Theory of Complex Homogeneous Bounded Domains

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Author: Yichao Xu

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Theory of Complex Homogeneous Bounded Domains

Mathematics and Its Applications

Managing Editor: M.HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 569

Theory of Complex Homogeneous Bounded Domains by Yichao Xu Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, P.R. China

^

SCIENCE PRESS BELTING/NEW YORK

l|S»l

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

A CLP. Catalogue record for this book is available from the Library of Congress.

ISBN 7-03-012335-2 (Science Press, Beijing) ISBN 1-4020-2132-1 (HB) ISBN 1-4020-2133-X (e-book)

Published by Kluwer Academic Publishers, RO. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic PubUshers, 101 Philip Drive, Norwell, MA 02061, U.S.A. Sold and distributed in the People's Republic of China by Science Press, Beijing. In all other countries, sold and distributed by Kluwer Academic Publishers, RO. Box 322, 3300 AH Dordrecht, The Netherlands. This is an updated and revised translation of the original Chinese publication. © Science Press, Beijing, R. R China, 2000

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All Rights Reserved © 2005 Science Press and Kluwer Academic Publishers No part of this woric may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permissionfromthe Publisher, with the exception of any material supphed specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in China.

Preface

In tracing the history of classification theory of homogeneous bounded domains in C^, one must mention the pioneering work by H. Poincare (ref. [161]). In 1907, he proved that thebicyhnder (|2:i| < 1, \z2\ < 1) and the hypersphere (l^^ip + |2:2p < 1) in C^ are not holomorphically equivalent to each other, but they are both connected and simply connected. Therefore, the famous Riemann Theorem in the geometric function theory of one complex variable is not true in C'^. In 1935, E. Cartan (ref. [15]) gave a complete classification of Hermitian symmetric spaces. He proved that any Hermitian symmetric space is a topological product of irreducible Hermitian symmetric spaces, and there are exactly four classes and two special kinds of irreducible Hermitian symmetric spaces. He also gave the realization of four classes of irreducible Hermitian symmetric spaces in the complex Euclid space (The irreducible symmetric domains are called "classical domains" by Hua (ref. [70] ). E. Cartan also gave two special kinds by the left coset spaces E6(_i4)/(SO(10) X T) of complex dimension 16 and E7(_25)/(E6 x T) of complex dimension 27. But he did not know that these coset spaces can be imbedded into the complex Euclid space. In 1956, Harish-Chandra (ref. [63]) introduced the so-called Harish-Chandra imbedding, and proved that every Hermitian symmetric manifold is holomorphically isomorphic to a complex symmetric bounded domain. But again he did not know the explicit expressions of these two special kinds in C^^ and C^^. On the other hand, H. Cartan gave a complete classification of homogeneous bounded domains in C'^ and C^, which was published in E. Cartan's paper (ref. [15]). Since all homogeneous bounded domains are symmetric in C, C^, C^, E. Cartan proposed a famous conjecture: Any homogeneous bounded domain in C^ must be symmetric.

vi

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

There was no significant progress until 1954—1955, when A. Borel (ref. [9]) and J. L. Koszul (ref. [113]) independently proved that if the holomorphic automorphism group Aut (D) acting transitively on a homogeneous bounded domain £> is a real semisimple Lie group, then D must be symmetric. In 1959, Piatetski-Shapiro (ref. [155]) found two counter-examples of non-symmetric domains and E. Cartan's conjecture was answered in the negative. In 1960, Piatetski-Shapiro (ref. [156]) introduced the notion of Siegel domains. In 1963, Vinberg, Gindikin and Piatetski-Shapiro (ref. [222]) proved that any homogeneous bounded domain is holomorphically isomorphic to a homogeneous Siegel domain. Therefore the classification of homogeneous bounded domains up to a holomorphic isomorphism is reduced to the classification of homogeneous Siegel domains up to an aifine isomorphism ( an affine isomorphism means a non-singular linear transformation with a translation). In 1943, the function theory on a homogeneous bounded domain was studied by Siegel (ref. [177]) for the first time. He considered the automorphic functions on the matrix domain Im (Z) > 0, where Z is an nxn complex symmetric matrix. This domain is holomorphically equivalent to the classical domain of the second kind. At the same time, L. K. Hua (ref. [70]) studied the classical domains case by case. In 1965, Koranyi (ref. [109]) studied the symmetric bounded domains by the Harish-Chandra imbedding. In order to study the classification and realization of homogeneous Siegel domains, we need to classify and realize homogeneous cones up to an affine isomorphism at first. In 1963, Vinberg (ref. [220]) realized a homogeneous cone by a subset in a non-associative algebra (so-called T algebra). Since the homogeneous Siegel domain of the second kind of rank AT is a sectional surface of a homogeneous Siegel domain of the first kind, Takeuchi (refs. [187], [188]) proved that the homogeneous Siegel domains of the second kind could be imbedded into some T algebras of special kinds in 1975. In 1976 and 1977, the author (refs. [229], [230]) constructed a small class of homogeneous Siegel domains in C^, which were called normal Siegel domains (the old name is N Siegel domains). Normal Siegel domains are defined by a special matrix set satisfying some conditions, which is called a normal matrix set of type N (the old name is N matrix system). We proved that any homogeneous Siegel domain is affinely equivalent to a normal Siegel domain. Hence the classification of homogeneous Siegel domains up to an affine isomorphism is reduced to the classification

PREFACE

vii

of normal Siegel domains up to an affine isomorphism. Then the classification of homogeneous Siegel domains up to an afSne isomorphism is reduced to the classification of normal matrix sets up to a special unitarily equivalent. But this classification problem has not been completely solved up to now. The purpose of this book is to introduce the theory of homogeneous bounded domains. In Chapter 1, we give some general results on bounded domains and Siegel domains, and compute the Lie algebra of the holomorphic automorphism group of a Siegel domain. In Chapter 2, we introduce homogeneous Kahler manifolds, homogeneous bounded domains and homogeneous Siegel domains in C^. The sufficient and necessary condition of the Lie algebra of the holomorphic automorphism group acting transitively on a homogeneous Kahler manifolds will be given as well. In particular, we introduce J Lie algebras, effective proper J Lie algebras and normal J Lie algebras. We give the Piatitski-Shapiro decomposition and the J basis of a normal J Lie algebra. Some properties are discussed for homogeneous Siegel domains. In Chapter 3, we introduce the notion of normal cones and normal Siegel domains, and prove that any homogeneous Siegel domain is affinely isomorphic to a normal Siegel domain. We will prove that the classification of normal Siegel domains up to an affine isomorphism is reduced to the classification of a normal matrix set of type AT up to a special unitarily equivalent. We give the exphcit expressions of Bergman kernel function and Bergman metric for any normal Siegel domain (ref. [233]) in the last section. In this book, we use normal Siegel domains to study the geometric property and the function theory for homogeneous bounded domains, which is not used case by case. In Chapter 4, we prove that the (generahzed) Bergman mapping is a holomorphic isomorphism from a normal Siegel domain onto a homogeneous bounded domain. Thus we obtain a class of canonical homogeneous bounded domains, which are called normal bounded domains (ref. [238]). We also deduce the Vinberg's and Takeuchi's realizations from normal Siegel domains. Then we conclude that Vinberg's realization is a real parameter representation of homogeneous Siegel domains of the first kind and Takeuchi's realization is also a real parameter representation of homogeneous Siegel domains of the second kind (refs. [220], [187]). In Chapter 5, we give the affine automorphism group Aff (V) acting on a normal cone V, the affine automorphism group Aff (i?(V,F)) and the holomorphic automorphism group Aut {D{V, F)) acting on a normal Siegel domain D{V,F) (ref. [230]). We also give the isotropy subgroup

viii

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Iso {a{D(y^ F)))^ where a is the Bergman mapping of D{V, F), In Chapter 6, we give the classifications of complex and real Hurwitz matrix sets. Then we give the complete classification and realization of square domains and a partial classification of dual square domains (refs. [231], [232], [235]). We prove that any quasi-symmetric Siegel domain (which is mentioned by Takeuchi (ref. [190])) is a square domain. In Chapter 7, we give a new proof of the classification and realization (to normal Siegel domains) of symmetric bounded domains. This realization contains two exceptional symmetric bounded domains in C ^^ and C^^. In the last section, we give some properties of the two exceptional symmetric Siegel domains, (refs. [227], [231]—[233], [239], [242]—[244]). In Chapter 8, we give the Cauchy-Szego kernel function and the formal Poisson kernel function of normal Siegel domains. We prove that the formal Poisson kernel function of a normal Siegel domain is a Poisson kernel function if and only if this normal Siegel domain is a symmetric Siegel domain. Hence the Stein-Vagi conjecture holds for the homogeneous Siegel domains (ref. [233]). In Chapter 9, the following results are proved: (1) The Vinberg, Gindikin, Piatetski-Shapiro theorem: Any homogeneous bounded domain is holomorphically isomorphic to a homogeneous Siegel domain; (2) the holomorphic automorphism group Aut {D) acting transitively on a homogeneous bounded domain £) is an algebraic Lie group and its isotropy subgroup is a maximal compact subgroup (ref. [222]). It follows that any homogeneous bounded domain is holomorphically isomorphic to a normal Siegel domain by Chapter 3. In this book, the symbol "•" indicates that the proof of a Theorem or Lemma is finished. The author would like to express his special thanks to Professors Shannian Lu, Zhongmin Shen, Tianze Wang and Ying Bi for carefully reading the manuscript.

Yichao Xu

Contents

Preface

v

Chapter 1. SIEGEL DOMAINS 1 1. Bounded Domains 1 1.1. Some Conceptions and Symbols 1 1.2. Bounded Domains 11 2. Siegel Domains 20 3. Holomorphic Automorphism Group of Siegel Domains 32 Chapter 2. HOMOGENEOUS SIEGEL DOMAINS 47 1. Homogeneous Bounded Domains 47 2. Homogeneous Siegel Domains 56 3. Normal J Lie Algebras 65 4. J Basis of a Normal J Lie Algebra 73 Chapter 3. NORMAL SIEGEL DOMAINS 101 1. Normal Cones and Normal Siegel Domains of First Kind ... 101 2. Normal Siegel Domains 124 3. Decomposable Normal Siegel Domains 139 4. Bergman Kernel Function of Normal Siegel Domains 145 Chapter 4. OTHER REALIZATIONS 151 1. Homogeneous Bounded Domain Realization 151 2. T Algebra Realization 164 Chapter 5. AUTOMORPHISM GROUP 181 1. Affine Automorphism Group of Normal Cones 182 2. Affine Automorphism Group of Normal Siegel Domains — 204 3. Holomorphic Automorphism Group 210 4. Other Results 232

IX

X

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Chapter 6. CLASSIFICATION OF SQUARE DOMAINS 1. Classification of Two Special Matrix Sets 2. Normal Cones and Dual Normal Cones 3. Classification of Square Cones 4. Classification of Dual Square Cones 5. Classification of Square Domains 6. Classification of General Square Domains Chapter 7. SYMMETRIC BOUNDED DOMAINS 1. Symmetric Bounded Domains 2. Semisimple J Lie Algebras 3. Classification of Symmetric Siegel Domains 4. Reahzation of Exceptional Cases Chapter 8. SZEGO KERNEL AND POISSON KERNEL 1. Cauchy-Szego Integral 2. Formal Poisson Kernel Function 3. Stein-Vagi Conjecture 4. Poisson Integral on Symmetric Siegel Domains Chapter 9. HOMOGENEOUS BOUNDED DOMAINS 1. Non-Semisimple Effective J Lie Algebras 2. Algebraic J Lie Algebras 3. Realization of Homogeneous Siegel Domains 4. Homogeneous Bounded Domains 5. Main Theorem References Index

237 237 253 265 278 282 284 291 292 303 313 319 329 330 339 344 355 361 361 371 378 391 406 411 425

Chapter 1 SIEGEL DOMAINS

In this chapter, we will discuss some properties of bounded domains in C^ from the Bergman kernel function, introduce the notion of Siegel domains and compute the Lie algebra of the holomorphic automorphism group on a Siegel domain.

1.

Bounded Domains

In this section, we will give some geometric properties of a bounded domain (or a unbounded domain, which is holomorphically isomorphic to a bounded domain) in C^ using the Bergman kernel function.

1.1.

Some Conceptions and Symbols

We will give some conceptions and symbols using in this book most frequently. (1) Let i4 be an n X m complex matrix, let A be the conjugate matrix of A, and let A^ be the transpose matrix of A, Sometimes an n X m matrix is also denoted by A^^''^^. A square matrix A = A^'^''^^ is denoted by A^'^h Usually, E or E^'^^ denote the unit matrix of order n. Let { a i , • • • ? <^n } be a basis of a vector space V. The vector a = X]iLi ^i^i has the coordinates x = (xi, • • •, Xn)^ it is also a 1 x n matrix. The matrix representation A = (a^j) of a linear transformation A is defined by A{ai) = Yl]=i ^ij^j^ i = 1, • • •, n. Let y = (yi, • • •, yn) be the coordinates of A{a). Then y = xA. Given two linear transformations

2

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Aj B onV with the matrix representations A = (a^j), B = (fe^j), respectively, then the product AoB has the matrix representation BA. Hence the mapping A^^ A gives a group isomorphism from the general linear group GL{y) onto its matrix representation GL(n,C), where A is the transpose matrix of A. GL (n, C) is called the general matrix group. Given a complex matrix A, then the real part of A is denoted by Re (A) and the imaginary part of A is denoted by lm{A). Hence A = RelA) + yf^Im{A), If a matrix A is a real symmetric matrix or a Hermitian matrix, by ^ > 0, A > 0, ^ < 0, A < 0, we mean that A is positive definite, semipositive definite, negative definite, semi-negative definite, respectively. (2) Let / ( a , /?) be a real symmetric bi-linear function on a real vector space V. Given a basis { a i , • • •, a^ } in V^, let Sij = f{ai,aj),

l

The nxn real symmetric matrix S = (sij) is called a matrix representation of the real symmetric bi-linear function /(a,/?) with respect to the n

basis { a i , • • •, Qin }. On the other hand, given two vectors a = ^

xiai

n

and /? = X] Vi^i^ ^^^^ i=l

f{a, /?) =. (xi, • • •, Xn)S{yi, '",yr,y

= xSy',

(1.1)

where x = (o^i, • • •, Xn) and y = (yi, • • •, j/n) are the coordinates of a and /?, respectively. _ Given another basis {/?i, • • •, /?„ } in F , let 5 = (/(A? /^j)) be a matrix representation of / with respect to the basis { ^ i , • • •, /?n }• Then 5 and S satisfy the relation S = PSP\ where an n x n non-singular matrix P — {pij) is a matrix representation of the basis transformation: n

Pi = Y1P''0^O^

i = 1,2, •••,n.

Now we give some notions on a real analytic manifold 9Jl of dimension 2n as follows.

Siegel Domains

3

(3) Given a point p in a real analytic manifold 9Jl of dimension 2n, the real vector space Tp{dJl) of dimension 2n consists of all tangent vectors on a point p G 3Jl, which is called the tangent space on p. A real analytic vector field X on 3JI is a set of tangent vectors {Xp e Tp{M) | Vp G 9Jt}, such that p -^ Xp^ Vp G SDT is a real analytic mapping. The vector space r(SPt) consists of all real vector field on 3DT. Let {(C7, -0)} be a real chart cover of a real analytic manifold 9Jl. Given a chart (C/, ip) of SPt, denote by ilj{p) —u the local real coordinates of p G [/, where u = (i/i, • • •,U2n) G R^^. A real vector field X has a local coordinate representation:

2=1

dui

du

where C('^) = (Ci(^)? * * * 7C2n('?^)) is a real analytic vector function on

my

Let d be a differential operator in R^'^. Then du = {dui, • • •, du2n), where u = {ui, • • •, U2n) G R^'^. (4) Newlander and Nirenberg prove that 3Pt is a complex manifold if and only if there exists a real linear transformation Ip of Tp{dJl), Vp G SDT with the following properties: (a) the mapping / : p —> Ip, Vp G 9Jl is a real analytic mapping on 9Jt;

(b)

l^ =

-idp,ypem;

(c) [Xp,Yp]+Ip[Ip{Xp),Yp]+Ip[XpJp{Yp)] = [Ip{Xp\Ip{Yp)i \^pem,\/Xp,YpeTp{m). I is called a complex structure of 9Jl. Obviously, condition (a) implies that / is a real analytic tensor field of type (1,1) on 3DT. Condition (b) implies that 72 = - i d . (1.3) Condition (c) implies that [Xo, Fo] + /[/(Xo), Yo] + /[Xo, I{Yo)] = [I{Xo)J{Yo)i VXo,lo G r(9Jl).

^ •^

(5) A complex manifold 971 of dimension n is a real analytic manifold of dimension 2n with respect to a complex structure / . Hence there is a real chart cover { (f/, ^|;) } of 9Jl, such that / is represented by a 2n x 2n constant matrix

4

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

on '0(C/). Denote by ip{p) = u = (x,y) the local real coordinates of '0(f/), where x = {ui, • • •, Un), y = (^n+i, • • •, ^2n). Thus ^l^dp) = z = X + yf^y is a local complex coordinates of '^(f/). Hence {{U^il)c)} is a complex chart cover of 9Jt. The relation between -0 and V^c is U 3p^u

= {x,y) <—> z = x + y/^y<^pe

U.

(1.6)

Denote

where

d ^1 f d 9zi

^

2 \9iXi

a \

_9__l/_9_

dun+i) ' dfzl

^

2\dui

a \ dun-\-iJ *

The totally differential operator in C^ is denoted by d, and d is the complex conjugate of d, where d2: = {dzi, • • •, d^;^), (i.2:i = dui + y/^dUn-\-i^

d;2^ = {dzi, • • •, d^^n), dzi = dui — y/^dUn-\-i-

(1.8)

(1.5) gives that d_ l(A.)d \ = _^^

j(/

d ]^ \ ^

d^ ,

(1.9)

Thus

Hence / can be expressed by

^ =^ E ( ^ ® ^ ^ . - 4 ® ^ ) j=i

•/

(1-11)

-^

(6) Take an arbitrary real analytic vector field XQ on a complex manifold 9DT with a complex structure / . XQ has a complex coordinate representation

xo = E « i W ^ = E ' ' * . ^ ) | - + E ; ^ ( ^ i | . i=l

2=1

(1.12)

i=l

where ^i(tA) is a real function, 1 < i < 2n, and miz.t)

= ^i(u) + V^^n+z(^),

1 < i < n.

(1.13)

Siegel Domains Let (1.14) We have

Xo = X + X.

(1.15)

Next, we give some notions and properties of a complex manifold 9Jl of dimension n as follows. (7) Let ([/, i/j) be a chart of SPt, where '^(p) = (?/i, • • •, U2n)i Vp G C/, is a real coordinate representation on ^l^{U), A complex value function / on SDt is called a holomorphic function if and only if the real value vector function F — ( R e ( / ) , I m ( / ) ) satisfies the Cauchy-Riemann equation system: 9Re(/)^aim(/) 9^i 5l^n+i '

aim(/) ^ dUi

aRe(/) dUn-\-i

.^^...^

The expression of complex coordinates is a/ = ^ = 0 . (1.16) oz Hence f{z^~z) only contains the complex variable z. Denote by Hol(9Jl) the holomorphic function space of 3DT. (8) A topological transformation group Auti?(9Jl) consists of all real analytic automorphisms on 9Jl. Given a point p G 2Jt, the isotropy subgroup at p is defined by (Isoi^)p(9Jl) - { a G Aut,^(an) I <j{p)=p},

(L17)

Let (i7, il)) be a chart of 9Jt, where i^{p) = (?xi, • • •, ?/2n)7 Vp G C/. Given cr G Auti^(9Jl), then (j{p) = (pi(i^), • • • ,52n('^)) i^ ^ ^^^^ analytic vector function on rp{U). (9) A real analytic automorphism a on 3DT is a holomorphic automorphism if and only if cr*|p o Ip = I^o a^\p on Tp(9Jt), Vp G 9Jt. The topological transformation group Aut (9Jl) consists of all holomorphic automorphisms on 9Jl. Aut (9K) is called the holomorphic automorphism group, and / is called an Aut (9Dt) invariant complex structure.

6

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(10) Let ([/, t/j) be a chart of the complex manifold 9Jl of dimension n, such that i(){p) = (^i, • * * ? ^^^n), Vp G [/". Given a E Aut (9Jl), then the local coordinate representation of a is a:

z = {zu '-jZn)-^w

= {wi, • • •, ix;^) = (fiiz), • • •, fniz)),

(1.18)

where ifi{z)j • • •, fn{z)) is a holomorphic vector function on ip{U). Thus the real coordinate representation of a is an:

U= {Ui, ' • •,U2n)

-^V = {Vi, • • ',V2n)

= ( 5 i ( ^ ) , • • • ,52n(^))^

(1.19) where aR € Auti2(9H)j -Ui = Re {zi)j Un+i = Im {zi), Vi = Re (wi), Vn-\-i = Im (to^), i = 1, • • •, n. The mapping (7-X7/2, V(7€Aut(aJl)

(1.20)

is an isomorphism from the topological group Aut (9Jl) into the topological group Aut 1^(971). (11) Let (SPt, g) be a 2n dimensional Riemannian manifold with a Riemannian metric g. Given a real chart ([/, ip) of 9Jl, then 2n

g =rfs^_ ^

g^j{u)dui ® duj — duG{u) ® du\

(1-21)

where G{u) is a 2n x 2n real positive definite symmetric matrix on ip{U). (12) If a 2n dimensional Riemannian manifold (OJt, g) has a complex structure / , such that g{I{Xo), /(lb)) = 5(^0, ib),

V Xo, FQ e rR(!OT),

(1.22)

g is said to be invariant with respect to the complex structure / . In this case, G{u) satisfies JOG{U)JQ = G{u)j where JQ is defined by (1.5). Given the block partitions of matrix G{u) as (1.5), then

«(") = ( -'it sg ) •

f^-^^)

Then H{z,z) = S{u) + y/^K{u)

(1.24)

is a positive definite Hermitian matrix, and g can be represented by g = ds'^ = Re {dzH{z, z) ® 'dz').

(1.25)

Siegel Domains

7

In fact 5 dz — dx + y/^dy,

dz = dx — yf^dy

(1.26)

give that dx=^ -{dz + lz), 2

dy=—j={dz-dz). 2 Y —1

(1.27)

Hence (1.21)—(1.24), (1.26) and (1.27) induce g = duG{u) ® du' = {dx, dy)G{u) ® (dec, dy)'

= Re {dzH{z,

z)®dz)

as desired by (1.25), where E is the unit matrix. The tensor field h of type (1,1) is called a Hermitian metric, where h = dzH{z, t)®l^

= g + %/^0,

(1.28)

and H{z^ J) is a positive definite Hermitian matrix. In this case, (971, h) is called a Hermitian manifold with respect to the Hermitian metric h. The formula (1.24) implies that the real part g of /i is a Riemannian metric, which is expressed in the following form: 1 ^ 3 = Re (/i) = - y j hij{z^~z){dzi ® dzj + dzj O dzi).

(1.29)

The imaginary part fi of /i is a (1,1) form: n

0 = Im(/i) = - \ / ^ ^

hijiz,z)dziAdz'j,

(1.30)

O is called a Kahler form with respect to the Hermitian metric h. (13) A Hermitian metric h is said to be Kahler on a Hermitian manifold 9JT, if the Kahler form n is a closed form. Namely, dO = 0.

(1.31)

A Hermitian manifold (9Jl, h) with respect to the Kahler metric h is called a Kahler manifold.

8

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(14) Suppose that (9Jl, h) is a Kahler manifold. A holomorphic automorphism a is called an isometric transformation, if a satisfies the condition cr*(/i) = h. The isometric transformation group I(SDt) consists of all isometric transformations of (9JI, h). In this case, h is said to be invariant under the isometric transformation group I{WV). Given an isometric transformation cr of a Hermitian manifold (SDt, /i), then aR is also a real isometric transformation on the Riemannian manifold (aJl, Re(/i)), where an is defined by (1.19). (15) It is well known that the isometric transformation group I(9Jl) of a Kahler manifold (9Jl, h) is a Lie transformation group on 93T. Hence lRm)

= {aR\yaeim}

(1.32)

is a real closed Lie subgroup of the topological transformation group Ant R{9JI), such that mapping (1.20) is a Lie group isomorphism. (16) Let (OJl, h) be a Kahler manifold with a complex structure / . Then {(TR),oI = Io{aRU

{cTRr{h) = h,

\/aRelR{m).

(1.33)

Denote by fi = Im (h) the Kahler form of /i, we have niX,Y)

= 9{X,I{Y))^{Re{h)){X,I{Y)),

\/X,YeT{m).

(1.34)

(1.31) and (1.34) imply that

n{Y,X) = -n{x, Y), vx, r € T{m); n{ix, lY) = n{x, Y), vx,y € T{m); n{ix, x)>o, yo^xe T{m).

(1.35) (i.36) (1.37)

(17) Given a point p € SPt, then the isotropy subgroup {lsoR)pm

= {ae

iRm

I a{p) = p }

(1.38)

in IR{WI) is a compact Lie subgroup. (18) Let (9Jl, h) be a Hermitian manifold of dimension n. By (1.28), an isometric transformation a: w = f{z) is a holomorphic automorphism satisfying dwH{w, w)®dw = dzH{z, 'z)®dz.

Siegel Domains The Jacobian of a mapping a is defined by / dwi

dwn \

dzi

dzi

dw dz "

(1.39)

dw\

dwn

dZn I

\ dZn

where the down index denotes a column number, and the upper index denotes a row number. Let z = {zi," - ,Zn) and w = {wi^ • • •, Wn) = {fl{z),---,fn{z))=fiz),

dw =

(1.40)

dz-;^. dz

Hence 'dw

fdw\'

(g).K.)(^) =.(..).

(1.41)

(19) Suppose that 9Jl is a complex manifold of dimension n with respect to a complex structure / . Given a complex chart (C/, V^) of 9Jl, then a vector field XQ G TR{^) has a local coordinate representation Xo = X + X,

X = i{z,^)—

&

d

= 2=1

Y.^i{z,-z)—, dzi

where ^(^, z) = (^1(2;, z)^-- • ^^n{z^z)) is a complex vector function. 1.1 A differential operator Lx acting on the tensor algebra A(9Jl) o/3Jl is called a Lie derivative, if

DEFINITION

Lxof = Xof, Lxo{Yo) = [XoM, Lxoidzi) = d^i{z,z),

/GHol(9Jl); YoeTRimy, 1 < i < n;

Lxoidzi) = d^i{z,z),

1 < z < n;

LxW

= %(V^),

^ G A(9Jl).

A complex vector field X on 9Jt is called a holomorphic vector field, if Lx{I) = 0. Given a complex chart {U,ip), then X is a holomorphic vector field if and only if (1.42) i=l

10

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where ^{z) = {^i{z)^ • • •, ^ni^)) is a holomorphic vector function on ip{U). (20) Given an isometric transformation a on a Hermitian manifold (3!Jl, h)j then aR is a real isometric transformation on the Riemannian manifold (3Jl, Re(/i)). Denote by i(SPt) and ii?(3Jl) the Lie algebras of Lie groups I(3Jl) and IR(WI), respectively. X e i(9Jl) if and only ii XQ = X+ X e ii?(9Jl). (21) Given a vector field X e i(93t), then the one parameter subgroup exp (tX), Vt G R, is a unique holomorphic solution of the complex ordinary differential equation system ^^=^i{w{t)),

\t\<e,l
(1.43)

with the initial value w{0) = z,

(1.44)

Now, a one parameter subgroup exp{tX), coordinate representation

Vt G R, has a local complex

m = fi{z, t),

fi{z, 0) = Zi,

l
(1-45)

Thus exp (tX)^ Vt G R, has a real local coordinate representation: Pi = Re(wi) = Re{fi{z,t)),

qi = Im{wi) = Im[fi{z,t)),

l
where z = a:-h^/—ly,

w =p+

y/^q.

The real coordinate representation of a one parameter subgroup exp (tX)^ Vf G R, is the unique real analytic solution (p, g) of the real ordinary differential equation system

with the initial value Pi{0)=Xi,

qiiO) = yi,

l
Hence the local real coordinate representation of exp(tX), Vt G R, is determined by a real analytic vector field XQ = X + X on the Riemannian manifold (9Jl, Re_(/i)) (see (1.12)—(1.15)). Therefore X G 1(071) implies ths^tXo = X + X einim).

Siegel Domains

1.2.

11

Bounded Domains

Now, we consider a special class of complex manifolds. Suppose that £) is a bounded domain in C^. Denote by H{D) and L'^{D) = L?{D^ii) the holomorphic function space and the square integrable function space on D, respectively, where —— j dzi f\"'

^ dzn Adzi A" ' A dzn

(1-46)

is the Euchd measure on R^^ = C^. Bergman (refs. [7],[8]) proved that (1) the function space L'^{D) fl H{D) is a Hilbert space with respect to the inner product

if. 9)= I f{z)W)^^\

(1-47)

JD

(2) the Hilbert space L'^{D) fi H{D) has a countable basis; (3) given an orthonormal basis (Ai(2),
(1.48)

the function oo

if(2,liJ) = 5 ] 0 i ( 2 ) ^ ^

(1.49)

is called the Bergman kernel function. Then K{z,w) is a holomorphic function on £> x ^ , and K{z,'z) is independent of the choice of any orthonormal basis, where 5 = {2 € C " I 2 € £)}; (4) f{z) = I K{z,w)f{w)n,

yzeD,

fe

H{D) n L2(Z>).

(1.50)

JD

Bergman proved that

h = ds'^ = ddlogii:(^,z) = dzT{z,z)Tz

= Y

^-^^^^k^dzi0d^j

(1.51) is a Hermitian metric on D, which is called the Bergman metric. Namely, the n X n matrix

12

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

is annxn positive definite Hermitian matrix, which is called the Bergman metric matrix. The Bergman metric h induce a Riemannian metric and a Kahler form g = Re{h) = ly^-^^^£k^{dzi®d^i+d^j0dzi),

n = lmh = -V^

T ^^^^^^dziAd^j,

(1.53)

(1.54)

respectively. Obviously, dO = 0, hence /i is a Kahler metric. Given two holomorphic vector fields

on D, then two real analytic vector fields

(1.56) i=l

2=1

where ^{z) = (Ci(^)r • • ^CnC^)) and r]{z) = {r]i{z),-- ,rjn{z)) are holomorphic vector functions. Hence 9{Xo,Yo) = I f^ mz)W) + WMz)f ij=l

^2^^^^^ >

(1-57)

* ^

where fi is a real skew-symmetric bilinear function, and g{Xo,IYo) = niXo,YQ),

(1.59)

n{IXo,IYo) = n{Xo,YQ).

(1.60)

Given a holomorphic automorphism a : w —* z = f{w) on D, then Mz) = Mm){det ^ ) ,

Mz) = Mm)ldet ^ ) , • • •

Siegel Domains

13

is also an orthonormal basis of H{D) fl LP'iD), Thus K{z,z)

=

dw^

•"^az

K{w,w)

(1.61)

and r(z,.) = ^T(..wf£'.

(1.62)

Hence the isometric transformation group on the Kahler manifold (JD, h) is equal to the holomorphic automorphism group Aut {D) on D. An Aut (D) invariant volume element of D is v = x{^yK{z,z)fi,

(1.63)

where A is a positive constant and /i is defined by (1.46). 1.2 A holomorphic vector field X on a bounded domain D in C^ is called a Killing vector field associated with the Bergman metric h = ddlogK{z^^)^ if DEFINITION

Lxo{h) = L^+xih)

= 0,

(1-64)

where K{z,'z) is the Bergman kernel function of D. Obviously, we have 1.1 A holomorphic vector field X on a bounded domain D is a Killing vector field if and only if LEMMA

a'R.(xy.,.)) ^

^^

aziOZj LEMMA

1.2

Give a holomorphic vector field

2=1

on a bounded domain D. Then Lxoiv) = 2\[^yK{z,z)Re

{xiogK{z,z)

+f ^ ^ ) ,

(1.66)

i=l

where XQ = X + X and v is an Aut (D) invariant volume element (1.63)

ofD.

14

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

LEMMA 1.3 Suppose that the Bergman metric h of the bounded domain D is a complete metric. Then the next four conditions are equivalent to each other:

(i)

a holomorphic vector field X is belong in aut (D);

(ii)

a holomorphic vector field X is a Killing vector field;

(iii)

a holomorphic vector field X satisfies the condition Lxo{v) = 0,

(1.67)

where XQ = X + X and v is defined by (1.63);

(iv)

a holomorphic vector field

satisfies the condition Proof By a well-known result for a complete Riemannian manifold in the differential geometry, conditions (i) and (ii) are equivalent to each other. By Lemma 1.2, the coordinate representation of Lxo{v) = 0 can be written by (1.69), hence conditions (iii) and (iv) are equivalent to each other. It is enough to prove that condition (ii) is equivalent to condition (iv). Namely, we will prove that a holomorphic vector field (1.68) satisfies (1.69) if and only if X is a Killing vector field. Assume that (1.69) holds, the operator 7:—^zz acting on formula azidzj (1.69) yields d'^Re{XlogK{z,z)) ^^ dzi&Zj

where ^ki^) is a holomorphic function on D, fe = 1, • • •, n. By Lemma 1.1, X is a Killing vector field. Conversely, if X is a Killing vector field, X € aut (D) by the first statement. Hence exp (tX) e Aut (J9), Vf E R. Given the Taylor expansion of w = (exp {tX)){z) at i = 0, then w = z + t^{z) + o(t), |t| < e.

Siegel Domains

15

Hence the Taylor expansion of the Jacobian matrix of exp (tX) at t = 0 is

- = E + t-^+oit),

\t\<s.

Therefore (1.61) induces that

K{z,z) = |det (^E + t^

+ o(t)) ^K{z + t^ + o{t),z +1| + o{t))

= \l + tJ2^+o{t)\\K{z,z)

+ 2Re{tXK{z,z))

+ o{t))

1=1

= K{z,z)(l

d^i

+ 2tRe [xiogK{z,z)

+ J ] ^ ] + o{t)),

where |t| < s. Comparing the coefficients oft, conclusion (1.69) holds. I The above lemma also prove the conclusion as follows: Suppose that n

Q

JD is a bounded domain in C^. If X = Yl^ii^)'^~ ^ aut(D), then {^i{^)j • * • 5 Cn(^)) is a holomorphic solution of the hnear partial differential equation (1.69). Conversely, if D is a complete bounded domain in C^, then all holomorphic solutions ^{z) = {^i{z)j • • •, ^n{^)) of the linear partial differential equation (1.69) give aut (D). We will give two apphcations of Lemma 1.3. First, we introduce the Bergman mapping and find a complex chart on which we obtain an explicit expression of any element in the Lie algebra aut(Z?). Second, we find a 1-form ^ on axit R{D) satisfying the condition 2d# = O on auti^(D). We give the following symbols, which are used in this book most frequently. Let ei, • • •, en are 1 x n matrices, which are defined by ei = ( 0 , . . . , 0 , 1, 0 , . . . , 0 ) ,

i = l,2,...,n,

(1.70)

where the i^^ component is 1, and the other components are zero. { ei,---,en } is a basis of C^. The symbol e^ can be used in many cases. For instance, the element in the i*^ row and the j * ^ column of an n x m complex matrix A is denoted by eiAe'j, e i E R " , e^-G R ^ . (1.71) (1)

The Bergman mapping.

16

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Given a point p in a bounded domain JD in C^, denote by K{z]z), T{z] 'z) the Bergman kernel function and the Bergman metric matrix of £), respectively. The Bergman mapping is defined by .:

, = g,ad,(log|£|)|^^^^,

(1.72)

where A is an n x n non-singular complex matrix. The Jacobian of the Bergman mapping is T{z]p)A. Hence the Bergman mapping is a local holomorphic isomorphism acting from a neighborhood U oi p into C^, such that a{p) = 0. Therefore (U,a) is an admissible complex chart, where G{Z) = y^^ Z EU. Suppose that D contains the origin. Let p = 0, and let A = r ( 0 ; 0)~2. Then (1.72) gives a complex chart (C/, cr) as above. The Bergman kernel function is Ko{y;y) =

\detT{z;0)\-\detT{0',0))K{z;z)

by the new coordinates y = cr{z)^ and the Bergman metric matrix is ro(y;y)=T(0;0)iT(;^;0)-iT(^;z)r(0;^)-iT(0;0)^ by the new coordinates y = (T{Z). Hence ro(y;0)-ro(0;y)-£;,

^

a

Let X = Yl ^ji^)^—

Vy G a(C/).

a' — ^(^)'^

^ aut(D). The differential operator

d -^zz acting on (1.69) gives

^{z)Tiz,z)e,

+ ^ [^,{z) T—: V 2=1

- ^ ) - . "l^^J

+- ^ 9zj

dzldzj -^

— dzi

-^

^--.V.-.n.

where Cj € M", j = 1, • • •, n are defined by (1.70). Then the computation ofe(2)r(^,z)-e(0)r(0,2)gives i{z) = (/? - grad^log ( | | | | ) \_^^B - mM{z))T{z;

0)-\

(1.73)

Siegel Domains

17

where ^=0 ^

(1.74) ^ y

d'\og{K{z;z)/KiO;z)) r^—r.—

eefc. F=0 '--^ ^

dzjdzk J,k=l

-^

Hence we have 1.4 (Bergman) Let K{z\~z) he the Bergman kernel function of a hounded domain D in C^. Suppose that the Bergman metric is a complete metric. Then any X G aut {D) can he expressed hy

LEMMA

'dZj-'-'^'dz-'^

--V^^-^^^^y'

where a = e(0)T(0;0)2, L{y) = {ejkiy)) = r ( 0 ; 0 ) ^ M ( z ) T ( 0 ; 0 ) - ^ P=

(1.76)

-r(0;0)55r(0;0)"i

The Taylor expansion of all elements in the matrix L{y) at the origin does not contain the constant and linear terms. In particular, X = yP—, Proof

P + p' = 0,

\/XGiso{D).

(1.77)

Now,

Kiz;zy

y=^-'<'^^m^)U^'-^'^-' By (1.73) and (1.74), the first statement holds. Since X = ^{z)-^ G iso{D) impUes ^(0) == 0, the unit connected oz component of the compact Lie group Iso {D) = Isoo(^) is contained in the unitary transformation group U (n) when we use the new coordinates in the chart (C/, a) as above. The last statement also holds. I (2) The 1-form * . Suppose that D is a complete bounded domain with respect to the n

Q

Bergman metric. Let X = ^ ^ii^)'^" i=l

OZi

_

^ aut (D) and XQ = X + X.

18

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Then

1=1

2=1

is a real linear function on the real Lie algebra a,ui R{D) by Lemma L3, where sMtniD) = { Xo = X + J( \\/ X e aut{D) }. We have (see the Note in §2.1 for below Theorem): T H E O R E M 1.5 Suppose that the Bergman metric of a bounded domain D inC^ is a complete metric. Then the linear function ^ is a real value function J such that "^ X,Y e aut (D),

2{d^){X + X,Y + Y) = ^{[X + X,Y + Y]) = Q{X + X,Y + Y), (1.79) where fi is the Kdhler form with respect to the Bergman metric of D. Proof

By the direct computation, we have

•'

1,3=1

= -v^(E/^..^+ E ^ ^ - ^ +E^^), z,j=l

-^

ij=l

-^

ij=l

-^

dzidzj

lj =l

thus

-^

lj =l

-^

2J = 1

_ _ y * ( x + X) + x * ( y + Y) = o, Y^{X

+ X) + X^{Y

+ Y) = 0,

Observing that 2<S(IX,Y] + (X,Y]) . ^

\dzi

ozj

ij=i

. ^ = 2X^{Y

+ Y)-

dziOZjJ

-^

\a2:i 2Y^{X

azo + X),

-^

dziOZjJ

-^

Siegel Domains we have

19

_ _ {X + X ) * ( y + Y)-{Y + F ) * ( X + X) = 2X^{Y + F) - 2Y9{X + X) = 2^{[X,Y] + [X,Y]) = 29{[X + X,Y + 7]).

Hence 2{d^){X +JC,Y + Y) ={X + X ) * ( y + F) - (F + F ) * ( X + X)=^{[X + X,Y + Y]), VX,yGaut(L>).

^{[X + X,Y + F])

By (1.58), niX^X,Y^Y)=-^±im-T^r,.f^ = X^{Y + F) - Y^{X + X) = 9{[X + X,Y + F]), VX,y G aut (D), Therefore (1.79) holds.

I

1.6 Suppose that the Bergman metric of a bounded domain D in C^ is a complete metric. Then (7*(*) = *, a G Aut (D).

LEMMA

Proof Given a holomorphic automorhpism a \ w — f{z) and an element X in the Lie algebra aut {D) of the Lie group Aut (D), let Y = (j*(X). Then

1=1

i,:?=l

•'

j=l

•'

Thus Viiw) = 2^^ji^)-Q^

= 2^Cj(^)^'

z = 1, • • • ,n.

Hence

E

1=1

dr]i{w) ^ Y^ d^j{z)dzkdwi i,],k=l

•'

y^ i,J,fc=l

= Elf-E<.w ^^j(^) . v ^ p

_^Wi_dzk^ :j2

(z\^-^J}!l-

20

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Now, K{w,w) = K{z^z)\det-;r—\ induces I aw I dlog K{w,w) dwi dlogK{w,w) ^ViH dwi .. , dzj dwi i=l

= E ^^•(^) 1,3=1

-^

^^^'^^U

j=i

dz.

^

dz.

y

Hence ^ViH 2=1

d\ogK{w,w) dwi

^dr]i{w) 2=1

9tf;i

n

+ s,

SZo

7= 1

-^

where dz

^=E 3=

Therefore dz 92;7

\dz J

dzj \dz J

^

dwi dzj \ dzk /

impUes that 5 = 0. It prove that cr*(*) = * .

2.

Siegel Domains

1.3 A subset V of the real Euclidian space R^ is called a cone with the vertex origin, if

DEFINITION

XxeV,

Vx G F,

A > 0.

The cone V is called a convex cone with the vertex origin, if \x + {l-\)yeV,

\fx,yeV,

0 < A < 1.

1.4 Let V be an open convex cone with the vertex origin in R'^ not containing any straight line. The point set

DEFINITION

D{V) = { z e C" \Im{z) e V }

(1.80)

Siegel Domains

21

in C^ is called a Siegel domain of the first kind over V or a tube domain over V in C^. 1.7 Suppose that V is an open convex cone with the vertex origin in W^ not containing any straight line. Let D{V) be a Siegel domain of the first kind over V in C^. Then D{V) is holomorphically isomorphic to a bounded domain in C^.

LEMMA

Proof It is obvious that V is linearly isomorphic to an open convex cone V, which is contained in Vo = {x = ( X i , '",Xn)eR''\xi>Or",Xn>0}-

(1.81)

Clearly, the Siegel domain D{V) is linearly isomorphic to the Siegel domain D{V). Since D{V) C D{Vo) and D{Vo) is the topological product of n upper half planes, V is holomorphically isomorphic to a bounded domain in C^. I DEFINITION 1.5 Let V be an open convex cone with the vertex origin in R^ not containing any straight line. Given m x m Hermitian matrices ifi, • • • ,Hn, such that

F{u, u) = {uHiu\ • •., uHnu'),

ueC^

(1.82)

satisfies two conditions (1) F{u, u) = 0 if and only if u = 0; (2)_F{u,u)eV, VUGC^, where V is the closure of V. The point set D{V,F) = { {z,u) e C^ X C ^ I Im (z) - F{u,u) e V }

(1.83)

in C^"^^ is called a Siegel domain of the second kind over V. Siegel domains of the first kind and the second kind are usully just called Siegel domains. 1.8 Let DiV^F) be a Siegel domain of the second kind over V in C^ X C"^. Then D{V^F) is holomorphically isomorphic to a bounded domain.

LEMMA

Proof

Given a linear automorphism x -^ xA on R'^, then Vi = VA = {xA

IM

xeV}

is also an open convex cone with the vertex origin in R^ not containing any straight line. Let

Fi{u,u) = F{u,u)A,

V^GC^.

22

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Then D{Vi,Fi) is also a Siegel domain. Moreover D{V,F) is linearly isomorphic to D{Vi,Fi) by the affine isomorphism z -^ zA, u -^ u. Hence, without loss of generality, we can assert that V C Vio, where VQ is defined by (1.81). By the definition of Siegel domain, lm{zi)-uHiv!

>Q,

i = l , 2 , . . . , n , \l{z,u)

Obviously, there are linear functions fl such that

eD{V,F),

{u), fl \u), • • •, f^

uHiu' = f2\fi'\^)\^

{u) on C"^,

i-l,2,.-.,n.

Denote by { f\'^\u)^ ^ < j < U, 1 < i < n} the maximal linear independent vector set of { fl'^\u), I < j < Si,l
because F{u^ u) = 0 imphes u = 0. Thus YlU = m. Hence t-

s-

j=i

j=i

i = 1, • • •, n, and the holomorphic isomorphism a: (Zi = {zi-V^){zi + ^^)-\ l
+ V^)-\

l<j
maps the Siegel domain ti

[{z,u) € C " X C"» I lm{zi) - Y, \fi'\u)?

> 0, 1 ^ • • • y. Dn of n unit spheres

i = 1,2, • • •, n. Hence cr(D(V, F)) C Di x i?2 x • * * x ^ n is a bounded domain. I 1.9 Suppose that V is an open cone with the vertex origin in R'^j and F{u^u) G R'^, VIA G C"^ is given by Definition 1.5. A point set LEMMA

D{V,F) = { {z,u) G C^ X C ^ I Im(z) - F{u,u) G V }

Siegel Domains

23

is an open convex set in R2('^+"^) if and only if V is an open convex cone with the vertex origin m R^. D{V,F) is a simply connected domain in C^ X C^ if and only if V is a simply connected cone. Proof If D{V, F) is a convex set, then { z G C^ | (^, 0) G D{V, F) } is also a convex set. Thus D{V) is a convex set. Hence V is also a convex set. Conversely, if V is an open convex cone, given two points (zi^Ui) G £71 s^ £m^ i =: 1,2 and a real parameter t G [0,1], then Vi = Im (zi) - F{ui, Ui) eV, i = 1,2, Im (tzi + (1 - t)z2) - F{tui + (1 - t)u2, tui + (1 - t)u2) = v + vo, where V = tVi + {l-

t)v2

= t{Im{zi) - F{uuui)) Vo = t ( l - t)F{ui

+ (1 - t){Imz2 - ^(^2,^2)),

- U2, Ui - U2).

By (2) in Definition 1.5, VQ = t{l — t)F{u\ — U2^ui — U2) G V, where V is the closure of V. Then V is also a convex set, hence v + V() eV. It follows that Vi = Im (zi) — F{ui^ Ui) eV, 0
{z, u) G D{V, F)

is a homeomorphism acting from D{V,F) onto R'^ xV x C^, where i; = Im {z) — F{u^ u) G V. Then the inverse mapping a~^ is {x,v,u) -^{x + y/^{v

+ F{u,u)),u),

Vx G R'',

veV,

uE C ^ .

Hence D{V^ F) is a simply connected domain if and only if F is a simply connected cone. I It is well known that any open convex cone is a simply connected cone in E^, therefore we obtain the following corollary. 1.10 Any Siegel domain D{V,F) in C nected and simply connected open convex domain.

COROLLARY

x C^ is a con-

24

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

The next symbols are used in this book most frequently. Denote by GL (n, B) the real general matrix group; GL (n, C) the complex general matrix group; U (n) the unitary group; O (n) the real orthogonal group; Sp (n) the real symplectic group, respectively. The Lie algebras of these groups are denoted by gl (n, R ) , gl (n, C ) , u (n), o (n), sp (n), respectively. The following H. Cartan Lemmas (refs. [14], [15], [16]) are well known. 1.11 (H. Cartan) Let D be a bounded domain inC^. by Aut (D) the holomorphic automorphism group on D and

LEMMA

Denote

Iso (D) = IsOp(D) = { a € Aut {D) I a{p) = p }

(1.84)

the isotropy subgroup in Aut (D) at a fixed point p E D. Then Aut (D) is a real Lie transformation group acting on D and IsOp(D) is a compact subgroup of Aut (D). 1.12 (H. Cartan) Given a point p in a bounded domain D in C^, the Taylor expansions at z = p of a^r £ Isop{D) are LEMMA

a : w = p+ {z — p)A + terms of {z — p) with higher degrees, T :w =p+ {z -p)B + terms of {z -p) with higher degrees, respectively. Then a = r if and only if A = B, where A, B G GL (n, C). 1.13 (H. Cartan) Let D be a bounded domain inC^, Denote by aut(D)^ isOp(D) the Lie algebras of Lie groups Aut(D); IsOp(D);

LEMMA

respectively. Suppose that the Taylor expansions atz=pofX Y = vi^)-^

= ^{z)-r-j oz

^ isop(-D) are

X = {z — p)A-^ + terms of {z — p) with higher degrees, oz & Y = {z — p)B-^ + terms of {z — p) with higher degrees, oz respectively. Then X = Y if and only if A = B, where A, B G gl (n, C). In particular^ X = 0 if and only if A = 0. Given a vector field X = C ( ^ ) ^ ^ aut (D), oz then Xj \/^X G aut (D) if and only if X = 0; X £ isop{D) if and only if X E aut (D) and ^{p) = 0, where p is a fixed point in D.

LEMMA

1.14

(H. Cartan)

Siegel Domains

25

Now, we consider a Siegel domain D{V^ F). Let Aut {D{V^ F)) be the holomorphic automorphism group on D{V,F). Let Aff {D{V, F)) = {ae

Aut (D( V, F)) \ a : {z, u) -^ {z, u)A + (a, b) } (1.85) be the afiine automorphism group of D{V^ F ) , where A e GL {n + m^ C), (a, 6) € C^ X C ^ . Clearly, AS{D{V,F)) is a closed subgroup of the Lie group Aut (D(V",F)), hence Afl{D{V^F)) is also a Lie subgroup. The Lie algebra aff (D(F,F)) of AS{D{V,F)) is a Lie subalgebra of a u t ( D ( y , F ) ) . We have 1.15 Suppose that D{V,F) is a Siegel domain in C^ x C^, Give two points a G R^^ b € C ^ and two real parameters t,6 eR^ Then AS {D{Vj F)) contains the mappings LEMMA

Pia,b) • ^ '

(y = z-2V^F{u,b) \ . [v = u-b

+

V^F{b,b)-a, (1-86)

and y = e2*2,

V = e*+^^%.

(1.87)

By Lemma 1.15, given a point {ZQ, UQ) G D{V, F), then VQ = Im (ZQ) — F{uo,uo) € V, and PiRe(zo),uo){^o,uo) = (V=lt;o,0) € D{V, F).

(1.88)

Since the mappings z ^^ z^ u ^' ue^^^^, V^ G M, are contained in Aff (Z?(F, F)), D{V^F) is an unbounded semi-circular domain. The function space H{D) consists of all holomorphic functions on D, where D is the closure of D. The function space B{D) consists of all bounded functions in H{D). We have 1.6 A subset S{D) in the boundary dD of a domain D in C^ is called the characteristic boundary of D, if DEFINITION

(1) given a function f G B{D), there is a point p in S{D), such that \f{z)\<\f{p)\,yzeD; (2) given a point ZQ G S{D), there is a function f G B{D), such that \f{z)\ achieves the maximal modulus at the unique point ZQ, that is, \f{z)\<\fizo)\,^zeD. The characteristic boundary is called Silov boundary in some papers, but we remark that the characteristic boundary was first considered by

26

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Hua, who defined the characteristic boundary for the four classes classical domains. We immediately have the following result for the characteristic boundary. LEMMA 1.16 Let Si be the characteristic boundary of a domain Di in C^j i = 1,2. If a is a holomorphic isomorphism acting from Di onto D2, and a is also a homeomorphism from the closed domain Di onto the closed domain D2, then

a{S{D,)) = S{D2).

(1.89)

1.17 Denote by S{D{V,F)) the characteristic boundary of a Siegel domain D{V,F) in C x C ^ . Then

THEOREM

S{D{V,F))

= { {z,u) G C^ X C ^ I Im(^) - F{u,u) }.

(1.90)

Proof Using the proof of Lemma 1.8, there is an n x n non-singular matrix A, such that Vi = VAcVo

= {x eR'^ \x = (a;i,---,Xn), Xi > 0, 1 < i < n },

(see the formula (1.81)). Since VA = {x = vA\yv e V}, VA is also an open convex cone not containing any straight line. Using (2) of Definition 1.5, F{u,u) eV.'^ue C"". Hence Fi{u,u) = F{u,u)A e VA = (VA) = F i , and Fi{u, u) = Oii and only iiu = 0. Therefore D{Vi,Fi) = { {z,u) G C^ X C ^ I Im{z) - Fi{u,u) G Vi } is also a Siegel domain, and the linear automorphism a: z —^ zA^ u ^^ u maps D(y,F) onto D(Vi,Fi). It is enough to prove that the characteristic boundary S'(D(Vi, Fi)) of S{D{Vi,Fi))

= { {z,u) € C " X C"^ I lm{z) = Fi{u,u) }.

In fact, if S{D{yi,F\)) is the characteristic boundary of D{y\,F\), then
Siegel Domains

27

Now, we find the characteristic boundary S{D{Vi,Fi)) of D(Vi,Fi) as follows. Given a point (^,0) G D{Vi,Fi), then Im(2:) G Vi C Fo induces that there is a holomorphic function n

ip{z, u) = Ylizi + v ^ ) " \

Im (^^i) > 0,

1 1 > 0 , l {z,u) is a holomorphic function on the closure of D(Vi,Fi). Clearly, \m{zi) > 0 implies \zi + \ / ^ | > 1, thus \il;{z^u)\ < 1, hence ^IJ{Z,U) e B{D{Vi,Fi)). Now \^p{z,u)\ = 1 if and only if Im(z^) > 0, \zi + ^ / ^ | = 1, 1 < i < n, hence z = 0. Now, (0, u) G D{Vi, Fi) implies tx = 0, therefore IIJ{Z^U) achieves the maximal modulus 1 at a unique point (z,u) = 0e dD{Vi,Fi) of D{Vi,Fi). Given a point (2:0,1x0) G C^ x C"^, where Im(2;o) = Fi{uo,uo), let /(z,'a) =t/;(P(Re(^o),wo)(^''^))- By (1.86), then f{z,u)

= ij){z - 2\/^Fi{u,uo)

+ V^Fi(?/o,^o) - Re(2^0),'?x - uo),

hence f{z^u) G 5(£)(Vi,Fi)) and f{zo,uo) = '0(0,0). Therefore given a point (^0,^0) e S'(Z)(Vi,Fi)), there exists / G 5(D(Vi,Fi)) such that / achieves the maximal modulus at a unique point (2:0,^0)On the other hand, ^g G B(D(Vi,Fi)), if g achieves the maximal modulus at {zi^ui) G 9(D(Vi,Fi)), then 1^(2:1,1x1)1 > \g{z,u)\,'i{z,u) G D{Vi,Fi). Let /i(^,?x) = g(P(_Re(^i),-ni)(2:,ix)) = g{z + 2y/^Fi{u,

m) + V^Fi{ui,ui)

+ Re {zi),u + ui).

Then h{z,u) G B{D{Vi,Fi)). Denote by d{Vi) the boundary of Vi. Let vi = Im (2:1) — Fi{ui,ui) G d{Vi). Then ( V ^ ^ i , 0 ) G 9(i)(Vi,Fi)), and fe(V^^i,0) = 5('2^i,^i) implies that \h{z,u)\ < | / i ( x / ^ ^ i , 0 ) | , V(^,ix) G 2:>(Vi,Fi). Given a real parameter t G C, where Im(t) > 0, then (t^i,0) G 5£>(Fi, Fi). Hence h{tvi^ 0) is a holomorphic function of t on the closed upper half-plane in C. Using the maximum modulus principle, there is a real number to such that |/i(tot;i,0)| > |/i(ti;i,0)|, Vt G C, lm(t) > 0. So IM*o^i,0)| > \h{^^viM

> \Kz,u)l

\f{z,u)eD{VuFi),

28

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Thus h{z,u) achieves the mammal modulus at (to'^i^O) G Hence g{z^u) achieves the maximal modulus at

{tovi + V^Fi{ui,ui)

+Re{zi),ui)

Therefore we prove that S{D{Vi,Fi)) D{VuFi).

S{D{Vi,Fi)).

e S{D{Vi,Fi)).

is the characteristic boundary of •

By Lemmas 1.7 and 1.8, a Siegel domain is holomorphically equivalent to a bounded domain. But a lot of bounded domains are not holomorphically isomorphic to a Siegel domain. By Theorem 1.17, we have dim{S{D{V,F))) < dim{dD{V,F)) for any Siegel domain D{V,F). We can construct some examples of bounded domaims, such that the characteristic boundary of any one of these domains is whole boundary. Hence no such a domain is holomorphically equivalent to a Siegel domain. (1) Obviously, the domain Dp = { izi,Z2) € C I |zi|2 + |^2p/^ < 1 }

(1.91)

is a bounded Reinhardt domain in C^, where p G (0,1). The characteristic boundary of Dp is S{Dp) = dDp = {{zi,Z2) € C2 I |^i|2 + \z2f/P = 1}.

(1.92)

We give a simple proof as follows: By the Lie group technique, Thullen (ref. [192]) and H. Cartan (ref. [15]) give the holomorphic automorphism group Aut (Dp) on Dp, which is

^^^^V-ie^i:Z±^ 1 — zia

^^^^yzT^^JxAES!) , \

(1.93)

^1^ /

where 0 < 9,ip < 27r, a G C, \a\ < 1. Obviously, any holomorphic automorphism of Dp is also a holomorphic automorphism of the closed domain Dp. Given a point (a, 6) G dDp, then |ap + |6p/^ = 1. If \a\ < 1, then (a, 6) is mapped to the point (0,6/|6|) by the holomorphic automorphism (1.93). The holomorphic automorphism wi = zi, W2 = -{\b\/b)z2 sends (0,6/|6|) to ( 0 , - 1 ) . If \a\ = 1, then 6 = 0 and the holomorphic automorphism wi = — (l/a)zi, W2 = z^ sends (a, 0) to (—1,0). Hence we have only two Aut (Dp) orbits in dDp. One orbit is a unit circle containing (—1,0), the bounded holomorphic function f{^i^Z2) = {^1 + 2)"-^ on Dp achieves the maximal modulus 1 at (—1,0); The other orbit is dDp-{

(e^^^O) I V ^ G R }

Siegel Domains

29

containing (0, —1). The bounded holomorphic function g{zi^ Z2) = (^2 + 2)~^ on Dp achieves the maximal modulus 1 at (0, - 1 ) . Therefore dDp is the characteristic boundary of Dp as that of the proof of Theorem 1.17. (2) By the same argument as above, given r G {1, • • •, n — 1} and a positive number p e (0,1), then the Reinhardt domain D{r^p): N | ' + --- + k r P + (|2.+l|2 + .-- + |z„|2)2/P
(1.94)

does not holomorphically equivalent to a Siegel domain, and the holomorphic automorphism group on D{r^p) can be obtained by ThuUen and H. Cartan's method easily. By Theorem 1.17, we have 1.18 Suppose that a is an affine isomorphism from a Siegel domain D(y^F) onto a Siegel domain D{Vi^Fi). Then

LEMMA

d i m F = dimVi = n,

dimD{V,F)

= dimD{Vi,Fi)

= n + m, (1.95)

and there exist two matrices A G GL(n,M) and B G GL(m, C), such that (1)

X ^ xA is a linear isomorphism from V onto Vi;

(2) z^zA,

u-^uB

is a linear isomorphism from D(V,F)

onto D{Vi,Fi),

F{u,u)A = Fi{uB,uB), Proof

(1.96) where

V« € C " .

(1.97)

Since cr is an affine isomorphism, so n + m = dim {D{V, F)) = dim {D{Vi,Fi)) = no + mo,

where dim (V) = n, dim (Vi) — UQ. NOW, cr can be represented by

{ wherea€C"o,

w = zA + uB + a, v = zC + uD + b,

beC""^,

(

_^(n,no)

(^(n,mo)

\

I € G L ( n + m,C).

30

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS By Lemma 1.16 and Theorem 1.17, Im {zA + uB + a) = Fi{zC + uD + b,zC + uD + b),

where {z,u) = (Re(z) + ^^F{%u),u) Im {xA + V^Fiu,

€ S{D{V,F)).

Thus

u)A + uB + a)

=Fi{xC + \^F{u,

u)C + uD + b,xC + \/^F{u,

u)C + uD + b),

\lx = Re{z) € « " , « € € " * . Comparing the terms of x of degree 2, we have Fi(a;C,xC) = 0. By (2) in Definition 1.5, xC = 0, Vx € R", hence C = 0. Comparing the coefficients of the terms of x of degree 1, we have Im(ic^) = 0, Vx € R". Thus A is a real matrix, and F(«, u)A + -^={uB 2\/—1

- IIB) + Im (a) = Fi{uD + b,uD + b).

Hence Fi{b,b) = Im(a),

2yf^Fi{uD,b)

= uB,

Fi{uD,uD)

=

F{u,u)A.

Now, C = 0 and

{B

^ ) e G L ( n + m,C)

implies n < no- For the same reason, the expression of the inverse a~^ of a gives no < n. Hence n — no, m = mo, A G GL (n,R), D G GL (m, C). Therefore the mapping a can be expressed by (w = zA + 2y/^Fi{uD, \v = uD + b,

b) + V^Fi{b,

b) + Re (a),

where F(uD,uD)

=

F{u,u)A.

Given the affine automorphism ^0

{

w = w - 2 V ^ F i ( ^ , b) + yf^Fiib,

b) - Re (a),

V = V—

on i)(Vi, Fi), then the affine isomorphism aoo a: w = zA^ v = uD acts also from D{V,F) onto Z?(Vi,Fi), where A and i ) are two non-singular matrices, and Fi{uD^uD) = F{u,u)A.

Siegel Domains

31

Now, we prove that the mapping x -^ xA is a Unear isomorphism from V onto Vi. In fact, (/=Tx,0) G D{V,F), Mx e V. Thus ao((T(V-lx,0)) = ( V ^ x A O ) e i?(Vi,Fi), where A E GL(n,R). Hence xA eVi. It proves that VA C Vi. Using the inverse mapping of CTQ O a, we have V\A~^ C V for the same reason. Therefore VA = Vi, and a; -^ arA is a hnear isomorphism from V onto ^1. • By Lemma 1.18, we have THEOREM

1.19

Let D{V,F) he a Siegel domain in C^ x C ^ . Then

A = 2z^ + ^ | - e aff {D(y, F)), oz ou The eigenvector subspace decomposition of aS{D{V^F)) transformation ad (A) is

(1.98) by the linear

aff {D{V, F)) = L_2 + L-i + LQ,

(1.99)

L-2 = | a — | V a € K " | ,

(1.100)

where

a'

„ r-^^, , , a '

L_i = { b ^ + 2 V ^ F ( u , b ) — I Vb € C " } , .& ^& & U = {zP-+uQ-\.P-e^iyi dx F{uQ,u) + Fiu,uQ)

= F{u,u)P,

(1.101)

(1.102) V« € C " ' | ,

and P is annxn real matrix, Q is anmxm complex matrix, and aff (V) «s ^/le Lie algebra of the Lie group Aff (V), which consists of all linear automorphisms of V. By Theorem 1.19, we have 1.20 Let D{V, F) be a Siegel domain in C " x C " . Then the affine automorphism group Aff (£>(V, F)) on D(V, F) is a semi-direct product of the following closed subgroups: THEOREM

G-2 = {w = z-a, G_i = {w = z - 2y/^F{u,

v = u,

VaGR"},

(1.103)

13) + ^ / ^ F ( / ? , /3), v = u-P, y 0 e C"^}, (1.104)

32

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS Go = {w = zA,

u = uB},

(1.105)

where x —> xA is an affine automorphism onV, B E GL (m, C) satisfies F{uB,uB) = F{u,u)A. Therefore Aff (i)(F,F)) consists of (w = zA- 2V^F{uB, {v = uB-/3,

p) + x/=lF(/3, /?) - a, y '

)

where a e R^, /3 G C"^ and B G GL(m,C). The affine automorphism X -^ xA of V satisfies F{uB,uB)

= F{u,u)A,

V^ G C ^ .

