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Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations
Alexandru Ioan Badulescu, Neven Grbac
To cite this version:
Alexandru Ioan Badulescu, Neven Grbac. Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations. Inventiones Mathematicae, Springer Verlag, 2008, 172 (2), pp.383 - 438. �10.1007/s00222-007-0104-8�. �hal-01822255�
GLOBAL JACQUET-LANGLANDS CORRESPONDENCE, MULTIPLICITY ONE AND CLASSIFICATION OF
AUTOMORPHIC REPRESENTATIONS
by Alexandru Ioan BADULESCU1 with an Appendix by Neven GRBAC2
Abstract: In this paper we generalize the local Jacquet-Langlands correspondence to all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one Theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).
Contents
1. Introduction 2
2. Basic facts and notation (local) 5
2.1. Classification ofIrrn (resp. Irrn0) in terms ofDl(resp. Dl0),l≤n 6
2.2. Classification ofDn in terms ofCl,l|n 7
2.3. Local Jacquet-Langlands correspondence 7
2.4. Classification ofD0n in terms ofCl0,l|n. The invariant s(σ0) 8 2.5. Multisegments, order relation, the functionland rigid representations 8
2.6. The involution 9
2.7. The extended correspondence 9
2.8. Unitary representations ofGn 10
2.9. Unitary representations ofG0n 11
2.10. Hermitian representations and an irreducibility trick 12
3. Local results 13
3.1. First results 13
3.2. Transfer ofu(σ, k) 17
3.3. New formulas 18
3.4. Transfer of unitary representations 19
3.5. Transfer of local components of global discrete series 21
4. Basic facts and notation (global) 21
4.1. Discrete series 21
4.2. Cuspidal representations 22
1Alexandru Ioan BADULESCU, Universit´e de Poitiers, UFR Sciences SP2MI, D´epartement de Math´ematiques, T´el´eport 2, Boulevard Marie et Pierre Curie, BP 30179, 86962 FUTUROSCOPE CHASSENEUIL CEDEX
E-mail : badulesc@math.univ-poitiers.fr
2Neven GRBAC, University of Zagreb, Department of Mathematics, Unska 3, 10000 Zagreb, Croatia
E-mail : neven.grbac@zpm.fer.hr
1
4.3. Automorphic representations 23
4.4. Multiplicity one Theorems forGn 23
4.5. The residual spectrum ofGn 23
4.6. Transfer of functions 24
5. Global results 25
5.1. Global Jacquet-Langlands, multiplicity one and strong multiplicity
one for inner forms 25
5.2. A classification of discrete series and automorphic representations of
G0n 30
5.3. Further comments 32
6. L-functions and0-factors 34
7. Bibliography 36
Appendix A. The Residual Spectrum ofGLn over a Division Algebra 39
A.1. Introduction 39
A.2. Normalization of intertwining operators 39
A.3. Poles of Eisenstein series 43
References 45
1. Introduction
The aim of this paper is to prove the global Jacquet-Langlands correspondence and its con- sequences for the theory of representations of the inner forms of GLn over a global field of characteristic zero. In order to define the global Jacquet-Langlands correspondence, it is not sufficient to transfer only square integrable representations as in the classical local theory ([JL], [FL2], [Ro], [DKV]). It would be necessary to transfer at least the local components of global discrete series. This results are already necessary to the global correspondence with a division algebra (which can be locally any inner form). Here we prove, more generally, the transfer of all unitary representations. Then we prove the global Jacquet-Langlands correspondence, which is compatible with this local transfer. As consequences we obtain for inner forms of GLn the multiplicity one Theorem and strong multiplicity one Theorem, as well as a classification of the residual spectrum `a la Moeglin-Waldspurger and unicity of the cuspidal support `a la Jacquet- Shalika. Using these classifications we give counterexamples showing that the global Jacquet- Langlands correspondence for discrete series does not extend well to all unitary automorphic representations.
We give here a list of the most important results, starting with the local study. We would like to point out that the local results in this paper have already been obtained by Tadi´c in [Ta6]
in characteristic zero under the assumption that his conjecture U0holds. After we proved these results here independently of his conjecture (and some of them in any characteristic), S´echerre announced the proof of the conjecture U0 ([Se]). The approach is completely different and we insist on the fact that we do not prove the conjectureU0 here but more particular results which are enough to show the local transfer necessary for the global correspondence.
LetF be a local non-Archimedean field of characteristic zero andDa central division algebra over F of dimension d2. For n∈N∗ set Gn =GLn(F) andG0n =GLn(D). Letν generically denote the character given by the absolute value of the reduced norm on groups likeGn orG0n. Letσ0 be a square integrable representation ofG0n. Ifσ0 is a cuspidal representation, then it corresponds by the local Jacquet-Langlands correspondence to a square integrable representation σ ofGnd. We sets(σ0) =k, wherek is the length of the Zelevinsky segment ofσ. If σ0 is not
cuspidal, we set s(σ0) =s(ρ), whereρis any cuspidal representation in the cuspidal support of σ0, and this does not depend on the choice. We set thenνσ0 =νs(σ0). For anyk∈N∗ we denote then byu0(σ0, k) the Langlands quotient of the induced representation from⊗k−1i=0(νσk−102 −iσ0), and ifα∈]0,12[, we denoteπ0(u0(σ0, k), α) the induced representation fromνσα0u0(σ0, k)⊗νσ−α0 u0(σ0, k).
