Sur une conjecture de Breuil–Herzig

Hauseux, Julien

doi:10.1515/crelle-2016-0039

Cited by 4 publications

(6 citation statements)

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“…, π) is locally nilpotent by the induction hypothesis, thus the Hecke action of z on H 1 ( N [Hau16b,(2.2)]) and we deduce that H n ( N (r+1) 0 , π) is locally z-finite. In this case, we put…”

Section: Sublemma Let π Be a Locally Z-finitementioning

confidence: 62%

“…In [Hau16b], we formulated the following conjecture. In cases (iii) and (iv), 'Otherwise' means that the conditions of case (ii) are not all satisfied.…”

Section: Presentation Of the Main Resultsmentioning

confidence: 99%

“…In order to do so, we use the derived functors of the left and right adjoints of the parabolic induction functor, namely the Jacquet functor and the ordinary parts functor ( [Eme10a]), both related to Emerton's δ-functor of derived ordinary parts ( [Eme10b]). We compute the latter on parabolically induced representations of G by pushing to their limits the methods initiated in [Hau16a] and expanded in [Hau16b].…”

Section: Introductionmentioning

confidence: 99%

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Parabolic induction and extensions

Hauseux

2018

Alg. Number Th.

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Let G be a p-adic reductive group. We determine the extensions between admissible smooth mod p representations of G parabolically induced from supersingular representations of Levi subgroups of G, in terms of extensions between representations of Levi subgroups of G and parabolic induction. This proves for the most part a conjecture formulated by the author in a previous article and gives some strong evidence for the remaining part. In order to do so, we use the derived functors of the left and right adjoints of the parabolic induction functor, both related to Emerton's δ-functor of derived ordinary parts. We compute the latter on parabolically induced representations of G by pushing to their limits the methods initiated and expanded by the author in previous articles. * This research was partly supported by EPSRC grant EP/L025302/1. arXiv:1607.02031v2 [math.RT] 25 May 2017Let J ⊆ ∆. We fix a totally decomposed 5 standard compact open subgroup N J,0 ⊆ N J and we define an open submonoid of L J by settingNote that Z J is generated by Z + J as a group and L J is generated by L + J and Z J as a monoid (cf. [Eme06, Proposition 3.3.2]). Moreover, any λ ∈ X * (S) associated to P J has its image contained in the maximal split subtorus S J of Z • J and satisfies α, λ > 0 for all α ∈ Φ + \Φ + J , thus the assumption of § 3.1 with N = N J and Z = S J is satisfied. We fix z ∈ Z + J strictly contracting N J,0 (equivalently Z J is generated by Z + J and z −1 as a monoid).We use the notation of § 2.3. The subgroup N J, I w ⊆ N J is stable under conjugation by B J,w J , and we have a semidirect product N J,Since N J, I w,0 is totally decomposed, we have a short exact sequence of topological groupsIn particular, N J, I w,0 is stable under the quotient action of B + J,w J on N J, I w . Lemma 3.2.1. For all n ∈ N, the inflation map is a natural B + J,w J -equivariant isomorphism H n N J, I w,0 , π N J, I w,0 I w ∼ −→ H n N J, I w,0 , πI w . Proof. The Lyndon-Hochschild-Serre spectral sequence associated to (20) is naturally a spectral sequence of B + J,w J -representations (cf. [Hau16b, (2.3)])Proof. We have a semidirect productLet n ∈ N J, I w,0 . Proceeding as in the proof of Lemma 2.3.1, we see that u n u −1 n −1 ∈ N J, I w so that u n u −1 = n n with n ∈ N J, I w . Thus bn b −1 = (b n b −1 )(b n b −1 ) ∈ N J, I w,0 , and since N J, I w,0 is totally decomposed, we deduce that b n b −1 ∈ N J, I w,0 .Lemma 3.2.4. For all n ∈ N, there is a natural such that the action of z on its kernel and cokernel is locally nilpotent.Proof. We use the notation of § 2.3. We let w J ∈ J∩ I w J −1 (I) W J and we put I w :and πI w be as in § 3.2. In the course of the proof of Proposition 3.3.2, we see that H n (N J,0 , πI w ) is locally Z + J -finite (as we saw it for H n (N J,0 , c-ind P − I I w J P J P − I σ) in the course of the proof of Proposition 3.3.1). Since σ is locally admissible, the L I∩ I w J (J) -representations H • Ord L I ∩P I∩ I w J (J) σ are locally admissible by [Eme10b, Theorem 3.4.7 (2)], thus locally Z I∩ I w J (J) -finite by ...

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Section: Sublemma Let π Be a Locally Z-finitementioning

confidence: 62%

“…In [Hau16b], we formulated the following conjecture. In cases (iii) and (iv), 'Otherwise' means that the conditions of case (ii) are not all satisfied.…”

Section: Presentation Of the Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parabolic induction and extensions

Hauseux

2018

Alg. Number Th.

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“…31 Recall that the hypothesis that b is decent is harmless, since any σ-conjugacy class in G(Qp) contains a decent element. 32 By the definition of a local Shtuka datum! Proof.…”

Section: The Main Resultsmentioning

confidence: 99%

“…for all w J ∈ W J . We partially compute the B I -representation Ord M α ∩P I (H 1 Ord P α (I w J )) using the isomorphism (2.10) and, once again, [32,Th. 2.6].…”

Section: 41mentioning

confidence: 99%

p-adic étale cohomology of period domains

et al. 2021

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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On the exactness of ordinary parts over a local field of characteristic p

Hauseux

2018

Pacific J. Math.

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Let G be a connected reductive group over a non-archimedean local field F of residue characteristic p, P be a parabolic subgroup of G, and R be a commutative ring. When R is artinian, p is nilpotent in R, and char(F ) = p, we prove that the ordinary part functor Ord P is exact on the category of admissible smooth R-representations of G. We derive some results on Yoneda extensions between admissible smooth R-representations of G.2010 Mathematics Subject Classification. 22E50.

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Sur une conjecture de Breuil–Herzig

Abstract: AbstractSoit G un groupe réductif p-adique de centre connexe et de groupe dérivé simplement connexe. Nous montrons que certaines “chaînes ” de séries principales de G n’existent pas et nous établissons plusieurs propriétés de la construction Show more

Cited by 4 publications

References 9 publications

Parabolic induction and extensions

Parabolic induction and extensions

p-adic étale cohomology of period domains

On the exactness of ordinary parts over a local field of characteristic p

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