(1.107)

N o t e Many results in this section are given in Piatetski-Shapiro's book (ref. [158]), in which he first introduced the notion of Siegel domains in 1961, and he gave some properties of Siegel domains. He also gave some examples of non-symmetric homogeneous bounded domains in the form of homogeneous Siegel domains.

3.

Holomorphic Automorphism Group of Siegel Domains Let { a i , • • • ,an } be a basis of a vector space V. The vector a =

n

^ Xiai has the coordinates x = (a;i, • • •, Xn)- The matrix representation 1=1

A = {aij) of a linear transformation A is defined by n

Ajaj)

= /^ajjOtj,

i =

l,"',n.

Let y = (yi, • • • ? Vn) be the coordinates of A{a). We have y = xA. By ref. [101], we have LEMMA 1.21 (Kobayashi) Suppose that D{V, F) is a Siegel domain in C^ X C ^ . Then D{Vj F) is a complete Riemannian manifold with respect to the Bergman metric.

By Lemmas 1.21 and 1.3, the Lie algebra aMt{D{V,F)) of the holomorphic automorphism group Aut {D{V^ F)) consists of all Killing vector fields on a Siegel domain D{V, F). 1.22 Let K{z^u]'z^u) be the Bergman kernel function of a Siegel domain D{V,F) in C^ x C"". Then

LEMMA

K{z,u;z,u)

= Kv{lm{z)

- F{u,u)),

V(^,u) G D{V,F),

(1.108)

Siegel Domains

33

where KY{X) is a positive value analytic function on the cone V. Proof

Given a point (ZQ? '^O) ^ i^(^? F), then the affine automorphism

{

w = z - 2\f^F{u,

u^) + y/^F{u^,

uo) - Re (2:0),

V = U — UQ

on D(V, F) sends {ZQ, UQ) to ( V ^ X Q , 0), where XQ = Im {zo)-F{uo, UQ) e V. By the property of the Bergman kernel function, we have K{zo,uo] zo, Uo) = K{V^xo,

0; - V - l x o , 0),

where K{\/^xo^O; —\/^XQ^O) — Ky{xo) is a positive value analytic function of x^ on V, and x^ = Im {ZQ) — F{UQ^ UQ) G F . I Now, we consider the Lie algebra aut {D{V^ F)) as follows: 1.23 (Kaup, Matsushima, Ochiai) Let D{V, F) be a Siegel domain inC^ xC^. Then aut {D{V^ F)) has the direct sum decomposition of eigenvector subspaces:

LEMMA

aut {D{V, F)) = L_2 + L_i +L0 + L1 + L2

(L109)

by the linear transformation ad(M^); where pj

pj

W = 2z— + « | - € aut {Diy, F)). oz ou The subspace Lj is defined by Lj = {x€ aut {D{V, F)) \ ad {W)x = jx },

(1.110)

\fj = 0, ± 1 , ±2, • • •, (1.111)

andLj = 0,jf / 0 , ± l , ± 2 , [Li,Lj]cLi+j, ^i,j€Z. The subalgebraaS{D{V,F))

is

aff(D(y,F)) = L_2 + L_i + Lo, where L-2.,L-\^LQ

(1.112)

are defined in Theorem 1.19.

The subspace L\ is contained in the set

(1-113)

34

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where A , ^ i , • • •,A^ arenxm complex matrices^ C'l,• • •,Cm are mxm complex symmetric matrices^ z — (^i, • • •, z^) G C^; u — (?xi, • • •,Urr^ G

The subspace L2 is contained in the set

{J2{zizB,)f^+J2i^,uD,)^}, i=l

(1.115)

2=1

where S i , • • •, 5^^ are n x n real matrices satisfying the conditions: CiBj = CjBi^

and Z)i, • • •, Dn are mxm

1 ^ i, j ^ ^5

(1.116)

complex matrices.

Proof Step 1 We prove that all vectors in aut {DiV^ F)) are polynomial vector fields. It is obvious that n + 1 vector fields — dzi'

— ' dzn'

W-2z— dz

u— du

are linearly independent in aut(jD(V,F)) by Theorem 1.19. These vectors linearly generate a solvable Lie subalgebra 6 of dimension n + 1, such that [ 6 , 6 ] is a commutative Lie subalgebra with the basis

(A...

A\

la^l'

'dZn)

By the Lie theorem in the Lie group theory, there exists a basis of the complexification aut (Z)(F, F ) ) ^ , such that the matrix representation of ad & consists of upper triangular matrices. Hence the matrix representation of ad [ 6 , 6 ] also consists of upper triangular matrices with all / d Y diagonal elements equal to zero. Therefore ( a d ^ — j = 0, 1 < z < n, where s = dim (aut {D{V, F))). Given ^ = E ^ i ( ^ ' ^ ) a ^ + E ^ f c ( ^ ' ^ ) a ^ ^ aut (D(y,F)),

then (ad — j X = 0, 1 < z < n. Hence all holomorphic functions dzi) ^ ai(z,i6), • • •, aniz-^u), bi{z^u), • • •, bm{z^u) are polynomials of z.

Siegel Domains

35

Now, y/^u—

G aut ( J D ( V , F ) ) . Using the homogeneous polynomial (JU

expansions of a{z^ u) and b{z, u) of u: oo

oo

where a(^)(z,t^) and h^^\z,u) are homogeneous polynomial vector functions of u with degree p. aut (2)(V, F)) has a vector y = ( a d v ^ t t — ) x + (ad V ^ « | - )

X

p=0

Obviously, the value of Y at any point ( \ / ^ x o , 0) is equal to zero, where XQ € V. Thus r € isO(^/rTa;o,o)P(^'^))- % Lemma 1.13, F = 0. Hence a{z, u) and 6(2;, u) are polynomial vector functions of «, such that the degrees are not exceed 1 for a{z,u) and 2 for b{z,u). Therefore X = {a{z) + uA{z))^^

+ {h{z) + t*B(^))|^ + E ( « ^ ^ ( ^ V ) ^ ' j=l

-^

where a(2:) is a 1 x n matrix function, A{z) is an m x n matrix function, h{z) is a 1 X m matrix function, B{z) is an m x m matrix function, and Cj{z) is an m X m symmetric matrix function, such that the elements of a{z), b{z), A{z), B{z), Cj{z) are all polynomials of z. Step 2 We prove that ad (W) determines the following direct sum decomposition of eigenvector subspaces: aut {D{V, F)) - L_2 + L_i + Lo + Li + L2 + • • •,

(1.117)

where ad {W){x) = kx, and [Lj^Lk] C Lj^k^j^k

MxeLk,

keZ

G Z. In particular,

aff {D{V, F)) = L_2 + L_i + LQ; L2k = {Y2k = « ^ ' ^ ' H ^ ) | ^ + ^ ^ ^ " H ^ ) ^ }, fe = - 1 , 0 , . •.;

(1-118) (1.119)

36

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

j=l

where fc = 0 , 1 , . - . , CJ"^^(^) = 0, d~^\z) upper indices represent the degree oi z. By step 1,

-^

(1.120) = 0, 5(-i)(z) = 0, and all

oo

X=

Y,yke^ut{D{V,F)), k=-2

By a direct computation, {a.d{W)rYk = kPYk,

p = 0,l,----

Thus CX)

(ad {W)rX

= Y,

k^Yk e aut {D{V, F)),

p = 0,1, • • •.

fc=-2 oo

Since X is a polynomial vector field, X =

"^ Yk is 3>finitesumk=-2

mation. By the Vandermonde matrix technique, Yk G aut(D(V,F)), k = - 2 , - 1 , •.., hence (1.117) holds. Finally, (1.118), (1.119) and (1.120) hold by a direct computation. Step 3 In the following, we will prove that the radical (i.e. the maximum solvable ideal) S of aut {D{y^ F)) is oo

s = {sn L_2) + {Sn L-i) + (5 n Lo) + 5 ] Lk.

(1.121)

fc=3 00

By the definition of radical, [W,S] C S. Thus S = Y^ (Lj n 5). It suffices to prove that Li fi 5 = L2 fl 5 = 0, L^ c 5, A: > 2. Clearly, the kernel of the Killing form B{x,y) — trada^ady is contained in S, where x,y e dMi{D{V,F)). Given Xk E Lk, k > 3, Pr, Qr ^ Lr, r > - 2 , then (adXfc)(adPr)Q5 C Lk-\-r-\-s, k + r > 1. Thus tradXfcadP^ = 0. Hence 5(Xfc,aut (D(y,F))) = 0. Therefore Xk e S and then L^ C 5 , fc > 3. Given Yp e Snip, p = 1,2, then '^

*=1

<-k/

^/

n

i=l

2=1

Siegel Domains

37

where Ai,B,Ci,Di are constant matrices. Given a point XQ = (xi,---,a;„) € V, then Zi =

d'

a'

^"a^'hai'^^]] at/

^

oz

1=1

7

1[

^ 2=1

^

Hence Y^ + Z^e 5niso (yi:ia;o,o)(-'^(^' ^))- Since ad S is a linear solvable Lie algebra acting on the linear space aut (i)(V,F)), the Lie Theorem implies that there is a basis of the complexification aut {D{V, F))^, such that all elements of ad S can be represented by upper triangular matrices. Thus a d l ^ , adZp, p = 1,2, are nilpotent upper triangular matrices. Hence ad (1^ + Zp), p = 1,2, are also nilpotent upper triangular matrices. But all linear transformations in ad(iso/^^/zi;a;Q o)(^(^'-^)) ^^^ semisimple, so ad(rp + Zp) = 0, p = 1,2. Thus Yp + Zp e C(aut (Z)(y,F))), p = 1,2. Hence

, i^ + 2p OZj

Yp + Zp = 0, p = 1,2, where

= 0, 1 < j < n. We conclude that

J

ZieSf] L-i, Z2 € 5 n i:_2, Therefore Yi=Y2 Step 4 Let

n € 5 n Li, F2 € 5 n L2.

= 0. Namely, 5 n Li = 5 n L2 = 0.

We prove that Lfc = 0, fc > 3.

Zi^i2'-it = ^^^—^d^

azi^ dzi^ Given Yk e Lk^ k>3, then Zi^i2••4,.^ {Y2k) E S H L2 = 0,

ad-^—,

l
iH
azi^ Zi,i,...i,_^ {Y2k-l) E S il Li = 0.

Thus Ifc = 0, fc > 3. Hence Lk = 0,k>3.

I

By step 3 and step 4 in the above proof, we have 1.24 (Kaup, Matsushima, Ochiai) Let D{V, F) be a Siegel domain inC^xC^. The direct sum decomposition of eigenvector subspaces of the Lie algebra aut {D{V^ F)) by ad (W) is

LEMMA

aut {D{V, F)) = L_2 + L_i + L0 + L1+ L2,

(1.122)

38

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and the radical o/aut (£)(y,F)) has the subspace decomposition S = {Sn L_2) + {Sn L_i) + (5 n Lo) C aff {D{V, F)),

(1.123)

where d i m ( L i ) + d i m ( 5 n L _ i ) = 2m,

dim(L2)+dim(5nL_2) = n. (1.124)

In particular^ i/aut (£)(V,F)) is a semisimple Lie algebra^ then dimLi = 2m, Proof

dimL2 = n.

(1.125)

It suffices to prove that dim (Li/S) = dim {L-i/S),

i = l,2.

2

Since L — aut ( D ( V , F ) ) / 5 =

X) {^i/^) is a semisimple Lie algebra, 2=-2

the kernel of the Killing form BL of L is non-degenerate. Given XQ G L^/S' and IQ ^ ^j/'S', i + j 7^ 0, then adXcadYb is a nilpotent linear transformation acting on L, the Killing form BL on L satisfies J5L(Xo,yo) = 0. Thus BiiLi/S^Lj/S) = 0, i + j j^ 0. Hence JBL is non-degenerate on Li/S x L-i/S, i = 0,1,2. Therefore dim {Li/S) = dim (L_i/5), 2 = 1,2. On the other hand, we have L-i/S ^ L.i/{S n L_i), L_2/S ^ L_2/(5 n L_2), Li/S = Li, 1/2/5 = L2, and dimL_2 = n, dimL_i = 2m by Theorem 1.19, hence (1.124) holds. When aut (D(V, F)) is a semisimple Lie algebra, we have 5 = 0. Hence (1.125) holds. I Theorem 1.19 and Theorem 1.20 give that exp (L_i + L_2) = { P(a,6) I a G E ^ 6 G C ^ },

(1.126)

where ^'^

(w = z + 2V^F{u, 1^ 1; = ix + 6.

b) + V^F{b,

b) + a,

Prom (1.113) and (1.114), Murakami (ref [133]) gave the next lemma. LEMMA

1.25

(Murakami)

Let D{V,F)

be a Siegel domain in C^ x

2

C^; and let aut (JC)(V,F)) = Yl ^i ^^ ^^^ direct sum decomposition of i=-2

eigenvector subspace of dMi{D{y,F))

by dA{W).

Then

Siegel Domains

39

(i)

y, = E ( ^ ^ " ' ) ^ + zA^ + E("C,«')3^ € L.

(1.128)

if and only if [Yi, L-i\ C I-i-i, i = 1,2, w/iere A, Ai, •• •, An are n xm complex matrices, Ci, • • -, Cm are m x m complex symmetric matrices,

(ii)

Y2 = Y1 ""'^^'^ + E ^ ' " " ^ ' ^ ^ ^2

(1.129)

i/ and only if [Y2, L-i] C L2-i, i = 1,2, and Im (tr (Dj)) = 0,

1 < j < n,

(1.130)

ly/iere Bi, • • •, 5^^ are n x n real matrices satisfying the conditions: CiBj = CjBi,

I

(1.131)

and Di, • • •, Dn are m x m complex matrices. Proof

It suffices to prove that 1^ G Li if and only if [Yi, L_i + L_2] C Li-i + Li_2,

i = 1,2

(1.132)

and ^y,+T^(^)L=V^a:o,.=o = 0.

^ = 1^2,

(1.133)

where XQ = Im{zQ) - F(^o,'^o), V(^o,^^o) ^ D{V,F). Clearly, (1.132) holds. Lemma 1.3 induces that Yi e Li if and only if LY.^YS'^) = 0. Thus (1.132) and (1.133) hold for Yi e L^ i = 1,2. Conversely, if (1.132) and (1.133) hold, we will prove that Ly.^Yi'^) ~ 0, i = 1,2. It is easy to show that (1.132) implies

Thus ^Yi-^Yi,x+x]('^) = 0Since all vectors of subspace L-1+L-2 Lx^xi'^) = 0. Thus

are Killing vector fields, we have

40

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Hence Ly.(_yr(t;) is invariant under the action of exp(L_i + L-2). In particular, I/y._,.jr(v) is invariant under the action of P'(Re(zo),uo)' where {zo,uo) G D{i^J). Therefore i^y,+y^(^)|(.,„)=(.o,no) = 0, z = 1,2. The statement holds. Now, we axe going to give an explicit expression for (1.133). According to Lemma 1.22, the Bergman kernel function of D{V^ F) is given by K{z,u;z,u)

= Kv{Im{z)

- F{u,u)),

^{z,u)

G D{V,F).

Replacing the left-hand side of (1.69) by Yi, we obtain h = Re ( n log i^y(Im {z) - F{u, u)) + ^

'

i=i

+E

d{zAe'j + uCju') duj

= ReiYilogKviIm{z) where |. =

dzi

- F(«,u)))|.,

Observe that

\^=,/^XQ^^=O.

dlogKv{lm{z) - F{u,u)) = 0. ou du Thus / i = 0. Hence Yi is a KiUing vector field. Therefore Yi is a Killing vector field if and only if \Y\,L-i + L_2] C LQ + -^-iReplacing the left-hand side of (1.69) by Y2, we obtain Yi

=

V^XQA—

/2 = Re {Y2 logi^v(Im {z) - F{u, u)) "

dzjzBje'j "^a^i«A4\| dzj ^2^1^ duj ) \ .

'^ ^

= Re {Y2 log Kv{lm. {z) - F{u, u)) n i=l

m

n

3=1 2=1

Observe that I2

and 9 ,

,, /

1

& —

= -Y,^iXoBi Z=y/

—\Xo,U=Q

i=l

ai^'°«^''(^<^-^'-^<»'»))l. =

^

IKvixo)

dKvix) dxk

x=xo

Siegel Domains

41

Hence

/3=ite (- tx.x„B.) ( - ^ = 1 — - ^ y n

m

+ Re (2V^^xoBje[

where I — - — - j

n

+ V^J^^J^^Xie^Ae^),

is the transpose of the matrix —

. Using the

condition Bi = Bi^ we have n

f2 =

-Y,XitT{Im{Di)). 2=1

By (1.69), I2 is a KilUng vector field if and only if [I25 L-i] C I/2-i? ^ = 1,2 and /2 = 0. Hence Y2 G aut {D{V, F)) if and only if [Y2, L-i] = Y2-U z = 1,2, and tr (Im ( A ) ) = 0, 1 < i < n. I Now, we are going to determine all Killing vector fields on a Siegel domain i ) ( F , F ) . THEOREM

1.26

Let D{V,F) he a Siegel domain in C

x C ^ . Let

aut (D(y, F)) = L_2 + i - i + Lo + Li + L2

(1.134)

6e the direct sum decomposition of eigenvector subspaces o/aut {D{V^ F)) by a ( F , F ) ) , (1.135) uz on where aff (i)(V^, F)) is determined in Theorem 1.19; and the subspaces Li and L2 are determined as follows: (1)

Yi G Li if and only if pj

pit

"^^

pk

Fi = 2.f^F{u,zA)^ 9^; + zA^du + 5;^(«C,«')^-, -f-f 5«i

[

(1-136)

(jf & 1

yi = 2F(«, z ^ ) - + ^/^zA—^ + v ^ E ( « ^ i " ' ) a ^ '

(1-137)

42

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

A is annxm complex constant matrix^ Ci, • • •, Cm are mxm complex symmetric constant matrices, such that pj

pj

{F{zA,a) + F(a,zA))|^ - F{u,a)A^

Tn

rx

- V^Y.^CiCjv!)~

6 Lo,

j=i

•^

(1.138) Va € C"*. Condition (1.138) is equivalent to the combination o/(1.139) and (1.140) helow: {F{xA,a) + F{a,xA))^

€ aff {V),

(1.139)

ox F{F{u, u)A, a) + F(tt, F{u, u)A) + F{F{u, a) A, u) + F{u, F(u, a) A) m

= v ^ X ] (^((«<
(1.140) (2) y2 6 L2 z/ and on/y if Y2 = J2z,zB,f^

+ J2zi^D,^,

(1.141)

where 5 i , • • •,Sy^ are n x n real constant matrices, Di, • • •,-Dn ^^^ mxm complex constant matrices, such that CiBj = CjBi,

l
tr (Im (Dj)) = 0,

(1.142)

1 < i < n,

(1.143)

and psf

pJ

2zBi— + uDi— € Lo, oz ou J2zi{F{aDi,p) + F{0,aDi))^^

1 < z < n.

(1.144)

i=l

pj

n

+ Y,{F{u,a)e'i0Di - F{u,p)e'iaDi + F{p,a)e'iuDi)— G Lo(1.145) And condition (1.145) is equivalent to conditions (1.146) and (1.147) below: ^•^^^'

J2xj{F{ccDj,0)

+ F{0,aDj)')^eaS{V),

(1.146)

Siegel Domains

43

where 1
i = 1, • • •, n,

n

Y,{F{u, u)e[){F{aDu /?) + F(/3, ccD,)) n

+ 5 ] ( ( F ( u , f3)e^F{aDi, u) + (F(/?, u)e'^F{u, aDi)) n

Y,{iF{u.a)e^F{0Di,u)

+

{F{a,u)e'^F{u,pDi))

n

+ Y,iiF{a,Pyi}F{u,uDi)

+ {Fi/3,a)e'i)F{uDi,u)),

ya,0e

C'". (1.147)

Proof (1)

By Lemma 1.25, it suffices to check [YuLilcLi^i,

i=

-l,-2,

where Yi is defined by (1.128). By a direct computation, [Yi, L-2] C L-i if and only if

1=1

We obtain zAiu' = 2^/^F{u,zA). Thus (1.136) holds. Obviously, (1.137) holds. Now, [Fi,L_i] C LQ implies (1.138) and tn

2F{u,F{a,u)A)

+ y/^FQ2{uCju')ej,a)

= 0, a € C ^

(1.148)

holds. Note that (1.138) and (1.102) imply (1.139) and (1.140). Let a = u in (1.140). Comparing the coefficients of all terms of u with degree two, we conclude that (1.148) holds. Therefore (1.138) is equivalent to (1.139) and (1.140). (2) For the same reason, it suffices to check [1^2?-C^i] C 1/2+2? i = - 2 , - 1 and (1.130), where Y2 is defined by (1.129), and (1.142) is equivn

alent to the uniqueness of the expression ^ ZizBi. Thus we only need 2=1

to prove that [12,^^-2] C LQ and [12,^^-1] C Li if and only if (1.144) and (1.145) hold, and (1.145) is equivalent to (1.146) and (1.147).

44

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS Now, [y2,L_2] C LQ implies that 2zBi— + uDi-— G Lo, oz ou

l
Hence (1.144) holds. By \Y2,L-i] C Li and (1.101), we obtain pj

n

Fi = 2Y^Zi{F{uDi,0L)

- 2F{u,a)Bi)—

pj

n

-

1=1

V^^^iaDi— du

i=l Qi

m

j=i

Using (1.136) and (1.138), and letting A = -V^Tj^e^aA,

Cj = - J]F(eA:,a)e^(A4efc + e'fce,J90,

we see that (1.145) holds, and F(/?A, OL) + F(/3, aDi) = 2F(/?, a) A The last condition can be induced by (1.144). By (1.102) and a direct computation, we see that (1.146) and (1.147) hold. I THEOREM 1.27 Let D{V,F) be a Siegel domain in C point XQ G V, there holds

x C"^. Given a (1.149)

where Lo = { zA— + uB-^ e Lo I xoA = 0 }, oz ou

Li = {Yi + [V^«^, ha^'^i]] I ^^^ ^ -^^ ^' i^2 = { r 2 + -

370 ^

az'

9' : ^ 0 ^ , i 2 ] ] IVF2 6 L 2 } ,

(1.150) (1.151) (1.152)

and dim (Li) = dim (L^),

i = 1,2,

dim (L2) < dim (LQ) — dim (LQ) < n. Proof

Obviously, 2

i=0

(1.153) (1.154)

Siegel Domains

45

By the definition of Li and Z/2, it suffices to prove that iso ( ^ . 0 , 0 ) ( ^ ( ^ ' F)) n aff {DiV, F)) = LQ. This is easy by Theorem 1.19 and Lemma 1.14. Hence (1.149) holds. By the definition of L^, i = 1,2, dimL^ = dimL^, i = 1,2. We will prove that dim L2 < dim LQ — dim LQ < n. In fact, letting zA-r- + uB-r- e LQ, we have xA-^r- e aff (V) and oz ou ox XQA = 0. Denoting by aff xo(^) the Lie algebra of the isotropy subgroup of Aff (F) at xo, we have dim(aff (F)/aff xo(^)) < d i m F ^ n . Since dim(Z/o/I/o) ^ dim{aS{V)/aSxo{V)), we get dimLo — dimLo ^ ^•

d'

a'

Now, xoTT ^ ^ - 2 implies that (ada:o7r-)i/2 C LQ- Given I2 ^ ^2 (see oz oz (1.141)), then the linear mapping ada^o^-: L2 -^ LQ gives uz

2=1

2=1

So £(a;oe9xoBi = 0. Thus

[x„|,yd

= 0. Z=y/—lXQ

,lt=0

Hence I2 ^ iso/^/zrj^^ o)(^(^5^))- Therefore a d l 2 is a nilpotent and semisimple hnear transformation. Namely, a d l 2 = 0. It follows that I2 belongs to the radical S in aut {D{V, F)). By Lemma 1.24, 5 fl L2 = 0. pj

So I2 = 0. Thus ad0^0T;-: L2 -^ LQ is a linear isomorphism, and oz 0/ ^^^ 0/ ^^^ ((ada:o^-)i^2) nLo = 0. Hence a d x o ^ - maps L2 into LQ/LQ. oz oz dim (L2) < dim (LQ) — dim (LQ).

Therefore

Theorem 1.26 tells us that if one wants to find the Lie algebra aut {D{V, F)) = L_2 + L_i + L0 + L1+L2 of Aut(J9(V,F)), where D{V,F) is a Siegel domain, he firstly needs to find the subspace LQ- By (1.102), the subspace LQ depends on the choice of basis in aff (V). But we cannot find a basis in aff (F), because we cannot find the expression of that cone V. Hence it is difficult to determine aff (V), Therefore we only know a few algebraic structure of

•

46

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

aff (F). When F is a homogeneous cones, aS{V) can be determined in Chapter 5. Note Satake (ref. [172]) gives another version of Theorem 1.26. There are two ways to generahze Siegel domains. One way is to change F{u, u) by a vector function Fc{u), where c is a positive constant number and (ref. [133]) Fc{Xu) = |A|i/^Fc(^),

VA G C,

ue

C^.

Another way is to change the condition F(u, u) = 0 ii and only ii u = 0 by the condition: A cone V contains the convex hull of the point set

{uHiu' \\/ueC'^,

i = l,-..,n}.

In order to consider this structure, we need to find the automorphism group of the characteristic boundary as a CR manifold, (see V. V. Ezov and G. Schmalz (refs. [49], [50])). On the other hand, we consider the generalized Siegel domain D{V^ F). The definition is as the same as that for a Siegel domain, except for the convex condition for the cone V. Namely, the cone V is only an open cone in R'^ not containing any straight line. In this case, if V is linearly isomorphic to an open cone V\ where the point coordinates in y are all positive numbers, then D{V^F) is also holomorphically equivalent to a bounded domain. Since the convex hull of V^ does not contain any straight line, so D{{V)^F) satisfies the Piatetski-Shapiro's definition, such that D{V,F) is a sub-domain in D{V,F). This is also a generalization of Siegel domains.

Chapter 2 HOMOGENEOUS SIEGEL DOMAINS

In this chapter, we will introduce the notions of homogeneous Kahler manifolds, homogeneous bounded domains and homogeneous Siegel domains. In particular, we will give a sufficient condition for the Lie algebra of the isometric transformation group acting transitively on a homogeneous Kahler manifold. We will introduce the notions of K Lie algebras, J Lie algebras, effective proper J Lie algebras and normal J Lie Algebras, respectively. We will prove that the Lie algebra of the holomorphic automorphism group acting transitively on a homogeneous bounded domain is an effective proper J Lie algebra. On the other hand, we will give a J basis of a normal J Lie algebra, and prove that the classification of effective proper J Lie algebras up to a J isomorphism is equivalent to the classification of normal matrix sets up to a unitarily equivalent.

1.

Homogeneous Bounded Domains

A real analytic manifold 3Jl is called a homogeneous manifold, if given two points pi,p2 ^ 9?^? there exists a real analytic automorphism a on 9Jl, such that a maps pi to p2Clearly, the real analytic automorphism group Aut (271) is a topological transformation group acting on 9Jt, where the topology of Aut (9Jl) is defined by the compact-open topology (i.e., the Zariski topology). Denote by IsOp(S!Jl) the isotropy subgroup at p G 9Jl, Is0p(2JI) is a closed topological subgroup of Aut (971). Given a connected transitive topological subgroup G of Aut (971), then the isotropy subgroup at p of G is 47

48

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

H = Iso p(9Jl) n G. Hence there exists a real bi-analytic isomorphism ip:

xH—^x{p),

VXGG

(2.1)

from a left coset space G/H onto 9Jl. Given g G Aut(SPt), then the mapping v4^ = (/?~-^ o gf o (^ is a real analytic automorphism on G/H^ where Ag: xH -^ {gx)H, ^g, xeG. (2.2) Given a normal subgroup ^ o of G, then HQ C H implies that xeG where Hi is also a normal subgroup of G. Given g E Hi, then the mapping Ag is a real analytic automorphism acting on G/H by (2,2). Clearly, gxH = x{x~^gx)H = xH, Vx G G imphes Ag = id. Hence g = e. Therefore HQ = Hi = {e}. A topological transformation group G acting transitively on 9JI is said to be acting effectively, if a normal subgroup HQ of G satifies the condition Ho C H, then HQ is the trivial subgroup, where H is the isotropy subgroup of G at p G 3JI. In this case, G is called an effective topological transformation group. Let (SDt, h) be a Kahler manifold of dimension n with respect to a complex structure / . A holomorphic automorphism a acting on (9Jl, h) is called an isometric transformation, if a*h = h. The isometric transformation group I(3Dt, h) consists of all isometric transformations acting on 971. It is well known that I(3Jt, h) is a real Lie transformation group acting on 9Jl. The isotropy subgroup Is0p(9Jt) at p G 9Jl is a compact subgroup of I(9K, h). A Kahler manifold (9Jt, h) is called a homogeneous Kahler manifold, if 1(971, h) acts transitively on 971. In this case, given a transitive Lie subgroup G of 1(971, /i), then 971 is holomorphically equivalent to a left coset space G/H, where H = Gnls0p(97t, h) is the isotropy subgroup of G at p G 971. By the Lie group isomorphism (1.20), we have a real Lie group GR and a compact subgroup HR, which are defined by GR = {aR\\/aeG},

HR = {aR e GR\ aR{p) = p},

(2.3)

respectively, where a and aR are defined by (1.18) and (1.19), respectively. Clearly, GR is a real effective transitive Lie transformation group on GR/HR, where an isometric transformation x is defined by x{yHR) = {xy)HR,

MV^GR.

Homogeneous Siegel Domains

49

On the other hand, ^0 '

(^RHR

is an isometric isomorphic from GR/HR has a GR invariant complex structure

—> aH onto G/H.

Moreover,

GR/HR

and a GR invariant Kahler metric

with respect to a GR invariant Kahler form

where f2 = Im {h) and Ip = ^pQO(p. Hence ^ is a holomorphic isomorphism acting from GR/HR onto SUt. Replacing the symbols J, h, Cl by / , /i, fi, respectively, we have

P = -id, [X, Y] + I[X, lY] + I[IX, Y] = [IX, lY], fi(X,y) = p ( X , / y ) , g = Re{h), n{Y,X) = -Q{X,Y), Q ( / x , / y ) = f2(x,y),

(2.4)

2.1 Lef Sj be a compact suhalgehra of a real Lie algebra 0, X G 6 , X ^ ^ , and F{JX, X) = 0, VX G ^ ; (8) F{X,[Y,Z]) + F{Z,[X,Y]) + F{Y,[Z,X]) = 0, \/X,Y,Ze<&. K Lie algebra 0 can be expressed by (0,55; J, F ) .

DEFINITION

50

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

The maximal ideal of 0 containing in S) is called the effective kernel of (3. If the effective kernel of (& is equal to zero, 6 is called an effective K Lie algebra. By the definitions of a linear transformation J, a skew-symmetric bilinear function and Definition 2.2, we know that J and F are not unique. We have 2.2 Two K Lie algebras (©,^; J , F ) and (6,i3; J i , F i ) is said to have same structure, if

DEFINITION

(J-Ji)(0)Ci3,

Fi = AF,

(2.5)

where X is a positive constant. 2.3 Let (©,i}; J, F) be aK Lie algebra. A subspace Slof& is said to be J invariant, if J{£) C 2. A J invariant subalgebra ( ideal) 01 is called a K Lie subalgebra {K ideal). In this case, ( © i , ^ n 0 i ; J, F) is also a K Lie algebra ( K ideal) of ( 0 , ^ ; J, F).

DEFINITION

2.4 / / there is a Lie algebra isomorphism p acting from a K Lie algebra ((S,i3; J, F) onto a Lie algebra &\, then p{S)) is a compact subalgebra. Let DEFINITION

Ji = poJop-\

Fi{p{X),p{Y))

= F{X,Y),

\/X,Y€<5.

(2.6)

Then (0i,i3i; J i , F i ) is also a K Lie algebras, and p is called a K isomorphism acting from 0 onto ©i. It is well known that the natural mapping TT:

GR^GR/HR,

7r{x) = XHR,

^XEGR,

(2.7)

is a real analytic open mapping. Thus conditions (2.4) on GR/HR can be transfered to conditions on GR by TT as follows. Denoting by 'S{GR) and ^{GR/HR) the real analytic function classes on GR and GR/HR, respectively, we have ^^{^{GR/HR)) C S^(Gi?), where TT* is defined by {7r''f){x) = f{7r{x)) = f{xHR),

WfediGR/HR),

XEGR.

(2.8)

Thus {7r*f){xh) = f{7r{xh)) = f{7r{x)) = {n*f){x),

^heHR,

X^GR,

where TT*/ on GR. Hence the function TT*/ on GR has the same value on any left coset XHR = {xh\h E HR}.

Homogeneous Siegel Domains

51

Given a real analytic vector field X G T{GR) on GR^ then X is a linear differential operator action on ^{GR). Suppose that X{7r*{d{GR/HR))) C 7r*{d{GR/HR)). Given / G

^{GR/HR),

then

((7r,(X))/)(7r(a;)) = (X(7r7))(a;), Hence the defined domain of mapping where

TT*

Va; € Gi?.

is a subspace

TT^{GR)

(2.10) of T(Gi?),

I X(7r*(5(GH/^H))) C 7r*(5(GH/ifii)) }• (2.11) Clearly, TT^{GR) is a Lie subalgebra of T{GR). An element in TT^{GR) is called a projective vector field. If a linear transformation Jg acts on the tangent space Te{GR), such that T^{GR)

= {X

(2.9)

G T{GR)

(7r*)e O J e = leHR O (7r*)e,

(2.12)

then there is a linear transformation J acting on the vector space where J\g = Jg = {Lg).0j,0{Lg):\ \/9 E GR,

T{GR),

(2.13)

We will prove that (TT* O J ) ( X ) = (J o 7r*)(X),

VX G T^(G^)

(2.14)

and J{T^iGR)) C r,(G'R). In fact, given g G GR and X G

T.,^{GR),

(2.15)

then

K ) * O Js(Xp) = (TTp)* O (Lj)* O Je O

{Lg)-\Xg)

= (Lg)* O / e H O (TTe)* O ( L < , ) ; : ^ ( X g )

= (Lg)* O /eH O 57^ ° MgiXg)

= IgH °

Mgi^g).

Hence (2.14) and (2.15) hold. Given X G T^{G), then 7r*(X) = 0 imphes 0 = /o7r*(X) = TT* o J{X). Thus J{X) G 5 . Hence the kernel of TT* in r^(G) is

Therefore

^ = 7r,-H0) = { X G T^{G) 17r*(X) = 0 }.

(2.16)

_ J ( ^ ) C i3.

(2.17)

52

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Given X 6 T^{G), then (/2+id)o7r*(X) = 0. Thus 7r*o(j2+id)(X) = 0. Hence (J^ + id)(X) e ^j. Moreover, [7r,(X),7r*(y)]+/[J7r,(X),7r,(y)]+7[7r*(X),/7r*(y)] = [/7r.(X),/7r*(y)] implies

[x,Y]+j[j{x),Y]+j[x,j{Y)]-[j{x),JiY)]e^, vx,yer^(G). Let F = 7r*(fi).

(2.18)

F is a skew-symmetric bi-Unear function on r,r(G), and F{X, Y) = {ir*n){X, Y) = niiT^X), 7r*(y)).

(2.19)

By (2.16)—(2.19), {T„{G),'^;J,F) is a K Lie algebra. Denote by (3r the Lie algebra, which consists of all right invariant vector fields on a Lie group GR. The Lie subgroup HR determines the Lie subalgebra S)r of <&r- We want to prove that (dr C T^(Gij). In fact, let Lx and Rx, x € GR, be a left translation and a right translation, respectively, we have TTo Lg = gon,ygeGR,

TT o Rh = n, ^h e HR.

(2.20)

Thus {TT^)ghiXgh) ^ {TT,)gh{{Rh)*Xg) = {Tr*)g{Xg), ^ k € HR, where X 6 ©r- Hence 7r*(X) € T{GR/HR).

QEGR,

Therefore

<S>r C T^{GR).

(2.21)

In particular, (©,)e = UGR)

^

TeH^GR/HR)

and (5e = {^r)e = Te{GR). Given X,Y i^r)e/{^r)e induceS

= (0.)e/(i3r)e,

e ©r, then TeHn{GR/HR) =

7r^{[Xe,Ye]) = [Tr^Xe.TT^Ye] + (^r)eHence ((0r)e5 {^r)ei Je, ^e) is 3> K Lie algebra, where Fg = Fg as (2.19). Therefore (©e^-fte; Je^^e) is a K Lie algebra.

Homogeneous Siegel Domains

53

Given X G 0 , then X is a left invariant vector fields on GR^ and {Lx)*{JX)g

=

{Lx)A^{Lg)ncJe{Lg)^

=JxgXxg = (JX)xg,

Xg

=

{Lxg)*Je\Lxg)^

{Lx)^Xg

yx,g e GR.

Thus J ( 0 ) C 0 . On the other hand, F satisfies Fg = {Lgy{Fe), ygeGR. (2.23) Hence {(5R, S)R; J, F) is a, K Lie algebra. Therefore ( 0 , S^; J, F) is also a K Lie algebra. 2.1 Suppose that H is a compact subgroup of a real connected Lie group G not containing any non-trivial normal subgroup of G in H. Denote by (& and 9) the Lie algebras of G and H, respectively. If there exists a G invariant complex structure I and a G invariant Kdhler form Jl on the left coset space G/H, such that G/H is a homogeneous Kdhler manifold, then (0,i3; J^F) is an effective K Lie algebra, where J and F are defined by (2.13) and (2.23), respectively. And

THEOREM

(9) (10)

[J,Ad(/i)]((J5)ci3,

^h^H',

F((Ad/i)X,(Ad/i)y) = F ( X , y ) ,

\/heH,

X,Y

e<6.

Proof Now, ad(/i)e = heh~^ = e implies that (Ad(/i))e is a linear transformation acting on Te{GR)^ where h E H. It suffices to prove that formulas (9) and (10) in Theorem 2.1 hold at e. Given h e HR and X,Y £(5R, then Fe((Ad(/l))eXe,(Ad(/l))eye) = (TT* O f7eif^)((Ad (/l))eXe, (Ad {h))eYe) =

neHniM^d{h))eXe,M^d{h))eYe)

= ^h-^Hniih^ o 7r*){Xe), (/i* o 7r^){Ye)) = {{Lhr)Fh-i{Xe^Ye)

=

FeiXe^Ye).

Hence formula (10) holds in Theorem 2.1. Now, (TT* O J e ) o ( A d =

{K

O ICHR)

{h))e

O TT* =

= {ICHR

O TT*) O ( A d

/l* O (TT* O J e )

=

{h))e

(TT* O A d (h))

=

ICHR O Je,

O (/l* O TT*) V/l

G

HR.

Thus 7r*([Je, Ad(/i)]) = 0, V/i G iJi?. Hence [Je, Ad(/i)e]((5i?)e C (i3ii)e, V/i G i^i?. Therefore formula (9) holds in Theorem 2.1. I

54

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

When the compact subgroup i? is a connected Lie group, conditions (3), (8), (5) of a iiT Lie algebra imply conditions (9) and (10) in Theorem 2.L Now, we discuss the homogeneous bounded domains. A bounded domain D in C^ is called a homogeneous bounded domain, if the holomorphic automorphism group Aut (D) acting on i ) is a transitive Lie transformation group acting on D. Since the Bergman metric matrix T[z^'z) satisfies (L62), a homogeneous bounded domain is a homogeneous Kahler manifold with respect to the Bergman metric, and the isometric transformation group is Aut {D). 2.5 A K Lie algebra (0,-ft; J,F) is called a J Lie algebra, if there exists a real linear function ip on (8, such that

DEFINITION

F ( X , Y ) = ^P{[X,r]),

V X , Y e (5.

(2.24)

The J Lie algebra can be expressed by {(8^S^;J^ip). The definitions of J subalgebra, J ideal and J isomorphism can be given by the same way as in Definitions 2.3 and 2.4. DEFINITION 2.6 An effective J Lie algebra ( 6 , i^; 7,-0) is called an effective proper J Lie algebra, if any compact J subalgebra of(8 is contained in 53. Hence So is the maximal compact J subalgebra.

2.2 Suppose that D is a homogeneous bounded domain in C^. Then D is a homogeneous Kahler manifold with the Bergman metric. Let aut(Z)) and isOp{D) are the Lie algebras of holomorphic automorphism group Aut (D) and the isotropy subgroup lsOp{D) at p E D, respectively. Let I and Q are the Aut {D) invariant complex structure and the Aut (D) invariant Kahler form of D with respect to the Bergman metric. Then there exists a real linear transformation J and a real linear function ip on aut(Z))^ such that (aut (i?),isOp(Z)); 7,-0) is an effective proper J Lie algebra, and THEOREM

[Ad {lsOp{D)), J](aut {D)) C isOp(D), i){[{kdh)X,{kdh)Y])

= ^([X,F]),

(2.25)

V/i G IsOp{D), X,Y e aut(Z)). (2.26)

Proof The real analytic automorphism group GR = Aut R{D) acts transitively on D^ and the isotropy subgroup at a point p in D is a compact subgroup HR = {IsoP)R{D). Denote by (5R = a.ut R{D) and S^R = {isoP)R{D) the Lie algebras of Lie groups GR and HR, respectively. The holomorphic mapping (/?: QRHR -^ gR{p), "^QR^GR maps GR/HR

Homogeneous Siegel Domains

55

onto D, and the natural mapping TT: QR -^ QRHR^ ^QR^ ©i? maps GR onto GR/HR, such that ^"^(jp) = HR. _ Given X,Y e aut(L)), then XR = X + X, YR = Y + Y e ^R, and there exists a hnear function * on (5i?, such that n{XR,YR)

= ^{[XR,YR])

(2.27)

by Theorem 1.5, where fi is the Kahler form with respect to the Bergman metric of D. Thus %{[Xo,Yo]) = np{Xo,Yol

\fXo,YoeTp{D),

(2.28)

By the holomorphic isomorphism (/?, (2.28) induces that ^eHn{[XoM)

= ^eHniXo,Yo),

^Xo,Yo

e T^HniGR/HR).

(2.29)

Since the natural mapping n acts from e to CHR, where e is the unit element in GR, there exists a real linear function if: and a skew-symmetric bi-Unear function LJ on Te{G), which are defined by ^e=7r*(*ei/«),

u;e=ir*ineHn)-

(2-30)

Hence there exists a real 1-form tp and a real 2-form w on G, such that ^Pg = L;i^Pe),

iVg^Lliue),

i^{[X,Y])=u;{X,Y),

^geOR,

yX,Ye<5.

(2.31) (2.32)

By Theorem 2.1, (aut (£)), isOp(jD); J, -0) is an effective J Lie algebra and (2.25), (2.26) hold. Now, we will prove any compact J subalgebra A C isop{D). Given a compact J subalgebra ^ , then iiT = exp {^) is a connected compact subgroup of Aut (£>), and the orbit K{p) of iiT is a compact complex submanifold. Since any bounded holomorphic function is equal to a constant on any compact complex manifold, and a coordinate function Zi of D is a bounded holomorphic function on the orbit K{p), we see that zi,-- - ,Zn are all constants. Thus the orbit K{p) — p. Hence K C lsOp{D). Therefore A C isOp(£)), and (aut (D),isOp(D); 7,-0) is an effective proper J Lie algebra. I Note In 1955, Koszul (ref. [113]) proved a general result: if a homogeneous complex manifold 9JI of dimension n has an invariant volume element t;, given a chart {U^(p)^ then V = K{z^ 'z)dz\ A • • • A dzn A dzi A • • • A dz^^

56

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and there exists a 1-form ^ , such that d^ is the imaginary part of the 2-form h = ddlogK{z,z)

= dzT{z,z)dz

= V

^^ —^^dzi{x)

iS)'dz'j,

Let aut (3Pt) and isOp(9Jl) are the Lie algebras of the holomorphic automorphism group Aut (9JI) and the isotropy subgroup IsOp(aJl) at p G 9Jl. Then there exists a hnear transformation J and a Hnear function t/^ on aut (9JI), such that (aut (OK), isOp(SDT); J, -0) is a J Lie algebra, where *(^) = trautim/iso(an)(ad {Jx) - Jad (x)),

Vx e (&,

and ^ = ip* oyb,

(/?: Aut (9Jt) -^ Aut (9n)/Is0p(aJl).

When £) is a complete bounded domain, we give a hnear function * on the Killing vector fields in Theorem 1.5 and Lemma 1.6, such that 2d* = Jl, where fi is the Kahler form with respect to the Bergman metric h. Kobayashi (refs. [100], [101]) proved that any Siegel domain is a complete domain. Therefore Theorem 1.5 holds for any Siegel domain. When £> is a homogeneous bounded domain, Theorem 1.5 is a special case of Koszul's result.

2.

Homogeneous Siegel Domains

In this section, we will give some properties of Siegel domains. Since any homogeneous Riemannian manifold is isometricly isomorphic to a left coset space G/H, where G is a real connected Lie group, and H is a compact subgroup, such that H does not contain any non-trivial normal subgroup of G, hence the structure of a homogeneous Riemannian manifold is determined by a special Lie group pair (G, H). Given a homogeneous Siegel domain D{V^ F ) , we will prove some properties for a connected transitive Lie subgroup G of the holomorphic automorphism group Aut {D{V, F)): (1) the affine automorphism group Aff (D(V,F)) of D{V,F) is a transitive Lie transformation group acting on D{V^F)^ and any connected transitive Lie transformation subgroup G of Aff (D) is an algebraic Lie group; (2) the isotropy subgroup i? of G at a fixed point {ZQ^ UQ) G Z?(V, F) is a maximal compact subgroup of G;

Homogeneous Siegel Domains

57

(3) any homogeneous Siegel domain is acted simply transitively by an affine automorphism subgroup G, the Lie algebra 0 of the Lie group G is a normal J Lie algebra. 2.3 (Kaup, Matsushima, Ochiai) Let D{V,F) be a Siegel domain in C^xC^. Then D(V, F) is a homogeneous Siegel domain if and only if the affine automorphism group AS{D{V^F)) is a transitive Lie transformation group acting on D{V,F). Thus D{V^F) is an affine homogeneous domain J and AfF (V) is a transitive Lie transformation group acting on V. Hence V is an affine homogeneous cone.

THEOREM

Proof We prove only that the sufficiency holds. Since Aut (D, F) is a transitive transformation group acting on D{V^ F ) , so real dimensions of Aut {D{V, F)) and Isop{D{V, F)) satisfy the condition dimR{Aut{D{V,F)))-dimR{IsOp{D{V,F)))

= 2(n + m),

where p = ( 1 / ^ x 0 , 0 ) and XQ is a fixed point in V. By Theorem L27, dimR{[sOp{D{V, F))) = dim (Z^ + L^ + Z^) = dim {L0 + L1 + L2). Thus 2(n + m) = dim L_2 + dim L_i + dim LQ — dim LQ = dim (Aff {D{V, F))) - dim {lsOp{D{V, F)) n Aff {D{V, F))). Hence the orbit M = Aff (jD(V,F))(\/^a;o,0) is an open subset in D{y, F). Since Aff {D{V, F)) is a closed Lie subgroup of Aut {D{V, F)), so the orbit M is also a closed subset of D{V, F). Hence M = D(V, F). Therefore the affine automorphism group Aff {D{V, F)) acts transitively on D{V,F). In the following, we will prove that Aff (V) is a transitive transformation group acting on V. By Theorem L19 and Theorem L20 and Aff (£)(V,F)) is a transitive transformation group acting on £)(V,F), the unit connected component Aff {D{V, F))^ of Aff {D{V, F)) is a: w = {z-2V^F{u,(3) where aeW,

(ieC^,

+ V^F{f3,P)-a)A,

v = {u-0)B,

(2.33)

Ae GL (n,R), 5 G GL (m, C), such that F{uB, uB) = F{u, u)A,

(2.34)

and the mapping T{A): X -^ xA is contained in Aff (V). Given two points x,y eV, there is a a € Aff (i)(V,F)), such that a{^^^x,Q) = (>/^j/,0), where a is defined by (2.33). Thus (3 = 0.

58

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Hence y = xA + y/^aA, so that y, xAeV and aA e MP' imply a = 0. Hence a is w = zA, v = uB, where y = xA is an affine automorphism on V, Therefore F is an affine homogeneous cone. I T H E O R E M 2.4 Let D{V,F) be a homogeneous Siegel domain in C^ x C^. Given a connected component Gi of the holomorphic automorphism group Aut {D{V, F)), Then there exists a representation element a ojG\ in the isotropy subgroup lsOp{D{V,F)) at p = (v^^a;o,0) G D{V^F). And there is only finite connected components of Aut (£)(V,F)).

Proof Let Auto = Aut(D(V,F))^ be the unit connected component of Aut {D{V, F)). Then Auto is a real connected Lie group acting tran-sitively on -D(F, F). Given a connected component Gi of Aut (-D(V, F)) with a representation element a and p = ( \ / ^ x o , 0 ) E D{V,F), if q = (7(p), there is a r G Auto such that T{q) = p. Hence r o a e IsOp(D(V,F)). Now, a and r o a are all contained in Gi. Thus there is a representation element a G Isop{D{V^F)) of Gi. Therefore the first statement holds. Clearly, Aut {D{V,F))/Auto is a discrete Lie group. Hence a component representation element set © in Isop{D{V^F)) is a discrete subset. The compactness implies that 6 is a finite set. I As an appUcation of Theorem 2.3, we have 2.5 Suppose that D{V^F) is a homogeneous Siegel domain in C^ X C ^ . Then the isotropy subgroup Isop{D{V,F)) at a fixed point p G D{V^F) is a maximal compact subgroup of Aut (D(V,F)). THEOREM

Proof Let Isop{D(y,F)) and Iso^(Z)(V,F)) are isotropy subgroups at two fixed point p^u E £)(V,F), respectively. It is well known that Is0p(-D(V, F)) and IsOu{D{V^ F)) are conjugate to each other. It suffices to prove this Theorem for a fixed isotropy subgroup IsOp(i)(V, F)), where p=(V^xo,0)GD(y,F). Suppose that the isotropy subgroup lsOp{D{V^F)) C K, where K is a maximal compact subgroup of Aut{D{V,F)). Denote by A the Lie algebra of K. Then Lo + T[ + T2 = isop{D{V,F)) C .^ by Theorem 1.27, where LQ C Lo._Thus A = ^ + Z[ + L2, where Lo C . ^ = AnaS {D{V, F)), and LQ is a compact subalgebra of aff {D{V, F)). Let Ko be a maximal compact subgroup of Aff (Z)(y,F)), such that Aff {D{V, F)) n ISOp{D{V, F)) = ASp{D{V, F)) C Ko. Let 5 be a bounded open subset of D{V^F) containing the point p = {\/^xo,0). Then the orbit Ko{S) = UkeKoH'^) is a bounded open subset of D(V,F). Let q be the gravity center of Ko{S). Now

Homogeneous Siegel Domains

59

D{V,F) is a convex set implies q e D{V,F), Now, k'Ko{S) = Ko{S), Vfc^ G KQJ where fc' is an affine transformation. So fc' maps a straight segment to a straight segment. Thus fe' maps the gravity center to itself. Hence k\q) = q^k^ G KQ. Therefore KQ is contained in the isotropy subgroup ASg{D{V,F)). We conclude that ASp{D{V,F)) c KoCASq{D{V,F)). By Theorem 2.3, AS{D{V,F)) acts transitively on D{V,F). Thus there is a a G Aff {D{V, F)), such that a{q) = p, and ASp{D{V, F)) c Aff q{D{V, F)) = a-\ o Aff p(D(F, F)) o a implies ASp{D{V,F)) = ASg{D{V,F)), Hence p - q. Therefore KQ = ASp{D{V, F)) and then K = Isop{D{V, F)), I LEMMA 2.6 Suppose that dJl is a connected and simply connected manifold. Let G be a Lie transformation group acts transitively on 3Jl. Denote by H the isotropy subgroup ofGatpedJl, such that H is a compact subgroup. Then G is a connected Lie group if and only if H is a connected Lie group.

Proof Let H^ be the unit connected component of H. Since if is a compact Lie group, so the quotient group H/H^ is a finite group. Hence the natural mapping TT acting from the left coset space G/H^ onto G/H is a covering mapping. It is well known that the connected and simply connected manifold 9Jl is homeomorphic to the quotient space G/H. Thus G/H is also connected and simply connected. Hence the natural mapping TT is identity, therefore H = H^. I 2.7 Suppose that D{V^F) is a homogeneous Siegel domain in C^ X C^. Denote by G the transitive connected Lie subgroup of the holomorphic automorphism group Aut(jD(y,F)) on D{V,F). Then the isotropy subgroup H of G at the fixed point p e D{V, F) is a compact connected Lie groupj and H is a maximal compact subgroup of G.

THEOREM

Proof By Lemma 2.6, if is a connected compact subgroup of G. By Theorem 2.5, H = Is0p(D(y, F)) n G (2.35) is a maximal compact subgroup of G because that IsOp(i)(V,F)) is a maximal compact subgroup of Aut (I?(V, F)). I In the following, we will prove that the unit connected component Affo = AQ{D{V,F)f of the affine automorphism group A^{D{V,F)) is a algebraic Lie group acting on the complex Euclidean space C^ x C"^.

60

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

2.8 Suppose that D{V^F) is a homogeneous Siegel domain in £n y^ (Qjm y^g ^^j^^ connected component Aff o of the affine automorphism group Aff{D{V^F)) is a connected subgroup of the affine transformation group Aff (C^+^) acting on C'^^^. Denote by N(Aff o) the normalizer 0/Aff 0 in Aff (C^+^). Then the unit connected component N(Aff 0)^ of N(Aff 0) is equal to Aff Q. LEMMA

Proof Since Aff 0 C N(Aff 0)^ and N(Aff 0)^ is connected, there is a unit neighborhood U of N(Aff 0)^, such that the orbit {a{p) \\/a eU} C D{V,F) is an open neighborhood of p € D{V,F). Given a E U C N(Affo)^ C N(Affo), then a{ASo)(^'^ = Affo- Thus ( a o Affo)(p) = (Aff 0 o a){p) C D{V^F). On the other hand, Aff 0 acts transitively on D{V,F) impUes a E AS{D{V,F)). Hence C/ C Aff Q. NOW, N(Aff 0)^ is generated by U, Hence Aff 0 C N(Aff 0)^ C Aff Q. Therefore N(Aff 0)^ = Affo. I Clearly, Affo C Aff (C^ x C"^) is not a linear group. Given an afhne transformation a: w = zA + a (2.36) in Aff (C^ X C^), where z.w.aeC'x C^, ^ G GL (n + m, C), then we can construct a non-singular linear transformation a: iw,wo) = iz,zo)(^^

j )

(2.37)

acting on C'^+^^'+i^ where ZQ^WQ G C The mapping a ^ a gives a Lie group isomorphism from Aff (C^ x C^) onto GL (n + m + 1, C). Hence a(AS (C^+^)) C GL (n + m + 1, C), and a(Aff (C^+^)) is a closed Lie subgroup of GL (n + m + 1, C). Sometime we denote a(Aff (C''+'^)) by the same symbol Aff (C^"^'^). Then the connected Lie group Affo C Aff (C"+"^) is a linear group. 2.9 Suppose that D{V,F) is a homogeneous Siegel domain in C^ x C^j Affo is the unit connected component of the affine automorphism group AS{D{V,F)). Then the center C(Aff o) = { i d } ; where id is the unit element of ASQ. Moreover, Affo and Ad (Affo) are two real connected algebraic Lie groups, where Ad is a Lie group isomorphism from Aff 0 onto Ad (Aff o). THEOREM

Proof

By Theorem 1.20, if f

w = zA-

2V^F{uB,

i

v = uB-P

/?) + \ / ^ F ( / 3 , p) - a,

Homogeneous Siegel Domains

61

is an affine automorphism in C(Aff o), where a € R", /? € C " , A € GL (F) = { a € GL {nR) \ a{V) = F }, B G GL (m, C) and F ( « B , «B) = F{u,u)A. By (1.86) and (1.87), Aff o has elements w = ze^* + i, V

Now, (T{T{Z, U))

= a{e^'z + C, e * + ^ ^ ^ ) = (e2*2A + e^l - 2v/=Ie*+^^^F(MB, /?) + \ / ^ F ( / ? , /?) - a, e*+^%B-/3)

and T(
= T{ZA - 2 V - 1 F ( « B , 13) + V-lF{^,/?) = {e^'zA - 2^f=\e^'F{uB,

- a,uB - 0)

(3) + ^/^e^'F{(3,

(3) - e^^a + C,

e*+^^%5-e*+-^^/3). Hence a o r = r o a, Vt, ^ € R, ^ € R", if and only if /? = e*+^/^^/3, V « , ^ € R , and ^A - 2 v / ^ e * + ^ ^ ^ F ( « B , /?) + \ / ^ F ( / ? , /?) - a = ^ - 2 v ^ e 2 * F ( « S , f3) + ^/^e^*F{P,

(3) - e^^a,

Vt, ^ G R, ^ € R".

Thus a = /? = 0 and A = £•(") is the unit matrix. Hence F{uB, uB) = F{u, u), and cr is u; = 2, i; = uB. It is obvious that Aff 0 contains _ r \

w; = 2: - 2 v-lF{u,7]) /^ +

V-lF{rt,v),

V = u — rj,

Vr? € C " . Now, r((T(2, u)) = T{Z, UB) = {Z-

2V^F{UB,

rj) + ^/^F{l^,

77), uB - r/),

a{T{z, u)) = {z- 2 V ^ F ( « , ri) + V ^ F ( r / , r?), (w - 7?)B). Thus T0(T = (70T, V77G C"^ if and only if F{uB,'r]) = F{u,r]) and r/ = 775, Vr? e C " . Hence 5 = £"("*) is the unit matrix. It proves that a = id. Therefore the center of Aff 0 is a trivial center. As a corollary, Aff 0 is isomorphic to Ad (Aff 0). The first statement holds.

62

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Since Lemma 2.8 and Chevalley's book II, Chapter 2, Section 1.2, the unit connected component N(Aff o)^ of the normahzer N(Aff o) is an algebraic Lie group. Thus Aff o is an algebraic Lie group. Hence the Lie algebra aff {D{Vj F)) of the hnear Lie group Aff o is an algebraic Lie algebra. By Chevalley's book II, Chapter 2, Section 1.4, aS{D{V,F)f is a complex algebraic Lie algebra, and the complex connected linear Lie group (Affo)^( C aS(D{VjF))^) is an algebraic Lie group, where (Affo)^ is generated by exp(aff (D(F,F))^). By Chevalley's book II, Chapter 2, Section 1.9, the mapping Ad: (Affo)^ -> Ad (Aff o)^ is a rational mapping. By Chevalley's book II, Chapter 2, Section 1.7, Ad (Aff o)^ is a complex connected algebraic Lie group. Hence the Lie algebra ad (aff {D{V, F)f) = (ad (aff {D{V, F)))f of the linear Lie group Ad (Aff o)^ is a complex algebraic Lie algebra. By Chevalley's book II, Chapter 2, Section 1.4, ad(aff (D(V,F))) is a real algebraic Lie algebra. Therefore Ad (Aff o) is a real connected algebraic Lie group. I 2.7 Let W be a real vector space of dimension m. A subgroup G of the general linear group GL (W) (a subalgebra (S of the general linear Lie algebra gl (W)) over W is called a triangular subgroup G {triangular subalgebra (3 ) , if there is a basis of W^ such that the matrix representation of any element in G (0) is an upper triangular matrix. DEFINITION

Obviously, a real connected linear Lie group G C GL (n, R) is a triangular Lie group if and only if its Lie algebra 0 C gl (n, R) is a triangular Lie algebra. The triangular Lie algebra 0 is a solvable Lie algebra and all eigenvalues of any element in © are real numbers. Therefore all eigenvalues of any element in G are positive real numbers. Given a compact connected linear Lie subgroup H of G, there is an H invariant inner product on W, Thus any element of fl" is a semisimple linear transformation and all eigenvalues are complex numbers, which module are equal to 1. If G is a real connected compact triangular group, then all eigenvalues of any element in G are n positive real numbers. Hence a compact triangular subgroup ff of G is equal to {e}. Therefore we have LEMMA 2.10 The maximal compact subgroup of a real connected triangular Lie group is the trivial subgroup. LEMMA 2.11 (Vinberg) A real connected linear Lie group is a triangular Lie group if and only if all eigenvalues of any element in G are real numbers. Hence the triangular Lie group is a solvable Lie group.

Proof It suffices to prove the necessity. Suppose that G is a real connected linear Lie group and all eigenvalues of any element in G are real

Homogeneous Siegel Domains

63

numbers. If G is not a solvable Lie group, G has a non-trivial semisimple Lie subgroup by Levi Theorem in the theory of Lie groups. It is well known that any non-trivial real semisimple Lie group has a non-trivial compact subgroup. Thus we find a contradiction by Lemma 2.10. Hence G is a solvable linear Lie group. Therefore G is a triangular Lie group by the Lie Theorem in the theory of Lie groups. I The proof of the next Lemma is omit, see Vinberg ref. [219]. 2.12 (Vinberg) Let G be a connected algebraic Lie subgroup of the general linear group GL {W)^ where W is a real vector space. Then any maximal triangular subgroup T of G is a connected closed subgroup of GL{W)^ and any two maximal triangular subgroups are conjugate to each other in G. On the other hand, there exists a connected maximal compact Lie subgroup K and a maximal triangular subgroup T, such that

LEMMA

G = T'K,

TnK

= {e}

(2.38)

is an unique semi-direct product decomposition up to an inner automorphism of G. Denote by 0 , % and ^ the Lie algebras of Lie groups G, T and K, respectively. Then we have a unique subspace direct sum decomposition up to an inner automorphism of (3: 0 = % + M.

(2.39)

By Lemma 2.12 and Theorems 2.7, 2.9, we have T H E O R E M 2.13 Let D(V, F ) be a homogeneous Siegel domain in C^ x C ^ , denote by Ant {D{V,F)f and AS{D{V,F)f the unit connected components of the holomorphic automorphism group Aut(D(V,F)) and the affine automorphism subgroup Aff {D{Vj F)), respectively. Then there exists a connected and simply connected closed subgroup TQ of the connected Lie group Aff ( J D ( V , F ) ) ^ , such that TQ acts simply transitively on D{V^ F)j and Ad To is a maximal triangular subgroup of the algebraic Lie group Ad (Aff (D(V, F))^. Moreover there exists a point p G D(V, F), such that Ko = IsOp{D{V, F)) n Aff {D{V, F)f (2.40)

is a maximal compact subgroup o/Aff (D(V,F))^, where Isop{D{V^F)) is the isotropy subgroup at a fixed point p. On the other hand, AS{D{V,F)f

= To'Ko,

TonKo = {e},

(2.41)

64

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and Aut {D{V, F)f

= To • {lsop{D{V, F))f

(2.42)

are two unique semi-direct product decompositions up to a conjugation of AS{D{V,F))^, respectively, where Isop{D{V,F))^ is the unit connected component ofIsOp{D{V^F)). Proof By Theorem 2.9, Aff {D{V, F))^ is isomorphic to the hnear algebraic group Ad (Aff (D(V, F))^). By Lemma 2.12, there exists a maximal connected compact subgroup Ki and a maximal triangular subgroup Ti of Ad (Aff {D{V, F))0), such that Ad{AS{D{V,F)f)

= Ti. Ki,

TinKi

= {Ade}

(2.43)

is a semi-direct product decomposition of Ad(Aff ( J D ( V , F ) ) ^ ) , where e is the unit element of Aff (Z)(V,F)). Moreover, this decomposition is unique up to an inner automorphism on Ad (Aff {D{V, F))^). By Theorem 2.9, AS{D{V,F))^ has only the trivial center, and the adjoint representation Ad is an isomorphism. Thus KQ — Ad~'^(i^i) is a maximal connected compact subgroup of AS{D{V^F))^ and TQ = Ad ~^(Ti) is a closed connected solvable subgroup of Aff (£)(V, F))^, such that AS{D{V,F))'' = To'Ko, TonKo = {e} (2.44) is a semi-direct product decomposition. By Theorem 2.5, there is a point p G Z)(V", F ) , such KQ is the isotropy subgroup at p. Obviously, aff (£)(V,F)) has a subspace direct sum decomposition aff {D{V, F)) =%o + (isOp(D(F, F)) H aff (D(V, F))), To n ISOp{D{V, F)) n aff {D{V, F)) = 0,

^* ^

where Ad TQ = T I is a real connected triangular Lie group acting on the vector space aff {D(y, F)). By Lemma 2.12, the semi-direct product decomposition Ti • Ki is unique up to a conjugation on Ad (Aff {D{V, F))^) by the inner automorphism and Aff {D{V^ F))^ has only the trivial center. Thus the decomposition (2.44) is unique up to the inner automorphism on Aff ( J D ( V , F ) ) ^ . Now, we will prove that TQ acts simply transitively on D{V^ F ) . In fact, Ko = IsOp{D{V,F))nAS{D{V,F)f and D{V,F) is holomorphically equivalent to AS {D{V,F)f/Ko. Now, AS{D{V,F)f

= To'Ko,

TonKo = {e}

is a semi-direct product decomposition. Thus Aff {D{V, F)f/Ko

= {To • Ko)/Ko = To/{To n Ko) = To.