The representationπ0(u0(σ0, k), α) is irreducible ([Ta2]). LetU0 be the set of all representations of typeu0(σ0, k) orπ0(u0(σ0, k), α) for allG0n,n∈N∗. Tadi´c conjectured in [Ta2] that
(i) all the representations in U0 are unitary;
(ii) an induced representation from a product of representations in U0 is always irreducible and unitary;
(iii) every irreducible unitary representation of G0m, m ∈ N∗, is an induced representation from a product of representations inU0.
The fact that the u0(σ0, k) are unitary has been proved in [BR1] if the characteristic of the base field is zero. In the third Section of this paper we complete the proof of the claim (i) (i.e.
π0(u0(σ0, k), α) are unitary; see Corollary 3.6) and prove (ii) (Proposition 3.9).
We also prove the Jacquet-Langlands transfer for all irreducible unitary representations of Gnd. More precisely, let us writeg0 ↔g ifg∈Gnd,g0∈G0n and the characteristic polynomials of gandg0 are equal and have distinct roots in an algebraic closure ofF. DenoteGnd,dthe set of elements g ∈ Gnd such that there exists g0 ∈ G0n with g0 ↔ g. We denote χπ the function character of an admissible representationπ. We say a representationπofGndisd-compatible if there exists g∈Gnd,dsuch that χπ(g)6= 0. We have (Proposition 3.9):
Theorem. If uis a d-compatible irreducible unitary representation of Gnd, then there exists a unique irreducible unitary representation u0 of G0n and a unique signε∈ {−1,1}such that
χu(g) =εχu0(g0) for all g∈Gnd,d andg0↔g.
It is Tadi´c who first pointed out ([Ta6]) that this should hold if his conjectureU0were true. The signεand an explicit formula foru0 may be computed. See for instance Subsection 3.3.
The fifth Section contains global results. Let us use the Theorem above to define a map
|LJ|:u7→u0 from the set of irreducible unitaryd-compatible representations ofGndto the set of irreducible unitary representations ofG0n.
Let now F be a global fieldof characteristic zeroandD a central division algebra overF of dimensiond2. Letn∈N∗. Set A=Mn(D). For each placev of F letFv be the completion of F atv and set Av =A⊗Fv. For every place v of F, Av 'Mrv(Dv) for some positive integer rv and some central division algebraDv of dimensiond2v overFv such thatrvdv=nd. We will fix once and for all an isomorphism and identify these two algebras. We say thatMn(D) is split at a placev ifdv= 1. The setV of places whereMn(D) is not split is finite. We assume in the sequel thatV does not contain any infinite place.
Let Gnd(A) be the group of ad`eles ofGLnd(F), andG0n(A) the group of ad`eles ofGLn(D).
We identify Gnd(A) withMnd(A)× and G0n(A) withA(A)×.
LetZ(A) be the center ofGnd(A). Ifωis a smooth unitary character ofZ(A) trivial onZ(F), let L2(Z(A)Gnd(F)\Gnd(A);ω) be the space of classes of functions f defined on Gnd(A) with values in C such thatf is left invariant underGnd(F),f(zg) =ω(z)f(g) for all z∈Z(A) and almost allg∈Gnd(A) and|f|2 is integrable overZ(A)Gnd(F)\Gnd(A). The groupGnd(A) acts by right translations on L2(Z(A)Gnd(F)\Gnd(A);ω). We call adiscrete seriesof Gnd(A) an
irreducible subrepresentation of such a representation (for any smooth unitary character ω of Z(A) trivial onZ(F)). We adopt the analogous definition for the groupG0n(A).
DenoteDSnd(resp. DSn0) the set of discrete series ofGnd(A) (resp. G0n(A)). Ifπis a discrete series ofGnd(A) orG0n(A), andvis a place ofF, we denoteπv the local component ofπ at the place v. We will say that a discrete seriesπof Gnd(A) isD-compatibleifπv isdv-compatible for all placesv∈V.
If v ∈V, the Jacquet-Langlands correspondence betweendv-compatible unitary representa- tions ofGLnd(Fv) andGLrv(Dv) will be denoted|LJ|v. Recall that ifv /∈V, we have identified the groups GLrv(Dv) andGLnd(Fv). We have the following (Theorem 5.1):
Theorem. (a)There exists a unique injective mapG:DSn0 →DSndsuch that, for allπ0 ∈DSn0, we have G(π0)v =π0v for every placev /∈V. For every v∈V,G(π0)v isdv-compatible and we have |LJ|v(G(π0)v) =πv0. The image ofGis the set of D-compatible elements ofDSnd.
(b) One has multiplicity one and strong multiplicity one Theorems for the discrete spectrum of G0n(A).
Since the original work of [JL] (see also [GeJ]), global correspondences with division algebras under some conditions (on the division algebra or on the representation to be transferred) have already been carried out (sometimes not explicitly stated) at least in [Fl2], [He], [Ro], [Vi], [DKV], [Fli] and [Ba4]. They were using simple forms of the trace formula. For the general result obtained here these formulas are not sufficient. Our work is heavily based on the comparison of the general trace formulas for G0n(A) andGnd(A) carried out in [AC]. The reader should not be misled by the fact that here we use directly the simple formula Arthur and Clozel obtained in their over 200 pages long work. Their work overcomes big global difficulties and together with methods from [JL] and [DKV] reduces the global transfer of representations to local problems.
