Comptes Rendus

Comptes Rendus

Mathématique

Eknath Ghate and Mihir Sheth

On non-admissible irreducible moduloprepresentations ofGL2(

Q

<https://doi.org/10.5802/crmath.85>

© Académie des sciences, Paris and the authors, 2020.

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This article is licensed under the

Creative Commons Attribution4.0International License.

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LesComptes Rendus. Mathématiquesont membres du Centre Mersenne pour l’édition scientifique ouverte

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2020, 358, n5, p. 627-632

https://doi.org/10.5802/crmath.85

Number Theory and Reductive Group Theory /Théorie des nombres et théorie des groupes réductifs

On non-admissible irreducible modulo p representations of GL ₂ ( Q ^p

⁾

Sur les représentations irréductibles non-admissibles modulo p de GL ₂ ( Q p

)

Eknath Ghate ^aand Mihir Sheth^∗,a

aSchool of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai - 400005, India.

E-mails:[email protected], [email protected].

Abstract.We use a Diamond diagram attached to a 2-dimensional reducible split modpGalois representa- tion of Gal(Qp/Q_p2) to construct a non-admissible smooth irreducible modprepresentation of GL2(Q_p2) following the approach of Daniel Le.

Résumé.Nous utilisons un diagramme de Diamond attaché à une représentation galoisienne modpsemi- simple réductible de dimension 2 de Gal(Qp/Q_p2) pour construire une représentation modpnon-admissible irréductible lisse de GL2(Q_p2) en suivant l’approche de Daniel Le.

2020 Mathematics Subject Classification.22E50, 11S37.

Funding.The authors acknowledge support of the Department of Atomic Energy, Government of India, under project number 12-R&D-TFR-5.01-0500.

Manuscript received 12th March 2020, accepted 9th June 2020.

1. Introduction

Letpbe a prime number,Qpbe the field ofp-adic numbers, andFp be an algebraic closure of the finite fieldFpof cardinalityp. The study of the admissibility of smooth irreducible represen- tations of connected reductivep-adic groups goes back to Harish–Chandra (see [6]). Building upon his work, Jacquet proved that every such representation over the field of complex numbers is admissible (see [8], see also [3]). This result was extended by Vignéras to smooth irreducible representations over any algebraically closed field of characteristic not equal top (cf. [12]). In

∗Corresponding author.

628 Eknath Ghate and Mihir Sheth

the note [1], the authors ask whether this is true for smooth irreducible representations over al- gebraically closed fields of characteristicp. It is known that every smooth irreducible represen- tation of GL₂(Qp) overFpis admissible (see [2]). However, Daniel Le recently constructednon- admissiblesmooth irreducibleFp-linear representations of GL2(F), forFa finite unramified ex- tension ofQpof degree at least 3 and forp>2, providing a negative answer to the question raised above (see [9]). In this paper, we follow Le’s approach and construct non-admissible irreducible representations of GL₂(Qp²) whereQp² is the unramified extension ofQpof degree 2. These re- sults support the viewpoint of Breuil and Pašk ¯unas that the modp(andp-adic) representation theory of GL₂(F) becomes more complicated as soon asF6=Qp(see [5], see also [11]).

LetG=GL₂(Qp²),K =GL₂(Zp²), andΓ=GL₂(Fp²), whereZp² is the ring of integers ofQp²

with residue fieldFp². Fix an embeddingFp²,→Fp. LetIandI1denote the Iwahori and the pro-p Iwahori subgroups ofKrespectively, andK₁denote the first principal congruence subgroup ofK. WriteNfor the normalizer ofI(and ofI₁) inG. As a group,Nis generated byI, the centerZofG, and by the elementΠ=¡_{0 1}

p0

¢. All representations considered in this paper from now on are over Fp-vector spaces. For a characterχofI,χ^sdenotes itsΠ-conjugate sendingginItoχ(ΠgΠ⁻¹).

