Structure. . . 132 26. Generalizations to Unbounded Operators . 141

27. Analyticity of Kghlerization in sp(r). . 145 28. An Example and Analytic Continuation: SU(1,1)...... 0 .. 0 .0 .0 .9 . 147

CHAPTER VIII. The Cayley Transform ...... 155 29. General Properties ...... 155 30. Cayley Transforms for Causal Groups. .. 158

CHAPTER Ix. Applications to Differential Equations . 162 31. Preliminaries...... 162 32. Regions of Stability ...... 164 33. Two Stability Criteria ...... 168

REFERENCES.. . . 0 0 0...... 179

BIOGRAPHICAL NOTE - AFTERWORD ...... 182 8

INTRODUCTORY REMARKS

The main ideas of chapters one to five on the classification 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 essential to establishing the naximality of certain 9

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 10

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 in B and A/'B. 11

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 closed half-space. 12

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: 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.

Proof. C A - C is an ideal in cy by the invariance. 13

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 =<(expX)w,w>>0. Now =

G: (expX)k-+

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

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

convex cone generated by the gw for g E G , it is clear that C 0 is a non-trivial G-invariant convex cone in V, and >0 Vu,v E C .

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 semisimple & 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 integration 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 15

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 decomposition, and rank is the dimension of any maximal abelian subalgebra of p. sp (n.,EI) dim k =n2, dim p =n 2+ n; ranko n ;charoflOV n 2 2 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 .

4 e VI (a real form of e7 ) dimk=79, dimp=5 ; rank1= 3 .

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

Corollary. A complex simple Lie algebra oy admits a noncompact real form having a causal cone iff the adjoint group corresponding to a compact real form of ct is not simply connected. 3. Structure of Hermitian Lie algebras.

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 hOchc * We have k = k 9ce , 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 subalgebra of q . LetC 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. All roots are real-valued on ih , and a root is compact iff it vanishes on c . Let B be the restriction 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 1 6...,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 combination of the a . By the compatibility of the ordering, SZ E c such that a(iZ) > 0 iff a. is 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 Ei &0 are compact.

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

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

ThusE implies a.+(a. +s)(Z)C = 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 19

noncompact 51,4 2 '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 20

shown that h + is the centralizer of or in h , and that the complexification of h + eSo 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 Z 0 =-(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+ /00. 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 21

the next several sections.) Let X = taX Eatr. The relations (1) imply ME 0

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

M ES0

= 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 representation, is ideal for our purposes. The exceptional algebras will probably require a more abstract approach, 22

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 quater-

nionic vector space as follows. Let 23

= (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-c j+dk E % a+bi+cj+dk E a

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

space. SU*(2n) is connected. 24

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, )

Let T be a nondegenerate anti-symmetric complex 25

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

Let H=(g E G: (ge,ge)

(a) a2 = I 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. (b) a2= -I . 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 denote T , in which case a is absent. 26

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. 27

There is a natural mapping of uniform types to semisimple 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 nondegenerate 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 = 6 1 +... +L into indecomposable types (which clearly exists) is unique.

Proposition 5.3. If N is uniform it is uniquely determined by ht & and 3

Proposition 5.4. If 6 is indecomposable then 6 is uniform and I is indecomposable.

We next give the simple construction recovering a 28

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) = 2 (-1) (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

Denote this type by (O) 29

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) .

G=TU(p,q) . 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 corresponding 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) 30

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

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 S= t(Q). I) (a) by A.l; type (

2) = / 0 ; (c) by A.l and A.2; type A()

3) C = 0 ; (c) because if (b) and W = e , D(e+ae) is a-invariant, contrary to indecomposability; type (0) 2 a2=-I . Let A = (Q) . I) C / C ; (a) by A.1; type (CC) 31

2) C = { / 0 ; (b) by A.2; type A(C,C) 3) = 0 ; (b) as dim V must be even; type 6(O,0).

G = (V

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) a2= -IT. 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). 32

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 ] - 33

The Semisimple Indecomposable Types 2 __ Type Conditions

A(c) -I

* &(C, -) A (C)

sym A(o) +I sym

C = 3/0

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

-I sym

A(C- C = -Z / 0 A(0,0)

alt

+I alt

A (C,-C) C = -C / 0 alt

A (C,-C) C = -C

Using the construction outlined before, it is now a simple matter to list the indecomposable types. We do so 34

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

Group Type Conditions Signature

GL(V, *) Am C / - (m+l,m+l) m even(i(m+1+6) ,2(m+l-s)) [>SU(pq) ] Am ( C , - )C- m odd (mlMl)

Am () '- O(V,a+' 6m(c,- ,z,-:) C / tC (2(m+l),2(m+l))

[= o(p,q)j Am(c,-c) 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- )) t4(C,-0)am (0,0) m odd (M+lM+l)

0 (Va,-) 3-0 C tm(0,0) [= So*(2n) 6e(C,-c)1 C 6m Am(oo)(c I - c I, I-0 even A m(C1 -0 '~ m L (0,90) odd m Sp(V,a+' 3/ C C [=Sp(n,IR)I E: (c,.-C) C C even t (0.0) m odd 35

The indecomposable types which can contribute to a

type for o(2,n) are quite limited, and listed below.

( ,-C, ,-C) m =0

0,1 am( C, -0) m =

S(e,-0) m = 0, e = -l; m = 2 with c = 1

=;h m = , c

S(0) in = 0, e = t1; in = 2, e = tl; in = 4 with

S= -l

Am (0,90) (0,0)=l1 36

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(d 1+dk): 1 j

compact roots} = ti(da 1 -y: 1j

and the positive roots we take to be the above with + only. Also,

simple positive roots} = t+= i(d1 -d 2 ),...,i(dn -dn) ,i2d

mEQ . For a= i(d.+dk We give the H., Xa, Y.for +0 kk (I < k 19n)

0 B A=0 =tIjt k I B where D = -iE. - E A= B = Ej + Ekj

For = i2 d (1 j n)

(0ID 01 0 B) XA XO =AY .--D 0 B 0j where H =-iE , A = B=E

We have then Z= 3=Z , h=h , and take 1 0) A 0 A diagonal} = Pi2d ,...i2dn} , r = 3-E

The cone in h obtained from Corollary 3.2 is

10 -D1 E _h: D has positive diagonal entries} 0Y =D

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

x A B D G E , X= C -At ' H -D t

we have

B(XY) = 2 31A D 1 + I B 1 1 H + C. G.. i,j i,j iuJ The Cartan involution is 9: X and B9(XY) = -B (Xt9y), 38

A B D G + C H 2 Ai Di 4 + 1 . BGK ;j :ii)= 4 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 EBR2 n and X Ecy. Then a(Xu,u) =

Be(XY) where c(Yv,v) 0 for all v EE2 n

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 1 w v4 = JBA(-D = B 6 (XY) where

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

Y E c. ~ ~ 2n Te Secondly, let QJ= 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)

= (x-v + w.y)2 2 0. 39

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 <-, -

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

< Hv, > 3k/ B 111/2) 1v12 ( IB 1 2kl/2) 11 v 112 > 4o

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

2 (3) JIBI J vt2~Svv) kIl v1 'Vv 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 -B 2 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 Setting gl(xy)= a(Kxy)V x,y C , we have

(4) ) = g(QB )_(2 <. 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

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

ecuivalent to 42

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

1 2 (11(|BI3 2 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 11I 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

is B2n with the standard inner product. The complex 43

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-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 C0= 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

C contains either J or -J . Assume J E C . Now 44

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 diagonalization 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 C and -C by Proposition 2.1. 0 0 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 - C0 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 C0 e > 0 , so X E7C0.

Finally, let C be closed and in 2- . We may assume 45

C CC 1 so that C 0 QC 1 =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 2 which is just - 0 CC5 . If now X 9' -t SvEE n 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 000

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

1+ ... + Im . By Theorem 8.1 it is clear that t 46

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 nondegenerate on E

Assume first m 1 , take v,w E E , and consider a(A (Nm-1v+w) ,NM-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 .

If in = 2 the expression is -1(SN2v,v) +21(N2v,w) , 47

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 b0 )iff 200 c aa iff -a 2 -bc > 0 , b < 0 , and c > 0 . (c-a)/O is c0. This in %0(0) iff -a-bc =0 ,b O, and 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) [(x 1 ,y1),...] -*-+ ,X1)...] ,

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

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

3) 1 [( . [ ,(l/X)y,...1 for X > 0 , and

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

Let be the collection of all Schrbdinger bases

in 2 on which G acts, i.e. 48

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) = [((Xa 1 ,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.)

1 replacing (c,d) , and P,N,Z replacing P',N',Z 49

respectively. Now each c. and d. is a linear combination 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 $ EE , 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

C 0.aInvariance under the generators 1) - 3) follows from 50

the trivial basis changes a -* b 1 ,b 1 -+ -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 = a 1 a2 -+c2 = a2 + ka (6) b -+d = b 1 - kb2 b 2 -+*d 2 = 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

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

course, the (N,Z) cannot be specified arbitrarily.) 51

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 ,Z 2 ) , and N3+Z3 min(N1 +Z1 ,N 2 +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

on Z2 of them Y acts as ,on N 2 like 52

and on n- Z2 -N2 copies like a representative of a

L (+,) , = - ' 0 . Sp(nR) includes SL(2,IR) on

each 2 and all permutations of the BA's . The average

of, say 2 0) with its transform 0 -) is 01 '

so a At(O) can be changed to a &+ (,-C) easily. Also, the average of ( 0) 9 (0 0) with its transform

0 0) e ( 0) gives (0 0) S (0 0) ,etc. In this way one can see that X is in the invariant convex cone

generated by Y ,provided N2>0 .

If N2 = 0 , Z2 < n (otherwise Y =0), and we must show how ( -1) e9(0 0) can give rise to an

e( (7 ) . This isdone by the Sp(2,ER) symmetry

(6). If Ya1 = -b1 'm = a 1 , Ya2 = Yb2 = 0 , we consider

the action on the basis (c1 ,d1 ,c2 ,d2 ) from (6):

Yc 1 = -d 1 - kd 2 Yc 2 = -kd -k d2

Yd 1 =c1 Yd2 = 0

The average for k = ti is

Yc1 = -d1 Yc2 = -d2

Yd 1 = c Yd 2 =0

as desired.

Thus X E C 2 also, so C1=02102 and E -+E is injective. This completes the proof.

We see from the proof that any cone in + may be 53

represented graphically by a set of lattice points (N,Z)} in ((x,y) E 2ZG2Z: x,y z 0, x+y ! n} , having the general form

Z z

* . or . .*. N - L---. /f Wo %W .' %zo -q N N

(in sp(5,E) ), for example.

The segments bounding the region containing the (N,Z) are either horizontal or have slope -1. (0,0) is always included (C0 C always), and (o,n) is included optionally according to whether 0 E C or not. It is clear anyway that sl(2,R) has 4 cones in C+

(closed) (open) and there are 6 cones in sp(2,R) which do not contain 0 54

They are listed this way to exhibit a duality. Letting C+ denote the cones inC,+ not containing 0, we have the mapping

o A 0 C E 0 + -+ C= (X E g: B,(XY) > 0 VY E C} E 13+ and perhaps there is a corresponding explicit duality in these graphs. Are there any "self-dual" cones?

Since the previous was written, Bob Proctor pointed out that the enumeration of the above graphs involves no complicated combinatorics. The positive cones not containing 0, in 1-1 correspondence with the above graphs not containing (0,n), are also in 1-1 correspondence with the set of nonempty and proper subsets of Sn = as follows. Notice that each row in a graph contains strictly fewer points than any rows below it, and the graph

is faithfully represented by the collection of positive

integers each the number of points in some row of the graph.

For example, in the sp(5,?t) examples above, the subsets are (5,4,3,13 and (6,4,2), respectively.

Thus there are 2 (2n_) positive invariant convex

cones not containing 0 in sp(n,R) . And it appears that

the geometrical duality above corresponds simply to comple-

mentation in Sn of these subsets, in which case there are n evidently no cones C satisfying C = C. 55

CHAPTER III. Invariant Convex Cones in su(p,q

10. Conventions for suap,k) .

We take p z--q 1 and n= p+q. As in section 3

A,B,C complex matrices; B and C skew- Hermitian of order p and q, resp.; B2A Tr C=0 A arbitrary; Tr B + p2= 2

/c J E k: B and C diagonal1

(q/(p+q))iI and if Z = c=JRZ 0 (p/(p+q))iI]

Let d. (4 = 1,...,n) be linear functionals spanning the dual of h :

B 0B.. for 4 5 p

SCc pl- for 4 > p

Then noncompact rootsj = (d.-d) : 1eS p, p+1 9k rn 56

(compact roots = t(dJ-dk): 1!j

= (d1-d2,..I. -dd-dp+1,dp+1-d ,...,d -d } and d -dp+1 is the only noncompact simple positive root.

For 1l';j:!p and p+li';k rbn , m= d -dk E Q+ .,and

H =-k X = Ejk + E k and Y = -iEjk + iEk. We take Z = (d-d .: 1 S i : q} so 0 1 p+i

(0 0 D = 0 0 0 : D real and diagonal (D 0 0,

Z=- $ 0 0 0 , 00 -I, Di 0 0

_0h~ 0 0 0 E h , 0O-Di,

Di 0 0 and h+ =h 0 Fi 0 E h O 0 D,

Thus Z=Z0 iff p=q ,and h =0 iff p=q=l

0 0 (-q/(p+q) Z Z = 0 -(q/p~q))iI 0

0 0 (p/(p+q) )iI) and the cone from Corollary 3.2 is the set of rays containing 57

-D 0

Z-z0 + K0 0 0 Eh ii 0 0 -D where all D 1

Define B(X,Y) = Tr(XY) for XY E. If X = y= ( ,GJ)we have

B(XY)=B- I Bij+ 2 Re A Cii i,j i,j i,j

The Cartan involution is 9: X - )X( , and

BB Q G 2) + c H . T.Bij +2 RejA D

Define the Hermitian form H(,-) on Cn by q H= I i=l ~j=1 for x,yE C , y,v E . H(Xv,v) is purely imaginary

VX E ,v E T n

Theorem 10.1. Let u E Dn and X Then iH(Xu,u) =

B9 (X,Y) where Y E u(p,q) (using the same formula for Be).

Tr Y = -iH(u,u) , and iH(Yv,v) 2 0 \ v CEn

Proof. Let u = 2)and X = (t . Then iH(Xu,u)=i{IviB v+ WA - wA v.- 7wC. . i j i, jIj

SG D = B,(X,Y) =j B 58

where G.. = -ivIv. , H.. = iwW , and D . = -iv.W

Clearly Y E u(p,q) and Tr Y = -iH(u,u) If v =j)CEc ,

iH(Yv,v) = j2 + 2 Re () + ||2

2 = + f > 0

11. Conjugacy of Positive Elliptics in u(p,q)

Let V be a complex Hilbert space with complex

structure i as usual and complex inner product < , >0 "Complex-linear" (or i-linear) in this section means

commuting with i . Let 9 be an Hermitian operator on V

with spectrum (+1,-l} and define H(r,-) = <;-,->0 . Let

T(-,-) = -ImH(-,-) , a (real) symplectic form on V . If we set

= Re<-,-> 0 + ic(-,-)

and J = -i = -i9 , then (%, < , >) is a complex Hilbert

space with complex structure J.

Our notation here is chosen carefully to correspond

with section 7; in particular, the J and < , > corre-

spond, and the earlier remarks on 11-1f apply. All

operators are bounded.

Let U(V,) = VVS: -+ V S cin i-linear invertible

operator preserving H (or )) , and c,= 3: Y -+Vi

B i-linear and skew for H1 . The following lemma is clear. 59

Lemma 11.1. For all A E I , c(A-,-) = Re(iH(A-,.)) is a symmetric real bilinear form on V for which i is

orthogonal. iH(Ax,x) is real for all A Et.

Theorem 11.2. Let B E c satisfy a(Bv,v) z k V v E V ,

for some k > 0 . Then there exists S E U(V, ) such that

SBS1 = JH1 where H 1 satisfies Hl) = ;H and is a

positive self-adjoint operator on (V, < , >) (and

(V, < , > ))). Furthermore, H1 is bounded as follows-:

