By Ichiro Satake

This publication is a finished remedy of the final (algebraic) conception of symmetric domains.

Originally released in 1981.

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**Sample text**

Let Ll= {a,,···, a 1) be the fundamental system oft with respect to such an order and put J 0 =J n X 0 • Then J 0 is a fundamental system of t 0 =t n X 0 and one has J=1r(J-J0 ). For each ae Gal(P/F), there exists a unique element Wu in W such that Ju=wuLl. One defines a new action of a on J by a[ul=w;;-'af. Then, for a;, a 1 eJ-J0, cine has 1r(a;)=1r(a1 ) if and only if a;ul=a 1 for some ae Gal(F/F) (cf. Satake [10], pp. 71-72). It is convenient to illustrate this situation by a Dynkin diagram of J with additional marks ("I'-diagram", loc.

2) and (6. 3). Thus the Jordan algebra A becomes aJTS with the trilinear product (6. 4). When A has a unit element e, one can recover the Jordan product from { } by (6. 5) {x,y, e} = {x, e, y} = {e, x, y) = xy. Returning to an (arbitrary) JTS (V, { } ), we define the trace form by (6. 6) ,(x,y) = tr(xoy), and from now on assume that ( V, { } ) is "non-degenerate" (or "semi-simple"), that is, (JT 3) , is non-degenerate. § 6. The structure group of a (non-degenerate) JTS 23 Under this condition, we define the adjoint* with respect to -r: by -r:(Tx,y) = -r:(x, T*y) for Te End(V).

If x'+ f=O, one has 0 = (x'+ f, e) = (x, x) + (y,y), so that x= y=O. Therefore U is formally real. 2) First by (iii) Q is open. J* by (8. 2). Then it is clear that Q• is a (non-degenerate, open convex) cone. To see that Q* is non-empty, we show that QcQ*. Let G1 =exp p·K1 be the global Cartan decomposition. Then, for g 1, g,E Gi, if one puts g,'g,= pk with PE exp p, kEK 1, then one has (u, g,e) = (ple,p'ize) > 0. Hence one has QcQ• and so QcQ•. One has also G,cG(Q*). Therefore, on Q•, there exists a G1-invariant Riemannian metric q (cf.

### Algebraic Structures of Symmetric Domains by Ichiro Satake

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