, Action de G sur S x S \ N s . -D'après
On définit alors l'action "diagonale" de G sur 5 x 5 par: g·(z,w) =(g ,
,
, LEMME IV.2.1. -Le groupe G opère sur la variété 5 X 5 \ N s
-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
,
, 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
,
, 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
= {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
) = 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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,
,
, ~y~~:e'~rit cone,:,-We' cho,?,se'to,', staclY' th~ "su,1:>~sel:r;'igro~p,:~f ,~le,men,ts ,-o~' ~he 'co?fpr~a:l :grotfp:y.rlli,èh: ,pr,eseryes :th,e' c'Ot;le. ~Te give, tv.(o âdciiti:cWal c\1araç~~iiz,at~on~', of ;,tl1is, ;seriügroup'., :rl:î~, first èharae;terizatiqn, re,~iizes' th~' 'se'mig:ro'u~, as,' t~'e:' :r,'e~l: l?pîntr;;,-,ôi,;tlle: :~ir'>,si-I~,NSrüL' s,~rnigrQup which, :stabilizes ~ the: ,assQ~iate(r"bo'unded' s,ytllniet,ric" dornain:.;, The; Becon<f:; , characterizatiQn c~I1ser~s, th~ syn;lrn~'tl:~~'spa~e' ,ol:~'ayl~~,:txpe,'~Ü'âc~à',:i,~, "tJi~ ):?rd~r''-~ : 'algebra~