( JIB 11/2 A '/2) 11|v |12 2

Zt (k3 2/|| BJJ1/2) 1| v|12
for all v E V .
Proof. The situation is the same as in section 7 except
that we have included an i which is intended to commute
with all operators we consider and be orthogonal for every
bilinear form. The proof is virtually the same.
We define g(0,-) = a(B-,-) and (3) still holds. We
decompose B = KA/-B 2 in the real space (V,9) in which i
is orthogonal. Bi = iB so K and -B also commute with
i , and K E1 l ut(,) . Letting ( = c(K-,-) =
Kr(-B2,>1 = g1 (,) + i(.,) is a complex
Hilbert structure on V with complex structure K
Any unitary equivalence
s: (x,< > ) ->(Y,< > 6o
satisfies SKS1 = J = -i4 and preserves Im H , but we want S to be also i-linear. This is clearly equivalent to S- 9S = iK if S is a unitary equivalence, which we now assume. Now g and iK are Hermitian with spectra
(+1,-fl in the complex spaces (V , >) and (V, < , >1) respectively. Thus we need an S taking the +l eigen- spaces of iK to the tl eigenspaces of Q . Clearly such an S exists iff the L2 -dimensions of these subspaces match.
Observe
<= Re H(,-) - iImH(-,) and
= Re H(iK-,-) - iImH(-,).
Thus if L C V is in the positive eigenspace of iK , H(,-) is strictly positive-definite on L . We need an infinite-dimensional "Sylvester's law of inertia," and the lemma below is sufficient for this.
The rest of the theorem is proven as in section 7.
Lemma 11.3. Let (V = ,1 e,V2, < , >) be a nontrivial direct sum of complex Hilbert spaces. Let 9 be the Hermitian operator on V equal to +I on Y 1 and -I on V 2 , and set H(-,.) = <;-9,.> . Let m. be the cardinality of the dimension of V1 . If S c V is a closed complex-linear subspace such that HIs is strictly positive-definite on
S , then card(dim S) s , . Proof. Let e ji}r be an H-basis of S , i.e.
H(e,,e4 ) = 5 By the open mapping theorem
M: A22(r') -.S :1X e is an embedding. Clearly vx = H(M(v),e,) 9v E 12(r) x E r.
If Card(dim S) > . , then Sv E Z2(r) where v/3'O and
(Pry M)(v) = 0 . (Otherwise the adjoint of Pr M has dense range from V to 12(r) , and the Gram-Schmidt process gives a contradiction.) Then
2 0 < Iv =H(Pr 2 o M(v),e)v4
piEr
=I3H(Pr, oM(v),Pr, OM(v)) S 0 as H : 0 on V.
12. Minimal and Maximal Causal Cones in su(p,q)
Let C- be the set of all nontrivial invariant convex cones in su(p,q) . For convenience we now exclude the case p = q = 1. We use the notation of section 10, and now indicate the connection with the notation of section 11.
The complex Hilbert space (V, < , >0) is (n =p+q) with the usual Hilbert structure. Writing x E W as
(Y) where y E TP, z E 9 = and H(-,-) is an
Hermitian form on V with signature (p,q) . J = ( 0D and J E iff p = q , but this is not a problem; recall 62
the closely related Z E y. Let G = SU(pq)
One computes that g = ( )E U(pq) iff AA -BB=I,
DD - CC* = I , and AC* = BD* , or equivalently
AA CC=I ,DD - BB = Iand AB = CD. It follows that gl = A* -C*
Now 0 E gl(n,C) commutes with J iff OZ = Za iff 0 has the form A . Our maximal compact K is U(0 all 1 ), UJU2 unitary with (det Ul) (det U2) = I 0 U 2 Definition. Let
C0= (XE %: iH(Xv,v) > 0 V nonzero v E Cn)
(Note Lemma 11.1) and
C 1= (XE : iH(Xv,v) >0 V nonzero v ECn such
that V1(v,v) = 0) .
Theorem 12.1. C0 and Cl are nonempty, open, and in C5
Any C E C, contains either C0 or -C0 but not both.
in the S0 and 'U are obtained by changing > to definitions ofC0 and Cl, and COC0C1 ' O 1 CO'C 1E C- , and any closed (open) cone in C, which contains
CO is contained in Cl (resp. )
Proof. It is clear that Z E C0 CC and C0OC1 E C-
C0 is open by the same argument as in Theorem 8.1; we will return toa1 later.
Let C E C. As before, we may assume Z E ' , so 63
again S B E C A C0 . Apply Theorem 11.2 and note that a conjugation by S E U(p,q) can be achieved by an element of
G . Diagonalizing further by means of unitaries, it follows -iX 0 that C contains ( ) where X and a are positive 0 ia diagonal matrices. Averaging over permutations of the X and aj (obtained from conjugations from K), we see Z E C.
Now reverse the argument to show C0 c C . The -ix 0 unclear step is that any ( ) as above can be obtained from Z by conjugations, convex combinations, and scalar ,-ix 0 multiplication. Now the effect of conjugating ( ) as
above, where all Xi+ a > 0 , by a,a 1 E A and averaging,
-iZ0 is as follows. Any can be so obtained from -ix 0 IC 0 where T anda are related to X and a as
follows. Take ij such that 1 S i p , I S j - q , and
c>0. Then X = X ,o =Ck for A2/ i and k/#1j,
and Y' = X + c , a = a + c . (We emphasize this only
depends on X. +a > 0.) It is now clear how any i 0 C) , , > 0 , can be obtained from Z , so C0 C That C0and 7 have the asserted forms follows exactly in the symplectic case. %/- Cl follows from the exclusion of the case p = q = 1 earlier. The situation is clarified by the
following
-ix 0 Lemma 12.2. If X = ( )E h , X E Cl iff i, 0 la + a1 > 0 . This is clearly equivalent to: 2c E E 64
such that X > c > -a V i,j . Thus either all X > 0 or all a > 0 , but not necessarily both. XE 0 if c = 0 may be taken.
Proof. Let y = E Cn satisfy y / 0 and V(y,y) =
Then p q iH(Xyy) = u + a 2 i=1 J=1 p q > c( 3 I 2 3 I-v2) = 0 i=l 3=1 the inequality being strict as y / 0
Thus C0 / C1 as, for example, in su(2,1) -3i 0 0 o 0 Li 0 E C 1 - CO , and other examples show 00 / 1 0 0 2i) 70'971E C is clear from Lemma 1.1.
To show C1 is open we need the fact, proven by classification of the orbits in the next section, that each ' -i X 0 X E C 1 is conjugate to a Y = () E h where all
%. + a. > 0 . Then Se > 0 such that Y - CZ E Cl also,
so Y E (Y- CZ) + C 0 QC 1 ,so Y , hence X ,'is in the interior of Cl
The last assertion about open cones follows just as
in the symplectic case from the maximality of Cl , which we
now prove. Let C EC, be closed and contain CO . Clearly then
-Co C* so -C% C* , hence C Q -C = C by Theorem 12.3
below and Theorem 1.4. 65
Theorem 12.3. C* = Th1
Proof. 2 -fl follows from the minimality of C0
E S and -C0 Ct so -C% , hence -C1 QC5 by duality.
If now Xg9 -r 1 sav/o such that 'i(v,v) = 0 and iH(Xv,v) > 0 . By Theorem 10.1 B9 (X,Y) > 0 for some
Y E 'C , so X 0 C also.
Corollary 12.4. Cl (resp. CO) is the topological interior of Cl (resp. C)
Proof. The assertion for Cl follows as before (Corollary
8.3) from the maximality of C1 . We cannot use this argument for (O)O , however. But ( o)o C C, , and if
X E (CO) 0 Q , X is conjugate to a Qx 0) such that all X.,a. 0 , again by the fact we prove in the 13, next section. But in fact all %Va must be positive as
(0)o is open and in ~CO
Remark. One can see already that C and C intersect nontrivially by examining particular diagonal matrices.
13. Orbits in Cl of su(p,q)
It is clear from our description of C in the last section that, in examining which types I can lie in C 1 we obtain only necessary conditions on A by considering the possible indecomposable types in C. Also, we must 66 examine all the orbits as it is not yet clear that all elements of C are elliptic. Still, the necessary conditions are quite restrictive on the possible' indecom- posable types.
Theorem 13.1. Among the indecomposable types for u(p,q), only A (C) and &(C) (C=-C) are in t .. (We define
Cd , CO , etc. for u(p,q) the same as for su(p,q) .)
Proof. Consider first a type Lm(c,-C) for C / -; , containing (A = S+N,V) where V = E+NE+... +NmE
E is two-dimensional, and there is a basis e,fj for E such that H(f,Nmf) = H(e,Nme) = 0 and H(e,Nmf) = 1
Let a,b E T . If m 1 , then only the cross-terms in the following expression contribute:
H(A(aNm-lf+be),aNm-lf +be) = -2iImba and this is of indeterminant sign.
If m = 0 , let Ae = Ce , Af = -Cf , H(e,e) =
H(ff) = 0 , and H(e,f) = 1 . Then H(ae+bfae+bf) =
2 Re ab , so let ab = ci , c E 3R.. Then H(A(ae+bf).,ae+bf) =
2ciReC , Re C / 0 , and c > 0 or c < 0 are possible.
Finally, consider e(C) for C = -C , V as above.
E is one-dimensional, A = S+N , and S has eigenvalue
C . The case m 3 is quickly ruled out just as in the
symplectic case. Next let m = 2 . If uv,w E E set 2 2 y = N2 v + Nw + u , and note H(y,y) = 2 Re H(u,N v) - H(w,N w) 67
2 2 2 and H(Ay,y) = (2 Re H(u,N v) - H(w,N w)) + 2i1Im H(N w,u)
Take u,v,w all nonzero such that H(y,y) = 0 . As the phase of w is rotated this is maintained but iH(Ay,y) takes both positive and negative values.
The remaining cases are indeed in Cl , and lie in u(l,l) or u(l)
To aid in understanding these types ( , we record a convenient isomorphism between sp(l,R) and su(1,1) . It is given by conjugating by a particular 2x2 unitary, coming from a "Cayley transform". In fact, we lose nothing in letting our 2x2 matrices have square matrix entries.
Let A , B , and C be real nxn matrices, B and
C symmetric, and let D=A +XiI ,XE ER. Then
D B B xiT o ), C -D C -A t +
Let U= (1/I) , so U~ = U3* = (lAI )h.
Then 'i 0 A-A +i(B-C) -A-A +i(B+C)' 2 t tI 0w u ) -A-A -i(B+C) A-A +i(C-B)) which is in u(n,n) . Letting X = 0 , note this embeds sp(n,R) in su(n,n) . a b %1 0 If n = 1 then + E sl(2,R) + RiI c -a i0oi corresponds,, under a Lie algebra isomorphism, to 68
xi 0 i(b-c) -2a+i(b+c)J i 0 di S [a i) -2a-i(b+c) -i(b-c) o Xi $ di,
Recall represents a positive elliptic type sc - 2 iff b < 0 , c > 0 , and -a - bc > 0 , and a positive
nilpotent type iff b 0 , c 0 , c-b > 0 , and
-a2 - bc = 0 . These conditions correspond for
d > 151 , and d > 0 su(1,1) to d > 0 , rdi9d-di 'E 9 d= 15| . Note K d is conjugate under SU(1,1) to -d'i 0 9 dii iff Idl > 1$1 0 d'1:1
We relate the types AO,(iX) (X E ]R) to particular matrix representatives. Recall the notation AL(C) means
E is spanned by e with H(e,e) = 1
In %. +(iX) for X < 0
and 6A(ix) for X > 0.
Both correspond to the lXl (ix)
I CAQ00 and C (vacuously). 6 (0); (0)
In 0 Cl. A6+(0) ; corresponds to, say ( t
Cn - (C U 0) . 6(iX) for X /0 corresponds to X i Ki0 1 -i i & 0 1%i, -iL
The above facts follows from the lemma below, proved
by computation. 69
xi 0 -di $ Lemma 13.2. Let X = + E u(1,1) . Then 0 sS di X E C0 iff d > 0 and d2 > %2 + 1S1 2 (a consequence is 2 d-X,d+X > 0) , X E07 0 iff d 0 and d 2 + |j2
X E C iff d > 0 and d > 1S , and X E iff d 2 0 and d 2 I$I
Finally, we list the types in for su(p,q)
Proposition 13.3. The types A in C are of the form
(7) A=$iX)+ .00 +A+(-iX ) +a- (icy,) +...L+A(icyq) where all X + a > 0 and EIX. = Ea . A is furthermore in C0 iff all X ,a > 0 .
The types A in C0 are of either of two forms:
1) (7) with all XVa1 0 (and such a a may also
be in Cl) , or
2)
(8) A (-iQ)+... +fl +(ix) +A(-iX) +... +A(-i)
2 times
for X=0 ,some 2 with 1 2::.q ,all
.,a. 0 and ZX = ja . Such a type is on
the boundaries of C0 and C .
The remaining types in C1 are those either of the
form (7) with all X + C 0 , tX. = 7.a , and some
X. = -a. 0 , or (8) for X 7 0 , some 2 with q , , , a 70
. and a. satisfying . 2 X -a i,j , and I 3 21 I p-2q-z
21X + I =I ~ i=1 J=1
Proof. It is clear from the classification of indecomposable
types in Cl and the computation in the proof of Lemma 12.2.
14. Classification of Open and Closed Cones in su(pq)
To classify some of the invariant convex cones in
we need a convexity result (Theorem 14.2), the proof of
which depends on the
Lemma 14.1. All orbits in C are closed.
Proof. Let Y E C1 , and suppose
X=
is in the K-orbit of Y , for Hermitian X,a . We need
some measure of the finite distance of the K-orbit of Y
from the boundary of C1 :clearly Tk > 0 such that
2 iH(Yv,v) kflvfl \-/v E n with H(v,v) = 0 , | - the n usual (K-invariant) norm on C . The same holds for X ,
and a simple computation shows that a consequence is
(Xii+ T j -/2 1$ 1 -k
1 i p, 1 :5j q. for i,j satisfying I
Now let Ad(gm)Y - 0 E as m - . We can write 71
gm according to G=KA+K. Let
coshe1 sinhf E G a =d 3E A+ ,0 20 . sinh e. cosh9)
Letting Ad(a)X = , then for 1 q
L = -ix. cosh2 e. - ia. sinh e. - 2i sinh 9 cosh9 Imp. and
M = ia.cosh2 e0. + ix sinh2 9 .+ 2isinh9.cosh9.Im..
Also,
iL4 -iM cosh 2%j(X4.4 + a.) - 2I$ | sinh 20a (sinh 2e .)2k
and note fxf2 + y| 2 > x * Y 21 xyE C
Now return to the Y= Ad (g)Y . The Be-norms of
the YM remain bounded by some fixed number as m > and Be is K-invariant. The above estimate shows that
the allowable A+ components in gm must lie in some
compact set. As K is compact the g have some convergent subsequence, proving the lemma.
We recall a result of Horn [11]: given an Hermitian
matrix A with eigenvalues X ,l.X n , the set of diagonal
entries of all conjugates UAU~ , U unitary, is precisely
the convex hull of the points (Xp(l),...,Xp( )) p
a permutation of (l,...,n) . We will only need the
(presumably) easier half of this theorem, where "recisely" 72
is replaced by "contained in".
Definitions. 1) Let V,W = (X ,..,j , ,..Y E .
say V is in the noncompact convex hull of W if V is
equal to a vector in the convex hull of the set
( r(l)''''r(p)'s(l)'''..''s(q)):rspermutationsof respectively)
plus a vector in the closed positive orthant of ]p-
Clearly this relation is transitive, and the noncompact convex hull of a point is closed.
