An application of a theorem of Emerton to mod p representations of GL 2

Hu, Yongquan

doi:10.1112/jlms.12080

Cited by 5 publications

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“…Further, we compute it when G is split with connected centre (see Theorem 1.4 below). Note that in cases (ii) and (iii), the source of the isomorphism can be non-zero ( [Hu16]). Theorem 1.3 (Theorem 5.2.4).…”

Section: Presentation Of the Main Resultsmentioning

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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Section: Presentation Of the Main Resultsmentioning

confidence: 99%

Parabolic induction and extensions

Hauseux

2018

Alg. Number Th.

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“…Unlike the classical case of complex coefficients, this exact sequence does not necessarily split (cf. [Hu17]). We have the following necessary condition for the nonsplitting of the above exact sequence.…”

Section: 2mentioning

confidence: 99%

“…As an application of the above results, we study extensions of G-representations. Suppose we have a short exact sequence of the form 0 −→ Ind G B (χ) −→ π −→ τ −→ 0, where τ is an irreducible, admissible, supersingular representation of G. Hu has shown that such an extension does not necessarily split (contrary to the case of complex coefficients; see [Hu17]). Using the calculation of H 1 (I 1 , Ind G B (χ)), we give in Proposition 5.15 a necessary condition for the short exact sequence above to be nonsplit.…”

Section: Introductionmentioning

confidence: 99%

The first pro-$p$-Iwahori cohomology of mod-$p$ principal series for $p$-adic $GL_n$

Koziol

2018

Trans. Amer. Math. Soc.

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Let p ≥ 3 be a prime number and F a p-adic field. Let I 1 denote the pro-p-Iwahori subgroup of GLn(F ), and H the pro-p-Iwahori-Hecke algebra of GLn(F ) with respect to I 1 (over a coefficient field of characteristic p). We compute the structure of H 1 (I 1 , π) as an H-module, where π is a mod-p principal series representation of GLn(F ). We also give some partial results about the structure of H 1 (I 1 , π) for a general split reductive group with irreducible root system.

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“…• Under the assumption of Buzzard-Diamond-Jarvis conjecture it was shown by Hu [Hu17] that for GL 2 (E) there are non-trivial extensions between supersingular and principal series representations which is not the case for GL 2 (Q p ) (we will come back to this point later in the introduction).…”

Section: Introductionmentioning

confidence: 99%

“…4.9] which states that the completed cohomology of Shimura curve (over a totally real field F ) realises the mod l local Langlands correspondences for GL 2 (F l ) where F l denotes the completion of F at the prime l (for the precise statement see Conjecture 1). The local factors appearing in Conjecture 1 is studied in [EH14], [Hel13] for primes l ∤ p and in [Hu17], [BHH + 20], [HW20] for primes lying above p. Therefore to salvage information about mod p local Langlands correspondence its natural to study the completed cohomology of Shimura curves. In fact in this article we will be also be concerned with the completed cohomology of ordinary and supersingular locus of Shimura curves.…”

Section: Introductionmentioning

confidence: 99%

Towards a mod-$p$ Lubin-Tate theory for $\GL_2$ over totally real fields

Banerjee¹,

Rai²

2022

Preprint

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We show that the conjectural mod p local Langlands correspondence can be realised in the mod p cohomology of the Lubin-Tate towers. The proof utilizes a well known conjecture of Buzzard-Diamond-Jarvis [BDJ10, Conj. 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field F and of mod l( = p) local Langlands correspondence as given by Emerton-Helm [EH14]. In the case of modular curves a similar theorem was obtained by Chojecki [Cho15]. assuming a Serre type Conjecture 2, in Theorem 1.2, we show that the conjectural mod p local Langlands correspondence lies in the mod p cohomology of the Lubin-Tate towers.In recent years quite a progress have been made to establish the local Langlands correspondence (both p-adic and mod p) but unfortunately the complete picture is not clear even for GL 2 . The mod p and p-adic local Langlands correspondence has been established for GL 2 (Q p ) by Breuil, Berger, Colmez, Dospinescu, Paskunas and others. But for GL 2 (E), when E( = Q p ) is a finite extension of Q p the correspondence is not known. Roughly speaking the aim of the mod p local Langlands is to give a correspondence between the n-dimensional continuous mod p representation of G E := Gal( Ē/E) and smooth admissible mod p representation of GL n (E). The picture on the Galois side is much simpler and it is the automorphic side which should be understood better in order to give such a correspondence.For GL 2 (Q p ) the classification of irreducible smooth admissible representation was started by Barthel-Livné [BL94] and completed by Breuil [Bre03] wherein he classified all the supersingular representation for GL 2 (Q p ) but for GL 2 (E) the classification is still incomplete mainly because of the mysterious supersingular representations.The following are some basic differences between the smooth representations of GL 2 (Q p ) and GL 2 (E):• Unlike GL 2 (Q p ) the complete classification of supersingular representations for GL 2 (E) is not known. The study of such representations were carried out by Breuil-Paskunas [BP12] wherein

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An application of a theorem of Emerton to mod p representations of GL 2

Abstract: Let p be a prime and L be a finite extension of Qp. We study the ordinary parts of GL2(L)representations arise in the mod p cohomology of Shimura curves attached to indefinite division algebras which splits at a finite place above p. The main tool of the proof is a theorem of Emerton.

Cited by 5 publications

References 21 publications

Parabolic induction and extensions

Parabolic induction and extensions

The first pro-$p$-Iwahori cohomology of mod-$p$ principal series for $p$-adic $GL_n$

Towards a mod-$p$ Lubin-Tate theory for $\GL_2$ over totally real fields

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