Let us explain now what are the main extra ingredients required for application of the spectral identity of [AC] in the proof of the theorem. The spectral identity as stated in [AC] is roughly speaking (and after using the multiplicity one theorem for Gnd(A)) of the type
Xtr(σI)(f) +X
λJtr(MJπJ)(f) =X
m0itr(σ0i)(f0) +X
λ0jtr(Mj0πj0)(f0)
where λJ and λ0j are certain coefficients, σI (resp. σ0i) are discrete series of Gnd(A) (resp. of G0n(A) of multiplicity m0i),πJ (resp. π0j) are representations ofGnd(A) (resp. ofG0n(A)) which are induced from discrete series of proper Levi subgroups andMJandMj0are certain intertwining operators. As forf andf0, they are functions with matching orbital integrals.
The main step in proving the theorem is to choose a discrete series σ0 ofG0n(A) and to use the spectral identity to defineG(σ0). The crucial result is the local transfer of unitary represen- tations (Proposition 3.9.c of this paper) which allows to ”globally” transfer the representations from the left side to the right side. This gives the correspondence whenn= 1 as in [JL] or [Vi].
The trouble when n >1 is that we do not know much about the operators Mj0. We overcome this by induction over n. Then the Proposition 3.9.b shows thatπj0 are irreducible. This turns out to be enough to show that the contribution of σ0 to the equality cannot be canceled by contributions from properly induced representations.
In the sequel of the fifth Section we give a classification of representations ofG0n(A). We define the notion of a basic cuspidal representation for groups of typeG0k(A) (see Proposition 5.5 and the sequel). These basic cuspidal representations are all cuspidal. Neven Grbac will show in his Appendix that these are actually the only cuspidal representations. Then the residual discrete
series ofG0n(A) are obtained from cuspidal representations in the same way the residual discrete series of GLn(A) are obtained from cuspidal representations in [MW2]. This classification is obtained directly by transfer from the Moeglin-Waldspurger classification forGn.
Moreover, for any (irreducible) automorphic representation π0 of G0n, we know that ([La]) there exists a couple (P0, ρ0) where P0 is a parabolic subgroup of G0n containing the group of upper triangular matrices and ρ0 is a cuspidal representation of the Levi factor L0 ofP0 twisted by a real non ramified character such thatπ0 is a constituent (in the sense of [La]) of the induced representation fromρ0 toG0n with respect toP0. We prove (Proposition 5.7 (c)) that this couple (ρ0, L0) is unique up to conjugation. This result is an analogue forG0n of Theorem 4.4 of [JS].
The last Section is devoted to the computation ofL-functions,0-factors (in the sense of [GJ]) and their behavior under the local transfer of irreducible (especially unitary) representations.
The behavior of the -factors then follows. These calculations are either well known or trivial, but we feel it is natural to give them explicitly here. TheL-functions and0-factors in question are preserved under the correspondence for square integrable representations. In general, 0- factors (but not L-functions) are preserved under the correspondence for irreducible unitary representations.
In the Appendix Neven Grbac completes the classification of the discrete spectrum by showing that all the representations except the basic cuspidal ones are residual. His approach applies the Langlands spectral theory.
The essential part of this work has been done at the Institute for Advanced Study, Princeton, during the year 2004 and I would like to thank the Institute for the warm hospitality and support. They were expounded in a preprint from the beginning of 2006. The present paper contains exactly the same local results as that preprint. Two major improvements obtained in 2007 concern the global results. The first one is the proof of the fact that any discrete series of the inner form transfers (based on a better understanding of the trace formula from [AC]). The second is a complete classification of the residual spectrum thanks to the Appendix of Neven Grbac.
The research at the IAS has been supported by the NSF fellowship no. DMS-0111298. I would like to thank Robert Langlands and James Arthur for useful discussions about global representations; Marko Tadi´c and David Renard for useful discussions on the local unitary dual;
Abderrazak Bouaziz who explained to me the intertwining operators. I would like to thank Guy Henniart and Colette Moeglin for the interest they showed for this work and their invaluable advices. I thank Neven Grbac for his Appendix where he carries out the last and important step of the classification, and for his remarks on the manuscript. Discussions with Neven Grbac have been held during our stay at the Erwin Schr¨odinger Institute in Vienna and I would like to thank here Joachim Schwermer for his invitation.
2. Basic facts and notation (local)
In the sequelNwill denote the set of non negative integers andN∗ the set of positive integers.
A multiset is a set with finite repetitions. Ifx ∈ R, then [x] will denote the biggest integer inferior or equal tox.
LetF be a non-Archimedean local field andDa central division algebra of a finite dimension overF. Then the dimension ofDoverF is a squared2,d∈N∗. Ifn∈N∗, we setGn=GLn(F) and G0n = GLn(D). From now on we identify a smooth representation of finite length with its equivalence class, so we will consider two equivalent representations as being equal. By a character of Gn we mean a smooth representation of dimension one of Gn. In particular a character is not unitary unless we specify it. Let σbe an irreducible smooth representation of
Gn. We sayσissquare integrableifσis unitary and has a non-zero matrix coefficient which is square integrable modulo the center of Gn. We sayσisessentially square integrableifσ is the twist of a square integrable representation by a character of Gn. We say σ is cuspidal if σhas a non-zero matrix coefficient which has compact support modulo the center ofGn. In particular a cuspidal representation is essentially square integrable.
For alln∈N∗ let us fix the following notation:
Irrn is the set of smooth irreducible representations ofGn,
Dn is the subset of essentially square integrable representations inIrrn, Cn is the subset of cuspidal representations inDn,
Irrun (resp. Dun, Cnu) is the subset of unitary representations inIrrn (resp. Dn,Cn), Rn is the Grothendieck group of admissible representations of finite length ofGn,
νis the character ofGndefined by the absolute value of the determinant (notation independent of n– this will lighten the notation and cause no ambiguity in the sequel).