2) If X = ( , call R = the p-norm i,j of X (although it is not a norm on T ).
Theorem 14.2. Let X E Cl , and let X represent the type
&= & (-'xl) +.. + 6+(-ix ) + + **+ !& q~)
as in Proposition 13.3. Let the diagonal elements of X
be (-ia 1 ,...,-iapib ,.,ib ) . Then (a1 ,...,aPlb ,..,bq) is in the noncompact convex hull of (X...,XpCI0,.,a).q)
The proof is dependent on the following
I thank Renee' Chipman for an accidental wish for a "good
flow" which helped to get this theorem and its lemma
unstuck. 73
Lemma 14.3. Let X E Cl - k . Then there exists Y in the orbit of X with strictly smaller p-norm such that the diagonal of ( )X is in the non-compact convex hull of the diagonal of ( .)Y
Proof. By Horn's theorem we may assume
-iX 3 X = Sia, where X and a are diagonal. (The p-norm is
K-invariant.) As in the proof of Lemma 14.1, . +a. > 1 3J 2ISij Vi,j . Set Y = iS /(X +a ) , and consider
A B' Ad(exp eP)X = Y(e) = B CI where
0 Y P=P ~YQECJ 0 We have
2 A = -iX + C(yS -Sy) + O(e)
2 C = ia + e(YS- SY) + O(e ) , and
B = S + ie(Ya+Xy) + O(e2
The diagonal entries of A and C are q -iX + 2ic j (IS I 2 /(Xi+7)) + O(e2 ) j=l p and ia - 2ie )+ i=l 74
and we have B.. = (1-e)$ + O(e 2
q Note that if 3 5 H2= 0 for some i , then the J=1 ith diagonal element of A is equal to -iX. for all e and likewise for the jth diagonal element of C if P siJ 2 = 0. Thus if e > 0 is small enough Y(e) satisfies the requirements of the lemma.
Proof of Theorem 14.2. It is not clear that the lemma above can be strengthened to produce a sequence converging to an element of k , but Zorn's lemma can be used. First define some notation. If Y,Z E o write
Y Z if the p-norm of Y is s the p-norm of Z, and the diagonal of ( _7)Z is in the noncompact convex hull of the diagonal of ( 7Y . Let
Q = LY E Ad(G)X: Y z X} . We apply Zorn's lemma to ( , ).
By Lemma 14.3 it is clear that a -maximal element Y of
must be in k , and if such an element exists the
K-diagonalization of Y exhibits the type of X , proving the theorem.
To prove the hypothesis of Zorn's lemma, first note that C is a bounded subset of as the Killing form
is constant on the orbit of X . Let ! be a simply ordered subset of 2 , considered as a net indexed by
itself. The closure of 4 is compact, so I has a subnet
i converging to Y . Y is in the orbit of X by 75
Lemma 14.1, and clearly the p-norm of Y is S the p-norm of any Z E 4 , hence any Z E by the cofinality of .
Finally, if Z E I express the diagonals of (0 DZ as a vector in the noncompact convex hull of the diagonals of (0i -i0) W , where W is any element of I where W ' Z The set of possible nonnegative vectors in these expressions is bounded as Q is bounded, so the expressions must have a convergent subnet. The limit expresses the diagonals of % 7)-z as a vector in the noncompact convex hull of the diagonals of (.)Y Thus Y E Q and Y 2 Z V Z E , proving the hypothesis of Zorn's lemma.
It is now not difficult to classify the open and closed invariant convex cones in oy. We need some notation.
Let
EB =001 ' 1 P" 160CTq E ]Rp Z.X.E= Bica and
X. +C. > 0 V i,j) 1- a and E0 = Ce E B1 : all components of e z 01 . Let $ be the group of permutations of the X's and permutations of the a's acting on E . Let C+ = CC E el: 0a C5 0 c0 C = CC E 0+: C is open) , C<= (C EC+ : C is closed) 3 (all 9-invariant open convex cones in E
0 containing (E) ) , and let
c = (all &-invariant closed convex cones in E
containing B0 ) 76
Let i be the collection of all bases ((e ,f
I = l,...,p, j = 1,...,q} ofTn such that H(e,e ) = ,
H(fJfk) j and H(e ,f ) = 0 V ,j. Given
P = (e ,f ) E i define
X -+ ,f )...,iH(Xfqfq)) ,(iH(Xefe),...,iH(Xepfep),iH(Xf
As noted previously, 1M 9 (X) E 1 if X E Cl , and
M (X) E(E1) if X E C1 .
0 C Theorem 14.4. Let 03 E a+ and a +CE. Let Ec. be the convex cone generated by (M(X): X E C. , p E fl for i = 3,4. Then EC E 0 , E E Sc , and these mappings
o: 60
co: aC are bijections.
Proof. We deal with the open cones first. To see that E is open we show that EC3 contains33 a neighborhood of each
M (X) , X E C3 , P E . By Theorem 14.2 M(X) is a nonnegative vector plus a convex combination of other vectors
S = ((rl) ' ' '*'sq cps() in E coming from the type of X . BE must contain a 3 5 neighborhood of each element of S as the intersection of C3 with h must be open in hE. B also contains 77
all positive vectors as C0C 03 . A simple picture now shows M (X) to be in the interior of EC . Thus EC 3 3 is open and clearly then F E i0 . 03 To show surjectivity, let D E d0 , and let
03 = (X E Cl: M (X) E D V y E U) . Clearly C3 is invariant, convex, and contains C0 . To show C is open, we may take X E h fl C3 .3 Ze > 0 such that the diagonal of
(X - eZ) is in D . By Theorem 14.2 M9 (X- eZ) E D
V pE ,so X -Z E C 3 , and X E (X- eZ) +0 0 gC03 proves X E (C3)0. Thus C3 E c0. That E = D follows 3+ 3 from Theorem 14.2.
If now C2 EC+ and E =D then clearly C2 0C3 By Theorem 14.3 E is simply the set of diagonals of
(_)(cn h) for any open cone C , so C2 and C3 clearly contain the same elliptic orbits and are equal.
Thus E is a bijection for the open cones.
Now let 0 E . The set S of diagonals of C4 &+ (7> f h) is closed and in E0 we will see that it is all of E . If MP(X) f9S for some X E C4 p E , the same is true for X + eZ E C4 l Cl for some e > 0 , and Theorem 14.2 may be applied to reach a contradiction. Thus ECE
To show surjectivity, take D E dc and let
C4 = (X E ~l: M (X) E D V cp E f} . Then 04 E C0 is automatic, and clearly E C D . Further, equality holds again by an application of 14.2 to X + eZ . Finally, suppose E = D for C2 E Cf. C 0 = 2 + 2 04 again, and E2 = implies 02 04 C2 fl h==~l C4 n -,so o C2 0 and C4 have the same elliptic orbits. Suppose then
A +I<.i)+ *** +i) Ae
where
a-A+ L+l) + 4. +A+iX) +fA a) +...+A aq)
as in Proposition 13.3, is a nonelliptic type in C4 . Then
+ = A+(-i%) + ... + A7-i ) + ae
is an elliptic type in C4 as C4 is closed, so A is
in C2 , too. But then A is obtained back from A by
adding on t+(0) + 1+(0) + ... + A(O) (which is in
Z0Q 02) the appropriate number of times. Thus C2 =04 ' and the proof is complete.
The classification of all invariant convex cones
in does not seem hopeless, and probably involves ideas
similiar to those successful for sp(n,IR) . Just as for
those groups, the set of elliptic orbits in the closed
maximal cone do not form a convex cone: we saw that a
convex combination of Sp(2,) transforms of
0 -1) 0 0 . 0/ - 1"\ 0 .O.\ )is ® (J(0. There is asimilar S 0 ( ) symmetry here, existing in any SU(2,2) subgroup of
SU(pq) . Tf (ee 2 'fl'f2 ) E , then one may check that 79
E = e - k(e 2 -f 2 ) E 2 = e2 + k(e1 +f 1 )
F = f1+ k(e 2 -f 2 ) F2 = f2 + k(e1 +f1 ) for k real, is also orthonormal in (14 ,H) . It comes from the nilpotent
0 k 0 k
-k 0 k 0
0 k 0 k
k 0 -k 0
in su(2,2) ,which has square zero.
Furthermore, one may calculate that averaging the transforms of the elliptic type A+ (-ik) + 6+(-i) + C~(-iX) + 6(id) (X + d > 0) for k positive and negative gives
61(-iX) + &+(-i%) + A(id) . There may be other SU(p,q) symmetries involved in classifying the invariant cones in su(p,q) , but based on the analogy with sp(n,]R) I suspect not.
Finally, there is a simple geometric picture that one can use to see how the various closed and open invariant cones in su(2,2) overlap. We consider the sliceX + X2 =C 1 + a2 = 1 in E = (X 9X1 2 ' 1 'a92
X .+ ' 0, X + X2 = a1 + a2 , perpendicular to (1,1,1,1), regarded as a z-axis.
Define x = X1- 2 and y = a1 -Ca 2 . As closed convex cones, E is generated by (1,0,1,0)., (1,0,0,1),
10 80
i(3 -1, ,1 , (-l 3, ,1) , (1,1,3.,-l) ,and (1,1,-1,3) . (In general, Cfo has pq generators and Cl has p+q generators.) They correspond in the (x,y) plane to
(x, y) :x = tI, y = tl) and (0,t2),(2,o)3 , respectively. We have
y
S2
Sl
The minimal cone corresponds to the inner square S the maximal to the outer S2 . Closed invariant convex cones correspond to closed convex sets containing Sl and contained in S2 which are invariant under reflections about the x and y axes.
The outer automorphisms of su(2,2) are generated by the inner as well as "time reversal" X ->X and
space reversal" X -+ X ) . The former takes C to -CO , and the latter has the effect in the diagram of a
reflection about x=y, or, up to inner automorphisms., a 9o
rotation. Thus there are clearly lots of invariant convex
cones also invariant under that second outer automorphism. CHAPTER IV. Invariant Convex Cones in so(2n)
15. Conventions for so(2n) .
We take n 3 ,and let A= [n/21 , the greatest integer i n/2 . As in section 3, fA A,B complex nxn matrices;
A A skew, B Hermitian k= X E : X real ET: A sk ew, B symmetric -E = p =X E :iX realj =iB - E:A,B skew
r0 D l E k: D diagonal h = ,D -I c = Z where Z = .
Let d. (j = 1,...,n) be linear functionals spanning the dual of h:
0 DI
Then noncompact roots) = ti(dJ+dk): I j K K Sk)
tcompact roots) = i(d-d): I j < k ! n)
and the positive roots we take to be the above with "+" only.
(i(da+d9)i(d -),...,i(d 1 -d 82
Let l'a j
where D=-iE - iEkkA=iEjk +iEkj ,and
B =-iE + 1Ekj0
We take Z0 =ti(d1+d2),i(d3 + d4),..,i(d2A-1 + d ) so
D -D22' '2-1,21-1 d2 ' h F D E h: D 0) D2 + -'=0 (if it exists) 0 D + ~ E h: -D 22' '' 21- 1 , 21- 1 =-D21, 2,t D 0
UD) if' Z0= E h where all D -1 n even;
D . = - for j Dnn if n odd,
so
Cc ,
Ot= o C 1.) E R c i
Thus Z=Z0 iff n even, and h+ / 0 always. The cone in h obtained from Corollary 3.2 is all ( D in n where diag D d = (-d,-d for
d > 0 if n is even, and diag D= (-d ,-d ,..,d
where 0 < e d. if n is odd. 83
Define B(X,Y) = 'Tr(XY) for XY E '/. If
A B C D TY - E , X = ,
B(XY) = - Re 3I A C 1 - Re B3B. .D . itj i, j The Cartan involution is X -7 , and
A B C D Be _9_,$ = Re Ai i + Re B .jDi A-.+-B..D.. C1 ,7 3 1] 2n Define the complex symmetric form (,O) on (C by
T(X,y) = x j=l
and the symplectic form a(-,-) on C 2n by n
y v = Re (y 5 - xfr)) j=l for x,y,u,v E Tn . In fact is precisely the complex-
linear endomorphisms of V2n skew with respect to T and 0 -I C . Also, 4(-,-) = Re (-,a-) , where C = ( -) 2n I 0 where C is complex conjugation on 2 . All elements of commute with the anti-linear a .
Theorem 15.1. Let w E V and 2nnX E o4. Then C(Xw,w)=
Be(XY) where Y E 0 , and C(Yv,v) 0 for all v E T2n
Proof. Note T(Xu,au) is automat'cally real for X E , u E C Let w = () and X = . Then Lt(u> A uEC . 84
a(Xww) = (fu) i + (AV) fl - (Au) ~r - (Bv)2~r}
5 =3 B.jBij ( (-ai U.u 4 -Viv.~ 4 ) +R+ Re AlaA..uv i v j- 5jil~. i,j i~j A B C D = B_5 )= Be,(XY)
where =uv -uv , D.= -v v -u ij Ji 13 13 ij Cij iVj i Clearly Y E . If v (x)E® 2 n
a(Yv,v) = -L xDi 3 yRj xC + 3 XCi y -Yi y.3 i,j i,j ij
2 2 = 1 + I1 + 1 2 + ||
+ 2 Re () - 2Re (<,y>)
I- J2 + +j 2 0
<-,-> being the usual complex Hilbert structure on T
16. Conjugacy of Positive Elliptics in s3* .
Let (H, < , >c) be a real Hilbert space and
J: H -+ H an orthogonal with square -I . Let V = H e iH
and extend J to V the obvious way so that iJ = Ji
Extend <,>0 to V so that (7., <, > ) is a complex
Hilbert space with complex structure i . We write
2 1x11 = 0 for x E V , and |IBII for the usual norm
with respect to < ,> for any linear operator B . In 85
this section all operators are bounded and commute with i
Finally, extend < , > on H to T(,-) on V by complex bilinearity so that T is a complex symmetric form on V . If C, is the anti-linear conjugation of V with
respect to the decomposition H 0 iH (S = +I on H
-I on iH) , then T(-,) = <-,0-> 0
Letting 4 = iJ , Q is an Hermitian operator on
(V, <, >0) with spectrum (+1,-i) . In fact, the +1
eigenspace of ; is V+ = x +iJx: x E H) and the -1
eigenspace is V_ = [x - iJx: x E H) . One can check that
T vanishes when restricted to either V+ or V.
Our notation is chosen to correspond with sections 7 and 11, and this section is a further restriction on the
situation in section 11. In particular, the J , , and
< , >0 correspond.
Let a = JS = CSJ , an anti-linear operator with
square -I . We have T(a-,c-) = T(-,-) , and note that
a switches V+ and V_ . As before, we define
a(-) = - Im
a real symplectic form on V . Furthermore,
(9) c(-,-) = Re T(-,C-)
Defining K-,-> = Re<-,-> + i(,) , (V, <,->) is a
complex Hilbert space with complex structure J
Define 86
SO(V) = (g: V/-+ VI g invertible, g commutes with i
and a , and T(g.,g.) = T(.,.)} and
(B: V-+V B commutes with i and a and is
skew for TI3= so(7N)
Note T could be replaced by the symplectic form a in the above two definitions by (9). Also note J E ol SO(v)
The condition defining the "maximal compact subgroup"
K of SO() is the same as in the previous sections, i.e. g E K iff gJ = Jg , which here is equivalent to g leaving the space H invariant. (Not the "K" in 16.2.)
Lemma 16.1. For all X E , (A-,-) = Re T(A-,cT-) is a real symmetric form on V for which i and a are orthogonal. T(Ax,ax) E MRV x E ZW
Theorem 16.2. Let B E y satisfy s(Bx,x) ' kVxE%' , for some k > 0 . Then there exists S E SO*(?V) such that
SBS~ = JH = H J , where H is a positive self-adjoint operator in (V,< , >) (and (7,< , >0)). Also,
3 1 3 1 2 2 (1B1 2 /2 )I12xI 2

