2022/03/28 by Peiyi Cui, Cui, Peiyi, Thomas Lanard +3
Engineering · #22E50 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2203.14788
openalex publication_date 2022/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a non-archimedean local field of characteristic different from 2 and residual characteristic p. This paper concerns the ℓ-modular representations of a connected reductive group G distinguished by a Galois involution, with ℓ an odd prime different from p. We start by proving a general theorem allowing to lift supercuspidal \mathbbFℓ-representations of GLn(F) distinguished by an arbitrary closed subgroup H to a distinguished supercuspidal ℚℓ-representation. Given a quadratic field extension E/F and an irreducible \mathbbFℓ-representation π of GLn(E), we verify the Jacquet conjecture in the modular setting that if the Langlands parameter ϕπ is irreducible and conjugate-self-dual, then π is either GLn(F)-distinguished or (GLn(F),ωE/F)-distinguished (where ωE/F is the quadratic character of F^× associated to the quadratic field extension E/F by the local class field theory), but not both, which extends one result of Sécherre to the case p=2. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to p=2. After that, we give a complete classification of the GL2(F)-distinguished representations of GL2(E). Using this classification we discuss a modular version of the Prasad conjecture for PGL2. We show that the "classical" Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil-Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the SL2(F)-distinguished modular representations of SL2(E).