Let k be an algebraically closed field with characteristic ℓ = p. We show that the supercuspidal support of irreducible smooth k-representations of Levi subgroups M ′ of SLn(F ) is unique up to M ′ -conjugation, where F is either a finite field of characteristic p or a non-archimedean locally compact field of residual characteristic p.
This paper concerns the ℓ-modular representations of GL 2 (E) and SL 2 (E) distinguished by a Galois involution, with ℓ an odd prime different from p. We start by proving a general theorem allowing to lift supercuspidal F ℓrepresentations of GLn(F ) distinguished by an arbitrary closed subgroup H to a distinguished supercuspidal Q ℓ -representation. Then we give a complete classification of the GL 2 (F )-distinguished representations of GL 2 (E), where E is a quadratic extension of F . For supercuspidal representations, this extends the results of Sécherre to the case p = 2. Using this classification we discuss a modular version of the Prasad conjecture for PGL 2 . We show that the "classic" 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 SL 2 (F )-distinguished modular representations of SL 2 (E).for all h ∈ H and v ∈ π.Distinguished representations are central objects in the study of the relative Langlands program. The distinction problem is closely related to the Langlands
Let $p$ be a prime number and $k$ an algebraically closed field with characteristic $\ell \neq p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups $\textrm {M}^{\prime}$ of $\textrm {SL}_n(F)$ is unique up to $\textrm {M}^{\prime}$-conjugation, where $F$ is either a finite field of characteristic $p$ or a non-Archimedean locally compact field of residual characteristic $p$.
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