(k /%11B11 / )!x1 Vx E {
Proof. Again we follow the proof in section 7. Defining
9( = (B,) , we polar-decompose B = K-B in (hg)
As i and a7are orthogonal for and commute with B K and J-B2 also commute with i and a , and again
K Ej n so(zr)
Defining g1(.,-) = C(K-,-) and ,>
9(,-)+ ic(,-) , (V, <, >) is a complex Hilbert space with complex structure K . It suffices to find a unitary equivalence
which commutes with both i and a . As before, i-linearity follows if S takes the +1 eigenspace 2- (resp. -1 eigenspace -_) of iK to %+ (resp. Y_) . Note a switches-6+ and ;_ also as a is anti-linear and commutes with K . Thus clearly all of and 4 have the same dimension, and the desired S is obtained by taking any unitary equivalence (L+,< ,>1) - (+'' and extending it to V by requiring it to commute with a
The rest of the theorem follows as before.
17. Noncompact Convexity in so(2n
Just as the orbits of interest in su(p,q) were associated with the non-simple u(1,l) and u(l) , we will have occasion to examine the orbits of so( 4 ) ~ sl(2,R) su(2) and so(2) ~ so(2) in classifying invariant convex cones in so(2n) . However, n 2 3 is assumed in what follows unless otherwise stated. 88
To relate the notations of sections 15 and 16, we 2n 2n let H = R with the usual inner product, 14 = J (0 t) (regarding R2 n = n 9BRn)C, S complex 2n conjugation in , and a = J . There are basically three forms on C invariant under SO(2n) , all closely related:
1) the complex symmetric form T
2) the symplectic form a(,) = Re T(-,C-) (both defi ned in section 15), and
3) the Hermitian form H0 (-,-) = -iT(-.,c.) with signature (n,n) alluded to in section 16.
As with the earlier groups, we define a class of bases that are transformed by G = SO(2n) . Call
= ((e ,f4 )n: T(ek'eL) ==0' ' 'k 5 T(e4 ,f = jk and ae = if.,
af. =ie the set of G-bases. Our standard basis is e =(e iej)/XI, f. = (e. - ien )/42 where 2n e ,...,e are the standard unit vectors in V . For Cp = (e.,f.) E , note cp is a basis for H in the sense of section 14 for su(p,q) , and our notation is consistent. We will need several observations about "matrix elements" '1(X-,) . Note that (a-,aa) = (,r) , so
YX E and V(e.,f.) E T we have a(Xe4 ,e.) = (Xf.,ff) 89
Also, if a = (ei + yfi)/N fW for YI = 1 , then c(Xaia) = 1(Xeile ) . (Such ta)l are quaternionic bases in the sense of section 4.)
In particular, if X = _ E and a is the -B ~ unit vector3. , then a(Xaa ) = -Bi . (Also, n a( ' Q), = 3D (IvI2 + 1w) 2) where D is j=l diagonal.) If e = ' i%2 , then T(ee) = -T(e,ae) = 0 and c(Xe,e) = -B - B +~ 1 1 2 2 2 ImB1 2 * (AB Any g E SO(2n) has the form g = - 9)where
AtB =BA and AtA +BB = I , or equivalently t -t A-B tt t - t - 1/A - AA + BB =I and ASB=B . We have gA= t) and g is in the maximal compact K 1 tU(n) iff g is a
real matrix.
Definitions. 1) Let
= C2 (X E 1(Xe,e) > 0 V e / 0 such that T(e,e) T(e,ae) = 0) A B 9 2) If X = (_A E 1 , call (-B,...,-Bn)E n the -B A G-diagonal of X . (It is essentially the projection
of X on the maximal torus h 3) Let
0 = (Xl'4''n) E En: o 3 X'.' il,.,n) j=l
and E 2 x,..X E ln:XV + VX 0 '9i /j 90
4) If V,W = (l,...,n E n, say V is in the noncompact convex hull of W if V is equal to an
element of E plus a vector in the convex hull of
p(l)j,...,p(n)) p permutation of (l,...,n)}
5) If X = Y + Z , Y E k , Z E p , call NFBTZT the p-norm of X
Lemma 17.1. X = E C2 if D is diagonal and iD. + D. > 0 V i / j .
Proof. Let e = ( +idI 3d/0 satisfy T(e,e) = T(e,ae) = 0 , where c,d,f,g E En . The conditions are equivalent to
c-g = f-d , c-c + f-f = d.d + g-g , and c-d + f-g = 0 2 2 In terms of vectors in ®n ,they become 1f c+iff1 = jd+igf1
and = 0 ( < , > the sesquilinear form on Cn
Recalling the isomorphism R~2 n , n and K :U(n)~ , there
clearly exists g E K and s > 0 such that
g(sQ) + is dg) = el + 2 ~f
Then 6(Xe,e) = s- 2 -((gXg1 )ff) . gXg~ 1 is a real
matrix, so a(Xe,e) > 0 follows from the remark above
(on the a(Xe,e)) and Horn's theorem.
Lemma 17.2. The G-orbit of any Y E C2 is closed.
Proof. 3k > 0 such that a(Ye,e) k e1 2 V e such that
(e,e) = t(e,ae) = 0 . (Note -i (e,Ce) = H0 (e,e) , and
the proof is the same as for su(p,q) .) The inequality 91
is also true for any X in the K-orbit of Y . We compute Ad(a)X for a E A+ . As
S 0
a0 S - cosh 9 -i sinh 0
i sinh 9 cosh 92 where S = * cosh 9 -isinh 8
i sinh 91 cosh 0
8. 0 , it suffices to multiply
cosh 8 -i sinh 9 -c e+ir cosh 9 -i sinh 9
i sinh 8 cosh ) e-ir -d i sinh 8 cosh 9 where c,d,e,r are real. We have c+d- 2jrJ - 2k by our choice of k and a remark earlier. The diagonal elements of this matrix are -ccosh2 9 - dsinh 9 - r sinh 29 and
-d cosh 2 8 - c s inh2 9- r sinh29 , and their sum is
-(c+d) cosh 29 - 2r sinh29 ! (-(c+d) +2fr}) sinh 29
: -2k sinh 2 .
Now argue as in the proof of Lemma 14.1, using the estimate above.
Theorem 17.3. Let Y E C2 . Then Y is conjugate to an X = (~-) as in the statement of 17.1, where the
G-diagonal of Y is in the noncormpact convex hull of the
G-diagonal of X . 92
The proof runs exactly as the proof of Theorem 14.2, and is dependent only on Lemma 17.2 and the following
Lemma 17.4. Let X E C2 - k. Then there exists Y in
the orbit of X with strictly smaller p-norm such that the G-diagonal of X is in the noncompact convex hull of
the G-diagonal of Y
Proof. By Horn's theorem we may assume X= (+ ) +(i( B)
where A and B are real and skew, and D is diagonal and satisfies the hypothesis of Lemma 17.1. If G and H are nxn real and skew,
G HiA+e(S+iM) iB-D+e(T+iL) .d .
2 Ad(exp eC(iH -iG ))X = i+~eTi + O(e ) QiB+D+e(-T+iL) iA+e(S-iM)j
where S,T,L,M are real and nxn
S = AG - GA + BH - HB
T = HA + AH-BG - GB
L = -GD - DG
M = HD + DH
Therefore we set
G =(B 4 /(D +D ) and H 1 =(-Aii/(D2+D)
so that
(B + CL) = (-B) 93
= (le)A. ,) and n (D+eT) ii = -D + 25 ((A )2+(B)2)/(D+D) . j=l As before, if T = 0 the corresponding O(e )
terms are zero V s . It is clear that (T. n E E as ii i=l a A and B are skew. Thus the desired Y is obtained if
e is positive and sufficiently small.
18. Minimal and Maximal Causal Cones in so(2n)
In addition to the definition of C2 in the last section, let
C1 = CX E : $(Xv,v) > 0 V v 3' 0 in T2n, n
0= (X E C1 : ja(Xe.,e ) > 1(Xe1 ,e1 ) V (e4,f1 ) E } j=2 and now let C be the collection of all nontrivial
invariant convex cones in .
Theorem 18.1. Ci (i = 0,1,2) are nonempty, open, and in
0 . Any C E C, contains either C0 or -C0 but not
both. CO 'YC,, and 02 are obtained by changing ">"
to " " in the definitions. Any closed (open) cone in
C- containing C0 is contained in 02 (resp. C2 )
1 C2 and that the C areProof. in C Clearly C r I
provided they are nonempty. In fact the inclusions are
proper: if X = , D diagonal, and diag D is 94
(n,1,...,1) then X E C - co , and if it is (-1,2,...,2) then X by Lemma 17.1. E C2 - 1 We show J E c 0 to prove Co 0 . Using G=KA+K, it suffices to see that n I (Jasi,as ) > a(Jasj.,a$ ) j=2 where a E A+ and = (C E R2n (i,, = 1,...,n ) , the S corresponding unde 32n n to an orthon ormal 2 basis of Tn . Setting N. = IcI2 + jd'I , we hav e n n N =1 and N= V i,A T=o Taking an a as in the proof of 17.2, we have n $(Jas.,a.) = 7 (cosh 29 +/
(where 9 0) , and, dropping the factor of 2 from
9 , we must show n n n (cosh 9 )NM < I(cosh e[ 1/2])Nt c.=1 i=2 .=l n (cosh9(1.+1/2] 1 - N) or
2 cosh 9 + .. + 2 cosh 8 + 1
1 2+ 2Z-1 22.+1 > 2(N1+N 2 cosh1 + ... + 2(N1 +N 1 )coshe + 2N
if n is odd. (If n is even the same with the 1 and
2N Il''N2~ removedeoe mustut beeson shown, but Ii-this is now clear- fromf 95
n N? = and n z 3.) In that case it suffices to show
2 cosh 9 + 1 > 2(N1 +N 2 ) cosh 8 + 2N3 where 0 S N ,N2,N3 and N 1 +N 2 +N3 5 1 . But the r.h.s. is
2(1- N 3 ) cosh 9 + 2N 3 and 1 > 2(1 - cosh 8)N3 is clear.
Thus C0 is nonempty. Cl is open by the same proof as that for C. in sp(n,R) . By Theorem 16.2, any X E C is conjugate to a (- D) , D diagonal, where all D. > 0 . D is clearly unique up to a permu- tation as D essentially gives the eigenvalues of X acting on C2 . By 17.3 X E iff T0 it is conjugate to a
0 Y = ( D)D, diagonal, where diag D is in (E ) Co is then open by the continuity of eigenvalues. We have already seen that any Y E C2 is conjugate to an
X as in 17.1, so C2 is open by the same argument as for
C1 in su(p,q) .
Now let C E C; we may assume J E . As before
CO being open implies SW E Cflco so J E C by the above conjugacy in CO . (Average over permutations of
the G-diagonal of (0 .) Now 0D -) (where
diag D = (lle,...,c) for 1 > e > 0) can be obtained
from averaging the transforms of J by a,a 1 E A , as in
Corollary 3.2. It may be seen without much difficulty that
the convex cone generated by permutations of (1,1,0,...,0)
is precisely E 0 , and likewise the convex cone generated
by permutations of the above (lle,...,e) is just (E )0.
Thus CO is the invariant convex cone generated by J , 96
and C0QC That %C ,and 02 have the asserted forms follows exactly as in the symplectic case. Finally, the last two statements follow, as in the proof for su(p,q) , from
Theorem 18.2. 0 =-C and C=-C
Proof. Be(XY) > 0 '/ X,Y E Cl follows from Theorem 16.2 and the definitions of C and Be,so -CQO. The converse proceeds in the by now familiar way from
Theorem 15.1. -02C C follows from the minimality of iCO
Conversely, if X 2' 02 then, as in the proof of 17.1, B there is some K-conjugate of X of the form Y =(- ) -X 1 a-B where B has ( ) in the upper left corner, and 2 0 L X 1 + X- 2Ima < 0. But now W=( ) where L is 1 2 .- L 0 all zeros except for in the upper left corner,
is inC , and BYW) = X + x2 - 2Imz < 0 . Thus Y ,
and hence X ,is not in -05.
Corollary18 .3. COC1 ,and C2 are the interiors of
their closures.
Proof. As we have shown that all X E C2 are conjugate
to a Y E h , the proof follows that of Corollary 12.4. 97
19. Classification of Orbits and Cones in so(2n)
We shall see that if A is a type in some C2 and
= +... + 6 M its decomposition into indecomposable 4 types, then each t lies in an so( ) or so(2) .
It will be useful to examine the orbits in so(4) , as the same definitions of C (i = 0,1,2) make sense there
(and only C1 for so(2)), and the C can be explicitly described.
The sl(2, E) component of so(4 ) is spanned by
0 0 -I 0 0 -i 0 -i 0 I and K:0i , 0O 0] and the su(2) component is spanned by
0 1 0 0 1 0 0 1 -1 0 0-1 10 and
1 0) 11 --0 0 0-1 O 11 0 1 oJ
We let B1 and B2 be -Tr(--) restricted to the sl(2,R) and su(2) components, respectively.
By the uniqueness of the decomposition it is clear
that if Y E sl(2,R) , Z E su(2) , and (e1 ,e2 'l'f 2 ) E ,
a(Ye1 ,el) = a(Ye 2 'e2 ) and 1(Ze1 ,e,) = -1(Ze 2 2 2,e
Lemma 17.1 shows that Ve / 0 such that 0 = T(ee) =
T(e,ae) , 6(Ze,e) = 0 and 1(Ye,e) > 0 'Y in the
positive cone of sl(2,fl) . Note also 98
-0 0 .0 0
the 4 x4 matrix being in the positive nilpotent orbit.
We see that C0 is empty and " 0 " , so to speak,
is the closed positive cone in sl(2, B) . C2 is the direct sum of the open positive cone in sl(2,IR) and
su(2)
The cone C with self-dual closure (and 18.2 is
valid here, too) is
r Y E open positive cone in sl(2,]R) 01= ~j,Z):} Z E su(2) , and B1 (Y,Y) > B2 (ZZ) That C is convex follows from an elementary estimate.
Theorem 19.1. Only the indecomposables 0(OO) 6A(010)
( and ( where C=-K=/O, lie in t
Proof. We consider first any of the types m( ) or
with m ' 3 . As in the earlier proofs we considered $(Aa,a) where a E Nm-E + E , and found it
to be of indeterminant sign. T vanishes on the c-invariant i-I N E + E so clearly no such type can be in tO2 For the other orbits we take A = S +N as usual.
To rule out the remaining types excluded by the statement
of the theorem, it is sufficient to let S = 0 , as that
orbit is in any closed invariant cone containing the
original type with S /' 0. 99
For 6 2 Q- ,,-C) (C / -) , (N(Ne+w),Ne+w) = found -2(W,N2e) . NE has dimension 4, so e / 0 can be with T(Ne,Ne) = T(Ne,aNe) = 0 . w + Ne has the same properties, and a(w,N2e) can take any value.
For tL(C,-C) (C = -C / 0) or 62(0,O) again let
S -+0 V = N2E+NE+E so let T = +Id. on N2E +E and
T = -Id. on NE . Then T E SO(6) and TNT = -N , showing that N 9 02 as 02 0-2 = For (3,C,,)( / -) again let S -> 0
E has a basis e,f,ae,af where T(e,Nf) = 1 , T(e,Ne) =
T(fNf) = 0 , and Me + Jf is T-orthogonal to
N(Cae +XTaf) . Let Te = f ,Tf = e , T(Ne) = -Nf
T(Nf) = -Ne , and extend it to V by requiring it to commute with a . Then T E SO(8) and TNT = -N , excluding this type.
The remaining types are in so(2) or so(4 ) so has a hyperbolic sl(2,R) component
is never in a 02 . L(3,-Q) (6(0,0)) correspond to a
nilpotent sl(2,IR) component and a nonzero (resp. zero)
su(2) component, so are always on the boundary of 2
From now on reserve the notation At(C,-) for the
case C = -C / 0 only; this is the only ambiguity in the
notation for the indecomposables left. We note that
6 (0,0) is on the boundary of C and 02 but (C,-C)
is on the boundary of C2 but not that of C1 or C . 100
This is quite analogous to the types &1(0) and 4 (ix)
(X 3' 0) for su(p,q) . Finally, just as for su(p,q) representatives of j((,-Q) are a sum within so(4) of a representative of +4(0,0) and a representative of the elliptic type &+(,-G) + A(,-) , which is on the boundary of C2 but not that of l or 0
Finally, as a consequence of the results of the last three sections, we list the types in C2 for so(2n) , n > 3.
Proposition 19.2. Types in C are those of the form
(10) & = A+(C,-C%) + ... + a+(CnCn)
0 A is furthermore inC' if (C11..In) E (B0) Other types in C2 have either the form
(11) A = o(0,0)'~ + + nnA+( 2 2 ) + or
(12) 6 = L~(C1 ,-C1 ) + (C 2 '~%2) + ... + n'~ n
0 for (-
Types in C0 have either the form (10) with
(IC1 I..,KnI) EEEor
(13) A=(0,0) + ... + 4(o06) + A(o 0) + ... + A(oJo 2 times n times
+ AQ 2 ++ 1 '~ 2Z+m l +) . . + ( n ~n 101
with (0,..0,1c 2z+n+iK''.Lnt) EE and 2Z + m 1*
Types in C have the form (13) with 2Z + m I
Finally, types in )02 have either the form (12)
with (- C11,|C2 ''''' no) E BE2 ' (13) with L I or
m 2 , or
= ,-C1)+ -(C)+... + , (14) A
where ( 3nj) EB2 ,$*
It is now easy to classify the open and closed invariant
convex cones in The proof involves no new ideas over
those for su(p,q) , but one must use the remarks about
the nilpotents above for the closed cones. The proof
uses the maps
M : -+"Ln:X -+ (47(Xe 1e 1),...,4(Xenen))
for cp = (e.,f.) E . We have seen that M(X) E 2 if
X E 02 ,m 0 if X E C2 , etc. (X) E (E2 )
Theorem 19.3. The open (closed) non-trivial invariant convex cones in so(2n) (n 3) containing C0 are in 1-1*
correspondence with the open 0 (resp. closed) cones in (E2 ) 0 (resp. E2 ) containing (Eo) (resp. Eo) and invariant under the finite group of permutations of the Euclidean coordinates of Bn 102
CHAPTER V. Invariant Convex Cones in so(2,n
20. Conventions and Preliminaries for so(2,n)
so(2,n) is simple if n 2 3 . Let 1 = [n/2]
We take A B skewl o = Ct A,B,C real; B 2xn; A,C
{XE9t: X skew ,=19XE 3 : X symmetric}
0 -d 0 0 ( Gd -d 0 EOJ
0
0 -d -d 0 t F((0 - c = Z where Z== : 0 0.
Let d4 (j = ,...,t) be the linear functionals spanning
the dual of h , whose definition is clear from the above
notation for h . Then
noncompact roots) = ti(d09d), ti(d-d): j =1.
if a is even; if n is odd one has the additional
noncompact roots tid 0 103
(compact roots) = (ti(dj+dx), ti(dJ-dR): 1 j < k )
if n is even; if n is odd one has the additional
compact roots tid ,...,tid . The positive roots we take
to be the above with "+" only, and
S = (i(d-d ),i(d1 -d),...,i(d2 -d ),i(d 1 )}
if n is even, and
6 = (i(d 0 -dl),i(d 1-d,...,i(d 1 -d ),id2 )
if n is odd. We forego listing the H., XM, Ya for
a E Q+ as we will not really need them.
We let Z0= i(d 0 +dl),i(dO-d1 )} , so Z = Zo always, and
0 0 a 00 0 0 0b a 0 0-t = Ob : a,b E R 00O
Then h =X E h:dj(X)=0 for 2 j Z1) and
h (X Eh: d0 (X) d1 (X) = 0) , so h+= 0 iff n = 3 The cone in h given by Corollary 3.2 is (in o(2,2))
generated by ((1 tAf l ''2 0 0 0 1 i I d H.= E 0 0 0 0 0gdd 0 0 (d1 -d2 IK
2)(d 104
for d 1 , or all
0 -d d 0
0 0- g j
for g d-1 , d 1I . The cone generated by this is all of the form (15) where d > 0 , (gj < d
We take B(X,Y) = Tr(XY) for X,Y E so if A B P QE
X B c Qt R
B(XY) = - A 4 P - CPjRi1 + 2 B Q ij ij i,j and B 9(X, Y) is the same with "+" replacing - in the above. (9(X) = -X )
Given V =(dod ,...,d E +l , for concreteness we associate -do) 0 0
(0 -d) (16) 0 Eh
0
In what follows we replace Z by the more comfortable J ,
so J E h corresponds to (1,0,...,0) as in (15).
Note g=y0 U E o(2,n) iff AtA - C C = I t( t D D -B B = I ,and AtB=CtD , or equivalently AAt -.BB Bt =IDDt= , DD -CC1=I - t == and AC=BD t ,so 105
A -C g~ = . g is in the maximal compact K (of -B tD
SO(2,n)) iff B = 0 , C = 0 , det A = det D = 1 . 0(2,n) preserves the symmetric form T on 2 n
z + x 2 z 2 -wp j=1 2 n where x,z E 2 , y,w E JR
We characterize the identity component G = S00 (2,n)
2 as follows. Let el = (l,0,...yJ) , e2 = (0,1,0,...,0) EM +fn and let p: 3 .--2+n2 be the projection. If g E 0(2,n) , then g E G iff det g = 1 and the linearly independent vectors p(g(e)) and p(g(e)) have the same orientation as e1 and e2 Recall that the reflections of h coming from conjugations by K (or G) are, for n odd, all permu- tations and sign changes of the d ,...,d , and all permutations and even numbers of sign changes of the d,..., d for n even. This finite group we call W
There is an analogue of Horn's theorem (see section 14) for any semisimple Lie group, one side of which we state here for K acting on its Lie algebra k via Ad . Let
: k - h be the orthogonal projection.
Proposition 20.1. Let X E h . Then I(Ad(K)X) is contained in the convex hull of the points c(X) for c E W
Proof. The proof is essentially word for word that of io6
Kostant's proof on p. 452 in [16], and is the standard
maximization of a continuous function over the compact K
argument. In fact, equality holds in the above, but we
do not need this.
We record how A acts on h by conjugation. If 0 1 0 0 -d 0 CCd 0 0 a =2 ,then Ad(ea) a0- 810 0- o 0d10 0 2 0 -f0
fo 0 0
0 -f 1 f 0J
where f0 = d cosh e cosh 92 - d sinh 0 sinh 92 f = d coshO9 coshO92 - d0 sinhO1 sinh02 . Note the *
changes sign if -a is used instead.
