Theory of Complex Homogeneous Bounded Domains

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

Printed on acid-free paper

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 , • • • ? 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 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).

(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=^-').

^{[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-EOr",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 1 > 0 , l < i < n and i>{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(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

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