La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires
2016; Volume: 1; Linguagem: Francês
10.24033/msmf.457
ISSN2275-3230
Autores Tópico(s)Advanced Topics in Algebra
ResumoLet E/F be a quadratic extension of non-archimedean local fields of characteristic 0 and let G = U (n), H = U (m) be unitary groups of hermitian spaces V and W . Assume that V contains W and that the orthogonal complement of W is a quasisplit hermitian space (i.e. whose unitary group is quasisplit over F ). Let π and σ be smooth irreducible representations of G(F ) and H(F ) respectively.Then Gan, Gross and Prasad have defined a multiplicity m(π, σ) which for m = n -1 is just the dimension of Hom H(F ) (π, σ).For π and σ tempered, we state and prove an integral formula for this multiplicity.As a consequence, assuming some expected properties of tempered L-packets, we prove a part of the local Gross-Prasad conjecture for tempered representations of unitary groups.This article represents a straight continuation of recent papers of Waldspurger dealing with special orthogonal groups.où φ désigne la transformée de Fourier de ϕ et J O est l'intégrale orbitale sur O. La somme porte sur l'ensemble des orbites nilpotentes de g t (F ).Bien sûr les mesures et la transformée de Fourier doivent être définies précisément, on renvoie pour cela au corps de l'article.Soit N il reg (g t (F )) l'ensemble des orbites nilpotentes régulières.On pose alorst) |N il reg (g t (F ))| On définit de même la fonction c σ .Posons m geom (π, σ) = T ∈T |W (H, T )| -1 lim s→0 + T (F ) c σ ∨ (t)c π (t)D H (t) 1/2 D G (t) 1/2 ∆(t) s-1/2 dt
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