By Ichiro Satake

This e-book is a entire therapy of the overall (algebraic) thought of symmetric domains.

Originally released in 1981.

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**Extra resources for Algebraic Structures of Symmetric Domains**

**Example text**

J). J. J) 0 x0 = G(Q) 0 Then, since (gx0 ). J) -orbit in U. J. JcE. J). J), K is compact. J); then K' cg,Kg,'. It follows that K is maximal compact, and every maximal compact subgroup of G(Q) is conjugate to K, q. e. d. 0 Now let Q be a self-dual homogeneous cone in U. Then, in the notation of Proposition 8. J)=K'P, where K' is also a maximal compact subgroup of G(Q). J)z,· When this equality holds, we say that the reference point x0 is "compatible" with ( ) . In the following, we fix such an x0 once and for all and denote it by e.

G•). The subspace of Y 0 generated by t coincides with the annihilator of A<"i in YQ, so that t can be identified with the root system of G• relative to A <•J. We introduce a linear order (compatible with the vector operations) in YQ, and define the notion of "positive" roots in the usual manner. A positive root a is called simple if it can not be written as a sum of two positive roots. The set of all simple roots L1 is called the fundamental system of t. :0 or all :s;;O, according as a is positive or negative.

J') satisfying the conditions (9. 5), (9. 6). Conversely, suppose there is given a Lie algebra homomorphism p0 satisfying these conditions. Then one has Po(Ta)e' =