, Action de G sur S x S \ N s . -D'après

G. Opère-sur-la-frontière-de-silov, S. Du-domaine, and D. , On définit alors l'action "diagonale" de G sur 5 x 5 par: g·(z,w) =(g

. Où-g,

, LEMME IV.2.1. -Le groupe G opère sur la variété 5 X 5 \ N s

. Démonstration, -Pour (z,w) dans D X D on pose

, z,w)r 1 = DetB(z,w

, 0.2) s'écrit alors sous la forme: Nv(g· z,g' w) = (Detg'(w))-l Nv(z,w)(Detg'(z))-1 (1.4.2) pour tout

G. Soit-maintenant-g-dans,

, 4.2) se prolonge à D x D, donc Nv(g· z,g· w) = (Detg'(w))-lNv(z,w)(Detg

, Nv(z, w) cF O. Donc Nv(g· z, g. w) cF 0, et comme g' z appartient à 5, DetP(g· z) cF 0, par suite

G. Notons,

, est une courbe causale dans C, et comme '1 est fermée EXp-l 0, est fermée donc triviale et par suite, est triviale. La structure causale de 'G/G(fl) étant globale on peut définir un ordre partiel sur cet espace de la manière suivante: y::: :r s'il existe une courbe causale dans, fl) dont x est l'origine et y est l'extrémité

I. Soit, = {g E cG 1 g' Xo >: xo} le semi-groupe associé à cet ordre et montrons que IC = r

, (X)h où h E G(fl) et X E C. Soit, la courbe définie par, exp(tX) . Xo , est une courbe causale et nous avons) = Xo et

S. Récipoquement, E. Ic, and . Une-courbe-causale, ) = g. XQ. Comme ci-dessus, on peut supposer que Exp( C) pour tout t, donc gXQ E Exp(Cl, et par suite 9 . XQ = (exp X) Xo où X E C, ou encore 9 = (exp X)h avec 11 E G(n), d'où 9 E r. o REMARQUE IV.4.5. -Dans [44], KANEYUKI montre, par la méthode semi-simple, que S x 5 est une compactification de G/c-1 G(fl)c. La méthode, basée sur la théorie des algébres de Jordan, que nous avons proposée ici est plus simple et a l'avantage de nous permettre de décrire avec précision l'image de, fl) dans sa compactification S x S, à savoir, pp.524-555, 1946.

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