Standard Module Conjecture

a r X i v :m a t h /0603395v 1 [m a t h .R T ] 16 M a r 2006STANDARD MODULE CONJECTUREV.Heiermann and G.Mui ´c Abstract.Let G be a quasi-split p -adic group.Under the assumption that the local coefficients C ψdefined with respect to ψ-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber,we show that the standard module conjecture is true,which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.Let F be a non archimedean local field of characteristic 0.Let G be the group of points of a quasi-split connected reductive F -group.Fix a F -Borel subgroup B =T U of G and a maximal F -split torus T 0in T .If M is any semi-standard F -Levi subgroup of G ,a standard parabolic subgroup of M will be a F -parabolic subgroup of M which contains B ∩M .Denote by W the Weyl group of G defined with respect to T 0and by w G 0the longest element in W .After changing the splitting in U ,for any generic repre-sentation πof G ,one can always find a non degenerate character ψof U ,which is compatible with w G 0,such that πis ψ-generic [Sh2,section 3].For any semi-standard Levi-subgroup M of G ,we will still denote by ψthe restriction of ψto M ∩U .It is compatible with w M 0.If we write in the sequel that a representation of a F -semi-standard Levi subgroup of G is ψ-generic,then we always mean that ψis a non degenerate character of U with the above properties.Let P =MU be a standard parabolic subgroup of G and T M the maximal split torus in the center of M .We will write a ∗M for the dual of the real Lie-algebra of T M and a ∗+M for the positive Weyl chamber in a ∗M defined with respect to P .There is a canonical map H M :M →a M ,such that |χ(m )|F =q − χ,H M (m ) for every F -rational character χ∈a ∗M of M .(We remark that this is not the classical definition of H M .)If πis a smooth representation of M and ν∈a ∗M ,we denote πνthe smooth representation of M defined by πν(m )=q − ν,H M (m ) π(m ).The symbol i G P will denote the functor of parabolic induction normalized such that it sends unitary representations to unitary representations,G acting on its space by right translations.Let τbe a generic irreducible tempered representation of M and ν∈a ∗+M .Thenthe induced representation i G P τνhas a unique irreducible quotient J (τ,ν),the so-called Langlands quotient.Typeset by A M S -T E X12V.HEIERMANN AND G.MUI´CThe aim of our paper is to prove the standard module conjecture[CSh],which states thatJ(τ,ν)is generic,if and only if i G Pτνis irreducible.We achieve this aim under the assumption that the local coefficients Cψdefined with respect toψ-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber.This property of the local coefficients Cψwould be a consequence of Shahidi’s tempered L-function conjecture[Sh2,7.1], which is now known in most cases[K].Nevertheless,the result that we actually need may be weaker(in particular,we do not need to consider each component r i of the adjoint representation r separately).So it may be possible to show it independently of the tempered L-function conjecture(see the remark in1.6).Our conditional proof of the standard module conjecture follows the method developed in[M],[M1],but using the description of the supercuspidal support of a discrete series representation of G given in[H1].The second named author would like to thank the Alexander von Humboldt-foundation for their generous grant helping me to stay in Berlin and complete this work.1.Let P=MU be a standard F-parabolic subgroup of G and(π,V)an ir-reducibleψ-generic admissible representation of M.The parabolic subgroup of G which is opposite to P will be denoted U.The set of reduced roots of T M in Lie(U)will be denotedΣ(P).We will use a superscript G to underline that the corresponding object is defined relative to G.1.1For allνin an open subset of a∗Mwe have an intertwining operator JP πν.Forνin(a∗M)+far away from the walls,it is defined by a convergentintegral(JUv(ug)du.It is meromorphic inνand the map JP|P|Pis scalar.Its inverse equals Harish-Chandra’sµ-function up to a constant and will be denotedµ(π,ν).1.2Put w=w G0w M0.Then wψ(u)λP(ν,π,ψ)(v).Remark that by Rodier’s theorem[R],i G Pπνhas a uniqueψ-generic irreducible sub-quotient.Fix a representative w of w in K.Let t(w)be the map i GP wV,whichsends v to v(w−1·).There is a complex number Cψ(ν,π,w)such thatλ(ν,π,ψ)=STANDARD MODULE CONJECTURE3 Cψ(ν,π,w)λ(wν,wπ,ψ)t(w)J(πν)(resp.