For any σ ∈ Dn, there is a unique couple (e(σ), σu) such that e(σ) ∈ R, σu ∈ Dnu and σ=νe(σ)σu.
We will systematically identify π∈Irrn with its image inRn and considerIrrn as a subset of Rn. ThenIrrn is aZ-basis of the Z-moduleRn.
Ifn∈N∗and (n1, n2, ..., nk) is an ordered set of positive integers such thatn=Pk
i=1nithen the subgroup L of Gn consisting of block diagonal matrices with blocks of sizes n1, n2, ..., nk
in this order from the left upper corner to the right lower corner is called a standard Levi subgroupofGn. The groupLis canonically isomorphic with the product×ki=1Gni, and we will identify these two groups. Then the notation Irr(L),D(L), C(L),Du(L),Cu(L),R(L) extend in an obvious way toL. In particularIrr(L) is canonically isomorphic to×ki=1Irrni and so on.
We denoteindGLn the normalized parabolic induction functor where it is understood that we induce with respect to the parabolic subgroup of Gn containing L and the subgroup of upper triangular matrices. Then indGLn extends to a group morphismiGLn:R(L)→ Rn. If πi ∈ Rni
for i ∈ {1,2, ..., k} and n = Pk
i=1ni, we denote π1 ×π2 ×...×πk or abridged Qk
i=1πi the representation
indG×nk
i=1Gni⊗ki=1σi
ofGn. Let πbe a smooth representation of finite length of Gn. If distinction between quotient, subrepresentation and subquotient of π is not relevant, we consider π as an element of Rn
(identification with its class) with no extra explanation.
If g∈Gn for somen, we sayg isregular semisimpleif the characteristic polynomial ofg has distinct roots in an algebraic closure of F. Ifπ∈ Rn, then we let χπ denote the function character ofπ, as a locally constant map, stable under conjugation, defined on the set of regular semisimple elements ofGn.
We adopt the same notation adding a sign0 forG0n: Irr0n,D0n,Cn0,Irr0nu,D0nu,Cn0u,R0n. There is a standard way of defining the determinant and the characteristic polynomial for elements of G0n, in spite of D being non commutative (see for example [Pi] Section 16). If g0∈G0n, then the characteristic polynomial ofg0 has coefficients inF, it is monic and has degree nd. The definition of a regular semisimple element ofG0n is then the same as forGn. Ifπ∈ R0n, we let againχπ be the function character ofπ. As forGn, we will denoteν the character ofG0n given by the absolute value of the determinant (there will be no confusion with the one onGn).
2.1. Classification of Irrn (resp. Irrn0) in terms of Dl (resp. Dl0), l ≤n. Let π∈Irrn. There exists a standard Levi subgroup L =×ki=1Gni of Gn and, for all 1 ≤i ≤ k, ρi ∈ Cni,
such that π is a subquotient ofQk
i=1ρi. The non-ordered multiset of cuspidal representations {ρ1, ρ2, ...ρk}is determined byπand is calledthe cuspidal support of π.
We recall the Langlands classification which takes a particularly nice form on Gn. Let L=
×ki=1Gni be a standard Levi subgroup ofGnandσ∈ D(L) =×ki=1Dni. Let us writeσ=⊗ki=1σi
withσi∈ Dni. For eachi, writeσi=νeiσui, whereei∈Randσui ∈ Duni. Letpbe a permutation of the set {1,2, ..., k} such that the sequence ep(i) is decreasing. Let Lp = ×ki=1Gnp(i) and σp = ⊗ki=1σp(i). Then indGLpnσp has a unique irreducible quotient π and π is independent of the choice of punder the condition that (ep(i))1≤i≤k is decreasing. Soπ is defined by the non ordered multiset {σ1, σ2, ..., σk}. We write thenπ=Lg(σ). Everyπ∈Irrn is obtained in this way. Ifπ∈Irrn andL=×ki=1GniandL0=×kj=10 Gn0j are two standard Levi subgroups ofGn, if σ=⊗ki=1σi, withσi∈ Dni, andσ0=⊗kj=10 σ0j, withσ0j∈ Dn0j, are such thatπ=Lg(σ) =Lg(σ0), thenk=k0 and there exists a permutationpof{1,2, ..., k}such thatn0j =np(i) andσj0 =σp(i). So the non ordered multiset{σ1, σ2, ..., σk} is determined byπand it is calledthe essentially square integrable support of πwhich we abridge asthe esi-support of π.
An element S =iGLnσof Rn, withσ ∈ D(L), is called astandard representation ofGn. We will often writeLg(S) forLg(σ).The setBn of standard representations ofGn is a basis of Rn and the mapS7→Lg(S) is a bijection fromBnontoIrrn. All these results are consequences of the Langlands classification (see [Ze] and [Rod]). We also have the following result: if for all π∈Irrn we writeπ=Lg(S) for some standard representationSand then for allπ0∈Irrn\{π}
we set π0< π if and only ifπ0 is a subquotient ofS, then we obtain a well defined partial order relation onIrrn.
The same definitions and theory, including the order relation, hold for G0n (see [Ta2]). The set of standard representations ofG0n is denoted here byBn0.
For Gn or G0n we have the following Proposition, where σ1 and σ2 are essentially square integrable representations:
Proposition 2.1. (a) The representation Lg(σ1)×Lg(σ2) contains Lg(σ1×σ2) as a subquotient with multiplicity 1.
