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Correspondances de Howe sur un corps p-adique by Colette Mœglin, Marie-France Vignéras, Jean-Loup Waldspurger PDF

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By Colette Mœglin, Marie-France Vignéras, Jean-Loup Waldspurger (auth.)

ISBN-10: 3540186999

ISBN-13: 9783540186991

ISBN-10: 3540481028

ISBN-13: 9783540481027

This e-book grew out of seminar held on the college of Paris 7 through the educational yr 1985-86. the purpose of the seminar used to be to provide an exposition of the idea of the Metaplectic illustration (or Weil illustration) over a p-adic box. The publication starts with the algebraic conception of symplectic and unitary areas and a common presentation of metaplectic representations. It maintains with exposés at the contemporary paintings of Kudla (Howe Conjecture and induction) and of Howe (proof of the conjecture within the unramified case, representations of low rank). those lecture notes comprise numerous unique effects. The e-book assumes a few historical past in geometry and mathematics (symplectic types, quadratic varieties, reductive teams, etc.), and with the idea of reductive teams over a p-adic box. it truly is written for researchers in p-adic reductive teams, together with quantity theorists with an curiosity within the function performed through the Weil illustration and -series within the concept of automorphic forms.

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Get Correspondances de Howe sur un corps p-adique PDF

This ebook grew out of seminar held on the college of Paris 7 through the educational 12 months 1985-86. the purpose of the seminar was once to provide an exposition of the idea of the Metaplectic illustration (or Weil illustration) over a p-adic box. The e-book starts with the algebraic thought of symplectic and unitary areas and a normal presentation of metaplectic representations.

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On ddmontre [R] que la propri6t6 (iii) reste (presque) vraie. Soit (X o, X 1, X2) (3(0, Y1, Y2) deux triplets de Lagrangiens, (iv) {il existe pc P , Yo = PXo, Y1 = PX1, Y2 = PX2 } ¢* { les deux triplets ont m~me invariant de Leray, et dim Xi~X j = dim Yif3Yj pour tout 0_<_i,j<_2,dim Xo¢'~Xl n X 2 = dim X o n X l n X 2 } On dira que q~ S2(X) est de classe p si la forme quadratique non d6g6n6r6e associ6e ~ q est de dimension (dimWM)/2 etde classe p. 55 Th6or~me. Si g l , g2 ~ S p ( W ) , il existe p, P l , P2 ~ P, S, S' : { 1....

Certains termes seront not~s M ~ ] dana chacun des cas. 2. Soient W=X+Y une polarisation compl~te. Identifions Y ~ X*. R~alisons ~ dana J(X*) (cf. 1). Un ~l~ment g de Sp(W) s'identifie g une matrice (~ ~I ' avec aEEnd(X), d~End(X*), bEHom(X*,X), cEHom(X,X*). Pour un tel g, fixons une mesure de Haar sur X/Ker(c), et d~finissons M~g]~End ~(X*) par M[g~f(x*)=~X/Ker(c ) ~(/2++/2)f(a*x*+c*x) pour f ~ X * ) , x*~X*. 2). En particulier, en normalisant convenablement lea mesures, on obtient lea formules plus usuelles: - p°ur a~GL(X)' et g=(a' a*0 0~ M[g~f(x*)=;detxall/2f(a*x*), - p o u r b~S2(X)~Hom(X*,X), et M[g~f(x*)=~(/2)f(x*), g=(~~1' dx 41 - pour b&Isom(X*,X) et g=( 0 bIJ b *-I 0 M[g]f(x*)= IX ~)f(b-lx) dx.

Alors ~ s = ~ s, d'ofi le s lemme. Ce lemme d4finit un endomorphisme M, ou M[g], de S. 3. Dans les d4monstrations des trois lemmes suivants, on traite le cas d'un corps local, le cas d'un corps fini ~tant plus simple. Lemme. Pour tous g&Sp(W), h~H, o_n a l'4$alit4 M[g] f~(h)= ~w(gh)M [g ]. D~monstration. Posons M=M[g], supposons h=~(w0), et soit s&S. Pour un r4seau L assez grand, on a f~(~ (gw0) )Ms= ~( ~ (gw0))~s = ~L qT(/2)~(~(gw0)~=(l-g)w)s dw. On a ~(gw 0) ~o(1-g)w=~(gw0+(1-g)w)~(/2) =~((l-g) (w-w0)+w0) ~(/2) = ~(l-g) (w-w0) ~(w0) ~(/2+).

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Correspondances de Howe sur un corps p-adique by Colette Mœglin, Marie-France Vignéras, Jean-Loup Waldspurger (auth.)


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