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13. 117] semi-simple, est quasi-semi-simple et il quasi-semi-simple. x si par et quasi-semi-simple, de pour x si [41, est ce tout de plus , tout que tout x est un [41]. 14. LEMME. conque Soit x ~ G On peut x = su la d6composition Alors rgx(G) = rgu{CG(s]) supposer que 8 G B G = B GAB X Alors S = CT[S]~ @l@ment la quent, de de CG[S) et par ee . 15. LEMME. B T et un Jordan. s et normalise ferm@e est a dono a donc est est ] 9 En dens N un un et, T un . O'apr~s par cons6- sous-groupe sous-groupe = dim partieulier, parabode Cs[Y)~ le qui normelise Borel , rgx(G) on peut et soit s' 6 sB u' G uB dill@rent T ce un qui par fait ~l~ment CTiX' ]o T un que de B ,]o = CsiU et rg remplaeer de CT{S']~ qui on x'= IN) = rg X 616ment B en X' s'u' {N) qui nor- = OTIS) ~ = S .

Donc de CG(Y) zg oO h = hxh 1 = Y qua d'a- h 6 H . II O'apr~s . Comma On en dE- NN(T) est H phisme CT(Y)~ tores maximaux xT'U It 6 T', v 6 U) implique -1 v x v -I ) [t[x outre . Cela ~ . d) ElEment dens d@fini Si qu'il Mais maximaux ca un contenu G § de = CgTg_l[y) ~ ( g T g -I ] . En y = xtv x 6 G supposer c anonique des -1 = x [ • T' = {x - l t x t - 1 t 6 T} , et cxT'U si est peut d6monstration [tv)x(tv) t- 1 ) , est est un une sous-vari6t@ , alors (e), et sous-tore de T . s xt 6 C / T eussi (y) .

I BvGl = 1 Une L = 1 COROLLAZRE. dans , est une base de -8]/Z dim . Si droite suite r6duit G Bf et triangulaires f13 = f31 O6finition que . I1 si et (s ~ ~) f maintenant a une m a t r i c e f g NG(B) et pour est Aut{SL{E)) Choisissons o~ sym6trique isotropes pour B~n G-66~ 2 alors , uG ~ et peuvent ci-dessus En , BG u ~tre reli~s enti6rement particulier BG u est par conteconnexe. 3). un 616merit unipotent v tel que 4B Nous pouvons ~i@ment unipotent potente de ~, ..... qui dens G est aormalise quel cheque que ~l~ment x-orbite r@ducti{.

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