A weight is a smooth irreducible representation ofK. TheK-action on such a representation factors throughΓand thus any weight is described by a 2-tuple (r₀,r₁)⊗det^m:=Sym^r0Fp

2⊗ (Sym^r1Fp

2)^Frob⊗det^m of integers with 0≤r₀,r₁≤p−1 together with a determinant twist for some 0≤m<p²−1 (see [4, Lemma 2.16 and Proposition 2.17]). Given a weightσ, its subspace σ^I1ofI₁-invariants has dimension 1. Ifχσdenotes the corresponding smooth character ofIand χσ6=χ^s_σ, then there exists a unique weightσ^ssuch thatχσ^s=χ^s_σ(see [10, Theorem 3.1.1]).

A basic 0-diagram is a triplet (D₀,D₁,r) consisting of a smooth K Z-representationD₀, a smoothN-representationD1and anI Z-equivariant isomorphismr:D1−→∼ D₀^I1with the trivial action ofponD₀andD₁. Given such a diagram such thatD^K₀¹has finite dimension, the smooth injectiveK-envelope inj_KD₀admits a non-canonicalN-action which glues together with theK- action to give a smoothG-action on inj_KD₀(see [5, Theorem 9.8]). TheG-subrepresentation of inj_KD0generated byD0is smooth admissible and itsK-socle equals theK-socle socKD0ofD0.

From now on, assume thatpis odd. Letρ: Gal(Qp/Qp²)→GL2(Fp) be a continuous generic Galois representation such thatpacts trivially on its determinant andD(ρ) be the set of weights, calledDiamond weights, associated toρ as described in [5, Section 11]. Breuil and Pašk ¯unas attach a family of basic 0-diagrams (D₀(ρ),D₁(ρ),r), calledDiamond diagrams, toρsuch that soc_KD₀(ρ)=L

σ∈D(ρ)σ(see [5, Theorem 13.8]).

For a finite unramified extensionF ofQp of degree at least 3, Le uses a Diamond diagram attached to anirreducibleρ: Gal(Qp/F)→GL₂(Fp) to construct an infinite dimensional diagram which gives rise to a non-admissible smooth irreducible representation of GL₂(F) (see [9]). His strategy does not work for a Diamond diagram attached to an irreducible Galois representation of Gal(Qp/Qp²) because such a diagram does not have suitableΠ-action dynamics. However, forF= Qp², we observe that a Diamond diagram attached to areducible splitρhas an indecomposable subdiagram with suitableΠ-action dynamics so that Le’s method can be used to obtain a non- admissible irreducible representation ofG=GL2(Qp²).

2. Reducible Diamond diagram

Letω2be Serre’s fundamental character of level 2 for the fixed embedding Fp² ,→Fp, and let ρ: Gal(Qp/Qp²)→GL₂(Fp) be a continuous reducible split generic Galois representation. The restriction ofρto the inertia subgroup is, up to a twist by some character, isomorphic to

Ãω^r₂^0+1+(r1+1)p 0

0 1

!

C. R. Mathématique,2020, 358, n5, 627-632

for some 0≤r₀,r₁≤p−3, not both equal to 0 or equal top−3 (see [4, Corollary 2.9 (i)] and [5, Definition 11.7 (i)]). Define the weight

σ:=(r₀+1,p−2−r₁)⊗det^p−1+r1p. Then the set of Diamond weights forρis given by

D(ρ)=©

(r₀,r₁),σ,σ^s, (p−3−r₀,p−3−r₁)⊗det^r0+1+(r1+1)pª

(see [5, Lemma 11.2 or Section 16, Example (ii)]). Fix a Diamond diagram (D₀(ρ),D₁(ρ),r) attached toρ, and identify D₁(ρ) withD₀(ρ)^I1 as I Z-representations via r. There is a direct sum decompositionD₀(ρ)=L

ν∈D(ρ)D_0,ν(ρ) ofK-representations with soc_KD_0,ν(ρ)=ν(see [5, Proposition 13.4]).

Now define

D₀:=D_0,σ(ρ)⊕D_0,σ^s(ρ) and D₁:=D^I₀¹.