Homogeneous Siegel Domains

65

Hence the orbit To(p) = {(T{P) | Vcr E TQ} in D{V,F) is an open subset and also a closed subset. It proves that To{p) = D{V,F). Therefore To acts transitively on D{V^F), Obviously, the isotropy subgroup of TQ is a trivial subgroup. Hence To acts simply transitively on D{V,F), We will prove that Aut {D{V,F))^ has a semi-direct product decomposition (2.42). In fact, D{V,F) is holomorphically equivalent to Aut {D{V, F))7(Is0p(D(F, F)) n Aut {D{V, By Lemma 2.6, the compact subgroup Isop{D{V,F)) is a connected Lie group. Thus ISOp{D{V, F)) n Aut {D{V, F)f Aut (D{V, F)f Clearly, To nlsOp{D{V,F))^ (2.42).

F)fy D

= Isop{D{V,

Aut{D{V,F))^ F)f,

= To . lsOp{D{V, F)). = {e}. Hence we obtain a decomposition I

N o t e A lot of results of homogeneous cones in this section were proved by Vinberg (refs. [157], [158]) in 1963. Let V be an open convex cone not containing any straight line. If the affine automorphism group Aff (F) acts transitively on V, it was proved that the unit connected component Aff (V)^ of Aff (V) is an algebraic Lie group, the isotropy subgroup Aff (y)^ n Iso(F) is a maximal subgroup of Aff (F)^, and there exists a unique triangular subgroup T{V) up to the conjugation of Aff ( y ) ^ such that Aff (V) = T{V) • (Aff (V)^ n Iso (V)) is a semidirect product decomposition. These results can be extended easily to a homogeneous Siegel domain.

3.

Normal J Lie Algebras

The conception of normal J Lie algebras was introduced by PiatetskiShapiro in 1962. It is very important because there exists a holomorphic automorphism subgroup T{D{V,F)) acting simply transitively on a homogeneous Siegel domain D{V,F), where the Lie algebra t{D{V^F)) of T{D{V^F)) is a normal J Lie algebra. In this section, we will introduce Piatetski-Shapiro's result of a normal J Lie algebra. 2.8 A real Lie algebra 0 of dimension 2n is called a normal J Lie algebra, if there exists a linear transformation J and a linear function ip, such that DEFINITION

66

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(1) all eigenvalues of the linear transformation a d X are real numbers ^ where X e (5; (2)

j2 = -id;

(3)

[J,ad JX] = J [ J , a d X ] ,

(4)

^([JX,jy])=:t/;([X,y]),

(5)

V^([JX,X])>0,

VX G 0 ; VX,yG0;

V O ^ X G 0 .

^ normal J Lie algebra 0 is denoted by (0; J, V^). 2.14 Suppose that D{V^F) is a homogeneous Siegel domain in C^ X C ^ . Denote by aff (D(V,F)) the Lie algebra of the affine automorphism group Aff (D(V,F)) on D{V,F). Then aff (£)(F,F)) has a subalgebra direct sum decomposition

THEOREM

aff {D{V, F)) = {isop{D{V, F)) n aff {D{V, F))) + 0 .

(2.46)

Moreover^ there exists a linear transformation J and a linear function ijj, such that ( 0 ; 7,-0) is a normal J Lie algebra. Proof Denote by aut {D{y, F)) the Lie algebra of the holomorphic automorphism group Aut {D{V,F)). By Theorem 2.2, there exists a hnear transformation J and a hnear function '0, such that (2.25), (2.26) hold, and (aut (I?(V, F)), isOp(Z)(V, F)); J, xj)) is an effective proper J Lie algebra. By Theorem 2.13, there exists a real subalgebra 0 of aff (I?(V, F)), such that a d 0 acting on aff (D(V,F)) is a real triangular Lie algebra, and aut (£)(V,F)) has a subspace direct sum decomposition: aut {D{V, F)) = 0 + isOp{D{V, F)),

0 n isOp(D(y, F)) = 0.

(2.47)

Clearly, there exists a linear transformation J, such that J ( 0 ) = 0 . Therefore (0; J, V^) is a normal J subalgebra. I Now, we give a detail discussion of a normal J Lie algebra. Clearly, any J invariant subalgebra (ideal) of a normal J Lie algebra is also a normal J Lie algebra. By (4) and (5) in Definition 2.8, there is an inner product {x,y)=^{[Jx,y\\

^x,y€<3,

(2.48)

iJx,Jy)

\/x,yG
(2.49)

such that = ix,y),

Homogeneous Siegel Domains

67

Let (0; JJV?) be a normal J Lie algebra of dimension 2p, Given an orthonormal basis ai,'"^a2p in 6 , then (ai^aj) = 6ij, I < i,j < 2p. 2p

2p

Hence (a,/?) = a;yVwhere a = ^ x^ai, (3 = J2 Vi^i^ ^ = (^i^ * * * 5^2p), y = (yi? • • • 5 2/2^)- Namely, the inner product ( a , ^ ) has a matrix representation E^ where E is the 2p x 2p unit matrix. Clearly, condition (2.49) gives that J J' = E, and condition (2) in Definition 2.8 imply that J^ = —E. Thus there exists an orthogonal matrix O, such that

Since an orthogonal transformation change an orthonormal basis to an orthonormal basis, there exists an orthonormal basis {eir--,ep,/i,---,/p}

(2.50)

in (S, such that the matrix representation of J is

Hence J{ei) = fu

J{fi) = -ei,

l
(2.52)

If J has the same matrix representation as (2.51) on two orthonormal basis, and the basis transformation gives a matrix representation P , then PP' = E,

PJ = JP,

(2.53)

Given the block partitions of the 2p x 2p matrix P as J, then

where AA' + P P ' = P ,

AB' = BA!,

(2.55)

Hence U = A + V ^ P E U (p),

(2.56)

where U {p) is the unitary group. It follows that given arbitrary unit vector a i , there is an orthonormal basis a i , 0^2? • • * ? OL2p in 0 . 2.15 (Piatetski-Shapiro) A normal J Lie algebra ^ is a solvable Lie algebra with the trivial center^ and there is an ideal of dimension one of (3.

LEMMA

68

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Proof By (5) in Definition 2.8, the center C{&) = 0. Hence (S is isomorphic to ad ((5). By Lemma 2.11, ad(J5 is a hnear solvable Lie algebra. Hence the first statement holds. By Lemma 2.11, (1) in Definition 2.8 implies that a d ( 0 ) is a real triangular Lie group. Thus there exists a common real eigenvector a of ad (0) with a real eigenvalue. Namely, [X,a]=b{X)a,

VXG©,

where b{X) is a real linear function on (3. Denote by 9^i the subspace of 0 with a basis only a. Then JHi is an ideal of dimension one. I Given a unit vector gi in an ideal IHi of dimension one, there exists an orthonormal basis 91^92, '",9pi fij2,'"

Jp

(2.57)

i = lr",P'

(2.58)

in 0 , such that J9i = fi,

Jfi = -9u

Now, {9i^9j) = ifiJj)

= Sij^

{9iJj) = 0,

l

we have Wu9j])

= Sij,

n9u9j])

= i'i[fiJj])

= 0,

l
(2.59)

and [fi,9i] = M9i,

lfi,9i] = [9i,9i] = 0,

2
(2.60)

Hence Xiipigi) = ^P{[fi,9i]) = V'([^5i,5i]) = i9i,9i) = 1 implies Ai=i/'(5i)-'7^0.

(2.61)

Now, we use the matrix representation in the following. Given a,/? G 0 , then the matrix representations / a n \

•••

CL2p,\ ' ' '

ai,2p \

/

CL2p,2p J

ftii \ hp,l

••• '''

h,2p b2p,2p

of two linear transformations ad {a) and ad (/?) are defined by 2p

ad {a){ai) = [a, ai] = ^

aijaj,

i = 1, • • •, 2p,

3=1

2p

ad (/?)(a^) = [/?, OLi] = ^ 3=1

bijaj,

z = 1, • • •, 2p.

69

Homogeneous Si^el Domains 2p

Given a vector ^ = J^ Xjaj G ©, then the coordinates of ^ is a; = (a;i, X2, • • •, X2p)- Denoting hy y = {yi, y2, • • •, j/2p) the coordinates of the vector a d ( a ) ^ € (S, we have the matrix relation y = xAa. By the direct computation, we have

Denote by Ag^ and Aj^, the matrix representations of linear transformations ad(pi) and a d ( / i ) , respectively. Clearly, -Ai 0 0 0 0

o(p) •Agi —

0

^(P) C

Vi

B D

Hence [91,91 + V^fi]

= -V^Xi9i,

[51, 9i + v ^ / i ] = 0,

2
By (3) in Definition 2.8, we have (ad(/i) + J o a d ( / i ) o J ) / ^ = 0 ,

2<j
thus (

( \\

0

0

0

An =

\\^

0 0 r?' Bi \

-B,)

/

0

\

0

\ - r A,) j

where ^,»? are 1 x (p — 1) matrices, A\, B\ are (p — 1) x (p — 1) matrices. By [/i,5i] = Ai5i, we have Aiad (51) = ad (/i)ad (51) - ad (5i)ad (/i). Thus \Ag^ — ^31 .A/j — Af^ Ag^ . By a direct computation, we have ^ = 0, ?? = 0. Hence /l^Ai 0

0 •^ffi —

.^/i

0

=

0

0 Ai

0

0

0 0

VI 0 -Bi

0 ^^^^

0 Bx 0 Ai;/ (2.62)

70

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Denote by SDt a J invariant subspace with respect to a basis {g2r " JQP^ /2) • • •) /p }• The expression of the matrix Af^ gives (ad (/i))(9K) C 9Jl, and the matrix representation of ad (/i) on 97t is

Thus Ceigi) = {x€(&\ [x,gi] = 0} = IHi + a n by (2.60). Since the Jacobi identity and J o ad (/i) = ad (/i) o J holds on 9?l, we have ^((exp (
(2.64)

Hence _d2

-^((exp (tad (/i)))a;, (exp (tad {h)))y) dt d = Ai-((exp(tad(/i)))a;,(exp(tad(/i)))y), at

Vt G R,

x,y

^m.

The matrix form is ^(e*^e*^') = Ax^(e*^e*^'). Hence + 2BB' + (B')^ = Ai(5 + B')Denoting C\ = Ai + yf—lBi, we have B2

Ai(Ci + C'l) = Cl + 2CiCr' + (CT? = (Ci + CT? + [Ci, Ci + CT']. Since Ci + Ci is a Hermitian matrix, there exists a unitary matrix C/, such that u\Ci + Cr')C/ = diag {piEi, • •., p,£;,),

Homogeneous Siegel Domains

71

where pi> P2> " - > Ps and Ei^-" ^Eg are unit matrices. Thus XiU\Ci + C~i)U = {u\Ci

+C'i)Uf

+

u\Ci,Ci+C'i]U.

Comparing diagonal elements in above matrix formula, we have Xipi = pf. Thus Pi = 0 or Pi = Xi. Hence s < 2. It follows that u\Ci

+ Ci)U

-

XiE 0

0 0

Therefore there exists an orthonormal basis of (8, such that (Ai/2)£ 0

Ci =

0 Ki 0 ) + ' \ 0

0 K2

where Ki and K2 are skew-Hermitian matrices. Denote by Ki = Kz + v ^ 5 i ,

K2 = K4 +

V^S2,

we have ( ( {Xi/2)Ei+K3 0 B = Si 0

V

0 ^4

Si 0 0 52 {Xi/2)Ei + K3

0

0

-KA

V 0 52

) J

By (1) of Definition 2.8, K3 = Si = 0, K4 ^ S2 = 0, B ' = B, and there exists an orthonormal basis {gi,- • • ,gp,fi,- • • ,fp} oi<&, such that Ai = V(5i)"^ 7^ 0, and (2.65) A/, -diag((Ai,(Ai/2)E,0),(0,(Ai/2)E,0)).

(2.66)

And (2.63) induces that a{x,y) = (exp(-ai))x(Se*^e*^' +e*^e*^'BOy' = 2xBy'. Hence the projection of [/2,gi] on 9li is Aipi, 2 • • • ) /p)>

(2.67)

72

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where (^i, • • •, in) denotes the vector space, which is generated by £i, • • •, 4 - Then [(&l,m] C M = OJti + (SJ and [/i, [Jx,y]] + [y, [fuJx]] + [Jx, [y,/i]] = 0,

\/x,y€

Since the matrix representation of ad (/i) on 9Jli is [/i, [Jx,y]] = -{\i/2)[y,Jx]

+ (Ai/2)[Jar,y] =

Tli.

{l/2)XiE, Xi[Jx,y].

Now, (2.62) imphes that [fi,m{x,y)] = Xim{x,y), where m{x,y) e 9JI and the matrix representations of a d ( / i ) on SWi and ©^ are {l/2)XiE and 0, respectively. Hence m{x, y) = 0. Therefore [Jx, y] = a{x, y)gi = {xy')gu

Vx, y e 97li.

(2.68)

2.16 Let (©; J, V') &e o normal J Lie algebra. Then there exists an orthonormal basis

LEMMA

{ ^i)---»5p)/i>••-,/? }

in ©, such that Jgi = fi,

J ft = -9i,

1 9i] = Ai5i, [h,gi] = (l/2)Ai5i,

[gi, 5i] = [31, fi] = 0, [/i,/i] = (l/2)Ai/,,

bt> ifj] = [/i. fj] = 0>

bi, /j] =

tPigi) = X-\

V(5j=V'(/i) = 0,

-SijXigi,

(2.69)

< q and

[gi.Qj] = [giJj] = [fu9j] = [fiJj]=0^

q + l<J
(2.70)

Let ^ 1 = (51),

3Jti =

{92r",9qj2r"Jq),

(2.71)

Then 01 = $Hi + J9li + 9Jli

(2.72)

is a J ideal with a unique ideal 9^i of dimension one. ©i Z5 called an elementary J Lie algebra. Moreover^ (&l is a J subalgebra. Proof It suffices to prove that (1) subalgebra.

©i is a J ideal; (2)

©i is a J

Homogeneous Siegel Domains

73

Now, 3Jl = 97li + (SJ is a direct sum decomposition of J invariant subspaces. Using [0i,9!Jl] C 9Jl as above, then [/ij^,y]] = [[fux],y] + [x,[/i,y]] =

[[fux],y],

where a: G OJli + ©i = 9H, y G 0 | . When X e 9Jli, we have [fi,x] = (l/2)a;. Thus [/i, [x,y]] = {l/2)[x,y]. Hence [x,y] e dJli. Therefore [2^1,61] C aJli; when x G 6 1 , we have [fux] = 0. Thus [/i, [x,y]] = 0. Hence [a;,y] G ©J. Therefore [(5i,(5i] C

ei

I By Lemma 2.16, we have

2.17 (Piatetski-Shapiro decomposition) Let {<&\J^i})) he a normal J Lie algebra. Then there exist some elementary J Lie suhalgebras ©1, • • • 5 ®isr of 0 , such that

THEOREM

0 = 01 + . . . + (5^

(2.73)

is a direct sum decomposition of elementary J subalgebras, and (3. = ^ + j ^ + mu

l
(2.74)

are the direct sum decomposition of subspaces, where (i) (lHi,JfHi) = (9li,ani) = (J5H,,OTi) = 0; (ii) {<Si,(dj) = 0,i^j; (iii) 9^ is the unique ideal of dimension one of 6^; (iv) 01 + • • • + ©i is a J ideal of 0 ; (v) 0* = ©i-f-i H h 0iv is a J subalgebra of 0 with an ideal 9ii-|-i of dimension one; (vi) [&*,9U] = [(S*,Jmi] = 0, Note

4.

[(&*,mi]cdJti,

l
(2.75)

All results in this section is belong to Piatetski-Shapiro.

J Basis of a Normal J Lie Algebra

In order to realize the homogeneous Siegel domains, we will give the detailed discussion of the Piatetski-Shapiro decomposition of a regular J Lie algebra. We will introduce a J basis and a formula of a J basis transformation of a normal J Lie algebra. Moreover we will give the J isomorphisms acting from a normal J Lie algebra onto a normal J Lie algebra.

74

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS We usually denote a real matrix set

{4J,

t = l,...,n,,-,

l
(2.76)

where Aj^ is an Uik x rijk matrix, t = 1, • • •, riij, l
ep G R^^^

e, G R^^^

(2.77)

where the matrices ep and Cq are defined by (1.70). A complex matrix set {gg\

t=lr-,nij,

l
(2.78)

where Q\J is an m^ x rrij complex matrix, t = 1, • • •, n^j, I
epGR^S

e,GR^^

(2.79)

where the matrices Cp and Cq are defined by (1.70). Given some non-negative integers riij, I < i^j < N] rrii, 1 < i < AT, where riij = riji and nu = 1. Let n =

^

nij,

iv m = ^mi.

l
(2.80)

i=l

The coordinates of a point 2; G C^ are arranged in the form Z = {Sl, Z2, 52, • • • , ^iV, Sjv),

^j = {Zlj, • • • , ^ j - l , j ) ?

(2-81)

where sj€C,

^i, = ( 4 ; \ - . . , 4 * ^ ) ) 6 C " - .

(2.82)

Let (1) ejj = z, where Sj = 1 and all other coordinates are equal to zero; (2) e\j = z, where z|- — 1, and all other coordinates are equal to zero. The coordinates of a point u e.C^ are arranged in the form u = iui,U2,---,UN), Let

uj = {uf,---,uf'^),

«,€C"^^-.

(2.83)

Homogeneous Siegel Domains

75

(3) Cj ^ = IX, where Uj=l and all other coordinates are equal to zero. If a cone V contains the point VQ^ where N

vo = J2ejj = {lAh'"Al).

(2.84)

i=i

we always put the fixed point {z,u) = (\/=4t;o,0) G C^ X C ^

(2.85)

in a Siegel domain D{VjF). DEFINITION 2.9 Let N be a positive integer. Let rijk, I < j < k < N + 1^ are the non-negative integers. Let Aj^ be an riik x %*fc matrix^ where t = 1, • • •, riij, l
an Uik X rijk real matrix; when k = N + 1^ A^j "^ is an rii^N+i x ^j,iv+i complex matrix. The matrix set { Ag,

t = 1, • • '.riij,

l
+ l}

(2.86)

is called a normal matrix set of type N, if the matrix set (2.86) satisfies four conditions as follows: (1)

//

riik = 0,

then

(2)

(A^yAt}

(3)

A^j Ajp = 2_^\^rA-eg) Aj^p 5 t = 1, • • •, riijf,

+ {Alfy4^

rtijTijk = 0,

l < i < j < f c < i V + l;

= 26st^^^^^

s,t =

lr-.nij;

nip

5 = 1, • • •, rijp]

r=l ^JP

(4)

(4J)M|* = j ; ( e « 4 e ; ) ^ g ,

t = l,---,nij,

s = l,---,n,p,

r=l

(2.87) where l
2.10

6 = { 4J,

Given a normal matrix set 6 of type N^ where l
Uij,

l

+ l}.

(2.88)

Denote by and rui x THJ complex matrices 4^^+i = QW,

l
l
(2.89)

76

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS Let {(3;J^ip) be a normal J Lie algebra, A basis 9jku

fjkt = JQjku

1 < * < njk,

l<j
+ l

of the normal J Lie algebra 0 is called a J basis, if the multiplication table is

fe,a]=0,

{^j,^k] = V^Sjk{^j

+ ^j),

(2.91)

where 1 < j , fc < N;

(ii)

(iii)

when I < q = j < N,

when I <j < q < N,

Lsj5SgrsJ — " , Lsi5SgrsJ ~ U, [sjpti^qrsl = ~' Opq / ^{^v^ig^sJ^jrvj V

IsjptJ SgrsJ = Opr J^^K^t^jq^sJ^jqv

~ ^pr /

V

^{^t^jq^sJsjqv V

V

(iv)

when I
LsjjsgrsJ =

y—^sgrs?

LSipt)SgrsJ = Ojr /

Isj^sgrsJ = ^V '~^OjrC,qrs

T^^sgrs?

^{^v^qi^tJ^qFVy V

LSipt? CgrsJ = <^jr / ^(^t;^Q^€t)Cgpt; + Opr 2^\^sA^j^tKqjv i;

(2.94)

V

V

where ^j = 9j + V^fj,

l<j
ijpt = 9ipt + y/~-^fjpt,

1 < * < Wjp,

l < j < p < i V ' + l. (2.95)

Homogeneous Siegel Domains

77

Moreover, there exist N real constants Ai, • • •, A^v 7^ 0; such that H9j) = ^ ,

W ) = 0,

i^iOjpt) = ^ifjpt) = 0,

l<j
l
^2^^^ l<j
+ l,

where

9jkt = -kdjku JQJ = fj^

fjkt = -hfjku

Jfj = -9j,

l
Jgjkt = fjkt^

l<j
+ l,

J fjkt = '^djkt (2.97)

is an orthonormal basis in the normal J Lie algebra (0; J,'0). THEOREM

2.18 Let {0;J^jp) be a normal J Lie algebra. Let 0 = 01 + . . . + 0 ^ ,

0 . = 9^ + jfH^ + ajt-

(2.98)

be a Piatetski-Shapiro decomposition of 0 , where Tli has a subspace direct sum decomposition

an, = 9Ji,,,+i + ... + ani,iv+i,

(2.99)

i = 1, • • •, AT. Then there exists a J basis of 0 , which consists of a basis Qi in 9li, i = 1, • • •, AT; a basis fi in J 9 ^ ; i = 1, • • •, AT; a basis 9ijl^9ij2^ * * * J 9ijnij 5 /ijl? Jij2^ * ' * 5 fijuij 5

Jj ^ *J9j^

Jijt — ^9ijt

(2.100) indJlij, l
V2

V^(Pi) = ^ ,

J{fi) = -9u

J{9ijt) = fiju

J{fijt) = -9iju

^ ( / j ) = 0,

^Pifijt) = 0, nij) = u, ^l^{gijt) n9ijt) == niijt) = u,

(2.101) where t = 1, • • •, riij, l
78

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Now, we prove that a normal J Lie algebra ((S; J, I/J) has a J basis by the induction for N. When iV = 1, (S is an elementary J Lie algebra (S of dimension 2p. By Lemma 2.16, there exists an orthonormal basis {g[^ • • •, g^, / { , • • •, /p } in C5, such that the multiplication table is [fi,9'i] = M9[,

[9'i,9'i\ = [9'iJl] = 0,

[fi,9'i\ = {l/2)\ig'i,

[fi,fl] =

{m\ifi

bUj] =

-SijXi9[,

[9'i,9'j\ = [fiJj] = 0, where J9l = fi

Jfl = -9'i,

l
Let 5i = n ^ P i ,

fi = j^fi,

9oi = —9i,

foi = ^fi^

2<^
The multiphcation table of the basis 5^1) fl'o2. •••)5op> /i) /o2» •••) /op

(2.102)

IS

f [/i,Pi] = V2gi,

[9i,9oi] = [9iJoi] = 0'

[fu9oi] = ^9oi,

[fuU

I [Poi' PoJ = [/oi > /oj ] = 0>

where 2
= ^/oi.

(2.103)

[flot) /oi] = -"^ij ^ 5 1 ,

J9oi = /oi.

-^/o* = -9oi,

^
Hence there exists a J basis in an elementary J Lie algebra. In this case, ^(pi) = V2\-^ > 0, ^ ( / i ) = 0, i^igj = V'(/oi) = 0, 2 /oi) = 2/A^. By Theorem 2.17, © has a normal J subalgebra (5* = (52 + ©3 + • • • + ©iv,

(2.104)

where 6 = ©i + ©^. When iV > 1, ©1 7^ 0. Give a J basis of the elementary J Lie algebra ©1 = 0^1 + JlHi + ajti as the case iV = 1 (see (2.102), (2.103)). By Theorem 2.17,

[dKi,<&i] = o, [mi,(&i] = o, [©t,ani]caTti.

(2.105)

Homogeneous Siegel Domains

79

In order to extend a J basis in 0 = ©i + ©J from a J basis in the normal J Lie algebra ©J, we need consider the matrix representation of adsrrii(®i) as follows. By inductive assumption, there exists a J basis in the normal J Lie algebra 0* = 02 + • • • + ^N' {9iJh

2
+ l}, (2.106) where 0 j is an elementary J Lie algebra, 2 < j < N. A multiplication table of this J basis is defined by (2.90)—(2.95), where 2 < j,q < N and {^*J, IS

+ l, giktJiku

l
l
2
2
+ l}

(2.107)

a normal matrix set of type N - 1. When k < N, Ajj are all real

matrices; whenfc= AT + 1, ^*'. "*" are all complex matrices. On the other hand, we give a J basis (2.102) of the elementary J Lie algebra 0 i of dimension 2p as the case N = 1, The multiplication table of this J basis is (2.103). Now, JQQ. = /o^., Jf^. = -g^., 2<j
and the skew-symmetric bilinear function ip{[x^y]) on the subspace 9Jli has the matrix representation rl;{[x,y]) = fixJy',

(2.109)

where // is a positive constant. If a basis transformation acts from the J basis (2.102), such that the linear transformation J and the bi-linear function '^([x, y]) has same matrix representations, then the matrix representation of this basis transformation is

( J ^ ^ W

U = P + V^QeU{p-l),

(2.110)

where U (p — 1) is a unitary group. Given x,Jx € ©i, let Ax and Ajx be matrix representations of adgni(a;) and adtOTi(Ja;), respectively. By the relations [9Jti,9Jli] C 9^i and 0 = [x,[y,z]] = [[x,y],z] + [y,[x,z]], ^y,z G OJli, Va; G (5*, then ^{[{aAx)y,z])+i){[y,{3Ax)z\) = 0, Vy,2 G OJli, Va; G (5*. Thus A^J + JMX = 0,

A j ^ J + JA'jx = 0.

(2.111)

80

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Given y G SPti, then [x, y] + J[Jx, y] + J[x, Jy] = [Jx, Jy] implies A^ + Aj^J + J{A^)J = JAj^.

(2.112)

Hence if we give the partitions of matrices Ax and Ajx by the first p rows and columns, we have

A -C-C'-B

B \ -A' ) '

_ f

C

^-^"^'[A

D

+ A'-D

-C

(2.113) where 5 , D are real symmetric matrices, and

K+^ij:o - y ^/zi(^ + E')-s

-E' ) '

^^-^^^^

where S = B+\f^D is a complex symmetric matrix and L = A+y/^C is a complex matrix. Clearly, all conditions of matrices Ax and Ajx are invariant under basis transformation matrix (2.110). Thus y/^{L P

L + L')-S

S -L'

Q\f

L

S

\(P'

-Q P J I VrT(L + L ' ) - 5 -L' )\Q'

-Q'

P' (2.115)

where U = P + V ^ Q € U (p - 1),

{

L = ULP' + yf^QL'U' + PSQ! - QSP', S = PSP' + QSQ' - ULQ' - QL'U'.

(2.116)

We will consider the case of a; G 02- Denote by

'^32 = y_C-C'-B

-A' ) ' ^^^^\A

+ A'-D

-C ) '

(2.117) where B, D are real symmetric matrices. Let L = A + ^/^C. L + L' is a complex symmetric matrix, and there is a unitary matrix U, such that L + L' = U{L + L')U' = 2A = diag (2Ao, 0), where Ao = diag(pi£;(^i),---,p«£;(^»)),

pi>p2>--->P«>0.

(2.118)

Thus there is an orthonormal basis in 9Jli, such that L + L' = 2A. Hence ^ + ^ ' = 2A, C + C" = 0. Therefore k +K ^92-[

_Q

B

\

_A + i^ ) '

,

(

C

^/2 - I A - L>

A+D C

Homogeneous Siegel Domains

81

where B and D are real symmetric matrices, K and C are real skewsymmetric matrices. Now [ad (/2), ad (^2)] = \/2ad(p2) implies [^^25^/2] — V^^^2- Hence [K,A]+AD

+ DA = 0, ^ [K, C] + [D, B] = V2K, Clearly, (2.119) implies TA + AT' = - V ^ A ,

[A,C]+AB

+ BA = V2A, 2.119

[B, C] + [K, D] = V2B-

[T.f]

2A2.

- -V2(r + r') - 4 A ^

(2.120)

where T = —B + C + y/^{D + K) is a complex matrix. Given the block partitions of T as A = diag (AQ, 0), then TA + AT' — - \ / 2 A imphes

and [T, T'] = - V 2 ( r + T') - 4A2 implies [Til, l b ' ] + Tnlh'

= -V2{Tn

+ Wi) - 4A§.

Hence s

i=l

Now, tr (T + T')[T,T']=0 gives tr (T + r ' f = - i = t r ([T, T'] + 4A2)(r + r ' ) = 4 ^

4pf.

Thus t r ( r + r')2

=tr \

_ ,

_ ,

^12

T22 + T22

)

= tr (Tn + 7b')2 + 2tr r i 2 T b ' + tr (T22 + Th'?^ It follows that Since Tn + Tn is a Hermitian matrix, and tr (Tn + Tn )^ is a square sum of the module of all elements in Tn +T11 , and all diagonal elements of matrix Tn + Tn are equal to — \/2, we have s

82

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Thus T22 + T22 = 0, T12 = 0 and Tn + Tn = -\/2E. - ^ E + Ko V2 0

Hence

0 Ki

where KQ and Ki are skew-Hermitian matrices. By (2.120), KQA

+ AK'Q = 0,

V2{T + T') - - 4 A 2 .

Thus Ao = E, KO + KQ = 0. Hence KQ is a real skew-symmetric matrix. Therefore / A

- -—

V

E 0 0 0 -E 0 0 0

E 0 ( -E \^ 0

0 0 0 0

\

n

K2 0 E 0 0 5o 0 X3 1/2 E 0 K2 0 72 V 0 - 5 o ) ( 0 Ks J J By (1) in Definition 2.8, ^l/j ^ ^ only the real eigenvalues, therefore K2 = 0,K3 = 0, 5o = 0. Given a basis transformation matrix

U = P + V^Q = diag(^ J- ^E,E) as (2.110) acting on 9Jli, there is a J basis in 9Jli, such that (2.121)

—pE

0

0

0

(2.122)

^h = E

V

0

0 0

where E is the ni2 x ni2 unit matrix. Now, [52, ® ^ ] = 0 . [/2, (5^]=0.

(2.123)

83

Homogeneous Siegel Domains Given a; e 63 = ^ 3 + • • • + ©iv, then

by (2.123), where c+^/-iJx

^^{L

L + L')-S

S -L'

(2.124)

Since all eigenvalues of matrices Ax and Ajx are real numbers, so (2.125)

L =

By the discussion above, there is a J invariant subspace decomposition ofOTli:

9Jii = ani2 + a[ni3 + --- + 9Jii,jv+i.

(2.126)

Denoting by dimSDTij = 2nij, 2 < j < iV + 1, there is an orthogonal basis 5iji, • • • ,9ijnij,fiji, • • •,/ijmj, 2 < j < iV + 1 in aJty. According to the same block partitions by ni2, • • •, ni,iv+i, '^12) • • •»wi,iv+i rows and columns on matrices Ag^, and A/^, we have

--^(ro) ^^^ = 7i

-Ai 0

(2.127)

0 Ai

(2.128)

where Ai = diag(0,---,0, £;("!*), 0,---,0),

2 = 2,3,---,Ar.

Now, we can determine the complex matrix A^^.^^ = Agj^^ + as follows, where

jpt

B jpt

ijpt ^^iPt

1%

(2.129)

=

-Bjpt + yf^{Ajpt

+ A'^pt)

-Ajpt

( C

^jpt

^jpt

^jpt

^jpt

^jpt

Ajpt —

\

^jpt

{N+1,N+1) jpt

\

84

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and Qjpt is an nu x nij complex matrix, 2 < z < j < i V + l. By (2.90)—(2.95), ^1^.^,, l
and when 2 < j < p < N,

Hence Bjpt = y/^{Ajpt + Aj^^) and when 2 <j

when^T^ j , p , AjptAq = AqAjpt,

(2)

Aq{Ajpt + A'jp^) = 0 ,

2
2<j
Aj [Ajpt + Ajp^) = Ajpt, Ap[Ajpt + Ajp^) = Ajp^,

By a direct computation, QjJ =0, s ^ j , t ^ p. Let Qjpt = Qjpt be an riij x nip matrix. Then Ajpt

^^^{Ajpt

+ A'^pt)

^^i?jpt =

0

(2.130)

-^iP*

where /O2

Uijpt

(2.131)

ijpt

0„

V

0^+1 /

where Qjpt, 2 < j < p < N, are all real matrices, and Qj,N+i,t, 2 < j < N, are all complex matrices.

Homogeneous Siegel Domains

85

Let A'i = '£Wte'set

(2.132)

t

be an riij x riij matrix, where A^, 1 < i < j < AT, are real matrices and Al\ "^ are complex matrices. Then Qjpt = Y.<et^'.

(2.133)

S

When 1 < j < AT, we have V

l^jpU ^Irs] = Sjr Y^{eyAl^jet)^ipy

+ 2 v ^ V

V

X^(^s(^lj)et)/ljt;. V

Therefore when iV = 2 and iV = 3, multipHcation tables (2.90)—(2.95) hold. Now,

[A^^r.^,A^] = ( ^^^^'f'^'^ -[A^J:^^]' ) ' where

,w..«= - X/^(A.^.'AZ:+2A,„, qrs + AqrsAjpi

+ ZAqrsAjpf

x qrs z ' ++ AL^AZTJ Ajp^Aqrs + .Agrs

Ajpf).

(2.135)

By (2.93), we have Hence SpriQjptQjrs

+ QjrsQjpt)

=

2SprSstE^^^P\

Therefore A^^Afj + I g ' ^ ^ . = 25,t^(^^>). By the multiplication table (2.93) and (2.94), when V

(2.136) 2
(^-^^^^

86

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

By (2.134) and (2.135), when q < j ,

V

V

V

and ^jpt,qrs ^^

=

V

^yAjptAqj-s ~r ZA-jpfA-grs

+ ^qrs-^jpt

+ ^^qrs^jpt

— 6jrV—l

/ A^v^ai^tji^qpv

\ A.jp^/iq'ps

+ ^qrs +

^jpt) ^qpv)

V

— dprV —1 2_^\^sAqj^t)\^qjv

+ ^qjv + ^qjv + ^qjv J-

V

Now, 2 < q < j < N, then Aqjs is a real matrix. By a direct computation, when q < j , then AjptAqrs = 0; when r ^ j , then AqrsAjpt = 0; when p ^ r, then AjptA'q^^ = 0 and AqrsA!^^^ — 0. Hence AgjgAjpt = /

^[eyA^^e^jAqpy, V

AqjsAjpt

= /

^[eyA^^e^jAqpy^ V

^jpt^qjs

+ AqjsAjpt

= 2^{^vAqjef)[Aqpy

Z^jp^Agps + zAqpsAjp^

= 2^\^sAq^ei){Aqj'i}

+ A^^^), + Aqjy + Aqjv + Aqjy ) .

V

The above formulas are equivalent to QqjsQjpt

= / ^\fivAfjje^)Qqpv^ V

QqpsQjpt — /

QqjsQjpt

~ /

^{^yA^^efjQqpy^ V

^[^sA^^e^jQqjy^

(2.137) where Qqjy is a real matrix. Hence (2.137) is equivalent to

Homogeneous Siegel Domains

87

Using Yl^e!f.er = E, (2.137) is equivalent to

K^7i = E(^-^i^^)^i'' ^'^IT = E('-'«K'V

(2-138)

V

Therefore { Afj^, I < s < n^j, 1 < i < j < k < N + 1} is SL normal matrix set with rank N. By Definition 2.10 and the induction^ there exists a J basis in a regular Lie algebra. I By Theorem 2.18, the structure constants of a J basis in a normal J Lie algebra are determined by a normal matrix set of type N. Clearly, the classification of normal J Lie algebras up to a J isomorphism can be induced by the classification of J basis in a normal J Lie algebra, because a J isomorphism transfers a J basis to a J basis. Hence we will study the uniqueness of the decomposition in Theorem 2.18. Let --pAi, • • •, -J=XN be the dual basis of / i , • • •, /N on the linear v2 v2 space 2:=(/ir..,/iv). Denote by Ao = {Xi,l
{^^^,l
(2.139)

the weight systems for the commutative Lie algebra ad {%). By the definition of the J basis, we have

[fj^^jrs] = -jK^JTsi [fji^qrs] = 0, [fjj^qrs]

=

^<3
l<s
l < j < g < r < i V + l,

1 < 5 < rigr,

J^^jriqrs,

l

1<S< Uqr.

By a direct computation, we have LEMMA

(1)

2.19 Some symbols are the same as above. The subspace T=(/i,...,/iv)

is a commutative Lie subalgebra of the normal J Lie algebra (0; J,t/?);

88

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(2) The normal J Lie algebra ( 0 ; J, ip) has the direct sum decomposition of eigenvector subspaces by ad (T); <& = J{9{) + <&i+m,

^ = ^i + E ^ ^ ' AeA

^(^) = V o + E ^ ^ ' xeA'

(2.140)

<5i = E ^^' (2-141) xeA"

where N VI = J2'^X,

= {9I,---,9N),

^A, = (Pi),

J ^ i = q 3 o = (/i,---,/iv), ^^i±^ = (/yi,---,^n,,),

l
l
(2-142)

2

KPxj-Xj = {9iji, • • •, 5ijni,),

l
2

^ ^ = {9hN-\-i,uhN-\-i,u ^
l
2

and qjA = { a: € 0 I [/, x] = A(/)a;, / e X }. LEMMA

2.20

(2.143)

Given a normal J Lie algebra (C5; J,-0); let (5 = (Si + • • • + (Siv,

^j=

^j + J^j + ^j

be a Piatetski-Shapiro decomposition of (5. Given an ideal IH of dimension one of (Sj there exists an index j G { 1,2, • • •, AT}, such that 9i = 9lj and [(5k,(5j]=0, fc = l , 2 , . . . , j - l . Hence rikj = 0, A: = 1,2, • • •, j — 1. Proof We prove firstly that any ideal of dimension one in (5 is equal to one of IHi,- • • ,9^iv. If ISH = iHi, this Lemma holds obviously. Suppose that 5H 7^ 9li. Given 0 ^ X e ^, then X = Y + Z, where Y E &ij Z E 0ihy the direct sum decomposition (J5 = (5i + (5*,

0* = 02 + --- + ©Ar.

If y 7^ 0, we have F = agi+bfi+Y' by ©1 = 9li + J^^^ +9Jti, where gi is a basis of 9li, / i = J51, [/i,gi] = V2gi and a, 6 G R, F ' G 9Jli. Now, [/i,(J5I]=0,thus [/i, X] = [/i,y] - x/2api + - ^ r G ^ .

Homogeneous Siegel Domains

89

On the other handj 91 is an ideal of dimension one, thus [/15X] = AX. Hence Xa = V2a, Xb = 0, XY' = - ^ \ XZ = 0. If A = 0, F ' = 0, a = 0, then Y = bfi ^ 0. Since [guX] G IHi n 9 t - 0 and [gu0i] = 0, we have 0 = [51,X] = b[giji] = -y^bgi. It follows that 6 = 0, Therefore Y = 0. A contradiction follow. If A 7^ 0, Z = 0, we have IH C ©1, iH / JHi. Since only one unique ideal 9li of dimension one exists in 0 i , we find a contradiction again. Hence Y = 0. Therefore d\ C 0 | . For ©2, ©3, • • •, ©AT, respectively as 0 i , there exists an index j satisfying 2 < j < N, such that ?H = 9lj by the induction. The first statement holds. Suppose that 9lj is an ideal of 0 , where I < k < j . Then [0^,91^] C SHfefl^lj = 0. By the multiplication table of a J basis, Ukj =0,1
LEMMA

Proof By the multiplication table of a J basis, if ^a is an ideal of dimension one of 0 , then nia = • • • = Ua-i^a = 0. Hence (i) implies (ii). If (ii) holds, then [0^, 0^] = 0, 1 < fc < a. By the multiphcation table of a J basis, we have [ « a , ©a+1 + • • • + 0 i v ] C 9Jla C 0 a ,

hence 0a is a J ideal. Hence (ii) imphes (iii). If 0a is a J ideal of 0 , we have [^a,©a4-l + • • • + 0iv] n 9 ^ a naJla = 0

by the multiplication table of a J basis. Now, [ « a , 0 1 + • • • + 0 a - l ] C ( m i l + . • . + 9 H a - l ) H 0 a = 0.

Thus JHa is an ideal of dimension one of 0 . Hence (iii) implies (i).

I

2.22 Given a Piatetski-Shapiro decomposition of a normal J Lie algebra {(d;J^il^): LEMMA

0 = 01 + ... + 0 ^ ,

0^. = mj + Jmj + Mj,

90

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

/ / a G { 1,2, • • •, iV }, IHfl is an ideal of dimension one of (&, then (&a is an elementary J subalgebra of 0 , which is determined uniquely by IHo, and there is a new Piatetski-Shapiro decomposition: 0 = (5a + ©1 + • • • + 0 a - l + 0a+l + - ' + ©AT-

(2.144)

Proof The elementary J Lie algebra (Sa is determined uniquely by IHoIn fact, it suffices to prove that [fa,X] = ^X,

X€(5

if and only if X G SPTa. By Lemma 2.21, [(5a,<»l + --- + ( 5 a - l ] = 0 . Let X = Y + Z + W e<S, where F € (5i + • • • + (3a-i,

Ze(5a,

We

(5a+i + (SN.

Then

gives that

Since [/«,Z] e (5a, we have Y = 0^W = 0, Hence X e (5a. Let X = aga + bfa + rria,

Qa € "^a,

fa ^ ^ ^ a ,

^ a ^ ^^a-

Then [fa, X] = aV2ga + -^rria = -7={aga + bfa + rua). Hence a = 0, 6 = 0 and X G 9Jla. The first statement holds. The second statement can be induced by Theorem 2.17 and Lemmas 2.20, 2.21 easily. I DEFINITION 2.11

Let (

S =

n\\

ni2

^21

^22

\ ^iVl nN2

n2N

(2.145)

• • • ^NN 7

be an N X N symmetric matrix^ where riij, 1 < i,j < N, are N'^ nonnegative integers, and riij = Uji. A permutation a = a(l)cr(2) • • • a{n) of

91

Homogeneous Siegel Domains

1, • • •, iV is called an admissible permutation with respect to S, if i < j , a{j) < a(i) implies n^(j)a{i) = 0. Obviously, given an AT x AT symmetric matrix S and an admissible permutation a with respect to 5, we find an N x N symmetric matrix /

^CT(1)<7(1)

^(j(l)a(iV)

^C7(l)a(2)

\

^a(2)c7(iV)

5^ = \ ^a(iV)c7(l)

^a(7V)a(2)

* * '

'f^a{N)a{N)

(2.146) J

2.23 Let (0; 7,-0) fee a normal J Lie algebra. Given two Piatetski-Shapiro decompositions of(S:

THEOREM

(J5 = ©1 + 02 + • • • + 0iv,

(»j = ^j + J^j + 9Jtj,

1 < j < AT,

(5 = 01 + 02 + • • • + 0M,

^ j = ^ j + J^j + Sl^,

1 < j < M, (2.147)

then M = N, and there exists an admissible permutation (j = a(l)cr(2).-.(j(Ar) o/1,2, • • • 5 AT with respect to a symmetric matrix S {see Definition 2.11)^ such that [©a(fc)5®a(j)] = 0, jf < k, a{k) < a{j), and ^3 = ®^0)'

1 < j < A^.

(2.148)

Conversely, given a Piatetski-Shapiro decomposition of a regular Lie algebra {(d^J^ip): 0 = 01 + 02 + . . . + 0 ^ ,

(5j= 9{j + J^j + OJlj.

(2.149)

/ / the permutation a = cr(l)a(2) • • 'a{N) is an admissible permutation with respect to a symmetric matrix S, then <5 = ©a(l) + ^a{2) + '" + ^a{N)

(2.150)

is also a Piatetski-Shapiro decomposition of (3. Proof The second Piatetski-Shapiro decomposition gives that IHi is an ideal of dimension one of 0 . By Lemma 2.22, there exists an index cr(l) e {1,2, •••,Ar}, such that ©i = ®a(i)5 and 0 has a PiatetskiShapiro decomposition: ® = ®a(l) + ©! + - • + (Sa(l)-l + ®a(l)+l + • • • + ©AT.

92

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

The orthogonal complement of ©a(i) = ©i is a normal J subalgebra ©1 + • • • + ^a{l)-l + <5a(l)+l + • • • + ©AT = ®2 + • • • + §M, and [6^(1)5 ®i H H ®o-(i)-i] = 0. Hence when the indices k < (7(1), [®cr(i)5©fc] = 0. Repeating the above process, the first statement holds by the induction. Conversely, if a is an admissible permutation with respect to a symmetric matrix 5, we will prove © = ©a(l) + 0^(2) + • • • + &a{N) is a Piatetski-Shapiro decomposition. In fact, if j < fc, a{k) < a{j), then n^(k)a(j) = 0, and [0^(1)^ ^fc] = 0^ k = 1,2,• •• ,(T(1) — 1. Hence !H^(i) is an ideal of dimension one. By Lemma 2.22, © = 0^(1) + 01 + • • • + 0a(l)-l + ®tr(l)+l + • • • + 0iV is a Piatetski-Shapiro decomposition of 0 . It is easy to prove that 0 = 0a(l) + 0a(2) + • • • + ^a{N) is a Piatetski-Shapiro decomposition by the induction. Hence the second statement holds. I DEFINITION 2.12

Let

e = {Af^,

l
+ l}

(2.151)

6 = {i*J,

l
+ l}

(2.152)

and

be two normal matrix sets oj type N and type N, respectively. & is called unitarily equivalent to &j if there exists an admissible permutation a with respect to a symmetric matrix S {see Definition 2.11), and real orthogonal matrices OjkeOirijk),

l<j
unitary matrices Uj eViruj),

l<j
such that N = N;

(2.153)

Homogeneous Siegel Domains

93

^ij = "
^

(2.154)

^i-E(^«O-«-(i)«t)<^^(iMfc)<«'4)O-0>W'

^^^;

(2-155)

S

^!f""' = E(«^O<^«<^0)e't)f^^'4'S:(>.0)-

(2.156)

Obviously, a unitaxily equivalent is an equivalent relation. By Theorems 2.18 and 2.23, we have THEOREM

Qjjj.

2.24 Let (©; J,t/?) be a normal J Lie algebra. Let

l<j
QjktJjku

l
l<j
+ l (2.157)

be a J basis associated with a normal matrix set of type N: e = { Ag,

l
riij,

l

+ l}.

(2.158)

Give another J basis 9jjj.

l<j
QjktJjku

l
l<j
+ l (2.159)

associated with a normal matrix set of type N: & = {Al},

l
Uij, l

+ l}.

(2.160)

Then 6 and & are unitarily equivalent. Conversely J give a J basis (2.157) of a normal J Lie algebra 0 associated with a normal matrix set (2.158) of the type AT, and a J basis (2.159) of a normal J Lie algebra (S associated with a normal matrix set (2.160) of type N, such that conditions (2.153)—(2.156) hold. Then © is J isomorphic onto 0 . Proof

By Theorem 2.18, denote by

^j = {9j)i J^j = ifj)^ ^ij

= {gijli ' ' '^gijriijjijl,

• • • , fijriij), (2.161)

and (2.162) Then (0; J,'0) has two Piatetski-Shapiro decompositions: 0 = 01 + • • • + ©iv,

&j= ^j + mj

+ Mj,

94

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and ~

_

^

_^

^

J

J

(2.164)

By Theorem 2.23, N = N and there is an admissible permutation a = a{l)a{2)' • • (T{N) with respect to a symmetric matrix S, such that ©, = <5<.0),

i<j
(2.165)

Since an ideal of dimension one is a unique ideal in an elementary J Lie algebra, we have ^^- = 91^^), where ipigj) = i^iQaij)) = V^i 9j = 9a{j) Obviously, iDDTj and 9^j + J^j

are orthogonal to each other, hence

ajt,-9Jl,(,),

l<j
(2.166)

Now, mj - mi,-,+i + . . . + Si,-iv+1,

(2.167)

9Jt.o-) - ^a{o)HJ)+i + • • • + 2JtaO),iv+i.

(2.168)

Hence

E

^-0),i^ = ^-0) = ^^' = E ^^•^-

(2.169)

Since cr = cr(l)or(2) • • • cr{N) is an admissible permutation of 1, 2, • • •, AT, so the condition ^
= 0,

j < i < N,

a{£) <

a{j)

holds. Denote a' = cr(l)(7(2) • • • a{N)a{N + l), where a{N + l) = N + 1. Obviously, the proof of Theorem 2.18 shows that (2.168) is determined by the matrix representations of ad(5^(j)_^i), •••, ad(piv) and ad(/^y)_^i), •••, ad(/iv) on 9H^y); (2.167) is determined by the matrix representations of ad (fl^j+i), • • •, ad (QN) and ad (/j+i), • • •, ad {/N) on 9Jlj. Hence (2.167) is determined by the matrix representations of ad (3^(^+1)), ••-, ad(g^(7v)) and ad(/^(,+i)), •••, ad(/^(^)) on dJlj = m^^jy Now, a ( j ) + t E {(7(l),a(2),...,a(Ar),(7(iV + l ) } , t = 1, -.., iV-hl-cr(j), one part of cr(j) + t lies in {(j(j + l), •••, cr{N), a{N + l)}, and the other part of a(j) + 1 lies in { cr(l), cr(2), • • •, a{j — 1) }. If there is an index a{j) +t E {cr(l), • • •, a{j — 1) }, denoted by cr(j) + t = cr(fc).

Homogeneous Siegel Domains

95

where fc G {1,2, • • •, j - 1}, then we have k < j , a{j) < a{k). Thus ^aOXfc) = ^c7(j),cr(j)+t = 0. Hence 2«aO) =

E

^crU)Mky

(T{j)<(T{k)
(2.170)

j
Now, ad(ge) = ad(5(7(^))) a d ( ^ ) = ad(/a(^)), l<£
rijk = n^(j)a{k).

l<J
have + l,

(2.171)

where a{j) > a{k) imphes that rijk = '^a(j)(T{k) = 0, Mjk = 9^aO>(/b) = 0. Hence _ (5jfci r • • 5 Qjkrijk' /jfci r • •, /jfen^fc) ? (2.172) and {9a{j)(7{k)l^ • • • J 9cT{j)a{k)n^(^j>,^^k^, f(T{j)a(k)^ "^ /^(j)^(fcWo)a(fc))'

(2-173)

are two J bases in the same vector space ^jk

= ^a{jMk).

j
+ l,

such that the matrix representation of J equals to

( - - ) and the matrix representation of the inner product {x, y) = '^{[Jx, y\) is the unit matrix. Denote the basis transformation formula from (2.173) to (2.172) by

~m = E ^^^^'^"^'^^MjMkJs, IM = E «1* ^•^^^'^V.OX.). (2.174) s

s

and

w.^^ s

'

. .NN

(2.175)

s

such that the matrix representations of ad(5fc) are equal to ad(^^(j)), j < k < N, Since (2.172) and (2.173) are orthogonal bases, we have OaUMk) = («i?''^"^'^^) € O (n^oXfc)),

1 < i < fc < AT,

96

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and

Using multiplication tables (2.90)—(2.95), when q < j , LSJpt) 9qrs\ — Ojr / ^[^v{^Qj)^t)^qpv

"I" v

1 ^rp /

V

i^jpti Jqrs\ =

^{^s^QJ^tJJqjV') V

^ ^rp

2_^\^s^qj^t)Jqjv^ V

where

Ijvt = 9jpt + V^fjpu

1 < t < rijp,

l<j
+ l,

and [sjpt^gqrsl = Ojr 2_^{^v^qj^t)^qpv

+ V— ^ Orp /

V

[^jpt5 Jqrs\ =

^{^s^QJ^tJJqjv^ V

~ Orp /^^^^

qj^*^-^^^^'

V

where

^jpt = 9jpt + y/^fjpu

l
rijp,

l<j
+ l.

By a direct computation, we have ^9i =

Yl^^t^^{qMJ)^s)0'^{qMp)^f^^^^ t

l
and ^If'^^ =

^{etO^{qMj)es)U^)A^;^^^^^ t

^ '^ Q < j ^ N. It proves that 6 is unitarily equivalent to 6 . Conversely, the basis transformations 9
fa{j) -^ fj^

j = 1, • • •, iV

and (2.174), (2.175) give a J isomorphism acting from 0 onto (5.

Homogeneous Siegel Domains

97

Using Theorem 2.24, we have 2.25 Let (0; J,!/^) be a normal J Lie algebra associated with a normal matrix set {A*^} of type N. Then the J automorphism group Aut ((5) of 0 consists of all mapping r as follows: LEMMA

r{9i) = 9a{i)^ r{fi) = /^(i),

i = 1, • • •, TV,

r

where a is an admissible permutation of 1,2, •••,7V with respect to a symmetric matrix S = (^ij), and Uij e O {riij.R),

1 < i < j < iV,

Ui^N+i e U {TH^N+I).

l
such that ^if

=

Yl

(^*^<^«^(J)^3)t^a(i),a(fc)'^!;[i)ty)f^a(j),a(fc), K^t^(T{i)(T{j)^sJ^a{i),(7(k

(2.177)

t=l

where 5 = 1,2, • • • n^j, l < i < j < A ; < i V + l. Therefore the classification of normal J Lie algebras up to a J isomorphism can be deduced by the classification of normal matrix sets of type TV up to a unitarily equivalent. Now, we study the property of a decomposable normal J Lie algebra. 2.13 A normal J Lie algebra 0 is said to be decomposable, if 0 has a direct sum decomposition of two J ideals of lower dimension. Otherwise, 0 is said to be indecomposable.

DEFINITION

Clearly, if a normal J Lie algebra (0; J^ip) is a direct sum of two J ideals ^i and ^2 of 0 , where ( ^ ; 7,-0), i = 1,2, are two normal J Lie algebras, then (.^1,^2) = H[J^u^2])

= i^ii^um

= ^(0) = 0.

Hence ^i and A2 are orthogonal to each other. 2.26 Suppose that 0 = ^i<S)^2 is a direct sum decomposition of two J ideals ^1 and ^2 of a normal J Lie algebra (0; 7,-0). Give Piatetski-Shapiro decompositions

THEOREM

^1 = 01 + . . . + 0 , ,

^2 = 0.+1 + • • • + 0iv.

(2.178)

98

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Then 0 = ^1 + ^2 = ®i + 02 + • • • + ®s + ^s+i + '- + &N

(2.179)

is also a Piatetski-Shapiro decomposition of (5. Proof

Since [^1,^2] = 0,

[(Si, (dj]=0,

hence nij = 0, l
l

Clearly, any ideal of ^ is also an l
By the same method as the proof in Theorem 2.18, we have 0 = 01 + • • • + 03 + 05+1 + • • • + 0iv is a Piatetski-Shapiro decomposition.

I

2.27 Let (0; 7,-0) be a normal J Lie algebra. Then 0 is decomposable if and only if there exists an admissible permutation a of 1, • • •, AT with respect to an N x N symmetric matrix

THEOREM

(

nil

•••

niN :

\ ,

(2.180)

riNi ••• riNN / such that the N x N symmetric matrix

(

^a(l)a(l) :

•••

'^a{l)a(N) \ :

^a(iV)(j(l)

••• '^(7{N)a{N) J

(2.181)

is a block diagonal matrix. Proof If (0; J, ip) is decomposable, then 0 = .fti (8).ft2? where {^; J, xp), i = 1,2, are normal J ideals of 0 . By Theorem 2.26, there exists a Piatetski-Shapiro decomposition 0 = 0 i + - - + 0iV, such that ^1 = 01 + • • • + 05,

^2 = 05+1 + • • • + 0iv

Homogeneous Siegel Domains

99

axe two Piatetski-Shapiro decompositions of Ki, ^2? respectively. Since [0q,ej] =0,l
^2 = ^s+i + ®s+2 + • • • + 0iv

are two normal J ideals of ( 0 ; J , ^ ) , and 0 = .Ri ® .^2 is a direct sum decomposition of ideals ^i and .^2- Therefore 0 is decomposable. I Given a normal J Lie algebra ( 0 ; 7,-0) with respect to an iV x TV symmetric matrix S = {nij)j we construct a diagram P ( 0 ) on a plane by N points 1,2, • • •, AT as follows: describe a segment from i to j when riij / 0; do not describe any segment from i to j when Uij = 0. It is easy to show the next theorem by Theorem 2.17. 2,28 Suppose that 0 = 0 i H \-(3N is a Piatetski-Shapiro decomposition of a normal J Lie algebra ( 0 ; J, t/^) with respect to an N X N symmetric matrix S = (riij). Then 0 is indecomposable if and only if the diagram P ( 0 ) is a connected diagram. THEOREM

By Theorems 2.27 and 2.28, we have 2.29 Any normal J Lie algebra (0; J.i/j) has a unique direct sum decomposition of some indecomposable normal J Lie ideals up to the order.

THEOREM

T H E O R E M 2.30 Let Di be a homogeneous bounded domain inC^. Suppose that there exists a connected Lie subgroup Gi of the holomorphic automorphism group Aut (D^) acting simply transitively on Di^ and the Lie algebra 0^ of Gi is a normal J Lie algebra (0^; Ji.i^i), i = 1,2. If ip is a J isomorphism acting from ( 0 i ; J i , ^ i ) onto (02; ^2,'^2); cind Gi is a simply connected Lie group, there exists a holomorphic isomorphism $ acting from Di onto D2, such that $* — ip.

Proof Since G\ is a connected and simply connected Lie group, there exists a Lie group isomorphism # acting from G\ onto G2? such that $ is

100

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

an isometric isomorphism acting from the homogeneous Kahler manifold Gi onto the homogeneous Kahler manifold 02- Since Gi acts simply transitively on A , ^ = 1,2, Di is holomorphically isomorphic to D2- • In Chapter 9, we will prove that any homogeneous bounded domain satisfies the conditions of Theorem 2.30. Using Theorem 2.30, the classification of homogeneous bounded domains up to a holomorphic isomorphism can be deduced to the classification of normal J Lie algebras up to a J isomorphism.

Chapter 3 NORMAL SIEGEL DOMAINS

In this chapter, we will introduce a special class of homogeneous Siegel domains, the so-called normal Siegel domains. We will first prove that any homogeneous Siegel domain is affinely isomorphic to a normal Siegel domain. Namely, we give a realization of homogeneous Siegel domains by normal Siegel domains. Then we will prove that the classification of normal Siegel domains up to an aifine isomorphism is equivalent to the classification of normal matrix sets up to a special unitarily equivalent. Finally, we will give the explicit expression of the Bergman kernel function K{z,u;'z,u) on a normal Siegel domain £)(V^,F) by the transitive affine automorphism group Aff {D{V^,F)), In this book, we use normal Siegel domains to study the geometric property and the function theory, which is not used case by case.

1.

Normal Cones and Normal Siegel Domains of First Kind

Suppose that D{V^F) is a homogeneous Siegel domain in C^ x C"^. By Theorem 2.3, the affine automorphism group Aff {D{V^ F)) acts transitively on a homogeneous Siegel domain D{V^F). The Lie algebra aff (£)(V,F)) of the Lie group Aff (D(T/,F)) has a subspace direct sum decomposition aff {D{V, F)) = L_2 + L-i + LQ (3.1) by Theorem 1.19, where L_2 = { a — I Va € M" },

101

(3.2)

102

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS L_i = { / 3 ^ + 2 v ^ F ( « , / 3 ) ^ I V/3 € C - }, /V

/V

Lo = { ^ ^ ^ + ^ ^ ^ I xAF(^^5,^) + F{u,uB)

(3.3)

/V

e aff (y), S G gl (m, C), = F{u,u)A,Mu

^3 ^^

E C ^ }.

3.1 The direct sum L_2 + L_i 0/ subspaces L_2 and L_i Z5 a triangular nilponent ideal o/aff (£>(V,F)).

LEMMA

Proof Now, [Li.Lj] C Li^j,\/iJ e { 0 , - 1 , - 2 } , where Lk = 0 when fc < - 2 , so [aff (D(F, F)), L_2 + L_i] C L_i + L_2 imphes that L_i + L_2 is an ideal of aff (-D(V, F)). Since [L_2 + L - l , i:-2 + ^-1] C L_2, [i:_2 + L_i, L_2] = 0, L-i + L-2 is a nilponent ideal. Given a basis { a i , • • •, a2m } in L - i and a basis {/3i, • * * 5 /?n } in L_2, we find a basis in L_i + L_2. Any linear transformation in ad (L_i + L-2) can be expressed by an upper triangular matrix with the all diagnoal elements equal to zero, hence L-2 + I/_i is a triangular subalgebra. I LEMMA 3.2 Let aff (D(V,F)) be the Lie algebra of the affine automorphism group Aff (£)(V,F)). Then the subset

{ X e aff (D(y,F)) I {didxf = 0} = L_2. Proof

(3.5)

Clearly, x G I/-2 satisfies (ada:)^ = 0. Conversely, given

such that (ad x)'^ = 0, then 0 = (ada;)2fa-^) = aA^ — ,

Va G E^. pj

Thus a^2 ^ 0, Va 6 R", it follows that A? = 0. Hence xyl— G aff (V), A^ = 0, therefore x -^ xe^^ = x{E + tA),t €R, are the affine automorphisms on V. If A ^ 0, there is a point x^ G V, such that XQA^O. Thus the subset {x = Xo+ t{xoA) I V< G R } C V.

Normal Siegel Domains

103

Hence V contains a whole straight Hne. This is a contradiction by the condition of V. Therefore ^ = 0. Now, uB-;r~ € LQ impUes au F{uB, u) + F{u, uB) = F{% u)A = 0. Since

and au we have

0 = (adx)»(V=l4) = 4F(/?,ffl| +2F(«B,/3)^ - V=T/5B^. Thus F{l3,f3) = 0, it follows that p = 0. Hence X = a—+uB—, oz ou

F{uB,u) + F{u,uB) = 0.

Given a vector ^G C " , 0 = (ada:)2(^^ + 2 V ^ F ( n J ) ^ ) =

2y^F{uB\p)f^+PB'^,

B^ = 0. By the condition F{uB,u) + F{u, txB) = 0, Vii G C"^, then F{uB,v) + F{u,vB) = 0, Thus F{uB,uB)

= -Fiu.uB'^)

\/u,v G C ^ .

