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## Overview

Originally published in 1981.

The

**Princeton Legacy Library**uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

## Product Details

ISBN-13: | 9780691615455 |
---|---|

Publisher: | Princeton University Press |

Publication date: | 07/14/2014 |

Series: | Publications of the Mathematical Society of Japan , #584 |

Pages: | 340 |

Product dimensions: | 9.10(w) x 6.00(h) x 0.80(d) |

## Read an Excerpt

#### Algebraic Structures of Symmetric Domains

**By Ichiro Satake**

**PRINCETON UNIVERSITY PRESS**

**Copyright © 1980 The Mathematical Society of Japan**

All rights reserved.

ISBN: 978-0-691-08271-4

All rights reserved.

ISBN: 978-0-691-08271-4

CHAPTER 1

Algebraic Preliminaries

§ 1. **Linear algebraic groups.**

Let *F* be a field of characteristic zero and let *V* be a vector space of dimension *n* over *F.* We denote by End(*V*) the algebra of all *F*-linear transformations of *V* and by *GL(V)* the group of units in End(*V*), i. e., the group of all non-singular *F*-linear transformations of *V.* When a basis of *V* is fixed, *V* is identified with the space of *n*-tuples *Fn*, and so End(*V*) and *GL(V)* are also identified with *Mn(F)*, the full matrix algebra of degree *n* over *F,* and *GLn(F),* the general linear group of degree *n* over *F,* respectively.

In the following, we fix an algebraically closed extension ** F** of

*F*once and for all. A subgroup G of

*GLn(*is called a (linear)

**F**)*algebraic group*defined over

*F,*if G is the set of common zeros of a (finite) system of polynomial equations in

*n2*matrix entries with coefficients in

*F.*If we embed G in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

G may be viewed as an affine variety in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The dimension of an algebraic group G is the dimension of the underlying affine variety. For instance, *GLn( F)* and

*SLn(*(= {g[member of]GLn(

**F**)**F**)|

**det**(

*g*)=1}) are algebraic groups of dimension

*n2*and

*n2*-1, respectively.

Let G and G' be algebraic groups defined over *F* in *GLn( F)* and

*GLn,(*respectively. A group homomorphism φ: G -> G' is called a (rational) homomorphism defined over

**F**),*F,*or for short, an

*F-homomorphism,*if φ is a restriction to G of a polynomial map with coefficients in

*F*of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. φ is called a (rational) isomorphism defined over

*F,*or an

*F-isomorphism,*if φ is a group isomorphism and both φ and φ-1 are

*F*-homomorphisms. A bijective

*F*-homomorphism is necessarily an

*F*-isomorphism. (This is not true in general when the characteristic is positive.) Linear algebraic groups which are F-isomorphic to each other may be viewed as matrix expressions of one and the same (affine) algebraic group defined over

*F.*

A subgroup H of an algebraic group G defined over *F* is called *F-closed* if H itself is an algebraic group defined over *F,* i. e., defined by polynomial equations with coefficients in *F* (for one and hence all matrix expression of G). When H is an *F*-closed normal subgroup of G, it is known (e. g., Borel [13]) that the factor group G/H has a natural structure of (linear) algebraic group defined over *F* (determined up to an *F*-isomorphism) such that the canonical homomorphism G -> G/K is an *F*-homomorphism; one then has the relation

dim G/H - dim G - dim H.

It is also known (Borel, loc. cit.) that, if φ: G->G' is an *F*-homomorphism, the image φ(G) is an *F*-closed subgroup of G', the kernel N = φ-1(e') is an *F*-closed normal subgroup of G, and one has an *F*-isomorphism G/N [congruent to] φ(G). The notions of a direct product G1 × G2 and a semi-direct product G1 · G2 of two algebraic groups G1 and G2 are defined in the natural manner.

For an algebraic group G defined over *F*, we denote by Gz the Zariski connected component of the underlying affine variety of G containing the unit element *e.* Then G2 is an *F*-closed normal subgroup of G of finite index, and the coset decomposition of G with respect to Gz coincides with the decomposition of the underlying affine variety of G into the union of irreducible components. Thus Gz is the unique irreducible component of G containing *e*; an algebraic group G is Zariski connected if and only if it is irreducible as affine variety.

For an algebraic group G defined over *F* in *GLn( F),* the subgroup G{

*F*) = G[intersection]

*GLn(F)*is called the group of "

*F*-rational points" in G. Clearly the group G(

*F*) is well-determined, independently of the matrix expression of G. More generally, if φ: G->G' is an

*F*-homomorphism, φ induces an (abstract) group homomorphism φF:G(

*F*)->G'(

*F*). It is known that, when G is Zariski connected, or when

*F*is algebraically closed, G(

*F*) is Zariski dense in G.