Define I to be the collection of all ordered bases
of B 2 +nwhich are transforms by a g E G of the
standard basis el,...e. 2 +n. Let Q be the set of all pairs (e,f) of linearly independent vectors in R
satisfying T(e,e) = T(f,f) = T(e,f) = 0 , and such that
the I2 components of e and f are oriented the same
as e1 and e2 . It is clear that G acts on 0 , and
given any (ef) E 0 2 r > 0 and g E K such that
g(re) = (1,0,1,0,...,0) and g(f) = (a,b,a,b,0,...,0)
for a,b E R with b > 0
Finally, we define two closed cones in R- 107
E1 = (d0 ,...,d2 ): d0 jd V f} , and
E0 = ((do,...,d2 ): d0d }l j=1
21. Noncompact Convexity in so(2,n)
The representation of 1on IR2+n which we use here is quite different from the defining representations of the three previous series of simple causal groups. (Their analogue for so(2,n) is the spin representation.) Here we do not have available Hilbert space arguments to prove ellipticity of the orbits in the interior of the minimal cone, so we rely more heavily on the classification of the orbits. In this section and the next n 4 for convenience.
Let C- be the collection of all nontrivial invariant convex cones in % . We define*
Cl = (X E : T(Xe,f) > 0 '1 (e,f) E } and
0 =x E C : 1 (Xvy,v2 ) > I 'r(Xv2j+1'2j+2) j=l
We will see in the next section that C0 and Cl are,
respectively minimal and maximal open invariant convex
cones in io8
Proposition 21.1. Let (dl,.d z ) E IR1+t such that d0 > |dj for j = 1,...,. Then X as in (16) is in
C1. If also d0 > d |,Ithen X is in C0 j=1 Proof. By 20.1 and the remark following the definition of
o , it suffices to consider a K-conjugate Y of X 0-d d 0 whose o(2,2) part is 0 where Idi Smax |d , O -d 4>0 d 0 e = (1,0,1,O0,...,0) , and f = (a,b,a,b,O,...,O) where
b > 0 . One computes T(Ye,?) = b(d0-d) > 0 , so X E C To prove the second statement, we first note that
J E CO . This is clear from Ad(G)J = Ad(K)Ad(A)J , the
computation of the A-action earlier, and Proposition 20.1.
But the X above is clearly in the invariant convex cone
generated by G-transforms of J , as a e > 0 such that
d3 = (|d L+e) , and (do d1,...,dz) = (1d11+e,d1,O,...,O) j=1 + ... + (IdzI+cO...,O0d,) . Thus X E C0 as C0 is
G-invariant.
Lemma 21.2. The G-orbit of an X as in (16) where
d > dj ( = l,...,L) is closed. o 3
Proof. Again we follow the proofs in Lemmas 14.1 and 17.2,
and give the necessary estimate. k > 0 such that
d - dI2 k 9 j > 0 , and this inequality also holds for
the (d0 ,h1 ,...,h) corresponding to any Y in the 109
K-orbit of X . We may assume 98lO82 z 0 , and using the earlier computation and the notation there,
f0 -f1 = d 0 cosh (81+8e2) - h1 cosh(a 1 +89 2 ) Sk cosh(8 1 + 92 )
Thus 91 +8 2 cannot grow too large, and likewise 81 and 92
Remark. As n z 4 , (vl,...,v2+n) E t implies
(v1 v3 ,v2 t v4 ) EQn ,and so if X E C1 we have
T(Xv1 ,v2 ) > jT(Xv3 ,v4 )j . It will turn out by classification that C1 consists of conjugates of the X considered in the previous lemma.
Definitions. 1) Given X E , let X = Y +Z , Y E k ,
Z E p . Call B(ZZ) the p-norm of X . Call the vector (do,...,d2 ) obtained from r(Y) E h as in (16) the G-diagonal of X
2) If VU EfR 1 +2 , say V is in the noncompact convex hull of U if V is equal to a vector in E0 (see section 20) plus a vector in the convex hull of (o(U): r E W.
Theorem 21.3. Let Y E Cl be in the orbit of an X as in the statement of Lemma 21.2. Then the G-diagonal of
Y is in the noncompact convex hull of the (d 0 .. ,de) there.
Proof. It is exactly the same argument as for su(pq) 110
and so(2n) , using the following.*
Lemma 21.4. Let X,Y be as in the statement of 21.3, and
assume Y 0 k . Then there exists Z in the orbit of Y
such that p-norm of Z is strictly less than the p-norm
of Y , and the G-diagonal of Y is in the noncompact
convex hull of the G-diagonal of Z .
Proof. The proof is computationally more difficult because
the noncompact root spaces are not as easily expressed in
terms of our representation. By Proposition 20.1 we may A B assume the k-component of Y is in h . Let Y = t
B / . For D 2xn we compute
I eD A B I -eD (17) ( )K )K t eD I)Bt C -eD-I
A + e(DB -BD ) B + e(DC - AD)9+0(c 2 B +e(DtA-CDt) C+e(DtB-BtD)
Assume ( 0) gives rise to (c0 ,c1 ,...,c ) as in (16). Break B and D up into 2x2 blocks plus possibly (n odd) a 2xl vector:
B=[(c 9..dB =. )
Then
DC - AD vyc 1 +zc0 -XcI+C0w c
DC) - A(C=V) 1 0o-x+wc 0 -cy-c-u-YzcyXWc11 -COu Now c0 > IcI (j = 1,...,) by the Remark in this
section, so x,y,z,w , etc. and u,v are determined ill
uniquely by
(c 0 +c )/2) (z+y) + ((00c1 )/2)(z-y) = -a
((00+01 )/2) (w-x) + (c0-cl)/2)(w+x) - -b
-((c 0 +c) /2) (w-x) + (c 0 -c)/2)(w+x) = C
((co+cl)/2) (z+y) - ((co-c 1 )/2)(z-y) = d , etc., and = -c0v , = c0u1. Then -a -b DC - AD = .(.
We will take e small and positive. Then
DB -BD p where P = -za - wb + cx + dy + ... -av + $u =
2 (c+c)/2)(z+Y)2+ (xw)2} + ((c 0 -c1 )/2){(y-z) + (x+w)2}
+ ... + c(u 2 +v 2 ) > 0 , and 0 Q
D B - BD
r 22 where Q = ya+wc-bx-dz =- c ) ( ) (-X)
+ ((c 0-c)/ 2 )(z-y)2+ (w+x)} , which is of indeterminant
sign, but it is clear that the G-diagonal of the e terms 112
of (17) are in -E
22. Classification of Orbits and Cones in so(2,n)
Just as the types in the maximal cone of so(2n)*
(n S 3) were composed of indecomposable types coming from so(4) , we will see that the corresponding orbits for so(2,n) come from the orbits of 0(2,2) ~ sl(2,JR) esl(2,.R) In fact, all but two of the indecomposable types for the series o(2,n) , n ' 1 (for the entire algebra), are in 0(2,2) or a subalgebra. (Of these two, one is in 0(2,3) and one in o(2,4) .)
Temporarily let C 1 be the minimal closed convex invariant cone containing J , and let C2 be the negative of the cone dual to C~1 under any Be(.,..) (By the G-invariance C2 is clearly independent of the particular e .) We have already seen that C2 contains C , which in turn contains all X as in (16) with d 0 Jd j for j > 0 . Also, we know already that
C 1 C , and that C 1 contains all X as in (16) with d . fd . As -(Eo)* = E (Euclidean metric), a 0 fl'j=1 necessary condition for X E C2 is that all G-diagonals of all conjugates of X lie in E.*
It is appropriate to first consider o(2,2) . One
sl(2,B) component is spanned by 113
0 -1 0 0 XY 1 0 y -x J = and all
1 0 y-X the other is spanned by
0 -1 0 0 XY 1 0 -Y X and all 1 0 1 x-y
Of the outer automorphisms from 0(2,2) , conjugation by
I 0 exchanges the two subalgebras (and exchanges J 0-10 0 0 1 0 0 0 1 and J2), whereas conjugation by "inverts"
01 each sl(2,1R) separately, exchanging the positive and negative cones. In Ad(G) the latter does not exist, but the former does in the sense that o(2,2) C o(2,n) and
n 4 . Note J = +(J1 +J 2 ) with this inclusion. Recall each indecomposable type for so(2,n) carries
a certain signature (p,q) , and we find (p+q)x(p+q) matrices representing such types.
S(o) , m even
A) in=0
1) e = +1 ; sig(l,0); (0)
2) e = -1 ; sig(0,1); (0)
B) m= 2 0 0 1 1) e +1 sig(l,2) ; 0 0 -l 1 1 0 114
In o(2,2) it is the sum of two nonzero nilpotents
in the sl(2,JR)'s , one with positive J. component,
the other with negative such. It cannot contribute to a type in C2 for so(2,n) 0 -1 0 2) e = -1; sig(2.1) ; 1 0 1j; the other sum of 0 1 0 two nilpotents ; it can contribute to C2
C) m = 4
Only e = -l gives a type for so(2,n) ; sig(2,3) ; 0 -1 0 1 0 1 0 1 0 4
0 1 0 -1 0. 1 0 1 0 -\2 0 CoWJN/7'0 OF 0 It is conjugate under K to an element with o(3) 0c - 0 3 component Pq 0 0 ,So cannot contribute. 0 0 0
& (O,0)j, m odd 0 -1 0 -l 1 011 0 Only m=1 is relevant; sig (2,2) 0 1 0 1 is a nonzero It nilpotent in one ;1 0 -1 0 sl(2,3) and 0 in the other; it can contribute.
M(,-K) (; = - ' 0)
A) mn=0 0 -X 1) e = I ; sig (2,0) ; )
2) e = -l ; sig(O,2) ; the same matrix. 115
B) m = 2
Only e = 1 is relevant; sig(2, 4 ) ; ( = ix)
K - % 0 0 1/ 0
0 0 0 0
0 0 0 -X 0
0 0 0
1//T 0 0 0 -x
0 1/20 1' X 0.
It is conjugate under S00 (2,4) to a matrix with unchanged X-entries and non- X entries multiplied by
an arbitrarily large factor, so clearly cannot contri-
bute to C2 '
C) xn=l
sig(2,2) . The e = +l cases are the two possibilities
for an elliptic element in one sl(2,R) and a nonzero
nilpotent in the other. One can contribute to C2 '
the other cannot. Let us say t 1 %,-O) can contribute.
A typical representative is
O-X00 0 -1 0 -1 X0 10 1 0 X + 0-X 0 1 0 1 0 XO l0 -10
The remaining cases % ,l -0 (0 / real) and
(0-0C,-C) ( / t) are in o(2,2) , and correspond to hyp.0e (0 or hyp.) , hyp. S nilp. , and hyp. 3 elliptic. 116
None can contribute to C2
Now a type involving AjQ-) above cannot contri- bute to C, as (e,f) = ((lO,-lO),(OlO,-l)) E 0 ,
and Xe = (oX,O,-X) , which is T-orthogonal to f . It
follows that 2(0) (nilp. G nilp.) and l(O0) (nilp. G 0) cannot contribute to Cl , as t (0,-) is
in the invariant convex cone generated by t2l(0) , and
61(0,0) is just t1 (C,-) for O = 0 , so to speak; the same (e,f) rules out L1(0,0)
By this analysis and Proposition 21.1, we see that C consists of conjugates of X as in (16) with
(d ,...,d ) E (E) 0 , and C2=1 Also, we see that C0 contains nothing else besides what we already knew it contained, namely, conjugates of X as in (16) with
(d ,...,5d) E (Eo) . Thus C_1 = C0o
Theorem 22.1. C0 and Cl are open cones in C , and any
C E C- contains either C or -C0 but not both. C0 and Cl are obtained by changing ">" to " in the
definitions. Any closed (open) cone in C containing 00 is contained in Cl (resp. Cl)
Proof. From the form of Cl and C0 exhibited in the
paragraph above, to prove C and C open it suffices
to show there is a neighborhood of J in C .By the
continuity of the eigenvalues there is a neighborhood N 117
about J such that VX E N X has one eigenvalue near i , one near -i , and the rest in a small disc about 0 .
By the classification of types in the ones near i must be on the imaginary axis and be negatives of each other. If then X(v +iv 2 ) = Xi(vl +iv 2 ) , 1 > 0 , then
Xv1 = -Xv2 , Xv2 = Xv 1 , '(vv2 ) = 0 , and T(v ,v ),IT (v2' 2)> 0 are evident if N is small enough.
X leaves the -complement of Fv1 + Fv2 invariant, and by the classification of types in o(n) and the continuity of eigenvalues it is clear that X E C0
Thus C0 is open, and any C E C with J E must contain U E Co, so J E C. (The sum of all W-conjugates of X E h is a multiple of J.) Thus C0 Q C as before. The statements about the form of C0 and 1 follow as before. Finally, we have already noted 00 = -c,"sol5 the remaining statements follow as before.
It is clear from our classification and Theorem 21.3 that types in Cl are composed of I1(0) (a IxI zero),*
L2(0) (nilp. e nilp.) , 6l(0,0) (nilp. e zero)
4(,-K) (nilp. G elliptic) , and 4% -K) (the 0 -X %s). Furthermore, representatives of 62(0) and X 0 t1(0,0) are conjugate to arbitrarily small multiples of themselves, so they can appear in a type in C only if they are followed by all t(O) 's . These types are in the minimal closed cone C0 . Also, representatives of i18
(C,-Q) are a sum of representatives of the elliptic types t (C,-C) and those of 61(0,0) (or 62(o))
Using the maps
X ET -+ (t(Xvl,v2,T(xv3,74) ...(x7' 21+1'9 2L+2)) for (Vl,...,v2+n) E , the proof of the following parallels the earlier proofs.
Theorem 22.2. The closed (open) non-trivial invariant convex cones in so(2,n) (n > 4) containing C0 are in 1-1 correspondence with the closed (resp. open) convex
0 0 cones in E1 (resp. (E1 ) ) containing E0 (resp. (EO) ) and invariant under the finite permutation and sign-changing group W acting on E1 119
CHAPTER VI. Global Causality of the Covering Groups.
23. Definitions and Isomorphisms.
If a Lie group G acts transitively on a manifold
M , then it is well known that the universal covering group
G acts on the universal cover M . In fact, if M = G/H for some subgroup H , and if 0 -+D -+GT+G ->0 is the exact sequence of groups, then there is the sequence of covering maps
G/H' -+ G/H G/H0 -+ G/H where H0 is the identity component of H , H = 7r6(H) ' and H' is the identity component of H . G/l' is simply connected, so M 1G/H' . (See for example [19, p. 33].)
Furthermore, the composition H - (H) -HO is also a covering map.
Recall that G is said to act effectively on M if the identity element e C G is the only element of G acting trivially on M . It is easily seen that if M = G/H, then the subgroup of G acting trivially on M is precisely the largest normal subgroup of G contained in H
The proof of the following is not difficult, and we will not use it in this chapter. 120
Proposition 23.1. With the above notation, if G acts
effectively on M , then G acts effectively on M if and only if r: HI -+ H0 is an isomorphism. In any case, the subgroup of G acting trivially on M is D A H'
We recall now some of the notions and definitions in
Chapter 2 of [193. A causal structure on a manifold M is a smooth assignment to each p E M of a non-trivial
closed convex cone in the tangent space T(M) , defined
in a neighborhood of each point by a finite number of
functions on points and the components of tangent vectors.
One say that a Lie group acts causally on M if the group
transformations preserve this structure. A causal structure on a Lie group is said to be invariant if it is invariant under both left and right translations. Clearly such a structure is completely determined by a causal cone in the
Lie algebra.
In [193 causal structures on groups acting causally
on causally oriented manifolds were defined implicitly
(Scholium 2.4). As we now have more detailed information
on the possibilities for causal structures on simple Lie groups, it seems worthwhile to make the following
Definition 23.2. Given a Lie group G with a causal cone
C in its Lie algebra, and a manifold M with causal
structure CP PEM on which G acts, say the group action
is compatible with the causal orientations, or briefly, 121
G acts temporally, if G acts causally on M and if
((d/dt) exp tX- y) t=o E C for all X E C , y E M . If t: GxM -+ M is the group action, these conditions are equivalent to requiring that the differential d (g,p) map the direct sum of cones
C x Cp into C p for all gE G , p E M . (Ofcourse
C = dL C = dR C .) g g g
Remark. In [193 the cones C were essentially defined by projecting the inverse images (d l)(Co) onto T (G) .
We now specialize to the cases of the simple groups considered thus far. We have sometimes regarded su(p,q) as a subalgebra of sp(p+q,JR) (section 11) and so(2n) as a further restriction of su(n,n) . Here we keep the
second embedding so(2n) e-- su(n,n) (detailed below)
and consider it in parallel with another map sp(n,fl) a--
su(n,n) , which was in fact already defined in section 13.
This last was accomplished by a linear operator that we
now call C?: 2 -+ X2n , originally denoted U . Regarding 2n sp(n,fl) acting as complex-linear operators on TV ,
we have
0 sp(n,3) C7 sp(nA1R) su(n,n)
with equality only if n = 1 122
We extend the original symplectic form a(t) on
S2n to a real symplectic form (-,) on T2n by a(ix,y) = 0 , c(ix,iy) = 7(x,y) x,y E 3 2n . This form is to be regarded as the c appearing in the treatment of so(2n) (section 17): &(x,y) = T(x,ay) = Re 0 2n <.,0>0the standard complex Hilbert structure on C . 2n One checks that c(v,w) = Re~iH(Cv,Cw)3 for v,w E C, where H(-,-) is the Hermitian form on X2n defined in section 12, and in fact
cso(2n) C,, so(2n) c su(n,n) for all n ' 1 . Thus we can compare directly the matrix elements (Xv,v) for X in sp(n,M) and so(2n) with the corresponding Re(iH(Yu,u)} for Y in sp(n,1R) , so(2n).
It is useful to have the explicit forms of these
new algebras and their corresponding analytic groups in
SU(n,n) . We have
t a,$ nxn comple x matrices, Sps(n.,ns)n 3ia skew H~ermitian, symmetric
A B and C -A t)
where 2-(A -At + i (B- C) )
and = (-A-A +i(B-C)) F G Also, Sp{(n.,): GF symmetric, FF-Gt 123
t the two conditions being equivalent to FG symmetric and
FF- GG = I Analogously,
3 z,5 nxn complex matrices, so(2n) = -3 C a skew Hermitian, S skew*
and
A B a
where a = (A+A+i(B+B)) and 5 = (-A+ +i(B-T)) Finally,
F G GF skew, SO(2n)1 t ~t -G F FF - GG=It the two conditions equivalent to FG skew and FF - GG* =I.*
Remark. By examining the maximal tori in sp(nR)
and so(2n)1 corresponding to the h for sp(n,B) and*
so(2n) , and comparing them with the h for su(nn) we have the following: the invariant positive cone in*
s p(n.,]R),and the "middle" (self-dual) cone C1 in
so(2n)1 are contained in the minimal cone 0 for*
su(n,n) . However, the maximal cone C2 of so(2n)1*
extends outside the maximal cone Cl of su(nn)
24. Causal Actions on Shilov Boundaries.
We recall the standard action of SU(n,n) on tne
group of unitaries U(n) by fractional linear trans- 124
A B formations [19, p. 351. For g = ( E SU(n,n) and C D U E U(n) , define
(p(g))U = (AU +B)(CU1+D)~ E U(n) p defines a (left) group action, and the action is causal.
(U(n) has the standard invariant causal structure determined by positive-definite Hermitians.)
Let
Us(n) = (U E U(n): U symmetric } and (only for n even!)
Uk(n) = (CUE U(n): U skews
One checks that Sp(n,E) I1and SO(2n)1 transform Us(n) structure Uk(n) , respectively. Restricting the causal of U(n) to these closed submanifolds, one finds that
the resulting cone fields on Us(n) and Uk(2n) are
non-trivial for all n > 1 . Thus Sp(n,E)1 and SO(4n)
act causally on Us(n) and Uk(2n) , respectively. (These two actions and the action of SU(n,n) on U(n) are
individually isomorphic to the actions of each group G
on the Shilov boundary of its associated Hermitian symmetric domain G/K .)
Theorem 24.1. SU(n,n) acts temporally on U(n) , with
the causal structure on SU(n,n) obtained from the maximal
cone C in suln,n) described earlier. 125
Proof. As SU(n,n) acts causally on U(n) and its causal structure is invariant, it suffices to check the second condition in Definition 23.2 for g = e ; we compute the differential of p at this point. For X = (A* }/in the Lie algebra, (U + eAU + B) (eBU + I + CC) = U(I +eL) +o(e2) ,where L = U~AU - C - BU + U 1B. We check that iL is positive-definite Hermitian if X E Cl
Given y E Cn set x =TUy ,and note
= i - i
-i + i
In fact, this expression is just iH(Xw,w) for w = and H(w,w) = 0 ,so 0 v y E Qn by the definition of l*