= 0. Hence ?xB = 0,Vtx G C ^ , it follows

that J5 = 0. Therefore x = a-;r- e L_2. az By Chapter 2, we have

•

3.3 Let D(V,F) be a homogeneous Siegel domain in C " x C ^ . Then there is a maximal triangular Lie subgroup T(£)(V, F)) in the affine automorphism group AS{D{V,F)), such that T{D{VjF)) acts simply transitively on D{V,F). And the Lie algebra 0 o / T ( D ( F , F)) has a subspace direct sum decomposition:

LEMMA

(& = L^2 + L-i + (dnLQ,

(3.6)

On the other hand, there exists a linear transformation J and a linear function ij) on 0 , such that ( 0 ; J,t/?) is a normal J Lie algebra.

104

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Proof By Theorem 2.13, there exists a maximal triangular Lie subgroup T{D{V,F)) acting simply transitively on D{V,F), such that the Lie algebra 0 of T{D{V^F)) has a subspaces direct sum decomposition: 0 = L_2 + I/-i + 0 n L o . • We will study the homogeneous Siegel domain D{V) of the first kind as follows. By Theorem 1.19, aS{D{V))

= L.2 + Lo,

(3.7)

where L_2 = { a G — i V a G R ^ } , oz pj

pj

LQ = { zA—

I ^ € gl (n,R), xA—

(3.8)

G aff {V) }.

(3.9)

3.4 Let D{y) he a homogeneous Siegel domain of the first kind. There exists a maximal triangular Lie subalgebra

LEMMA

(S = L_2 + Lo n 0

(3.10)

of aflf {D{V)), such that (3 is a normal J Lie algebra. Therefore there exists a J basis {9j,fj,

l<j
gjktjjkt,

l
l<j

+ l}, (3.11)

such that L-2 = {9j, l<j
fjkt, l
+ l).

(3.12)

By Lemma 3.2, L-2 = {x€(5\

(ada:)2 = 0 } .

Denote hjt = 9j,N+i,t + ^/^fj,N+l,t,

1 < * < nj^N+i,

l<j
By the multiplication table of a J basis (see Definition 2.10), gj G L-2,

l<j
fjkt € L-2,

l
By (2.143), l
~^^

I < j < k < N.

Normal Siegel Domains

105

In order to find L_2, it suffices to determine the element

3,k,t

J

j,t

Since the projection of [x, [a:,^^]] in Qp component is 2yp, (a^dx)^ = 0 impHes t/p = 0, 1 ^jthjt)'

Now, y/^[fk,x] =

Y^ihthkt+^thkt)^ t

hence N l
"^"^

k=l

~^

Since dim^D(F)=n,

d i m ^ F = n,

dim^L_2 = n

and dim (0) = 2iV + 2

Y^

njk = dim^(i?(F)) = 2n,

l<j
we have dim (L_2 n q j . ) = nfc^iv+i,

Vfc,

and [xj Jx] = 0 impUes x = 0. Thus there exists an orthogonal basis pkt^ 1 < * < rik^N+i in 1^-2 n ^ ; , , such that pku Jpku 1 < * < rik^N+ij is an orthonormal basis in ^ ^ , where Pfc^ = yZi^ksu9k,N-\-hu

+

yksufk,N-hl,u)'

Therefore we find a new part J basis in ^ ^ . Without loss of generahty, let pkt = /fc,iv+i,t. We have

l
it follows that (3.12) holds.

~^^

fe=l

^

•

106

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

3.5 Let D{V) be a homogeneous Siegel domain of the first kind in C^. Then there exists a maximal triangular subalgebra

THEOREM

(& = L-2 + Lon(3

(3.13)

in aS{D{V)) and a linear transformation J, a linear function t/; on 0 , such that (0; J, -0) is a normal J Lie algebra. The Piatetski-Shapiro decomposition of 0 is

0 - 01 + • • • + 0iv, 0^. = y{j + j ^ . + Mj,

dJlj = ^

Mjk^

^ ' ^

And there exists a real linear isomorphism 6 on D{V), the image 9{D{V)) is also a homogeneous Siegel domain of the first kind. The coordinates of 9{D{V)) are denoted by

The normal J Lie algebra 0 has a J basis { 9j^ fj, 9jku fjku ^
1 < j < fc < iV };

(3.16)

L-2 has a basis { Qj = V2^,

fjkt = -?iY, l
(3.17)

Lo n 0 has a basis

-V2fj = Aj = 2sj^ + J2^PJ^.+Y1 CfSq OZ'Q'i J

P<J

^J

hv4-^ OZqr)

J
1<J
(3.18)

-^^

and

-^-^.-^n^<''^^. ''

i
ss

where I
'"'

^^ip

'

"

j
(3.19)

^

Normal Siegel Domains Proof

107

By Theorem 2.18, n = N+

^ njk- By Lemma 3.4, L_2 l<j
has a basis {9j, l<j
fjku l
l<j
+ l},

Denote Z = {Si,Z2,^2, • • •,Zjsf,SN,^iv+i) = {Z^ ,' " •>Z ) the coordinates of point Z in a Siegel domain D{V) of the first kind, where Sj e C, Zj = (Zij, • • •, Zj_ij), Zij E C^'-^. Thus L-2 has a basis ^ —pr, 1 < j < n >. Hence

9j = V^o^j -Q^,

/jfct ^ Q^jfct ^ ,

where QJ, a^-fct G R^. Now, the fact that {5j, hku 1 < < < n,fc, 1 < j < A: < iV + 1} is a basis of L_2 impUes that {a^.OLju. 1 < j < f c < A r + l } is a basis of R^. There is a Unear isomorphism Q\ z — ZP such that pj

pj

-—- = PTT- Hence 9JL-.2) has a basis oZ oz { 5^ = V 2 — , fijt = —^,

l
+

on the homogeneous Siegel domain 0{D{V)) = D{0{V)). of generality, let z = Z. Then

1 /^ a'

„ a'\

& pJ

lj

Without loss

^ & 0/

where ^^^ a^-fct ^ R^, B^, Bjkt e gl(n,R), ^Bj — , xBjkt^ e aff (V). We will compute Pj, ajkt, Bj, Bjkt in the following. Since Theorem 2.18 and Lemma 3.4 hold and [Z/-2, -£^-2] = 0, we have [fm, fjk,] = - E

e*(Im ( A j ) ) e ; / y , = 0.

108

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Hence Im {A^^) = 0. It follows that the matrix set { Af|, 1 < t < riij, 1 < i < j < f c < i V ' + l } i s a real normal matrix set of type N. Given

PyQ^S

p,q,s

then

d' By the multiplication table of J basis, we have

d

d

-

^^^'^'^ '72^^'''dz^

^2^y''dz^r p,s P3

thus

^^^l; = 2^^^^+EE^: ds _ _ 'PJ* 1

d Q{S)

+EE^.-

A . Z ^ ^OIS

d (,) •

x^'

Hence / j = —-j= (Pjw- + AjY where Aj is defined by (3.18). Since / j , ^ j ^ - € 6 , we have Aj G LQ Pi 0 . Again, the multiplication table of J dz C/0 basis and

,9' a' a' r.^ a'I ' n ^Ai'^ife* = P ^ ' « i f e * ^ + ^-S^'=*pr =/^^ifc*^

a^'^^'^'j

raz'^^'^^a^ ' ~"""'dzi

'-""''dz

imply

.a' P,9>«

a dz.it)

^fefc-

. a V-. ,rk, + „2h^kt^ + V -Y,Y.^pksesA p<3

•jk

j
r,s

d

»•.«

(r) jp

P3

j
s

Normal Siegel Domains

109

Comparing the coefficients in the above formula, letting z = (3, we have 9jkt = otjkt-Q-+Xfjl, where xf^ is defined by (3.19). Since gjkt, ocjktQ- ^ 6 , we have xf^ G LQ n 0 . Now, rij^N-^i = 0, 1 < jf < iV. In fact, let

^, = (afV.-,af^---)),

l<j
Then X]ji+i € Lo n 0 , Vt, j , imply

j
s

^^jp

Hence exp {tXj) is a one parameter subgroup: Si

^ Si^

Sj -^ Sj +

^^.^^^.

'^ 7^ 3-) 2Zj^N-\-l^jt,

+ Y,{zp,N^iA;f^'^-es)t,

Zjp -^ Zjp + Y^{^jA'^^^^z'j,^N^^es)t,

l

s

Zkp^Zkp,

k.p^j,

VtGR.

Since V is an open convex cone in M*^ not containing any straight line, and a one parameter subgroup {exp(tXj), Vt G M} is a set of affine automorphisms on F , the image set hes in V, hence the cone V contains a straight line (exp{tXj))z when TIJ^N^I > 0. It is a contradiction. Therefore we prove that rij^jv+i = 0, 1 < j < AT. Finally, we consider the subspace LQ = LQ H 0 with a basis { Aj, l<j
Xlf,

l
Uij,

l
By a direct computation, the multiplication table is

pq

dzy-

^"^pq

(3.21)

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

no

^,3 Y.^^rA%e[)

d pi

d

d + < Sgj ds.

,x.

25piSgj5rt^-

^^^ti

d Yj(^t^ip^r)

if

i
dz.ip

djper{A^j) ^^

if i < q,

•iq

(3.22) (3.23)

[Ak,xf;>] = {5,^-5ki)Xf;>,

(3.24)

[ 4 ^ 4?] = ^^i Y.i^rA%e',)X^^ - 5j,

Y>rAi04(3.25)

(r)

d' & On the other hand, /• = —-7={Pj-^+Aj), gijt = aijt-^+Xy'

.{t)

as above.

Using the multiplication table of J basis, we have f3j — 0, a^t = 0. Hence

fi-

J_=^j,

'V2^

l<j
Qjkt = Xfl

l
Conversely, we can assert that <& = L_2 + LQ H (5 is a normal J Lie algebra. Namely, <& = L-2 + LQ C) <S is a maximal triangular Lie subalgebra of aJf {D{V)). In fact, it suffices to prove that all eigenvalues of linear transformation aAX^-' acting on 0 ^ are real numbers. Given an eigenvector

k
r{t)

if [X^j^X]

= XQXJ then AQ E E by a direct computation. Hence all

eigenvalues of linear transformation adX|- are real numbers.

I

By Theorem 3.5, without loss of generality, there is an affine automorphism subgroup T{D{V)) acting simply transitively on the homogeneous Siegel domain D{V) of the first kind, where the Lie algebra t(-D(V')) of T{D{V)) is a normal J Lie algebra with a basis (3.17), (3.18), and (3.19). The coordinates in D{V) is expressed by (3.15). We will compute a generated element set exp(t(J9(V'))) of T{D(V)) firstly, and determine all T{D{V)) open orbits in C"^ secondary. Then we get 2^ open orbits i^(V^^^...^g^) in C^, where e^ = 1 or —1, and prove that only two orbits V^,... ^^ and VL^... _i do not contain any straight line, where

Normal Siegel Domains

111

Vj... 1 is affinely equivalent to K.^... _i. The Siegel domain -D(Vi,...,i) i^ called a normal Siegel domain of the first kind. Finally, we prove that any homogeneous Siegel domain of the first kind is affinely equivalent to a normal Siegel domain D(Vi... J , and T(£)(V'i... J ) acts simply transitively on D(Fi... J . Now, we consider a basis {Ai,l
X^f,

l
Uij,

l
in t ( y ) , where

j
d ^ij

-"^^ij

d

-^

v^v^

f)^. + ^ ^ T l t T + ^

i<j

-^

... , d

Z-/ ^PjA^^^'t

(3.27) i
s

^^ip

j
*^

The coordinates of V are denoted by X — v'liX2,T2,

• • • ,Xjy,rjy),

Xj = {Xij, • • •

,Xj—ij)

(3.28)

Xij = {x^\---,x'^*^^)eR^'i, n€R. Given a vector b EMP', where b = (&11,62,622, • • •, b^.bNN),

bj = ( 6 i j , • • •, & j - i , j ) ,

(3.29) then the coordinate expression of N

exp (t(6fcfc^fc + X^ 5;^(6fc,e;)xif)),

V t e R,

(3.30)

is y = x^^\t). This is a unique analytic solution of the differential equation system

^ ^

= 2bkkrk{t) + 2 J2

hixklity,

l=k+l

^

^

= bkkXpkit) + E l=k+l

EXpi{t)A;ib'kies, s

l
112

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

^ ^

= bkkXki{t) + n{t)bki+ X ; J2^kpAfiXij,{t)'es k
+ E

s

T.^kpe'sXpimKy,

k
k
s

with an initial value x{0) = x, where fc G {1,2, • • •, N}. Let a = {aii,a2,a22r" .CLN^^NN) ^ R"^, where akk = e'^^^ > 0,

aki = {e'^'' - l)bkkhh

k
(3.31)

and let Rfl{x) = f^e',xjk{A^^y

(3.32)

be an Uij x riik matrix. Thus the afSne automorphism (3.30) on V is N

Okia) = exp {t(bkkAk + J^

Yl ^kie'sXif)),

l=k+l

A: G { 1,2, • • •, AT }.

s

(3.33) Hence the coordinate expression of 6k{a) is N

N

Sk = alkVk + Y^ {dkMri

+ 2 J ] ] akkdkix'ki

l=k+l

l=k+l

+ 2 Y. ««<'^(^)4p, k
yik = CLkkXik+ Y

^i^^kii^y^

l
l=k-\-l j-l

Vkj = dkjTj + akkXkj + Y l=k-\-l

Uij ~ *^iji

N

^kiRlj\x)

+ Y

^kiR^^\xy,

k<j
/=j+l

^5 J ^ At.

(3.34) Since a^fc > 0, we obtain an open convex cone W^ = { X G R^ I ri > 0,. • •, r^ > 0 } C M^

(3.35)

where a G W^. The equation (3.31) has a unique solution b G R'^. Therefore we obtain a unique solution 9k{(i).

Normal Siegel Domains

113

Denote by (&k a Lie subalgebra with a basis A^^ ^li J * ~ I5 * * * ^^fc/? / = fc + 1, • • •, N^pk gives a connected Lie group Gk with the generated element set exp(0^) — {dk{o)\Mae W^ }. We give the matrix expression of ^^(a) as follows. Let ^jk = ^33 + ^ J j + i + • • • + ^jfc,

l<j
We construct an r^.^ x r^.^ real symmetric matrix \

Xjk

(3.36)

Cjkix) =

V ^;^ where 1 <j
^J?i,.(^)'

rk

E^^jk)

When j = fc, Cjj(a;) = (rj).

" j = »"jfc + Sjk = Ujj + rijj+i -\

h UjN,

l<j
Denote Cj{x) =

[

E(rj,k-i)

(3.37)

l<j
CJN{X),

0

0

\

(3.38)

0 0 £;(*>*) / where Rjk = {Rkl+i{a), •••, R'j^lia)), 1 < j < k < N md Pjkia) = E,

(3.39)

l
Let I ^33

^jJ+1

\

^3+hN (a)

PM =

(3.40)

V be an n^- x n^- matrix, I < j < N. By the direct computation, we obtain Pj{a) = PjN{a)Pj^N-i{a) • • • Pji{a). Obviously, we have

(3.41)

114

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

LEMMA

3.6 Use the symbols as above. Given a vector a 6 Wj^, then Cj{6{a)x) = Pj{a)Cj{x)Pj{a)\

I < j < N,

(3.42)

where e{a) = eM^r^-iia) THEOREM

T{V).

(3.43)

3.7 The connected Lie group

T{V) = { e{a) = OMO^-M Proof

•••0iia)€

• • • ei{a) l^iasW^

}.

(3.44)

Given a vector a = (aii5a2,a22,---5%?tt^^) ^ W^jv?

where an = 1^ i ^ k and a^j = 0, i < j^ i ^ kj then 9{a) = 0k{o). Hence T(V) has a generated element set {^^(a) | Va G W^, 1 < fc < i\r}. Given two vectors a, 6 G W^^, then CMa)e{h)x)

=

P^{a)Pj{b)Cjix)Pj{byPj{ay

by (3.42). Let Pj(a)Pj{b) = Pj{c), thus c = (cii,C2,C22, • • • , c ^ , c ^ ^ ) € W^, where j-i

Cii = aubii > 0,

Cij = Oijftjj + aubij + 2_^ Oji-R]^ (&)•

Hence Cj{e{a)e{b)x) = Cj(^(c)a;). Therefore ^(a)^(6) = ^(c). Now, v^ = (1,0,1, • • •,0,1) 6 W^ and P,(vo) = £^, hence Cj{d{vo)x) = Cj{x), l<j
then e{b) = e{a)-K

•

Now, we consider all T{V) open orbits in R" as follows. Given two open cones W = {yGR"|si^0,---,SiV7^0} (3.45)

Normal Siegel Domains

115

and W = {xeR''\det

{Ci{x)) ^ 0, • • • ,clet {C^{x)) ^0},

(3.46)

by a direct computation, then we have LEMMA 3.8

Let

(3.47) y = (^1,y2,52, • • •,VN^^N)^

Vj = (yij^ • • •'Vj-ij)^

where xij, yij G R^^^, r^, Si G M. The mapping TT; N

ri = Si+ J ^ s^^yuy^i,

1 < j < AT, (3.48)

^u = y^ + S ^T^yn^fhyy^

i
is a one-to-one real analytic mapping, such that 7r{W) C W and

Cjix) = cMy)) = PM^jiyr'PM'^ i
(3.49)

where Aj{y) = diag (s,-, 5,+i£;(">.>+i), • • •, s^£;(">^)), LEMMA 3.9

Let

1 < j < iV.

(3.50)

irk:

si = ri- r^^xikx'ik,

l
*' = '•'• , „ , ypq=Xpq- Tfc XpkR^^'ix) , ypg = a:pg,

1

' ^ ' ^ ^ I < p < q < k, 1

(3.51)
be a real analytic mapping, and let TT' = 7ri7r2-"7r^.

(3.52)

Then TT^iW) C ty, and Cj{x) = Pj{n\x))Aj{7r\x)r'Pj{7r\x)y, where 1 < j < N. Proof

Since det (Cjy(x)) = rn 7^ 0, we have / C,jv-i(7r^(a;)) C,(a:) = Q,(7r,(a;)) f '' ^ "' ''

0 \ ~ ^_^^ j g,(7r,(a;))',

(3.53)

116

THEORY OF COMPLEX HOMOGENEO US BO UNDED DOMAINS

where j = 1, • • •, iV and /

/

^JN

\

\

?^') .rix) ( B^.

E

,

QjiT^Ni^)) = Qjix) =

l<j
\ R^lM I r^E

\o

/

Clearly, det {Cj{x)) 7^ 0 implies det (C,,jv-i(7r^(a;))) / 0, 1 < j < i V - 1 . Let

^U^)

E

,

Qjk{x) =

l<j
V ^ei,.(^) / \ 0

VkE

hence QJN{X) = Qjix), l<j
/ When rfc 7^ 0,

Cj,fe-i(7rfc(a;))

Cjk{x) =

QjkMx))

where l<j
0

0

r;'E

QjkMx))',

Let

QMM^))

= diag(Q,fc(7rfc(a;)),E("^.'=+i+-+"^-^))

be an rij x Uj upper triangular matrix, where rij = riij + • • • + rijj^ ^ ^ j ^ N. The elements of matrix Qjki'^ki^)) are all analytic functions with real variables r^, Xjk^ • • •, ^fc-i,fc- Now, Q3Ni7r^{x))Qj^N-i{7r^_^7r^{x)) • • • Qjj(7r^-7r^+i • • • T^A^)) = QjN{^'i^))Qj,N-i{7r\x))... Qjj{7r\x)) = Pj{7r\x)), l<j
I < j < N, Thus the

y = 7r'(rr) = (51,^2, ^2, • • •, y ^ , 5 ^ )

have the conditions 5i • • • 5^ 7^ 0, and 7r'(W) C VF.

I

Normal Siegel Domains LEMMA

3.10

117

Using the symbols as above two Lemmas^ we have

Namely J TT is a bi~analytic isomorphism acting from W onto W. Proof

By Lemma 3.8,

CMv)) = PMHyr'P3{y)'^

Vyei^,

I<J
By Lemma 3.9, Cj{x) = Pj{7t'{x))Aj{7r'ix))-^Pj{7r'{x)),

\fxeW,

l<j
Since 7r^{W) C W, we have y = TT^{X) e W, \/X e W, and Cj{x) = Pj{y)Kj{y)-^Pj{yy. Hence Cj{^{y)) = Cj{x). It follows that 7r{y)^ x. Namely, 7r7r'(x) = x^ \fx e W. So TTTT' = id holds on W and W = TTTT^W) C 7r{W) C W, Thus 7r{W) = W, Hence TT is a bi-analytic isomorphism from W onto W. For the same reason, TT'TT = id holds on PF, hence TT' is an analytic isomorphism from W onto W^. Therefore TT' = TT"-^ and TT is a bi-analytic isomorphism. I By Lemma 3.10, the bi-analytic isomorphism TT acts a connected component of W onto a connected component of W. We will prove that any connected component of W is a T(F) open orbit and we will give the explicit expressions of these connected components. The subset W in R^ has exactly 2'^ open connected components. Let S = i5i,52,---,S^),

<5,-6{l,-l}

(3.54)

yj = {yijr--,yj-i,j),

,^^^.

and

2/ = (si,y2,s2,---,yjv>«N)'

W, = {yeR''\5isi>0,---,5^s^>0}, where Wg is a connected component of W. Now,

W = [jWg, 5

^'

R" = U(W^,) = U ^ ^ ' S

(3-56)

6

where W^ is the closure of the open subset W^ in R^. Hence W = 7r{W) has exactly 2^ connected components W^ = 7r(W^), V5, and W = niW) = [J7r{W,) = [jW,,

(3.57)

118

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Let Vs = (r,X2,r2,'' where TJ = 6j, l<j
• ,a:^,r^), 2<j<

(3.58)

N, thus v^ G W^ and

— ^6^ hence ^^ G VF5 fl W^. We have

3.11 The following symbols are the same as above. If S is defined by (3.54), then W^ = 7r(W^) is a T(V") open orbit, where W^ contains the point v^.

LEMMA

Proof Let U^ be a T{V) open orbit, such that 6 G UaBy (3.42), Cj{e{a)v,) = P,•(a)C,•(^;JP,(a)^ Va G W^, where W^ is defined by (3.35). Given y = {si.yi.s^r " .VN^^N) ^ ^'^^ Vj = {yij^ • • • 5Vj-ij)^ where Sj = Sja'^j, yj = Sjajjaj, hence Pj{a)Aj{v,)-'Pj{ay By (3.48) and (3.49), yeW^.

=

Pj{y)Aj{yr'Pj{yy.

Let 7r(y) = x G W?^, thus

CM^X) = PMHyy'Pjiyy

= CMv)) = Cjix).

Hence 0{a)Vg = x eW^. Therefore the T(V') open orbit U^ contains the point Vg, and U^ C W"^. Conversely, y = n^x) eW^.^xeW^, Thus SjSj > 0,1 < j < AT. Let ajj = {SjSj)2^ aj = Sj{SjSj)~'^yj. We have a G W^ and

Cj{x) = cMv)) = PMHyY'PM'

=

PM^M)PM'

Thus X = 9{a)v^ G U^. Hence it is contained in the T{D(y)) Therefore W^(ZU^. Namely, <W,) = W, = { e{a)v, \^aeW,}

= U,,

open orbit.

I

Obviously, we have 3.12 A T(F) open orbit W^ is linearly isomorphic to the T ( y ) open orbit W^s by y = —x.

LEMMA

By Lemma 3.12, it suffices to consider the T(V) open orbit W^ up to a linear isomorphism, where 5 = (1, ($2, * * * 5 ^N)-

Normal Siegel Domains

119

3.13 Use the symbols as above. When 5i = 1 and there exists j G {2,•••jiV}, Sj = —1, the T{V) open orbit W^ contains a straight line. LEMMA

Proof Without loss of generality, we can consider the case j = 2. Namely, 5i = 1, ($2 = - 1 . Given a vector a G W^, where a n > 0, ^22 = • • • = %jv = 1, ^2 ^ R^i2^ ^^ _ 0^ .. .^ a^ = 0, then x = 0{a)vs G Wg. Since Cj{x) = Cj{0{a)vs) = Pj{a)Cj{v^)Pj{ay, thus r i = a i i - a 2 a 2 , r2 = 62 = - 1 , rj = Sj, a;2 = — a2, a^j = 0 , 3 < j < iV.

3 < j < iV,

Hence X = ^(a)!;^ = (a^i - a2a2, - a 2 , - 1 , 0 , ^ 3 , • • • ,0,^^) G W^. If a2 is a constant vector satisfying the condition 03 (a2 v = 3 and a n = Vst^ - 3t + 1 > 0,

a2 = t4*^^

VtGR,

then X = (1 - 3t, -taf\

-1,0,53, • • • ,0,5^) G T^„

Vt G R.

Hence W^ contains a straight line. DEFINITION 3.1

A

•

set

F^ = { X G R^ I Ci{x) > 0,. • •, C^(a:) > 0 } C R^

(3.59)

is called a normal cone associated with a real normal matrix set of type

iV;{4j}. LEMMA

3.14

Use the symbols as above. Let SQ = {l^-- - ,1). Then

V^=W,^={xER^\Ci{x)>0,---,CM>0} = { a; G R " I det (Ci(a;)) > 0, • • • ,det (CJx))

> 0 }. (3.60)

Proof

Given x €Wg , there is an a G V^, such that x = 9{a)vso. Thus

Cj{x) = Pj{a)Cj{vso)Pj{ay

= Pj{a)Pj{a)' > 0,

1 < j < iV.

Hence W,^ C V;^. Conversely, if Cj{x) > 0, Va; G F^, 1 < j < iV. By (3.45), X eW.

Let y = n^x) G W. We have x = 7r(y) and

Cj{x) = C^y))

= Pj{y)Aj{y)-'Pj{y)'

> 0,

1 < j < iV.

120

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Thus si > 0, • • •, Sj^ > 0. It follows that y € W^ . Hence x = •K{y) € <WJ = Ws^- Therefore V^ C W,^. Namely, V^ C W,^. Denote V^ = {xeR''\det

(Ci(x)) > 0, • • •, det (C^(x)) > 0 }.

Clearly, F^ C V^. Conversely, since Vj^ is a connected open subset of W and ?;^ ^ Vj^ ? we have V^ C W^ = Vj^. Hence Vj^ =V^. I LEMMA 3.15 Use the symbols as above. Vj^ is an open convex cone in R^ with the vertex origin not containing any straight line, and V^ is acted simply transitively by T ( y ) .

Proof Given two vectors a, ^ G V^, a 7^ ^, where Cj{a) > 0, Cj{P) > 0, 1 < j < iV, then Cj{ta + {l-t)P)

= tCj{a) + {l-t)Cj{P)

> 0,

\fte

[0,1],

l<j
Hence to + (1 —1)/3 G V^. It follows that V^ is an open convex cone. If V^ contain a straight line, given two points a and /3 in this straight line, where a ^ f3^ then to + ( l - t ) / ? e V^,

VtGE.

Hence t[Cjia) + ( r ^ - l)Cj{(3)] > 0,

VO / t G R.

Letting t —^ +00, we have Cj{a) — Cj{P) > 0; letting t —> —00, we have Cj{a) - CjiP) < 0, hence Cj{a) = Cj{0), l<j
E = Cjiv,) = CM<^K) = PMPM'Since Pj{a) is an upper triangular matrix and diagonal elements are all positive numbers, we have Pj{a) = E. Hence a = v^. Therefore 6{a)vQ = VQ imphes a = VQ. It follows that the isotropy subgroup of VQ in T{V) is { id}. • Now, we consider the normal Siegel domains of the first kind. DEFINITION

3.2

Given a real normal matrix set { A g , l

Normal Siegel Domains

121

of type N. Denote by V^ the normal cone associated with a normal matrix set { Aj^ } of type N. Then the point set D{V^) = {z€C^\lm{z)€V^} = { z 6 C " I Im{Ci{z)) > 0,- ••,Im(C^(2)) > 0 } = { ^ G C " I det (Im(Ci(2))) > 0, • • •,det{Im{C^{z)))

>0} (3.61)

is called a normal Siegel domain of the first kind. Now we prove that the classification of homogeneous Siegel domains of the first kind up to a holomorphic isomorphism induces the classification of normal Siegel domains of the first kind up to an affine isomorphism. 3.16 Suppose that D{V) is a homogeneous Siegel domain of the first kind. Then there exists a linear isomorphism r : V —^ Vj^^ where V^ is a normal cone associated with a real normal matrix set {A^p 1 < t < Uij^ 1 < i < j < k < N}, such that T{D{V)) = D{Vj^) is a normal Siegel domain of the first kind. THEOREM

Proof A homogeneous Siegel domain D{V) is an affine homogeneous Siegel domain by Theorem 2.3, and the holomorphic automorphism group A\itiD{V)) = lso{D{V)) • T{D{V)), where T{D{V)) is a maximal triangular subgroup of the affine automorphism group Aff {D{V)) by Theorem 2.13. Prom the J basis in the Lie algebra t{D{V)) of the Lie group T{D{V))j there exists a normal Siegel domain D{V^)) with the maximal triangular subgroup T{D{V^)) of the affine automorphism group Aff {D{V^)) as above, such that Theorem 3.5 holds. Now, T{D{V)) and T(D(V^)) are connected and simply connected Lie group implies that T(D(F)), T{D{V^)) act simply transitively on D(V), D{V^), respectively. The J isomorphism

Qijt -> X^j\

fijt -> — 7 ^ , 1 < t < riij,

H-

l
(see (3.17), (3.18) and (3.19)) induces a Lie group isomorphism from T{D{V)) onto T(D(V)), and this J isomorphism induces a holomorphic isomorphism r from D{V) onto D(V^). On the other hand, t{D{V)) = L-2 + Lo n t(D(V)), where L^2 = {xe

t{D{V)) I (ad(x))2 = 0 }

122

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

by Lemma 3.2, and

w = {wij"'jWn) {si,Z2,S2j-',z^,Sj^) Then

is a point coordinates in D{V). Denote hy z = = {yij-',yn) the point coordinates in P(V^).

r.(L-2)^{^.l
L^ 3

Qyj.

Qy. -^

L^ J

^'^ dVi^ -^

—

•"

where aij is a constant, 1 < i, j < n. Therefore r is an afSne isomorphism from D{V) onto D{V^). I Now we prove that the classification of normal Siegel domains of the first kind up to a holomorphic isomorphism induces the classification of real normal matrix sets up to a unitarily equivalent (see Definition 2.12). 3.17 Denote by D{V^) and D{V-.) the normal Siegel domains of the first kind associated with a real normal matrix set {Afj} of type N and a real normal matrix set {-B||} of type N, respectively. If a holomorphic isomorphism ip acts from D{V^) onto D{V^), there exists an affine isomorphism

THEOREM

r : Sj-^

S^y), l<j
Zij -^ Z^{i)a{j)Oa{i)a{j),

l < i < j < N

(3.62) acting from D{Vj^) onto D{V^)j where Oij € 0{nij)^ I < i < j ^ N) and a is an admissible permutation of 1,2,••-jiV. Namely, when i < Ji(^{j) < ^(i); then n^(j)a(i) = 0. Proof Let ip: w = f{z) be a holomorphic isomorphism acting from D{V^) onto D{V^). Then g -^ i) o g o ^ - i , g e Aut {D{V^)), gives a Lie group isomorphism from Aut(D(V^)) onto Aut (£>(F^)). By Theorem 2.13, denoting by Ad(T(£)(V^))) a maximal triangular subgroup in Ad(Aut(D(V;^))), A d ( ^ o T{D{V^)) o ^j;-^) is also a maximal triangular subgroup in Ad{A\it{D{V~))). Hence there is a holomorphic automorphism rj G Aut {D{V~)), such that {r, o i^)TiDiV^))iv

o V)-i = T(Z)(F-)).

Normal Siegel Domains

123

Denote by t(D(V^)) and t{D{V^)) Lie algebras of connected Lie groups T(Z?(V^)) and T(JD(V'~)), respectively. Without loss of generality, we can assert that the holomorphic isomorphism -0 satisfies the condition ^poT{D{V^))o^|;-'^ =T{D{V^)). The mapping y?: r ^ i/^or OT/;-^ gives that M^{D{V^))) d ^-,

^ij

= t ( D ( K ) ) , where d —r-r,

l
- ^hi Q + h

i
l
(t) +2^2^

T

Cf^ip

l
^PJ^pi^t

M

j
*P

1 < t < Uij, 1 (V^)), and z = (si, Z2, d — ,

~ l
d —J-,

l
l
^^ = 2 t . | + 5 : « ; ^ ^ + 5 : « , p ^ , p
8

^

_ a

i
l
^

^ ^

™ , d

^^? = ^''^w+*i^+T,Y:^<^^>'. i
r

^^ip

j
^^

where 1 < s < riij^ 1 (K)) I (ady)2 = 0 }.

Hence i/?* maps the vector space (^,

l
—^,

l
l
124

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

onto the vector space

Since

a^.

Z^ /^c. /9^. "^ Z^ z ^ /)c.

SO the Jacobian matrix of the mapping ij;: w = f{z) is a constant matrix. Hence the mapping ^ is an afRne isomorphism. By Lemma 1.18, there is a Hnear isomorphism r: y = xA from V^ onto K , and T{D{VJ^)) = D{T{V^)) = D{V^). So r: w = zA is a complex hnear isomorphism from D(F^) onto D(F~). Thus r: w = zA induces a J isomorphism from t(D(F^)) onto t{D(V~)). Hence r maps a J basis onto a J basis. By Theorem 2.24 and Definition 2.12, the real normal matrix set {A^j} of type N and the real normal matrix set {Aj^} of type N are unitarily equivalent. Hence there is a linear isomorphism r^

where a is an admissible permutation of 1,2, • • •, iV, such that T ' ( V ^ ) = v.. Therefore T'{D{V^)) = DiV^). I By Theorem 3.17, we have 3.18 The classification of normal Siegel domains of the first kind up to a holomorphic isomorphism is equivalent to the classification of real normal matrix sets of type N up to a unitarily equivalent (see Definition 2.12 ).

THEOREM

2.

Normal Siegel Domains

In this section, we will introduce the conception of normal Siegel domains, and prove that any homogeneous Siegel domain D{V^F) in C^ X C ^ is holomorphical equivalent to a normal Siegel domain. We consider a homogeneous Siegel domain D{VjF) C C^ x C^ in the following. By Lemma 3.3, there is a connected afRne automorphism

Normal Siegel Domains

125

subgroup T{D{V,F)) acting simply transitively on D{V,F). by t{D{V, F)) the Lie algebra of T{D{V, F)), we have t{D{V, F)) = L_2 + L_i + 4 ,

Denoting (3.63)

where LQ is a real Lie subalgebra t ( y ) = { xA^

I 2 7 1 ^ + uB^

G 4 } C aff (F),

(3.64)

and exp(t(V)) generates a linear automorphism group acting simply transitively on V. By Theorem 3.17, there is a real hnear isomorphism 9: y = xP acting from V onto V^, where V^ is a normal cone associated with a real normal matrix set { Al^ } of type N. Hence the affine isomorphism 9^: w = zP, V = u acts from D{V^ F) onto a homogeneous Siegel domain D{V^,F). Without loss of generality, we consider only this homogeneous Siegel domain D{V^,F), where V^ is a normal cone. By Section 3.1, the Lie algebra autD(V^, F) of Aut {V^,F) has a direct sum decomposition of the subspace: aut (Vj^,F) =: L_2 + L - i + LQ + Li + L2, where L_2 = { a — I Va € R " }, uz

(3.65)

L-1 = { / ? ^ + 2 V ^ F ( « , / ? ) ^ I V/? G C^" },

(3.66)

Lont{D{V^,F))

= {A-j, l<j
Xlf,

l
N}, (3.67)

where Aj = z{Aj)o—

p<j

+ uQj—,

l<j
t

3
(3.68)

^

is an n X n matrix, the symbols ejj and efj are defined by (2.80)—(2.85), Qj is an m X m complex matrix, such that F{uQj,u) + F{u,uQj)

= F{u,u){Aj)o,

Vix G C ^ ,

(3.70)

and

1
S

^^ip

J
''

(3.71)

126

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where 1 < f < n^, 1 0, • • - , 0 ^

> 0 }.

(3.72)

Thus the closure of V^ is F ^ = { a; G E " I Ci{x) > 0, • • •,C^{x)

> 0 }.

(3.73)

The point coordinates in V^^ is x^iri,X2,r2,---,Xj^,r^),

r^ € R, (3.74)

Xj = {xij, • • •, Xj-ij),

Xij = ix\]\

•••, x\j'^^) G R'^*^

where n>0,---,r^>0

(3.75)

and when rj = 0, Xij =0, I
(3.76)

where ifj, 1 = 1, • • •,n, are mxm Hermitian matrices and the Hermitian vector function F{u, u) satisfies conditions (1)

F{u,u)eV^,

(2)

F{u, u)^0

VWGC'",

if and only if u = 0.

Let F{u,u) = (Fniu,u),F2{u,u),F22{u,u),-••

,F^{u,u),F^f^{u,u)), (3.77)

where Fa = uHiu', Fj{u, u) = (Fij(u, u), F2j{u, tt), • • •, Fj-ij{u,

u)),

Fijiu, u) = {F^piu, u),.--, F^p^\u, u)), 4*)(n, u) = uH^fu', i < j . _ (3.78) Thus F{u,u) € Vj^ implies Fjj{u,u) = uHju' > 0, Vtt € C"*. Hence Hj >0, 1 < j < N. On the other hand, Fjj{u, u) —0 implies Fij{u, u) = 0,

l
Fjk{u, u) = 0,

j
(3.79)

127

Normal Siegel Domains By (3.67)-(3.70), jp

P<j

j
*

= F{uQj, u) + F{u, uQj),

t

(3.80)

l<j
Now, given an afRne isomorphism r: z ^^ z^ u D{V^,F), where P € GL (m,C), then

V = uP~^ on

F ( ^ , ^) = F{vP, vP) = {vPHi'P'v', • • •, vPHnP'v')

= F{v, v),

and r maps D(V^,F) onto a homogeneous Siegel domain DiV^F), Since H^ > 0, where Fj^^{u^u) = uH^u\ there is an m x m nonsingular complex matrix P , such that PH^^' = diag(0,E^^^)). Without loss of generahty, we can assert that H.

0

0

0

E^'^N)

Given the block partitions of matrices /fi, • • •, H^_^ as iJ^, denote

iJ,= ( ^ ^ ^ ^ M > 0 ,

l
If P € GL (m, C) satisfies the condition PH^P ^=(P2

= iJ^, then

Pa)'

^3GU(mJ.

0

^1 ^

Hence

/ /O

\

£'2 ^ ^ ^ - 1 ^

PHj,_^P

>0.

=

> 0 imphes Di = 0. Let Pi =

- (0

£»;)

£;

Then det (Pi) 7^0 and 0 {PiP)H^_,{PiP)'=

I

(o 0

0 D

128

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where D >0. Without loss of generality, we can assert that 0 0 ^iv - I 0 £;('

' . ) ) ' ^.-.= ( ( o E^^^-^^) '].

By (3.80), where j = N -1,N,

we have

FNN ( « Q N - 1 . « ) + ^JViV (U, UQ^_, ) = 0,

Hence Q^_,H^+H^Q'^_^=Q,

(3.81)

Q^_,H^_,+H^_,Q'^_^=2H^_,.

Given the block partitions of the matrix Q^^^i as H^^ then

«-=(°^)' and Qj^_^H^ + HJ^Q'^^

= 0 implies 6 = 0, d + (/ = 0, Q^_^H^_^ +

HM-IQ'N-I = 2Q;v-i implies dD + D^ = 2D. So d + d' = 0 and [d,Z)] = 2r>. Hence t r D = 0. Now, if^_i > 0, £> > 0, so trl> = 0 implies D = 0. Hence

By the induction, there is a non-singulax complex matrix P, such that PHjP

= diag {0('^o)^Q{mi)^ . . . ^ Q{mj-l)^;E{mj)^Q(mj+^)

. . . ^ 0^"*^)),

i < j < N. Without loss of generality, we can assert that Hj = diag (0('"o), 0('"i\ • • •, O^"'J-'\E^"'^\O^"'^+'^

•••, O^'^^v)),

i<j 0, there exists Fu{u, u) = uHjV^ = 0,

l<j
By (3.79),

uHlfu' = 0, l
l
129

Normal Siegel Domains

It follows that F{u^ u) = 0. By the condition of Siegel domain, we have u = 0. We get a contradiction because u ^ 0^ therefore mo = 0. This statement holds. The above discussion shows that there exists a non-negative integer partition for a positive integer m: m = mi+m2-\

h m^,

(3.82)

such that Hj = diag (0(^i\ • • •, o(^^-i\ ^ ( ^ ^ \ O^^^+i) • • •, 0 ( ^ N ) ) ,

(3.83)

1 < j < iV, u = {ui,. •., u^),

^Uj G C ^ ^

1 < i < iV

(3.84)

and (3.85)

Fjjiu^u) = Uju'y By (3.79), we have

F^{u, u) = uHlfv! = Re

(3.86)

{mQlf^),

where Q L is an mi X rrij complex matrix. Hence / 0 ••• 0 y

0

•••

0

0

Q^

••• 0 \

•••

0

2

Q

Vo

(t)

0

0

0/

^(t) and the Next, we determine the conditions of the complex matrix Q\j Lie algebra LQ D t{D{Vj,,F)). By (3.80), we have

Fii{uQj, u) + Fu{u, uQj) = 25ijFjj{u, u), Fik{uQj,u) + Fik{u,uQj) = {6ij + 5kj)Fik{u,u), hence QjHi + HiQj = 2SijHj,

l
130

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Given a block partition of a matrix Qi as Hi, then

Qi = diag (Q^l---, Q S U '

Q'S + ^

QiH'S + Hf^Q'j = {Sij + Ski)H'S,

= 25^^-^' 1 < ^ < n,fc,

l
Hence Let We have Let

X,=diag(xl^U?,---,if?). We have

^.•^S^ -

HI'^KJ =

0.

Hence F{uKj,u) + F(u, uKj) = 0. By (3.4), « ^ ^ ^ = E " ^ - ^ ? ^ ^ e aff (D(F,F)) nisop(D(y,F)), where p = (\/^Vo,0) is a fixed point in £>(V^,F). Thus ^J+«^i^' where

p
j
Now, (3.71) gives FkkiuQijt, u) + Ffcfc(«, wQijt) = 25kiFlj\u, u), F^f{uQijt, u) + F^f{u, uQijt) = SstFjjiu, u), Fil\uQijt,u) + F{^\u,uQijt) = 5pietAl^Fgj{u, u)' + 5piFj,{u, n)(Ag)'e; + Sgi{Fpj{u, «)A^'4).

Normal Siegel Domains

131

Thus

Hence

f K,ijtl K,ijti Q ijt K,ijtj

Q^ \

KijtN I

where Km\. is a skew-Hermitian matrix, such that KijtiQf^ - Q^^Kijtj = 0

(3.88)

and

Qf^Qlf + Q^fQfh^^^E^""'^^ Q^ Q^ = Y:i^sA^S-'r)Q'^l (^(^ = YMr^^^Q' Clearly, (3.89) implies that if m^ = 0, then Since Y^UkKijtk^^

Iv'i j libn

=

(3.89) 0,l
e aff (L»(V,F))nisop(D(y,F)),

we have

where ^ij

-^"^ij a^. + * J „ (t) + Z . Z ^ ^ P J ^ p i ^ t . (5) ij

i
P
«

pi

s

(3.90)

132

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and {All

Qth

l
(3.91)

is a normal matrix set of type AT by a direct computation. Denote rijk

(3.92) r=l

and (

^0

\

(3.93)

Rj{u) =

thefc*^column and thefc*^row block of Rj{u)Rj(u) + Rj{u)Rj{uy is

When k < I, thefc*^column and the /*^ row block of Rj(u)Rj{u) + Rj{u)Rj{uy is R'il\u)Rl^\u)

+ R'i^\n)R'f\uy

= 24?(Re ^UkQtlui'et)

= 2Re

iR^^\u)Ri^\u))

= 2R'j^^\Fki{u,u)).

Hence C,(F(u,«)) = R e i ? , ( t . ) ^ M ' = DEFINITION

\RJ{U)R~(U)'

+ ^R^Rj{uy.

(3.94)

3.3 Given a normal matrix set & of type N, where

e = { Af^, Qg\

l
(3.95)

ifCi{z),---,C^{z) are defined by (3.36), (3.37); Ri{u),-• • ,R^{u) are defined by (3.92), (3.93), then D(F^,F) = { ( 2 , « ) e C " x C ' " | Im {Cj{z)) - Re {Rj{u)Rj{u)) > 0, 1 < j < iV } (3.96)

Normal Siegel Domains

133

is called a normal Siegel domain of the second kind associated with a normal matrix set & of type N. The normal Siegel domain of the first kind or second kind is called the normal Siegel domain. THEOREM 3.19 Let D{V^F) be a homogeneous Siegel domain in C^ x C ^ . Then there exists a normal Siegel domain D(V^,Fo) and an affine isomorphism r: z -^ zP, u -^ uQ, such that r{D{V,F)) = D{Vj^,Fo).

Proof Given a homogeneous Siegel domain £)(V,F), there exists an affine isomorphism a: x -^ xP from V onto a normal cone V^ by Theorem 3.16. The affine isomorphism a: z -^ y = zA^ u -^ u acts D{V^F) onto a homogeneous Siegel domain D ( F ^ , ( F ) P ) , where F is a function vector and P is a matrix. By the above discussion, there exists an affine isomorphism ^: y -^ y, u -^ uQ = v, such that '0((j(£)(V, F))) = D{Vj^,Fo) is a normal Siegel domain. I 3.20 Let D(V^, F) be a normal Siegel domain in C^ x C"^. Then there exists a maximal triangular subgroup T{D{V^^F)) acting simply transitively on D(F^,F)^ where T(£>(V^,F)) is a subgroup of the affine automorphism group Aff (£)(V^,F))^ such that the Lie algebra t{D{V^^F)) ofT{D(Vj^^F)) has a subspace direct sum decomposition: THEOREM

t{D{V^,F))

= L.2 +

L.i+Lont{D{V^,F)),

where L-2, L-i, LQ fl t(-D(V^,F)) are defined by &_ i-2 = { a ^ I Va € R" j ; 'dz

(3.97)

L-i has a basis

^" = v^7^ +p =EEK<«'.): ftj l S PJ +^'£-+EEMQ5P)^*)

(3.98)

134

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where 1
has a basis

^i - 2.,a;- + E - p i a ^ + E^.>a;- + «^a^'

(3.99)

where I < j < N and

4^=24^£:+^i7^+EE(^-^;^'^i)-

+ E E(^*44)A+E--(4)'^+-i^'^' i
dz\J

j
O^^V

where I
'

5ii« = Xlf,

Ji —

/^

j'

fijt = - ^ ,

^i,jv+i,t = PJ*^,

(^^^ (3.100) t{D{V^,F))

1 < t < Uij, l
/i,jv+i,t = P f \

(3.101)

l
Proof By (3.65) and (3.66), we know that L_2 and L_i are given by (3.97) and (3.98). By the discussion before Theorem 3.19, LQ H t(V(V;^,F))hasabasis {Aj = Aj + Bj,xif

= xlf +

Bif,l
where LQ contains vector fields Aj, l<j
X^f, l
l
and Bj = Y^UkKi^^^^,

B^ = Y^UkKijtk^

k

€

Loniso,iDiV,F)).

k

Hence the Lie algebra t{D{V^,F))

%•

has a part of a J basis:

9* = ^«+E'"-K'«"§^'

'•"-Wy ^'--Til^'^?-"^''^)'

Normal Siegel Domains

135

1 (*)l__A., pW [Ak. Pn = -^jkPf\

d

[P^\PJ'^] = ^Srt^ f

= 2V2Srtgj^

I[Ak. A, Pf^] p W i= _ _ A-SjuPf^ . , pW

(3.103) the matrix representation of adL_i(Aj) is a real diagonal matrix. Since the eigenvalues of adL_ {Aj) = adi,_ {Aj) + adL_ [Bj) are real numbers and [adL_^(^j),adL_^(^j)] = 0, implies that the eigenvalues of adL_^(5j) are real numbers too. But i C , i = 1^-",N, Hermit ian matrices, and

are skew-

implies that all non-zero eigenvalues of adL_ (Bj) are pure imaginary numbers. Hence pKj = 0, V/? G C ^ . It follows that B^ = uK^—- = 0 and fj = — i ^ A j , Similarly, the eigenvalues of matrix representations v2 of adL_^ ^-^ij) ^^^ ^^^_i (^27 ) acting on L-\ are all real numbers, and [adL_^ {Xf)), adL_^ (-^i^ )] = 0, thus the eigenvalues of adL_^ (SJj^) are all real numbers too. But Kijtk, Vz, j , t, k, are skew-Hermitian matrices, and ^dL_.(Bjj))(/3|;^+2v^F(r.,^)^) = 2 v ^ F ( n 4 \ ^ ) - - / 3 4 ^

5«

implies that all non-zero eigenvalues of ad^.^ (^i? ) ^^^ P^^^ imaginary numbers. Hence /3id*^ = 0, V/? € C ^ . It follows that Bf} = uXif—

=

0 andflijt = ^i*^. Finally, we will prove that gj,N-\-i,t = Pf\ /j,7V+i,t = Pf\ l<j
N

Pf

Ft

l
136

THEORY OF COMPLEX HOMOGENEOUS BOUNDED

DOMAINS

and N

L_i = { a: eaS{D{V^,F))

1

c

\ [J^/^.x] = --^x

},

AT

we have L _ i = ^ SPti^iv+i by the multipUcation table of a J basis. Let i=l

thus L/j'fl'fe,iV+l,rJ ~

nz9k,N+\,r'>

V2'

[jJ^Jk,N-\-l,r\

/p:Jk,N+l,r

V2'

give

i,t

i,t

i,t

i,t

Hence there exist airt^birt^Cirt^dirt^ such t h a t 5,^+x,. = Y.iairtPJ'^ t

+ birtP}\

It follows t h a t { P^^\P^^\

/ , , ^ , , = 5](QHi^^*^ + t

dirtP}\

1 < t < m^ } is a basis of dJli^N+i-

Since \ / ^ i x — G isOp(D(V,F)), and L _ i is a J invariant subspace, (J ill

and

^1 J/^(*\v/:iT.f =j^(^ 5?xJ

-jp^'\

te(^v^«"^ du

we have •^^^'^ =E^^^rPt^+E^'^^rPt\ r r

JPt^

= r

-Y.^itrP^^+Y.^^^rPr^r

Now, 0 = ^{[li'\P^^])

= i^{[JP^'\

JP^])

= 2 Y^iXisrmr r

Let \i = {Xitr), IJ'i = ifJ'itr)' Thus Xi/jt^ = /x^A- and

-

Ktmsr).

Normal Siegel Domains

137

Hence A^A^ + /i^/i^ = E, It follows that Ui = Xi + ^f^fii € U(mi), 1 < i < N. By Ui e U(mi), 1 = l^'-jiV, we can construct a J basis transformation z -^ Zj u --^ tAdiag(f/|,---jC/^). Without loss of generality, we can assert that

JPP = P^,

JP^ = ~-P^

and t

t t

t

Since

we have Ai -Bi

Bi\fO Ai ) \ E

-E\fAi Q )\^Bi

Bi\ _f 0 Ai ) - \ E

~E 0

where Ai = {airt),Bi = (6^^^)- Hence Wi = ^i + ^f-iBi € U(mi). By PFi, 1 < i < iV, we can construct a J basis transformation z -^ z^ u—^u diag (WFi, • • •, W^). Without loss of generality, we can assert that mi^N^i has a J basis { g,^^^,^^ = P^''\ f,^^^,^^ = Pt\ ^
THEOREM

Sj - > % ( j ) ,

Zij -^ Za{i)(T{j)Oa{i)aiJ)j

^ i " ^ U^(i)U^(i),

(3.104)

where Oij € O (n^j), t/^ € U (mi), and 9 is an admissible permutation of 1,2, • • •, iV. Namely, i < j , e{j) < e{i), then ne{j)0(i) = 0. (2) If ai is a holomorphic isomorphism acting from a normal Siegel domain D(V^,F) onto a normal Siegel domain D{V~,F)j then there exists a holomorphic automorphism a^ € Aut (D(F~ ,F))j such that a = 02 o o\ is an affine isomorphism from D(VJy,F) onto D{V~jF),

138

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(3) A homogeneous Siegel domain D{V^F) is holomorphically isomorphic to a homogeneous Siegel domain D{Vi,Fi) if and only ifD{V^ F) is affinely isomorphic to D{Vi^Fi). Proof (1) It can be proved evidently. (2) Similar to the proof of Theorem 3.17, we can prove that if D{Vj^jF) and D(V~, F) are holomorphically equivalent by ai, there exists a holomorphic automorphism (72 G Aut (D(y~ , F ) ) , such that the holomorphic isomorphism a = a2 o ai : {z,u) -^ [x.v) from D{Vj^^F) onto D(V^,F), where cr* is a J isomorphism from t(i)(V^,F)) onto

cr*(L_2) = I/-2,

cr*(L_J = L_^.

Since the maximal triangular subgroup acts simply transitively on a normal Siegel domain, without loss of generality, we can assert that a{vQ^O) = (^050), where v^ and VQ are two special points in D{Vj^,F) and Z)(V^,F), respectively. Let a : x = f{z^u)^ v = g{z^u)^ where

Then aJL-2)

-

df(z.u)

= L-2 implies that —^r-^— is a real constant matrix P oz

and —-^— = 0. Hence f{z,u)

= zP + h{u), g{z,u) = g{u), where

uZ

g{0) = 0. It follows that d^_pd^ dz dx^

^ _ dh{u) d' du du dx

dg{u) ^ du dv'

Now, cr*(L_J = L_^ implies that

= a Thus ——

is a complex constant matrix and g{0) = 0. It follows that

g{u) = uQ, where Q is a complex constant matrix, and a{z, u) = {zP + h{u), uQ), and a ^ l ^ = 2^/^{F{uQ,aQ)

- F(u,a)P),

Va G C ^ .

Normal Siegel Domains

139

The right-hand side of the above formula contains only a, but the lefthand side of it contains only a, hence

= 0 and F{uQ, aQ) =

F{u,a)P. Therefore h{u) = h{0) = \ / ^ ( v o - v^P). Clearly, if X G LQ n t{D{V^,F)) and L-i = {YE

t{D{V^,F))

I [X,Y] = -iY},

i = 1, 2,

then X = ad hz— + u—\. Thus \ oz ouJ (7* 2z— + u-^ = 2x-^ + V \ oz ouJ ox dv' Hence h{0) = 0. Therefore ^o = '^o-^^ ^^^ a is an affine isomorphism X = zP, V = uQ. Statement (2) of this theorem holds. (3) By Theorem 3.19 and (2) of this theorem, statement (3) of this theorem holds. I By Theorems 2.24 and 3.21, the classification of homogeneous Siegel domains up to a holomorphic isomorphism deduces the classification of normal matrix sets by a unitarily equivalent (see Definition 2.12).

3.

Decomposable Normal Siegel Domains

In this section, we will consider the decomposable normal Siegel domains. By the definition of decomposable normal J Lie algebras (see Definition 2.13) and Theorems 2.26, 2.27 and 2.28, we have 3.4 A normal Siegel domain D is called a decomposable normal Siegel domain^ if D is holomorphically equivalent to a topologic product of two normal Siegel domains of lower dimensional Otherwise, D is called an indecomposable normal Siegel domain. DEFINITION

3.22 Any normal Siegel domain can be decomposed to a topologic product of some indecomposable normal Siegel domains. This decomposition is unique up to an order.

THEOREM

Proof Suppose that D{V^^F) is a decomposable normal Siegel domain associated with a normal matrix set {Afj Q L } of type N. Let S = (riij) be an iV X TV symmetric matrix. If T(D(Vj^,F)) is a maximal triangular subgroup acting simply transitively on Z)(V^,F), there exists an admissible permutation a : i\i2 " -i^ of 1,2, • • •, AT, such that S^ =

140

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

i'^ijik) ^^ ^ block diagonal matrix. Without loss of generality, we can cissert that 5 is a block diagonal matrix, such that a diagram of any diagonal block gives a connected diagram. Thus the normal J Lie algebra (t(D(V^,F)); Jji/?) of T{D{V^,F)) has a direct sum decomposition of some minimal J ideals: t ( i ) ( y ^ , F ) ) = ^ i + . . . + .ftt.

(3.105)

It follows that there exist connected normal subgroups Gi, •••, Gt in T{D{Vj^,F)), such that Gi is generated by exp (.^), i = 1, • • •, t and T{DiV^,F))

= GiX'-'xGt

(3.106)

is a direct product, because the center C (T {D{Vj^,F))) of T{D{V^,F)) is a trivial subgroup. Now, ^ , i = 1, • • •, t, are all normal J ideals, which axe denoted by {^'^J^ijj), 1 < i < t. Hence Gi determines a normal Siegel domain £)(V^.\ F ( ^ ) ) , 1 < i < t, according to Section 3.1, Section 2.4 and Definition 3.3. Since T{D{V^,F)) acts simply transitively on J D ( V ^ , F ) , there is a linear isomorphism r, such that r ( D ( y ^ , F)) is a topologic product as follows: r ( D ( F ^ , F ) ) = £'(y^;),F(i)) X ••• X D ( F « , F W ) .

(3.107)

Since the decomposition of a normal J Lie algebra by the minimal J ideals is unique up to an order, the above decomposition is also unique up to an order. I The above theorem shows that we only need to consider the indecomposable normal Siegel domain. But we need to know if the indecomposable normal Siegel domain cannot be decomposed as a topologic product by two domains of lower dimensional. Since any Siegel domain is holomorphically isomorphic to a bounded domain, the following discussion can answer this question. 3.5 A domain D in C^ is said to be decomposable, if D is holomorphically isomorphic to a topologic product of domains Di and D2 of lower dimensional. Otherwise, D is said to be indecomposable. DEFINITION

3.23 Let D EC^ be a homogeneous bounded domain. Then (1) If D is a topologic product of domains D\ and D2 of lower dimension, then the domains Di and D2 are both homogeneous domains. (2) Any homogeneous bounded domain D can be decomposed to a topologic product of some indecomposable homogeneous bounded domains. This decomposition is unique up to an order.

THEOREM

Normal Siegel Domains

141

Proof (2) Since any homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain, using Theorem 3.22, statement (2) holds. (1) Without loss of generahty, we can assert that D is a homogeneous bounded domain, which contains the origin point and the Bergman metric matrix T{z,z) of D satisfies T(0,0) = E. li D = Di x D2 is a direct product of bounded domains Di and D2, then the point coordinates in Di and D2 can be expressed by x = (xi,-• • ,a;r) and y = (yi5 •••? 2/5)5 respectively, and the point coordinates in D can be expressed by 2: = (zi, '",Zn) = (xi, • • •, x^, yi, • • •, ys), where r + s = n, Zi = Xi, 1
= Ki{x,x)K2{y,y),

T{z,z) =

dmg{Ti{x,x),T2{y,y)), (3.108) where K{z,'z), Ki{x^x), K2{y^y) are Bergman kernel functions, and T{z,l)j Ti{x^x), T2(y,y) are Bergman metric matrices of D, Di, Z)2, respectively. Since r ( 0 , 0 ) = E, we have ri(0,0) = Ei, T2{0,0) = E2, where E, Ei, E2, are all unit matrices. The Jacobian of the Bergman mapping K{z,z) (3.109) w = grad^log K{0,z) z=0 IS

dw = Tiz,0), (3.110) dz and there exists an e > 0, such that det {T{z, 0)) > 0, \z\ < e. Hence the Bergman mapping w = {u,v) = grad^log

Kiz,z)\

KiO,z)

z=0

gives an admissible coordinate transformation at the neighborhood of the origin point in J9, where u and v are both admissible coordinates at the neighborhood of the origin point in Di and D2, respectively. By these new coordinate systems as above, the Bergman kernel function of D is ^

I

diij I - 2

K{w,w) = d e t - I oz I

K{z,z)

= \detT{z,0)\-^K{z,z),

the Bergman kernel functions of domains Di and D2 are Ki{u,u)

=

\detTi{x,0)\-^Ki{x,x),

K2{v,v) = |detr2(y,0)|-2ii:2(y,y),

(3.111)

142

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

respectively. And the Bergman metric matrices of domains £),£>!, D2 are f{w,w) = T{z,Or'T{z,z)T{0,zr\ fi{u,u)

= Tiix,0)-'Ti{x,x)Ti{0,x)-\

niv^v)

=

(3.113)

T2{y,0)-'T2{y,y)T2{0,y)-\

respectively. Let ^ = 0. Then w = 0, thus f{w,0)

= E,

Ti{u,0) = Ei,

f2{v,0) = E2

and f{w,w)

=

dmg{fi{u,u),f2{v,v)).

(3.114)

Now, -D = Di X D2 is a homogeneous bounded domain, aut {D) is the Lie algebra of the holomorphic automorphism group Aut (D) on D. Given ^ = ^Vii'"',v)Q—+ i=l

5 ^ T]i{u,v)—e *

i=r+l

aut (D),

'

denote r]{u, v) = {rii{u, v),---,

r)r{u, v)) = {ii{z), •••, ^r{z))Ti{x, 0),

C{U, V) = (77r+l(M, V),---,

r]n{u, v)) = {^r+1 {z), • • • , ^n{z))T2iy,

0).

Then

Denote / drt{u,v) du

V

dv

dC{u,v) du

\ A{u^v)

B{u^v)

C{u,v)

D{u,v)

(3.115)

dv

I

where all elements of matrices A{u^ v)^ B{u^ v), C{u, v)^ D{u^ v) are holomorphic functions. Now, B{u^ i;) is an r x (n — r) matrix, by Lemma 1.3 and the proof of Lemma 1.6, Re ( 7/(t^,t;)grad^(log^i(^;?l))' + C(^^,^^)grad^(log^2(^;^))'

+E j=i

9vj{u) dui

,

\:^dCj{v) + ^ dvi )

0.

j=l

(3.116)

Normal Siegel Domains

143

Using the differential operators -^r—^, we have OWkWp

B{u,v)T2{v,v)

+ Ti{u,u)C{u,v)

=0.

