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Représentations distinguées et conjecture de Prasad et Takloo-Bighash

Abstract : The framework of Marion Chommaux's PhD thesis is a very active and requiring area of representation theory of p-adic groups, called the "local Langlands program''. A highly investigated branch of this program is the ``relative Langlands program'', of which one of the key players is Dipendra Prasad. Together with Takloo-Bighash, he proposed in 2011 a conjecture concerning distinction of discrete series of inner forms of p-adic general linear groups, with respect to the centralizer of the invertibles of a quadratic extension of the base field: this conjecture is in terms of subtle Galois invariants. In her doctoral work, Marion completeley solves this conjecture in the case of Steinberg representations, and proves it for level zero cuspidal representations of split inner forms. In this latter case, it is remarkable that she obtains a counter-example to a more general version of the conjecture. The techniques employed are diverse. In the first chapter on Steinberg representations, the main tool is the Bernstein-Zelevinsky geometric lemma, but some analytical invariants such as L-functions also play a role. In the second chapter the essential ingredients are Bushnell-Kutzko's type theory (more precisely the admissible pairs of Bushnell-Henniart) and the geometry of the Bruhat-Tits building. Once the classification of level zero cuspidal representations is obtained, Marion skilfully reduces the verification of the conjecture to a result of Fröhlich and Queyrut on the Galois side.
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Marion Chommaux. Représentations distinguées et conjecture de Prasad et Takloo-Bighash. Théorie des groupes [math.GR]. Université de Poitiers, 2019. Français. ⟨NNT : 2019POIT2287⟩. ⟨tel-02529153⟩

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