(b)Ifπis another irreducible subquotient ofLg(σ1)×Lg(σ2), thenπ < Lg(σ1× σ2). In particular, if Lg(σ1)×Lg(σ2) is reducible, it has at least two different subquotients.
For Gn, assertion (a) is proven in its dual form in [Ze] (Proposition 8.4). It is proven in its present form in [Ta2] (Proposition 2.3) for the more general case of G0n. Assertion (b) is then obvious because of the definition (here) of the order relation, and since any irreducible subquotient ofLg(σ1)×Lg(σ2) is also an irreducible subquotient ofσ1×σ2.
2.2. Classification of Dn in terms of Cl, l|n. Let k and l be two positive integers and set n=kl. Letρ∈ Cl. Then the representationQk−1
i=0 νiρhas a unique irreducible quotientσ. σ is an essentially square integrable representation ofGn. We write thenσ=Z(ρ, k). Everyσ∈ Dn
is obtained in this way andl,kandρare determined byσ. This may be found in [Ze].
In general, a set X ={ρ, νρ, ν2ρ, ..., νa−1ρ}, ρ∈ Cb, a, b∈N∗, is called asegment,aisthe lengthof the segmentX andνa−1ρis theendingofX.
2.3. Local Jacquet-Langlands correspondence. Letn∈N∗. Letg∈Gndandg0∈G0n. We say thatgcorrespondstog0ifgandg0are regular semisimple and have the same characteristic polynomial. We shortly write theng↔g0.
Theorem 2.2. There is a unique bijectionC:Dnd→ Dn0 such that for allπ∈ Dnd
we have
χπ(g) = (−1)nd−nχC(π)(g0) for allg∈Gnd andg0 ∈G0n such that g↔g0.
For the proof, see [DKV] if the characteristic of the base fieldF is zero and [Ba2] for the non zero characteristic case. I should quote here also the particular cases [JL], [Fl2] and [Ro] which contain some germs of the general proof in [DKV].
We identify the centers ofGnd andG0n via the canonical isomorphism. Then the correspon- denceCpreserves central characters so in particularσ∈ Dundif and only ifC(σ)∈ Dn0u.
If L0=×ki=1G0ni is a standard Levi subgroup ofG0n we say that the standard Levi subgroup L=×ki=1GdniofGndcorrespondstoL0. Then the Jacquet-Langlands correspondence extends in an obvious way to a bijective correspondenceD(L) toD0(L0) with the same properties. We will denote this correspondence by the same letterC. A standard Levi subgroupLofGn corresponds to a standard Levi subgroup or G0r if and only if it is defined by a sequence (n1, n2, ..., nk) such that eachni is divisible by d. We then say thatLtransfers.
2.4. Classification of D0n in terms of Cl0, l|n. The invariant s(σ0). Let l be a positive integer andρ0∈ Cl0. Thenσ=C−1(ρ0) is an essentially square integrable representation ofGld. We may writeσ=Z(ρ, p) for somep∈N∗and someρ∈ Cld
p. Set thens(ρ0) =pandνρ0 =νs(ρ0). Let k and l be two positive integers and set n =kl. Let ρ0 ∈ C0l. Then the representation Qk−1
i=0 νρi0ρ0 has a unique irreducible quotientσ0. σ0 is an essentially square integrable represen- tation of G0n. We write thenσ0 =T(ρ0, k). Everyσ0 ∈ Dn0 is obtained in this way andl,kand ρ0 are determined byσ0. We set thens(σ0) =s(ρ0). For this classification see [Ta2].
A setS0={ρ0, νρ0ρ0, νρ20ρ0, ..., νρa−10 ρ0},ρ0 ∈ Cb0,a, b∈N∗, is called asegment,aisthe length of S0 andνρa−10 ρ0 is theendingofS0.
2.5. Multisegments, order relation, the function l and rigid representations. Here we will give the definitions and results in terms of groupsGn, but one may replaceGn byG0n. We have seen (Section 2.2 and 2.4) that to eachσ ∈ Dn one may associate a segment. A multiset of segments is called a multisegment. IfM is a multisegment, the multiset of endings of its elements (see Section 2.2 and 2.4 for the definition) is denotedE(M).
If π∈Gn, the multiset of the segments of the elements of the esi-support ofπis a multiseg- ment; we will denote it byMπ. Mπ determinesπ. The reunion with repetitions of the elements of Mπ is the cuspidal support ofπ.
Two segments S1 andS2 are said to belinkedifS1∪S2 is a segment different fromS1and S2. IfS1 andS2 are linked, we say they areadjacent ifS1∩S2= Ø.
LetM be a multisegment, and assumeS1 andS2 are two linked segments inM. Let M0 be the multisegment defined by
-M0= (M∪ {S1∪S2} ∪ {S1∩S2})\{S1, S2}ifS1andS2are not adjacent (i.e. S1∩S26= Ø), and
-M0= (M ∪ {S1∪S2})\{S1, S2}ifS1 andS2 are adjacent (i.e. S1∩S2= Ø).
We say that we made anelementary operationonM to getM0, or thatM0 was obtained from M by an elementary operation. We then sayM0 is inferior toM. It is easy to verify this extends by transitivity to a well defined partial order relation<on the set of multisegments of Gn. The following Proposition is a result of [Ze] (Theorem 7.1) forGn and [Ta2] (Theorem 5.3) forG0n.
Proposition 2.3. If π, π0 ∈Irrn, thenπ < π0 if and only ifMπ< Mπ0.