(3.117)

Let tt = 0 in (3.117). Then fi{0,u) = Ei implies B{0,v)f2iv,v) + C{u,v)' = 0. Letu = 0again. ThenB(0,v)f2{v,v)+C{0,v)' = 0. Hence C{u,v) = C{0,v). By B{0,v)f2{v,v) + CiO,v)' = 0 and f2(0,v) = E2, let v = 0. We have B{0,0) + C{0,v)' = 0. Hence C{u,v) = C{0,v) = - B ( 0 , 0 ) is a constant matrix. Let ?J = 0 in (3.117). Then B{u,v) + fi{u,u)C{u,0)' = 0. Let U = 0 again. Then B{u,v) = - C ( 0 , 0 ) ' = B{0,0). Let J5(0,0) = P be an rx{n—r) constant matrix. The above discussion gives B{u, v) = P, C{u, v) = -P\ (3.118) Hence r]{u, v) = rj{u) - vP\

C^{u, v)=uP

+ l{v),

where r]{u)^ Q{v) are holomorphic vector functions. Suppose that P = 0, VX G aut {D). Then

hence exp {tX) = (exp (ty))(exp (tZ)), Vt G R, is a one parameter subgroup. The coordinate expression of exp {tX) is x ^ / ( x , t),y -^ g{y^ t), Vt G R, where x —^ f{x^t) is a holomorphic automorphism, t G R, hence Y G aut(£)i). It proves that Y G aut(Di), Z G aut(D2) and aut(£)) C aut(£)i) -l-aut(Z)i). Obviously, for any holomorphic automorphism aij u ^^ ai{u), V ^^ V induces a holomorphic automorphism, hence aut (Di) + aut {Di) C aut {D). Since the unit connected component Aut (D)^ of Aut (D) is generated by exp (aut {D)), we have Aut(D)^ = Aut(Di)^ X Aut(D2)^ is a direct product. Now Aut{D)^ acts transitively on D impUes that Aut(J9i)^ acts transitively on Di, z = 1,2. Therefore Z?i and D2 are both homogeneous bounded domains. If there is an X G aut (D), such that P 7^ 0, then (3.117) and (3.118) give fi{u,u)'P = P'T2{v,v), so ?; = 0 implies fi{u,u)

'P = P,

P • f2{v,v)

=

P

144

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Thus Ti{u^u) has a constant eigenvector a with the eigenvalue 1, V |ix| < e, \v\ < e. Hence there exist two unitary matrices U and F , such that Ufi{u,u)U'

= dmg{Ei,fio{u,u)),

v'f2{v,v)V

=

dmgiEi,f2o{v,v)),

where Ei is an sxs unit matrix and Tio{u^u), T2o{v,v) do not have any constant eigenvector a with the eigenvalue 1, V|?/| < e, \v\ < e. On the other hand, diag (El, fio(^, u))UPV = UPV,

UPYdiag {Ei, f2o{v, v)) = UPV,

Without loss of generality, let u = {ui,U2), v = (^'i,t;2), ui, vi G C^. Then fi{u,u)

= dmg{Ei,fio{u2,u^)),

^2(1;,^) = diag(£;i,f2o(^2,^)),

= {exp{\ui\'^))f{u2,U2),

K2{v,v) = {exip{\vi\'^))g{v2,V2)'

and Ki{u,u)

Given a vector field

where A, 5 G gl (5, C), then X satisfies (3.116) if and only \i B = - A \ Hence Qf

X = viA-

— d^

uiA -;:—• e aut (D) OUi

OVi

by subsection 1.2. Denote

&, = { \Y = r,iu)^}

e3 = { \Z =

av)^}.

Then aut (D) = S i + 62 + ©3- Now, given A = Ei, then

and

in aut (D) imply that T]{u) = TJ(U2),

Civ)

= C(V2),

»?0(«) = UlA,

Co(v) =

ViB

Normal Siegel Domains

145

by a direct computation, where A, B are skew-symmetric matrices, hence aut (D) has a direct sum decomposition of ideals: aut (D) = &[ + ©2 + 6 3 , where

l

oui

ovi

| y l € g l ( s , C ) , Hi+Wl

©'2 = { . ~ M | ; ^ } ,

du\

dv\

= 0, i = l , 2 | ,

e^ = {cM^}>

and exp(©i) c SU(2s,C) nlsoo(I>). Therefore given (tti,0,^1,0) ^ 0, there is an e > 0, such that \u\\ < e, \vi\ < e, and there is not any holomorphic automorphism a G Aut{D), such that a{0) = (•ui,0,ui,0). It proves that Aut (D) acts transitively on D if and only if s = 0. I

4.

Bergman Kernel Function of Normal Siegel Domains By (3.60) and Definition 3.1, (3.94) and Definition 3.3, we have

3.24 A normal Siegel domain Z>(V^, F) in C " x C " has four expansions as follows:

LEMMA

D{V^,F)

= { Im (Ciiz)) - Re {Rj{u)Rj{u))

> 0, 1 < j < N }

= { det (Im {Cj{z)) - Re {Rj{u)E~{u)')) > 0, 1 < j < N } = { Cj{Im{z) - F{u,u)) >0,l<j0,l<j
{Fn{uu),F2{u,u),F22{u,u),---,F^{u,u),Ff^j,{u,u)),

Fj{u, u) = (Fijiu, « ) , • • • , Fj-ij{u,

u)) (3.120)

and Fu{u,u) = UiT^i, m (r, ^ Fijiu,u) = ( R e ( u i Q g ^ ) , - - - , R e ( « 4 ; * ^ V , . ) ) . By the above two sections in this chapter, we have

(3-121

146

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

3.25 Let D{V^,F) be a normal Siegel domain in C^ x C ^ . Then there exists a triangular group T{D{Vj^^F)) acting simply transitively on D{V^,F)j where

LEMMA

TiD{V^,F))

= { aia)P^,^^^) | Va € W^, 6 G E^, t^o € C ^ }, (3.122)

Wj^ ={a

= (aii,a2,a22,--*,%5%iv) ^^"^ I «ii > O r - - , % i v > 0 } (3.123) and P{b,uo) ^^ defined by w = z - 2y/^F{u,

UQ) + \ / ^ F ( t i o , uo) -b,

v = u- UQ]

(3.124)

cr(a) is defined by (Wjv) = a{a){zju) =

{a'{a)z^a'\a)u)j

where cr'ia) is a real linear automorphism satisfying the condition Cj{a'{a)z) = Pj{a)Cj{z)Pj{ay, ( ^30

^jJ+1

(3.125)

*iiv

\

(3.126)

Pjia)

V

a^^£'("^^^) /

and a" (a) is a linear automorphism acting on C " satisfying the condition Rj{a"{a)u) =

(3.127)

Pj{a)Rjiu).

Hence lm{Cj{w)) -

'Rie{Rj{v)Rj{v))

= Pj{a){lm {Cj{z)) Proof

Given a point

(ZQ^UQ)

(3.128)

Re{Rj{n)R^'))Pj{aye D(V^,F), then

P{Reizo),uo)i^OjUo) = (V^a:o,0), where XQ = Im^o — F{uo^ ^o) ^VN- By Lemma 3.11, there is an element a E PF^, such that a^{a)xQ = VQ, thus ^(a)P(Re(zo),uo)(^0,^o) = ( v ^ t ^ o . O ) ^

D{V^,F),

(3.129)

In order to prove that T{D{V^^F)) is a Lie group acting simply transitively on D{Vj^,F), we need to find a eW^^, b eR"" anduo £ C"^, such that <7{a)P^b,uo){^/^v„0) = {V^v„0).

Normal Siegel Domains

147

Now,

thus

By the condition cr(a)P(5^^Q)(\/^'i;o, 0) = (V^^-UQ, 0), we have a^\a)uo = 0, (7\a){v^ + F{uo,uo)) = v^, (j\a)b = 0. Thus 0 = Cj{a\a)b) =

Pj{a)Cj{b)Pj{ay

imphes Cj{b) = 0. Hence 6 = 0. And 0 = Rj{a"{a)uo) = Pj{a)Rj{uQ) impUes RJ{UQ) = 0, hence uo = 0. Since cr^ajv^ =VQ, we have E = Cj{v,) = C , ( a ' ( o K ) = Pj{a)Cj{v,)Pj{ay

=

Pj{a)Pj{ay,

thus a = VQ. Hence

if and only ii a = v^^ b = 0^ UQ = 0. It follows that cr(i'o)P(o,o) = id. Therefore the Unear group T(£)(V^,F)) acts simply transitively on

D{v^,n THEOREM

• 3.26 Suppose that

DiV^,F)

= { iz,u) e C " X C - I Im(C,(^)) - Re {Rj{u)Rj{u)')

>0,

l<j
is a normal Siegel domain associated with a normal matrix set

of type N. Then the Bergman kernel function of D{y^,F)

is

N

K{z,u',^,u)

= CQ\\dei{lm{Cj{z))

-Re{Rj{u)Rj^^^

(3.131)

where co is a positive constant, and //i, • • •, /x^ G R Z5 the unique solution of the linear equation system k

Y,N^3k 3=1

= -{rik + n'k + rrik),

l
(3.132)

148

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where nk = nik + n2k + -' + rikk, n'k = ^fcfc + rifc,fc+i + • * * + rikN, Proof Let K{z,u;'z^u) We have

be the Bergman kernel function of

l
K{z, u; z, u) = K{Z, U; Z, U)|det ^^^' ^^ '^ d{z,u) by (1.61), where (z^u) -^ {Z,U) is a holomorphic automorphism on D(V^,F). Given (2:0,1^0) ^ -D(V^,F), there is an element a € W^, such that a'{a)xQ = t;Q by the proof of Lemma 3.11, where x^ = Im(2:o) — F{uo,uo)eVj^. Thus

by (3.129). The Jacobian determinant of the mapping P{Re{zo),uo) is equal to 1; the Jacobian determinant of the mapping a'(a) is equal to N

%k

^ ^y (3.125) and (3.126); the Jacobian determinant of the map-

k=l N

ping a^^{a) is equal to TT a^j^ by (3.127), thus the Jacobian determinant fc=i N

of the mapping a{a) is 1 1 ^ ^ ^ ^'' ^^' Hence N k=i

By (3.125), Cj{a'{a)x,)

=

Pjia)Cjix,)Pj{ay,

Since Cj(cr^(a)a:o) = CJ{VQ) = £J and Pj(a) is an upper triangular matrix, we have det Cj{x,) = (det P,(a))-2 = (a^fa^^^,

• • • al^^Y^

Hence N

^^ij\ogdetCj{x^) i=i

N

= -2Y^iijnjk^ogakk i
= 2 ^(rifc+nj^+mfc) log a^fc. fc=i

Normal Siegel Domains

149

Therefore N

K{zo,uo]Zo,uo) = Co JJ(detCj(Im(zo) -

F{uo,uo)))^^,

where (2:0,^0) eD{V^,F).

I

By Theorem 3.26 and a direct computation, we have 3.27 Let D{Vj^,F) be a normal Siegel domain associated with a normal matrix set {A*^, Qtj} ^f^VP^ ^ - Then the Bergman metric ofD{V^,F)is

THEOREM

ds^ = dd log K{z^ u; z, u) N

=

-Y,HtT[Cj{xr'Cj{F{du,du))] (3.134) N

- 5;]/i,tr [C,(a:)-iC,(da:)C,(a:)-^C,(S)], where X = — 7 = ( z -'z) - F(u, u), and dx = —-^zdz — F{du, u), 2v—1

dx =

j^dz 2\/—1

- F{u, du).

In particular, the value of ds^ at the fixed point {z^u) = (y/^VQ^O) is ds\

= - ^{uj

+ n'j + mj)\dsj\'^ + - Y^irij + n'j + mj)\dzjk\'^

N

(3.135) where |. represents the value at the point {z,u) —

[\f^v^^^).

Given a normal Siegel domain, then the Bergman kernel function and the Bergman metric can be computed by Theorems 3.26 and 3.27 easily. As an example, we will give the Bergman kernel function and the

150

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Bergman metric on quasi-symmetric Siegel domains (see Chapter 6 and ref. [170]). We know that a quasi-symmetric Siegel domain is aifinely equivalent to a normal Siegel domain D{Vf^,F) satisfying the conditions riij = p > 0,

I
< j < N,

mi

= 1712 = • • • = "^iv = T > 0.

Hence (3.132) and (3.133) give that fc J ] M , = - ( 2 + (iV-l)/9 + r ) / p . Thus 2 + (iV-l)p + r Hence the Bergman kernel function is K{z, u; z, u) = codet (Im {Ci{z)) - Re

{Ri{u)Ri{u)))

2+(N-l)p+T P

(3.136) where CQ is a positive constant, and the Bergman metric is ds2 = 2 + ( i V - l ) p + r^^ [Ciilmiz)

-

Fiu,u))-^Ci{F{du,dii))

+ Ci{lm{z) - F{u,u))-'^Ci(-^dz • Ci(Im (z) - F{u, «))-^Ci (

^dz

F{du,u)\ - F{u, du))].

(3.137) Since any symmetric Siegel domain is a quasi-symmetric Siegel domain, it is easy to obtain the Bergman kernel function and the Bergman metric on any indecomposable symmetric Siegel domain (see Chapter 7).

Chapter 4 OTHER REALIZATIONS

In this chapter, we will prove that the Bergman mapping on a homogeneous bounded domain is a holomorphic isomorphism. By Chapters 3 and 9, it suffices to prove this assumption for a normal Siegel domain. In fact, we will prove that the (generalized) Bergman mapping a acting on a normal Siegel domain D ( F ^ , F ) is a product of three holomorphic isomorphisms ai, a2-, o^:

such that D, = {a,oax){D{V^,F)l

a{D{V^,F))

= D

are both homogeneous bounded domains in Section 4.1. Therefore we give two realizations of homogeneous bounded domains, where the image domain D is a. Bergman represented domain. D is called a canonical homogeneous bounded domain. In Section 4.2, we will introduce a reahzation of homogeneous Siegel domain of the first kind by Vinberg using a T algebra and a realization of homogeneous Siegel domain of the second kind by Takauchi using a special T algebra. But Vinberg's and Takauchi's realizations are all real parameter expressions.

1.

Homogeneous Bounded Domain Realization

4.1 Let D be a bounded domain or an unbounded domain, such that D is holomorphically equivalent to a bounded domain in

DEFINITION

151

152

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

C". Denote by K{z,^) phic mapping

the Bergman kernel function of D. The holomor-

a:

w = grad jlog 5 / ^ ' - I L _ ^ (4-1) K{zo,z)\ z=zo acting on D is called a Bergman mapping on D, where ZQ E D is a fixed pointy and P is an arbitrary constant non-singular complex matrix. The original Bergman mapping is defined by «;=(grad^log^^) \

K[Zo^Z)/

Tizo,zo)-K

(4.2)

z=zo

where T{z, t) is the Bergman metric matrix, and ZQ is a fixed point in D. Clearly, given a Bergman mapping a, where (y{z{)) = 0, then the Jacobian matrix J^- of a is Mz)

= ^=T{z,z^)P.

(4.3)

Since ^o ^ ^ is not a zero point of detr(2,;^o), detr(2;, ^o) is not equal to zero at a neighborhood of 2:0, thus a induces an admittable chart at a neighborhood of ZQ. By Lemma 1.4, we have 4.1 Let D he a hounded domain in C^. Given a point zo G D, there exists an admittahle chart at a neighborhood of ZQ by the Bergman mapping a: THEOREM

w = gmd^\og^^^\ K[Zo^Z)

T{zo,zo)K

(4.4)

\z=zo

such that the maximal connected isotropy subgroup {Isozo{D))^ acting on D is contained in the unitary group U (n). It is easy to prove that 4.2 Let K^{z^'^\z^'^^) be the Bergman kernel function of a bounded domain D. in C^. Given a fixed point ZQ e D., i = 1^2^ then

THEOREM

(1) ifr: z^^^ = fi^^^"^) is 0, holomorphic isomorphism acting from D^ onto D^, such that

T(4") = 4''. then the Bergman mapping at : w^'> = grad-Tjylog

'^ ,.'

^

t(i)=zk'^

Other Realizations

153

acting on D., i = 1,2, satisfies the relation

Hence a^or oa^

is a linear isomorphism.

(2) if T is also a one-to-one real analytic mapping on the boundary dD^ of D^, then T(dD^) — dD^^ If the Bergman mapping a^ is a holomorphic isomorphism acting from the closure of D^ onto the closure of D^, then the Bergman mapping a^ is also a holomorphic isomorphism acting on the closure of D^. By the above theorem, if the Bergman mapping on a bounded domain D is a holomorphic isomorphism, and a bounded domain D is holomorphically isomorphic to D, then the Bergman mapping on D is also a holomorphic isomorphism. Several questions arise on the Bergman mapping a: w = f{z). (1) Does a preserve the domain? Namely, is (T{D) a domain? (2)

Does a preserve the orthonormal basis? Namely, if (pi{z), (p^{z), •••

is an orthonormal basis in the function space Hoi (D) fl L^(£>), is ip^{z)det Jaiz) = ^^iw)j ip^{z)det Jaiz) = ^{w)^

•••

also an orthonormal basis in Hol(cr(D)) fl L^(cr(Z)))? (3) Does (7 preserve the Bergman kernel function? Namely, when cr{D) is a domain, is

KiD)i^^^)

= KioM'^)^'^)

=

K^{z,z)\detT^{z,zo)r''

the Bergman kernel function of (T{D)1 (4) Is a locally one-to-one? Is a one-to-one? In general, any question as the above does not have an obvious answer. However we can prove that the Bergman mapping a is a holomorphic isomorphism acting on homogeneous Siegel domains. Hence the Bergman mapping is also a holomorphic isomorphism on any homogeneous bounded domain by Chapter 9 and Theorem 4.2. In order to prove that the Bergman mapping on a normal Siegel domain D is a holomorphic isomorphism, we construct three mappings as follows.

154

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

4.3 Let D{V^,F) be a normal Siegel domain. Given the mapping (p^:

THEOREM

j
(4.5) Uj = vj,

l<j
then (p is a holomorphic isomorphism acting on -D(V^, F). Let a-^ = (p^^. Denote by

y = (ti,y2,t2r--,y;v.^iv). y^ = (yiir--,y,„i,,), tjeC, y,. GC^^^ the points coordinates. Then the mapping ip^ has the matrix representation: Cj{z) + V^E

= Pjiy)Aj{y)-'Pjiyy,

Rj{u) = i?,(^),

1 < j < iV, (4.6)

where I

3

PM

y3,3 + 1

'''

tj+iE

••• i?52i,iv(2/)

t^E

V

\

VjN

(4.7)

J

and Aj{y) = diag (tj, <,+IE("^.^+I), • • •, <^E("i^)).

(4.8)

The image domain D,=a,{D{V^,F)) is defined by

D,=a,{DiV^,F)) = {{y,v)\E-Xj{Xj)

> 0 },

(4.9)

where X,

(4.10) "l2

^22

Other Realizations

155

and a^}=E-

2^^(P,(2/)')-^A,(2/)P^(y)-\

ag =

^/2{Pi{y)')-^^^{y)PM-'R^{vl

(4.11)

Proof It is easj^ to show that a^ is a holomorphic isomorphism acting on D(y^^F). We want to find the exphcit expression of the image domain D^. Denote /

Cj{z)

Rj{u)

w^ =

Rjiu)'

v/^£;K)

\ ^/^Rjiu)'

y/^Rj{u)

\

0

0

(4.12)

v^£;("^^) /

and E P= I 0 0

- I m Rj {u) E 0

- R e Rj (u) 0 E

(4.13)

Since ( lmCj{z)-ReRj{u){Rj{u))' F(ImW,)P' =

V

0

o \ (4.14)

0

E

0

0

0

EJ

ImCj(2) — Rei?j(M)(i?j(tt))' > 0 is equivalent to lmWj>0,

l<j
(4.15)

In terms of the Cayley transformation ^j = {Wj - ^/^E){Wj

+ ^/^E)-\

1 < j < N,

(4.16)

we have

= E-

{Wj + V^E)-'^{Wj

= 4{Wj + \/^E)-\lm

- ^^E){Wj

+ V^E){Wj

Wj){Wj - y/^E)-'^

> 0,

-

V^E)-^

1<3
Therefore D{V^, F) can be represented by E - ZjZj

> 0,

l<j
(4.18)

156

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

By a direct computation, _ ^j =

/ a^

ai2

ai3

<2

«22

«23

<3

«33

\ «'l3

\ »

(4-19)

/

where

a,, = (Cjiz) - V=lE){Cjiz) + V^Er\ Oi2 = {Cj{z) + V^E)-'^Rj{u) = - v ^ a i s , «22 = —2-^ji^yi^ji^)

+ V^Ey^Rjiu)

= -yf^a^s = -033. (4.20)

Given a unitary matrix by the block partitions as Zji ^

( y/2E

Uj = —\ ^2 \

0

0 0

0

E E

-^T^E ^f^E

then

where Xj=( Hence E — ZjZj

%'\

^""'^

) .

(4.21)

> 0 is equivalent to E - JCjX'j > 0,

l<j
By (4.6), D^ has expression (4.9).

(4.22) I

4.4 The following symbols are the same as Theorem 4.3. The Bergman kernel function of D^ is LEMMA

K^^{y,v^y,v)^h^^U-^^^^^^^^

(4.23)

where bo is a positive constant number. Proof

Since the Jacobian determinant of a^ is equal to 1, N

^D (y^^;y^^) =

K{Z,U;Z,U)

= coY[det{ImCj{z)-ReRj{u)Rj{u)

YK

157

Other Realizations By (4.14) and (4.15), det (ImCjiz)

- ReRj{u)Rj{u))

= detlin(Wj)

By (4.16) and (4.17), = |det {\{Wj + v / = l E ) ) r -2 M e t (Im {Wj)).

det {E - ZjZ-) Hence

det(ImCj(2)-Rei?j(«)i?j(u) ) = d e t ( - ( W ^ + V ^ i ^ ) )

d e t ( E - Z y Z j ).

By (4.21), 6.ei{lmCj{z)-K&Rj{u)Rj{u))

= |det-(Wj + \ / - l E ) r d e t (E-X^X^ ).

By (4.6), we have /P,(y)A^(y)-ip,(j/)' det {Wj + ^f^E)

= det

Rj{u)'

V

^^^Rj{u)'

Rj{u) 2^r^E

0

v^i?,(«)\ 0

2v^£;

/

= (2V^)2-idetP,(y)A,(2/)-ip,(y)' = (2v/::T)2-inLir^'It proves that (4.23) holds. 4.5 mapping a^:

THEOREM

Xij "jj -= tj\ "j

I

Use the same symbols as those in Theorem 4.3. Given a l<j
Xii

N W.

k=j+l

(4.24)

158

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

acting on D^ = a.^(D{V^,F)), such that Aj{x) = Ajiyr\

then a^ is a holomorphic isomorphism,

Pjix) = Pjiyr\

R^{w) = Pj{y)-'Rj{v),

(4.25)

and D, = a,{D,) =: ^{{x,w)\E-

{a,oa,){D{V^,F)) , ZjZj > 0, 1 < i < iV }

(4.26)

is a homogeneous hounded domain, where E-2y/^Pj{x)'Kj{x)-^Pj{x)

V2Pj(xyAj{x)''^Rjiw)

\

^/^Rj{w)'Kj{x)-'^Rj{w)

)

Zi = V2Rj{w)'Kj{x)-'^Pj{x) I

I )

+K'{P^{x)Ri{w)yk^{x)-\P^{x)R^{w))K, (4.27)

and

A = diag ((1 - V^)E,

^'^JF^E)

.

(4.28)

Proof (4.26) can be deduced by (4.25) easily. And then the mapping (Tj is a holomorphic isomorphism acting onto a^{D{Vff,F)). By (4.9), (4.10) and (4.11), then (4.26) holds. It is easy to prove that D^ is a bounded domain. In fact, the first row of Zj is (1 - 2V^x..,

-2^^x.^.^,,

• • •, -2V^Xj^,

V2w.).

By (4.26), the element in (1*'', 1*'*) position of matrix E — ZjZj positive number. Hence

is a

|l-2^/=Ta;..|2 + 4|x,..^J2 + ••• + 4|x.,|2 + 2 K | 2 < l . Therefore \x..\ < 1, |x.J < 1, |w.| < 1.

I

4.6 The following symbols are the same as in Theorem 4.5. The Bergman kernel function of D^ is

THEOREM

N

K^ {x, w; X, w) = \ \ [ det {E - ZjZj'f^, where b^ is a positive constant number.

(4.29)

159

Other Realizations Proof

Now, det

d{y,v) d{x,w)

By (4.25), det

d{x,w) n.

d{y

Lemma 4.4 deduces that this lemma holds.

I

4.7 Let D{V.^,F) be a normal Siegel domain in C^ x C^. Then the Bergman mapping a at the fixed point p = (y/^v^jO):

THEOREM

w = gTad(^^y;)log

K{z^u;z,u)

T{V^v,A-y/^v,,o)-K (4.30)

is a holomorphic isomorphism acting on D{V^yF), where

The image domain D = a{D{V^^F)) is a homogeneous bounded domain, and (J3 = cr o a~'^ o (j-i (4.31) is defined by i-l

£.. = V—la. H S%%

V

"^

(ax.. + 7 ax d

^ t

"

^ A^

k=l

k

x,x^ kk

),

ki^ki^^

(4.32) *

tt.

fc=l

r

k=l

where a. = y n^ + n^ + m^,

1
Proof We will give the explicit expression of the Bergman mapping acting on D{V^,F) at the point N

P={\^VQ,0),

V^

=Yl^i i=i

160

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Since dlogK. -\. = - J ^ M . t r (Cfe(0) + di:

s/^E)-'Ck{eJ,

fe=i

dz>j^

k=i

where K = K{z,u;J,u),

using (3.135), we have

= ^diag(a2,2a2£;("i^),a2,..., 2a2£;K;v),... ,2a^_^£;^"A'-i.^\a2 ,4a2£;('"i), • • • ,40^ E^"^^)). Hence a: {z, u) -^ (^, //) can be written as

^S' = - ^ E i " . t r (Cfc(^) + v ^ E ) - i C f e ( e W ) ,

(4.33)

By Theorem 4.3, the holomorphic mapping cr o aj- 1 acting on D^ can be written as

e,, = V^a, -'^J2n,tr{Pk{y)')-'^k{y){Pk{y))-'Ck{e,,), ^* k=l

^ ? = -^E/z,tr(F,(y))-iAfe(y)(Pfc(y)')-^Cfc(e;;)), "• fc=i

r?f) = ^=--E^^*'^(^'^(2/)')"'Afc(y)(Pfe(y))-'i2fc(^)i2/c(erO • (4.34) By Theorem 4.5, the holomorphic mapping OS = a o cr^^ o a2^

Other Realizations

161

acting on D^ can be written as 2 * e« = V ^ a , - - ^M,trPfc(ar)'Afc(x)-iPfc(a:)Cfc(e,J 2 *

/I

^

a.

**

^ -^ k

kk

ki

ki^^

k=i

^i^ =

-^EM.trP,(a;yA,(x)-ip,(x)Cfc(eW) ^'

k=i

(4.35)

= 2V2a.x.. + -^

^ya^x-'yx^'^x,.Al^., a

Z^

*

2 »7i =

k kk Z-^

fc=l

ki

k]

kti

r

rr'

^

^=^y;/x,trPfc(a;)'Afc(a;)-^i?fc(«;)i?fc(erO

Clearly, a^ is a holomorphic isomorphism. Hence a = G^O a^o a^ is a product of three holomorphic isomorphisms. Therefore cr is a holomorphic isomorphism. Finally, we prove that the image domain D = a{D(V^^F)) is a homogeneous bounded domain. In fact, \{e,ASe!,,...,en^^jqe^^

= \{e,Qt^e[,. • • ,e^^Qt^e',)f

by the definition of a normal matrix set of type N. Thus \^kj^ki^t\ = IZ^^kj^sAj^i^tl

< \xj^.\,

s

WkQt}e[\ =

\Y,w^\sQth't\<\w,\. s

Hence r

r

\I14MQtM\
By Theorem 4.5,

1 _ |1 _ 2V^x./ - 4 J2 K\' - 2|«^.f > 0. 1=3 + 1

=1

162

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Hence 1 _ |1 _ 2 x / ^ a ; . / > 4 ( - ^ « ; , ) ' ,

1 - |1 - 2^/=lxj'

> 4\x.f.

It follows that I - oinr. |2 > 1 - 4|x..|. 1 - 4\x.f |2 v^ > h|1 - o. 2 Vy ^ T^ a ; ^ ,|2r ^> /(1 2\x..\')

Therefore \xjj\ > \xji\^. The same reason impUes that \x..\ > |—p w-l^. Hence 2\x..\ > \w.\'^. By \Xjj\ < 1 and (4.35), we have 2

\U
" ' fc=i

2>/2 *~^

"* fc=i

Therefore £) is a bounded domain.

I

The next lemma gives a matrix property. 4.8 such that

LEMMA

Suppose that A and B are complex symmetric AA' < E,

BB' < E.

matrices, (4.36)

Then det {E - AB') ^ 0. Proof

(4.37)

It is well known that there exists a unitary matrix U, such that

Other Realizations

163

where Ao = diag (Ai, • • •, Xp),

0 < Ap < • • • < A2 < Ai

(see Yichao Xu ref. [242]: "Linear Algebras and Matrix Theory" 1992, Section 11.5 (Chinese)). Now, E-AA'

= E-

UAQU'UAOU'

= U{E

- ^)u'

> 0,

thus 0 < Ap < • • • < A2 < Ai < 1. Let

(

B\\

B\2 \

Bi2

B22;

where Bn is a p x p matrix. Then / Bn Bn det {E-BA)=dLBt{E-[ \ B12 B22 ( E- BnAo 0 \ = det = det (E - BnAo) = det {E -

V -BiaAo

Ej

AQBU).

Now, E - BB' > 0 imphes BnB^'

<E<

Ao ^

Thus (Ao5n)(Ao5n)' < E. Hence det {E - AQBU) ^ 0. Therefore det {E - A'B') ^ 0.

I

4.9 Let D{Vj^,F) be a normal Siegel domain in C " x C"*. Then the Bergman mapping a at the point p = ( V / ^ V Q , 0) is a holomorphic isomorphism acting on the closure D{V^,F) of D{Vj^,F), and a maps the characteristic boundary onto the characteristic boundary. THEOREM

Proof

Given a homogeneous bounded domain D, then det T{z- 0 = XK{z; 0,

ZED,

^e{DUdD),

where A is a positive constant number. It suffices to prove that the Bergman kernel function KQ{x,w;^,r}), y{x,w) € Dg, (C,??) € Dg, does not have any zero point in the closure of a normal Siegel domain D{V^,F). In fact, det {E - Zj{x, w)Z~(J~r})) ^ 0, and E - Zj{x,w)Zj{x,w)

>0,

E - Zj{i,ri)Zj{i,ri)

> 0,

164

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where {x,w) € D^, (^,r/) G {D^ 1)80^). have any zero point by Lemma 4.8.

2.

Hence Ko{x,w]^,rj)

does not •

T Algebra Realization

When Vinberg, Gindikin, Pjatetski-Shapiro gave the reaUzation of homogeneous bounded domains by homogeneous Siegel domains, they wished to give the realization of homogeneous Siegel domains. In 1963, Vinberg gave a realization of an affine homogeneous cone by a subset in a non-associated algebra, the so-called T algebra. Then Takauchi gave a realization of homogeneous Siegel domains of the second kind by a subset of some special class of T algebras. In this section, we will introduce the definitions of T algebras and give the relation between normal Siegel domains and T algebras, then we know that this realizations are all real parameter representations of normal Siegel domains. Given a positive integer N and a real vector space SPt of dimension 2n, let 3Jl be a direct sum decomposition of subspaces as follows:

l
where dim(ajl..) = n . ,

l
(4,39)

and ^ii = l.

ri,.=n.,,

Yl

^i3=^^'

(4-40)

l
The positive integer N is called a rank of 9Jl. Suppose that 9Jt has a multiplication with an involution anti-automorphism a -^ a^, such that (a*)* = a,

(a6)* = 6*a*,

and 9Ji*=an,„

\
(4.41)

It is easy to show that there exists a unique vector e^ in QJlii, such that 6^ = Ci ^ U,

6^ == e^.

Given a vector a in 9Jt, then «=

E l
'^^i'

«.i^^ii'

(4.42)

Other Realizations

165

where %=Aie.,

l
The trace of a vector a is defined by N

tv{a) = Y,\'

(4.43)

i=l

We have tr(a*)=tr(a),

^aem,

tr(e.) = l,

1 < i < iV.

4.2 The following symbols are the same as the above. A non-associated algebra Tl is called a T algebra, ifWl has a multiplication satisfying the conditions:

DEFINITION

(1) m,^m,, = o,jj^k,dJi,^m.,cm,,; (2)

e,a,^. = a..Sj = a,., Va,, e 9Jly;

(3)

tr (ab) = tr (ba), tr {{ab)c) = tr {a{bc)), \/a,b,ce

(4)

tr (aa*) > 0, and tr (aa*) = 0 if and only if a = 0;

(5)

Let

M;

l
Then a{bc) = {ab)c,

{ab)b* = a(66*),

Va,6,c € dJl^.

By (1), (2), (5) in Definition 4.2, SJIQ is an associative subalgebra of dJlj and QJIQ has a subset %(m) = { a G a J t o l « =

X^

«, o CLii = AiCi, Ai > 0, 1 < i < iV }.

l
(4.44) and 9Jl has a subset V{m) = { aa* G OH I Va € 2:(9n) }. DEFINITION

4.3

(4.45)

Given two T algebras

m= Yl ^ii' ^'= l
H l
K^

(4-46)

166

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

a one-to-one mapping a acting from VJl onto SPt' is called a T algebraic isomorphism^ if a satisfies the conditions (1)

N' = N;

(2) a{DJl..) = ^a(i)cr{j)^ ^^^^^ a{l)a{2)'" a{N) is an admittable permutation o / 1 , 2 , • • •, AT. Namely, when i < j , a{j) < a{i), then ^(T{j)cT{i) = 0 ;

(3)

a{a*) = a{ay.

4.10 (Vinberg) Given a homogeneous cone V in 91^; there exists a subset V{VJl) of V, such that F(9Jl) is a realization of V up to a T algebra isomorphism. THEOREM

We give a new proof of the above theorem using the relation between a T algebra and a normal cone. Given a T algebra 9Jl, then tr(a*)=tr(a)

(4.47)

by the definition of a T algebra, thus tr(a6*)=tr(a6*)*=tr(6a*). Hence there exists an inner product (a,6) = tr(a6*),

^a.bedJl

(4.48)

on DJl by (4) in Definition 4.2. Now, a6* = 0,

or

6*a = 0,

VaG9Jl,.,

6G9JI,,,

{i,j)^{p,q).

Thus tra6* = tr6*a = 0,

Mae 9Jt..,

b G OJl^^,

(i, j ) ^ {p,q).

Hence l
satisfies Since dim (DJl..) = 1, there is a basis e^^ = e. in SDT.. satisfying K ? ^ 4 ' ^ ) = ( e . , e j = tr (e,e*) = tr (e^) = tr ( e j = 1.

Other Realizations

167

Given an orthonormal basis eW,...,e;^\

l
(4.49)

in dJl.., then

(ej;),6g)) = ((6(;)r,(ewr) = 5.,. Hence 9Jl,, = 3Jl*. has an orthonormal basis e(p = ( e W r , . . . , e > ^ = ( e > V .

(4-50)

Hence we obtain an orthonormal basis of SDt as the above. The multiphcation table is

^?45=<^ipE(«^^:K)4^

(4.51)

r=l

where A^^ is an riiq x rijq real matrix, 1 < s < n^•^ 1 < t < n-"pq-i Now, (e(^)eW)* = e^e^ implies that

U

U

Hence the existence of the involution anti-automorphism is equivalent to the conditions

^? = E^&<^-

(4.52)

t

Now, {e^(^)} is an orthonormal basis, so S,t = (e(^),e(*)) - tr (eW(e(*))*) = tr (e^^^e^*)) = A'% Hence Af.=e,eR''i^,

Vt

(4.53)

It is easy to show that (2) in Definition 4.2 is equivalent to

Aj/= £;(%•), 4 = e;,

(4.54)

and (3) in Definition 4.2 is equivalent to

4 = E(^li)'<«-

(4-5^)

168

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Hence (4.52) and (4.55) imply A%

= (Afk)'.

(4.56)

Since MQ is an associative subalgebra, so condition (5) in Definition 4.2 is equivalent to

^?(4^^2^) = (e;;)e(.^,Vi?,

l
Hence

where 1 < r < n^^., 1 < 5 < rijk, I
J^

1 < r < n..,

1 < z < j < A^,

Zl bpq^pg is a vector in aJlg, thus

^
s

E E ^ S K^.'i - E(et^:>'J4i] A'fe = 0, 1 < / < A: < iV, i<j
S

t=l

where

Afc=ib'ii\---Mr\ 1 < ^ < «i.> 1 < i
(4.57) holds. When / = j ,

U

U

by (4.53) and (4.54). The above discussion gives the proof of the next Lemma. 4.11 If a real vector space 9Jt of dimension 2n has a direct sum decomposition of subspaces:

LEMMA

i
and there exists a basis eWeOn,,,

l < i < n , , =dim(9Jl,,),

e^ = e„

l
Other Realizations

169

satisfying the condition e^ = e^ in 3Dt..^ I
(4-59)

«;^JS = < ^ P . E ( ^ ^ ^ : K ) 4 ^ r=l

where n.^ x rijq matrices

satisfy the conditions A^i = E^^^i\ 4 A'^ = J2K<^S,

= e[, A^=e„l
Vs,t,

(4.60)

l
(4.61)

t

where l^j,

1 < r < n.^., l<s<

rijk,

l
Y, e.Aje;e,A:^e; + J ] euAp[e,A^ie', u

= 26st5rv.

(4.63)

u

where \fs^t^r^v^ 1 < i < j < k < N. defined by (ef;))* = ejf,

The anti-automorphism * is

l<s
(4.64)

l<s<

(4.65)

The trace tr is defined by tr (ej;)) = K.

n,., l
The structure constants give that there exists a real matrix set {Af.,

l
l
such that A'^ = {A^y,

^& = E ^ ? < « -

(4.66)

4 1 = (^*^^)' = j;e'3e.(Ay)',

S

(4.67)

S

4 = E^<'s = E <««^?' ^^i = (4)' = E(^?)'^:^s

r

r

(4.68)

170

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Clearly, when the indices ijjjk are different from each other, (4.61) and (4.62) are equivalent to (4.66), (4.67), (4.68); when there exist two equal indices in the index set i,i, fc, (4.60) implies (4.61). On the other hand, (4.62) is equivalent to

t

where 1 < r < n.^., 1 < 5 < rijkj l
t

t

where the first formula can be induced by (4.68); the second formula holds obviously. It suffices to check the four cases I < i < j < k^ i
where 1 < r < n^^., 1 < s < n.^, l

By (4.68),

(J^YAti = Y^{e,A^^e[)A%, l
(4.70)

t

By (4.66), (4.67) and (4.68),

U

V

t

U

Hence we obtain the relation (4.70) again. By (4.66), (4.67) and (4.68),

r

V

t

u

Hence we induce the relation (4.70) again. Finally, (4.63) gives that

u

w

u

w

The above formula is equivalent to the relation (A^.^YAf + {A'^^yA":^ = 25srE,

l
(4.71)

Other Realizations

171

By a computation as the above, we have 4.12 Multiplication table (4.51) ofaT algebra Wl (see (4.69), (4.70), (4.71) gives a real normal matrix set

THEOREM

6 = {^J,

l
l
of type N. Conversely J given a real normal matrix set & of type N, there exists a unique T algebra VJl with an orthonormal basis e^V as Lemma 4.11, the multiplication table is (4.59), where & is defined by (4.60), (4.66); (4.67) and (4.68). Hence the T algebra M is defined by a real normal matrix set & of type N. 4.13 Given a T algebras DJli associated with a normal matrix set &i of type Ni^ i = 1,2, then SDTi is isomorphic to SDt2 if and only if &i is unitarily equivalent to 62•

THEOREM

Proof Since the different orthonormal basis transformation in the same subspace 9!Jl.^. is determined by an n.^. x n.^ real orthogonal matrix, this theorem holds evidently. I 4.14 Given a homogeneous cone V{dJV) in R^ as the above, the coordinates of a point x is denoted by

THEOREM

(4.72) then F(SDt) has a real parameter representation: N

y =X^.+

V

if II

/

^^

X,,x'.., ^3

^

l
13 ^

—

—

'

(4.73)

N

Vij = ^33^i3 + E

E^^^^!>;.^-

1 0 , - . . , a : ^ ^ > 0 . Proof

(4.74)

By Lemma 4.11 and theorem 4.12, give a vector

l
t

«»• = EE-i^i?EE4'
t

j
s

ij
172

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where

Since the above summation can be partitioned by aa* i<j
i=j
j
i<j=k

j
i—j=k

using (4.60), (4.66), (4.67) and (4.68), we have

aa* = y

(xl + y

/—•

\

i=l

x.^x'Y''^

Z—i/

**

^3

^3 J

+

/

j=i+l

+ E

y

It

y -J

l
Z—•

x^^x^) f e ^ + e ^ ) J3

tj

\

tJ

J*

/

t

E K<^.4^;e!;)+^,.<e.4i<e(:)).

l
r,t

Since

t

t

where e., e^, e^ are defined by (1.70), thus

i=l

l
t

where ^ and x satisfy (4.72) and (4.73). Given a point x in the subalgebra OHQ of a T algebra with the coordinates y, there exists a basis {eSP, l 0, • • •, a:^^ > 0. Since y in (4.73) is the point coordinates in the homogeneous cone y(9Jt) = { a a * | V a E ! m j , (4.73) gives a real parameter representation of F(9H).

I

Let Fo = {^ e 1 ^ I X,, > 0 , . . . , x ^ ^ > 0} C R"".

(4.75)

By Theorem 4.13, (4.73) gives a real analytic mapping cr, such that

173

her Realizations

Let ( ^.. '

30

• R^^ (x

)

Poi^) =

J

^NNE

\ Then a can be written as Cj{y) = Pj{x)Pj{x)', 3). Therefore we have

I < j < N (see Chapter

THEOREM 4.15 There exists a suitable orthonormal basis in a T algebra Tlj such that the coordinate expression y of a homogeneous cone V{dJl) as the above consists of a normal cone.

Now, we consider a normal Siegel domain £)(V^,F) of the second kind. We will prove that £)(V^, F) is a section of a complex hyperplane and a special normal Siegel domain D(Vjy_^J of the first kind. Namely, we will give a relation between a special normal matrix set of type N and a real normal matrix set of type AT + 1. Using this relation, we will give Takauchi's reahzation of a homogeneous Siegel domain of the second kind, and prove that this realization is a real parameter expression of a homogeneous Siegel domain of the second kind. Given a real normal matrix set of type N: QR ^ I j^tk 1 < f < ^

l
(4.76)

we have an AT x TV symmetric matrix n,

5 -

(4.77)

V

n NN

J

Given a normal matrix set of type N: e^ = e^D{Qlf, l
l
(4.78)

where Q^^^ is an m^ x mj complex matrix, we have an (A^ + 1) x (AT + 1) symmetric matrix: /

^11

n.

^12

^22

\

n. *' *

^2iV

^2,iV+l

(4.79)

So = rir. \

^l,iV+l

n 2,iV+l

n. ^iV,iV+l

n N,N+1 ^iV+l,iV+l

/

174

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where n^+i,^+i = 1, n.^+i = 2mj, l<j
Let

be an n.^_^j x n^yv^i real matrix, where I
l
®;+i={<' l
(4.81)

of type N + 1. By a direct computation, we have 4.16 The following symbols are the same as the above. A real matrix set ©^ '^^ is a real normal matrix set of type N + 1. Conversely, given a real normal matrix set (4.81) of type N + l, ifn^^^^^, • • •; n^^_^^ are all even numbers, and LEMMA

^r=(% \

%]' ij

ij

("2)

/

where 5(*) and C^^ are both rrii x rrij real matrices, 'n^,^_^^ = 2mi, i = 1, • • •, N! Let QW = B » + V ^ C W , (4.83) then the matrix set 6'^ = { 4 € 6 j ^ ^ ,

l
Qlf, l
l
is a normal matrix set of type N. THEOREM 4.17

D{V^,F)

Let

= { {z,u) I ImCjiz)

> 0, 1 < j < N } (4.85) be a normal Siegel domain of the second kind associated with a normal matrix set 6 ^ of type N as (4.78). Given a real normal matrix set &^_^^ of type N + 1 as (4.80) and (4.81), denote by Z)(Vj^_^J a normal Siegel domain of the first kind associated with a real normal matrix set ©^^^^ where the point coordinates in D{Vj^_^^) is

V,, u, G C ^ S

1
- ReRj{u)Rjiu)

u = {u,,'",u^),

v = (^i,---,^iv)-

Other Realizations

175

Give a real hyperplane U: s^^.^l,

u.=-V^v^,

then n n I>(V;v+i) ^«

j = l,---,N,

(AM)

D{V^,F).

Proof Let 1 / £;(mO

.TIE

J

GUK^^J,

l
(4.87)

Thus (4.88) Since { A*'', Q^*) } satisfies the definition of a normal matrix set of type N, 6 ^ ^ , = { ^*^ ^*'^"^\ 1 < t < n,., 1 «. 1SJ<JV + 1,

(4.89)

where

-^j+i,iv-hiv'^j+i,iv+i/

Cji(2:^_^J =

Hence (4.89) is equivalent to ImC,.(z) - ( I m ( 5 ^ ^ J ) - i C , , ( I m z , ^ J C , , ( I m z ^ ^ J ' > 0, Imsj^^i > 0 ,

j =

l,---,N.

Letting u = — y—Iv, we have Im (v) = Re (u),

Im (v) + \ / ^ I m (u) = Re (u) + \ / ^ I m («) = u,

176

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and V2{lmz.j^_^^)Uj = {Imv. + \/^lmu., =

Imv^ -

\f^lmu.)

{u.,u.), r /

= (5^e;(Imz;, + V=Ilm«J(Q5;^), r

r r

r

hence letting v = y—Ttx, C,,(Imz^^,)C.,{Imz^^J

= Re{R.{u)R.{u)),

j = lr-,iV.

Let Sj^_^^ = 1. The section 11 fl i)(V^^J is defined by ImC^{z)-ReR.{u)R.{u)

> 0,

j = l,...,iV.

Therefore H n DCF^^J = D{V^,F),

I

In order to reahze homogeneous Siegel domains of the second kind by Theorem 4.17, we need to add some conditions to a T algebra, such that a homogeneous Siegel domain of the second kind can be realized by a subset in this T algebra. Given a T algebra 9JI with rank iV + 1, the direct sum decomposition of subspaces

m=

Yl

^^i

(4.90)

l
with an orthonormal basis e(*), 1 < t < n... in 9JI,., such that

The multiplication table is

«?4?='^ipE(^^^:/<)4^'

(4.91)

where {^2'

^
+ l}

Other Realizations

177

is a real normal matrix set of type N + 1. Suppose that n.^_^j is an even number 2mi, 1 < i < iV, and

^*^

-[-ImQif

ReQ(J) j '

^^"^^^

where {A'^,

Q«,

l
l
is a normal matrix set of type N. Let

^^ = ( -£;(-)

o(-) ) ' ^ ^ ^ ^ ^-

(^-''^

Hence j.^«,iv+i ^ ^t-^+i j ^ . ,

^'= E

(^^,-+I+^^+M)'

1 < i < j < iV.

(4.94)

2Jto= E

(4.95)

l
^^i

'^
are subspaces of a T algebra 9Jl = ^ UJl.^.. A linear automorphism J acting on dJV is defined by jJt)

_ Jmi-\-t)

jJrrii+t)

_

(t)

7^W

- ^(^i+t)

jArrii-i-t) _ _

(4.96)

(t)

where 1 < t < mi. Hence

jm.^^^,=m.^^^,, jm^^,^.=m,^,,,

I<J
We have an inner product (a, 6) on 9Jl', the matrix representation of (a, 6) is the unit matrix E. Since the linear transformation J has the matrix representation (4.93) on 5[)t.^_^j, 9Hj^_^j ., (Ja)* = Ja*,

(a,6) = (Ja, J6),

^a,be

m\

(4.97)

By a direct computation, (4.94) is equivalent to the condition aJb = J{ab),

Va em^.be

Wl'.

(4.98)

Conversely, given an orthonormal basis in dJl.j^_^^ and an orthonormal basis in 3Jt^_^j ^. as the above, since j2 = -id,

J2n,,^+i=9Ji._^+i,

jm^^,^.=m^^,^.,

I<J
178

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and (4.97), (4.98) hold, the matrix J has a matrix representation Ji on 9Jl^.^_^i and 97l^+i,, (see (4.93)), such that J2 = -E,

Jjfj = E.

(4.99)

It is well known that there exists a basis transformation from one orthonormal basis to the other orthonormal basis, such that the matrix representation of Ji on 9Jt^ jy+i ^^^ 9Jl^_^^ ^. as (4.93), because Ja* = {Ja)\ When i < j , hence U

3,N+l

/

-J * ^

xj

J'

t

t,N-\-l^

r r

Therefore (4.94) holds. 4.18 (Takauchi) Give a T algebra 9Jl of rank N+1 with the direct sum decomposition of subspaces as (4.90). Give subspaces (4.95) and a linear isomorphism J acting on SDt', such that

THEOREM

(1) jm,^^^,=m,^^^„ Jm^^,.=m^^,^,; (2)

(Ja)* = Ja*,

Va € M';

(3)

J2 =

(4)

{Ja, Jb) = ia,b),

(5)

a{Jb) = J{ab),

-id; Vo, 6 6 !H'; Va € SJto, & e OK'.

Then a homogeneous Siegel domain can be realized by {ia + x){a + x)*-

XX* eV{m)},

(4.100)

where F(9Jl) is a subset in QJt', which is defined by {{a + x){a + x)* = aa* + XX* \Vae

^^ l
M.^, x e

J]

^i,N+i }•

l
(4.101)

Other Realizations

179

We give another proof of the above theorem as follows. 4.19 There exists a suitable basis of a special T algebra 9Jl as the above, such that the coordinate expression is a normal Siegel domain

THEOREM

{ (2,«) e C" X C " I Imz - F{u, u)€V} = { {z,u) G C" X C"* I F{u,u) = XX*, a G Von,,., fe^ (4.102) Imz = {a + x)(a + x)*, xeY^dJl^,^^^

}.

i

Therefore Takauchi ^s realization is a real parameter expression of a normal Siegel domain of the second kind. Proof

Given a vector

^ = E^?^
(4.103)

where Xi = (xf \ .. • ,xj''^'^+^^) = {Reui,Imui),

m e C^%

(4.104)

then

i<j

i

Now, XiXi = {Reui^Im.Ui){ReuijIm.Uiy = uu\

(4.106)

and

= Re«iQ(;^, (4.107) hence arx* = 5;;(«,«9e, + i

J ^ 5;](Re(tx,Qj;V,.))(ef^+4^)l
(4-108)

r

Given two vectors a€%{m),

xe Yl ^i,iv+i> l<j<JV

(4-109)

180

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

ax* = xa* = 0. By Theorem 4.14, V{im) is defined by (4.101). Therefore a homogeneous Siegel domain can be realized by (4.100). Now, X = (Re(u),Im(«)), hence z = a + \f-ilm

{z) = a + V^{a

+ x){a + x)*,

Va € M",

where x G Z)ili ^i,iv+i ? ^ ^ '^(SH), so that Takauchi's reaUzation for the homogeneous Siegel domain of the second kind is a real parameter representation (see (4.44) and (4.45)). •

Chapter 5 AUTOMORPHISM GROUP

In Chapters 2 and 3, we proved that any homogeneous Siegel domain £)(V,F) is affinely isomorphic to a normal Siegel domain D{Vj^,F), and the holomorphic automorphism group Aut (D(y^, F)) on a normal Siegel domain Z>(y^,F) has a semi-direct product decomposition of the subgroups:

Aut{D{V^,F))

=

T{D{V^,F))-lso{DiV^,F)),

where T(Z)(V^,F)) is a linear Lie transformation group acting simply transitively on D(y^^F), where Ad (Aut (JD(V^, F))) is an algebraic Lie group acting on the Lie algebra aut (Z)(V^,F)), Ad(T(i?(V^,F))) is a maximal triangular subgroup in Ad(Aut (Z)(F^,F))). The Lie algebra i{D(V^,F)) of T{D{V^,F)) is a normal J Lie algebra. Any homogeneous manifold 3Jl is holomorphically isomorphic to a left coset space G/H, where G is a connected Lie group and i? is a closed subgroup in G, such that the normal subgroup of G in J? has only the trivial subgroup, hence a Lie group pair (G, H) determines a lot of the properties of a homogeneous manifold SDt. Therefore it is very important to determine the holomorphic automorphism group on a normal Siegel domain. Since any homogeneous bounded domain D is holomorphically isomorphic to a normal Siegel domain D{V^^F), we know that the holomorphic automorphism group Aut (D) is isomorphic to the holomorphic automorphism group Aut {D{V^^F). In this chapter, we will determine the maximal hnear automorphism group Aff (VJ^) on a normal cone V^, and determine the maximal holomorphic automorphism group Aut (jD(y^, F)) on a normal Siegel domain

181

182

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

In 1978, J. Dorfmeister ("omogene Siegel Gebiete". Habilitationsschrift, Universitat Miinster) also gave the holomorphic automorphism group Aut {D{V, F)) of a homogeneous Siegel domain D{V, F) by a difference representation independently, but some conditions are not clear, because he did not discuss which homogeneous Siegel domains satisfy these conditions.

1.

Affine Automorphism Group of Normal Cones

Given a normal cone V^, we will determine the Lie algebra aff (Vj^) of the linear automorphism group Aff (F^) on V^ in this section. By Section 3.1, there exists a Lie subgroup T(V^) of the linear Lie transformation group Aff (V^), where Ad(T(F^)) is a maximal connected triangular subgroup of the algebraic Lie group Ad (Aff (V^)) acting on the Lie algebra aff (V^). Hence T(V^) acts simply transitively on V^. The Lie algebra t(y^) of T{V^) has a basis

^i = 2 r , | - + ; ^ . , , ^ + 5 ; x , , ^ , p<j

l<,-
(5.1)

j
4'=-g>|--,^.i:i:K.-s*A ^ij

p
i
s

^*^pi

ip

(5.2)

3
where l
= {Sik-6ij)xfl

[ ^ ^ ^ S l = Si. Ei^nAf/,)xlf U

l
(5.3)

l
l<j
- 5,, Y.^e^Af,e[)xf^.

(5.4) (5.5)

U

where 1 < 5 < n^.^, 1 < t < rip^, l<j
and the

Automorphism Group

183

Clearly, the connected Lie group T(F^) is generated by exp(t(y^)). By Section 3.1, T{V^) = {a'{a)\yaeW^}, (5.7) where t r ^ = { a: G M^ I r, > 0, • • •, r^ > 0 },

(5.8)

and Cj{a\a)x)

= Pj{a)Cjix)Pj{ay,

l<j
xeV^,

(5.9)

where a € W^^, and W^, Pj{a)j Cj{x) are defined in Section 3.1. 5.1 Let V^ be a normal cone associated with a real normal matrix set {Aj^} of type N in R^, and let the Lie group T(V^) be defined by (5.7) J (5.8) and (5.9). Then a T(V^) invariant volume element ofVj^ is

THEOREM

N

V = (p{x)dx^^^A'. .Adx(^) = Ao ]][det {Cj{x)Y^dx^^^A- • -Adx^^^, (5.10) 3=1

where x = ix^^\ - • • ^x^'^^) € V^; A^ is a positive constant, and k

2 E ^ i ^ ^ = -^^-^'fc'

l
(5.11)

The positive integer numbers rik, n^ are defined by k

N

^fc = S^ife' 3=1

Proof

4 = 53^fci,

l
(5.12)

j=k

If V = ip{x)dx^^^ A . •. A dx(^)

(5.13)

is a T(V^) invariant volume element on a normal cone V^, where v^ = (l,0,l,...,0,l)€V;^,(^(^o)>0,then <^((7^(a)a:) = (^(a:)(det ^ ^ ^ ) ~ \

(5.14)

Va G W^. Given a point XQ G V^, there exists a point a G W^, such that cr'(a)t;o = x^^ and

184

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

by Section 3.1. Clearly, (5.9) and (5.14) imply that N

d^«(^)=n''r"*

k

k=l

is a constant. Letting x = v^, we have CJ{VQ) — E, where E is the unit matrix. Thus N

det{Cj{x,))

= det {Cj{a\a)v,))

= det{Pj{a)f

= [14"'^. i=j

k

Since —n^ — ^jt = 2 ^ ^j'l^jk, we have N

N

N

'PM = ^K) 11(11 "il'^P =
fc=j

j=l

where ^{VQ) > 0, Cj(xo) > 0, VXQ G Vj^, so (5.10) is a T(V^) invariant volume element on Vj^. I 5.2 Let Vj^ 6e a normal cone associated with a real normal matrix set {A\^} of type N in R^, and let the Lie group T(V^) be defined by (5.7), (5.8) and (5.9). Then Vj^ has a T(V^) invariant Riemannian metric

THEOREM

N

d'^logifix) = -Y^u.tT{Cj{x)-^Cj{dx))^,

(5.15)

where z/^, • • •,i/^ are defined by (5.11). Proof Denote by d^log(/?(x) — dxS{x)dx' a T(Vj^) invariant Riemannian metric, where S{x) is a real positive definite symmetric matrix on V;,. By (5.14), d^ log ^{a\a)x)

= d^ log (p{x) - S log det ^

V a G M^^. Since dyS{y)dy^ = dxS{x)dx\ £ ^ 5 ( . ) ( ^ ) '

= d^ log (/P(X),

Vy = cr'(a)a:, we have = .W.

,5.16,

Given a point XQ eVj^, there exists a point a G W^, such that cr\a)vQ = XQ. Let a; = I'Q in (5.16). Thus

Automorphism Group

185

By (5.10), N

(flog^ix)

=

dHog[X,lldet{Cj{x)p]

N

N

= dY^iy.tv{Cj{xr'Cj{dx))

=

-Y,^.tT{Cj{x)-'Cj{dx))\

Hence (5.15) holds. In particular, d'^logip{x)\j:=v^ =

-Y^u.tvCj{dxf

= 2 X/(^^ + ^i)(rfn)^ + Ylirii +

n'i){dxij){dxijy

i<j

is a positive definite quadratic form of dx. Thus S{VQ) > 0. Hence S{XQ) > 0, \/XQ e V^. Therefore d'^log(f{x) is a T{V^) invariant Riemannian metric of V^. I T H E O R E M 5.3 The unit connected component of the linear automorphism group Aff (VJ^) on a normal cone Vj^ is equal to the unit connected component of the linear isometric transformation group on Vj^ with a T{V^) invariant Riemannian metric (5.15).

Proof Let Aff (Vj^)^ be the unit connected component of Aff (V^), and let K{z^^) be the Bergman kernel function of the normal Siegel domain D{V^). Give X ^ xAin Aff (F^)^, where det (A) > 0, then z -^ zA is also an affine automorphism on D{Vj^), and K{ZA,JA)

= det (Ay'^Kiz.z).

(5.17)

By Theorem 3.26, N

K{z,z)

=

c,lldet{Cj{Im{z))p, 3=1

where CQ is a positive constant, and k

By (5.11), Hj = 2u., I < j < N. Hence there exists a positive constant c[j, such that K{z^'z) = c^^(p{Imz)'^. Now, D{V^) =

{zeC''\ImzeVn}.

186

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Hence

= <det(A)-V(^)2,

\/xeV^

by (5.17). Prom ip{x) > 0 and det (A) > 0, we have (f{xA) > 0, Vx G Vj^. Thus ip{xA) = det iA)-^(p{x). (5.18) It proves that there exists a T(V^) invariant Riemannian metric ds^ = dxS{x)dx' on V^ satisfying S{xA) = AS{x)A^ by Theorem 5.2. Hence X -^ xA is an isometric transformation. I Now, a normal cone Vj^ is Unear homogeneous cone, thus V^ is a complete Riemannian manifold. Hence X e aut(V)^) if and only if X is a Killing vector field. Namely, Lx{ds^) = 0, where Lx is the Lie derivative (see Definition 1.1) and a u t ( y ^ ) is the Lie algebra of the isometric transformation group Aut {V^) acting on V^. Now, we give a exphcit expression of a basis in aff (Vj^). LEMMA

5.4 Let x — {x^^^^ • • •, x^^^^ he a point coordinates in Vj^, where

n = dim (Vj^). A linear vector field X — xB—

G aff (Vj^) if and only if

Xlogip{x) + tr{B) = 0. Proof

(5.19)

Given a linear vector field

if Xlog<^(a;) + t r ( B ) = 0, then Lxids"^) = Lxid'^log(p{x))

= 0. Hence

XeaffiV^). Conversely, X e aff (F^) implies that exp (tX) G Aff (F^), Vt € R. It is well known that exp(fX) can be written as y = xexp{tB), Vt G R. By (5.10), (p{xexp (tB)) = det (exp {tB))~^ip{x),

\t\ < e.

Given a power serier expansions by two sides for the above formula at t, \t\ < e, then (fix + txB + o{t)) = (1 - <(tr {B)) + o{t))(p{x). Comparing the coefficient of every linear term of t in the above formula, (5.19) holds. I

Automorphism Group

187

Now, the triangular linear group T(V^) acts simply transitively on V^. Let ISOQ(V^) be the isotropy subgroup at the fixed point

There is a semi-direct product decomposition of subgroups: ASiV^)

= T{V^)^Iso,{V^).

(5.20)

Let t(V^) and isOo(V^) be the Lie algebras of T(V^) and IsOo(V^), respectively. There is a direct sum decomposition of subalgebras: BS{V^) = t{V^) + iso,{V^).

(5.21)

Hence it suffices to compute the Lie algebra isOo(F^). By Lemma 5.4, isOo(^^) = {X = xB^ LEMMA

5.5

I v,B = 0, X\ogip{x) + tv{B) = 0 }. (5.22)

Given an n x n real matrix B, then

if and only if v^B = 0, tr (B) = 0, J 2 z / . t r {Cj{x)-^Cj{xB)) Proof

By Lemma 5.4, xB—

tr (B) = 0, and xB— 5.1,

= 0, Va; € V^. (5.23)

e aff (F^) if and only if Xlog(^(x) +

e isOo(V^) if and only if t;oB = 0. By Theorem N

^{x) =

\J{&et{Cj{x)p, 3=1

and N

X\ogip{x) + tr (B) = Y^p.ix{Cj{x)-^Cj{xB))

+ tr (B) = 0,

x£

V^.

3=1

Let X = t;o in the above formulas. Then v^B = 0 impUes t r ( B ) = 0. Hence N

J2i^.tT{Cjix)-'Cj{xB))

= 0.

I

188

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

LEMMA

5.6

Give linear vector fields

y(*) - 2x^^^— +r — drj dx^*^ ij

+S^^x^'^eJA'^y— P3 P<» .xi 5'

(s)

* x->

.*«

+ EE-^*4^+E-^.4 t
p^

s

j
(5.24)

d' ^p

on R'^, where I < t < nij, 1 < i < j < N, and aij G R^^^, then 6xp(^X](aijeJ)K^- ), V^ G R, is a one parameter subgroup of the Lie t

-^

group G L ( n , R ^ ) . The coordinate representation of this one parameter subgroup is

Sj = rj + 2xija[j6 + riaija^jO'^, Hpq = Xpq,

P-iQ. "r 3i

vpj = xpj+^J2^?^^i(^pi)''

^ < ^' (5.25)

Vpj

= ^P3 +^Y1 ^^^^^J^%'

Vjp = Xjp + e Y^{aije's)xipAlJ, Proof

i
W^. „y =- f{x,0) is the unique analytic Clearly, exp{6^{aije't)Y;j'):

solution of the differential equation system dsk dB dyij _

dJd

d^ ~*'"'^'

"'

" ' •"

dO

^'^ de = Ei^W(^S)'' ^<^' (5.26)

- ^ = E 4 " i i ^ S ' i
de

E("y^^)^^P^iJ'

^
with the initial value Sfc(O) = r^, 1 < fc < iV, yij(O) = a^y, 1
189

Automorphism Group Give a natural number N and an N x N matrix / mi ^21

rii2 ^22

\ riNi

n]s[2 '' • riNN )

n2N

(5.27)

where n^j axe all non-negative integer numbers satisfying Uii-X,

I
N,

riij = Uji,

l
< N,

(5.28)

and N N (5.29) dim(V;^) = n = ^ m = J]]n-. 2=1 i=i When iV > 3, let indices t^i^j^k satisfy conditions I
l
riik,

l
Ujk,

(5.30)

where ep G R''^^ Cq G M''^^ 5.1 Given an element uij > 0 in the N x N symmetric matrix S {see (5.27)), S is said to satisfy condition (Z(ij)), if an index / G {z + 1, • • •, j — 1} satisfies condition (n^/, nij) ^ 0, then DEFINITION

Tlpi ^=- Tlpi ^=^ "^pji P "^ ^^

(5.31)

'^ip ^ '^Ip ^^ "^jp-) J ^ P^ and for any q, i < q < j , '^iq = riqi = riqjj riiq = niq = nqj,

i < q < I, I
(5.32)

5.7 If nij > 0 and symmetric matrix (5.27) satisfies condition (5.31), then

LEMMA

Y^iA^Jne'set

+ e[es)A'^ = 26stE^^^^\

p < i,

u J2 A^Jie'set + e;e,)(A^)' = 26stE^^r>^\

(5.33) p
190

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS Y^{Ap{e',et

+ e',e,)A^i = 25stE^-^^\

i
U

and A^iA^y

+ 4 ( y l g ) ' = 25,tE,

^ ^ • ( 4 ) ' = E(^^^fc^^tKi

(5.35)

k
(5-36)

k
(5.37)

r

^'kMTjY = T.(^sA]^A)^Z r

Proof Obviously, riu = Uji, j < /, implies that (5.35) holds. Given aij G R^^^, o^ija^ij = 1, whenp < z, then ripi = ripj and R\j{aij) is an Upi x ripi matrix. Now, R\J {aij)R\j {aijY = \otij\'^E = E implies ^!f (<^u) ^ O(npi). Hence u

It proves that (5.33) holds. Let Bll\aij) = E^^^ij^t^

^^^^^ I < i < p < j < N.