In particular, SU(n,n) acts temporally with the causal structure obtained from the minimal cone -C , and by the last remark in Section 23 we have the
Corollary 24.2. For all n > 1 , Sp(n,F) 1 acts temporally on Us (n) (with the essentially unique causal structure on Sp(nAR)1 ) . For the causal structures on SO(2n)1 obtained from the self-dual cones l , So(2n) acts
temporally on Uk(n) (for n 2 4 even) and on U(n) (for all n 3)
The case above of SO(2n) , n even, can be
strengthened. 126
Theorem 24.3. For n 2 , So(4n) acts temporally on
4 Uk(2n) , with the causal structure on SO( n) obtained from the maximal cone C2
Proof. As in the previous proof, we must show that iL = i(UlaUJ-& LT+Uis) is positive-definite Hermitian if U is unitary and skew and C~L E C2 of so(2n)
To simplify the argument we use the fact that any
U E U (2n) may be written as ULDU1 where 0 -1 0 -1
1 0 1 0
(as in [12, p. 6]) and U1 is unitary. (This can be proven by the usual diagonalization arguments, but abstractly one knows that K acts transitively on the
Shilov boundary [15, p. 270].) Then
(18) = i+i - i<3 1 w,w> + i<$ 1 w,Dw> where w=Utva=U 1 U1 , $1 =U 1 Ut-1 . Then
- 1> E 2 also, and we drop the subscript.
Given any w E n let e = (;tw) ,and check that
T(e,e) = T(e,ae) = 0 (as in Section 17), so (xe,e) 0
V X E 02 . By the identity of the last section
(19) iH< )e,t 0 127
Now e = 27) for some v , and in fact (19) equals (18) with a,,v replacing % , 1 w respectively, which completes the proof.
Before stating the consequences of the above for global causality we consider S00 (2,n) and its action on the projective quadric
E R 2x2+n:T(X,X) = 0]}
[x] meaning the line determined by x . (See [19], Section II.4.) The tangent space to [x] E Q can be identified with the projective space of vectors y ER2n
such that T(y,x) = 0 , modulo IRx. T factors to a
linear conformal structure l1 on 0. with signature (1,n).
As S xSn is a double cover of Q a system of forward cones can easily be chosen.
Theorem 24.4. S0 (2,n) acts temporally on Q , with the
causal structure on the group coming from a closed maximal invariant convex cone in the Lie algebra.
Proof. Let C1 be the cone defined earlier. (See Sections 20 and 21 for notation.) Then X E iff T(Xe,f) z 0 V (e,f) E C0.
This condition has a pleasant geometric interpretation.
Given any [e] E Q and X E T,4 Xe E Te(R2+n) as usual,
and with care we can write [Xe] E T[el(Q) , as T(Xe,e) =0
always. Now the projective space of f E B such that 128
(f,f) = T(f,e) = 0 and (e,f) E 0 , modulo Re , gives one of the cones C (not convex) in Tej(Q) of T -isotropic vectors. Then T(Xef) 0 V (ef) E Q implies T 1 ([Xe],w) 0 V w E C , which implies (as T has signature (1,n)) that [Xe] is a tangent vector in the convex cone generated by C . Thus [Xe] is a forward- pointing tangent vector at [e] for all [e] E Q,.
Definition. A manifold M with causal structure is called
globally causal if there exists no closed non-trivial piece-
wise C1 curve in M the differential of which lies in the causal cone at each point.
Theorem 24.5. For all n 1 , Sp(n,JR) , SU(n,n) ,
S ) , and S00 (2,n+2) are globally causal with respect
to their causal structures coming from a maximal causal
cone in their Lie algebras. SU(p,q) (p > q 1) is
globally causal with respect to the minimal cone(O
causal structure, and S0(4n+2) (n - 1) is globally*
causal with respect to the causal structure coming from
the self-dual cone ~1 .
Proof. The first four series of groups act temporally Sn, on U3(n) , U(n) , U(2n) , and RxS , respectively,
by the initial remarks in section 23 and Theorems 24.1-4. U(n) and 2RxSn are globally causal by Corollary 2.3.1 and Scholium 2.11 in [19].
To identify U s(n) and U,(2n) , we set 129
t Ms (n) = (U E U s(n) : U E SUI(n)} = UU : U E SU(n)} and
lMk(2n) (UDUt: U E SU(2n)}
D as in 24.3. They are coset spaces, isomorphic to
SU(n)/so(n) and SU(2n)/Sp(n) , respectively, and are simply connected because SU(n) is simply connected and the factor spaces are connected. Thus*
Us(n) Ms(n) x R and Uk(2n) Mk(2n) x PR, and the spaces on the r.h.s. are contained component-wise in SU(n) x R' 3U(n) . Clearly then Us(n) and Uy(2n) are also globally causal.
We can assume that any closed time-like curve g(t) (0 r t s 1) in any of the covering groups G starts and ends at e EG . As the action of G on the relevant manifold S is temporal, g(t)p (V p E S) is a closed time-like curve in $ , contradicting the global causality unless g(t)p = pV t , so g(t) = et' t . The last statement follows in the same way by
Corollary 24.2, and the next-to-last by the embedding su(p,q) C--+ su(p,p) , under which the minimal cone of su(p,q) goes into the minimal cone of su(p,p) .
I thank Peter Greenberg for these observaticns. 130
corollary 24.6. If G is any of the above (24.5) groups with the causal cone C indicated, then there exists a
semigroup S in G such that S n S~ = (e and
gSg 1 =S for all g E G , generated by exp X: X E C .
Proof. If exp X ... exp Xn = e , Xi E C , the expression*
gives a piecewise C time-like curve in G , so all .X
must be 0. This suffices to get s ns~ = (el, and gSg~ 1 =S is clear.
Remarks. In the theory of Hermitian symmetric spaces the
first four series of groups are referred to as of "tube
type", the latter two of "non-tube type". (The ey domain
is also of tube type, the e6 domain not.) Only in the V case of tube type domains does the Shilov boundary unravel and have a globally causal universal covering. (For
example, the Shilov boundary for SU(2,1) is isomorphic
to S .) Thus proving the global causality with the
maximal causal structure may require some new ideas in the
non-tube type case. In the former cases one can ask whether the semi-
group of Corollary 24.6 is closed and in fact the entire
set of forward displacements of the universal covering of
the Shilov boundary coming from the group action. I believe that it is, but this question may require a
consideration of the classification of the conjugacy
classes in the groups. [ 2] also gives a treatment of 131
this, a simple extension of the Lie algebra classification.
It is interesting that here the Cayley transform (see section 29 ) appears naturally. Based on the analogy with SL(2,R) and some other cases, I would conjecture that any forward displacement is a product of two exponentials as in 24.6, one of which is some Exp tZ , Z E Z(k) , t 0 , the other in the range of the Cayley transform restricted to the forward maximal cone C . (Any such transform is also an exponential of an X E C .) 132
CHAPTER VII. Khlerization in Causal Lie Algebras.
25. Uniqueness of the Complex Structure.
We have seen that each X in the interior of a maximal open invariant convex cone in the Lie algebras thus far considered, is conjugate to an element in an open cone in the maximal torus h of k . In particular, such
X are elliptic, i.e., in some maximal compact subalgebra.
In fact, this maximal compact is uniquely determined by
X . Such uniqueness does not hold in general for non- positive elliptics, but indications are that it holds for at least a "generic" set (Proposition 25.3). We call this mapping of elliptics to maximal compacts Khlerization.
Uniqueness in the symplectic case, from the standpoint*
of linear quantum fields, was first shown by Weinless in his thesis [201. His proof involved holomorphic functions
in the upper half-plane and particularly some results of
Goodman [8 ] on such functions. Here we give two fairly straightforward proofs of the above claim. The first is an argument in terms of real Hilbert spaces, finite- or
infinite-dimensional, applicable to the open invariant cone
in sp(n,JR) , the minimal open cone in su(p,q) , and the
Although, this is a term perhaps best reserved for only
the symplectic groups. 133
"middle" cone C in so(2n) . The second proof is more non-linear, involving the action of G on the symmetric space G/K , but covers all the maximal open cones.
Consider first sp(t) , where (%,0) is a real
Hilbert space possessing an orthogonal J with square -I
(We use the notation in section 7, and again consider only bounded operators.) The extension appropriate for our purposes of the notion of "maximal compact subalgebra" of sp() to infinite dimensions* is as follows. Such a
subalgebra is all those elements skew with respect to a
complex Hilbert structure on 7< , topologically equivalent
to and having the same imaginary part 2(,-) as the*
original one. A complex structure in sp(Y) is a
J E sp(7) such that J = -I and c(J-,-) is a
positive-definite symmetric form on Y defining an
equivalent norm. Denote the set of complex structures
by P . One checks that I' C SpQ(V) also, and that P is
in 1-1 correspondence with the afore-mentioned set of
complex Hilbert structures. Furthermore, the correspondence
extends to the set of maximal compacts: given Jl C P ,
the corresponding maximal compact _k is all X E sp()
commuting with J. That k in fact determines J is
"Quasi-compact" in [18], but we shall refrain from intro-
ducing this term as the compactness of the corresponding
group in the finite-dimensional case is not used here. 134
clear from the observation that no two distinct complex structures commute: if say Jlcommuted with the original
J , then J~ J = D is positive-definite symmetric in*
(Va ) and has square I , hence equals I and J = J
(This argument is a special case of 25.2.) By the unitary equivalence of Hilbert spaces with the same dimension, all maximal compacts in sp(V) are conjugate under Sp(4) , and I is a single orbit in the positive cone in sp(W) .
Now U(V,g) and SO(1) (see sections 11 and 16) are further restrictions of this situation. To pass to u(v1,) from sp(7) , one takes an i E sp() such that i = -I , iJ = Ji q , and c(i-,-) not positive- or negative- definite (in finite dimensions the (real) signature being
(2p,2q) , p,q 2 1) , and defines u(t) = (X E sp(i): Xi=iX}*
(All i E sp(n,E) such that i2 = -I and c(i-,-) has a particular signature, also form a single sp(nE) -orbit.)
This embeds u(p,q) C sp(p+q,]R) , and one sees that
maximal compact subalgebras of su(p,q) are in 1-1 corre-
spondence with r'l u(p,q) , a single u(p,q)-orbit, which
lies in a minimal open invariant cone in su(pq) iff
p = q . If p / q , P fl u(p,q) projects injectively
onto an su(p,q)-orbit, each element of which spans the
center of some maximal compact.
As shown in section 16, so(Y) is a further
restriction on u(rt$) in the case when the dimensions of 135 the eigenspaces of 9 are equal. In finite dimensions one has so(2n) c su(n,n) c sp(2n,R) , as follows. so(2n) is defined by taking i,a E sp(2n,JR) which arise as in section 16, and letting
so(2n) = (X E sp(2n,R): Xi = iX, Xa = aXt
The intersection of the open positive invariant cone in sp(2n, E) with so(2n) is the cone. C above, and furthermore maximal compacts of so(2n) are in 1-1 correspondence with P f so(2n) , a single SO(2n)-orbit.
The purpose of the preceeding two paragraphs is to make the uniqueness argument of 25.2 applicable to the indicated cones in su(p,q) and so(2n) . We see that maximal compacts in these two algebras are contained in unique maximal compacts in a symplectic algebra, so uniqueness for sp(nr)Ts open cone C implies uniqueness for the cones obtained by intersecting C% with su(p q) and so(2n) .
There is a simple parametrization of the set of positive complex structures P for sp(Y) . Let i(t) denote the algebra of bounded operators on V , and At
the transpose, with respect to go(., = i(J,) , of*
A E g([) . We have
Sp (?) = (gE () : gtJg = J) and
sp (V) = [X E 2(V) : X J7+ JX = 0) = (JH: H E .(V) , Hu = H) 136
If K E P then K = JD where D = D is positive-definite symmetric (with respect to 0 ) . K2 = -I implies J= DJD, so D E Sp(Y) .
In the finite-dimensional case we have in fact
D E exp p . D is the exponential of a unique symmetrix X , and D = JDJ~ is equivalent to -X=J1 XJ ; together with X = X this implies X E p . Reversing the steps, we have
r = tJexp X: X E p}
Returning to the general case, note that (positive- definite) symmetric forms 3(-,-) on V are in linear 1-1
correspondence with (positive-definite, resp.) symmetric
operators g on (ti) , a ;(-,.) = S (%-,-) - 0 0
Proposition 25.1. A positive-definite symmetric form
can be the real part of a topologically equivalent
complex Hilbert structure on V with imaginary part c(,-), if and only if E Sp(/)
Proof. If R E P and s(-,-) = Z(R-,-) , then 1(R-,-) =
= (J%-,-) , so 9 = J R E Sp() . Conversely, 2 g E Sp(t") implies R = JS C Sp(?) and ,R = J(J7)
= J2 =
Theorem 25.2. Suppose X E sp(V) and I(Xv,v) : kt (vv) v E ,for some k > 0. Then X commutes with a
unique complex structure. 137
Proof. The existence of one such complex structure is clear from Theorem 7.1, and so it suffices to assume JX= XJ.
If now XK = KX for some K E r, write K = JD and
X = JH as above. The hypothesis on X implies that H is positive-definite symmetric in(, ) . Further, we have (JH)D = D(JH) as JH = HJ , so H2 commutes with D
Thus H commutes with D by the functional calculus, and D(JH) = (JH)D = J(HD) = J(DH) . As H is surjective,
JD = DJ also. Now J DJD and D is positive-definite, so D2 = I,hence D=I and K = J .
Mostly for the sake of convenience, let us restrict to the finite-dimensional case for the rest of this section. It is not difficult to see that X E sp(n,E) is elliptic iff X is diagonalizable over W with purely imaginary eigenvalues. The condition that X is in a unique maximal compact subalgebra is fairly simple.
Proposition 25.3. Let X = JH = HJ E sp(n,B). Then H is in a unique maximal compact subalgebra iff H and -H have no eigenvalues in common.
Proof. Let R2n = V .e.. 9 Vm be the eigenspace decomposition of H , where H = on V. , a. E E.
The V.a are S0 - and a-orthogonal and J-invariant, so all dim V are even. Let JD , D positive-definite
symmetric and symplectic, be a candidate for an additional
complex structure commuting with JH . As above, D 138
commutes with H2 , hence with jHJ , so the sufficiency is clear as in the proof of 25.2. Conversely, if say a= 0, then any D E exp p on V is possible, and there is no uniqueness. Secondly, if say al = -a /'O: , it suffices to see that there are lots of complex structures in sp(2,2R) commuting with the elliptic
0 1 0 0 -1 L = -1 0 0 0 1 J These can be determined, and are a two-parameter family: L commutes with all
0 a x b a 0 b x JD = -x b 0 -a b -x -a 0 2 2 2 where x > 0 and 1 + a + b = x . (The eigenvalues of
D are x t x 2 -l , each with multiplicity two.)
We next give the second proof of uniqueness mentioned at the start. The idea is to note that each maximal
compact subgroup leaves fixed a unique point in G/K and also that one-parameter groups generated by an X in
an open maximal cone also have this property. The first statement is a general fact about symmetric spaces, and
is sharpened in the Hermitian symmetric case as follows
(Theorem 3.2, Chapter IX [ 9 3). Let G be a simple Lie group and K a maximal compact subgroup, such that G/K
is Hermitian symmetric, and 'Tk,p as usual. Let x0 be 139 the identity coset in G/K and J E Z(k) such that ad(J): p -+p is a complex structure (see section 3).
Then Ad(exp'7rJ) is the Cartan involution for I = k + p, and the geodesic symmetry (reflection) about x0 is the action of exp(wJ) on G/K . x0 is the unique fixed point of the one-parameter group generated by J .
Now consider the standard actions of SU(p,q) ,
Sp (n, R) 1,and SO(2n)1 (see section 23) on the appro- priate bounded domain in some Cm by fractional linear
transformations [14, p. 215]. To prove the second assertion above, it suffices to consider X in the maximal torus h
SU(p,q) Take (p j ,...,aq) ER such that*
%. + a. >0 i,j . If Z is any pxq complex matrix then clearly
r.- 1l 0 r-tial 0 etixI K . t j=z V t E F 0 Kt~Xee ikPi0tic 0 e
iff Z =0,
Sp(n,T) Let (d,...,dn) E :n such that all d > 0 If Z is any nxn symmetric complex matrix then
-tid1 0r -tid 1 0 (20) 0 -tidn) e0 .. e'-tidn = Z \/ t E
'I- e e
iff 7 = 0 14o
SO(2n) Let (d 1,..dn) E Rn such that dj+dk > 0 y 5 /'k . If Z is any nxn skew-symmetric complex matrix, then clearly (20) holds V t E R iff Z = 0
Finally, consider so(2,n) . The action of S00 (2,n) on its bounded domain is a bit of a mess [21, p.16 8 1, but the K = SO(2) x SO(n) -action is simple in a particular representation. In fact it is equivariant with the linear adjoint action of K on p [151. So, let d0 > dI
(j = l,...,[n/2]) give an element in a maximal open cone cos e -sin e as in (16), set R,=( cs ) ,and let B be 2xn. sin 0 Cos 0 Then
Rdo 0 0 B R0 0 0 C
0 Rdj B 0 0 R~d C 0 where C = Rd BRd . One checks easily that
dd(BRd = = (RdB)t=0
iff B = 0 As a result, we have proven
Theorem 25.4. Let be one of the simple causal Lie
algebras sp(n,]R) , su(p,q), so(2n), so(2,n) . Then each*
X E +in an open maximal invariant convex cone is
contained in a unique maximal compact subalgebra. 141
26. Generalizations to Unbounded Operators.
In this section we extend the discussion in section 7 to some unbounded infinitesimally symplectic operators.
Using the notation there, we consider real-linear, possibly unbounded S: . -- 6 in (V,< , >) having dense domain
.5S qV and satisfying c(Sx,y) + c(x,Sy) = 0 V x,y E f3
We raise the question: when can we regard S as a skew- adjoint operator on some complex Hilbert space (2, , >) having a "dense intersection" with 76 and such that
Im< , > = Im < , >1 on that intersection? On this question mention should be made of [3] and [18].
When S satisfies a positivity condition on its domain, we find a canonical V , and / A 2 contains a certain "finite-energy" subspace, in complete analogy with the positive-mass Klein-Gordan wave equation in
Minkowski space. However, the formulation of a uniqueness statement like Theorem 25.2 is not yet clear.
We first recall the Friedrichs extension. Let
(2,c , >) be a real or complex Hilbert space and H an
Hermitian operator on V with dense domain 5H , bounded from below by k > 0 :
(21) k V v EaH
(If 2 is complex we assume H is complex-linear.) H
extends to a surjective self-adjoint A: JA ->31 , also
bounded below by k , as follows. Let ( H1 be the 142