Ifπ < π0, then the cuspidal support ofπequals the cuspidal support ofπ0.
Define a functionlon the set of multisegments as follows: ifM is a multisegment, thenl(M) is the maximum of the lengths of the segments in M. If π ∈ Irrn, set l(π) = l(Mπ). The following Lemma is obvious:
Lemma 2.4. If M0 is obtained fromM by an elementary operation then l(M)≤ l(M0)andE(M0)⊆E(M). As a function on Irrn,l is decreasing.
The next important Proposition is also a result from [Ze] and [Ta2]:
Proposition 2.5. Letπ∈Irrk andπ0 ∈Irrl. If for allS∈Mπ andS0 ∈Mπ0 the segmentsS andS0 are not linked, thenπ×π0 is irreducible.
There is an interesting consequence of this last Proposition. Let l∈N∗ andρ∈ Cl. We will call the setX ={νaρ}a∈Zaline, the line generated byρ. Of courseX is also the line generated byνρ for example. Ifπ∈Irrn, we sayπisrigidif the set of elements of the cuspidal support of πis included in a single line. As a consequence of the previous Proposition we have the
Corollary 2.6. Let π ∈ Irrn. Let X be the set of the elements of the cuspidal support of π. If {D1, D2, ..., Dm} is the set of all the lines with which X has a non empty intersection, then one may write in the unique (up to permutation) way π=π1×π2×...×πmwithπi rigid irreducible and the set of elements of the cuspidal support ofπi included in Di,1≤i≤m.
We will sayπ=π1×π2×...×πmis thestandard decompositionofπin a product of rigid representations (this is onlythe shortestdecomposition ofπin a product of rigid representations, but there might exist finer ones).
The same holds for G0n.
2.6. The involution. Aubert defined in [Au] an involution (studied too by Schneider and Stuh- ler in [ScS]) of the Grothendieck group of smooth representations of finite length of a reductive group over a local non-Archimedean field. The involution sends an irreducible representation to an irreducible representation up to a sign. We specialize this involution to Gn, resp. G0n, and denote itin, resp. i0n. We will writei andi0 when the index is not relevant or it is clearly un- derstood. With this notation we have the relationi(π1)×i(π2) =i(π1×π2), i.e. “the involution commutes with the parabolic induction”. The same holds fori0. The reader may find all these facts in [Au].
If π ∈ Irrn, then one and only one among i(π) and −i(π) is an irreducible representation.
We denote it by |i(π)|. We denote|i| the involution of Irrn defined byπ 7→ |i(π)|. The same facts and definitions hold fori0.
The algorithm conjectured by Zelevinsky for computing the esi-support of |i(π)| from the esi-support of π when π is rigid (and hence more generally forπ ∈Irrn, cf. Corollary 2.6) is proven in [MW1]. The same facts and algorithm hold for|i0| as explained in [BR2].
2.7. The extended correspondence. The correspondenceC−1may be extended in a natural way to a correspondence LJ between the Grothendieck groups. Let S0 = iGL00nσ0 ∈ B0n, where L0 is a standard Levi subgroup ofG0n and σ0 an essentially square integrable representation of L0. Set Mn(S0) = iGLndC−1(σ0), where L is the standard Levi subgroup ofGnd corresponding to L0. Then Mn(S0) is a standard representation of Gnd and Mn realizes an injective map from Bn0 into Bnd. Define Qn : Irrn0 →Irrnd byQn(Lg(S0)) =Lg(Mn(S0)). If π10 < π02, then Qn(π01)< Qn(π20). SoQn induces onIrr(G0n), by transfer fromGnd, an order relation<<which is stronger than <.
Let LJn :Rnd→ R0n be the Z-morphism defined onBnd by setting LJn(Mn(S0)) =S0 and LJn(S) = 0 ifS is not in the image ofMn.
Theorem 2.7. (a) For all n∈N∗,LJn is the unique map from Rnd to R0n such that for all π∈ Rndwe have
χπ(g) = (−1)nd−nχLJn(π)(g0) for allg↔g0.
(b)The map LJn is a surjective group morphism.
(c)One has
LJn(Qn(π0)) =π0+ X
πj0<<π0
bjπ0j
wherebj∈Zandπ0j ∈Irr0n. (d)One has
LJn◦ind= (−1)nd−ni0n◦LJn.
See [Ba4]. We will often drop the index and write only Q,M andLJ. LJmay be extended in an obvious way to standard Levi subgroups. For a standard Levi subgroup L0 ofG0n which correspond to a standard Levi subgroupLofGnd we haveLJ◦iGLnd =iGL00n◦LJ.
We will say that π ∈ Rnd is d-compatible if LJn(π) 6= 0. This means that there exists a regular semisimple element g of Gnd which corresponds to an element of G0n and such that χπ(g)6= 0. A regular semisimple element ofGndcorresponds to an element of G0n if and only if its characteristic polynomial decomposes into irreducible factors with the degrees divisible byd.
So our definition depends only on d, not onD. A product of representations isd-compatible if and only if each factor isd-compatible.
2.8. Unitary representations ofGn. We are going to use the wordunitaryforunitarizable.
Letk,lbe positive integers and set kl=n.
Letρ∈ Cland setσ=Z(ρ, k). Thenσis unitary if and only ifνk−21ρis unitary. We set then ρu=νk−12 ρ∈ Clu and we writeσ=Zu(ρu, k). From now on, anytime we writeσ=Zu(ρ, k), it is understood thatσandρare unitary.
