# Algebraic Structures of Symmetric Domains by Ichiro Satake

By Ichiro Satake

This publication is a finished therapy of the overall (algebraic) idea of symmetric domains.

Originally released in 1981.

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**Example text**

Thus a Cartan involution 8 of g (relative to G) is always liftable to an R-automorphism ofG, called a (global) "Cartan involution" of G and denoted by the same letter 8. In the following, we distinguish the actions of 8 on G and g by writing them in the form g• (g EG) and 8X (XE g), respectively. A similar convention will also apply to the complex conjugations q 0 and (J. It is clear that, conversely, if we have a compact R-form U of Ge with complex conjugation q such that q0 q=qq0, then 8=q[G (or q[g) is a Cartan involution of G (or g).

C(a)+n is a minimal parabolic subalgebra of g. From the theory of algebraic Lie algebras (Chevalley [2], [4]), it is easy to see that all Lie subalgebras of g containing b~ are algebraic and hence of the form br for some I'cJ. It is known that any algebraic group defined over F contains an (absolute) maximal torus j defined over F. It follows that there is an F-torus Tin C(A) such that 7=TFismaximalinC(A)F. Then Tis (absolutely) maximalinG and A coincides with the F-split part of T. Let X be the (absolute) character module of j and Xo the annihilator of A in X.

First let us recall some definitions and basic identities. A finite dimensional (non-associative) algebra A over Fis called a Jordan algebra if the following two conditions are satisfied : Chapter I. 22 Algebraic Preliminaries (J 1) xy =yx, for all x,yeA. Y) = x(x~) For aeA, we define T.. eEnd(V) by Tax=ax, where Vis the underlying vector space of A. Polarizing the identity (J 2), one obtains (ab) (cd) + (be) (ad)+ (ca) (bd) = a( (bc)d) +b( (ca)d) +c((ab)d) (a, b, c, de A). , T,a] + [ Tc, T •• ] = 0, (6.