Deformation rings and parabolic induction

Hauseux, Julien; Schmidt, Tobias; Sorensen, Claus

doi:10.5802/jtnb.1046

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…Further note that under the assumptions of Theorem 4.2, the deformations of χ (hence of Ind G B pχq) are well-understood: according to [HSS18, Proposition 4.17], Def Ind G B pχq pAq is in natural bijection with Hom PropOq pΛ, Aq for any A P ArtpOq, the converse map being given by ψ Þ Ñ Ind G B pψ ˝χuniv q, and this bijection is functorial in A P ArtpOq. In this statement, Λ denotes the Iwasawa algebra associated to the torus T (see [Sch11,Section 19.7] for a precise definition) and χ univ : T Ñ Λ is the so-called universal deformation of χ (see [HSS18,Proposition 4.17] for the explicit formula defining χ univ ). To completely understand deformations of parabolically induced representations, we now have to answer the following open question.…”

Section: Deforming Parabolically Induced Representationsmentioning

confidence: 99%

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

Abdellatif¹,

Dávid²,

Romano³

et al. 2020

Preprint

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This paper surveys what is known about (conjectural) p-adic and p-modular semisimple Langlands correspondences in the non-supercuspidal setting for the unramified quasi-split unitary group Up1, 1qpQ p 2 {Qpq. It focuses in particular on the potential of deformation theory to relate these correspondences.this semisimple correpondence behave under deformations, i.e. when the objects it involves are lifted to p-adic representations/parameters?This paper explains what is known so far in this direction for non-supercuspidal objects. As above, let G " Up1, 1qpQ p 2 {Q p q. First, in Section 2, we review basic definitions about representations of p-adic groups in the context of the group G, including the definition of a non-supercuspidal representation of G. We finish the section with the classification of irreducible smooth non-supercuspidal representations of G over F p , where F p denotes an algebraic closure of the residue field of Q p . In Section 3 we introduce Langlands parameters and describe where non-supercuspidal representations fit into the semisimple Langlands correspondence Kozio l attached to representations of G [Koz16]. In Section 4 we begin to explore how the semisimple Langlands correspondence behaves when lifted to characteristic 0. To do so, we describe recent results of Hauseux-Sorensen-Schmidt [HSS18, HSS19] about the behaviour of parabolic induction under deformation, and we explicitly determine which representations of G their results apply to on the automorphic side.Finally, in Section 5 we use recent work of Kozio l-Morra to study how the Galois side of the correspondence behaves under deformation [KM18]. Our method here is to use the theory of Kisin modules to better understand the deformations of interest. The distinction between supercuspidal and nonsupercuspidal representations has a counterpart on the Galois side of the correspondence, but we do not need to make this distinction for the main results of Section 5. We can afford to be more general in this section because the results on the Galois side are uniform. (In contrast, the construction and deformations of supercuspidal representations on the automorphic side involves different techniques than for their non-supercuspidal counterparts.) We point out open problems related to deformations on both sides of the Langlands correspondence that are the subject of work in progress.

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Section: Deforming Parabolically Induced Representationsmentioning

confidence: 99%

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

Abdellatif¹,

Dávid²,

Romano³

et al. 2020

Preprint

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“…In this article, we study the behaviour of extensions and deformations under the first step (extension to a larger parabolic subgroup and twist by a generalised Steinberg representation). For the second step (parabolic induction), this has been done in [Hau18b,HSS16] when char(F ) = 0 and [Hau18a] when char(F ) = p.…”

Section: Introductionmentioning

confidence: 99%

Functorial properties of generalised Steinberg representations

Hauseux¹,

Schmidt²,

Sorensen³

2019

Journal of Number Theory

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Let G be the F -points of a connected reductive group over a nonarchimedean local field F of residue characteristic p and R be a commutative ring. Let P = LU be a parabolic subgroup of G and Q be a parabolic subgroup of G containing P . We study the functor St G Q taking a smooth R-representation σ of L which extends to a representation e G (σ) of G trivial on U to the smooth R-representation e G (σ)

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“…These computations have also been used to study the deformations of parabolically induced admissible smooth mod p representations of G in a joint work with T. Schmidt and C. Sorensen ( [HSS16]).…”

Section: Introductionmentioning

confidence: 99%

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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Deformation rings and parabolic induction

Cited by 4 publications

References 19 publications

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

Functorial properties of generalised Steinberg representations

Parabolic induction and extensions

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