Now, ifσ∈ Dlu, we set
u(σ, k) =Lg(
k−1Y
i=0
νk−21−iσ).
The representationu(σ, k) is an irreducible representation ofGn. Ifα∈]0,12[, we moreover set
π(u(σ, k), α) =ναu(σ, k)×ν−αu(σ, k).
The representationπ(u(σ, k), α) is an irreducible representation ofG2n (by Proposition 2.5).
Let us recall the Tadi´c classification of unitary representations in [Ta1].
LetU be the set of all the representationsu(σ, k) andπ(u(σ, k), α) wherek, l range overN∗, σ ∈ Cl and α ∈]0,12[. Then any product of elements of U is irreducible and unitary. Every irreducible unitary representation π of some Gn, n∈ N∗, is such a product. The non ordered multiset of the factors of the product are determined byπ.
The fact that a product of irreducible unitary representations is irreducible is due to Bernstein ([Be]).
Tadi´c computed the decomposition of the representationu(σ, k) in the basisBn ofRn.
Proposition 2.8. ([Ta4]) Let σ = Z(ρ, l) and k ∈ N∗. Let Wkl be the set of permutationsw of {1,2, ..., k}such that w(i) +l ≥ifor all i∈ {1,2, ..., k}. Then we have:
u(σ, k) =ν−k+l2 ( X
w∈Wkl
(−1)sgn(w) Yk i=1
Z(νiρ, w(i) +l−i)).
One can also compute the dual ofu(σ, k).
Proposition 2.9. Let σ=Zu(ρu, l)andk∈N∗. Ifτ =Zu(ρu, k), then
|i(u(σ, k))|=u(τ, l).
This is the Theorem 7.1 iii) [Ta1], and also a consequence of [MW1].
2.9. Unitary representations of G0n. Let k, l ∈ N∗ and set n = kl. Let ρ ∈ Cl0 and σ0 = T(ρ0, k) ∈ Dn0. As for Gn, one has σ0 ∈ D0un if and only if ν
k−1 2
ρ0 ρ0 is unitary; we set then ρ0u=νk−21ρ0 and writeσ0 =Tu(ρ0u, k).
If nowσ0 ∈ D0lu, we set
u0(σ0, k) =Lg(
k−1Y
i=0
νk−
1 2 −i σ0 σ0) and
¯
u(σ0, k) =Lg(
k−1Y
i=0
νk−12 −iσ0).
The representationsu0(σ0, k) and ¯u(σ0, k) are irreducible representations ofG0n. If moreoverα∈]0,12[, we set
π(u0(σ0, k), α) =νασ0u0(σ0, k)×νσ−α0 u0(σ0, k).
The representationπ(u0(σ0, k), α) is an irreducible representation ofG02n(cf. [Ta2]; a consequence of the (restated) Proposition 2.5 here).
We have the formulas:
(2.1) u(σ¯ 0, ks(σ0)) =
s(σ0)
Y
i=1
νi−s(σ0)+12 u0(σ0, k);
and, for all integers 1≤b≤s(σ0)−1,
(2.2) ¯u(σ0, ks(σ0) +b) = ( Yb i=1
νi−b+12 u0(σ0, k+ 1))×(
s(σ0)−b
Y
j=1
νj−s(σ
0)−b+1
2 u0(σ0, k)), with the convention that we ignore the second product ifk= 0.
The products are irreducible, by Proposition 2.5, because the segments appearing in the esi- support of two different factors are never linked. The fact that the product is indeed ¯u(σ0, ks(σ0)) (and resp. ¯u(σ0, ks(σ0) +b)) is then clear by Proposition 2.1. This kind of formulas has been used (at least) in [BR1] and [Ta6].
The representationsu0(σ0, k) and ¯u(σ0, k) are known to be unitary at least in zero characteristic ([Ba4] and [BR1]).
One has
Proposition 2.10. Let σ0=Zu(ρ0u, l)andk∈N∗. If τ0=Zu(ρ0u, k), then (a)|i0(u0(σ0, k))|=u0(τ0, l)and
(b)|i0(¯u(σ0, ks(σ0)))|= ¯u(τ0, ls(σ0)).
Proof. The claim (a) is a direct consequence of [BR2]. For the claim (b), it is enough to use the relation 2.1, the claim (a) here and the fact that i0 commutes with parabolic induction.
2.10. Hermitian representations and an irreducibility trick. Ifπ∈Irr0n, writeh(π) for the complex conjugated representation of the contragredient ofπ. A representationπ∈Irrn0 is called hermitianifπ=h(π) (we recall, to avoid confusion, that here we use “=” for the usual
“equivalent”). A unitary representation is always hermitian. If A = {σi}1≤i≤k is a multiset of essentially square integrable representations of some G0li, we define the multiset h(A) by h(A) = {h(σi)}1≤i≤k. If π ∈Irr0n and x ∈R, then h(νxπ) = ν−xh(π), so if σ0 ∈ D0l and we write σ0 =νeσ0u withe∈Randσ0u∈ D0ul , thenh(σ0) =ν−eσ0u∈ D0l. An easy consequence of Proposition 3.1.1 in [Ca] is the
Proposition 2.11. If π ∈Irr0n, and A is the esi-support of π, then h(A) is the esi-support of h(π). In particular, π is hermitian if and only if the esi-support A of πsatisfiesh(A) =A.
Let us give a Lemma.
Lemma 2.12. Letπ1∈Irrn01 andπ2∈Irr0n2 and assumeh(π1)6=π2. Then there exists ε > 0 such that for all x ∈]0, ε[ the representation ax = νxπ1×ν−xπ2 is irreducible, but not hermitian.
