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Construction de Séries Discrètes p-adiques: Sur les Séries by Prof. Paul Gérardin (auth.)

By Prof. Paul Gérardin (auth.)

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Additional info for Construction de Séries Discrètes p-adiques: Sur les Séries Discrètes non Ramifiées des Groupes Réductifs Déployés p-adiques

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D~finit une section w ~- d~signe la projection de composantes dans w s sur V= ~ ~• + sP § (w)s P § (w') = sP , w,w' - 9 En caract~ristique diff~rente de 2 , le lemme 9 implique l'isomorphisme 9~ . 9 . *~~ , ~ ~que pros 0 + s== s . En caract@ristique 2 , ce re@me lennne donne : P

12. : F(eZx) =~(z) F(x), si zek, F(s(v)(x) = ~ ( v ) F(x) ,si v~Q (~= I si la caract6ristique n'est pas 2) , F(tx) = a r ( t ) ~ ( t ) ~ fonction sur P S ,t(u)F(s(u)x)du , oG S t est la d6finie par : - en caract6ristique diff6rente de 2, si S ~,t , t~T (u) P~(t) = Q~'(t) C(t) (cf. h. I0 d)) : = Ir ( [u,u A(t) C(t) -I ] /2) ~P*Ce)(u) ; p - en caract6ristique 2, si P~(t) = Qe(t) C(t) : 52 S ,t (u) ~n prenant pour =~(uC(t)-IA'(t)~(uC(t)-1)-1~r([uC(t)-IA'(t),u~)~(s f la mesure de Dirac ~ l'orlgine de P on obtient un vecteur g~n~rateur de eette representation irr~ductible de DT par trans- lations ~ droite : %,e si zs (ezs (v)ts (u)) =q~(z) ~(v) ~ ( t ) v~Q, t 6T, u & P .

Iii) S i c e k , on d~finit un automorphisme d(c) de H par la formule s*(~)d(c) 0 c -I = T'(uv) S~(w (c 0 )) si W = au + v. I1 se relive darts Bo(H*) en : (D(c)f)(u) = ~(c -I u) , oG la transformation de Fourier est prise relativement U'- On a D(c) 2 = I. Ces deux assertions se v@rifient imm@diatement, comme la suivante : (iv) pour t # I , r(t) = t(~ t) dCcCt)) t(~t) o~ ~ t est le caract~re quadratique ~t(u) = ~(C(t) -I u) ~(A(t) C(t) -I u) -I u'(A(t) C(t) -I u 2) . 28 Avec les notations pr~c~dentes, l'automorphlsme RT,(t) = Rl~,~((t) = T~t) D(C(t)) T(~t) si t ~ I, R~(1) = I rel~ve r(t) dams Bo(H ).

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