It follows from [5, Theorem 15.4 (ii)] that (D₀,D₁,r) is an indecomposable subdiagram of (D₀(ρ),D₁(ρ),r). Set

τ:=(r0+2,r1)⊗det^{p−2+(p−1)p} and τ⁰:=(p−1−r0,p−3−r1)⊗det^r0+(r1+1)p. The graded pieces of the socle filtrations ofD_0,σ(ρ) andD_0,σ^s(ρ), with the convention that we ignore a weight if a negative entry appears, are as follows (see [5, Theorem 14.8 or Section 16, Example (ii)]):

D_0,σ(ρ) : σ τ⊕τ^s (p−4−r₀,r₁−1)⊗det^r0+2 D_0,σs(ρ) : σ^s τ⁰⊕τ^0s (r₀−1,p−4−r₁)⊗det^(r1+2)p. We have from [5, Corollary 14.10] that

D₁=χσ⊕χτ⊕χ^s_τ⊕χ^s_σ⊕χτ⁰⊕χ^s_τ0. (1) For anI Z-representationVand anI Z-characterχ, we writeV^χfor theχ-isotypic part ofV.

3. An infinite dimensional diagram and the construction LetD₀(∞) :=L

i∈ZD₀(i) be the smoothK Z-representation with component-wiseK Z-action, where there is a fixed isomorphismD₀(i)∼=D₀ofK Z-representations for everyi ∈Z. Follow- ing [9], we denote the natural inclusionD₀−→^∼ D₀(i),→D₀(∞) byιi, and writev_i :=ιi(v) for v∈D₀for everyi ∈Z. LetD₁(∞) :=D₀(∞)^I1. We define aΠ-action onD₁(∞) as follows. Let λ=(λi)∈Q

i∈ZFp×

. For all integersi∈Z, define

Πv_i:=











(Πv)_i ifv∈D₁^χσ, (Πv)_i+1 ifv∈D₁^χτ, λi(Πv)i ifv∈D₁^χτ0.

This uniquely determines a smoothN-action onD₁(∞) such thatp=Π²acts trivially on it. Thus we get a basic 0-diagramD(λ) :=(D₀(∞),D₁(∞), can) with the above actions where can is the canonical inclusionD1(∞),→D0(∞).

Theorem 1. There exists a smooth representationπof G such that (i) (π|K Z,π|N, id)contains D(λ),

(ii) πis generated by D₀(∞)as a G-representation, and (iii) socKπ=socKD₀(∞).

630 Eknath Ghate and Mihir Sheth

Proof. LetΩbe the smooth injectiveK-envelope ofD₀equipped with theK Z-action such that pacts trivially. The smooth injectiveI-envelope inj_ID₁ofD₁appears as anI-direct summand of Ω. Letedenote the projection ofΩonto inj_ID1. There is a uniqueN-action on inj_ID1compatible with that ofI and compatible with the action ofN onD₁. By [5, Lemma 9.6], there is a non- canonicalN-action on (1−e)(Ω) extending the givenI-action. This gives anN-action onΩwhose restriction toI Zis compatible with the action coming fromK Z onΩ.

Now letΩ(∞) :=L

i∈ZΩ(i) with component-wiseK Z-action where there is a fixed isomor- phismΩ(i)∼=ΩofK Z-representations for everyi∈Z. We wish to define a compatibleN-action onΩ(∞). As before, denote the natural inclusionΩ−→^∼ Ω(i),→Ω(∞) byιi, and writev_i :=ιi(v) forv∈Ω. LetΩ_χdenote the smooth injectiveI-envelope of anI-characterχ. Thus, from (1), we havee(Ω)=inj_ID₁=Ωχσ⊕Ωχτ⊕Ω_χ^s_τ⊕Ω_χ^s_σ⊕Ωχ_τ0⊕Ω_χ^s_τ0. Ifv∈(1−e)(Ω), we defineΠv_i:=(Πv)i

for all integersi. Otherwise, we defineΠv_i :=(Πv)_i if v∈Ωχσ,Πv_i :=(Πv)_i+1ifv ∈Ωχτ, and Πvi:=λi(Πv)iifv∈Ωχ_τ0. By demanding thatΠ²acts trivially, this defines a smoothN-action on Ω(∞) which is compatible with theN-action onD₁(∞), and whose restriction toI Zis compati- ble with the action coming fromK ZonΩ(∞). By [10, Corollary 5.5.5], we have a smoothG-action onΩ(∞). We then takeπto be theG-representation generated byD₀(∞) insideΩ(∞). If follows easily from the construction thatπsatisfies the properties (i), (ii) and (iii).

Theorem 2. If λi 6= λ0 for all i 6= 0, then any smooth representation π of G satisfying the properties(i),(ii), and(iii)of Theorem 1 is irreducible and non-admissible.

Proof. Let π⁰ ⊆ π be a non-zero subrepresentation of G. By property (iii), we have either HomK(σ,π⁰)6=0 or HomK(σ^s,π⁰)6=0. We consider the case HomK(σ,π⁰)6=0; the other case is treated analogously. There exists a non-zero (ci)∈L

i∈ZFpsuch that Ã

X

i

c_iιi

!

(D_0,σ(ρ))∩π⁰6=0.

We claim that Ã

X

i

ciιi+j

!

(D₀)⊂π⁰ for allj∈Z. (2)

We first show that¡ P

iciιi¢

(D0,σ^s(ρ))⊂π⁰. Note that¡ P

iciιi¢

(D0,σ(ρ))∩π⁰6=0 is equivalent to¡ P

ic_iιi

¢(σ) ⊂π⁰. Since ¡ P

ic_iιi

¢(D^χ₁^σ)⊂ π⁰ and π⁰ is stable under the Π-action, we have

¡ P

ic_iιi

¢(D^χ₁^sσ)⊂π⁰. By Frobenius reciprocity, we have a non-zeroK-equivariant map Ind^K_I

ÃÃ X

i

ciιi

!

³ D^χ₁^sσ´

!

→π⁰ (3)

whose image is¡ P

ic_iιi

¢(I(δ(σ),σ^s)), whereδis the bijection on the set of Diamond weightsD(ρ) defined in [5, Section 15], andI(δ(σ),σ^s) is theK-subrepresentation ofD_0,δ(σ)(ρ) with cosocle σ^s (and socleδ(σ)). In our setting,δmapsσtoσ^s and vice versa (see [5, Lemma 15.2]). Thus I(δ(σ),σ^s)=σ^s and so¡ P

ic_iιi¢

(σ^s)⊂π⁰. LetR¡ (P

ic_iιi)(σ)¢

be theK-subrepresentation of the compact induction c-Ind^G_{K Z}¡

(P

ic_iιi)(σ)¢

defined in [5, Section 17]. By [5, Lemmas 17.1, 17.4 and 17.8], we have

Ind^K_I ÃÃ

X

i

c_iιi

!

³ D^χ₁^sσ´

!

⊂R ÃÃ

X

i

c_iιi

! (σ)

! , and by Frobenius reciprocity, there is a non-zero map

c-Ind^G_{K Z} ÃÃ

X

i

c_iιi

! (σ)

!

→π⁰ (4)

which restricts to the map (3). So the imageQ of R¡ (P

ic_iιi)(σ)¢

in π⁰ under the map (4) contains¡ P

iciιi¢

(σ^s). Since socKQ ⊂socKπ=socKD₀(∞) and the Jordan–Hölder factors of

C. R. Mathématique,2020, 358, n5, 627-632

R¡ (P

ic_iιi)(σ)¢

are multiplicity free (see [5, Lemma 17.11]), soc_KQ is isomorphic to a subrep- resentation of the direct sum of the weights inD(ρ). Therefore by [5, Lemma 19.5], soc_KQ =

¡ Piciιi¢

(σ^s), and by [5, Lemma 19.7],Q contains a copy of theK-representationD0,σ^s(ρ). But

¡ Pic_iιi

¢(D_0,σ^s(ρ)) is the uniqueK-subrepresentation ofπisomorphic toD_0,σ^s(ρ) and withK- socle¡ P

ic_iιi

¢(σ^s). Thus¡ P

ic_iιi

¢(D_0,σ^s(ρ))=Q⊂π⁰. Now, since¡ P

ic_iιi

¢(σ^s)⊂π⁰, a symmetric argument shows that¡ P

ic_iιi

¢(D_0,σ(ρ))⊂π⁰. Thus Ã

X

i

ciιi

!