Proof. For allx∈RletAx be the esi-support ofνxπ1 andBxbe the esi-support ofν−xπ2. Then the setX ofx∈Rsuch that Ax∩h(Ax)6=∅or Bx∩h(Bx)6=∅ is finite (it is enough to check the central character of the representations in these multisets). The setY ofx∈Rsuch that the cuspidal supports of Ax and Bx have a non empty intersection is finite too. Now, if x∈R\Y,axis irreducible by the Proposition 2.5. Assume moreoverx /∈X. Asaxis irreducible, if it were hermitian one should haveh(Ax)∪h(Bx) =Ax∪Bx(where the reunions are to be taken with multiplicities, as reunions of multisets) by the Proposition 2.11. But ifAx∩h(Ax) =∅and Bx∩h(Bx) =∅, then this would lead toh(Ax) =Bx, and hence toh(π1) =π2which contradicts
the hypothesis.
We now state our irreducibility trick.
Proposition 2.13. Let u0i ∈ Irrn0ui, i ∈ {1,2, ..., k}. If, for all i ∈ {1,2, ..., k}, u0i×u0i is irreducible, thenQk
i=1u0i is irreducible.
Proof. There existsε >0 such that for all i∈ {1,2, ..., k} the cuspidal supports of νxu0i and ν−xu0i are disjoint for all x ∈]0, ε[. Then, for all i ∈ {1,2, ..., k}, for all x∈]0, ε[, the representation νxu0i ×ν−xu0i is irreducible. As, by hypothesis, u0i×u0i is irreducible and unitary, the representation νxu0i×ν−xu0i is also unitary for allx∈]0, ε[ (see for example [Ta3], Section (b)). So Qk
i=1νxu0i×ν−xu0i is a sum of unitary representations. But we have (in the Grothendieck group)
Yk i=1
(νxu0i×ν−xu0i) = (νx Yk i=1
u0i)×(ν−x Yk i=1
u0i).
If Qk
i=1u0i were reducible, then it would contain at least two different unitary subrepresen- tationsπ1and π2 (Proposition 2.1). But then, for somex∈]0, ε[, (νxQk
i=1u0i)×(ν−xQk
i=1u0i)
contains an irreducible, but not hermitian, subquotient of the form νxπ1×ν−xπ2 (by Lemma 2.12). This subquotient would be non-unitary which contradicts our assumption.
3. Local results
3.1. First results. Let σ0 ∈ D0un and set σ =C−1(σ0)∈ Dund. Write σ0 = Tu(ρ0, l) for some l ∈ N∗, l|n and ρ0 ∈ Cun
l. As C−1(ρ0) ∈ Dund l
we may write C−1(ρ0) = Zu(ρ, s(σ0)) for some ρ ∈ Cund
ls(σ0). We set l0 = ls(σ0). Then we have σ = Zu(ρ, l0) (means one can recover the cuspidal support of σ from the cuspidal support of σ0; it is a consequence of the fact that the correspondence commutes with the Jacquet functor; the original proof for square integrable representations is [DKV], Theorem B.2.b).
Let k be a positive integer and set k0 = ks(σ0). Let H be the group of permutationsw of {1,2, ..., k0}such that s(σ0)|w(i)−i for alli∈ {1,2, ..., k0}. For the meaning ofWkl andWkl00 in the following, see Proposition 2.8.
This is Lemma 3.1 in [Ta5]:
Lemma 3.1. If w ∈H, then for each j ∈ {1,2, ..., s(σ0)}, the set of elements of {1,2, ..., k0} equal to j mods(σ0)is stable under w, andwinduces a permutation wj of{1,2, ..., k}defined by the fact that, ifw(as(σ0) +j) =bs(σ0) +j thenwj(a+ 1) =b+ 1. The mapw7→(w1, w2, ..., ws(σ0))is an isomorphism of groups fromH to (Sk)s(σ0). One has w∈ H∩Wkl00 if and only if for all j, wj ∈Wkl. Moreover, sgn(w) =Qs(σ0)
j=1 sgn(wj).
We have the following:
Theorem 3.2. (a) One has
LJ(u(σ, k0)) = ¯u(σ0, k0).
(b)The induced representation u(σ¯ 0, k0)×u(σ¯ 0, k0) is irreducible.
(c)We have the character formula
¯
u(σ0, k0) =ν−k0+l2 0+s(σ
0)−1
2 ( X
w∈H∩Wk0l0
(−1)sgn(w)
k0
Y
i=1
T(νiρ0,w(i)−i s(σ0) +l)).
Proof. (a) Letτ0 =Tu(ρ0, k) and setτ=C−1(τ0). For the same reasons as explained forσ, we haveτ=Zu(ρ, k0).
We apply Theorem 2.7 (c) to ¯u(σ0, k0) and ¯u(τ0, l0). We get (3.1) LJ(u(σ, k0)) = ¯u(σ0, k0) + X
π0j<<¯u(σ0,k0)
bjπ0j
and
(3.2) LJ(u(τ, l0)) = ¯u(τ0, l0) + X
τq0<<¯u(τ0,l0)
cqτq0 We want to show that all the bj vanish.
Let us write the dual equation to 3.1 (cf. Theorem 2.7 (d)). As |i(u(σ, k0))| = u(τ, l0) (Proposition 2.9) and |i0(¯u(σ0, k0))|= ¯u(τ0, l0) (Proposition 2.10), we obtain:
