Correspondance de Jacquet-Langlands et distinction : cas des représentations cuspidales de niveau 0

Coniglio-Guilloton, Charlène

Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 2, pp. 163-216

Soit 𝕂/𝔽 une extension quadratique modérément ramifiée de corps locaux non archimédiens. Soit GL m(𝒟) une forme intérieure de GL n(𝔽) et GL μ(Δ)=(Mm(𝒟)⊗𝔽𝕂)×. Alors GL μ(Δ) est une forme intérieure de GL n(𝕂) et les quotients GL μ(Δ)/ GL m(𝒟) et GL n(𝕂)/ GL n(𝔽) sont des espaces symétriques. En utilisant la paramétrisation de Silberger et Zink, nous déterminons des critères de GL m(𝒟)-distinction pour les représentations cuspidales de niveau 0 de GL μ(Δ) qui sont l'image d'une représentations cuspidale de niveau 0 par Jacquet-Langlands, puis nous prouvons qu'une représentation cuspidale de niveau 0 de GL n(𝕂) est GL n(𝔽)-distinguée si et seulement si son image par la correspondance de Jacquet-Langlands est GL m(𝒟)-distinguée.

Let 𝕂/𝔽 be a tamely ramified quadratic extension of non-archimedean locally compact fields. Let GL m(𝒟) be an inner form of GL n(𝔽) and GL μ(Δ)=(Mm(𝒟)⊗𝔽𝕂)×. Then GL μ(Δ) is an inner form of GL n(𝕂) and the quotients GL μ(Δ)/ GL m(𝒟) and GL n(𝕂)/ GL n(𝔽) are symmetric spaces. Using the parametrization of Silberger and Zink, we determine conditions of GL m(𝒟)-distinction for level zero cuspidal representations of GL μ(Δ) which are the image of a level zero cuspidal representation of GL n(𝕂) by the Jacquet-Langlands correspondence. We also show that a level zero cuspidal representation of GL n(𝕂) is GL n(𝔽)-distinguished if and only if its image by the Jacquet-Langlands correspondence is GL m(𝒟)-distinguished.

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