We

u

have Hip = Hpj = Uij and ^ g ( ^ g ) ' + <^'(^g)' = 2(5„„£;. Thus Bll\aij)Bll\aijy = E. Hence B ^ V y ) € O(nip). Therefore E = Bll\aijyB^p\aij) imphes that (5.34) holds. Now, we prove last two formulas. Given l
E(^«<^*)^24 = J2('sAU)Y,ie,Al\e',)A^^i r

r

=

V

J2^t{A^kMevAte[A^l

Since Uki = Uku Yl^'r^ti^^^i)' is an Uki x riki orthogonal matrix, so Y.{A'i^iy{<e, + e'Js)A^, = 25svE. Hence Y.et{A]^iy<e,A^A r

= ^sv

r

Therefore ZiesA^kiO^'kAt r

holds. Given k
= ^kl and A^ G 0(ny) imply that (5.36)

Automorphism Group

191

Since Uij ^ 0 implies Af^ e 0{nip), we have Y^ et{A]^Je's^uA]^-e[ = 6su implies ZiesAZe'Ml^iAf^

= 4 ^ . Therefore (5.37) holds.

•

r

5.8 Let V^ be a normal cone in R'^ associated with a real normal matrix set {Aj^} of type N. If the symmetric matrix S is defined by (5.27), and an index (i^j) satisfies condition {7j{i,j)) {see Definition

LEMMA

5.1), then ^^P {J2i^ij^t)uj

) ^^^ ^^ presented by

Ck{y) = Pk{oc)'Ck{x)Pk{oc),

(5.38)

l
where a = {an,ar^,a22r"^Oij^,aNN), OLJJ = 1, I <j
Given k
.

0 \ E

...

?(fc) E^;>iaij)

...

0

Pk{a) = E

Ej

V

By the block partitions of the symmetric matrix Pk{ayCk{x)Pk{oi) — Ck{x) as Ck{x)j all blocks are equal to zero except the f^ row blocks and f^ column blocks, where j * ^ row blocks are denoted by D k , • • •, Djj^, where

D^^ = I^:;\ayR^^\x)'

= i?g)(2/,, - x,,)\

k+

l
Dg) = 4 ) ( a ) ' 4 ) ( a . ) + i ? i ? { x ) ' 4 ) ( a ) + Rf{a)'nRf{a) Df^ = Rf{a)'R^^{x)

= = Rfjiyjj,

{sj-r^)E, - xj,),

j
192

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

by (5.25) and Lemma 5.7, therefore this lemma holds. 5.9 Let matrix set {Ajj} of algebras o/Aff(V^) Then isOy (Vj^^ has

THEOREM

I

V^ be a normal cone associated with a real normal type N. Denote by BS{V^) and isOt;Q(V^) the Lie and its isotropy subgroup IsOt;^(V^); respectively. a direct sum decomposition of subspaces i s o „ j y ^ ) = o ( y ^ ) + y(F^),

(5.39)

where o ( ^ ^ ) = { T.^ijLij^^

I L',,. + Li,- = 0 },

L[„Af^ + Af^L^k = Y^{erU^e[)A\l

(5.40) (5.41)

r

1 < t < Uijj I 0^ ^^^ ^^^ symmetric matrix S = (uij) satisfies condition (Z{iJ)) {see Definition 5.1), where Y^j is defined by (5.24). Proof

Let x = {r^.x^.r^,---.x^^.r^,), and y = isT^,y2jS2r" & Give X = xB— G isOt;Q(V^), where y = xB is

Sk = Yl ^i^ik + J2 ^i^^ijk^ l
(5.42)

i<j

Vij = Yl ^k^^Jk + 5Z ^^^^§' k

^VN^^N)-

l
(5.43)

k
aik € M, bijkj aijk € E^^^, and L^j is an Uki x Uij real matrix. By Lemma 5.5,

i

A:

i

5;^i/.tr(C,(a:)-iC,(a:B))=0,

k
,g^^^.

Va: € V;,.

Since v^ e V^^ there exists an ^ > 0, such that x = v^ — z ^ V^^ \z\ < 6, and oo

Cj{x)-'

= {E + C^{z))-'

= Y.^j{zr, p=0

Cj{y) = CjixB) = Cj{zB)

Automorphism Group

193

is an expansion of the homogeneous polynomials at the point x = 0, so N

Y^ z/.tr {Cj{xyCj{xB))

= 0,

p - 0,1,2, • • •,

\x\<e.

(5.45)

When p = 0,1, we have ^ ( n f c + nk)sk = 0,

"^irik + n'j^)rkSk + 2 J ^ ( n i + n[)xijy[j = 0.

k

k

i<j

Substituting (5.42), (5.43) and (5.44) into the above formula, we have Y

an = ^{'rik + n'k)aik = ^ i

k

aijk = ^ ( ^ f c + ^fc)&2jfc = 0,

k

k

{rik + n'^)aik + (ni + n\)aki = (n^ + n'j^)hijk + 2{ni + n[)aijk = 0, (5.46) ( n , + n ; ) L g + ( n , + 4 ) ( L g ) ' = 0, {kJ)^{iJl L , , + L ^ ^ 0, (5.47) where Lij = V'ly Since

i,j

-^

i<j

i<jk
k

i<3

k

-^

^^^^'

and V^ has a linear isomorphism Ti -^ X^ri^

I
Xij —> XiXjXij^

1 < ^ < j ^ ^5

where Ai,---,A^ ^^ 0 are independent real parameters, thus aff(V^) contains 2^A^A^. Viaij-g:^ + z^2L^"^^^j^k i<j

^

i<j

k

i<j

^iMjk-Q-

k

^-^

i<j

k

Hence X = xB-^r- is a sum of the following vector fields in aff {V^): ox d_

^'^'•^'aT' ^^-^^ 0-

^^'^^^

194

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS ^y%fc^'

^''^^J'^'d^.'

^ij^Z ^Z-'

i<j,k^i,j,

i<j,k
J^riau—+

i,j j^kj,

5'

V•^

i
^

(5.51) "''

d

h'

(5.50)

J2^ijLij—, i<j

V. —

(5.49)

.,A d'

l
(5.52)

3
^'

\^

r^^ ^' J

(5.53)

By (5.44), (5.46) and (5.47), we can give explicit formulas of (5.48)— (5.53) as follows. (1) aij = 0, 1 < ij < N. In fact, the first formula of (5.44) holds for vector fields (5.48) and X, hence aij = 0. (2) aijk = bijk = 0, i < j , k ^ i^j. In fact, the fourth formula of (5.46) holds for the first vector field in (5.49), where i < j , k ^ i^j. Hence hijk = 0. Since the second formula of (5.46) holds for the vector field X, hence aijk = 0(3) V^i = 0, i < j , fc < /, z,j 7^ kj. In fact, the first formula of (5.47) holds for the vector field (5.50), where i < j , k < I, i^j ^ A;,/, hence V^^ = 0. (4) By (1),(2),(3) as above, (5.46) and (5.47) can be written as ^iji — —2aiji ^aiji = — 2aijj ^^ijj = —~

Uj + rij ,T^ijj'i ^-'

^ij "•" ^ij — ^?

(^0.04j

Tli ~T" TIA

(n, + <)(40' + (nfc + 4 ) 4 = 0. (5)

We can give the necessary and sufficient condition for the vector

field (5.51) in aff (V^). In fact, if X = ^ XijLij-—

G isOt;Q(Vj^), where

Automorphism Group

195

L[. + Lij = 0. We know that (5.2) gives X^J € t(F^) and [X,xS] aff (F^). By a direct computation,

8

l
p
s

S

€

^^Ip

O^pl

q
P^

where r

Now, Y = [X,X^]-^{esLpge't)X^l>

€ iso„^(V;^). By the results of (1),

s

(2), (3) as the above and (5.54), (5.55) for F ,

L\l = E ^ t y t e s , l
41 = E(^p')'^t«- P
Li = {Di,y, p
lip -T

^^

lip

rip + n^p^ ^^^

hence (5.41) holds. In this case, given a real number 9, then exp {0X) can be written as 5i = r^, 1 < i < iV, yij = XijOij^ I
a' where I < j < N, Thus X = Yl^ij^ij'E— ^ isOt;^(F^). Hence the subspace o(V^) is defined by (5.40) and (5.41). The subgroup O (V^) of IsOt;Q(V^) is also determined by O {VN) = {si = ri,l
N, vij = XijOij, l
(5.57)

196

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(6) Now we discuss the vector field (5.52). In order to determine the vector field Xij € aff (V)^), we need to use vector field (5.2). Now,

t

i
*^

By (5.54) and (5.55), we have Xij = 0. Hence Xij = Yli^ijj^i)Xij t

• _

(7) Finally, we give the necessary and sufficient conditions for Yij G aflF(Fj^), where Yij is defined by (5.53). Now, Yij £ aff (F^), hence ~ _ '' ~

2{ni + n'i) , d Uj + n'j '''"'''^ dvj

_ ^ , V ^ rii ^ ''''''" dxij + - ^ ^^'^'^' dxij

by (5.54) and (5.55). Let aijj = a. Then isOv^{Vj^) contains the vector field Y, = Y, + l : ( a e ; ) X < ' ' = 2 a 4 ( A - ^

l
'••'

l
s

A

)

+ (., - n ) « | -

'-'Xii

+ E-^'4^+EE"^^M ' By (5.54) and (5.55) again, we have ^ij% ^

^1

^ijj

^

^)

s

6i.j = 2Qf,

6j.-.- = -2—^^ ja, rij + rij

I
Automorphism Group

197

and

't-i ~r i^i

Hence rij + n'-

*•' Sr^

* ^Xi^

-^EE(-'.)-^«:^ Clearly, a = 0 implies Yij = 0. If a ^ 0, Yij G aff (Vj^), without loss of generality, let |a| = 1, and denote

Then [yy,X^*^] € aff (F^) induces that

4^(4?)'= £^, (4)'4=^, j
{B^yB^=E,

Ki,

Thus ^ y , B}^ , C^i are square matrices. Hence nii = '^ij,

I < i;

riii = nij,

i
nu = Uju

j < l-

In the case i < I < j^ rtu = nij ^ 0 imphes riij ^ 0 and aA^^ ^ 0. Now, ni + n[-nj-nj

= ^nii+

^

l
i
nu + ^nu-Y^nijj
l
^ i
mj-^riji 3
is equal to zero. Let a = et G M^^^, and let Yij be denoted by Y-^-\ If i
198

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Thus -Yij = 2 a 4 — + na—-^ + E ^ ' ^ ^ i ' drj dxij f^ " dxiO l
Hence [Xj^'\?y] = - E(et<^a^e'J?if^ implies that m + n'^ = m + n[, i 0, the vector (3 = es{C\i)' is a unit vector and

\x\fM

= Y>tPf^s^!P

€ aflF(FJ.

Thusr^/ Gaff(F^). Hence npi = npi,

p
nip = npi,

i
riip = nip,

I < p.

In particular, mj = nij = rtu, i 0 implies that a = etC^i is also a unit vector and Yij G aff (V^). Hence npi = npj,

p
nip = npj,

l
nip = njp,

j < p-

Therefore if Yij e aff (V^), the symmetric matrix S = (npq) satisfies condition (Z(i, j)) (see Definition 5.1), and -Yij = ZK^^O^i^- • t

(8)

-^

Conversely, if the symmetric matrix S = (npq) satisfies condition

(Z(2,j)), we can assert that Y^y G aff (Vj^). In fact, when AT = 2, exp (^li2 ) ^^^ be written as s^=r^,

s^=r^+

2exf^ + O'^r^, yu = xu + Or^eu

and V^ = {x = {r^,xi2,r^)

G R X R'^i^ x R | r^ > 0, r^r^ - xx2x\2 > 0}-

Thus s^ =r^ > 0, s-^s^—yuy 12 = ^1^2"^12^12 > 0. Hence exp(^112 ) ^ Aff ( F J . Therefore Y}^^ G aff ( F J . In terms of the induction for iV, given a normal cone of the rank M < N, such that M x M symmetric matrix 5 ^ = (npq 0 satisfies the condition (Z(i', j ' ) ) (see Definition 5.1), then Yfl G aff (Vj^), where

1
<M.

Automorphism Group

199

Now, we consider a normal cone V^ = {xeR''\C,{x)>0,'",C^{x)

> 0 },

where anNxN symmetric matrix 5 ^ = i'f^pq) satisfies condition (Z(z, j)). In order to prove Y^. ^ G aff (Vj^), we give the proof by the following three steps. (8a) By Lemma 5.6, if i > 1, then Ck{y) = Pj{vo + eef^yCk{x)Pj{vo

+ Oef^),

1 < fc < z,

is the coordinate representation of e x p ( ^ ^ y / - ), hence Ck{y) > 0, 1 < k 0. In fact, conditions Ci{x) > 0, •••, C^{x) > 0 only depend on the coordinates x — {ri, x^+i, n + i , • • •, x^, r ^ ) , Xk = {xik, '' •, Xk-i,k), where i
{x\Ciix)>Or-',C^{x)>0}

is also a normal cone. Moreover exp {OY^. ^) is a linear automorphism on VAT-Z+I , the coordinate representation can be written as Sk = rk, Sj=rj

i
k^

j,

+ 2x[p + rie^,

Vpq = Xpq,

i
P^qj^j^

Vpj = Xpj + ^Y^ ^^S^tAll,

i
s

yij = Xij -\- uriCti Vjp ^^ '^jp ' "*^ip^ij 5

J ^ P-

By induction, there exists a one parameter subgroup exp(^l^^- ) in the affine automorphism group Aff (Viv-i+i)- Thus Cj{y) > 0, i < j < N. Hence Ck{y) > 0, 1 < fc < iV. It follows that e^p{eY§^) G Aff (V;^), therefore y.J*^ ^aff(V;^). (8b) When i = l<j
then

n{x)Cj{x)n{xy ^ ( ^r^'^(^) ^^o^ ) ,

200

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where I

/

E

XkN

\

\

-r-'

Tk{x) = ?(fc)

V 0

E

/

Give a real analytic isomorphism a: (ti = ri- r'^xiNx'ij^,

1<1
Zpg = Xpq - r-^ J2i^pN^m^'qN)es,

l
S

I ZiN = XiN,

l

where z = {t^,z^,t^,• • •,ZN-i,tN-\), Zj = {zij,• • •,Zj-ij) and

cr%)= 4MI

tk+iEi""""^^)

...

Rtl,N-,i^)

k = l,...,N-l. Then Ck{x) > 0 implies cl!^~'^\z) >0,l 0, rn > 0 imply Ck{x) > 0, 1 < k < N. Hence V^ = {x€ R " | C f - i ) ( ^ ) > 0, • • •, dj^_-,'\z)

> 0 , C M = ( r , ) > 0}.

The dimension of a normal cone VN-1 = { 0 € R " ' I C;^-i)(z) > 0, • • •, Cf_-/Hz) > 0 } JV

is equal to n' = n — Yl i^jN. Moreover,

•^

ij

P
TO

«

d' i
s

3
d' dz IP

Automorphism Group

201

By the induction for Fjv-i, the {N ~ 1) x {N - 1) symmetric matrix SN-1 = (%g) also satisfies condition (Z(i, j)), where 1 < i, j < iV — 1. Hence Y^^ € aff (VN-I). Therefore Y^^ e aff (V^), (8c) Finally, we consider the following case: the N x N symmetric matrix 5 ^ = (ripq) satisfies condition (Z(l, AT)). If there exists i G { 2, • • •, iV — 1}, nn = UIN = niN > 0. By the definition of condition {Z{1^ N)), the indices (1,1) and (I, N) also satisfy condition (Z(1,0) and (Z(l,Ar)), respectively. Hence Y^'l F^JJ € aff(y^). By a direct computation,

r,s

If nil = riiN = 0, i = 1, - • 5 iV — 1, niiv > 0, then

where 1 < j < iST, hence V^={xeR''\C,{x)>

0,cf~^^ >0,l<j
C^{x) > 0 }.

Clearly, V^ = I4 x ^ - 2 is a topologic product, where

{x|Cf-i)(x)>0,...,c}f_-/)(x)>0} defines a normal cone VN-2^ and {x\C^{x) > 0, C^(a;) > 0 } defines another normal cone ^2- Moreover, exp(^lYiy) can be written as

hence exp ( ^ F / J ) e Aff (V^). Therefore F / J ^ ^ff (F^).

•

By Theorems 5.9, 5.3 and 3.6, we have two main theorems in this section. T H E O R E M 5.10 Let F^ be a normal cone. The unit connected component Aff (F^)^ of the affine automorphism group Aff (V^) on V^ is generated by all elements as follows: (1) the subgroup

TiV^) = {cria)\Cj{a{a)x) = Pjia)Cj{x)Pj{ay,

1 < j < iV, Va € W^}, (5.59)

202

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where W^^ = { a € R^ I a n > 0, • • • ,aAriv > 0 }; (2)

(5.60)

the subgroup SO (V^): Si = ri,

1 < i < n,

yij = XijOij,

l
(5.61)

where OikAlpjk

= J2{erOije[)A^,},

1
r

and Oij € SO (riij), l
Va,^,i, GlR^w.,

(5.64)

where ri^,iq{otij,,i^) is defined by

ypu=^puj

P<%

Vip^iq ~ '^ip,iq "•"

P,U^iq,

'^ip^ip.iq^

s y^,ig ~ ^l^iq '^Z^^ip.l^ip^iq^iph tp,i

^p < ^ < %7

s

s

As a corollary, we have the following 5.11 Let D(Vj^) be a normal Siegel domain in C^ of the first kind associated with a normal matrix set {Ajj} of type N. The coordinate representation of a point in D{V^) is

THEOREM

z = (si, Z2, S2r-i

ZN, SN),

Zj = (zij, • • •, Zj-ij),

Zij e C^'^.

Automorphism Group

203

Then the unit connected component Aff {D{V^))^ of the affine automorphism group Aff ( J D ( V ^ ) ) on D{Vj^) is generated by all elements as follows: (1) The subgroup z —>z-\-a, (2)

aeW.

(5.66)

The subgroup

TiDiV^))

= { a{a) | C,(a(a)z) = P,(a)Q(^)P,(a)', l<j
(3)

The subgroup SO {D{V^)):

Si —> Si,

I
Zij —> ZijOij,

1
(5.68)

where

O'ikAlpjk = J2(^rOije[)Al}, l
(5.69)

r

and Oij e SO (riij), I
Vai^,,^GR"^P'^s

where Ti^.i^ioti^.i^) is defined by

Zpu

^ip,iq

'^im^

V
^ ^ip,iq +

P,U^iq,

^ip^ipjiql

s

s

(5.71)

204

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

2.

AfRne Automorphism Group of Normal Siegel Domains

In last section^ we obtained a generator set of the unit connected component of the affine automorphism group on a normal Siegel domain D{V^) of the first kind. In this section, we will consider the sanae problem for a normal Siegel domain D{V^,F) of the second kind. Let D(V^, F ) be a normal Siegel domain in C^ x C"^ associated with a normal matrix set {Aj^, Qf)} of type iV. By Theorems 1.26 and 3.20, the Lie algebra aut (J9(V^, F)) of the holomorphic automorphism group A u t ( D ( F ^ , F ) ) on D(V^,F) has a direct sum decomposition of subspaces: aut {D{V^,F)) = L_2 + L_i + L, + Li + L2. (5.73) Let aff (2)(V^, F)) be the Lie algebra of the affine automorphism group Aff {D{Vj,,F)) on D{V^,F). Then aff {D{V^,F)) also has a direct sum decomposition of subspaces: aS{DiV^,F))

= L-2 + L^i+h,

(5.74)

L_2 = { c i — | V a € R " } , dz

(5.75)

where

L-i = {(3^

+ 2 v ^ F ( « , / 3 ) ^ I \//3 6 C - } ,

L„ = {.4 + . 4 } ,

(5.76)

(5.77)

a' xA—

€ aff (Vj^), B egl (m, C), and F{uB,u) + F{u,uB)

= F{u,u)A,

V-u € C"*.

(5.78)

Moreover there exists a triangular subgroup T{D{V^,F)) acting simply transitively on £)(V^,F), and the Lie algebra of T(Z)(V^,F)) is t{DiV^,F)) where t{D{V^,F))

= L-2 + L-i+t{D{V^,F))nL„

fl L^ has a basis

p<J

^-^

j
-^^

"^

(5.79)

Automorphism Group

205

ij

i
P
s

dzlp'

^^pi

j.
c'z.p

c«, (5.81)

where 1 < t < n^j, 1 < i < jf < AT. In order to find aff {D{Vj^jF)), it suffices to find a basis of the subalgebraLo5.12 Let D{Vj^,F) be a normal Siegel domain in C^ x C"^ associated with a normal matrix set {Afj, Qij} ^f ^VP^ ^ - Then the subalgebra LQ o/aff ( £ ) ( V ^ , F ) ) has a direct sum decomposition of subspaces

THEOREM

L,=t{DiV^,F))nL,+o{DiV^,F))

+ yiD{V^,F)),

(5.82)

where t{D{V^,F))

n Lo = {Aj, Xlf,

l
i<j

^

N},

(5.83)

i

and Lij is a real skew-symmetric matrix, I + Qf.'Kj = J2ierLije'M^\

(5.86)

r

The subspace y(-D(V^,F)) has a basis

i
s

^-^

j
-'^

-'

(5.87) where I < t < Uij, Moreover z|- G aff (Z?(V^,F)) if and only if the N X N real symmetric matrix S = {Tipq) satisfies condition (Z(i, j)) {see

206

THEORY OF COMPLEX HOMOGENEO US BO UNDED DOMAINS

Definition 5.1), such that

Qpj Qij = \S^n^yi^i)Qyi

-> P < '^^

(5 gg^

when

(5.89)

s

S

and rrii = mi = rrij,

i < I < j,

nu ^ 0,

Proof By (5.77) and (5.78) and Theorems 5.9, 5.10 and 5.11, it suffices to consider some vector fields X in LQ , where

^-^

i<j

i<j

ij

t

^

and Lij is an riij x riij real skew-symmetric matrix, I < i < j < N, satisfying (5.85), Kij is an m^ x rrij complex matrix, I < i < j < N, aij e R^^^ such that if aij ^ 0, then N x N real symmetric matrix

WG S = (ripq) satisfies condition (Z(i, j)) (see Definition 5.1). Hence Y^ aff (D(V;^)), where Y^^ is defined by (5.24). Clearly, D{V^^F) has linear automorphisms a: Si -^ Xfsi,

Ui -> XiUi,

I
Zij -> XiXjZij,

I
where A^, • • •, A^ are N non-zero real independent parameters. Hence

a^X) is

Therefore X is a sum of three type vector fields in L^, these are

where

Y^ZijLi—, i<j

-"'^

Y^peaSiDiV^));

Automorphism Group

207 & UiKij — ,

i^j.

(5.92)

(1) A vector field of type (5.92) is equal to zero. In fact, let uB = {v^,--- ,Vj^), where Vi e C"**, Vk = 0, k ^ j and Vj = UiKij. Then F{uB,u) + F{u,uB) = 0 by (5.77) and (5.78). Thus Vju'^ + Ujv'j = 0. Hence UiKijvfj + UjKiju'i = 0. Now, i j^ j , hence Kij = 0. This statement holds. (2) A vector field of type (5.90) satisfies (5.85) and (5.86). In fact, let

l.,'^^Lij^

+

l^u,Ku^^=zA-+uB-.

Then {F{uB,u) + F{u,uB))—

= Y,Fij{u,u)Lij

—

.

^-^

i<j

Comparing the coefficient of ^—, we have UiKau'^ + UiK^^[ = 0. It follows that iiTii is a skew-Hermitian matrix. Comparing the coefficient of —77V, we have F^\uB,u)

+ F^\u,uB)

-

Fij{u,u)Lije't,

where

V = {v^^' " ^v^)j Vj = UjKjj. Thus

It follows that

Ku + Xi = 0, -R^Q^ + Qf^Kjj = j;(e,L,,eJ)Q|,., Hence (5.86) holds. Therefore we obtain the subspace o(J9(y^,F)) c aff(D(y^,F)). (3) When Y^^ € aff (r>(F^)), a vector field (5.91) gives that

Kii = J2(^iA)Qfr t

208

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

In fact, given a unit vector a y G C"'^, let

J

t

where v = uB = ('^ir**?^)? t'fc = 0, A; 7^ j , i^j = UiKij, hence the vector V satisfies F(ie, 'u)^ = F{uB^ u) + F(?x, tiB) = F(t;, tx) + F(u, v). Comparing the coefficients of -r— and Si-— in the formula OSk

OZij

-^^ = EK^*)^.?' we have Fjj{v,u)+Fjj{u,v)

= 2Fij{u,u)a^ip

Fij{v,u)+Fij{u,v)

=

Fii{u,u)aij,

Thus

t

Hence

t

Therefore

It follows that Q^i^Qf- + Ql?Qif = 2(5st-B and rui = rrij. The other coefficients give the relations (5.89) by a direct computation. Now, the above identities can be induced by the conditions rrii = rrij and QlfQif + QfjQif = ^^stE. Moreover, when rnj > 0, Z^f G L^ by a direct computation. Since

[ 4 ^ ' 4^] = E(^*^I/^3)4^ i
(5-93)

r

and

[Xl!^> 4^] = - E(«*^i/<)4^'^ < ^ < •^•'

(5.94)

Automorphism Group

209

when nii> 0, we have Z>[\ Z^^ G L^, Therefore mi — mi — rrij,

I

By Theorem 5.12, we obtain a main theorem in this section. T H E O R E M 5.13 Let D{Vj^,F) be a normal Siegel domain in C x C^ associated with a normal matrix set {Al^^ QijS ^f ^VP^ ^ * ^^^ coordinate representation of a point in £)(V^,F) is (z^u), where z = {si, Z2, 52; •••; ^iV; 8N\ ^j = i^ijj '", ^j-ij)j ^ = ('^1; • • •; ^ N ) - Then the unit connected component Aff (D(F^,F))^ of the affine automorphism group Aff

{D{VN,

F))

= TiDiVN, F)).lso ^^^^ „, (DiVN, F))nAS

(£)(Fjv, F)), (5.95)

on D{Vi^,F) is generated by all elements as follows: (i) (ii)

T{D{V^,F)),

which is defined by Lemma 3.25.

A Lie group SO{D{V^,F)) {see Lemma 3.21).-

Sj -^ Sj,

Zij -^ ZijOij,

I
Ui ^ UiUi,

1
(5.96) where Oij € SO(ny), Ui G U(mi), such that

0',,Al^0^k = Y^{erOije[)A^f,

{UiYQfS = E^^-^^i^*)^-

r=l

r=l

(5.97) (iii) A generator set consists of some one of parameter exp{Zij{pij)), pij e R'^^3^ riij ^ 0. The definition is Sj — ' Sj + Sif3ij0lj + 2zijp^ij, Zij

^n

>• Sifjij

Zpj

p
>• «i + E N^'s'^iQij^

A: / j ,

> Zpj + ZpiiL— yPij)')

^01 "*• Z.^ Pij^s^iq-^ij 1

Zpq —>Zpq,

Ui

-\- Zij^

Sk —>Sk,

subgroups

J- S P "^ ^5

J < 9)

p<j
Up

j

' %'

P^ .?' (5.98)

where Zi0ij)

= J2PijtZif, Pij = (Aji, • • • ,/?ijni,) and Z^*^ is defined by

(5.87), and the NxN

real symmetric matrix S = (upq) satisfies condition

210

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(Z(i, j)) {see Definition 5.1), such that if nu ^ 0, rrii — rrij^

rrii = mi = rrij^ i < I < j^

s

Qif^^' = T.i^tA^i<)Qip^

i
(5.99)

S

3.

Holomorphic Automorphism Group

Give a normal Siegel domain D(y^^F) associated with a normal matrix set [Af^, qf)} of type N, and let aut {D{V^,F)) be the Lie algebra of the holomorphic automorphism group Aut (Z?(Vjy,F)) on D{V^^F). In Chapter 3, Aut (i?(Vj^,F)) has a semi-direct product decomposition of subgroups: kni{D{V^,F))

= T{D{y^,F))^\sOj,{D{V^,F)),

(5.100)

where p = (V^t^o,0) G D{V^,F), and IsOp(J?(V^,F)) is the isotropy subgroup at p in J D ( V ^ , F ) . NOW, a representation element of a connected component in Aut {Diy^^F)) is also a representation element of a connected component in the compact subgroup lsOp{D(y^^F)). In order to find Aut (i)(V^,F)), it suffices to find the isotropy subgroup Isop(D(F^,F)). In Section 1.3, there exists a direct sum decomposition of subspaces aut {D{V^,F))

= L_2 + L_i + L,+Li

+ L2,

(5.101)

where aff (D(V^,F)) = L_2 + L - i + L^ is given by Section 5.2. In this section, we will find a basis of the subspace L\+ L2- We know that the n. .

definitions of Aj, Xij{aij) and Zij{aij)^ ^aijER , 1 < i < j < AT, in (5.80), (5.81) and (5.87), respectively. Now, we introduce some polynomial vector fields on C^ x C"^ as follows:

2B. = ,.(A+^£-)+EE^;''(^«'+»-«!."'^) '*' ' i
s

,

(5.102)

Automorphism Group

211

+EE(^'i'^ae;)(xI?+»i«a''|-) duk' k
s

+

EE^i:herAie^){xS+u,Qt/^) duk'

k
i
r,s

i
s

^"fc^

+E-J?([4^^i?^])' (5.103) Zi(ai) = J^ReCa^eOlf) + 5;imKe;)y/*) t

+ EE4^<<|;+2^-.^(-^+».|-)

+v^EE(%^S^)<^^+v^EEK^g^W)^: p
s

i
s

+^EE(-^5S^'Qj^^^')i^^^ +v^EE(-^QSWW)4?A' ^i ^ C-. a (5.104)

212

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

The multiplication table is d_ ,Aj Idsk

= "^Sjk

(5.105)

dz,klit)'

dsk'

dz kl k—l

rsf

, ^ Xij {aij)] = XI ^*'= a ^ '

dsk i<j

i=l N

d ^

""kl

i<j

&

i
t<3

t

+EE(«^'^«*)««(^'i)'^+ E Ei^^'^kXi) dZil i
t

k
^^kl

t

i<j

*

(5.106)

dz^kj

j=k-j-i

ik

d

(5.107)

^^ki

Kj

^

+ E E(/^'^i«t)«''<-^+E/^'i(^^)'a^-

(5-108)

Using Section 1.3, we have (5.109) if and only if ^ , 1 - 2 ] C LQ and (1.142)—(1.147) hold. Thus dsi

,Y,

., [ dz ,n

= 2Wi G L

= 2<^€Lo.

(5.110)

Hence

n = E^^w+w^/)+E4?(^i'^+(^i'^)')'

(5-111)

r(s) .(«) where W/, (W^^)' are sums of all terms with type * — in Wi, W^/, 9M

respectively. Since Wi G L^ and W^/ € LQ, where

Wi-Yl ^^i^^-+E ^pi(4i)+E ^PMJ) 3

P<3

P<3

Opj

P<3

(5.112)

Automorphism Group

213

fc

p
p
(5.113)

+E-^4''£7^^o, a«fc

+ Y^ ZpkL. p
/?g\/?gf) ^ 0 imply 7(t)

7(t) p r

By (5.110), we have d E ^^r [IT' ^-^ - ^^] + E E 4? [|:, <^ - (
k
s

(5.114) and

i

^^kl

.

i<j

t

^^kl

(5.115) (1)

In the case of a normal Siegel domain of the first kind.

5.14 Suppose that D{V^) is a normal Siegel domain of the first kind, (5.112) and (5.113) are LEMMA

(i)l_

Wi = Yj^iJ^i + Yl^p^^P^ PJ dz p<j

•'

At) _

(5.116)

(5.117)

N

Then Y^^ =2J2

KiBi G L2, where Bi is defined by (5.102).

2=1

Proof

By (5.114) and (5.115), Y^ G L2 if and only if

k

k
s

= E ^^.-/^ [^.M + E E 4? [IT' ^'^K-g?)+z,,ii3W)], k
s

(5.118)

214

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

'^i?'=E»'[i;^.»'.l+EE4*' kl

i<j

- Z^^i3^i[

'dz.

t

dz kl (5.119)

{s)'-^j\ + / , SjesLf. ^^ dz,kl

+EE4*(t)

d

iij)^

.,fe) .XiMW)^^^m') dz^'^

By (5.105)—(5.108) and Theorem 5.12, formulas (5.116)—(5.119) give that Ay = 5ij\i, i 7^ j , Lf) = 0 , i< j ,

by a direct computation, where Xu = \i. (5.117) can be represented by Wi = XiAi,

It follows that (5.116) and

W,f = XijiXjet) + ZijiXiet) = XjX^f + XiZ^f,

respectively. Therefore

i

k
s

k
s

i
s

5.15 Suppose that D{V^) is a normal Siegel domain of the first kind, (5.112) and (5.113) are

LEMMA

W, = ^X,,ia^;^.)

+

P<3

k

^Z,jif3^^),

(5.120)

P<j

p
{p,k)^{ij)

(5.121)

+E E z.'C)+E^.*i!2"dzpk p
^^-EE(4M)^?^^2, i<j

t

where T^j is defined by (5.103).

215

Automorphism Group Proof

By (5.114) and (5.115), Y^ G L2 if and only if

Wi

d

= E«[£.H^EE^

'kl

k
s

d

^EE^^^'^-^'^S^^+^-^^S^) k p<j + EE4?[|:'E^^'''A] k
s

d

+ E E 4 ? £:>E E ds k
s

P<0 (P.j)^{k,l)

^(pj)^ ipj)^ (^P.W)+^p.(0)

(5.122) d

d

(5.123)

By (5.105)—(5.108) and Theorem 5.12, formulas (5.120) and (5.121) give that Wi = ^Xijioij) + ^ Zpiiapi) by a direct computation. It follows that (5.123) can be represented by

As) r a w,kl +EE4^«^4f ^ i<j

- (c^kie'sXAk + AO

t

d ,^ .\

v-„ / a

where

Cij = Siaij + X]E(^P*^*)"PJ'^S + E E^^^P^t^'^PJ^^iP' p
t

i
t

i
t

+EE(^^p^S"ip)^*' j
t

p
t

+EE("^p^^i'4)e'j
t

216

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

By (5.105)—(5.108) and Theorem 5.12, formulas (5.120)—(5.123) give that k
Therefore Y, = Z Ei<^iA)W i<j

^{s)

e L^.

I

S

5.16 Let D{Vj^) be a normal Siegel domain of the first kind in C^ associated with a real normal matrix set {Ajj} of type N. Then the Lie algebra of the holomorphic automorphism group Aut (J9(V^)) on D{Vj^,F) has a direct sum decomposition of sub spaces THEOREM

aut(D(V;^))=L_2 + L o + ^ 2 ,

(5.124)

where L2 has a basis Bi^," ',Bi^,T^%

l
Ui^i^,

\
A sufficient and necessary condition is Z 5 , €L„

l
and when ni^i^ ^Q, i ^i^,

l
(5.126)

-- -, iy-i, i < iv, (5.127)

and 0,

n.lUil' Proof

iu
(5.128)

i +

By Theorem 1.26, a vector field in Li can be written as Fj =

p{z) ^ - , where every component of p{z) is a homogeneous polynomial of oz degree two at z, and Y^ G L2 if and only if [L-2, ^^2] '^ ^o- Hence Idsi

,Y,

^dz ijit)'

=

2W^^eL„ iJ

(5.129)

and (5.130) By (5.130), (5.129) gives r d

E^4^'H+E4j^[^'<]=^^' 1^^^^' (5.131) I

l,j,t

Automorphism Group

217

Clearly, let (A^, • • •, A^) be an arbitrary real vector, then D{Vj^) has a linear automorphism Si —> Xfsij

I
Zij -^ XiXjZij^

T^
N.

Hence Ai —^Ai,

Xij[aij)

—> A^

^ijKPij) ~^ AiA^- Zij[fjij)^

AjXij[(iij), Zij Lij-^—

-^

ZijLij-^—,

By (5.112) and (5.113),

i

i

p<j

p<j

i

+ E E >^^^^p^^ a ^ + E E E >^mXiXjz^A, I p<j

^-^

z<j

t

k

+ E E E X-'Xk\i\jZ^X,,{Jg^) + Y. Xi^X^A^

+EEEv.-^AA,-4j^^p.(C) i<j

t

i<j

t p
p
^^p^

Now, the coefficient of A? is

+ E E -S^^.^(«g^)+E E 4?^^.(/^g?) ^ ^2; the coefficient of A^Aj is

y/^^) = «iX,,(a«) + *,Z,,(/?g>) + Y.11 ^iotkzf^k

+ E E ^g^^p.(«^?)+E E ^g'^p^K^^) + E E 4?^..(^grt+E E 4^^ip(/?.¥)

218

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

i
p
t

i
t

t

Therefore

2=1

l
= EE^^^p^(4?)+EE«^^^^(4i) +E

E

E^J?(^p^("S?)+^.'^(/^£'^))^^2-

y^« = AiBi G L2 by Lemma 5.14, and y^(^^) = EtC^^i^D^? ^ ^2 by Lemma 5.15. Let

where

^^ = EE^p>g)+EE^..(/^?)' p < i Pi^i

p<j JT^i

< = E E {xMa^t') + zM0lf)). p
By Lemma 5.15, Y^ also has a Unear combination of some T^j. Thus the basis in L2 consists of some Bi and some TV . By (5.102), Bi e L2 if and only if Z^'^ e L^, I = i + 1,-" ^N. By (5.103), I^y e L2 if and only ii zf^ e L,, k = i + l,-" .N, zf^ G 4 , fc = j + 1,... ^ iV. On the other hand, [X^IB,]

[xlf,T^]

= 5,^Tfl

[zflB,]

= SikT^l

= 25stBj,

[Z^f ,TJ^] = 2SstBi.

(5.132) (5.133)

Hence if Bi G L2, T^'^ e L2, j = i + 1,-• • ,N; ii T^f E L2, Bj G L2, Z\f e Lg, therefore Bi e L2.

Automorphism Group

219

The above discussion gives that if all vector fields with type Bi in L2 are Bi., 1 < j < s, then all vector fields of type T^^^- ^ in L2 are only T^^^, 1 < t < rii^^i^, 1
In the case of a normal Siegel domain of the second kind. 5.17 Let £)(V^,F) be a normal Siegel domain associated with

a normal matrix set {Alj^Qlj} of type N. Then the subspace L2 has a basis. The basis vector has two forms: Bi^SiY^

UjKij — , •'

j

where Kij and Kj^'' Proof

(5.134)

are skew-Hermitian

matrices.

By (1.141) of Theorem 1.26,

1=1

i=l

where Ci,---,Cn are real matrices, D^.-^Dn and

are complex matrices,

^^(i)^C'i^Gaut(D(V^)), 2=1

where -D(Vj^) is a normal Siegel domain of the first kind. By Theorem 5.16, there exist some indices 1 < i^ < • • • < is < AT, such that

Y, = E ^ - B , + E E ' - S J ; ! + E - ' " » A ^ , u
j

i=l

t

where Bi. is defined by (5.102), i ; ^ . is defined by (5.103), axe complex matrices. Now, d

dz^y•'^•JI^-^O' 'i ''

d

Laz([^ji)''^M^'^°'

D„--,Dn

220

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS r d

where z = ( z ^ , • • • ,z(")) G C " , —p-.Y^ dz^^' By Theorem 5.12,

&

G L^ implies uDi— 9ti

where Kij is a skew-Hermitian matrix, and KijQ^J = QAKik.

i<j

G L^

Hence

A;

t

where bj^ bij E R, i, j > 1, and Kij, K^^^ are skew-Hermitian matrices. Clearly, DiVj^^F) has a linear automorphism: r : si —^ Xfsij Zij -^ Ai Aj Zij, Ui —> XiUi, where A,, • • •, Aj^ are N independent non-zero complex parameters. Thus T*(Bi) = A?Bi, r^iT^f) = XiXjT^f. Hence

i<j

j

t

i

j

-^

Therefore

bjB,+., 5: a,K' a„fc' z^ Y: "y" ^^+E ^ E -'^^^ ij ^ z^ "ij z^ "*"* atifc'

are all in 1/2? where iCij, if^*'' % Kj^, if^*-' ^ are all skew-Hermitian matrices. I LEMMA

5.18 The following symbols are the same as the above. Then d' Y^= Bi + Si'Y^ UjKij-— E 1/2

if and only if Kij = 0 , I < j < N, and Z^feL,, V

j = i + l,---,N,

(5.136)

Automorphism Group

221

E(^i?)'«^'' + <^n)Qf^ = 0' ^<^^,v<mj,l<j
(5.137)

t

Y^iQfkYKev

+ e'veu)Qfj^ =0,

l
(5.138)

t

Proof

& '^ . d' The sum of all terms of type *-— is Y] z^^^uDj-r- in 11, where OUj

-^ OU

j^i

^ = ( ^ ( i ) , . . . , ^ H ) . By (5.102),

j=l

Ki

^

s

^

+ S S ^u^'^iQlf ^ + *i E ":'-^y i i
s

By (1.141)—(1.145), Y^ e L2 implies that (i) tr (Im Dj) = 0, 1 < j < n. Namely, ^ tr (Kij) = 0. (ii)

given any a, /? G C " , then L^ contains

(F(C(^,a),/?) + F ( / 3 , e ( ^ , a ) ) ) ^ (5.139) + (e(F(u, a ) , ;5) - anu,

/?),«) + e(i^(/?,«)' « ) ) ^ '

where

i

= s,{u, + Y: U,K,,) + J2Y1 ^f/^"^^'+E E ^^^Q^j

l
i
By the definition of F{u,v) in (3.77), (3.78), (3.85), (3.86), then

s

^i{z,u\a)

= SiUi{E + Kii),

^k{z, u; a) = SiUkKik + ^

4k'^iQik^

^ < ^•

s

By a direct computation, (5.138) imphes Kij = 0, I < j < N. The first statement holds.

222

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Suppose that Re{aiQf^^') j^ 0, Z^^^ e L^, 1 < t < n^fc. Then the sufficient and necessary conditions are

s

s

EQfk^Mi+T.(f^iQ\H)Q\'^ s

= 0, i
s

These formulas are equivalent to (5.137) and (5.138) obviously. LEMMA

I

5.19 The following symbols are the same as the above. Then

^?+E^i;^E-^^r^4'^^ s

^'-'"'^

k

if and only if K^''''^ = 0 for all indices and 5^, Bj G 1/2. Proof

The sums of terms with type *7r- in B^ and T^y are denoted by az

^

{BiY and {T^^Y, respectively. Clearly, Y^ e L2 implies some {T^j^Y E aut {D{V^)), By Theorem 5.16, {BiY, (BjY ^ aut {D{V^)). By Lemmas 5.17 and 5.18, then (ijs) d^ s

duki

k

K>.^?'+E 4;'E'«=«/"•'duki

= 2Bj G L2? = 2Bi e L2.

W D.l _ rp{t) r ^ ( t ) 0 . 1 _ rpit) Hence when Bi, Bj G L2, then [Xl'/.Bi] = T^\ [Z\]\Bj] = T^-^ G L2

by a direct computation.

Therefore J2^ij Yl'^kKj^'^ -— = 0, and s

k

T^j^ G L2 if and only if B^ Bj G L2.

^^k

•

Now, we give the main results in this chapter. 5.20 Let D(V^,F) be a normal Siegel domain in C^ x C ^ associated with a normal matrix set {AI^,Q\J} of type N. Denote by aut(D(V^,F)) the Lie algebra of the holomorphic automorphism group Aut {D{Vj^jF)). Then aut (£>(V^, F)) has a direct sum decomposition of subspaces (5.73), where sS{D{V^,F)) is given by Theorem 5.12. Then L2 has a basis:

THEOREM

Bi^r-'.Bi^,

T^%

l
l
(5.141)

Automorphism Group

223

where l
(5.142)

Li has a basis: YI'\

yf,

l
(5.143)

The basis elements satisfy the sufficient and necessary conditions as follows: index set 1 < i^ < - -- < ip < N is a maximal set, such that (1) l
ZIIEL,,

l
(5.144)

(2) zl2^Lo,

when

nu^^O,

i ^i^,--•

,ia-i,

i
(5.145)

(3) " v i = 0,

V < i,

i^ itr+i,•••,ip;

(5.146)

(4) J2(Quly«e^

+ e;e„)Q2 = 0 ,

l
l
(5.147)

r

Proof By Lemma 5.18, Bi e L2 if and only if Z|. € L^, 1 < t < n^j, j = i + 1^"' ^N, and

t

By Lemma 5.19, Bi, Bj e L2 if and only if T^f G L2, 1 < < < riij. Thus there exist some indices 1 < i^ < • • • < ip < iV, such that L2 has a basis { Bi^,'-,Bi^,

T^X,

l
where Z^GLO,

l
ia<j
Hence (5.144) holds, and

t

E ( Q S ) ' ( e ; e „ + e;e„)Ql*),. = 0 ,

i. < j .

224

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

the above first formula gives (5.147) holds. Now, we will prove that (5.146) holds by using the above formulas. In fact, if there exists an index j , where ia < j ¥" ^a+i, * * *, V ^^^ ^ v j > 0? we have [X^J^^^J = T^l] e L2. By Lemma 5.19, Bj G L2. Since Bi , • • •, Bi^ are Bj type vector fields in L2, we get a contradiction. Hence rii^j = 0, ia < j , j ^ W i ^ " ^ V- Therefore (5.146) holds. Now, we consider (5.145). Give i ^ i^, -- -, ia, i < ia and nu^ > 0, such that Z ^ e LQ. SO [ z g , B i J = T^} e L2 implies 5^ G L2 by Lemma 5.19. It is a contradiction from i 7^ i^, • • •, v , i < i^. Hence we prove that if only Bi , • • •, Bi^ G L2, then (5.145) holds. Conversely, if an index set 1 < z^ < • • • < ip < AT is the maximal set, such that (5.144)—(5.147) hold, by (5.144) and (5.146), Z^^Jj G L^, i(j < j , then Bi^, • • •, Bi^ G L2 (see Lemma 5.16). Now we prove that Bj G L2 imphes j G {i^, • • •, ip}. In fact, if Bi e L2, i^i^,- - ,ip, then conditions (5.144)—(5.147) hold for indices i, i^, • • •, ip. It is a contradiction because i^, Z2, • • •, ip is a maximal set. Hence this statement holds, and a basis in L2 is obtained. The next step is to consider the subspace Li. By Theorem L26, the polynomial vector field Y, = 2 v ^ F ( « , z ^ ) ^ + zA^

+ 5 ] ( « C j « ' ) e j ^ 6 Li

(5.149)

if and only if Lg has a vector field y„ = (F(^A,a) + F ( a , z ^ ) ) — m Q,

F{u,a)A-, (5.150)

-V^J](aCy)e^ —, where a G C ^ and yl is an n x m complex matrix, Cj is an m x m complex symmetric matrix. In this case, (5.150) can be written as

j
s

5' \ . ^^^

/.fzt^

A—rrfit)x ^'

+E(»r - ^ f ) (^^- - -4)+E ^.^ - ^/=I^l?)4 1. +EE(»S - v^4S)(zi? -«^«3^^). ''duk'

(5.151)

Automorphism Group

j
225

s

3

(5.152) and (t)r^ t^

d'

Y^

(it)

/—r (zt)x

d'

- 2 ^ ^ ( e f C,.')e,|E ( « f - "V S-u = ^^^^^ " ^ ^"^f ^)-i ^^'^duj j
s

^

(5.153) where 1 < < < m^, 1 < i < iV, ^(^*) ^(^0 ^W ^(i*) L(^*) ^W

^

^jfc ' ^7'fc ^^^ ^^^ ^^^^ skew-symmetric matrices and iQ* \ K^^ are all skew-Hermitian matrices. Now, Fii{u, v) = u^,,

F^{u, v) = \uiQ[p^j

+ \ujQf/v[,

where u = {u^,- • • , u ^ ) , v = {v^,- • • ,v^)j Ui, Vi e C^'.

(5.154) Comparing the

coefficients of —— in (5.151), where v = 'zA^ we have 25,,elV, = 2{af

- v ^ c f >)., + 2 J ] Y S ^ k
Hence iik ^i^ then Ait) _

j
(it) __ ^

"

^

^

^

s

s

(it) __ (it) _ ^

Ait) _

Jit) _ ^

(^^^^\

226

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Let (5.156)

Bil = (aj;? ^=Ic£)), •Us + '^'^~^^ils

i < I,

where 5 is a column index and t is a row index and Ajij Bu are Uji x rrii and nu x rrii complex matrices, respectively. Now Vi = Jlai + ^2 ^ji^ji + X ] ^^^^^^ j
(5.157)

i
and Aji 7^ 0, Z--^ € LQ, 1 < t < n^. Comparing the coefficients of -— in (5.151), where 1 < fc < Z < iV, we obtain the equivalent condition Ltl^ = L^P = 0,

k,l^i.

(5.158)

And Aki = J^e',akQ'j^^,

Bki = ^e',aiQ^^^ ,

r

l
(5.159)

r

satisfy the conditions l i f - x / ^ ^ t f = -{cr,e',)E + J2i^tQiW

4)e;e.,

L^^ - v ^ Z g ) = -iaie[)E + J^ietQ^^'QiM)e',es,

(5.160)

(5.161)

r,s

Q^iW

= T.etAie',Q^l

p
when 4 f € L„. (5.162)

r

Clearly, when zlf e L^, mi = rrik, Qj^i E U (m^). Thus

E(^tA%e',)Q^^ = E(«*
Automorphism Group and Rfi{es)

227

= Yl ^^^s(^pi)' is an ripk x ripi real matrix satisfying

Hence s

Therefore Er(et^pfce'JQj^ = QpiQki • It follows that (5.162) holds. Note that (5.157) can be written as

vj = w+E

E ^S^pQ^+E E ^^pQ^' p
(5-163)

3

^

and (5.152), (5.153) can be written as

*

i
s

(5.164) and

i
j

p
s

^

s

j

^

(5.165) respectively, where Kj ^, Kj ^ are skew-Hermitian matrices. Hence

Comparing two sides of (5.163) by (5.164) and (5.165), we have

i f f ) + V^Kf'^ = -2e',a, - {aie',)E, r

228

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and

E(^J^^)'«^- + <^-)Qi^ =0' P < ^'

(5.166)

r

E^y(<^- + ^^«")(^?)' = 0' '
(5-167)

r

Hence

= Y, Ziiai) € Li,

Vai G C"^%

(5.168)

i

where ^^(Q:^) is defined by (5.104). Let Si -^ Xfsi, Zij -^ XiXjZij, Ui -> XiUi be a linear automorphism, where A^, •••, A^ are N non-zero independent real parameters, thus ZiipLi) -^ XiZi{ai). Hence we obtain a basis in the subspace Li by Y^ , Y^*\ I < t < rrii. The condition is Z^^ G L^, p = z + 1, • • •, AT, and (5.166), (5.167) hold. Finally, we need to prove that when mi^ ^ 0, then Li has a basis Y^ , ^ ^ J 1 ^ t < n^^, 1 < cr < p. In fact, since

we have

€

Y}*\YI*^

LI,

I
<mi„.

Conversely, if y/*\ F/*) G i i , 1 < t < rm, since [Y}'\Y}'^] = Bi, Bi G L2, we have i € { h , • • • > V}• Therefore Li has a basis Y^^ , Y^J, 1
5.21

The subspace Li =0 if and only if L^ = 0.

Proof If Li 7^ 0, there exists 0 7^ Zj(ai) € Li by the proof of Theorem 5.20, and p i ( a i ) , I Z i ( a i ) , V ^ t ^ — I I = {aia[)Bi, Hence L2 7^ 0. Conversely, if L2 ^ 0, there is an index i, Bi ^ L2j then [P^'\Bi] = 2Y}'^ G L I . Hence Li / 0. • COROLLARY

5.22 p

dimLi = 2 y ^ m 2 ^ < 2m,
dimL2 =

Y^ l
ni^i^ < n.

(5.169)

Automorphism Group

229

5.23 Using the symbols in the following as Theorem 5.20. Let J D ( V ^ , F ) be a normal Siegel domain in C^ x C^ associated with a normal matrix set {AI^^Q\J} of type N. The coordinate representation is

THEOREM

D{V^.F)

= { {z.u) G C" X C - I lm{Ck{z))-Re{Rk{u)Rk{u))

> 0 }, (5.170)

where Z = {SI,Z2,S2,'",ZN,SN),

Zj = {Zij,"-

,Zj-ij),

U=

(iXi, • • • , IfcTv).

(5.171) If DiVj^^F) satisfies the conditions in Theorem 5.20. Let

(5.172) where /? = (/3i, • • •, PN) be a complex vector as (5.171), and Pi = 0, i ^ j , Pj € C"*J, j e {ii, • • •, ip]. Let a he a real vector as (5.171), and z = a, Zij =0,l
Ckiz)Ck{a)r'Ck{z),

Rkiv) = {E-

Ckiz)Ck{a))-'Rk{u),

l {x,v):

Pkix, v) = Pk + V^Aj

( - iV^

+ 2ujWj'){PkQk + Q'kPk)

+ PkiQ'k + 'Q~k)Pk + Pk{Q~k + 'Q'k)PkQk + Q'kPkiWk + Q'k)Pk + \Pk{QkQ'k + Q~kQ'k)Pk), (5.174) 1 < A; < i i , and Aj = Aj{z,u,/?)

= [1 - 2V^lujPj

- V-l\Pjrsj]-\

(5.175)

Proof In this proof, we use (5.31)—(5.37) and (5.99), (5.147) frequently.

230

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Proof (1)

By (5.102), (5.103) and Theorem 5.20, given X e L2, then

i=l

l
Let {exp{eX)){z,u)

t

= {x,v), MO G M. Then

Ck{i{x)) = Ck{x)Ck{oi)Ck{x\

Rkivix, v)) =

Ck{x)Ck{a)Rk{v)

by a direct computation from (5.102) and (5.103). Now, (x, v) is a unique holomorphic solution of the differential equation system

with the initial value x{0) = z, v{0) — u. By the induction, we have /(Pr\ Ck[-^)=p\{Ck{x)Ck{cc)rCk{xl Rk{^)=p\{Ck{x)Ck{a)fRk{v). Hence

°° e^ ^ rdFx <^'^(^) = E

-^'^^{'dP)

1=0 = ^^ "

eCk{z)Ck{a))-^Ck{z).

p=0

Therefore ^fc(^) = E

^ ^ ' ^ l ^ ) L=o = ^^ " eCk{z)Ck{a))-'Rk{u).

(5.176)

p=0

It prove that statement (1) holds. Proof (2)

By (5.104) and Theorem 5.20, given X € Li,

& & Zi{0i)=^{z,u)—+'n{z,u)—eLi,

l
t=

ti,---,PN,

where ^ = (0,-• • ,0,/?j,0,-• • ,0), 0j € C"*^ Let {exp{ex)){z,u) {x,v),\/eeR. Then Pk{^ix,v))

= Pk{x,v)Qk + QkPkix,v) + V^Pk{x,v){QJi

=

+ Q^')Pk{x,v) (5.177)

Automorphism Group

231

by a direct computation from (5.104), where k < j . Hence (x^v) is a unique holomorphic solution of the differential equation system !

^

= n(«.,.))

(5.178)

with the initial value x(0) = z, v{0) = u, where /3 is change to 9P, such that |/?p = 1 and ^ G E. Now, we will prove that the unique holomorphic solution of the differential equation system (5.178) is Pk{x, v) = P + v ^ A , -

( - d{V^

+ 20T){PQ

+

Q'P)

+ dP{Q + Q')P + e''P{Q + Q')PQ

(5.179)

^e^Q'P(Q^Q')P^'^lM^M)L)^ where A = (1 - 2 V ^ ^ r T = Uj0j',

yf^e'^Sj)-'^,

P=Pk{z,u),

(5.180)

Q = QkW).

Obviously, we have Rk{u)'Rkil3) + Rk{l3yRk{u) = 0. Hence QQ = QQ = Q'PQ = QPQ' = QPQ' QQ'Q = 2Q, Q'{QP

QPQ = 2TQ,

+ PQ')Q

= 0,

= 0,

Q'PQ = SjQ'Q,

Q{PQ

+ Q'P)Q'

(5.181)

= 0

by using Theorem 5.20. Now, V^Pkix,

v)Q = AjiV^

V^Pkix,

v)Q = AjV^PQ

V^Pkix,

+ 2dT)PQ - AjdPiQ + +

Q')PQ,

AjOV^Q'PQ,

v) = Aj(v/=1 - ehj)QP

+ AjOV^lQPQ

(5-182)

+ A,(^v/=I + e'r)QQ'P + A , ^ 5 ^ 5 ^ . Using (5.180), (5.181) and (5.182), we have ^"^^^^'^^ = Pk{^{x,v)) by du a direct computation. It proves that statement (2) holds Note The exphcit {Zi{ai)), exp {9Bi) and exp {OTJ^j ^) explicit formulas of exp {Zi (c are given by Min-ru Chen and Yichao Xu.

232

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

4.

Other Results

In this section, we consider the connected component of the holomorphic automorphism group Aut {D{V^,F)) and the isotropic subgroup on the homogeneous bounded domain reahzation (see Section 4.1). (1) The connected component of the holomorphic automorphism group G = Aut (Z)(V;,,F)). Let G^ = Aut (jD(Vj^,F))^ be the unit connected component of G. We will choose a representation element a^ in a connected component G^ of G with a very simple expression. By Theorem 2.4, it suffices to compute the representation element a^ in a connected component G^ n jff, where H = Iso(i)(F^,F)) is the isotropy subgroup at a fixed point ( ^ ^ i ; o , 0 ) G D{V^,F) of G. Since CTQ oG^ o {a^)'^ = G^, Ad {a^) is an automorphism acting on the Lie algebra aut {D{Vj^,F)) of the Lie group G^. By Theorem 2.13, Vinberg (ref. [219]) proved that G^ = T^ - {Iso{D{V^,F))^ is a semi-direct product decomposition of subgroups, where Ad (TQ) is a maximal triangular subgroup in the algebraic Lie group Ad(G^), and two maximal triangular subgroups in Ad (G^) are conjugation to each other. Hence there exists a representation element a^ in G^, such that Ad (cr^) is an automorphism acting on the Lie algebra to of a maximal triangular subgroups T^. Clearly, there exists a holomorphic automorphism a^ e G^, such that the holomorphic automorphism a = a^ o a^ e G^ induces a J automorphism Ad{a) on t^, such that Ad(or)(L_2) = L_2,

Ad(a)(L_i) = L_i.

By the proof of Theorem 3.21, cr G Aff {D{V^,F)) n Iso (D(F^, F)). By Theorem 3.21, we prove that 5.24 Let D{V^,F) be a normal Siegel domain in C^ x C"^ associated with a normal matrix set {A*j^, Qij} of type N. The coordinate representation of a point in D{Vj^,F) is {z,u), where

THEOREM

Z = {S1,Z2, S2, • • • , ZN, SN),

Zj = {Zij, • • • ,

Zj-ij),

Given a connected component G^^^ of the holomorphic automorphism group G = Aut (Z)(V^,F))^ there exists a representation element a in G^^\ such that a is a representation element of a connected component

Automorphism Group

233

of a real Lie group 0{D{V^,F)) Namely^ Sj^s,.,

( see (5.96), (5.97) in Theorem 5.13).

Zjk^z,.,^Oi.i„

u,.-^u,.Ui^,

l<j
l<j
where i\i2 "-IN is an admissible permutation o/ 1,2, • • •, AT with respect to a symmetric matrix S = {jiij), that is, if ik < ij, then n.. . = 0 . And r±E, y diag (±1, £^),

whenn,,^seven•, when rijk is odd^

^^^^^^

where E is a unit matrix, Ojk G 0{njk), Uj G \J{mj), satisfying the conditions njk = n,.^,^, l<j,k
A% = E(^*0^.^.<)0^.v<^^l^p5

(5.186)

s

Qfk = T.(^tOi,i,e',)Ui,Q^lu-'.

(5.187)

S

(2) The isotropic subgroup of the homogeneous bounded domain reahzation. By Theorem 5.20, we have T H E O R E M 5.25 Let Iso (Z)(V^, F)) be the isotropic subgroup of the fixed point p= {y/^vo,0) e D{Vj^,F), Denote by isOp{D{V^,F)) the Lie algebra oflsOp{D{Vj^^F)), Then isOp{D{V^^F)) has a direct sum decomposition of subspaces:

isOpiD{V^,F))

= o iD{V^,F))

+ y{D{V^,F))

+ L,+L„

(5.188)

where (i)

oiDiV^,F)) = {J2zijLij-^ i<j

+ Y,^.K,-^} ^^

i

(5.189) *

is a subalgebra, and Lij is an rtij x uij real skew-symmetric matrix satisfying the conditions

L'^^Af^ + Af^L^k = Yl{erLi^e',)A\l

(5.190)

234

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Ki is an rrii x mi skew-Hermitian matrices satisfying the conditions

Kl'Qf^ + Qf^Kj = Y,ierLije[)Q\;\

(5.191)

r

(ii)

y(D(V^,F)) is a subspace with a basis X^-Z^,

l
(5.192)

where X^l, Z^f € L,. (iii)

L^ is a subspace with a basis Y^-P^\

Y^ + P^^,

l
(5.193)

where i ^ i r * • ? Vp is defined by Theorem 5.20. (iv) L2 is ^ subspace with a basis Bi. + ^ - ,

i;:t+2-77r'

1 ^ *^ "vv,

1 < ^ < r < p, (5.194)

where io-i? * *' liup is defined by Theorem 5.20.

Given a normal Siegel domain D(V^, F ) , by Section 4.1, we prove that the Bergman mapping a:

(x^v) = (grad,,^log ^ . ^ ' ' ^ ' ^ ' ' ' j _,h=-V^i.,,^=o

(5.195)

is a holomorphic isomorphism acting from D ( y ^ , F ) onto a canonical homogeneous bounded domain D = a(D(V^,F)), where a{^/^VQjO) = (0,0) € D . Denote by {z,u) = (^(i),---,;^(^+^)) and {x,v) = (t£;(i), • • • ,it;(^+^)) the point coordinates in the domain D{Vj^jF) and D, respectively. The coordinates (a;, t;) = {w^^\ • • •, i(;(^+^)) in D = a(D(V^, F)) can be written as X

(r^,

2^25 ^25 * * ' ? '^N5 ^N/5

/ (1)

(»^n)\

*^i — V ' ^ i j 5 * • * J

/

\

*^j~ij)5

/ (1) 1)

(^i)\

(5.196) as the point coordinates (z.u) of D(V^,F).

