On voit facilement que ce morphisme est effectif et strict. Notons aussi que r : Z(G 4 ) ? Z(G 6 ) s'identifiè a lélévation au carré sur G m , donc la condition (P LAT SU RJ) de l'hypothèse est satisfaite. On suppose que SL 4 (k ) ? Im ? ? H. On a donc r ? ? ? H résiduellement absolument irréductible. Par contre, la représentation adjointe sur sl 6 est réductible ,
anneau de valuation d'un corps p-adique contenant les valeurs propres de Hecke de ?. Soit une uniformisante de O. On suppose que r ? ? µ H est p-distinguée (au sens de la Définition 4.2). C'est le cas si le paramètre de Satake en p ? ,
Pour le premier théorème, on se donne une famille de Hida µ : T G ? I telle que ? µ soit résiduellement irréductible et satisfasse (Z p ? REG) et (IRRAD G,? ) pour un ? régulier. On a déjà vu dans la section 1 que les analogues des conditions (H1) et (H2) sont satisfaisaites pour le c-groupe G n. On obtient donc le théorème 6. Si de plus on se donne µ H : T H ? I H passant par ? H , telle que ? µ H est absolument irréductible et Im ? µ H contient SL 4 (k ) on obtient le Théorème 7. En fait, encore une fois, la classe C des triplets (H 1 , r 1 , s 1 ) se réduitréduità H et G avec l, Ces algèbres ont 5, resp. 3 variables ,
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