Algebraic Structures of Symmetric Domains

Algebraic Structures of Symmetric Domains

by Ichiro Satake
     
 

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This book is a comprehensive treatment of the general (algebraic) theory of symmetric domains.

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 paperback editions preserve the

Overview

This book is a comprehensive treatment of the general (algebraic) theory of symmetric domains.

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 paperback editions preserve the original texts of these important books while presenting them in durable paperback 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 Series
Pages:
342
Product dimensions:
9.10(w) x 6.00(h) x 0.80(d)

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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



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(F) is called a (linear) 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(F) (= {g[member of]GLn(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,(F), respectively. A group homomorphism φ: G -> G' is called a (rational) homomorphism defined over 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 φF to G of an F-homomorphism φ:G->G' (which is uniquely determined by φF). 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 φF: 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 φF, 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, φF(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(R)=Rx and the R-homomorphism φ: x[??]x4, one has |N|=4, |N|=|N1|=2, and φ(G)(R)=Rx, φR(Rx)=Rx+ (the multiplicative group of positive real numbers).

When F=R or C, an 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=R; e. g., G=Rx and 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 λg1(g1 [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 [X1, X2]= X1X2-X2X1. 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 gln(F) of GLn(F) is identified with the tangent space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which in turn can naturally be identified with Mn(F). The bracket product [X1, X2] in gln(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 gln)(F)=Mn(F). It should be noted that not all Lie subalgebras of gln(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 C, the Lie algebra g and the tangent space 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 φF: 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)=fºφ (f]member of]F[G']). One defines the differential dφ: 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 dφ is a Lie algebra homomorphism, and the correspondence φ[??]dφ is clearly functorial. In the case F = R or C, this (algebraic) definition of differentials coincides with the analytic one. In particular, for 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 (dφ)(1)=X. In this book, whenever there is no fear of confusion, the differential dφ 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 gF'=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(F) associated with 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, jn), 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 from Algebraic Structures of Symmetric Domains by Ichiro Satake. Copyright © 1980 The Mathematical Society of Japan. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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