Central Characters for Smooth Irreducible Modular Representations of <span class="mathjax-formula">${ GL}_2({Q}_p)$</span>

Central Characters for Smooth Irreducible Modular Representations of GL

(Q

)

LAURENTBERGER

To Francesco Baldassarri, on the occasion of his 60th birthday

ABSTRACT- We prove that every smooth irreducible Fp-linear representation of GL2(Qp) admits a central character.

Introduction

LetPbe a representation of GL₂(Q_p). We say thatPis smooth, if the stabilizerof any v2P is an open subgroup of GL2(Q_p). We say that P admits a central character, if everyz2Z(GL2(Q_p)), the centerof GL2(Q_p), acts onPby a scalar. The smooth irreducible representations of GL₂(Q_p) over an algebraically closed field of characteristicp, admitting a central character, have been studied by Barthel-Livne in [BL94, BL95] and by Breuil in [Bre03]. The purpose of this note is to prove the following theorem.

THEOREMA. IfP is a smooth irreducibleF_p-linear representation of GL₂(Q_p), thenPadmits a central character.

The idea of the proof of theorem A is as follows. IfPdoes not admit a central character, and if f ˆ p 0

0 p

, then forany nonzero polynomial Q(X)2F_p[X], the map Q(f):P!P is bijective, so that P has the

(*) Indirizzo dell'A.: UMPA, ENS de Lyon, UMR 5669 du CNRS, Universite de Lyon, 46 AlleÂe d'Italie, 69007, Lyon, France.

E-mail: [email protected] URL: perso.ens-lyon.fr/laurent.berger/

structure of a Fp(X)-vectorspace. The representationP is therefore a smooth irreducible Fp(X)-linear representation of GL2(Q_p), which now admits a central character, sincef acts by multiplication byX. It remains to apply Barthel-Livne and Breuil's classification, which gives the structure of the components ofP afterextending scalars to a finite ex- tensionK ofF_p(X). A corollary of this classification is that these compo- nents are all ``defined'' over a subring RofK, wher eRis a finitely gen- eratedFp-algebr a. This can be used to show thatPis not of finite length, a contradiction.

Note that it is customary to ask that smooth irreducible representa- tions of GL₂(Q_p) also be admissible (meaning thatP^Uis finite-dimensional forevery open compact subgroupU ofG). A corollary of Barthel-Livne and Breuil's classification is that every smooth irreducible Fp-linear representation of GL₂(Q_p) that admits a central character is admissible, and hence theorem A implies that every smooth irreducible F_p-linear representation of GL₂(Q_p) is admissible. In particular, such a rep- resentation also satisfies Schur's lemma: every GL2(Q_p)-equivariant map is a scalar. Our theorem A can also be seen as a special case of Schur's lemma, since p 0

0 p

is a GL₂(Q_p)-equivariant map.

There are (at least) two standard ways of proving Schur's lemma: one way uses admissibility, and the other works for smooth irreducibleE-lin- ear representations of GL2(Q_p), but only if Eis uncountable (see propo- sition 2.11 of [BZ76]). In order to prove theorem A, we cannot simply ex- tend scalars to an uncountable extension ofF_p, as we do not know whether the resulting representation will still be irreducible.

We finish this introduction by pointing out that a few years ago, Hen- niart had sketched a different (and more complicated) argument for the proof of theorem A.

1. Barthel-Livne and Breuil's classification

LetEbe a field of characteristicp. In this section, we recall the explicit classification of smooth irreducible E-linear representations of GL₂(Q_p), admitting a central character.

We denote the centerof GL₂(Q_p) by Z. Ifr50, then Sym^rE² is a rep- resentation of GL2(Fp) which gives rise, by inflation, to a representation of GL2(Zp). We extend it to GL2(Zp)Z by letting p 0

0 p

act trivially.

Consider the representation

ind^GL_GL²₂^(Q_(Z^p_p)Z⁾ Sym^rE²: The Hecke algebra

EndE[GL2(Qp)]ind^GL_GL²₂^(Q_(Z^p_p)Z⁾ Sym^rE²

is isomorphic toE[T] whereTis a Hecke operator, which corresponds to the double class GL2(Zp)Z p 0

0 1

GL2(Zp). If x:Q_p !E is a smooth character, and ifl2E, then let

p(r;l;x)ˆind^GL_GL²₂^(Q_(Z^p_p⁾_)ZSym^rE²

T l (xdet):

This is a smooth representation of GL₂(Q_p), with central character v^rx² (wherev:Q_p !F_pis given bypⁿx07!x0, withx02Z_p). Letm_l:Q_p !E be given by m_lj_Zp ˆ1, and m_l(p)ˆl. Iflˆ 1, then we have two exact sequences:

0!Sp_E(xm_ldet)!p(0;l;x)!xm_ldet!0;

0!xm_ldet!p(p 1;l;x)!Sp_E(xm_ldet)!0;

where the representation Sp_Eis the ``special'' representation with coeffi- cients inE.

THEOREM1.1. If E is algebraically closed, then the smooth irreducible E-linear representations ofGL₂(Q_p), admitting a central character, are as follows:

(1) xdet;

(2) Sp_E(xdet);

(3) p(r;l;x), where r2 f0;. . .;p 1gand(r;l)2 f(0;= 1);(p 1;1)g.

This theorem is proved in [BL95] and [BL94], which treat the case l6ˆ0, and in [Bre03], which treats the caselˆ0.

We now explain what happens ifEis not algebraically closed.

PROPOSITION1.2. IfPis a smooth irreducible E-linear representation of GL₂(Q_p), admitting a central character, then there exists a finite

extension K=E such that (PEK)^ss is a direct sum of K-linear repre- sentations of the type described in theorem1.1.

