. Dans-l-'´-equation, Associé Associéà l'espace simplicial BG (avec un fibré ? ) il y a un (géométrique) Les G -opérateurs pseudodifférentiels elliptiques définissent desélémentsdeséléments [? D ] ? K * ,? (BG ) demanì ere naturelle. Dans ce groupe (géométrique) il y a défini un caractère du Chern tordu (Dual) (113) ch ?, donné par ch ? (x) = T d(? C ) ?1 ch(x). Dans la formule (112) l'accouplement de droite est l'accouplement naturel entre H ? * (BG ) et H * ? (BG ) et on a donc une formule comme celle qui appara??tappara??t dans le théorème classique de Atiyah-Singer, ([AS68b])

G. Soit, . Groupo¨?dégroupo¨?dé-etale, P. Soit-d-un-g-opérateur-pseudodifférentiel-elliptique-sur-une-g-variété-propre, and P. , Soit c ? H * ? (BG ) de dégrée total 2q, alors on a (114) ? F (c), ind F a

. Or, on sait que l'image de Ind ? D sign dans K 0 (C * r (G )) est un invariant par homotopie (grâcè a un résultat de Mishchenko et Kasparov, p.94

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