In general, let *G* be an (abstract) subgroup of *GLn(F)* and let G be the Zariski closure of *G* in *GLn( F).* Then G is an algebraic group defined over

*F*and one has

*G]*subset]G(

*F*). When we have the equality

*G*=G(

*F*), we call

*G*an

*F-group*and G the

*associated algebraic group.*Let

*G*and

*G'*be

*F*-groups with the associated algebraic groups G and G'. By definition, an "

*F*-homomorphism" of

*G*into

*G'*is the restriction φ

*F*to

*G*of an

*F*-homomorphism φ:G->G' (which is uniquely determined by φ

*F*). The notions of "

*F*-isomorphisms", "

*F*-subgroups", etc. of

*F*-groups are defined in a similar manner. For an

*F*-group

*G*with the associated algebraic group G, Gz=Gz(

*F*) will be called the "Zariski connected component" of

*G; Gz*is an

*F*-group with the associated algebraic group Gz. It should be noted that some properties of

*F*-homomorphisms of algebraic groups mentioned above break down in general for

*F*-homomorphisms of

*F*-groups. To see this, let φ

*F: G->G'*be an

*F*-homomorphism of

*F*-groups coming from an

*F*-homomorphism φ: G->G' of the associated algebraic groups, and let

*N*=Ker φ

*F*, N = Ker φ. Then it is clear that

*N*= N [intersection]

*G*, so that

*N*is a normal

*F*-subgroup of

*G.*However, if N1 denotes the algebraic group associated with

*N,*one has only Nz [subset] N1 [subset] N. On the other hand, φ

*F(G)*is Zariski dense in φ(G), but may not be equal to φ(G)(

*F*). Thus we have only an injective homomorphism G/N->φ(G)(

*F*) and, going to the Zariski closures, the induced

*F*-homomorphism G/N1->φ(G), which may not be injective. For instance, for

*G=G'=GL1(*and the

**R**)=**R**x**-homomorphism φ: x[??]x4, one has |N|=4, |N|=|N1|=2, and φ(G)(**

*R***)=**

*R***x, φ**

*R**R*(

**x)=**

*R***x+ (the multiplicative group of positive real numbers).**

*R*When *F= R* or

**, an**

*C**F*-group

*G*in

*GLn(F)*is a closed subgroup of

*GLn(F)*in the usual topology, so that

*G*has a unique structure of real (or complex) Lie group of the same dimension. We denote the identity connected component of

*G*in the usual topology by

*G*°. Then clearly and by a theorem of Whitney [1] the index (Gz:Gº) is finite. In particular, in the case

*F=C,*one has

*G° =Gz*, and so

*G*is Zariski connected (i. e., the associated algebraic group G is Zariski connected) if and only if

*G*is connected in the usual topology. This is false for

*F=*; e. g.,

**R***G=*and

**R**x*SO(p, q) (p*> 0,

*q*> 0) are Zariski connected, but one has (

*G: G*°)=2.

For a Zariski connected *F*-group *G* we denote by *F[G]* the "affine ring" of the associated algebraic group G over *F,* i. e., the ring of polynomial functions on G (viewed as an affine variety in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) with coefficients in *F. F[G]* is then an integral domain, finitely generated over *F,* and *G* acts on *F[G]* by the left translation λ*g*1(*g*1 [member of] *G*):

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By definition, a *derivation* of *F[G]* is an *F*-linear transformation *X* of *F[G]* satisfying the condition

(1.2) X(f1 · f2) = (Xf1) · f2 + f1 · (Xf2) (f1, f2 [member of] F[G]).

We denote by Der *F[G]* the Lie algebra of all derivations of *F[G]* with the bracket product [*X*1, *X*2]= *X*1*X*2-*X*2*X*1. A derivation *X* is *left invariant* if it commutes with all left translations λg1, i. e.,

(1.3) X · λg1 = λg1 · X for all g1 [member of] G.

The set of all left invariant derivations of *F[G]* forms a Lie subalgebra g, called the *Lie algebra* of the F-group *G* and denoted by Lie *G.* The dimension of g= Lie *G* (as vector space over *F*) is equal to the dimension of the associated algebraic group g. When an *F*-group *G* is not Zariski connected, we define Lie *G* to be Lie *Gz.*

Let *Te(G)* denote the "tangent space" over *F* to g at the unit element *e,* i. e., the vector space over *F* formed of all *F*-valued derivations of *F[G]* with respect to the homomorphism *F[G]]*??]*f]*??]*f(e)]*member of]*F*. If we set

(1.4) Xe(f) = (XF)(e) for X [member of] g,

then the map *X]*??]*Xe* gives an *F*-linear isomorphism of g onto *Te(G),* by which the Lie algebra g of *G* is often identified with the tangent space *Te(G).* For instance, the Lie algebra gl*n*(*F*) of *GLn(F)* is identified with the tangent space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which in turn can naturally be identified with M*n*(*F*). The bracket product [*X*1, *X*2] in gl*n*(*F*) (as derivations) coincides with the usual bracket product in *Mn(F)* (as matrices). Thus, for an *F*-group *G* in *GLn(F),* the Lie algebra g may be viewed as a subalgebra of gl*n*)*(F)=Mn(F)*. It should be noted that not all Lie subalgebras of gl*n*(*F*) correspond to *F*-subgroups of *GLn(F)*; one corresponding to an *F*-subgroup of *GLn(F)* is called an "algebraic" Lie algebra (cf. Chevalley [2], [4]).