PROOF. Barthel and LivneÂ's methods show (as is observed in § 5.3 of [PasÏ10]) thatP is a quotient of

Sˆind^GL_GL²₂^(Q_(Zp^p_)Z⁾ Sym^rE²

P(T) (xdet);

forsome integerr2 f0;. . .;p 1g, character x:Q_p !E, and poly- nomial P(Y)2E[Y]. Let K be a splitting field of P(Y), write P(Y)ˆ(Y l1) (Y l_d), and let P_i(Y)ˆ(Y l1) (Y l_i) for

iˆ0;. . .;d. The representationsP_{i 1}(T)S=P_i(T)S are then subquotients

of thep(r;l_i;x), foriˆ1;. . .;d. p

We finish this section by recalling that if l6ˆ0, then the representa- tions p(r;l;x) are parabolic inductions (when lˆ0, they are called su- persingular). Let B₂(Q_p) be the upper triangular Borel subgroup of GL₂(Q_p), letx₁andx₂:Q_p !Ebe two smooth characters, and consider the parabolic induction ind^GL_B2(Q^2(Q_p)^p)(x₁x₂). The following result is proved in [BL94] and [BL95].

THEOREM 1.3. If l2En f0;1g, and if r2 f0;. . .;p 1g, then p(r;l;x)is isomorphic toind^GL_B2(Q^2(Q_p)^p)(xm_1=lxv^rm_l).

2. Proof of the theorem

We now give the proof of theorem A. LetPbe a smooth irreducibleF_p- linear representation of GL2(Q_p). We haveP^(1‡pZp)Id 6ˆ0 (since ap-group acting on aFp-vectorspace always has nontrivial fixed points), so that ifP is irreducible, then (1‡pZ_p)Id acts trivially onP. Ifg2Z_p Id, then g^{p 1}ˆId on P, so that P ˆ _v2FpP^gˆvId. Since P is irreducible, this implies that the elements ofZ_p Id act by scalars.

If f ˆ p 0 0 p

, then forany nonzero polynomial Q(X)2F_p[X], the kernel and image of the map Q(f):P!P are subrepresenta- tions of P. If Q(f)ˆ0 on a nontrivial subspace of P, then f admits an eigenvectorforan eigenvalue l2F_p. This implies that PˆP^fˆlId, so that P does admit a central character. If this is not

the case, then Q(f) is bijective for every nonzero polynomial Q(X)2Fp[X], so that P has the structure of a Fp(X)-vectorspace, and is a Fp(X)-linear smooth irreducible representation of GL2(Q_p), admitting a central character.

Let EˆF_p(X). Proposition 1.2 gives us a finite extension K of E, such that (P_EK)^ss is a direct sum of K-linearrepresentations of the type described in theorem 1.1. The Fp-linearrepresentation underlying (PEK)^ss is isomorphic to P^[K:E], and hence of length [K:E]. We now prove that none of the K-linearrepresentations of the type described in theorem 1.1 are of finite length, when viewed as F_p-linear representations.

LetSbe one such representation, and letl2Kbe the corresponding Hecke eigenvalue. We now construct a subringRofK, which is a finitely generatedF_p-algebra, and anR-linear representationS_Rof GL₂(Q_p), such thatSˆS_RRK.

Ifl2F_p, then theorem 1.1 shows that

Sˆind^GL_GL²₂^(Q_(Z^p_p⁾_)ZSym^rF²_p

T l _FpK(xdet);orSp_FpFpK(xdet);orK(xdet):

We can then takeRˆFp[x(p)¹], andSRˆ ind^GL_GL²₂^(Q_(Z^p_p⁾_)ZSym^rF²_p=(T l) _Fp R(xdet), orSp_FpFpR(xdet), orR(xdet), respectively.

Ifl2=F_p, then by theorem 1.3, we have

Sˆind^GL_B2(Q^2(Q_p)^p)(xm_1=l;xv^rm_l):

We can takeRˆFp[l¹;x(p)¹], and letSR be the set of functionsf 2S with values inR.

In the first case,S_Ris a freeR-module, while in the second case,S_Ris isomorphic as anR-module toC¹(P¹(Q_p);R) and hence also free. In either case, iff2Ris nonzero and not a unit andj2Z, then f^j‡1SRis a proper Fp-linear subrepresentation of f^jSR, so that the underlying Fp-linear representation ofS_Ris not of finite length. SinceS_RS, the underlying F_p-linear representation of S is not of finite length, which is a contra- diction. This finishes the proof of theorem A.

Acknowledgments. I am grateful to C. Breuil, G. Chenevier, P. Col- mez, G. Henniart, F. Herzig, M. Schein, M.-F. VigneÂras and the referee forhelpful comments.

REFERENCES

[BL94] L. BARTHEL- R. LIVNEÂ,Irreducible modular representations ofGL2of a local field, Duke Math. J.,75, no. 2 (1994), pp. 261-292.

[BL95] L. BARTHEL- R. LIVNEÂ,Modular representations ofGL2of a local field:

the ordinary, unramified case, J. NumberTheory,55, no. 1 (1995), pp. 1- [BZ76] I. B27. ERNSÏTEIÆN- A. ZELEVINSKIIÆ, Representations of the groupGL(n;F), where F is a local non-Archimedean field, Russian Math. Surveys,31, no. 3 (1976), pp. 1-68.

[Bre03] C. BREUIL, Sur quelques repreÂsentations modulaires et p-adiques de GL2(Qp): I, Compositio Math.,138, no. 2, (2003), pp. 165-188.

[PasÏ10] V. PASÏKUÅNAS,The image of Colmez's MontreÂal functor, preprint, 2010.

Manoscritto pervenuto in redazione il 23 agosto 2011.