completion of in <, , and extend <.,->
to a real Hilbert structure on [H (H] e is*
*the so-called "finite-energy" subspace. Let - A = H** H Hl[ and A = H. A consequence of the proof is e =
V x E 4 A, y E [H] , and it is not hard to see that
HPI= [49AI Typically, the inclusion 4 A a 4A] is proper.
Lemma 26.1. (LDA],'>C) is unitarily equivalent to*
(,<-,->) via the densely defined unitary map
U = (A ) o A: a(A
112 A is the range of (A)2where .Thus a*
A= [A iff A is bounded.
Proof. By a remark above, if x,y EA
xly> e= = =
Let "A = A-1 ([4A]) and A = AfrA . Then
surjective. A 9A1 -[A] is
Lemma 26.2. In ([AlJ<.r)e A is dense (-also in
(t,,.>)) and A is self-adjoint and bounded from below also by k
Proof. That 4A is dense and A Hermitian is easily
shown. To see that A 1 is self-adjoint, let y E5(A,*
[.aA], i.e. S y E [KA] such that = 143*
1 ,y x E> A . Let A 1 x = x A , and obtain =
e for all x E [-A . If we assume only x* E PA then = , so y E A=A , and
Ay = y E .WA] implying y E a as desired. A~A 1 Finally, C k eV x CEA Kx,x>*
k V x E [A , which is clear as A S 1/k
Now take S: ->?/ as in the first paragraph of
this section, where (0,<-, > = g(,-) + ic(-,')) is a
complex Hilbert space with complex structure J . Assume
that 2(Sv,v) 2 k V v E J , for some k > 0 . We
define H = J-1 S so that H ~~ s and note that H is symmetric in (t',&) and bounded from below by k
Now extend H to a surjective (real) self-adjoint
operator A: PA -+V a Ia Friedrichs as above, and S -&. As previously, likewise, so that JA = S: A V 1 define the surjective restriction A1: PA A , and
likewise let P% = (-1 AS1-=S 1- -sA S Then Lemma 26.2 applies, and the following is the analogue*
for S, , proven the same way.
Lemma 26.3. In ([PA]s is dense (-also in and S is skew-adjoint.
Note that we have A '% A VA]as1QZIQin
general aA and P1 overlap non-trivially, and equality
holds in any of these inclusions iff A is bounded. We apply the polar decomposition in ([PAl1'It> ) for 144
closed operators in a real space, writing S = RT , where
R is orthogonal, T is non-negative self-adjoint, and R
and T commute with all orthogonals commuting with S
(R is orthogonal as ker S = 0 and range S= [AP )
We have S= -S{ -TR. = (-RKU)(RTR) by 26.3, so by 2_ uniqueness R =-I, R(frf)l=frS =6Tand RT=TR.*
T is also surjective, so T 1 : aisA ~.
bounded and positive-semidefinite. 7(Rx,Ry) = c(x,y)
Vx,y E (BA is shown using the equation JAx = RTx
ECx J3 . Now define
31 (x,y) = (Rx,y) V x,y E [RA]
We have
1 3 (X,y) = ;0 (Jo Rxy)
(21) = 9 0 (AT'x Y) = \XYEFA]
so is a positive-definite form on [.A I Let 2
be the completion of [.a with respect to H .b R space
and a extend continuously to 7! so that
( + a~e,-) ,-<'> ) is a complex Hilbert space with complex structure R
Vx C -A1 Proposition 26.4. 1) fxI IT~I|IxH W
2) In (%,< , >1) , % is dense and T is Hermitian
(complex-linear) and bounded from below by 1/||T-1' . 145
3) Making the Friedrichs extension (in the complex case) as earlier, T and S extend to surjective self-adjoint and skew-adjoint operators T , I respectively, so that
RT = TR = S
Proof. 1) follows from (21) and 3) follows from 2). The density of-s is not difficult, and the bound on T follows so: V x E as ,
1 1 %j(Tx,x) =,11 :(l/jT-Ij 6)S
= (l/i T 1I )S1(x~x)
27. Analyticity of KIdhlerization in sp().
As in section 7 let (%, 0 (,-) +ic(-,-) = <,->) be a complex Hilbert space with complex structure J . We return to bounded operators only, and recall the definitions of and i(i) from section 25, using only the 90-norm topology on V/
Let
CO = (X E sp(V): 1(Xv,v) k V v E t for some k > O}
an open convex cone in sp(r) , and r c= c . To define an analytic structure in P we use the following.
Lemma 27.1. P is closed in -(.)-
Proof. We saw in section 25 that 146
P = J(D E Qj: Dt = D, DJD = J, and D positive-definite).
The first two conditions are obviously preserved on passing to limits, and the third follows from D = JDJ
If X E C0 determines a complex structure J as in 25.2, we have X = J1 H = HJ , where H is symmetric and positive-definite with respect to X= c(X,)
Then clearly
(22) (lX=)
the square root being well-defined in (v, *
Theorem 27.2. The mapping : C-+' defined by (22) is real-analytic.
Proof. This is a local property, and as ( is constant on
rays it suffices to check analyticity at X E C0 such that 20 0 < I+ X < I with respect to X (lYltX (YEit),t > 0)
is independent of t .) Now if Y is in a small neigh- 2 borhood N of 0 in sp() , I +(X +Y) M is longer
symmetric in (Y,%E) , but JM11 1 may be assumed, and
the power series
M2 - -(X+Y) = I-M = I - M1/2)( 2 -11
is valid for lIMlt 1. m is also analytic in Y , and
thus 2(jX+Y) is a power series in Y for Y E N The theorem is no doubt true for the maximal open cones in the other algebras, too. It would follow if one could show that the unique fixed point in G/K of Ce 3 (see section 25, second proof) depended analytically on X
28. An Example and Analytic Continuation: SU(1,1)
It is not only customary but also usually quite worthwhile to illustrate general theory in Lie groups by considering in detail the case of SL(2,R) [10]. For this group we can analytically continue the map v in the last section to the whole Lie algebra minus zero (the range space no longer being merely complex structures), give an interpretation in the symmetric space, and complex- analytically extend y to SL(2,) - t I) . Some indications for SU(2,1) will also be given, but an overall picture of K'hlerization there has not yet emerged.
It will be useful to give SL(2,R) some global coordinates in order to visualize the various regions of ellipticity, etc. (Compare with the picture in [7 1.)
This we do by the Cartan decomposition G = K exp p , where cos 9 - sin e A B
sine0 cost B -A
Conjugation by K is merely orthogonal rotations in the
A,B-planes, with respect to the coordinates 9,A,B
Setting R = j2+B , the coordinates for the subgroup Dijsatisfy cosh R cos e = 1 , and a two-dimensional 148
slice of the group is as follows.
h2 U -I h /
=/2
h
R he
ano mwodo"nw go" e 2 e=-7/2
=-7r -I
The four conjugacy classes containing%5) ( =1(0(-1 21 , and are each homeomorphic to R2-(0) , and form boundaries between the simply connected elliptic regions e1 ,e2 and the two hyperbolic regions h1 ,h2 homeomorphic to Sl xF2 .The orbit of complex structures
(I expp is clearly just the plane 9 = 7/2 Turning to the Lie algebra, in Theorem 9.1 we saw that (a b> 2 X = K-a) is elliptic iff det X = -a -bc >0, in which case b and c must be non-zero and of opposite sign. 149
The positive cone C0 is where -b,c > 0 , and with our coordinates C0 exponentiates into e . One can see that g = ( b)(ad-bc =1) is in e U e 2 iff ITr gj= Ia+dI < 2 , and we have
a b e = c d) E SL(2,R): ta+d < 2, -b,c > 0}
e2= {{(b)CSL(2,JR): fa+dj < 2, -b,c < 01
These restrictions are closely related to formulas for Y, , which can be computed. X = (_bE) C commutes with the unique positive complex structure
r(X) = (1/' -a2 -bc)( _b
and C e1 is in the unique maximal compact determined by
(a-d 2b a )(2c -a+d)
Now y(X) (XE C.) is just the unique complex structure
in the one-parameter group generated by X , so clearly
< could be analytically extended to sl(2,R) - 0 with
range S2 , by mapping each X to the ray in sl(2,BR)
containing X . However, this prescription does not give must insight into all the other cases, where dim K > I .
We pass next to the bounded domain, where G = SU(1,1)
acts on DI = Lizi < 1) and nonsingularly on the entire
Shilov boundary = S = (fzf = 1). The complex structures
go over to 150
2 2 1 = N0)D: c < 0, D E T, c = D +l} which act as reflections about points in D . Let
= su(1,1) , recall ia B2 2 0aEBy: a - |B12 > 0, a < 0} B -ia and note that the corresponding elliptic regions are
A B E = E G: |Re Al < 1, Im B > Im A BA and
A B E2 = E G: IRe Al < 1, Im B < Im A} the two latter conditions being equivalent to Im A + Im B < 0,
Im A + Im B > 0 respectively.
In what follows we define Khlerization on the group; this is a bit more general as the corresponding map on the Lie algebra can be recovered by evaluating Y on exponentials. A B\ One computes that T = ) E determines B A ic P (23) l= D)ED1 'Ti-ic where c = (Im A)/L , D = B/L , L = 1-(Re A) 2 and the
unique fixed points of are
zT = i (interior)
and
wT = iD (exterior) 151
But zT)wT are not analytic functions of T near a nilpotent boundary. To remedy this, we assign to zT two points y,mT on 6 which will be analytic functions of
T as T crosses a boundary, and seem to have a chance to generalize to higher dimensions. We will see that these points are closely related to the ordinary polar coor- 2 dinates on S . At least in the tube-type cases V dim S = fdimp , so two points on would seem appropriate to determine a point in the bounded domain.
Of course this assignment is not made completely invariantly, but relative to prior choices of a maximal compact K , a maximal abelian c c p , and a Weyl chamber
Oz+ c a(. We take
+ K-=-g)- s O} and then A+ = exp c moves the real line in D' to the left with the unique fixed points 1.
Given now T E E , ZT = S(0) for a unique trans- vection S from 0 . S is conjugate under K to a unique a E A+ , so let M = a2(i) . As T moves to a boundary mT heads toward -1; only with the square power does mT extend analytically. If T E K, zT = 0 and is undefined; if T 9 K , let 4T be the unique point on 1 S on the same ray from 0 as zT . ("Ray from 0" is a
K-invariant notion.)
With T = ) E E , 152
T = -i(B/IBJ) (if T 0 K) (24) 21Bj ImA + i((ImA)2- B 2 (IMA) + |B1 The a E A+ above is Q) where r = JDj/M , s= (c+l)/M., M = -2-2c , c , D as in (23).
mT now extends to all of G - ( I) , and IT to
G- K . We interpret 2 T"T for T hyperbolic, unipotent below, but first indicate the connection with 2 i polar coordinates (p,9) on S2 Set 2T = e and cos e = (-Im A)/( (Im A) +|BI2) , sin 9 = IBf / ( (ImA) +(B )
2 9 for 0 S 9 wr . Then mT = ie . Clearly the mapping 22 K1 : G ( I) -+ S factors to "(1: (G/I) - ( I - RP2 in the obvious way. %%B1 ) is the set of all points with latitude 9 > +45* If T E G is on the boundary of E, 2T is the unique fixed point of T ; if on the boundary of E2 , the negative of that point. mT = -1, +1 respectively in the two cases.
Any t hyperbolic T E G has two unique fixed points on S , and is a sort of finite "flow" from one to the other. Then 2T is one of the two points midway between these two fixed points: the midpoint on the "left" side of the flow if T is in the hyperbolic region with
-T in its closure (example below), and the opposite midpoint if T is in the other region. mT measures the 153
relative distance of the two fixed points from each
other, but we will not
make this precise.
T With our polar coordinates, it is clear that
x+iy z+iw Ey2 7 S2 IV,=(y,z,w)/jVT?x-iy z-iw for x,y,z,w E FR satisfying
2 2 2 (25) x Y 2+ -_z - =2= , Iy + Izj + Iwj / 0 and it is clear why tI E G is excluded. The mapping extends complex-analytically to
C. 2 ~~~X+iy z+iw)- YZw K : SL(2,C) - (tI) -P : x+[y z~iw z-iw x-iy for x,y,z,w E X satisfying (25). At least for
SU(ll) - (I) , K1 is clearly equivariant with respect to conjugation in G and some representation of G by analytic transformations of S2 leaving the "polar regions" o > 45, e < -45 invariant (hence did not really depend on the original choice of K and ct+) , but this has not been worked out.
We conclude this section with a few remarks for
KXhlerization in su(2,1), where most of the real work related to this chapter has gone! From [ 2 3 one sees that 154
all orbits in su(2,1) come from elements of u(l) e u(1,1), except for the six-dimensional "principal nilpotent" N
3 orbit, where N2 0 , N = 0 as in [21. It seems that
Y. should extend to all of su(2,1) , minus the union of all the four-dimensional elliptic orbits of real multiples -2i 0 0 of Xe 0 1 0, which we recognize as lying on the 0 0i boundary of a closed maximal invariant cone. If this is the case, Y would be singular on a closed five-dimensional submanifold of SU(2,1) . Note that orbits of elements like 0 0 0
S+I are six-dimensional, and their multiples to i make up a seven-dimensional union of orbits composing the rest of the boundary of the maximal cones. 155
CHAPTER VIII. The Cayley Transform.
29. General Properties.
If H is a self-adjoint operator on some complex
Hilbert space, the Cayley transform C(iH) = (I+iH)(I-iH) 1
U is a unitary operator, and H can be recovered by the equation
.U-I H = -..
We have seen that, among the algebras enumerated in section 2, only sp(n,]R) , so(2n), and su(n,l) have maximal compact subalgebras isomorphic to all skew-Hermitian operators on some complex Hilbert space. However, sp(n,]R) and so(2n) are distinguished as there this complex space can be naturally taken to be a complexification of the representation space of the algebra's fundamental
(defining) representation. We will see that in the former two cases (but not in the case of su(nl)) C- extends to an open subset of the noncompact algebra, mapping into the corresponding group, and has several interesting algebraic and "causal" properties. We discuss features specific to sp(') and so(-() in the next section, but in this section consider the mapping
(26) ('(X) =I+X n_ 156 in the general context of a Lie group acting on a finite- dimensional vector space.*
So, let the Lie group G and its Lie algebra act on V
Proposition 29.1. C- maps a neighborhood of 0 in into
G iff tanh (given by the power series) maps a neighborhood of 0 in into
Proof. As C- is nonsingular near 0, the condition is equivalent to C- (eY) Eat for all Y in a neighborhood of 0, and the equivalence follows from the identity
2 e+-e- eY/2 -Y/2 I eY -+e-Y = tanh Y/2 e Y+ I e Y/2 +e-Y/2
As the series for tanh has only odd powers, it is
clear that C- locally maps 7 into G if G is a subgroup of GL(V) defined by conditions of commuting with a family of operators and/or preserving some bilinear forms.
Let us now consider only those G satisfying 29.1, a fairly wide class. Nevertheless, C-tX): t EIRj is
rarely a one-parameter subgroup of G (even with a different parameter than t ).
Proposition 29.2. The mapping t -C-(tX) is (locally) a
a homomorphism iff X3 = 0 . If X / 0 , Cj(tX)l is a
(local) group iff X3 is a multiple of X .*
Proof. If s,t,u are small, one checks that 157
c(tX)Cs(sX) = S(ux) iff (t+s-u)X = stuX3 . If X3= cX u is uniquely determined by
u=t+s u=1+ cts
S is defined on an open subset of , and the following is easily shown.
Proposition 29.3. Let (dC)X be the differential of a at XE 7 . If C is defined at X,
[dL -o (dc)X](y) = 2(I+X)Y(-X) 1 for all Y E , where L (g E G) is left translation by g
Remark. Let RX = -dL 1(d)X be the linear dC a:cj -1 map above. Then
Ad(S(X)) = (Rp) RX
so R is something of a "square root" of the adjoint action.
We record some easy identities, in the context of S as a partially defined map from gl(V) to gl(V) 1) (-X) = C(X)1
2) ,-s(X 1 =-()
3) 2
4) S(tanh(X+Y)) = C(tanh X)S(tanh Y)
hold whenever all C-transforms make sense, and in 4), 158 when X and Y commute.
30. Cayley Transforms for Causal Groups.
We consider simultaneously the algebras of bounded operators sp(V) and so(V) and their corresponding groups Sp(V) and SO(Y) as defined in sections 7 and 16.
In each case the group acts as continuous automorphisms of the symplectic space (%4(,-)) , and additionally, elements of SO(V) commute with the i and a which anti-commute, have squares -I , and preserve a . The propositions in this section are valid for both classes of groups; we let G denote the group, JJ1 the algebra, and T the topology on V(
Definitions. Let*
C = (X Eoj: <(X-,-) is a positive-definite form
defining the topology T j and r = LX E Cl: X2 =-I .
Proposition 30.1. 0(X) E G for X E whenever it is defined, and 2 is causal with respect to the cone C1
Proof. Let v E V (i = 1,2) and X E such that
(I-X)~ - exists. Thus there exists unique w such that w. - Xw. = V. Then 159
V(,I(X)v,(X) V2 ) = (w1 +Xw2 ,w2 +Xw 2 )
=7(w1 - Xw 1 ,w2 - XW2 ) = d(v 1,v2 ) as X is skew with respect to a , so the first statement
is clear. The second follows from the computation of the differential of C in the last section and the identity
a((I+X)'Y(I - X)V v,v) = c(Y(I-X)~ v, (I -X) v) ,
X,Y Ej. ,vE7.
Remark. (Refer to the remark in the last section.) The
proof of 30.1 shows that the RX , although not automorphisms
of 1, nevertheless preserve the causal cone. In the
case of sl(2,2R) , the R turn out to be "conformal
automorphisms", that is, positive multiples of automorphisms;
this is a low-dimensional "accident". In fact, if
X = and B(.,.) is Killing form, a lengthy
computation shows that
B(Rx(Y),Rx(Z)) = (1/( -a2-bc)2 2)B(Y,Z)
for all Y,Z E sl(2,R) ; of course a 2 +bc is a multiple of B(X,X) .
It is clear that 0(J) = J and exp(7/2)J = J for
all J E P . A generalization of this relationship between
0 and exp holds for all of C . (By Theorems 7.1 and
16.2 C is defined on all of C1 .) 16o
Proposition 30.2. Let X E C so that X = JH = HJ for a unique J E I' and H positive-definite with respect to 3(-,-) = C(J-,-) . Then
0(JH) = exp(JT) where JT=TJ, H=tan-T, and r >T>O in (Vs )
Proof. I X are invertible, and*
2 2 2 c(X) = (I+X)/(I-X) = (I+X )/(I-X ) + 2X/(I-X ) 2 = (I-H2 )/(I+H ) + J(2H/(I+H2))
Take T positive and symmetric in (,& 5) such that w > T > 0 and (I-H2)/(I+H2) = cos T . Then sin T = 2H/(I+H2) and a(X) = exp JT. Also,
2 tan 2 T = (-(I-2 sin2 T)/I+(2 cos T-I)) = (I -cos T) /(I + cos T) = H2
so that H = tan iT.
Proposition 30.3. Let J E r be arbitrary. The image of C under is
(g E G: a(gv,v) RF(Jv,v) V v E Y , for some k > 0 .
Proof. Use the identity
C((g-I) (g+I) v,v) = 2a(gw,w)
where w = (g+I)~ U1 . The inequality above for a g E G implies first that g+I has dense range, and that (g+I)f exists quickly follows.