When *F= R* or

**the Lie algebra g and the tangent space**

*C,**Te(G)*can also be identified with the ones defined in the theory of Lie groups. In these cases, one has the "exponential map" exp : g ->

*G*. For

*X]*member of]g, exp

*tX*(

*t]*member of]

*F*) is the unique one-parameter subgroup of

*G*such that one has in matrix expression

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or equivalently,

(1.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where *X* on the right-hand side is interpreted as a derivation of *F[G],*

Let *G* and *G'* be Zariski connected *F*-groups and let g=Lie *G* and g' = Lie *G'.* Given an *F*-homomorphism φ*F*: *G*->*G'* (coming from an *F*-homomorphism of the associated algebraic groups, φ : G->G', one has an *F*-algebra homomorphism φ*; *F[G']*->*F[G]* given by φ*(*f*)=*f*ºφ (*f]*member of]*F[G']*). One defines the *differential d*φ: g->g' by setting

(dφ(X) = X·φ* for X [member of] g = Te(G),

(where *X* is viewed as an *F*-valued derivation of *F[G]*). Then *d*φ is a Lie algebra homomorphism, and the correspondence φ[??]*d*φ is clearly functorial. In the case *F = R* or

**, this (algebraic) definition of differentials coincides with the analytic one. In particular, for**

*C**X]*member of]g, the "one-parameter subgroup" φ(

*t*)=exp

*tX*(

*t]*member of]

*F*) is defined by an

*F*-homomorphism φ of the additive group of

*F*into the

*F*-group

*G*and the relation (1.5) or (1.6) is equivalent to saying that (

*d*φ)(1)=

*X*. In this book, whenever there is no fear of confusion, the differential

*d*φ will also be denoted by φ.

Let *F'* be an extension field of *F,* contained in ** F.** Then from an

*F*-group

*G*in

*GLn(F)*one obtains an

*F'*-group

*G'*in

*GLn(F')*by putting

*G'*=G(

*F'*) where G is the algebraic group associated with

*G*(and hence also with

*G').*The

*F'*-isomorphism class of

*G'*is uniquely determined by the

*F*-isomorphism class of

*G.*We call

*G'*the

*F'*-group obtained from

*G*by

*scalar extension F'/F*and write

*G'=GF'.*The Lie algebra of

*GF*, can naturally be identified with the Lie algebra g

*F'*=g[cross product]

*FF'*obtained from g=Lie

*G*by scalar extension (in the sense of linear algebra). Conversely, suppose there is given an

*F'*-group

*G'*in

*GLn(F').*Then

*G=G']*intersection]

*GLn(F)*is an

*F*-group and one has

*GF']*subset]

*G'*.

*G*is called an

*F-form*of

*G'*if

*GF'=G',*or equivalently, if

*G*and

*G'*have the same associated algebraic group. When this is the case, we say that

*G'*has an

*F*-form in

*GLn(F)*(with respect to the given matrix expression) and write

*G=G'(F)*(or

*G'F*). Let G be the algebraic group in

*GLn(*associated with

**F**)*G'*and let I(G) be the ideal corresponding to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i. e., the ideal in the polynomial ring

*F*]*xij*(1 ≤ i,

*j*≤

*n*),

*y*] consisting of all polynomials annihilating G. Then in order that

*G'*has an

*F*-form in

*GLn(F)*the following two conditions are necessary and sufficient:

(i) G is defined over *F,* i. e., I(G) has a basis consisting of polynomials with coefficients in *F.*

(ii) Every coset in G/Gz contains an *F*-rational point.

When these conditions are satisfied, *G*=G(*F*) is an *F*-form of *G'* (relative to the given matrix expression).

*(Continues...)*

Excerpted fromAlgebraic Structures of Symmetric DomainsbyIchiro Satake. Copyright © 1980 The Mathematical Society of Japan. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

## Table of Contents

- FrontMatter, pg. i
- Preface, pg. vii
- Contents, pg. ix
- Instructions to the Reader, pg. xi
- List of Symbols, pg. xv
- Chapter I. Algebraic Preliminaries, pg. 1
- Chapter II. Basic Concepts on Symmetric Domains, pg. 44
- Chapter III. Unbounded Realizations of Symmetric Domains. (Theory of Wolf -Korányi), pg. 89
- Chapter IV. Equivariant Holomorphic Maps of a Symmetric Domain into a Siegel Space, pg. 165
- Chapter V. Infinitesimal Automorphisms of Symmetric Siegel Domains, pg. 209
- Appendix: Classical Domains, pg. 267
- References, pg. 291
- Index, pg. 319