Automorphism Group

235

By (1.77), given a vector field X in iso/^^i^ QJD{VJ^,F),

n-\-Tn

then

pj

= E«»(^.-)5^. (5.197) we obtain a vector field
".(E e«(^.'')Jj)) = -E>«'"(^i<-M^vo))5^(5.198) Since some of complex variables 5^, •: •, 5^ appear only in the linear

terms of vector fields Bi, T^j\ Y^^\ y/*\ and any linear vector field

we have

<^'Moi'l)d) = -^'''^^\si'3i)' d d\'\ , ^-,/d d\'

("'^'

5.26 Let DiV^^F) he a normal Siegel domain associated with a normal matrix set {Afj, Q L } of type N. Then the Bergman mapping a induces a Lie algebra isomorphism acting from is0p(D(V^,F)) onto (7*(isOp(Z)(V^,F))) = isoo(cr(i)(V^,F))), where a = a"oa' and a' is defined by THEOREM

a':

{x,v)=gr^dr-,^Aog

^(fl^^^^^)

I

,

(5.20O)

and a" is defined by a":

ri->2ri,

Xij-^ Xij,

Vi-^ Vi.

(5.201)

Let { Vi5 • • • ? Vp } be an index set, which is defined by Theorem 5.20. Then the basis vector o/(j*(isOp(£)(V^,F))) is one of following elements.

i<j

*'

i

i<j

-'

i

a.{xf^{z,u) - Zf^{z,u)) = Xlf{x,v) - zlf{x,v),

(5.202) (5.203)

236

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where Zy/{z,u)

e aut {D{Vj^,F)),

a4Bi + —) = V^Aiix,v),

(5.204)

where z G { v , - • •, v^ }.

''*^i^S^ + IlO) = ^ ^ ( 4 ^ ( ^ ' ^ ) + 4^(^.^))>

(5-205)

U

where l
=

Uij, i < j , ij

G { Vi, • • •, Vp }•

-Via

+^na.^ + EE-^.o?4+EE^SX'^. p
s

^

i
s

^

(5.206) where i G {i^i, • * * ^ Vp }; Q^i ^ C ^ ^ And Ai = Ai{z,u), X^f = Xlf{z,u), Z^f = Z^fiz.u) are presented by the coordinates {z,u) {see (5.80), (5.81), (5.84), (5.87), (5.24)). By Section 5.1, we can write the generator set of the maximal connected isotropy subgroup Isoo(cr(D(Viv, P)))^ acting on a{D{VN, F)) immediately. In fact, 1SOO{(T{D{VN,F)))^ is contained in Afr(C^ x C ^ ) , and generated by one parameter subgroups exp(tX), where t G M and X is one of (5.202)—(5.206). We omit the coordinate expressions of all one parameter subgroups in here.

Chapter 6 CLASSIFICATION OF SQUARE DOMAINS

We will give the classification and realization of a class of normal Siegel domains in this chapter. Namely, we will give the classification and realization of square domains. These domains contain symmetric Siegel domains and quasi-symmetric Siegel domains. On the other hand, some other classes are considered in this chapter.

1.

Classification of Two Special Matrix Sets

In this section, we will give some results of two special matrix sets for the classification of square domains and dual square domains. The main part discusses the case of square matrices. We obtain only a few results for non-square matrices. Let { 4 J , t = 1 , . . . , n,,-, 1 < i < J < A: < AT } (6.1) be a real matrix set, where Ajj is an riik x Ujk matrix, t = 1, • • • ,nij. The p*^ row and q^^ column element of Afj is denoted by the product of three matrices: 673

(6-2)

44'

where Cp € R''*'=, e, 6 R"^* Let {Q\}\t = l,---,nij,

l
(6.3)

be a complex matrix set, where Q ^ is an rui x rrij complex matrix, t = 1, • • • ,nij. The p*^ row and q*^ column element of Q\J is equal to 237

238

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

the product of three matrices:

epQl?e;,

(6.4)

where e p G R ^ S e^GR^^. Given non-negative integers n^j, I
n—

\ J

n^j, m = Y ^ m i .

'^
(6.5)

2=1

The coordinates of a point 2: G C^ are arranged in the form z = (si,^2,52r--,^iv,5iv),

Zj = {zij,'",Zj-ij),

(6.6)

where sjeC,

Zij = {z^r".z^''^)eC--,

(6.7)

Let (1) Cjj = z, where Sj = 1 and all the other coordinates are equal to zero. (2) e\j = z, where z^j = 1, and all the other coordinates are equal to zero. The coordinates of a point u E C^ are arranged in the form u = {uuU2,..

^,UN).

Uj = {uf\ .. '.uf"'^)

G C-^.

(6.8)

Let (3) CA = u, where Uj = 1 and all the other coordinates are equal to zero. Any normal cone V^ contains the point v^, where N

Vo = ^ejj = {lAl.'".OA).

(6.9)

3=1

Then p = ( v ^ ^ o . 0 ) G C^ X C ^

(6.10)

is a fixed point in a normal Siegel domain D{Vj^^F) (see Chapter 3). 6.1 Let ^ = {Ai,-" ,An} and 03 = { S i , • • •, J3n } be two mxr complex matrix sets. 21 and 05 are said to be complex equivalent, if there exist U eV (m), V e\J (r), O G O (n) = 0 (n, M), such that

DEFINITION

n

Bi = Y^{ejOe'i)u'AjV, 3=1

l
(6.11)

Classification of Square Domains

239

21 and 53 are said to be unitarily equivalent^ ifO = E is the unit matrix. Namely, there exist U G U(m), V G U(r), such that Bi = U AiV, i = 1, • • •, n; when m = r and O = E,V = U, ^ and 03 are said to be unitary similar. Let 21 = {Ai^" ' ,An} and 03 = { S i , • • •, B^ } 6e two nxr real matrix sets. 21 and 03 are said to be real equivalent, if there exist Oi G 0 ( m ) ; O2 G O (r), O3 G O (n), 5z/c/i i/iai Bi = J2{ejOse^O[Aj02,

1 < i < n.

(6.12)

21 and 03 are said to be orthogonal equivalent, ifO = E is the unit matrix. Namely, there exist Oi G 0 ( m ) , O2 G 0{r), such that Bi = 0[Ai02, i = 1, • • •,n; when m = r and O = E^Oi = O2, 21 and 03 are said to be orthogonal similar. (A) The case of a complex m x m matrix set. Denote by Qi, • • •, Q^^, m x r complex matrices, such that Q[QJ + Q'jQi = "^SijE,

1 < i, j < n.

(6.13)

Obviously, m > r. When r = m, condition (6.13) is equivalent to the condition QiQ'j + QjQ'i = 2SijE,

l
(6.14)

(6.14) is called a complex Hurwitz matrix equation. Clearly, the condition (6.13) is invariant under a complex equivalent (6.11). We give the canonical form of { Qi, • • * 5 Qn } up to a complex equivalent as follows: 6.1 Suppose thatmxm complex matrices Qi, • • •; Qn satisfy condition (6.13). Then (1) (Hurwitz) there exists natural number M, such that

THEOREM

m = 2^^^M,

(6.15)

where [x] is the integer part of a real number x; (2) there exist U,Ve. U(m), such that U'QJV

= Rj,

1<

j
240

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and the canonical form is Ri = E, Ej 0

0 -Ei

i?2i = \/—1 Ej 0

0 -Ei

-Ej

0 (6.16)

R2J+1 = 0

Ei

V V -EJ

0

/

where Ej is an ruj x nij unit matrix, ruj = 2~^m, j = 1,2,- • •, [ ^ ^ ] . In particular, when n is an even number, R\

0

(6.17)

Rn = V^ R

0 /

where R^

0

-E^^~P)

V

pe{0,l,2,...,M}.

(6.18)

Hence, when n is an odd number, the canonical form Ri, -", Rn is unique; when n is an even number, the canonical form depends onp = 0, 1, '", M. The total invariant up to a unitarily equivalent is a nonnegative integer p. Proof By the definition of a unitarily equivalent, without loss of generality, let Qi = E. Hence Q2r"')Qn are skew-Hermitian matrices. Without loss of generality, let

^2 = ^^(^o

J(^))-

When n = 1,2, this theorem holds; when n > 2, Q2QJ + QjQ2 — 0, j > 2, imply 0 Aj Qi -A'j 0

Classification of Square Domains

241

and AJA'J, + AkA'j = 28jkE^'^\ ~X^Ak + A^A^- = 2SjkE^'\ 2~^m = mi is a positive integer. liU.VeV(m) and UQiV = Qi, UQ2V = Q2,

Hence r = s =

then Qj -> UQjV implies Aj —> UiAjU2' In terms of the induction for m, (6.15) holds, and the canonical form is Ri,R2^ • • •, RnNow, we prove that the integer p is the total invariant. In fact, if n is an even number, (6.16), (6.17) and (6.18) give that R,=Ri%'-'Rn.i% = i-y/=l)^dmg{A,''',A),

(6.19)

where / E^P^ ^ - 1^ 0

0 -E^^-P^

If {Qi, • • •, Qn} has two canonical forms i?i, i?2, • • *, Rn-i} Rn ^tnd i?i, ^2? • • •, J ^ - i ? Rnj where i?n and i?n are determined by the integer p and PJ respectively, we have R, = R1R2 • • • Rn-iR^

= ( - v ^ ) ^ diag ( I , • • •, I ) ,

(6.20)

where 2_

f E^P^

^~V

0

0 -E(^-P^

and ^ is unitarily equivalent to A. Hence there exist C/, F € U (m), such that ^^ B4 = URiV, 1 < i < n - 1, i?n = C^iJnV: It follows that U{R\R2 • ' • Rn-iR,^)U

= R1R2 ' ' ' Rn-lRn •

Hence R^ is unitarily equivalent to R^. Therefore p = p. It follows also that Rn = Rn. I T H E O R E M 6.2 The following symbols are the same as the above. Suppose that an m X m complex matrix set {Qi, Q2j * • •; Qn} satisfies condition (6.13), and i?i, i?2; "j Rn is the canonical form up to a complex equivalent (6.11). When n is an odd number, the canonical form is unique; when n is an even number, p >: "y? cbTid the total invariant is pe{[M±l],...,M}.

242

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Proof By Theorem 6.1, it suffices to consider that n is an even number. Give two canonical forms i?i, • • •, Rn-ij Rn and Hi, - • •, Rn~i, Rn- If there are O € O (n), and C/, V G U (m), such that n

n

Ri = J2iejOe'i)URjV, where {Ri,•••,Rn} Rr%---Rn-iX

l
^{ejOe'jURjV,

is defined by (6.16), (6.17) and (6.18), then = Yl

When i < k,

Rn =

{enOe[)---{ej„Oe'ju'Rj,R'j^---Rj^_,R'j„U.

J2 (^jt^^OC^jfe^^fc) = ^ik = 0 and R^^Rv + R^Ru =

RuRv + RvRu = "^^uvE give that ^1^2 • • ' Rn~lRn E

KOe',)

= (det 0)U'RI%

• • • {ej^Oe'J

V

Ri,%^---Rj^_,%U

' • • i?n-iKu.

When detO = 1, p = p; when detO = —1, p = M —p. If 0 = diag(l,...,l,-l),

C7 = r

= diag(C/i,...,C/i),

C/IGU(M),

then Rn = —U RnU. Hence 0

-E^M-p)j-^^y

0

E(^-P))^

by (6.19) and (6.20). The above discussion proves that if two canonical forms have the invariant p and p = M — p^ respectively, then they are complex equivalent to each other. Hence if restricted by a condition p> ^j then the canonical form is unique. I (B)

The case of a complex m x r matrix set, where m> r.

6.3 If anmxr complex matrix Qi satisfies (6.13), then the canonical form up to a complex equivalent (6.11) is

THEOREM

Ein) ) •

(6.21)

243

Classification of Square Domains THEOREM 6.4

Iftwomxr complex matrices Qi, Q2 satisfy (6.13); then the canonical form up to a complex equivalent (6.11) is

1 ^

0 0

E(P)

0

Pi

,P2 =

0

(£;-A2)^

o(p)

0 V^A^-J) 0 0

0 1 0 V 0

0 0 0 0 0 V^E(")

0 0 0 0 0 0

\

5

0 (6.22)

where A = dmg{ai,'",aq)

e GL{q,]

0 < \ai\ < 1,

l
and p + q + u + v = rj2p + 2q + u + v<m, u>v. When u > v, the canonical form is unique; when u = v, two canonical forms are unitarily equivalent if and only if one is A and the other one is A or —A. Proof

Without loss of generality, let L K

By Q1Q2 +^2<5i = 0, we have K + K = 0 . By Q2Q2 = E, we have L L + K K = E. Now, we consider an m x m unitary matrix U = diag (Jj\, V), where t/ € U (m - r), thus

UQiV^Qu

v'Q^y-{"vZ

Without loss of generality, let K = v^diag(0(P),A(<'), £;("),-£;(")), where A = diag(ai,-• • ,ag) is a real matrix and aj ^ ±1,0. Now, t l = E-'K'K = d i a g { E ^ \ E - K ^ , 0 , 0 ) > 0givesE-K^ > 0. Thus0 < |aj| < 1. Now L'L = diag {E^\E - A^, 0). Given the block partitions of L hy p,q,u + v columns, then L = (Li,0), Li Li = diag {E^\E - A^). Thus L2 = Lidiag(£;^), {E - h?)~^) satisfies T2L2 = E. Hence there exist two unitary matrices U and V^, such that / U'LV,

0

= 1 E^^ V 0

0

0 {E-A^y^

244

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

The first statement holds. Finally, if there exist two canonical forms Pi, P2 as (6.22) and

1^

_E'(pi)

Pi =

0 E

0 0 0

,P2

1 °

V 0

0 0

0 0 0 0 0

{E-k\Y 0

^/Zi£;(ui)

0 0

0

0 0 0 0 0 0

-V^

(6.23) such that {Pi, P2} is complex equivalent to {Pi, P2}, there exist O € O(2), C/ 6 U(m), F G U(r), such that Pi = E{^jOe[)U'PjV,

i = 1,2.

Hence Pi P2 = ( d e t O ) y ' P i P 2 F . It follows that diag (0(Pi\ V^A^^'\

VAIIE("I\

-\/^^("i))

= (det 0)F'diag {O^P\ V ^ A ( « \ V ^ E ^ " ' ,

-V^E^''^)V.

Hence pi = p. When detO = 1, gi = q, Ai = A, tti = u, vi = v, hence Pi = Pi, P2 = P2; when det O = —1, qi = q, ui = v, vi = u and 0

/O

1\

Vi

0/

Ai =

1 0 0 -1 complex equivalent to Pi, P2.

Conversely, let O =

there exist C/, F , such that Pi, P2 are

When m > r, n > 2, we do not know any property for the canonical form of a complex matrix set Qir " iQn up to a complex equivalent (6.11), even the relation of m, r, n. (C) The case of a real m x m matrix set. Give an m X r real matrix set { Ai, ^ 2 , • • •, A^ }, such that A'iAj + A'jAi = 2SijE, Obviously, m>r.

l
(6.24)

When m = r^ (6.24) is equivalent to the relation AiA'j + AjA[ = 2SijE,

l
(6.25)

Classification of Square Domains

245

(6.25) is called a real Hurwitz matrix equation. It is easy to show that (6.24) is invariant under a real equivalent (6.12). We give the canonical form up to a real equivalent. 6.5 Suppose that an m x m real matrix set { ^ i , ^2? • * * ? ^n } satisfies the condition (6.24)^ there exists Oi, O2 G 0(m), such that

LEMMA

(6.26)

where Qk = Xk + y/^Yk^ 1 < k < n — 2 are the corriplexjkew-symmetric matrices satisfying condition (6.13). If there exist Oi, O2 G 0(m)^ such that &,E02 = E,

0',[l

- f ) 0 . - { l

t ' ) '

then P Q 02 = 0, = , _Q p where U = P — V—IQ G U (mi) and m\ = 2~^m is a positive integer. On the other hand, 0[Aj02 -> 0^(0^^^02)02 implies Qj -^ U QjU, 1 < j < n - 2. Proof Without loss of generality, let Ai = E = E^, A2, •••, An be

(

r\

T^ \

mi X mi unit matrix and mi = 2~^m. Since AI^Aj + A'jA2 = 0, we have A^ = ( Xj-2

[YJ.2

Yj-2 \

-Xj-2)'

o < „• < „

^-^-"

and Qj = Xj + ^/^Yj, j = l,2,---,n — 2, are complex skew-symmetric matrices satisfying condition (6.13). If Oi, O2 G O (m), such that

we have

C'2 = 0 i = ( j ^

^),

U=

P-V^QeU{mi),

246

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and Aj -> 0[Aj02,

j > 2, imply Qj -^ U'QJU, l<j
I

6.6 Suppose that an mi x mi complex skew-symmetric matrix set {Qi,-" iQriQ } satisfies condition (6.13), there is U € U(mi), such that U'QjU = BiRj, l<j<no,

LEMMA

and there is a positive integer M, such that mi =2

^ 1

(6.27)

M,

where Ti 0

Bi =

0 -Ti

0 Bi+i

Ti =

-Bi+i 0

Z = l,2,-..,

(6.28)

Bi is a real orthogonal matrix, where the last block matrix in Bi is no=8k-3,

B2k = E;

no=8fe-2,

B2k = E;

n,=Sk-l,

T2k = E;

n,=8k,

T2k=(^l

no=8k

Q ) , 0 E

+ 1, B2k+i = /

no=Sk

+ 2, B2k+i =

V

no=8fc + 3, B2k+2 = E; no=8fc + 4, B2k+2 = E,

P = 2-^M-

-E 0 0 E

-E 0

(p)

0

0 0 E

-E 0

(M-p)

p = 2-^M,

p and M are all even numbers. Proof Now, Qi + Q'i = 0, 1 < i < Uo, Q'iQj + Q'jQi = 26ijE, 1 < i,j < WQ, QiQi, i = 2, • • •, TZg, are all mi x mi complex skew-Hermitian matrices, QiQi = E. Hence iQiQi) (QiQj) + (QiQj) [QiQi) = ^Si^E,

i
Classification of Square Domains

247

By Theorem 6.1, there is C7 G U (mi), such that (U^QiU) {U'QiU) = U Q^QiU = Ik,

1 < i < n^.

Without loss of generahty, Qi = BiRi, 1 < i < n^, where Qi + Q[ = 0, i = 1, • • •, no and Qi = 5 i G U (mi). Thus BiRj - R'^Bi = 0. Hence B1R2J = R2jBi,

5ii?2j-f 1 + i?2j+l5l = 0 ,

j = 1, 2, • • • .

By induction for n^, we have Qi = Bi, Bi = I J^ ( B°+i

rp 1, T/ =

~^0^^ ) ' "^^^^^ / = 1,2,...,fco.When n, = ^k, + 2, Bk^^i =

0 \ / T (M-p) ; when no = ik^+S, B^^+i = I ^ ^ 0 CIfco+i / ^ when rio = 4fc + 4,

0 -U

\ ^, ) ' °^ ^

(p,M-p)

fco+2

5A., ^0+2

Since Qi + Q^^ = 0, we have ^2^+1 = -^2^+1,

B2q = B2q,

T2q = T2q,

^2^+1 = -B2q-{-\'

When UQ = 8fe —3, B2fc is a symmetric unitary matrix; when UQ = 8fc + l, J52A:+i is a skew-symmetric unitary matrix; when n^ = %k — 2, B2k = (P)

C^k'

0 {M-p) 1 i^ ^ symmetric unitary matrix; when UQ = 8k + ^2fc

2, B2A;+i = I ^^"^^ (M-p) 1 i^ ^ skew-symmetric unitary matrix; \ 0 ^2fc+i / when no = 8A: — 1, T2k is a symmetric unitary matrix; when no = 8fe + 3, T2A;+i is a skew-symmetric unitary matrix; when no = 8fc, T2k =

(

0

^{PM-V)

(M-p,p) ^2k

'

^^

\

j is a symmetric unitary matrix; when no = ^

/

(

Q

^(pM-v)

\

(M-p,p) '^^'^^ 1 is ^ skew-symmetric unitary ^2fc+l ' 0 / matrix by the induction. Given U G U(mi), such that Rj = U RjU, I < j < TIQ, then the mapping Qi -^ U'QiU, 1 < i < ^o gives that the last block matrix A in eight cases as the above maps from A to ^ ^ U'^AUQ.

248

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

This lemma holds by the canonical form of a complex symmetric matrix and a complex skew-symmetric matrix up to the equivalent relation A -^ U^^AUQ (see author's book: Linear algebras and theory of matrix, 1992 (in Chinese)). I 6.7 Suppose that an m x m real matrix set {Ai^-- - ^An} satisfies condition (6.24). Then there exists a natural number M; such that m = 2^'^^M, (6.29)

THEOREM

and the canonical form up to a real equivalent (6.12) is

P^'*^ - { - v ^ B j i J , ,

0

) '

where mi = 2'~^m is a positive integer and iJi, • • •, Rn-2 ^^^ the rui x mi canonical form as (6.16)^ (6.17) and (6.18). When n is an odd number, j = 1,2, • • •, ^ ^ ; when n is an even number, j — 1,2, • • •, ^ ^ . Then mi X mi matrix Bi is defined by

where when n = 8k—1, B2k = E; when n = 8k, B2k = E, andp > 2~-^M; when n = 8k + 5, B2k+2 = E; when n = 8k + 1, T2k = E; when n = 8k + 2 and p = 2-^M, B2k-i-i = ( E;

0

T2k = ( ^

^ ) ; when n = 8k + 3,

) ' ^^^^ n = 8fe + 4 and p > 2~"^M, Ssfc+i =

diag(l p 0 I A p 0 ] )^ where p and M are all even integers; when n = 8k + 6, p = 2~^M, B2k+2 = E. Proof If n = 1 and n = 2, this theorem holds obviously. If n^ = n - 2 > 1, without loss of generality, let Ai = Pi, A2 = P2 and mi = 2"~-^m is a positive integer. By Lemmas 6.5 and 6.6, canonical form (6.30) holds, and

Classification of Square Domains

249

(1) (2)

when n ^ 8fc, 8k + 4, the canonical form is unique; when n = 8fc, if there is another canonical form

p

_f

0

-^^BiR2j

^2i+2 - (^ _ ^ / Z i B , ^ 2 ,

0

(6.31) where (6.31) is determined by the natural number p. If (6.30) is real equivalent to (6.31), there exist O G 0 ( n ) , Oi, O2 € 0 ( m ) , such that •Pi = EiejOe'i)0[Pj02,

l
Since

5ii?j = {-lyRjBi,

BiRj =

where Bi, Bi € 0 ( m i ) , B[ = -Bi, and

{-lyRjBi,

B[ = - S i , then Bj = Bf =

-E

n

PiP^---K=

E

(e,i0ei)...(e,„0e;)0iP,,/^,.--i^„0i.

Hence P1P2 • • • Pn-iK

= (det 0 ) 0 1 PiP^ • • • Pn-iP^Oi

as the proof of Theorem 6.2. By a direct computation,

X-

RIR2

0 P1P^..P^^(V^)'^

0

-2

RIR2

R1R2

0

Rn- -2

0

Pn- -2

R1R2

Rn- -2

Hence

= (det 0)Oldiag (^(^), -E^^-P\

• • •, £;(^), - £ ; ( ^ - ^ ) ) 0 i .

Therefore when det 0 = 1, ^ = p; when det 0 = —l,p = M — p. Conversely, when p = p^ Pj = P/, 1 < j < n; when p = M — p, there is O = diag (1, • • •, 1, —1), such that {Pi, • • •, P^} is real equivalent to {Pi, • • •, Pn} as the proof in Theorem 6.2. Therefore when n is an even number, if we add a condition p > 2~^M, then the canonical form is

250

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

unique. In particular, when n = 8fc + 4, this theorem holds for the same reason. I Let 2 x 2 matrices 0 1

J2 =

-1 0

J3 =

0 1 1 0

6.8 Suppose that an n x n real matrix set

THEOREM

{Ai,---,An} satisfies condition (6.24). Then n = { 1,2,4,8}, and the canonical form is unique up to a real equivalent (6.12). canonical form is (1) n = l, Pi = l; (2) n = 2, Pi = Ei^\ P2 = J2; (3) n = 4,

(4)

n = 8,

Pz = diag

P4 = diag

/

0 0 P7 = 0 V ^1 Proof

Pi = E(8), 0 E

0 0 -Ji 0

P2 = diag(J2,-J2,-J2,J2),

-E 0

0 J2 J2 0

0 -E

0 -J2 -J2 0 0 Jl 0 0

The

0 0

0

E 0

^5 =

0 E

-E 0

/ 0 0 J2 0 \ 0 0 0 J2 Pfi^ J2 0 0 0 \ 0 J2 0 0 y

/ 0 0 0 0 ,^8 = 0 -J3 V J3 0 y

0 Js 0 0

-J3 \ 0 0

0

/

Since m = n, there is a natural number M, such that n =

2^~^^M by (6.29). When n > 8, we have 2^"^^ > n, hence 1 < n < 8. By a direct computation, n = 1,2,4,8. By Theorem 6.7, when n = 1, n = 2, this theorem holds.

Classification of Square Domains

251

When n = 4, the canonical form is £;(2) ^1 - f

0

0

\

£;(2) ) '

a

/

^2-1

0

-£;(2)

^(2)

0

Let

0 . = d i a g ( l , 1,1.-1),

0=^^[%

'l)

Then Ej=i(ejC^oei)0'PjO = Pi, 1 < i < 4, this theorem holds. When n = 8, the canonical form is Pi = P\^ P2 = P3, P3 = —Pr? ^^4 = P5, P5 = P2, Pe = P4, Pr = ^8, P^ = -PG:^ Clearly, there is an orthogonal matrix OQ, such that Yl'j=i{^j^o^i)^'-^j^ = Pi, 1 ^ ^ ^ 8, this theorem holds. 6.9 Suppose that {Ai, -" ,An} is annxn satisfying condition (6.24). Then COROLLARY

n

n

Y, ^M'uev + e'^eu)A^ = Yl M^^v j=i

Proof

real matrix set

+ e ^ e ^ ^ = 25uvE,

(6.32)

3=1

Let n x n matrices Hi = 2_^^j^iAj<,

Oi = / ^ ejCiAj.

Then { 5 i , • • •, Bn} and {Ci, • • •, Cn} satisfy (6.25), hence they satisfy (6.24). Therefore (6.32) holds. • 6.10 Suppose that {Pi, • • • ,Pn} is the canonical form in Theorem 6.8. Then

COROLLARY

n

PiPj = Y'^eiPie'j)Pi,

1 < i, j < n

(6.33)

1=1

if and only if n = 1,2,4. Proof When n = 1,2,4,8, Pie[ = e'^, Pie'- - - e ^ , i f = -fJ, i > 1. Hence this corollary holds when n = 1,2. When n = 4 and i = 1 or jf = 1 or i = j , this corollary holds also. When 2 < i, j < 4, z 7^ j , since P2P3 = —P4? this corollary holds also. But when n = 8, this corollary is invalid. I (D) The case of a real m x r matrix set, where m> r.

252

THEORY OF COMPLEX HOMOGENEOUS BOUNDED

DOMAINS

Using the proof as Theorems 6.3 and 6.4 for an m x r real matrix set {Ai, • • • 5 An}, where n = 1 or n = 2, we have THEOREM

6.11 If a real matrix Ai satisfies (6.24), then the canonical

form up to a real equivalent (6.12) is I j^ 6.12 Suppose that Ai, A2 are two real matrices satisfying (6.24). Then the canonical form up to a real equivalent (6.12) is

THEOREM

(

0 0

{i\

o(p) 0

0 0 {E-KK')\ 0 K

0

0

0 0 0 0 0 -£;(«o)

£•(90)

0

\

(6.34)

where

K = diag

-£•(91)

(^1 (E?^^)

,fts

0

0 £;(9.)

-£'(^^ 0

(6.35)

0 < 6 i < ••• <65 < 1, P + 2 j ^ q i = n, 2p + 2 9 o + 4 j ] ] 5 i < m , i=0

i=0

and the canonical form is unique. In the case m > r, n > 2, we do not know any property for the canonical form of an m x r real matrix set {Ai, • • •, An} satisfying (6.24) up to a real equivalent (6.12). (E) As an apphcation to a normal matrix set, we have the next lemma (in the next lemma, (2) and (3) are given by Sudo (refs. [183], [184])). 6.13 The following symbols are the same as the above. The non-negative integers riij, I < i < j < N, rui, I < i < N, satisfy six conditions as follows: (1) riik = 0 implies nijrijk = Q, i < j < k; (2) nijTijknik ^ 0 implies max {riij, rijk) < nik, i < j < k; (3) rijk = 0, rtijUiknunjiriki ^ 0 imply max {uij+riik, nij-^nkuriik + Uju riji + riki) < riii, i < j < k < I. (4) mi = 0 implies Uijirij = 0, i < j ;

LEMMA

Classification of Square Domains

253

(5) Uijiriimj / 0 implies max(nij,2mj) < 2mi, i < j ; (6) rijk = 0, nijriikmimjmk 7^ 0 imply max {uij+Uik, nij+2mk, nik+ 2mj, 2mj + 2mk) < 2mi, i < j < k. Proof Conditions (1) and (4) are definition conditions of a normal matrix set of type N. By Lemma 4.16, the proof of conditions (5) and (6) is the same as that of conditions (2) and (3) in this lemma, where ^i,iv+i = 2mi, I
rtii X riik real matrices Cjl = Y^s'^tl^t^s, then {Aj^YAl^ = E imphes njk < riik] {BJjYBlj = E imphes riik < Ujk, hence condition (2) holds. When n,fc = 0, then {A%yA% = 0. Hence (Afj, A*^)^(^|j,^*[) = E implies nji+riki < nu; {CtiycH implies ( C ^ , 4 ) ' ( C ^ , 4 ) = E, so riik + riji < nn; {Bfj,Ally{Bfj,Al[,) = E implies riij + Uki < nn; {Cg, Bf^'{Cll Bfj) = E implies riij + riik < nn. Therefore condition (3) holds. I

2.

Normal Cones and Dual Normal Cones

Suppose that V is an open convex cone in R^ not containing any straight line with the vertex origin. The dual cone of cone V is defined by V^ ={feR''\fx'>0,\/xeV{0} }, (6.36) where V is the closure of V. Therefore V* is also an open convex cone in R^ not containing any straight fine with the vertex origin. We have LEMMA

6.14

{V*y =V. LEMMA

6.15

(6.37)

(Koecher)

(fy{x^,'-',x'')

= iPy{x)=

I {exp{-xu'))du, Jv*

^xeV

(6.38)

is a positive value analytic function on V, such that ^^,^A9^1og^,(x)

is a Riemannian metric on V.

,

^^

(6.39)

254

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Proof

Now,

dx^ ( /

Jv

^Jv*

(exp(—W))^!/]

^

dx'dx^

= / i/*ix^(exp(—xt4'))dt^ / (exp(—x'u'))^^ Jv'' Jv* — I u^{exp{—xu^))du /

Jv*

Jv*

u^{exp {—xu^))du.

Hence ds*^ = (

{exp{-xu^))duj

— f / {exp{—xu^))du]

/

{udx')'^{exp{-xu^))dv m ( /

udx\exp{—xu'))du]

>0

by the Cauchy inequality. Therefore ds*^ is a Riemannian metric.

I

Sometime a Unear mapping y = xAis denoted by A. LEMMA

6.16 Suppose that y = xP is a linear isomorphism from V onto VP = {xP\yxeV}.

(6.40)

Then (yp)*=F*(p-iy.

(6.41)

On the other hand, a cone V is linearly isomorphic to a cone V if and only if V is linearly isomorphic to V . Proof

By the definition of V* (see (6.36)), {VP)

={/Gi?^|/y'>0,VyG(VP)-{0}}.

Since {VP) = VP, so ye (VP) - {0} if and only if a: = yP-^ and _

(vpy = {/ G R^ I f{xpy

eV-{0},

>o,\/xev-{o}}

= { / G R^ I / P ' E y* } = (F*)(p-l)^

•

LEMMA 6.17

AS{V') = {A'\yAeAS{V)}.

(6.42)

Classification of Square Domains

255

Proof A mapping a e GL (V) is denoted by cr : y = xA. Since A e Aff(y) if and only if A'^ e AS{V), so Lemmas 6.15 and 6.16 imply that (7 € Aff {V) if and only if r G Aff {V*), where r : y = x{A')-'^, I COROLLARY

6.18

Denote by aff (V) and aS{V*)

the Lie algebras of pj

Lie groups Aff (V) and Aff (y*), respectively. Then xA-^ e aff (F) if ox pj and only if xA^— G aff {V*). (JJb

LEMMA

6.19 Suppose that ^ : y = xP is a linear automorphism

onR'^,

Aff (VP) = p-^Aff {V)P.

(6.43)

Then Proof Suppose that cr G Aff (V) and r G Aff (VP), where a: y = xA Midr: v = uB.li^oa = To^, then B = p-^AP, I LEMMA

6.20

The mapping G' \

X* = -grad 3. log (/?^ {x)

(6.44)

is a local one-to-one analytic mapping from a cone V into the cone V . Given an affine automorphism a: y = xA, ^y{xA)

= (det^)-Vv(^);

{xA)* =x\A!)-\

(6.45) (6.46)

Suppose that Aff (V) is the affine automorphism group acting transitively on a cone V. Let V^.if) = J{exp{-fx'))dx,

V/GF*.

(6.47)

Then iPy{x)ip * (x*) = const.

(6.48)

Proof As Lemma 6.15, the Jacobian of the mapping (6.44) is nonsingular, hence the analytic mapping a' is a local one-to-one mapping. Now X* = u{exp{—xu^))du( / {exp{—xu^))du] implies x*y^ > 0,Vy G F - {0}, so x* eV\

Since

(Py(x) = / {exp {—xu^))du Jv*

256

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

and V — uA' contained in Aff (V*), so (Py{xA) = / {exp {-xAu'))du Jv*

= / {exp Jv*

= / {detA)~^{exp{-xv'))dv Jv* (xA)* =y* = -gvad y log (py{y)

{-x{uA'y))du

= {detA)~'^
= -(grad,log(det^)-Vv(a;))(^')' = -{gradJogipy{x)){A')-' = x*{A')Hence (6.45) and (6.46) hold. If Aff (V) is a transitive transformation group on V, given any two points XQ, X eV, there exists a matrix A € Aff {V), such that x^A = x, then

When V is a simply connected cone, as the case of complex bounded domains, we also do not know whether mapping (6.44) is a real analytic isomorphism. But this fact is true when F is a homogeneous cone (then V is hnearly equivalent to a normal cone) in the next lemma. Now, we consider the dual cone V^ of a normal cone V^. LEMMA

6.21 Suppose that Vj^ is a normal cone, N

^^{x) = ]\l^^{xr^

(6.49)

is defined by (5.10) and (5.11) in Theorem 5.1, where k

Aj{x) = det Cj{x),

2^

lyjTijk = -{nk + n^k)-

(6.50)

Then we have (1)

the analytic mapping a^: X = -grad log cp^ {x)

is an analytic isomorphism from Vj^ to cr^(V^); (2) a\xA) = x{A-^y, V ^ G Aff {V^); (3)

G'{X)X' = XX^ = n = dim V^, ^/x

eV^.

(6.51)

Classification of Square Domains Proof

257

(1) Given a mapping ai: N

ri = ti+

^

{tiY^Ziiz'^, N

by (3.48) in Lemma 3.8, then ai is an analytic isomorphism from WN = {Z = iti,Z2,t2,. • • /^^,tiv) e M^ I ti > 0,. • • , 0} onto V^. Given two mappings (72: Si = t^ , j-2 Vij = — [U'tj) Zij + / ^\^)

/ ^

1=2

E Pi

. ..V^

\UUi • • 'Hi-itjJ

i
(Pi) (P2) ^ ^ ^ ^(Pi-i)

Z^PO (AP1JAP23 . . . APl-d

y

Pi

and 0-3: 1 2'"

1 '"^ • -t/-^ • 2 1=1

Xij = {rii + n'i)yij + ^ ^ ^ ( ^ ^ + ^/)^/ ^yl?yi3^?i^ 1=1 t where x = (ri,X2,r2,---,Xiv,riv),

Xj = (xij, • • • ,Xj_i,j),

y = (5i,y2,s2,---,yiv,57v),

yj =

iyijr"^Vj-ij)^

then C,(a;) = P,(z)Q,(^)-ip,(z)', Q,(y) = Qjiz)'',

Pj{y) = Pj{z)-'

by Section 4.1. By Theorems 4.3, 4.5, 4.7, (72 and as are two analytic isomorphisms, and the explicit expressions of a^^ and a2 are the same. Hence 0-3 o 0-2 o aj"-^ is an analytic isomorphism sending x to x. The explicit expression

258

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

IS

1 1 *~^ 2 ((«i + n'i)si + - ^{ni

+ n'i))sYhiiy'ii

1=1 N

=

-J2^jtvP-j{yyQj{y)-'Pj{y)Cjieii)

= - f : .,.rC, W-'C,(e„) = - 2 ! 2 ^ = F,

im+K)y^+E("^+«^)*r'

E ?/!5^A.^f^'t

l
s

for all indices. It proves that a = (73 O (J2 O (J J" .

Hence a' is an analytic isomorphism. Statement (1) holds. (2) By (6.45), (py{xA) = (det A)-Vv(^), Vy = a:A € Aff {V), thus aiogy^(t/) ^ aiQgy^(x) ^ Y^aiogy?^(a?)ga;j %

%

^

5^j

%*

Hence (T'(X^) = a^x{A~'^y. Statement (2) holds. (3) Now, Aff (V^) acts transitively on V^ and the fixed point VQ — (1,0,1,--,0,1) G V^. Given a point x eV^, there is a matrix A G Aff (V^), such that v^ = xA. Now \v[ = £4(xA)' =

X(A-1)'AV

= xx',

Vx € y^.

By (6.49), (6.50) and (6.51), % = -(grad log(Pj^{x))x=y^ = (grad ^i;jlogdetC;(a:))a,=t;oj

Since aX^t/fclogdetCfc(x) ^ 1 ^^r = 2^i^j^^Cj{x) 'Cjieu), —(-

= }^i^ktrCkix)

^Cfc(ey),

Classification of Square Domains

259

so ^0 = (2(^1 + ^ i ) ' 0, 2(^2 + ^2), • • •, 0, -(TIN + n'jv)). 1 iV Hence v^v^^ = 7:^ (^j + ^j) = ^- Statement (3) holds.

I

6.22 The following symbols are the same as the above. We have (p^ = c(fy , where c is a positive constant, and a' = a, x = x ,

LEMMA

Proof It suffices to prove ^j^{x) = c(py {x). By (6.45) and (2) of Lemma 6.21, we have

Since Aff (Vj^) is a transitive transformation group on V^,

is a constant. Hence ip^ = c(py . LEMMA

6.23

The following symbols are the same as the above. {xy=x,

Proof

I

yxeV^.

(6.52)

By (6.48), letting x = (xi, • • •, Xn) G M^, we have (p^{x)(p* (x*) = const.,

hence d{\og(p^{x)+log(p* (a:*) : ^ = 0.

d{xy Since (a:*)* = —grad log(p* (x*), we have

(^*Y _

^'''> ~

^^Kj^

dixi)*

_ ^

~^

aiogy^(x) dxj

dxj

_ _^

d{xir~

dxj

2^^''^> dixi)*-

dx' By Lemma 6.20, x^'x' — n, thus ^[xifxi — n. Hence Xl(^i)*o/ ^\* + Xi = 0. Therefore (x*)* = Xi,l
260

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

6.24 The following symbols are the same as the above. Let V^ be a normal cone. Then a^ is an analytic isomorphism from V^ onto V^, and the affine automorphism group Aff (V^) is a transitive transformation group on V^. On the other handj

LEMMA

{a')~^ :

X = -grad logy?* (y).

Proof Given a point x eV^, there is a matrix A G AfF {V^), such that X ^VQA, where VQ is a point in V^ H V^. Hence the orbit Aff (V^)(^o) = Given a point y e (T'{VJ^), there is a point a; € V^, such that y = a'{x) = a:*, where x = v^A and ^ € Aff (F^). Thus y = x* = (VQA)* = vl{A-^y = v^{A-^^y by (6.46). Hence a'{V^) = Aff{V*^){v^). Therefore (T'(V^) is an orbit of Aff (V^), which contains VQ. Now, y* = {x*y = xeV^, (^O^'(K^) C V^. Given a point ye F ; , then X = (o'')~^(y) e l ^ , thus a^x) = y. Hence o''(V^) = V^, and Aff (V^) acts transitively on Vj^. I Next, we give the explicit expression of a new normal cone V^, and prove that W"^ is linearly equivalent to the dual cone V^ of a normal cone V^. 6.2 Suppose that V^ is defined by a real normal matrix set { Aj^ } of type N, Let DEFINITION

riij = nN+i-j,N+i-h

l
(6.53)

Given riik x rijk real matrices rijk

BS = E ^ ^ + f - r i v + i - i 4 e „

1 < t < n,,-,

(6.54)

p=i

then the matrix set {Bjj} is also a real normal matrix set of type N. A normal cone V^ is called the adjoint cone of a normal cone V^ associated with the real normal matrix set {Bj^} of type N. 6.25 The adjoint cone F ^ of the adjoint cone V^ of a normal cone Vj^ isVj^.

LEMMA

Proof

By definition 6.2, we have riij = niv+i-j,iv+i-i = n^j,

l
<j < N

Classification of Square Domains

261

and ntk _ Y^ TyP,N+l-i

/

_ V^ .qk f

/

_ Atk

PA

6.26 Let Vj^ he a normal cone associated with a real normal matrix set {^f^} of type N, and let the coordinates in V^ he presented by

THEOREM

X = (ri,a:2,r2, • •

-.XN^VN),

XJ

= {xij, • • -.Xj-ij),

ri G R, Xij e R

.

Then V^ is linearly isomorphic to a linear homogeneous cone

% = { t/ GM" I CM > 0,---,CM

>0}

(6.55)

by the linear isomorphism Si = r^+i_i, 1 i
(6.56)

where Si G R, t/y G R"*-', and y = isi,y2,s2,---,yN,SN),

yj = iyij,---,yj-i,j)

(6-57)

is a coordinates in Vj^, and ( siE("w) {A^^^y

A^} 12

^S-i

S2E^''^i)

.0)

y'^0 ^ 2/2.7

, (6.58)

CM =

(4]-l)' (4]-l)' i = i,---,iv,

4 - E 4 ^ ^„(*)i4«fc i'

i<^
(6.59)

Proof The normal cone V^ is defined by a real normal matrix set {Bf-} of type iV, where

•^N+l-k,N+l-j

~ Z ^ Wjik ^Ar+l-fc,jN+l-j-

262

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Now, V^ =

{Ci{y)>0,---,CN{v)>Ol

where ViN

\

TNE

j

Ci{y) \ ViN ^l,iV-i By the Unear isomorphism ^: ^i •" ^iv+i-i? 1 S * S -iV;

.„ l < i < i < i V , y.^ — x^^^_^^j^_^^_

where x = (ri,0^2,^2,••• ,a;iV5rjv) € R^ is a point coordinates in V^, hence _ P a ( y ) P ' = C,(x), l
0

•••

E("'J^)

\

0 0

/

P = 0

Vl

^("i,i+l)

0

Suppose that V^ is defined by a real normal matrix set {A*j^} of type N. Section 5.1 gives that the Lie algebra aff {V^) of the affine automorphism group Aff (V^) has a basis Aj\x) — ^rj—

+ } ,Xpj^—P3

p<j

+ J^Xjp dx 3P

(6.60)

3
4^(-) = 2xg)|: + ^ . A + EE(-pi^^'^*) ^ d^';pi

(6.61)

d i
s

^-^ip

L{x) = Y^XijUj^,

j
^

L[^Af^ + 4^L,fe = 5 ; ( e . L i , e O A j

(6.62)

U

for all indices and when the symmetric matrix S = (riki) satisfying condition Z(i, j ) (see Definition 5.1),

^^5"w-^^g'|:+-S5+EE^;;'M^y^ p
s

(6.63)

+i
3
3P

Classification of Square Domains

263

where 1 < t < riij, and (ri,0:2,^2, • • • ^XN^r^) is the coordinates in V^. By (6.56) and (6.57), the Lie algebra aS{V^) of the afSne automorphism group Aff {V^) has a basis

r

Now, Y.^^\x) G aff (V^) if and only if the symmetric matrix S = (rtpq) satisfies condition Z(i, j ) (see Definition 5.1). In this case, the symmetric matrix S = {ripq) = {nN-^-i-q N-\-i-p) satisfies condition Z{N + 1 —j^N + i-i). 6.27 Let Vj^ be a normal cone associated with a real normal matrix set {Ajj} of type N. Then (1) The linear isomorphism

THEOREM

T:

ri = 2si,

1 < i < AT,

Xij = yij,

l
(6.64)

maps the dual cone V^ onto the adjoint cone V^ of V^^j where x is the coordinates in V^, and y is the coordinates in V^. (2) Denoting by y = {si^y2,S2r " •^VN^^N) the coordinates in VJ^, the Lie algebra Aff (V^) has a basis

4*=2.A + g . , | - + g . , | - ,

(6.65)

(6.66)

i<3

^^^

r

264

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

for all indices, and when Y>P € aif (V^),

% i
^*» s

Ki

s

OVii

Hi j
^^^

(6.68)

I
Now, Vj^ is a linear transitive cone by Lemma 6.24. V^ is also a linear transitive cone. Hence Aff (V)v)(t'o) = ^N- Therefore r(V^) = Vj^.

I

6.3 Suppose that V is an open convex cone not containing any straight line m R'^ with the vertex origin. The dual cone V of V is called the self-dual cone, if

DEFINITION

y* = V,

(6.69)

6.28 Suppose that V is an indecomposable self-dual cone not containing any straight line in R'^ with the vertex origin. If V is a linear homogeneous cone, then V is linearly isomorphic to an indecomposable normal cone V^ associated with a real normal matrix set {Alj} of type N, where n^^- = p € { 1,2,4,8 }, l
Proof By Section 3.1, there exists a linear isomorphism y = xP^ such that V = V^P. Now F* = y , thus (F^P)* = V^P. By Lemma 6.16, ( ^ . v ^ r = V ^ ; ( ^ ' ' y , hence

vl = v,pp'. Let PQ = PP\ thus V^ = VJ^PQ, where P^ is a positive definite symmetric matrix.

Classification of Square Domains

265

By the discussion of Section 3.1, we can choose a hnear isomorphism T acting from V^ onto V^, the coordinate representation is y = xPi =

ri-^n,

l
Xij

where P2 G GL (n,R). Since niX^f)

i
^ Xij -lij 5 l
s

*P

1 < * < ny, 1 < z < j < iV. By (6.67),

(y,f(x))*=x(;)(y),

l
l
Hence F^f (a;) G aff (V^), 1 0. By Definition 5.1 and Theorem 5.9, Y^^ € aff(V;,), 1 < t < Uij, 1 3, then riij = p > 0, 1 < i < j < N, where p E { 1,2,4,8 } by Theorem 6.8. I The above lemma gives that if F is a self-dual cone, V is linearly equivalent to a square cone and a dual-square cone as in the following two sections. Hence the classification of self dual cones is contained in the classification of square cones and of dual-square cones.

3.

Classification of Square Cones In this section, we will give the complete classification of square cones.

DEFINITION

6.4 A normal cone V is called a square cone, if

nij = n2j = '" = rij-ij

= a^ > 0,

2 < j < N,

(6.71)

266

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

A square cone V^ is denoted by V{a2^ • • •, cr^).

Give a square cone V^ associated with a real normal matrix set {A*^} of type iV, where Afj is an riik x Ujk real matrix, then riik = rijk = (Tk gives that {A^j} is a cr^ x ak square matrix set. Given Oij G O(nij), "i-
4 ' = T.(^rOije'tnj,A:}Ojk,

(6.73)

r=l

In the case of the square matrix, the definition conditions of a real normal matrix set of type TV give that ak x ak real matrices Aj^, I
{AtfyA^^ + {Al^YA^} = 26stE,

^J^fi = E(e.4-6;)Arf.

(6.74)

(6.75)

r=l

Condition [A^j ) A^p — ^ ^ ( ^ t ^ i j ^rJ^jp^ r=l

can be induced by (6.74) and (6.75) easily. By Theorem 6.8, the canonical form of a normal matrix set {^*j}, Tin = '" = rij-ij = cr^- > 0, j = 2, • • •, TV, can be found in the next lemma. LEMMA

(i)

6.29 When N = A, the canonical square cones are ^2 = 1,

A}i = l,

A\\ = l,

4 3 = 4 1 = ^^

l
* * where a a^ x a^ canonical form P^, • • •, F^^ is defined by Theorem 6.7;

267

Classification of Square Domains (ii)

(72 = ^3 = 2, \lk

0

2fc

4=^' 4(iii)

-£;

1 < i < j
0-2 = 2, (73 = 4,

A\l = E,Ai^=(^l

o),k

= 3,4,A'A = A^ =

Pt,l
where / 0 E P2 = 0 \ 0

Pl=E,

-E 0 0 0

/O P3 =

0 E

-E 0

0 0

F4 =

0 0 0

0 \ 0 E

-E

0J

0 0 -E 0

0 E 0 0

(iv) (72 = 3, (73 = 4, ^1^ = E, A\\ = 4 1 = / 0 E j^2k 12 — 0 (v)

0 0 0

0 0 0

0 \ 0 E

-E

QJ

iZk •^12

-E\ 0 0 0 / Pk,l
0 £;

-E 0

''

(72 = (73 = 4, 4*3 _ p(4) ^12 — -ft )

/|t4 _

p(o-4)

1 < * < 4,

1 < i < j < 3.

P r o o f Now, Aj2, 1 < t < 072, are (73 x (73 real square matrices, ^4^2) 1 < f < (72, AI3, ^23) 1 < * < crs, are (74 x (74 real square matrices. Let On = E. There exist Oi3,023 G 0((73), such that Oi3A*i|023, 1 < t < (72, are canonical form Pf, 1 < t < (72, by Theorem 6.7. Without loss of generality, let Af2 = Pt, 1 < t < (T2. Let O12 = E, O13 = O23 = E. There exist O14, O34 G O ((74), such that O'l^Af^Osi, 1 < t < (73, are canonical form Pt*) 1 — * — ^3- Without loss of generality, let A*^ = P^, 1 < i < (73. By (6.74), A\^ G O ((74). Let Ou = O34 = £^, O24 = (^12)'We have O'^^A^^O^i^ = E, 0^4^1034 = A\^. Without loss of generality, let A\i = E, Af^ = P ; , 1 < t < (73. By (6.75), Alt

= J2ierPse't)Pr{Af^y, r=l

1<S<(72,

1 < t < a^.

268

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

When s = 1, we have Pi = E impUes Af^ = Pt*^ 1 ^ * ^ ^3- Hence the above formula can be changed into

r=l

If (72 = 1, case (i) of this lemma holds. If a2 > 2, we have A\l = E, ^

n

) * -^^^^^ ^3 i^ ^^ ^^^^ number. Let s = 2. We have

Hence _ _ ^ . . . , = ^ . . . , ^ f c > fc = 2 ,

^

If as = 4j > 4, then 2j + 2 < CT3. Let A; = 2, hence P2J+2 = ^27+1^2 By Theorem 6.7, P^ = E, P2 = ( ^ p* p*

~0^ ) ,

_ f B1R2J-1 _ /"

0

0

-v^BiP2,

thus i?2j-i = \ / ^ i ? 2 j - It gets a contradiction. Thus j = 1. Hence (73 = 4 and jR2 = -y^Ri = -V^E. Therefore p = 0. If (73 = 4j - 2, i > 1, 2j + l< (73. Let fc = 2, P^j^^ = P^jP^ Hence i?2j-i = —\/—Ii?2j-2- It also gets a contradiction. So j = 1. Thus CT3 = 2. Hence (72 > 2 implies as = 2 or as = 4. And i?2 = — V^-li?i. Therefore p = 0. If (72 > 2 and as = 2,

Thus (72 = 2. Hence |13 _ / 1 0 \

,23

/ 0

^ 1 3 "" -^23 "" ^y 24 __ 4:^4 __ 4yi24 2 4 _ ^/

0

^13 - ^23 - I £;

-£^

0

269

Classification of Square Domains Now,

A\i = E, AJiA'^ = Y,{erAlle[)Ali

t = 1,2.

r=l

Let t = 1, Ali = All

We have A}} = E,A^f=(^l

(f ) , 1 2, (73 = 4, then i?2 = -y/^Ri, hence p = 1. Without loss of generality, let A^^ = -Pt, 1 < t < cr2, where Pt, I < t < a2, are 4 x 4 matrices, therefore a^ = 2,3,4. When a2 = 2, as = 4, let Pi = E, P3 = / 0 -E £; 0 F2 = 0 0 \ 0 0

0 0 0 -E

0 \ 0 £; 0 J

E

"^ 0

0 0 0 E

0 0 -E 0

, a-

0 E 0 0

-E \ 0 0 0 J

By the same discussion,

where P ; = P l , P2=P3, Let O12 = £;, 0i3 = O23 = E,

^3=^2,

/ 1 0 0 - 1 Ou = 024 = O34 = O l \ 1 0

Pl=-P4.

0 1 \ 1 0 i o 0 -1 /

We can assert that 113 -^12 — ^1

23 ^12 —

0 E

-E 0

A'A = 4 1 = Pt, 1 < * < 4,

up to a real equivalent (6.12). Now, A\i = E,

Ali = ^{er(l r=l

-^ ^

Hence case (iii) of this lemma holds.

) 4)PrPl = Ps. ^

270

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

If (72 = 3, (73 = 4, without loss of generality,

/ 0 -1 1 0 0 0 \ 0 0

0 0 \ 0 0 0 1 - 1 0 /

0 E

433 _ ^12 —

-E 0

A\% = 4 1 = Pt, 1 < f < 4,

A\\ = E,

/ 0 -1 1 0 0 0 \ 0 0

A r,t

^?l = E('r

( l

0 0 \ 0 0 0 1 e[)PrPl = P2, - 1 0 /

" ( f ) e^PrPl = P3,

up to a real equivalent. Hence case (iv) of this lemma holds. Finally, if cr2 = as = 4, by Theorem 6.8, without loss of generaUty, 4 1 = Pul
Af, = Af^ = P^'\

l
A\2 — E, up to a real equivalent. Hence case (v) of this lemma holds. I 6.30 Suppose that V{a2,cr^-,"' I^N) i^ ^ square cone, N > 4. Then there exist three cases as follows:

LEMMA (i) (ii)

(72 = • • • = (7iV-2 = 1, Cr^v > (7^_i > 1. (72 = • • • = (7^ = 1, a^^i = '" = a^^r] = 2,

(7^_, = 4, where 1<^ 4.

-3,

crj+r/+i = -"

=

7?>0. W^/ien ^+ ri = N - 1, aN >2;

( i i i ) (72 = • • • = (7^ = 1, (7^+1 = 3 , (7^+2 = • • • = (7^_i = 4, (77V > 4, w;/iere 1 < ^ < AT - 3.

Proof Suppose that V{a2^ era, • • •, a^) is defined by a real normal matrix set {Ajj} of type N. Given a positive integer number M, where 4 < M < iV, then the subset {Al^,

l
M -3

defines a cone V{aM-2iO-M-i^o-M)- By Lemma 6.29, this cone is one of F ( 1 , ( 7 M - I , ^ M ) , F ( 2 , 2 , ( 7 M ) , F ( 2 , 4 , ( 7 M ) , F ( 3 , 4 , ( 7 M ) , F ( 4 , 4 , ( 7 M ) .

Hence ((72, (73, • • •, (7iv) is equal to one case in this lemma.

I

Classification of Square Domains

271

6.31 The canonical forms of square cones up to a linear isomorphism a: ri ^ ri, 1 0 }.

THEOREM

(2)

N = 2, Via2) = { (ri,a;i2,r2) € M | ( ^7

(3)

^^^

)>^^-

N>S,V{l,---,l,<7^_„aN), A}^ = E,

l

45v-i = -Pt, l
l
} is the a^ x CTN canonical form in (6.30).

(4) N>4, F ( l ~ ^ , $ 7 ^ , 4 7 ^ , (TN), where ^ + r? + C + 1 = AT, r?, C > 0, 1 < C < iV - 3. When ^ = 0, (Jiv > 2; when C > 0,CTiv> 4.

A}} = E, . 2fc _ /

1< l
-E 0

l
Af^ = Pu (5)

i<j

+ r], +

ri-\-l<j
AT > 4, F ( l ~ ^ , 3 , 4 ~ ^ , UN), where e + C + 2 = Ar, e, C >

1, ON > 4.

A\^ = E, Al^ = P4,

i<j
l
l<j
j = ^ + 1, t = 1,2,3; when j = ^ + 2,-• • ,N - 1, t = 1,2,3,4, and {Pi, P2, Ps, P4} is the ON X tr^v canonical form in (3) of Theorem 6.8. Proof By Lemma 6.30, it suffices to prove the case of iV > 4. When iV = 4, it is Lemma 6.29. When iV > 4, we use the induction. Without loss of generaUty up to a real equivalent (6.12), let A ! ^ , 1 < t < aj, l
272

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

6.8, and (y^j^AlfOjN = E,2<j Alf = E,l<j

A\f = Alf = ... = AJ%,

2<1
(i) The case 1^(1,•••,!,crj^_i,<7iv). Without loss of generality, let A\^j^_i = Pt in (6.30), thus >1^5v_i = Pt, l
l<s
l
r

Letl = N-l,

Alj^ = E{erA'if-'^e't)PrPl.

When aj = 1, Aj^^ = E =

r

Pi; when aj = 2,

' 0 -E E 0 <

hence Aij

\

0 -E' E 0A J?

)

^2;

when Cj = 3,4, Al^ = P5. Hence cases of (4) and (5) of this theorem hold. I The above theorem gives the classification of a real normal matrix set {Aj^} of type Nj where A^^ is a square matrix for all indices. The total invariant is (T2, • • •, CTNJ except AT > 3, V(l, • • •, 1, cr^_i, a^), where (7^_^ is an even number. In this case, there is another invariant p, where the positive integer p is defined by (6.18) and Theorem 6.7. THEOREM

6.32

Any square cone is real equivalent to canonical square

cones: Vi = {rie

R2+^2 I n

> 0 },

V{a2) = { (ri,a:i2,r2) G E | ( J/^

^ ^ J ) > 0 }.

273

Classification of Square Domains WhenN>3,

Vj, = V{l,---,l,a^_^,a^), / 5

1 < (TJV-I ^<^N' ^«

P

\ H^N-\,N^t

>0,

V X t where S is an {N — 2) x {N — 2) real symmetric matrix^ and P is an (AT — 2) X (<7^_i + cr^) real matrix. Pi, • • •, P^- _ ^^^ ^iv x (^N canonical forms (6.30). When N > 4, V^ = F ( l , - • • ,1,2,-• • ,2,4,-• • ,4,criv); where a2 = ••• = cr^ = 1, cr^+i = '" = a^j^r) = 2, a^+r^+i = ••• = cr^+r/+c = 4, ^ + r? + C = ^ - l , A ^ - 3 > ^ > 1 , r / , C > 0 . (1) C > 0; cTiv > 4, and V)^ is defined by

(T,r,). /n particular, if ^ = 1, V^^ is defined by S

T

Ti \

r' ^ Ta > 0, r; r; ^1 / where S is a^x^ real symmetric matrix, H is anrjxr] Hermitian matrix, Hi is a2( + ^ Hermitian matrix, and HiJ = J Hi, J = diag

0 1 \ f 0 -1 0 ;'*"'\^ -1

1\ f 0 0 J ' \ - E

E 0

T, Ti, T2 are complex matrices. (2) When ( = 0, aN >2, Vj^ is defined by H>0,

S>ReTH-^T\

In particular, when ^ = 1, the above formula is equivalent to S

T

r' H

>0.

When iV > 4, V;^ = V(l, • • •, 1,3,4, • • •, 4, GN), where a2 = " - = a^ = 1, cr^^-2 = •••=: (T^_^ = 4 , CTiv > 4, 1 < ^ < A/' — 3, Vj^ 25 defined by

-1

It

Ti

0

vJ

>0,

274

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where r eR^ v is a complex vector of dimension 2{N — 2 — ^) + ^ , Hi is a {2{N — 2 — ^) + ^) Hermitian matrix satisfying condition HiJ = J Hi, y e R^, Ti is a^x {2{N - 2 - ^) + ^ ) complex matrix. Proof By Theorem 6.31, we consider only one of the following square cones: (1) y ( l , . . . , l , 2 , . . . , 2 , 4 , . . . , 4 , ( 7 A r ) ; (2) y ( l , . . . , 1 , 3 , 4 , . . . , 4 , a A r ) . The proofs of the remaining cases are the same. (1) Let / 5«) U V Ci{x) = U' A(2^) C

\ v

a B

Give the block partitions of A by 2,2, • • • rows and columns, the diagonal block is aE^ and the non-diagonal block has the same expression as L, where a b L = —b a Give the block partitions of C by 4, • • •, 4, cr^v rows and 2,2, • • • columns, every block has the sanae expression as L, where a, 6, c, d are matrices. Given the block partitions of 5 by 4,4, • • •, 4, a^ rows and columns, the diagonal block has the form aE^^\ the upper triangular non-diagonal block has the form d \ b c -h a d —c —c -d b a \-d c -b aJ

1«

Let

x/2 V E^'^ V^E(')

Y

J„ = diag (£;(«, t/, •••,[!),

the block partitions of these matrices are the same as above. Then every block of matrices A, C changed to Ui

a -b

U\ =

a+

0

V-lb 0

16

a—

and every block of matrix B changed to

/a a

u

b

—b a —c —d \ —d c

cc d a —b

d \ —c b a

(

V

u

V

—V

u 0 0

0 0

0 0 u

0\ 0

—V

u j

V

275

Classification of Square Domains Hence Ci{x) > 0 if and only if / S

T T' if T'l % T' 0 \T{ 0

Ti T2 Hi 0 0

T Ti \ 0 0 >0. 0_ ^ H T2 T!^ Hi J

Therefore when C > 0, this theorem holds; when C = 0, this theorem holds for the same reason. Finally, if C = 1) (r.Ti)

(n 2

T2

{T,Ti)

Hi

is a real number. This theorem holds. (2)

Let S a U Ci{x) = I a' r5+i£;(3) v U' V B

Give the block partitions of Ci{x) according to ^, 3, 4(iV — ^ — 2) + crjv rows and columns, where B is as the above, and V = (Ai, • • •, Ajv-|-i), such that Aj has the form

Hence U'

where u^v axe complex numbers. Let ot = (e f g)^

) = {wfw),

276

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where w = -y={e + y/^g).

/

u 0

S b b' r^+i u 0

Hence Ci{x) > 0 if and only if

Ti vJ

Ti vJ

u 0

T\ vJ

\

H+\

>0.

Hi

V \0 vJ J

-' V'

H,

/

By Theorem 6.32, we give the complete classification of normal Siegel domain of the first kind over a square cone as follows. 6.33 A Siegel domain of the first kind over a square cone is holomorphically equivalent to a Siegel domain of the first kind over a canonical square cone in Theorem 6.32. The total holomorphic invariant is the total affine invariant of a square cone.