(D0)⊂π⁰. Therefore

à X

i

ciιi

!

¡D^χ₁^τ¢

⊂π⁰ and Ã

X

i

ciιi

!

³ D^χ

sτ 1

´

⊂π⁰. Sinceπ⁰is stable under theΠ-action, we have

à X

i

ciιi+1

!

³ D₁^χsτ´

⊂π⁰ and Ã

X

i

ciιi−1

!

¡D₁^χτ¢

⊂π⁰. In particular,

à X

i

c_iιi+1

!

(D_0,σ(ρ))∩π⁰6=0 and Ã

X

i

c_iιi−1

!

(D_0,σ(ρ))∩π⁰6=0.

By the same arguments as above, we find that Ã

X

i

ciιi+1

!

(D₀)⊂π⁰ and Ã

X

i

ciιi−1

!

(D₀)⊂π⁰. The claim (2) is now proved by repeatedly using theΠ-action.

For (d_i)∈L

i∈ZFp, let #(d_i) denote the number of non-zero d_i’s. Among all the non-zero elements (c_i) ofL

i∈ZFpfor which¡ P

ic_iιi¢

(D₀)⊂π⁰, we pick one with #(c_i) minimal. We may also assume thatc06=0 using (2). We now show that #(ci)=1. Assume to the contrary that #(ci)>1.

Since¡ P

iciιi¢

(D^χ₁^τ0)⊂π⁰andπ⁰is stable under theΠ-action, we have Ã

X

i

λiciιi

!µ D^χ

s τ0 1

¶

⊂π⁰.

Since¡ P

iλ0c_iιi

¢(D^χ

s τ0

1 ) is also clearly inπ⁰, subtracting it from the above, we get Ã

X

i

(λi−λ0)ciιi

!µ D^χ

s τ0 1

¶

⊂π⁰. Writing (c_i⁰) :=((λi−λ0)c_i), we see that

à X

i

c_i⁰ιi

!

(D_0,σs(ρ))∩π⁰6=0.

Following the same arguments as in the previous paragraphs, we get that¡ P

ic_i⁰ιi¢

(D₀)⊂π⁰. However, the hypothesisλi6=λ0for alli6=0, and the assumption #(c_i)>1 imply that (c_i⁰) is non- zero and #(c_i⁰)=#(c_i)−1 contradicting the minimality of #(c_i). Therefore, we havec₀ι0(D₀)⊂π⁰. Soι0(D₀)⊂π⁰. Using (2) again, we get thatL

j∈Zιj(D₀)=D₀(∞)⊂π⁰. By property (ii), we have π⁰=π.

The non-admissibility ofπis clear becauseπ^K1⊇socKπand socKπis not finite dimensional

by property (iii).

632 Eknath Ghate and Mihir Sheth

Remark 3. If the diagram (D₀(ρ),D₁(ρ),r) is defined overFp² and (λi)∈Q

i∈ZF^×_p2, then the representationπin Theorem 1 has a modelπ0overFp². Furthermore,π0is absolutely irreducible and non-admissible if the (λi) satisfy the hypothesis of Theorem 2. In fact, for any fieldC containingFp², the methods of this paper produce an absolutely irreducible non-admissible smoothC-representationC⊗F_p2π0ofG.

Now letCbe an arbitrary field of characteristicpwith algebraic closureC. From the discussion in the previous paragraph, the representationC⊗F_p2π0is a smooth irreducibleC-representation which has a modelC⁰⊗F_p2π0overC⁰, whereC⁰=CFp² ⊂C. By [7, Lemma II.5], there exists a smooth irreducibleC-representationπCsuch thatC⊗F_p2π0is aC-subrepresentation ofC⊗CπC. SinceC⊗F_p2π0is non-admissible,C⊗CπC is non-admissible and henceπC is non-admissible by [7, Lemma III.1(ii)]. Thus we obtain a smooth irreducible non-admissible representation ofG over any fieldCof characteristicp.

Acknowledgments

We thank Daniel Le, Sandeep Varma, M.-F. Vignéras and the referee for useful comments on earlier versions of this paper. The second author thanks Anand Chitrao for helpful discussions on diagrams.

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