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Introduction aux groupes arithmétiques by Armand Borel

By Armand Borel

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THÉORÈME. Soient G un Q-groupe et conditions suivantes sont équivalentes: r un sous-groupe arithmétique de G. Alors les . 15) de G. (a) G est unipotent. 3), et on a à montrer que GRJr est compact. Vu 8. 2, il suffit de faire voir que si 7t : G -+ GL(V) est un Q-morphisme, L un réseau de V Q et (gj)' (vJ des suites d'éléments de GR et L vérifiant 7t(gj) ,Vj -+ 0, alors Vi = 0 pour j assez grand. Pour cela on procède par récurrence sur dim V.

Alors G est semi-simple, donc X(GO) ~ { l}. Soit g E GQ unipotent. Il existe donc VI E V Q - { O} fixe par g. Si VI Ii' est pas isotrope. V est . somme directe du sous-espace VI engendré par VI et de son complément orthogonal V~, qui est stable par g. Il existe donc VII E V~ - {O} fixe par g. 3) G Q est formé d'éléments semi-simples. y Z2, donc au quotient de BLz par son centre. Par suite G Q contient des éléments unipotents f: 1. Si n = -2, alors GO est un tore de dimension un, donc GQ est formé d'éléments semi-simples.

Soit n = dim V ~ 3. Alors G est semi-simple, donc X(GO) ~ { l}. Soit g E GQ unipotent. Il existe donc VI E V Q - { O} fixe par g. Si VI Ii' est pas isotrope. V est . somme directe du sous-espace VI engendré par VI et de son complément orthogonal V~, qui est stable par g. Il existe donc VII E V~ - {O} fixe par g. 3) G Q est formé d'éléments semi-simples. y Z2, donc au quotient de BLz par son centre. Par suite G Q contient des éléments unipotents f: 1. Si n = -2, alors GO est un tore de dimension un, donc GQ est formé d'éléments semi-simples.

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