Naturally many mathematical questions arise concerning the applicability and causality of S for the other causal groups and structures. For the su(p,q) series 0 just does not map into SU(p,q) , only U(p,q); tr(-) = 0 implies det C(-) = 1 holds in general only for 2x2 matrices. C, does map so(2,n) into S00 (2,n) but is not causal; it suffices to examine (dC)x(Y) for X,Y in the two-dimensional maximal torus h of o(2,2) . Finally, an example in so(4 ) shows that 0 is not causal with respect to the minimal invariant cone, and causality seems doubtful for the maximal cone, too. However, it does not yet seem unlikely that causal mappings from algebra to group exist. (The exponential mapping is not causal, but note [19, p. 31 ].) Hopefully the isomorphism sp(2,E) ~ so(2,3) can suggest a definition for the so(2,n), and the failure for su(2,1) should be examined more closely.
Finally, our C- seems to have but loose ties with the Cayley transform for Hermitian symmetric spaces [153.*
However, the conformal mapping z -1+ in T , which takes the unit disc to the right half-plane and fixes i, is a*
Cayley transform for Sl(2, B)X/SO(2) . 162
CHAPTER IX. Applications to Differential Equations.
31. Preliminaries.
Let (H0e,->) be a real Hilbert space possessing an orthogonal J with square -I . Define the symplectic form a(-,) = as usual. The norm I.t=<-,.>1/2 gives the only topology on Ho that we use. In this final chapter we consider linear differential equations of the form (27) d (27) TX(t) = A(t)X(t) with periodic coefficients, i.e. for some T > 0
A(t+T) = A(t) , t E [0,co) where X(t) E Ho , and A(t) is a strongly continuous curve in sp(H0 ) . Recall that the latter means that for all v E Ho , t -+ A(t)v is a continuous curve in H .*
The usual existence and uniqueness statements hold for such equations. However in the infinite-dimensional case a solution to (27) is not necessarily (Frechet) differentiable; we define the solutions of (27) to be those X(t) satisfying the integrated form of the equation
(28) X(t) = X(O) + JO A(s)X(s)ds
Recall (section 25) that u(H ,0) and so(H0 ) can be regarded as subalgebras of sp(H0 ) 163
(27) is equivalent in the usual way to*
d (29) -U(t) = A(t)U(t) , U(o) = I where t ->U(t) is a curve in Sp(H0 ) , as the coefficients
are in sp(H0 ) . As in Floquet theory, U(-) satisfies
U(t +T) = U(t)U(T) , and as in [4 ], our basic reference for this chapter, we call U(T) the monodromy operator.
One advantage of the form (29) in the finite-dimensional
case is that (29) lifts to
(30) d U(t)A(t)U(t)/-'
where t -+U(t) is a curve in the universal covering group
Sp(H0 ) of Sp(H0 ) . (We identify the two Lie algebras.) Of course the solution of (30) covers that of (29). We
shall see in the next section that classical stability "bands" and regions correspond to certain regions of
ellipticity in Sp(H0 ) In accordance with general convention, we say (27) is
stable if each of its solutions (on [O,co)) is bounded,
and strongly stable [4 , p. 200] if for some e > 0 , all T equations X1 = A1 (t)X , where |fA(t) - A(t)|I|dt < s
are stable. It remains true in the infinite-dimensional
case that stability and strong stability are equivalent to
corresponding properties of the monodromy operator U(T)
described presently. (The uniform-boundedness principle is
crucial in this reduction.) We say U r K(H0 ) 164
[A: H 0 -+ H0 A real-linear, boundedj is stable if U exists and 2 M > 0 such that j|Un|| s: M for all integers n . U E Sp(H0 ) is called strongly stable if all
U E Sp(H0 ) in some norm-neighborhood of U are stable.
Lemma 31.1. ([ 4, pp. 219-20]) (27) is stable (strongly stable) if and only if the corresponding U(T) is stable
(strongly stable, respectively).
32. Regions of Stability.
We continue with the notation of the previous section, and in this section restrict H to be finite-dimensional.
For concreteness take H0 = R , and define the symplectic form as in section six.
Now the spectrum of any U E Sp(H ) is symmetric with respect to the unit circle (the Poincar6-Lyapunov
theorem), so U stable implies c(U) is a subset of
S= |zi I)= , invariant under complex conjugation.
Lemma 32.1. U E Sp(H0 ) is stable iff U is in some maximal compact subgroup, i.e., is elliptic.
Proof. The sufficiency is obvious. If U is stable, U
clearly must be diagonalizable over T . Extend c to
02n by complex bilinearity. By the invariance of a under
U , one sees that a U-invariant subspace on which U=e is C-orthogonal to one where U = e , unless e = e~ . 165
As the eigenvalues come in conjugate pairs, it is easy to find the proper basis of B 2n so that U is conjugate under Sp(n,E) to some
0 -D (31) exp( ) , D diagonal.
Given a stable U , the diagonal elements of D in
(31), say d. where -7 < d ! w7, j = 1,...,n, are unique up to a permutation. The eigenvalues of U (with tid tid multiplicity) are e ,...,e n . The classical termin- ology is that eid ..,eidn are the multipliersof the first kind, e ,...,en the multipliers of the second kind.
Definition. Let the systems of coefficients A(t).,A1 (t) give strongly stable equations. We say A(t) and Ajt) belong to the same stability region if and only if there is a continuous map
A2 : [0,T] x [0,1] -sr(H9: (t,s) ->A 2 (t,s) such that A2tO) = A(t), A2(t,1l) = Al(t), A2(0,s) = A2(T,s) and that the equation
x(t) = A2(t,s)X(t) is strongly stable V s E [0,1] . (One could require all these functions A, A1 , A2 to be merely piece-wise continuous in t , but the present definition suffices here.) 166
The stability regions were determined by I.M. Gelfand and V.B. Lidskii in [ 7 3. They are determined by an integer m , called the index, corresponding to the number of times (in one direction or the other) U(t) (Os9 t!T)
"winds around the group", and a pattern of the eigenvalues of U(T) , out of a total number of 2n such patterns, described as follows. U(T) can be the monodromy operator of a strongly stable equation (equivalently, U(T) itself is strongly stable (Lemma 31.1), i.e., has a neighborhood in Sp(H0 ) consisting of elliptic elements) iff the e obtained above have the property that no y E S1 is a multiplier both of the first and second kinds, that is,
there are no "repeated multipliers of unlike type". tid. Equivalently, no e J can be real, and, with the above restriction on the range of the d, , Vi,j = 1...,n d = -d. / 0 does not occur.
Corollary 32.2. An elliptic U E Sp(n,R) is strongly stable iff U is contained in a unique maximal compact
subgroup.
Proof. It follows from Proposition 25.3, the use of the
Cayley transform C , and the obvious fact that [J,X] =0 iff JC(X) = o(X)J for J a complex structure and X elliptic. Note that each strongly stable U is in the range of C , as -1 is not an eigenvalue.
Arranging the d . is order of ascending absolute 167 value, the "pattern of eigenvalues" above is just the sequence d1 /fdl ,**,dn/Idnl . Clearly there are 2n such patterns. We say that there are 2n elliptic regions in Sp(n,R) . As the mapping
11l+ ir iJR-+4S - (-) : ir - preserves the linear orders of' iF and Sl - (-1) , it is clear that these regions are diffeomorphic under Cl to the various regions in sp(nR) each containing elements uniquely Kghlerizable.
In [ 7] it is proven that each elliptic region is connected and simply connected. The region of conjugates of (31) where all d . > 0 , corresponds under C to the open positive cone in sp(n,JR) , hence being. convex, is contractible. One may ask, are the other elliptic regions contractible?
2 Note that the X E sp(n, B) such that X = -I (see the remarks in section 25; there are n+1 such orbits) are stable, but also strongly stable only in the two cases where X or -X is a complex structure, i.e. c(X.,-) is positive- or negative-definite.
As the elliptic regions in Sp(n,]R) are simply connected, the components of their inverse images in
Sp(n,R) project back to Sp(n,R) diffeomorphicly. The components of the inverse image of an elliptic region are parametrized by the index m E 2 above. The following is 168
clear from the theorems of [ ] and Corollary 32.2.
Proposition 32.3. Stability regions for equations of the form (29) are in natural 1-1 correspondence with the open regions in Sp(n,JR) consisting of those group elements uniquely rdhlerizable, i.e., those which are each contained in a unique (essentially) maximal compact subgroup, as follows. Lift (29) to (30), and associate a strongly stable A(t) with the region in Sp(n,3R) containing U(T)
It becomes clear why there should be stability "bands" for one-dimensional equations like the Mathieu equation
(32) Y"(t) + (e + 5 cos t)y(t) = 0 , T = 2; they just reflect the geometry of SL(2,R) . One has a mapping of (e,s)-pairs to U(T) E SL(2,R) , or rather
U(T) E SL(2,1R) , by solving the equation. The inverse image of the picture in section 28, under this mapping,
is just the diagram appearing in textbooks containing
sections on the Mathieu equation, for example, [ 1, p. 562].
33. Two Stability Criteria.
We return to the notation of section 31, and consider
the equation (27) with t -> A(t) strongly continuous and
norm-bounded: IA(t)1I < M Vt . Further, let l= H 0 e iH0
be the complexification of H0 with complex structure i,
and extend <.,.> on H. to a complex Hilbert structure 169
<.,-> on V' . We extend all operators on H0 to i-linear operators on V . Our goal in this last section is to prove Theorems 33.4 and 33.6. The latter is a generalization to
infinite dimensions of a theorem of M.. G. Krein [173. In the applications to quantum fields that we have
in mind U(T) becomes the "scattering operator", and we are interested in finding easily expressed conditions on
A(t) sufficient to guarantee the stability, i.e., unitari-
zability, of U(T) . Also, in these applications t-+A(t) will not be continuous in the norm topology.
We restrict now to "positive-energy" equations, that is, coefficients A(t) where
(33) a(A(t)v,v) 0 Vv E Ho , Vt E [O,TJ and for some k > 0 T (34) 1(Aavv,v) >k V v E Ho , where Aavv = A (t) vdt,
the integral of a continuous function in H0 . (The Bochner integrals we use are all integrals over an interval of -valued functions.)
Under somewhat more general hypotheses than these, one can prove
Lemma 33.1. ([4, p. 222]) Let [a,b] be a closed interval containing 0, and for 2 E [a,b] let U(T;X) be the monodromy operator for the equation 170
(35) (d/dt)X(t) = XA(t)X(t)
if
(36) (U(T;X) + I) exists as a bounded operator VX E [a,b], (35) is strongly stable
V X E [ab]
We recognize that (36) is equivalent to U(T;X) being a Cayley transform of some Y E sp(H ) , here
necessarily in the cone C of section 27, as the proof
shows.
Now let B be the Banach space of continuous functions
from [0,T] to V , with the norm
= sup Ilf(t)II (f E B) tE[0,T] and define
T C = dt for f,g E 3$
Let (%c be the completion of 3 in <,-% (1c' stands for curve.)
Lemma 33.2. ([4, p. 224]) As in Lemma 33.1, (U(T;X) +I)l exists if and only if for any f E 'V there exists a unique
solution X(t) in ? depending continuously on f (in
the norm jK-11 ) of the boundary value problem
(37) (d/dt)X(t) = xA(t)X(t) , x(0) + X(T) = f 171
We will solve the integrated form of this equation ((39) below), but first need some more notation.
Let A(t) = JH(t) ; then H(t) is a non-negative
Hermitian operator in (c . Define <-,-> on
8 xB by
T = Sdt , and let R = (f E / :H(t)f(t) = 0 Vt) . Then clearly
R = (f E C: 0 , and is - -closed in C Also if f E C
() jT f)|dt2 co 0 JHt lt
Define the function h on [-T,T] by h(t) =
0 t :rT). Let w = w/T . Then eio(2k+1)t - (-T t < 0) is an orthogonal basis of L2([0,T]) , and on [-T,T] it is a basis for that subspace of functions satisfying f(t +T) = -f(t) a.e. We have the L2-expansion
h(t) = (1/7i) 7 (1/2 k+l)elW(2R+-)t
etc. always denote a sum over all integers.)
For- f E C define
T (Kf)(t) = h(t-s)A(s)f(s)ds 0
Clearly K: C -+ 3 is continuous and vanishes on . Tn fact, 172
Lemma 33.3. ker K =
Proof. Kf = 0 implies fe-iW(2k+l)sH(s)f(s)ds = 0 for 0 all k E 2Z. Now s -+H(s)f(s) is a continuous function under our assumptions (t -+4H(t) strongly continuous and norm-bounded), so H(s)f(s) = 0 and f E .
We have introduced K because if X = 1 , (37) is equivalent to
(39) ((I- K)X)(t) =if
Thus it suffices to find conditions under whicn IlKi < 1 , or just invert I- K in the algebra of bounded operators in S , in order to deduce strong stability of (27).
In fact K is continuous with respect to all the other norms, too. Let f E e and g E 2V . Then for all t
2 2 <(Kf)(t),g>1 = 5Th(t-s)dsf 0
IT5{jH 1 2 s)f s) 'I lHl/2 (s)Jgjj ds)2 0
jJ ( H1/2J(s) 2(s)llds)l(I H /212ds) 0 0
e, g 12( IH(s)||lds) , so
Kf 11|1 1 T H(s) Ilds)1/2
Now 2/19 is a Banach space with the quotient norm
. By an earlier remark, <-,->s factors to a positive-definite form <.,->, on 2/19 , a kind of 173
"average-energy" norm. Let ( be the completion
of IS/9 in this norm. Then the inclusion J: S/9c-.
is continuous with norm
(41) 1 j I I ( ||H(t) Ildt)1/2 0 by (38). By (40) and (41), it follows that IJKlul s T IH(s) l|ds . By this estimate and the remark following 0 33.3, the following is evident.
Theorem 33.4. Suppose t -+A(t) E sp(H0 ) is strongly continuous, norm-bounded, periodic with period T , and satisfies (33) and (34). Then the equation
(d/dt)X(t) = A(t)X(t)
is strongly stable provided
jA(t) ldt < 2
We need additional notation in order to state our
second stability criterion. There are several other
continuous operators closely related to K that one can define. By 33.3, K factors to
K: (W/RI| q) - 8| |
and this extends to
K: (/c.. ÷(,jj I 174
Denote by K the composition
K K1KV---}0 /2 I +V
T 12T Then I (' HI)|sl/ 2 , IKJI| l fH(s)|ds 0 0 and of course range Kl S/9.
Lemma 33.5. Ki is self-adjoint.
Proof. To show 6 = (fg E /) , one uses h(-t) = -h(t) , A(t) E sp(H0 ) , and does an inter- change of order of integration.
Now h|KII < 1 is sufficient to force I - K to be invertible as a bounded operator in (3,ll|-||l) , as then 00 00 2 (I-K) = K2 = I+K+K K , where the sum converges 2=0 2=1 in the S-norm as the terms K * K can be factored
K
Recall the definition (34) of Aa . We are assuming
Hav > 0 where Aav = JHav , and
(Aav)2 = (JHav)(JHav)
2 (H 1/2 JH 1/2)(H1/2 JH1/2)Hh/2 H1/2av av av av av av so (Aav) 2 has nonpositive trace whenever it is trace class. (Compare with the Killing form in sp(n,R) .)
Unlike the criterion of Theorem 33.4, the following test is invariant under Sp(H0 ) . 175*
Theorem 33.6. Suppose that A(t) has an absolutely convergent Fourier expansion on [0,T] , and (Aav)2is trace-class, in addition to the assumptions of Theorem 33.4.
Then (note 33.5)
(42) tr(K) = -(T2/4) tr (Aa)2 and (d/dt)X = A(t)X is strongly stable if
-T2 tr (Aav) < 4
Remark. It is hoped that the assumption of an absolutely convergent expansion for A(t) can be weakened substantially or eliminated, as it is only used to justify a certain rearrangement of a series, as shall be seen.
Proof. We first show that Y is unitarily equivalent to a closed subspace of V. (proper if R / 0) , and deal with the transformed K1 . Define
T0 : I3/I -+ 5: f(t) +9 -+ H/2(t)f(t)
Then T0 extends to a unitary equivalence*
T: (I,<-,->) -+ (V,<.,->c) where V . 'Y is closed. One sees that f E V iff
H1 /2(t)f(t) = 0 a.e., and
(K2f)(t) ((TK 1 T )f)(t)
= Hl/2(t)h(t-s)JHl/ 2 (s)f(s)ds 0 176
for f E V .K2 extends to Vc by this expression and
vanishes on Vi ; denote this extension also by K2 . Then
tr(K ) = tr(K 2 ) Now H1 /2(t) can also be expanded absolutely, so let
2 H1/2(t) j ei wktLk , Lk: V ->V , satisfy
(43) <|Lk0
Let z E ( have norm 1; then wg(t)=(l/fT)e iW(2Z+l)tz
has norm 1 in /C . We have
(K2w )(t) = (ft/wr) 3 (1/2m+l)(Lk-mJLmzz)eiw(2k+l)t
k.,m
an absolutely convergent sum. Then is equal to (T 2) (1/2m+l) LkmJLm-,z, (1/2n+l)LnJL z>
k,kin n We wish to rearrange the summation in n,m,k,z in
order to faciliate massive cancellation. To show absolute
convergence, note 1n1111IILI (1/1 2m+1rn112n+-lf) I LkInII-ZIL-nI-ILJ k,Z, m, n 1
(/2n+1r 2(n+r)+K1Lk-r1LO Lk L ,,n,r
is less than a constant times
(45) I (3 ILKIL I) 2 r i Now 3 iLkl1lILk-riI) < co by (43), so clearly the sum of r r 177
squares (45) also converges.
Thus we can rearrange (44) arbitrarily. It becomes
(T2 /7 2 ) I I(1/(2n+l) (2(n+r)+l))7 r n k,9 However,
74(1/(2n+l)(2(n+r)+l)) = 0 n 2 2 if r /' 0 1 and 7 (1/(2n+l) /4 . To see this note n h(t)(T) = 1/4 , and multiply out the earlier expansion for h(t) .
The above then reduces to
(T2/4) I
Now L =Lk and (LkL- ) =Hav ,so
2 I K2 wK 2w> = <(T /4)
(T /4)3) (HjJL_ 9(H/ 2JL)zz>*
Letting z run over an orthonormal basis (za of gV we have
tr(K4) = (T2 /4) <(H/2JL_ )(H 2 JL ) ,Z >*
-(T2 /4)
-(T 2 /4) tr ((JHav) 2) -(T2/4)tr(Aav) 2 as claimed. 178
2 By Lemma 33.5, IK)! tr(Kl) , and so the last statement follows from the remarks after that lemma. This completes the proof. 179
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BIOGRAPHICAL NOTE - AFTERWORD
Steve Paneitz was born in Lawton,.Oklahoma on March 24,
1955, and lived in Red Oak, Iowa for about ten years until his family moved to McPherson, Kansas, where he attended the public high school. He attended the University of Kansas from 1973 to 1976, and subsequently moved to Boston to study at MIT. For the next two years he has a postdoctoral
fellowship from the Miller Institute for Basic Research in
Science at the University of California at Berkeley.
"The s t a r t lin g nearness of so many galaxies in t h e astronomy of the Other Men could be explained on t he theory of the 'expanding universe'. Well I knew that this dramatic theory was b u t tentative and very fa r from satisfactory...
"I, too, now sought to capture the infinite spirit, the Star Maker, inan image spun by my own finite though cosmical nature. For now -it seemed to me, it seemed, that I suddenly outgrew the three-dimensional vision proper to all creatures, and that I sa w with physical sight the Star Maker. I saw, though nowhere is cosmical space, the blazing source of th e hypercosmical ligh t, a s though it were a n overwhelmingly brilliant point, a star, a s un more powerful than all suns together. It seemed to me that this effulgent star wa s the centre of a four-dimensional sphere whose curved surface was the three-dimensional cosmos...
Olaf Stapldon, Star Maker (1937)
Would it have been worth while, To have bitten off the matter with a smile, To have squeezed the universe into a ball, To roll it toward some overwhelming question...
T.S. Eliot, "The Love Song of J. Alfred Prufrock"