THEOREM

By Chapter 5, we can compute the holomorphic automorphism group of a Siegel domain of the first kind over a square cone easily. By Lemma 6.28 as a corollary of Theorem 6.32, we give the complete classification of self-dual cones as follows. 6.34 Suppose that V is an indecomposable self-dual cone not containing any straight line m R^ with the vertex origin. If V is a linear homogeneous cone, then V is linearly isomorphic to one of the canonical normal cones V^ as follows: (1) N = l, Vi = { a ; € R | a ; > 0 } ; (6.76)

THEOREM

(2)

N = 2,

V2 = F(ni2) = {(n, X2, r2) € R X R"'^ x R | n > 0, ( ^J

^^^ \> 0 } ; (6.77)

(3)

AT = 3, ni2 = ni3 = n23 = 8,

V3 = V{d,,%) = {x =

{ri,xi2,r2,xiz,X2z,rz), ri€

X12 xiz r2E R{X23) ar'ia i?(a;23)' r^E

n

X 12

>o},

^5 '*^ij

(6.78)

277

Classification of SquareDomains where X23 =

{xi,'",xs), (

x\ -X2 -X3

X2 Xi

X3 X4

—X4^

Xi

—X4

Rix23) =

X3

-x^ -xe -X7 \

-xs x^

X^ X8 -X7

-X8

-X3 X2 XX

-X2 -X7

—XQ

X4

XQ

-X8 X7 -X6 ^5

X5

XQ

X7

-x^ xs xs --X7 xi X2

-Xs -x^ xe

^6 X7

X\ X4

-X2 -X3 —X4

X3 —X4 Xl

-X3

X2

Xs \ X7 —XQ

-x^ X4 ^3

-Xs Xl J

(6.79) (4)

riij =

l,l
1^N=V{lr-

1) = { X G M

N(N+1) 2

\s>oh

(6.80)

where S is an N x N real symmetric matrix by ^^^^^—- independent real variables; (5) riij = 2, l
=

{xe

,Ar2

Ci{x) > 0 },

(6.81)

where ( Cx{x) =

r\

iCl2

XlN

X 12

r2E

R{X2N)

\ x\^

\

(6.82)

R{X2Ny

where Xij = {uij,Vij) €

Ri^ij) = (6)

Uij = 4 ,

Ui

_7. ~Vij

""tj

(6.83)

Uij

l
V^ = F(4, • • • ,4) = { a; € R^(2iV-i) | ^^(^) > Q j ^ /

ri ri X 12

Ci{x) =

\ XiN where y =

a;i9 ^12 r2E R{x2Ny

••• •••

\

XIN

RiX2N) TNE

(6.84)

(6.85)

J

{yi,y2,y3,y4), (

yi 2/2 2/3 y4 -2/2 2/1 2/4 - 2 / 3 Riy) = - 2 / 3 - 2 / 4 2/1 2/2 V - 2 / 4 2/3 - 2 / 2 y i

(6.86)

278

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Conversely J the canonical cones in Theorem 6.34 are all indecomposable self-dual cones.

4.

Classification of Dual Square Cones

DEFINITION

6.5

A normal cone V is called a dual square cone, if

rii^i+i = ni^i+2 = '" = rii^N = pi>0,

I
(6.87)

A dual square cone V is denoted by V{pi^ • • •,Pj^_i)' By Section 6.2, given a normal cone Vj^ associated with a real normal matrix set {A^^, I < t < nij, l
LEMMA

Proof

If V^ is a square cone, then n\i = n2i = '" = rii-i^i = pi> 0,

2
Thus the adjoint square cone V^ of Vj^ satisfies the condition riij = nN+i~j,N-\-i-i = pN-^i-i = (Ti>0,

I
It follows that ^2,i-f 1 = rii^i-\-2 =

• = riiN = Piv+i-i = (^i, I
Hence a normal cone Vj^ is a square cone if and only if its adjoint cone V/v is a dual square cone. I By the above lemma, the classification of dual cones deduces the classification of adjoint cones in the case of square cones. By Theorems 6.26 and 6.31, we have 6.36 seven cases:

THEOREM

The canonical dual square cone has one of the following

Classification of Square Domains (1) (2) (3)

279

Fi = { r i G R | r i > 0 } ; ^2 = { (T-1, xi2, r2) € R X R"i2 x R | r2 > 0, rir2 - xux[2 > 0 }; 0-2 < (73, V{(T3,a2) = {

(ri,a;i2,r2,a;i3,a;23,''3)|

t

E^i2^/ I t \

riE^"'^

43

a;23

^3

a;i3

>0}, I /

where {Pi,---, Pa^ } is a a^y- (TZ canonical form (6.30) (4) V{aN, 1, • • •, 1), o-jv > 1, crjsi

S (5)

>0,

riv-5^c^i5"^a->0,

VrAr > 0,

and

a^eR^'^

V{ /

riE"

•••

xi^N-2E

Pi,N-i

x'^N

\

\

xiiv

•••

XN-2,N

XN^I.N

TN

I

where E = E ^ , and { Pi, • • •, P^- _ } isa^^xa^ and

canonical form (6.30),

t

(6) Vi<7N,4,---,4,2,---,2,l,---,l). The condition is the same as V{1, • • •, 1,2, • • •, 2,4, • • •, 4, anf), hence Bo

.

\Si

^0

K S2

Co

-s'A

Eo

S2

pi^)

h

>0,

S's ^ TN

/

where 5i, 82, S3 are three complex vectors, and A,=A2®E^^\

B^=B2®E^^\

Co = C2 ® E^^\

E^=E2®

E^^\

D,=D2®E^'-^\ Fo = F2 ® E^"-^\

280

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

(8) is the Kronecker product, and B^, D2, C2, E2, F2 are block matrices according to 2,2, • • • rows and columns^ the general blocks of matrices B2, D2 have the form I

>—r-,

J, where a, b are real numbers, and

the matrix D2 has 6 = 0. The general blocks of matrices C2, E2, F2 have the form a b —b a where a, b are complex numbers; (7) F ( a „ , 4 , . . . , 4 , 3 , l , - - - , 1 ) . The condition is the same as V(l, • • •, 1,3,4, • • •,4, a^).

/^(^) ^0

\c:

\Si

s[]

Bo Co Do Eo % S2

pi'-^)

s'. t.

Ss

TN /

>0,

where 6\, 62, 5z are three complex vectors, and a —h C^ = C2®E^"-f\

E^=E2®E^^\

F^=F2®E^^\

and a € M, 6 € C and C2, E2, F2 are the same as case (6). Proof We will only give the proof of cases (6) and (7) as follows: (6) F ( a j v , 4 , . . . , 4 , 2 , - - - , 2 , 1 , . - - , 1 ) . This cone can be written as a real positive definite symmetric matrix

X =

/ ^(e<Tiv) B B' i)(wjv) C

\ Qi

C E

9i (f^

\

E'

F^^^N)

q'^

0.2

93

fN J

>0,

where A, B, C, D, E, F are block matrices, such that the block partitions by a^ rows and columns. Hence A,B,C,D,E,F can be written as Kronecker products:

A = Ai®E^''^\

B = Bi®E^^\

D=

Di®E^^\

C = Ci®E^"-^\

E = Ex®E^^\

F = Fi®E^"-f\

281

Classification of Square Domains respectively, where Bi = (Bij)^ Ci = {Cij)^ Z?i = (Aj)? -Qij

rij

Sij

Qij Tij

Pij ~Sij

Sij

—'"'ij

^ij

Tij

( Pij

-bij

Bij = 1

\

aij J

)

^ij

~ \

Dii = CiiE('\ A i = ( 2 ^

Pij

Qij

Pij 1

-Qij

^ y

\

i + 3-

The matrices E\^ F\ and C\ have the same form. Let two unitary matrices

/^<^. \

E

^.[11) - A / ^

0 E

E 0

\

r, _( E -V^E \ ;

When C > 0, let C/ = (l/V2)diag(Po, • • • ,-Po>2); when ( = 0, let U = (l/v/2)diag(Qo,---,Qo,2). Using X -^ U'XU, statement holds by a direct computation. (7) F ( ( 7 ^ , 4 , - . . , 4 , 3 , l , - - - , l ) . This cone can be written as a real positive definite symmetric matrix /

X =

A^C<^N)

B

B'

£)i<^N)

C E

a

E'

piCffN)

Q2

QZ

\Qx

9i \ 92

>0,

^Z

rN )

where (a b c

5 = \0

—b —c 0 \ a 0 —c 0 a b c -b a )

,£;(^),

i ) = r^+i£;(^^),

the matrices A, C, E, F are also block matrices as case (6), and qij q2, Qs are real vectors. Now, C > 0, it is the same as case (6) acting the unitary matrix U, hence this statement holds. I By Theorem 6.36, we have 6.37 A Siegel domain of the first kind over a dual square cone is holomorphically equivalent to a Siegel domain of the first kind

THEOREM

282

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

over a canonical dual square cone. The total invariant is the total affine invariant of a dual square cone. When a Siegel domain is a normal Siegel domain of the first kind over a canonical dual square cone, the holomorphic automorphism group can be found easily by Chapter 5.

5.

Classification of Square Domains

In the above section, we give the classification of square normal Siegel domains of the first kind. In this section, we will give the classification of square Siegel domains of the second kind. 6.6 Given a square cone V^^ a normal Siegel domain D{Vj^jF) is called a square normal Siegel domain^ if DEFINITION

m i = 7712 = • • • = niN = m^j > 0.

(6.88)

A domain D in C^ is called a square domain^ if D is holomorphically isomorphic to a square Siegel domain. 6.38 A canonical domain of a square normal Siegel domain of the second kind up to an affine equivalence is one of the following domains.

THEOREM

(1)

N=l, D{Vi,F) = { {z,u) € C X C I lm{z)-uu'

(2)

> 0 }.

N = 2,

D{V{a2),F)

= {{s,zi2,S2,ui,U2)

G C X ^ 2 X C X C o X C^o |

where (72

RM

= Y^e'j.U2Rrr=l

(3)

N>3,

DiVil,---,l,a^_„a^),F), A}} = E, Aj^-i

= Pu

Q\f=E, Q & - 1 = Rt,

Qf^ = Rt,

l
l
l
l
283

Classification of Square Domains

(4) N>A, £>(F(1T^,5~^,C^,(T;v),i^), ^ + 77 + C + l = iV,77 + C > 0 ,

l
When C = 0, (Tjv > 2; w/ien C > 0; •''Ar > 4, and A|j= = £;,

l
^?f = f ^ ~0^ ) ' ^ 4,

I>(y(T7^,3,^r^,<Tiv)), C + C + 2 = iV,e, C > l , a i v > 4 . Al^ = E, l
l<j
Q^

0_ i£;

Qfl = Rt,

- ^ ^ E 0

,

l<j
l
-\,

284

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

The matrix set {Pt} is the canonical form in Theorem 6.8; and the matrix 5ef { iii, • • •, i?^ } is the canonical form in Theorem 6.1. Proof

Since mi = • • • = TUN = CTQ > 0, Qfj, I
N, are a^ x a^ matrices satisfying the condition Q\J Q\J + Q\J Q\J = 26stE. Clearly, there exist a^ x a^ unitary matrices UI^-"^UN^

such

that UiQ\^_^-^Ui-^i = E , i = ! , • • • , i V - 1 . Without loss of generality, let Q^,'=

E,i

= l,.--,N-1.

liU[Q\%,Ui+i

UN = U € U(c7j. Now, QlfQf,^ = EerA}^e[QfJ Qif

= Q\%,

Ui, ••-,

= Q^^ implies

= E, 1 < i < j < N, Biid Q^^Qf^ = Y.{^uAp't)Q^£

implies

U

that if r = 1, QJ2 = Qtl hence Q ^ = Q^l •••. ^ i - i , . = Q ? ; if * = 1, Qi? = Ee^Age'iQi^^ Since Ap\ = e „ QJ;^ = Q^^ Hence U

this theorem holds up to a unitarily equivalent: Q^^ -> U Q^^U = Rt, l
6.

Classification of General Square Domains

6.7 Given a square cone Vj^, a normal Siegel domain D(V^, F ) is called a general square Siegel domain, ifV^ = V(cr2, • • •, (TN) is a square cone. A domain D in C^ is called a general square domain, if D is holomorphically isomorphic to a general square Siegel domain. DEFINITION

Now, we will give the classification of a few special cases of general square Siegel domains as follows. Clearly, if rrii = 0, then riijiUj = 0, i = 1, • • •, j — 1. Now, riij > 0, ruj = Oj so rui = 0 imphes m^ = rrii^i = • • • = m^ = 0. Hence there is an index r E {1, 2, • • •, iV}, such that m\m2 - "fUr > 0^ rUr-^-i = • • • = rriN = 0. Since Qf) Qf) = E,mi>

ruj,

m i > 7722 > • • • > ^ r > 0 = rUr+l = • " = TUN-

(6.89)

We consider two cases as follows. Case 1. mi = ' " =mr > 0 = mr+i = • • • — m^v, 1
Le^ r G { 1, • • •, AT}

and

mi = ' " = rrir > 0 = mr+i = • • • = mjsf.

(6.90)

Classification of Square Domains

285

A general square Siegel domain has the canonical domain up to an affine equivalent as follows: (1) If T = \, the canonical domain D{V,F) is {iz,u) G C " X C^^Imiz) where V = (2)

- {uiu[A---,0)

€ V},

V(a2,•••,CTN)-

If T = 2, the canonical domain £>(V, F) is {{z, «i, ti2) e C " X C""' X €"^2 I Im {z) - F{u, u) G V},

where V = V{a2, • • •, CTN) and F{u, u) = {uiWi, Re («ii?it4), • • •, Re (uii?(T2^)) «2«2) 0) • • • > 0). The matrix set {Ri • • • i?
Qif=E,

l<j
Qfr =Rt,

(4) then

l
/ / (c72,•••, 1, r? + C > 2, Qf^ = E,

l<j
where ^ + T) + 1 < k
/ / (cr2, • • •, (Tr) = ( T T ^ , 3 , ^ 7 ^ ) , C > 1, ^/»en Q5.i) = J5, l • • • > rrir > 0, mr+i = • • • = rriN = 0, cr^ < 2 < r,

mi.

The condition CTT < 2 < r gives that

In this case, A}} = E,

l
Since r

there exist C/i G U (mi), C/2 ^ U (m2), such that

u'.Q>2 = ( ° ) . Without loss of generaUty, let Ui = E, U2 = E. We have Q12Q23 ~

( Q^ ) ^ ^^'^^- ^^^K ^ ) ^^ = ( ^ ) ' ^^ = diag([7l"\f/2), where [/2 is an arbitrary unitary matrix, hence there exist [72? Us, such that U2Q23US = I ri )• It is easy to show that there exist Ui G U(mi), 1 < i < r, such that U^Q^jUj = ( 771 )• Therefore, without loss of generality, let Qy^ = I ^ ] , li Ui e U(mi), I < i < r and UiQi/Uj = Q ) / , then [/^ = diag {U^^ , C/^^^^i , * * *, C^ir ). 1 < i < T. Now, n^j = 1, 1 < i < j < ^. If n^j = 2, 1 < z < j < T, C + 1 < j < r, and gg^Q^.^,) = E(et^r,'^e;)QSfc\ letting r = 1,5 = 2, we have ( ^j

Q^'j^ = Q J ^ \ 1 < i < j < k < T, because A^^^ - E. Thus

Q S ? = f g g ) j . Hence Q^^ :- ( ^?2) ) , 1 < i < fc, ^ + 1 < ^ < r, where Qk = Qk-i k ^^^^ ^^^ order m^-i — m^. If r = 2,5 = 1, since ^ff ^i = ^2^ Qif ( E ) "^ ^*^^^' where E is an mfc x rrik unit matrix.

287

Classification of Square Domains

^^ € = ^,J, Q^^jJ = I

j . Hence if the block partitions of

Qj are rrij — rrij+i, • • •, rrir-i —rrir^mr rows and rrij-i — rrij^ ruj — mj+i, '", rrir-i — rrirj rrir columns, we have /

0

0

^j-i,j

\

^30

Qj

ij+ij

(A

^i+ij+1

Kj

J'

T—1,T

V ATJ

^T,j+1

where Apg is an (mp—mp^i) x (mg —mq+i) complex matrix, and rrir+i = 0. If r = . = 2, since A?f = ( J "^ ) , A^', -Qi;)=(_°^).

Hence ( 4 )

= -e[. Thus Q\fQf^ =

(,°J

= (_«^ ) .

Therefore

0

Qk Since

)

=

(

-

.

)

•

Q\f Q\f + Q S ? QS]^ = 0, gSf Qlf = £;,

+ K'J = o,

KJ

Q'^QJ

we have ( A,-ij

0

0

\ - i j 0)

Aj+ij

Qi = V

0

-^T.r—1

-^rr

where Afcfc + A^^, = 0, A^^+i = -Afc+i,^, j

/

By a unitarily equivalent Qj —> Uj^iQjUjj where Ui = diag{Uiir - ,Uir) €V {ai), we have Apq -> Uj_ipApqUjqj ^ +

i = 1, • • •, iV,

l
=

E,

288

THEORY OF COMPLEX HOMOGENEOUS BOUNDED

DOMAINS

Now, we can obtain the canonical form from the matrix QT- Let

^2 = v/=l

/ 0(Pj) 0 0

0 A[f^^ 0

0 0 £•(">)

y 0

0

0

0 0 0

\

-E^^i^ J

By Theorem 6.4, without loss of generaUty, let /

Ar-l,r=

0 0 E(P^^ 0 \ 0 (^-A?)^

0 0 \ 0 0 , 0 0/

Arr = Bi-,\

where Ar = diag (Air, • • •, A,^r) € GL (9r,K), 0 < |Ai7-| < • • • < |Ag^r| < 1) PT + QT + UT + VT = rur- Thus j^

I

•'^T—1,T —1

-AT—1,T

-^r—l,r

V

L,_l = ( A,_2,r-1

0)

-^TT

Hence E=

f ^r-2,T-l^r-2,T-l

0

0

/ Bii

Bi2

\

0

\ B21 B22 J '

where Bn = A^__i,j._i — Ar-l^rA^-\^T'>B2\ = - ^ T - l , T ^ r - l , r - l " ^TT^r-l,T5 B12 = Ar,T-lAr-l,T

+ Ar-l^rArr,

0 Qr

By Qr-1

0 -E a

AT-1,T-1

B22 = A^^ - A^_i^^Ar-l^T'

, we have Ar-2,r-iAr-i,T

0

,

Ar-2,r-l

= 0. Hence

- ( /? 0 0 ) ,

where /?'/? = ^ + a a ^ Now,

thus

c/,>=diag (in^-\ w^^r^ w^^r^ w^rr^ C/,_i,T_i = diag {Wr-i,i,W^_%,

W^%)

satisfy the conditions Wr-1,2 = Wrl,

Wr-l,Z = {E - Al)'^Wr2iE

- A?)"?.

Classification of Square Domains

289

Hence where UJ-\^T-2I W^T-I,I are arbitrary matrices. Without loss of generality, let /

a = 5;iiV_i,

/?=

0

0

0

E^r-.)

0

0

0

(£;-A2_i)l

0

\ We have Ar-l,T-l

(Bt\U

0

0

0

0

0

0

0

= V

-\/^AT

/

0 f;(p--i)

0 0

0 0

\

0

(E-A2_I)I

0

>i^_2,^_i = By the induction, /

0 Afc-i,fc= ^^'=^ \ 0

0 0 (E-A2)l

0\ /5W 0 fBl^^ o],Akk={o'o 0/ V 0 0

0 0 -\/^Afe+iy

where j < k
6.40 If a general square normal Siegel domain D{V{a2,---,aN),F)

(6.91)

satisfies the conditions m\>m2>

•• •> TTIT > 0 = mr+i — • •• = TUN,

rur < mi,

(6.92)

and {
(6.93)

then the canonical domain is defined by a normal matrix set of type N: A}} = E,

l
^'' = {1 ~o)^ e+i
QiA-[

rs

h

l
e+ l<J
290

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where

I ^^jj+i

Q.=

•~^jj+i

B], 0 Ajj = I o' 0 0 0 / 0 V

0

^j+ij+i

0 0 —/^Aj+1 0 0 0 {E-t^^^\ /

Arr = 5 ; ^ ,

0 ^r-l,T = E^^^ \ 0

0 0 0 0 (E-A2)

0 0 0 0 0 0

Note The examples of continuum canonical domains were given by Piatetski-Shapiro in 1961. He proved that when n < 7, there are only finite canonical domains; when n = 7, canonical domains depend on a continuum. Theorem 6.39 appears that canonical domains contain some independent real parameters. Hence we obtain a lot of continuum canonical domains, which are not holomorphically equivalent to each other. On the other hand, the holomorphic automorphism group can be found by Chapter 5 on the above canonical domains easily.

Chapter 7 SYMMETRIC BOUNDED DOMAINS

The clcLSsification of symmetric bounded domains and a part realization of indecomposable symmetric bounded domains were given by E. Cartan in 1935. He proved that a Hermitian symmetric space is a homogeneous Hermitian manifold, the isometric transformation group is a real semisimple Lie group, and the isotropy subgroup is a maximal compact subgroup. Hence a Hermitian symmetric space is a Riemannian symmetric space as a real manifold. He gave the classification of Hermitian symmetric spaces by the classification of real Riemannian symmetric spaces, and he proved that every Hermitian symmetric space is a topological product of some indecomposable Hermitian symmetric spaces. The canonical manifolds of indecomposable Hermitian symmetric spaces consist of four classes canonical domains in complex EucUdean spaces, and two exceptional Hermitian manifolds by left coset spaces. The four classes canonical manifolds can be realized by matrix spaces as follows: (1) ^i{n,m) = { Z = (zij) I E - Z'z' > 0 }, where all elements of an n X m complex matrix Z consist of all independent variables; (2) ^n{n) of 91/(n,n);

= { Z = (zij) \ Z' = Z, E-

(3) 9l///(n) = { Z = (zij) \ Z' = -Z, subdomain of 91/(n, n); (4)

Z'z' > 0 } is a subdomain E - Z^'

> 0} is also a

9l/y(n) = { z e C^ I \zz'\ < 1, 1 + Izz'p - 2zz' > 0 }.

The explicit expression of the holomorphic automorphism group on one of the above domains is given by Hua (ref. [70]). Since this holomorphic automorphism group is a classical group, Hua called these do291

292

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

mains in (1)—(4) classical domains. Sometime, these domains are called Cartan domains. The exceptional canonical manifolds are S)y(16) = E6(-i4)/SO(2) X SO (10), and S)y/(27) = E7(_25)/SO(2) x Eg. In 1956, Harish-Chandra proved that a Hermitian symmetric manifold is isometrically isomorphic to a symmetric bounded domain in a complex Euclidean space by the Harish-Chandra imbedding. But HarishChandra did not know the explicit expressions of the above two exceptional classical manifolds as E. Cartan. In this chapter, we will give a new proof of the classification and the realization of Hermitian symmetric spaces in terms of the results of Vinberg, Gindikin, Piatitski-Shapiro (i.e. any homogeneous bounded domain is holomorphically isomorphic to a homogeneous Siegel domain, see ref. [223] or Chapter 9) and author (i.e. any homogeneous Siegel domain is affinely isomorphic to a normal Siegel domain, see Chapter 3). This realization is a natural realization of Hermitian symmetric spaces. The Hermitian symmetric spaces 2)^^(16) and S)y/(27) can be holomorphically imbedded into the complex Euclidean space as two normal Siegel domains. They are called the exceptional symmetric normal Siegel domain, and denoted by Rv{lQ) and Rvi(27) in C^^ and C^^, respectively. Finally, we will give some properties of these two exceptional domains, and give the Cartan realizations from the symmetric Siegel domains.

1.

Symmetric Bounded Domains

7.1 (E. Cartan) A Riemannian manifold M is called a Riemannian symmetric space, if M. satisfies two conditions: (1) given p E Ai, there exists an isometric transformation Gp, such that p is an isolate fixed point of a^;

DEFINITION

(2) K ) 2 = id. The isometric transformation a^ is called a symmetric transformation at the fixed point p. When a Riemannian symmetric space M is a Hermitian manifold, M. is called a Hermitian symmetric space. 7.1 (E. Cartan) (1) Any Riemannian symmetric space M is a homogeneous Riemannian manifold; (2) Any Hermitian symmetric space M is a homogeneous Hermitian manifold. In these two cases, the subgroup G of the isometric transformation group acts transitively on M, where G is generated by all symmetric transformations. LEMMA

Symmetric Bounded Domains

293

Proof Suppose that F is a simply closed curve segment, which connects the terminals p and q^ where p, q € M. Given a point a G F, there exists an open geodesic sphere G{a,€a) C Mj such that F c Ua€rG(a,ea)Since F is a compact set, there exists a finite point sequence p = ao, ai, • • •, tts-i, as = ^ in F, such that G(ai,Eai)nG(ai+i,£a._^,) 7^ 0, 0 < i < 5. Given a point bi E G(ai, SaJnG(ai+i, £a. ), 0 < i < s, we obtain a point sequence p = ao, 60, ai, • • •, ag-i, 6s_i, as = 9, such that two neighbor points are connected by a geodesic segment. Therefore there exists a point sequence p = po? Pi? * * ^ Pt-i? Pt = 9 in F, such that the geodesic segment F^ has two terminals pi and Pi+i, i = 0, 1, •••,t — 1. Denoting by qi the middle point in F^, the symmetric transformation (jg. sends Pi to Pi+i. Hence the isometric transformation a = crqt-i(^qt-2''' ^qi^qo sends p to q. Therefore the group G acts transitively on M^ and A< is a homogeneous Riemannian (Hermitian) manifold. I Suppose that a bounded domain D C C^ containing the origin is a symmetric bounded domain. Then J9 is a homogeneous bounded domain by Lemma 7.1. Denote by K{z^~z) and T{z^'z) the Bergman kernel function and the Bergman metric matrix, respectively. By Section 1.1, the Bergman mapping y = gradjlog

K(0,^)

T(0,0)-i

(7.1)

2=0

gives an addmitable local coordinates y in a neighborhood [/ in £) at the origin, such that the Bergman metric matrix T{z,z) =

T{0,0)-hT{z,0)Toiy,y)T{0,z)T{0,0)-h

of a domain D satisfies the condition

by the new coordinates y. Denote by aut(Z)) and isoo(-D) the Lie algebra of the holomorphic automorphism group Aut (D) and the isotropy subgroup Isoo(i^) at origin, respectively. By Lemma 1.4, any element X in aut(D) can be represented by

^^^^^^§~z = (" + "^(y) + y ^ ) | ; '

(7-2)

where P is an n x n complex constant matrix, and a = ^(O)r(0,0) 2 G C", 'd^{logKo{y,y)

+ logKo{0,y))^

f(y) = - E ( ' ' ""^lagf "'''"iv-o)''^''^

(7.3)

294

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

is a n n X n complex matrix. Hence F(0) = 0, and any element of F{y) is a holomorphic function with the Taylor's expansion at the origin without a constant term and Unear terms. 7.2 Give a linear automorphism a: a{y) = yU in Aut(I?). Then U e\J{n), and F{yU) = U'F{y)U, (7.4)

LEMMA

Proof Since the Bergman kernel function Ko{y,y) and the Bergman metric matrix To{y^y) satisfy the conditions: da(y) Ko{y,y) = d e t - - — dy

Koi(T{y),a{y)),

and ro(0,0) = £;, hence dy is a unitary matrix. Since Ko{y,y) = Ko{yU,yU), we have eiF(yU)e'j = eiF{a{y))e'j d\logKo{(7{y),'^)

-

logKoiO,'^)) a{y)=0

da{yi)da{yj) Now, logKoia{y),a{y)) dlogKo{a{y),a{y))

= logKo{y,y),

thus

^ y > d\ogKo{y,y)

d'^logKo{<7{y),(T{y)) ^^d'^logKo{y,y) d^(^d^ ^ dy;dy-q

/ dyp \ / dyp \ / dyq \ \da{y)i)\d(j{y)j)'

Hence

FiW) = F(.iy)) = ( ^ ) f

W(^)

Therefore (7.4) holds.

= VFiy)V. I

Clearly, we have LEMMA

7.3 Give a linear automorphism a: a{y) = yU in Aut (D), and X = ia + aFiy) + yP)—es.ut{D).

(7.5)

Symmetric Bounded Domains

295

Then a^{X) = Y = {aU+aUU-^F{yU-^)U+yU--^PU)—

3' dy

e aut(D). (7.6)

Given two points p, q in the symmetric bounded domain D, if d is a symmetric transformation at the fixed point p and the holomorphic automorphism r sends p to g, then r o a o T~^{q) = q, and r o a o T~^ is a symmetric transformation at q. On the other hand, if D is a symmetric domain and r is a holomorphic isomorphism, then T{D) is also a symmetric domain. Now, we consider the isotropy subgroup isoo(i?) at the origin as follows. By Section 1.1, X € isoo(i5) implies that X^yP^, dy

P + P = 0,

(7.7)

hence the unit connected component of Isoo(-D) is a compact subgroup in U(n). The symmetric transformation a at the origin has a local coordinate representation w = yU, (7.8) where [/ is a unitary matrix. Since a^ = id, C/^ = E. Thus the eigenvalue Ao of U satisfies the condition AQ = 1. Hence AQ = ± 1 . Therefore there exists a unitary matrix PQ, such that

Since the origin is the isolated fixed point of cr, r = 0. Hence U = —E. Therefore we have 7.4 By the local coordinates (7.1), the symmetric transformation GQ at the origin acting on the symmetric hounded domain D has the coordinate expression w = --y. (7.9)

LEMMA

7.5 (1) The matrix F{y) is a complex symmetric matrix satisfying two conditions:

LEMMA

F(0) = 0,

F{-y)

= F{y).

(2) / / X = {a + aF{y) + yP)^

G aut (D),

(7.10)

296

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

then pj

{a + aF{y)) — , oy

pj

yP—

e isoo(i?).

oy

Proof Clearly, F(0) = 0 by the definition. By Lemma 7.2, let U =-E in (7.4), thus F{—y) = F(y), 1 < i, j < n. Hence the first statement holds. Since w = —y is 3. holomorphic automorphism on domain D, X = {a + aF{y) + yP)—e aut (D) implies Y = {-aoy aut (D). Since F{-y) = F{y), ^{X + Y) = yP^,

^{X-Y)

aF{-y)

+ yP)— € oy

= {a + aF{y))^Gmt{D).

I

7.6 The Lie algebra a.ut{D) of the Lie group Aut{D) on a symmetric bounded domain D has a basis

LEMMA

Xj = ej{E + F{y))^, Zp = yKp-^,

y^ = ^fZle.{E

- F{y))^,

acting

l<j
l
(7.11) where Ka is a skew-Hermitian matrix^ I < ot < s and s = dimlsoo(i)); F{y) is a complex symmetric matrix satisfying condition (7.10). Proof

Given X = {a + aF{y) + yP)—

e aut (D), then

oy

{a + otF{y))—,

yP—

oy

e aut (D)

oy

by Lemma 7.5. Given a basis Z0 = yKp—,

l
in isoo(-D), then aut (D) has a part basis Xi = {ai + a-iF{y)) — , oy

1 < i < 2n,

pj

Zp = yKp—,

l
such that Xi, 1 < i < 2n, Zp, 1 < (3 < s^ consist of a basis of aut (D), where a i , • • •, a2n are 2n linear independent vector in R^"^. In fact, if

Symmetric Bounded Domains

297

there axe real numbers Ai, • • •, X2nj such that Yl ^i^i — 0? then X^ XiXi = i

i

0. Therefore Ai = • • • = A2n = 0. The statement holds. The 2n x 2n real matrix Re(ai) Im(ai)

(

;

:

Re (a2n) Im {a2n) is non-singular. Let Q~^ = {qjk}- Then 2n

2n

fc=l 2n

k=l 2n

Y^ qjklm {ak) = 0,

J ^ 9n+i,fclni (a^) = ej,

l<j
fc=i fc=i

Therefore 2n

2n

^ i = X ] (IjkXk, Yj = Y^ qn+j^kXk, 1 < j < n.

I

7.7 Tfte following symbols are the same as the above. There exists a skew-Hermitian matrix Lp, /? = 1, • • •, s^ such that

LEMMA

s=.

nv) = -.Y.I''0'vKg, S

S

':• ':' = 5 ^ eqLpe'iepKjse'j = ^ P

(7.12)

(7.13) e^L^ejepAT/jei,

1 < i, j ,

p,q
/3=1

And E e . n . K ^

= E e . ^ f e K ^ .

VU.

(7.14)

T^e multiplication table of the Lie algebra aut (D) with the basis as Lemma 7.6 is %

dyj

I dyp

298

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

S

= Y^{eiilm{L0)e'j)Z^, 0=1

[Xj, Zff] = 5^(e,(Re {Kp))ei)Xk

+ X;fe(Ini iK0M)Yk,

(7-15)

fc fc fc fc

[Za. Zp] = Yl ^^c.pZ5. 1 < a, /3, (5 < 5, 6

where

CU = CL = Cl5^'^,

l
(7.16)

and

Proof

By Lemma 7.6,

lx..-.l=-V=TE(^^^)4-^e. l^..-.l=E(%-^)^-e. where

By Lemmas 7.5, 7.6 and 1.14, [Xi,Xj],

[Xi,y,],

[Yi,Yj\e

isoo{D).

Thus [Xi,Xj] - [Yi,Yj\ = 2 6 € i s o o p ) .

Symmetric Bounded Domains

299

Now, Lemmas 7.5 and 1.13 imply that 6 = 0. Therefore (7.14) holds and the first two formulas in (7.15) hold, where L^ is an n x n skewHermitian matrix, and Kp is an n x n skew-Hermitian matrix, /? = l...,5.

On the other hand, eiF{y)e'^ is a homogeneous polynomial of y with degree two, the third and fourth formulas in (7.15) hold by Lemma 7.6. Now, ejF{y)e'j^ = ySjky'•> where 5^^ is a constant complex symmetric matrix, and (7.15) gives that ^

fdejF{y)e'^

p

deiF{y)e'^^

%

J^

, ^

,..

^

%

Thus ^Y^eiSjpy'e,

=2 ^

P

deF(v)e' * 'Q ^^P = Y^{eiL(3e^j)yKp

P

^^

0=1

impUes that (7.12) holds. Hence

/3=1

^ '

and Sjp = S^jp = Spj = S ^ imply that (7.13) holds. Finally, (7.15) and (7.14) imply (7.17) holds. Now, isoo(i5) is a compact Lie algebra, there exists a basis Z/?, 1 < /? < dim(isoo(I?)), such that

where C^o G R, hence (7.16) and the last formula in (7.15) hold.

I

Lemmas 7.5 and 7.6 give that LEMMA

7.8

y\Xi,Yi]

= 2^/=4J/— € aut(D),

graxij^(logKo(y,y)) = (y - yF{y)){E - ny)F{y)r\

(7.18) (7.19)

300

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Proof Since a homogeneous Riemannian manifold is a complete Riemannian manifold, a symmetric bounded domain is also a complete Kahler manifold. By Lemma 1.3 and (1.69), Xi, • • •, X„ and Yi,--- ,Yn in Lemma 7.6 satisfy ei{E + F{y)){gradylogKo{y,y)y + ei{E + F{y)){gxB.d^\ogKQ{y,y))' ^ ^

dyk

dyk

)'

(7.20) ei{E - F{y)){gradylogKo{y,y)y - ei{E - F(y))(grad^logKo(y,y))'

(7.21) By (7.20) and (7.21), (7.17) gives that grad^(logXo(j/,y))' = -F(2/)graxi^(logKo(y,y))'-E ^ ^ ^ p

%

('^•22)

is equivalent to (7.20) and (7.21). So dF{y)e' dF{y)e' {E - F{y)F{y))gvady{logKo{y,y)y = F{y)-^^ —^ = \j^F{y)L'^K'^y'

-\Y,L^Kpy[. 13=1

0=1

Thus grad^(logifo(y, y)) = I y^{W^pF{y)

- yK0L0)iE -

¥(^Fiy))-\

Hence I

1 ^

grad^(logKo(y,y))

= -^

Y^iyK^L^),

Since the Bergman metric matrix To{y,y) satisfies the condition To(y, 0) = ro(0,y) = E and grady(logKoiy.y) - logKo{0,y))\y=o = y, we have s

s

Y^{KpLp) = J^{L,3K/3) = -2E, /?=1

0=1

(7.23)

Symmetric Bounded Domains

301

where E is the unit matrix. Therefore (7.19) holds, and (7.23), (7.17) imply (7.18) holds. I As a corollary, we have 7.9 Let D be a symmetric bounded domain^ the holomorphic automorphism group Aut (D) on D has a one parametric subgroup

LEMMA

y-^yexp

(V^e),

V^ G R

(7.24)

by the addmitable coordinates (7.1).

By Lemmas 7.6 and 7.7, we have 7.10 The complexification aut {D)^ of the Lie algebra aut (D) of the holomorphic automorphism group Aut (D) has a basis

LEMMA

Y*=Xj

+ V^Yj

Z^ = yKp—, dy'

= 2ejF{y)—, 'dy

l<j
(7.25)

l
The multiplication table of the Lie algebra aut (D)^ with the basis (7.25) IS

[X;, X*,] = [Y*, Y,*] = 0, [X*, Zp] = Y.^eiKpe'„)Xl

[X;, Y,*] = 2 J2

e^he'^Z^,

[Y*, Zp] = ^^(e.F^je;)^*,

k

(7.26)

k

7.11 (E. Cartan) Let D be a symmetric bounded domain. Then the holomorphic automorphism group Aut (D) is a real semisimple Lie group, the isotropy subgroup Iso (Z?) 0/Aut (D) is a maximal compact subgroup.

THEOREM

Proof

Let ^ be a vector space with a basis -X^l5 • * * 5 J^n, Yl, • • • , I n -

302

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Then there is a subspace direct sum decomposition aut (D) = ^ + q},

^ = iso {D).

(7.27)

It suffices to prove that (7.27) is a Cartan decomposition. Namely, the KiUing form B{x,y) of aut(i)) is a non-degenerate bihnear function satisfying the conditions B{^,^)

= 0,

5(x,y)|^<0,

B{x,y)\^>0.

Hence aut (D) is a semisimple Lie algebra. In terms of the multipUcation table (7.15) of aut (D) in Lemma (7.7),

[A,A]c^,

[^^^]c%

[^,^]c.ft.

Thus B{^^ ^ ) = 0. By the multiphcation table (7.26) of the complex Lie algebra a u t ( i ) ) ^ , B{X*,X*k) =

B{Y*,Yk*)=0,

and s

B{X*, Y,*) = -2j2

ejiLpKp + K^Lp)e',

13=1

by a direct computation. By (7.23), B{X;,Y,*)

= S5jk.

Now, we compute the Killing form on aut (D). Since Xj = ^iX* + Y*),

Yj = ^{X*-Y*),

Zp = Zp,

we have B{Xj,Xk)

= BiYj.Yk)

= 4Sjk,

B{Xj,Yk)

= B{Xk,Yj)

= 0.

Thus the Killing form B{x,y) is a positive definite symmetric bilinear function on the subspace ^ . It is well known that the Kilhng form B(a:, y) is a negative definite symmetric bilinear function on the compact subalgebra ^. Hence aut (D) = .ft + ^ is a Cartan decomposition, and the Killing form B{x,y) is a non-degenerate bilinear function on the Lie algebra aut (D). Therefore aut (D) is a semisimple Lie algebra and iso (D) is a maximal compact subalgebra. I We have the following theorem by Section 2.1 and Theorem 7.11.

Symmetric Bounded Domains

303

7.12 Let D be a symmetric bounded domain and Aut (D) be the holomorphic automorphism group on D. Then there exists a linear transformation J and a linear function "^ on the Lie algebra aut (D) of Aut {D)j such that (aut (J?), iso (£)); J, -0) is a semisimple effective proper J Lie Algebra. And There exists a normal J subalgebra (t(Z)); J,'0); such that aut (D) = t{D) + iso (D) is an Iwasawa decomposition. THEOREM

7.13 (E. Cartan) A symmetric bounded domain is a topologic product of indecomposable symmetric bounded domains.

THEOREM

Proof Give a decomposable symmetric bounded domain D, such that D = Di X D2 is a, topologic product, where y = {u^v) is the point coordinates at the origin by the new local coordinates (7.1), where u E Di and V e D2 are also the new local coordinates (7.1) in Di and D2 at the origin, respectively. By the proof of Theorem 3.23, the domains Di and D2 are both homogeneous bounded domains. Given X G aut (D), X = Y + Z, where Y G aut (Di) and Z G aut {D2), d' & Now \f^z— G isoo(i^) by Lemma 7.8, hence \f^x— G isoo(£^i) and yf-\.y-^

2.

G isoo(i^2)- Hence D\ and D2 are symmetric domains. I

Semisimple J Lie Algebras

We consider the complexification of a J Lie algebra. Let 0 ^ be the complexification of a real Lie algebra 0 with respect to a semi-involution cr, where o is defined by (j{X + ^f^Y)

= X-

V^y,

VX, Y e(5.

(7.28)

7.14 Let (5^ be the complexification (5^ of a real Lie algebra (& with respect to a semi-involution a. Then (3 is a J Lie algebra if and only if (& satisfies the two conditions as follows: (1) There is a real subalgebra S) and two complex Lie subalgebras ©"^ and 0 ~ in (&^^ such that

THEOREM

(5^ = ©+ + (S-,

0+n0-=i3^,

a(0^) = 6^;

(7.29)

(2) there is a linear function t/; on (&, such that p{X,Y)

=

^:V^i;i[X,a{Y)])

is a semi-positive definite Hermitian form on (5^ with the kernel ^ and p([©^(&^]) = 0.

,

304

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

Proof

If (0,i3;«/, V^) is a J Lie algebra, the subset ©^ = { X ± V ^ J X I VX G 0 } + i}^ C (5^

(7.30)

is a complex subspace of 0^, and 0 + fl 0 ~ = S)^. In fact, there exists an element h{X) G 9)j where /i(X) depends on X € 0 , such that - 1 ( X ± ^/^JX)

= (TJX) ± V ^ J ( T ^ X ) +

V^h{X).

Hence \ / ^ ( X ± \ / ^ J X ) G 0 ^ . Therefore 0=^ is a complex subspace of 0 ^ , and 0 + n 0 ~ = S)^. It is easy to prove (7.29) because J^ + id = 0, (mod^). Clearly, if 0 ^ is a complex Lie algebra, then 0 ^ = a{0^) is also a complex Lie algebra. In the following, we prove that 0"^ is a complex Lie algebra. Given X, F G 0 , h, h' G 9), [X + V=4( J X + h),Y + / = ! ( J F + /lO] = [X, Y] - [JX, JY] + ^f^J{[X,

Y] - [JX, JY\)

+ % Y\ + V ^ J [ / i , Y\ + [X, h!\ + \ / ^ J [ X , U\,

(modi3^).

Hence 0"^ is a complex Lie subalgebra. Therefore condition (1) holds. A linear function ^ on 0 can be extended to the complex linear space 0 ^ . The definition is ^{X + V ^ Y ) = '^(X) + V ^ ^ ( y ) , V X , y G 0 . Hence p{X\ Y) = TV^tA([X', o{Y')\\ X', Y' e 0 ^ is a Hermitian form on 0"^, and p([X ± x/=4 J X , a{X ± \ / ^ J X ) ] ) = 2^([JX, X]),

X G 0.

Condition (2) holds. Next, we prove the necessity. Namely, if conditions (1) and (2) hold, then there exists a Hnear transformation J and a hnear function t/?(a:), such that ( 0 , ^ ; Jjip) is a J Lie algebra. In fact, if a real Lie algebra 0 satisfies conditions (1) and (2). Given X G 0 , X ^ i3, if there exist X + y/^Y, X + y/^Z G 0 + , then y^{Y -Z)e 0 + . It follows that a{-y^{Y - Z)) G 0 " . Hence y/^{Y - Z) G 0 + n 0 - = ^f^Sj, Therefore Z = Y, (modi)). Since there is a real subspace SDt, such that 0 = SPt + ^ is a subspace direct sum decomposition of 0 , the mapping J: X —> F is a real linear transformation on 9Jl, where X + \f^Y G 0"^, X , y G SJl. Obviously, J can be extended to 0 , such that J{^) C S). By the definition of J, we have 0=^ = { X ± ^ / ^ J X I VX G 0 } + i3^.

Symmetric Bounded Domains

305

Since 0"^ is a complex vector space, we have J^ + id = 0, (modi^). Since 0"^ is a complex Lie algebra, J([X, Y] - [JX, j y ] ) = [JX, Y] + [X, J F ] , J([/i,r]) = [/i,JF],

(mod^),

(modiD),

VX, F G 0 ,

V/iei3, X E 0 .

Now, there is a real linear function '0 on 0 , such that '0([0^, 0"^]) = 0. Thus ^([0^,i3^]) = 0. Hence V^([0,i3]) = 0, and ^([X + V^JX.Y + V^JY]) = 0 implies t/)([JX, F]) + t/;([X, J F ] ) = 0, V X , F G 0 . Since - V ^ t / ^ ( [ X + %/^JX,(7(X + V ^ J X ) ] ) > 0,

VX G 0 ,

and ^([X + x/=Te/X,a(X + v ^ J X ) ] ) = 0 if and only if X G -^, so ^([JX, X]) > 0, VX G 0 and V^([JX, X]) = 0 if and only if X G i}. The necessity holds. I In this section, we will prove that if D is a homogeneous bounded domain in C^, such that the holomorphic automorphism group Aut (D) on domain D is a real semisimple Lie group, then D is holomorphically isomorphic to a symmetric normal Siegel domain (see refs. [9], [114]). Denote by aut (D) and iso (D) the Lie algebras of Lie group Aut (D) and the isotropy subgroup Iso(D), respectively. By Section 2.1, there exist a linear transformation J and a linear function '^, such that the J Lie algebra (aut (D), iso (D); J, ip) is an effective proper J Lie algebra. Suppose that 0 = aut (D) is a real semisimple Lie algebra and S) = iso(D) is a compact Lie subalgebra of 0 . It is well known that the Killing form B ( X , F ) of a semisimple Lie algebra 0 is a non-degenerate symmetric bihnear function on 0 , hence there is a unique element A G 0 , such that i/^(X) = B(A,X),

VX G 0 .

(7.31)

The conditions t^([0,i3]) = 0 and t/^([JX,X]) > 0, VX G 0 , X ^ i3 imply that S) = C^iA) = { X G 0 I [AX] = 0 }.

(7.32)

In particular, AeS}. Hence the compact Lie subalgebra S^ has the nonzero center Cs^{S)). Now, we prove that if Sjo is a Cartan subalgebra of the compact Lie algebra i), such that A e S^Oj then i^o is also a Cartan subalgebra of the Lie algebra 0 . In fact, if
306

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

the Lie algebra 6 . Namely, 9)o is a special maximum compact Cartan subalgebra of a real semisimple Lie algebra 0 . Now, {S)o)^ C i3^ C 0 ^ . The root subspace decomposition of the complex semisimple Lie algebra 0 ^ by the Cartan subalgebra {S)o)^ is

Moreover, we have a direct sum decomposition of subspaces:

aGAo

a€A±

where the root system A is a finite subset in the real linear space {S^O)R = \ / ^ i 3 o j and a{A) = 0, Va G AQ,

a ( ^ ) 7^ 0, Va € A±.

(7.35)

AonA± = 0, A + n A _ = 0, - A o = AQ, - A ± = Azp, A = AoUA+UA_. (7.36) Hence we obtain the direct sum decompositions of subspaces:

^±=

J2 ®«'

0^ = i3^ + m±,

[i3^, Tl±] C m±

(7.37)

a€A±

by((Ao + A ± ) n A ) c A ± . 7.15 The following symbols are the same as above. There exists a positive direction in the real linear space {^O)R = \/^iDo? such that A± C A*, where A"^ is the positive root system, and A~ = —A"^ is the negative root system by this positive direction. In particular, the complex Lie algebras 0 ^ are the parabolic subalgebras of(5^. LEMMA

Proof

Now, (7.37) implies ((AQ + A^) fl A) C A^,

[9H±, m±] c [0^, 0^] c 0=^ = i3^ + an± implies A± + A± C AQ U A±. Now, we prove ((Ao U A±) + A±) n A C A±.

(7.38)

In fact, given a, /? € A±, ii a + (3 e AQ, then —a — ^ G AQ. Hence -p = a + {-a - /?) G (A± + Ao) fl A C A±. We get a contradiction. It proves that if a + ^ G A, then a + 0 e A±, Hence (7.38) holds.

Symmetric Bounded Domains Let C =

E

307

/^ ^ \ / ^ i 3 o . We will prove TBiC.a)

> 0, Va E A±,

5(C, a) = 0, Va G AQ. In fact, given a, /? G A+, if a 7»^ /?, then the root chain /? - pa, • • • , / ? - a, /?,/? + a, •••,/? + gfa satisfies that /? — (p + l)a, / ? + ( ? + 1)Q^ ^ A U {0}. Now, we prove P + ka ^ 0, —p < k < q. In fact, if there is an index k such that /3 + ka = 0, then /? = —fea G A, hence fc = 1 or fc = —1. Since P + a is a positive root, we have k = —1. It is a contradiction. Since /? + fca 7^ 0, —p
, = P + 9 > y + g. Hence 5 ( a , a) < 0 implies B(/?', a) < 0.

t)(OL^ Ot)

Hence

B(C-a,a)=

J]

5(/?,a)<0.

Therefore B{C^,a) < B{a,a) < 0, Va G A+, and 5(C,a) > 0, Va G A_. Finally, given a G AQ, /? G A+, there is a root chain /? - fea G A+, -p < k < q, where ^ - (p + 1)Q:, /? + (g + l ) a 0 A. Let p' = p - pa. Then ^By( a' , aV) = P + 9- Hence E^fc—p-g L r, n 5(/?' + ^0^, oc) = 0. Therefore B(C,a) = 0, V a G AQ. Using i5eA+

we can define a positive direction in 9)^, such that the positive root system A+ c A+. Namely, a > 0 if S(C, a) < 0. I 7.16 Let (©,i3; J,-0) be an effective proper J Lie algebra, where 0 is a real semisimple Lie algebra, and S) is the compact Lie subalgebra. Then (1) f) is a maximum compact Lie subalgebra of (5; (2) there exists an Iwasawa decomposition (5 = S) + & of semisimple Lie algebra 0 , where ad (©) is a maximal triangular subalgebra on the linear space 0 , and ( 6 ; J,'0) is a normal J Lie algebra; (3) 0 does not have the compact factor.

THEOREM

Proof Given a positive root system as Lemma 7.15, if 9)^ is the maximum compact Lie subalgebra, where i3 C i^', then we have [9)j9)^] C i^', so aeA'

308

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

where A' C A+ U A_. Let A'^ = A' n A±, so A' = A'+ U A',. Now, for the semi-involution a, a{{S)^)^) = {9y')^ imphes -A'_|_ = A^_ and {Sr)^f = (^')^ n 0 + + {9)^f n &-. Since J\^± = T V ^ i d , we have J{S^') C S)^. Thus 9)' is a compact J Lie algebra. By the proper condition of the effective proper J Lie algebra, [i3',i3'] = [^3?^]- Since the center € of i3' satisfies [€,A] = 0, € C S). Hence S)^ = 9). Therefore ij is a maximum compact subalgebra. The next step proves that 0 has no compact factor. In fact, 0 = Yli=i ®i is an ideal direct sum, where (Si is a simple ideal. Let

where 6 ' is a semisimple ideal without compact factor and 0^^ is a compact semisimple ideal. If J 0 " C 0", then 0 " C ,^ by the proper property. It suffices to prove that 0 ' and 0^' are two J invariant subspaces. Since J\(^± = =F\/^id, we have J 0 a = 0Q:, Va € A+UA_. Hence the Cartan decomposition 0 = ^ + ^ has JSj C Sj, J ^ C ^ . It follows that there exists a Cartan decomposition 0^ = S)i + ^ij such that s

s

i=l

2=1

Hence J 0 i C 0 i , i = 1, • • •, s. Therefore 0 has no compact factor. I T H E O R E M 7.17 Let D be a homogeneous bounded domain in C^^ and Aut (D)^ be the unit connected component of the Lie group Aut (D). Denote by 0 = aut (D) and S) = iso (D) the Lie algebras of the holomorphic automorphism group Aut{D) and the isotropy subgroup Iso{D)j respectively. If Aut (D) is a semisimple Lie group^ then (1) Iso (D) is a maximal compact subgroup of Ant (D), and Aut (D)^ has the Iwasawa decomposition:

Aut {Df = T{D)' (Aut {Df n Iso (D)),

T(D) n Iso{D) = {e},

where T(D) is a solvable connected and simply connected subgroup acting simply transitively on D, such that Ad (T(D)) is a triangular group acting on aut(£>). (2) there exists a linear transformation J and a linear function ij), such that (aut (D),iso(D); J, ^ ) is a semisimple effective proper J Lie algebra. The J subalgebra {t{D)]J^ip) is a normal J subalgebra, where t{D) is the Lie algebra ofT{D),

Symmetric Bounded Domains (3) there exists a normal Siegel domain D{VN^F)J holomorphically isomorphic to D{VN^F).

309 such that D is

Proof By Section 2.1 and Theorem 7.16, (1) and (2) in this theorem hold. By Chapter 3 for the normal J Lie algebra (t(D); 7,-0), there exists a normal Siegel domain D(Viv,F), and a subgroup T{D{VN,F)) of the afSne automorphism group Aff {D{VN^ F)) acting simply transitively on D(VN, F), such that the Lie algebra t(D(V)v, F)) of T(J9(Viv, F)) is J isomorphic to the Lie algebra t{D) of T(Z)). Since D{VN^ F) is a connected and simply connected manifold, T{D{VN^F)) is also a connected and simply connected Lie group, and T{D{VN^F)) is isomorphic to T(JD), hence D{VN^F) is holomorphically isomorphic to D. I 7.18 A homogeneous Siegel domain £)(V,F) is a symmetric Siegel domain if and only if the holomorphic automorphism group Aut (Z)(V,F)) is a real semisimple Lie group. THEOREM

Proof The sufficiency, is Theorem 7.12. We prove the necessity as follows. By Chapter 3, it suffices to prove this necessity for a normal Siegel domain D{VN,F). If Aut {D{V]s[^F)) is a real semisimple Lie group, by Section 4.1, the Bergman mapping a: {z^u) -^ {x^v) is a holomorphic isomorphism from D{VN^F) onto a homogeneous bounded domain a{D{VN^F)). By Lemma 1.24, the radical S of the Lie algebra aut{D{V,F)) is equal to zero, thus dimLi = 2m, dimL2 = n (see (1.124)). Hence Bi, • • •, BN G aut(Z)(V)v,F)) by Theorem 5.20. The Lie algebra aut ((j(D(V)v,F))) contains elements

by (5.196). It follows that a{D{VN^F)) is a homogeneous circular domain. Namely, a{D{VN,F)) is a symmetric bounded domain. Therefore D{VN^ F) is a symmetric Siegel domain. I 7.19 / / a normal Siegel domain D{VN^F) normal Siegel domain in C^ x C^^ then

COROLLARY

dimLi = 2m, dim 1^2 = ^-

is a symmetric (7.39)

7.20 D{VN,F) is a symmetric normal Siegel domain if and only if the homogeneous bounded domain a{D{VN,F)) is a homogeneous bounded circular domain^ where the holomorphic isomorphism a is the Bergman mapping acting on D{VN,F).

COROLLARY

310

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

By Theorem 5.20, we have 7.21 Let D{VN,F) be a symmetric normal Siegel domain in C^ X C ^ . Then the Lie algebra diut {D{VN,F)) of the holomorphic automorphism group A\it{D{VN^F)) has a subspaces direct sum decomposition aut {D{VN, F)) = L_2 + L_i +L0 + L1+ L2, (7.40) THEOREM

where L_2 = { a — I Va € R " }, oz

(7.41)

L-i = { / 3 ^ + 2 ^ ^ F ( « , / 3 ) ^ I V/? € C " },

(7.42)

Lo = {Ai,l
Xg\ d'

E

Y^

Zlf,

l
& (7.43)

where

L',^Af^ + Af^Ljk = Y.{erU^e[)A\l (7.44) r

and Li = { Y^'\Y^'\ L2 = {Bi,

l
l
l
riij,

l
Proof Clearly, Li and L2 are given by Corollary 7.20. By Theorem 5.20, Z^f e aut {D{VN, F)) for all indices. I 7.22 (Classification Theorem) Suppose that D{VN,F) is an indecomposable symmetric normal Siegel domain. Then D{Vj^^F) is one of normal Siegel domains as follows: (1) N = l, (2) AT = 2, mi = m2 > 0, E t QuiK^s + e's^rWuY - 0, 1 < r, s < mi. (3) N > 3 and there exist two numbers p G {1,2,4,8} and r > 0 such that riij = p^ 'i-
THEOREM

Y,QlMes

+ e',er){Qljy - 0,

1
1
Symmetric Bounded Domains

311

Proof By Theorem 7.18, aut (D) is a semisimple Lie algebra. Thus the radical S of aut (D) is equal to zero. Hence dim Li = 2m and dim L2 = n by Corollary 7.21. When iV > 3, Theorem 5.20 gives that S i , "^.BN e aut (Z)(VAr, F)), and Z^f e aut {D{VN, F)), l 0. Since i < j < k^ riik = 0 implies nijUjk = 0, hence UijTijk > 0 implies riifc > 0. Thus riij > 0, 1 0 implies rrii > 0. In terms of the induction, we can prove that riij = p > 0, 1 0, 1 < j < N and (7.45) holds. By Lemma 6.8, n^j = p G { 1,2,4,8 }, 1 0, I
LEMMA

Viv = {a; G R^ I Ci{x) > 0}.

(7.46)

7.24 Let D{VN,F) be a normal Siegel domain associated with a normal matrix set {Ajj, Qfj} of type N. If Uij >0, l
LEMMA

D{VN,F)

= {{z,u) G C^ X C ^ I lmCi{z)

-ReRi{u)Ri{u)

> 0}. (7.47)

Now, we give some properties on the Siegel domains. 7.25 (Hano) Let JD(V, F) be a homogeneous Siegel domain. If the holomorphic automorphism group Aut {D{V, F)) is a unimodular Lie group, then Aut (£>(V, F)) is a real semisimple Lie group and -D(V, F) is a symmetric Siegel domain. LEMMA

Proof It is well known that Aut {D{V, F)) is a unimodular Lie group if and only if tr (adX) = 0, VX G aut {D{V, F)). Now,

tr(ad(2.|^+4)) = - 2diin (L_2) - dim (L_i) + dim (Li) + 2dim (L2) = - 2(n + m) + dim (Li) + 2dim (L2) = 0. Since dim(Lj) = dim(L_i) - dim(L_j fl 5), i = 1,2, by (1.124), where S is the radical of the Lie algebra aut(D(y, F)), thus dim(Li) < 2m,

312

THEORY OF COMPLEX HOMOGENEOUS BOUNDED DOMAINS

dim (L2) < n. Hence dim (Li)+2dim {L2) = 2{n+m) implies dim {Li) = dim (L-i), i = 1,2. It follows that L_i fl 5 = 0, i = 1,2, and 5 C LQ by step 3 in Lemma 1.29. Given X = zA— + uB— € 5, oz ou then d_ „^ .d' , X ] =ejA—eSnL^2 idzi

= 0,

hence ^ = 0 and X — uB-p—, Since ou

[A+2V=If(»,e,)|,X = etB—

- 2V^F{uB,et)g^

G 5 n L _ i = 0,

we have B = 0. Hence 5 = 0 and Aut(Z)(V,F)) is a real semisimple Lie group acting transitively on D{V,F). By Theorem 7.18, D{V,F) is a symmetric Siegel domain. I 7.26 (Vey) Let D{V,F) be a homogeneous Siegel domain in C^ X C ^ , T be a discrete subgroup of Aui {D{V, F)). IfT is a lattice, namely, the measure of left coset space D{V, F)/T is a finite number, then Aut (-D(V, F)) is a real semisimple Lie group and D{V, F) is a symmetric Siegel domain. THEOREM

Proof Since F is a discrete subgroup of Aut {D{V, F)), there exists an open neighborhood U of the unit element e in Aut {D(V^ F)), such that U nT = {e}, and the closure of C/ is a compact subset. Given a right invariant measure fi on Aut (D(V,F)), then the measure of a compact set [/ is a finite number, and f^igU) = pioMU),

V5 € Aut {DiV, F)),

where g -^ p{g) is a real representation of Aut(-D(V,F)) of dimension one. By the natural mapping TT: Aut {D{V, F))/r /io:

Aut {D{V, F)) -^ Aut {D{V, F))/T,

has a measure tio{7r{gU)) = fji{gU),

Wg e

Aut{D{V,F)).

Symmetric Bounded Domains

313

Since F is a lattice, fi{gU) = fioiAgU))

< /xo(Aut iD{V, F))/T) < +oc.

Hence p is a bounded representation of Aut {D{V^ F)). Since e =

p{Avit{D{V,F)f)

is a Lie group of dimension one, 6 is a compact subgroup in R*. It follows that p{g) = h ygeAut{D{V,F)). Hence fi{U) = lJi{gU), V3 G Aut {D{V^ F)). Therefore // is a left invariant measure on Aut (i)(V, F)). It follows that Aut (-D(V, F)) is a unimodular Lie group. This theorem holds by Theorem 7.25. I The last theorem and proof pubhshed in Vey's paper (ref. [216]). Though his theorem prove that if a discrete subgroup F is a uniform lattice, then Aut(D(F, F)) is a real semisimple Lie group. Namely, the left coset space D(y^F)/T is a compact manifold. But his proof only used the fact that the measure of the left coset space -D(V, F)/V is a finite number. Namely, F is a lattice. On the other hand, it is well known that Vey's result can be proved by using the theory of infinite dimensional representations of a semisimple Lie group. Vey Theorem told us that the automorphic function theory cannot be established when the homogeneous Siegel domain is non-symmetric. It appears that the function theory on the non-symmetric homogeneous bounded domains is more complicated than the case of symmetric.

3.

Classification of Symmetric Siegel Domains

In this section, an indecomposable canonical symmetric normal Siegel domain is called a classical Siegel domain. We will give the complete classification and the realization of classical Siegel domains. By Theorems 7.22 and 6.34, we give the classification and realization of the classical Siegel domains of the first kind as follows: THEOREM

7.27

(1) Ri(A/', N):

The classical Siegel domains of the first kind are N>3,

nij = 2, l
< N,

D{VN) = { ^ G C ^ ' I ImCi(^) > 0 },

(7.48)

314

THEORY OF COMPLEX HOMOGENEOUS BOUNDED

DOMAINS

where (

s,

212 ^"i^

i?(22Ar)

•R(22JV)'

SNE

I

2i2

C^{z) = \

and Zij = {xij,yij)

AN

(7.49)

)

€ C^, (7.50)

(2) R2(^•).•

iV > 1, n y = 1, 1 0 },

matrix by

(7.51)

n{n +1)

independent

variables;

(3) R3(2i\r);

N>3,

D{VN)

nij=4,

= {ZG

l
N,

C^(2iv-i) I i^Ciiz)

> 0 }, ZiN

Ci{z)

S2E

n2

=

\ ZiN

R{z2Ny

\

R{Z2N)

'••

SNE

X2

Xs

0:4

-"^2

XI

x^

-X3

—X3 —Xji

—X4 X^

Xl —X2

X2 Xl

(7.52)

(7.53) )

where x = (0:1,0:25^1:3,0:4) € C^; / R{x) = \

(4) R4(2 + ni2).D{V2) = { isi,Z2,S2)

Xi

(7.54)

N = 2, € C2+"i2 I i m s i > 0 , l m ( ^^

^^^^ ) > 0 } ; (7.55)

(5) R6(27).-

iV = 3, ni2 = ni3 = n23 = 8,

DiVs) = {Z^

Im

(Sl, 212, S2, 213, 223, S3), si h2

212 S2E

213 R{z23)

^3

i?(z23)'

S3^

Si G C, ^;jj G C^

I >0},

(7.56)

315

Symmetric Bounded Domains where Z2s = ( x i , - • • , ^ 8 ) ; I

R{Z2S) =

X\

X2

Xs

X4

X^

XQ

XJ

X^

—X2

X\

X4

—Xs

XQ

—X5

—Xg

Xr

—X^

—X4

Xi

X2

X7

Xs

—X^

—XQ

—X4

Xs

—X2

X\

Xg

—X7

XQ

—X^

Xi

X2

Xs

X4

-0:5

\

—X7

-XQ

-Xs

—XQ

X5

—Xs

X7

—X2

Xi

—X4

—X7

Xs

X5

—XQ

—XS

X4

X\

—Xs

XQ

X^

-X4

X2

Xi

-Xs

-X7

-Xs

\

Xs j

(7.57) 7.28 If an m x m complex matrix set {Qi,-' the conditions:

LEMMA

Q'jQk + Q'kQj = ^SjkE,

iQn}

satisfies

l<j,k
(7.58)

then ^ Q'j {e'^Sy + e'yeu)Qj = 0, l
\/^E;

(3) n = 4,m = 2, Ri =

1 0 0 1

R3 =