Soient G un groupe p-adique se déployant sur une extension nonramifiée et Rep 0 Λ (G) la catégorie abélienne des représentations lisses de G de niveau 0 à coefficients dans Λ = Q ℓ ou Z ℓ . Nous étudions la plus fine décomposition de Rep 0 Λ (G) en produit de sous-catégories que l'on peut obtenir par la méthode introduite dans [Lan18]. Nous en donnons deux descriptions, une première du côté du groupe à la Deligne-Lusztig, puis une deuxième du côté dual à la Langlands. Nous prouvons plusieurs propriétés fondamentales comme la compatibilité à l'induction et la restriction parabolique ou à la correspondance de Langlands locale. Les facteurs de cette décomposition ne sont pas des blocs, mais on montre comment les regrouper pour obtenir les blocs "stables". Ces résultats corroborent une conjecture énoncée par Dat dans [Dat17a].Abstract. Let G be a p-adic group which splits over an unramified extension and Rep 0 Λ (G) the abelian category of smooth level 0 representations of G with coefficients in Λ = Q ℓ or Z ℓ . We study the finest decomposition of Rep 0 Λ (G) into a product of subcategories that can be obtained by the method introduced in [Lan18]. We give two descriptions of it, a first one on the group side à la Deligne-Lusztig, and a second one on the dual side à la Langlands. We prove several fundamental properties, like for example the compatibility to parabolic induction and restriction or the compatibility to the local Langlands correspondence. The factors of this decomposition are not blocks, but we show how to group them to obtain "stable" blocks. These results confirm a conjecture given by Dat in [Dat17a]. * σ ) ss ∼ −→ (G * gσ ) ss . * σ ) ss,Λ . Définissons le système S de classes de conjugaison par S σ = {s} et S τ = ∅ si τ = σ. On pose alors S (σ,s) := S. 2.2.2. Définition. On appelle C Λ l'ensemble des couples (σ, s) où σ ∈ BT et s ∈ (G * σ ) ss,Λ et on définit une relation d'équivalence ∼ sur C Λ par (σ, s) ∼ (τ, t) si et seulement si S (σ,s) = S (τ,t) . On obtient aisément le lemme suivant 2.2.3. Lemme. Soient σ, τ ∈ BT, et s ∈ (G * σ ) ss,Λ , t ∈ (G * τ ) ss,Λ . Alors, soit S (σ,s) = S (τ,t) , soit S (σ,s) ∩ S (τ,t) = ∅. Exprimé en termes de classes d'équivalence, ce lemme nous dit que [σ, s] la classe d'équivalence de la paire (σ, s) est donnée par [σ, s] = {(τ, t) | t ∈ S (σ,s),τ }. Posons S min Λ := {S (σ,s) , [σ, s] ∈ C Λ /∼}. L'ensemble S min Λ est constitué des ensembles 0-cohérents minimaux et par la proposition 2.1.6 on a M,Λ Rep T Λ (M ).
Let F be a non-archimedean local field and G the F -points of a connected simply-connected reductive group over F . In this paper, we study the unipotent ℓ-blocks of G. To that end, we introduce the notion of (d, 1)series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and d-Harish-Chandra theory. The ℓ-blocks are then constructed using these (d, 1)-series, with d the order of q modulo ℓ, and consistent systems of idempotents on the Bruhat-Tits building of G. We also describe the stable ℓ-block decomposition of the depth zero category of an unramified classical group.
The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of Rep R ( G ) \operatorname {Rep}_R(G) , the category of smooth representations of a p p -adic group G G with coefficients in R R . In particular, they were used to construct level 0 decompositions when R = Z ¯ ℓ R=\overline {\mathbb {Z}}_{\ell } , ℓ ≠ p \ell \neq p , by Dat for G L n GL_{n} and the author for a more general group. Wang proved in the case of G L n GL_{n} that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of G L n GL_{n} and a unipotent block of another group. In this paper, we generalize Wang’s equivalence of category to a connected reductive group on a non-archimedean local field.
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
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