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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

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.

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