λMα(ν,π,ψ))equals the restrictionP∩Mα|P∩Mαof J Mα4V.HEIERMANN AND G.MUI´CProposition:Letπbe an irreducible generic tempered representation of M. Assume that the tempered L-function holds forπrelative to any Mα,α∈Σ(P).Then Cψ(w(·),wπ,w−1)is regular in(a∗M)+.Proof:Letλ∈(a∗M)+.Denote by sα α,sα∈R,the orthogonal projection ofλon a Mα∗M.Then sα>0.By proposition1.4applied toτ=π,Cψ(w(λ),wπ,w−1)= α∈Σ(P)C M wαψ(−sα wα,wπ,w wα).By[Sh,3.11,7.8.1and7.3],C M wαψ(swα,wπ,w wα)=∗ i L(1−is,wπ,r M wα,i)is polynomial,this proves the proposition.21.6Remark:In fact,what is really needed to prove the above proposition is a result that may be weaker than the tempered L-function conjecture:suppose for simplicity thatπis square integrable and choose a standard parabolic subgroup Q=NV of G and a unitary supercuspidal representationσof N,N⊆M,and ν∈a∗N,such thatπis a sub-representation of i M Q∩Mσν.Then,by1.4,C Gψ(s α,π,w)=α∈Σ(Q)−Σ(Q∩M)C Mαψ(s α+ν,σ,wα).This is,up to a meromorphic function on the real axes,equal toα∈Σ(Q)−Σ(Q∩M)L(1−iαs,σν,r M wα,iα) 1−q−1+iα(s+ α∨,ν ),STANDARD MODULE CONJECTURE5 So,what we have to know,is that this meromorphic function is holomorphic for s<0.The fact thatσνlies in the supercuspidal support of a discrete series means by the main result of[H1],thatσνis a pole of order rk ss(M)−rk ss(N)of Harish-Chandra’sµ-functionµM.This can be translated somehow to the assertion thatνcorresponds to a distinguished unipotent orbit[H2].So,what remains,is a purely combinatorial problem in the theory of rootal hyperplane configurations, which can be stated independently of representation theory,although its validity may depend on the fact that the labels iαandνcome from a”generic setting”.As, in particular,one does not need here to consider each component r i of the adjoint representation r separately,this result may be weaker than Shahidi’s tempered L-function conjecture.2.In this section we make the following assumption on G(see the remark in 1.5for what we actually need):(TL)If M is a semi-standard Levi subgroup of G and ifπis an irreducible generic tempered representation of M then L(s,π,r i)is regular forℜ(s)>0for every i.We give a proof of the following lemma only for completeness:2.1.Lemma:Let P=MU be a F-standard parabolic subgroup of G andσan irreducible supercuspidal representation of M.If the induced representation i G Pσhas a sub-quotient,which lies in the discrete series of G,then any tempered sub-quotient of i G Pσlies in the discrete series of G.Proof:If i G Pσhas an irreducible sub-quotient,which is square-integrable,thenis by the main result of[H],σis a pole of order rk ss(G)−rk ss(M)ofµandσ|AM unitary.It follows that the central character of i G Pσis unitary,too,which impliesthat the central character of any irreducible sub-quotient of i G Pσis unitary.Inparticular,any essentially tempered irreducible sub-quotient of i G Pσis tempered.So,ifτis an irreducible tempered sub-quotient of i G Pσ,then there is a F-parabolic subgroup P′=M′U′of G and a square-integrable representationσ′of M′,such thatτis a sub-representation of i G P′σ′.The supercuspidal support ofσ′and the W-orbit ofσshare a common elementσ0.By the invariance of Harish-Chandra’sµ-function,σ0is still a pole ofµequal to the order given above.As this order is maximal and the central character ofσ′must be unitary,this implies that M′must be equal to G,and consequentlyτ=σ′is square-integrable.22.2Theorem:Let G be a group that satisfies property(TL).Let P=MU be a6V.HEIERMANN AND G.MUI´CF-standard parabolic subgroup of G andσbe an irreducibleψ-generic supercuspidal representation of M.If the induced representation i G Pσhas a sub-quotient,which lies in the discrete series of G(resp.is tempered),then any irreducibleψ-generic sub-quotient of i G Pσlies in the discrete series of G(resp.is tempered).Proof:First assume that i G Pσhas a sub-quotient,which lies in the discrete series of G.Let(π,V)be an irreducible,admissibleψ-generic representation of G,which is a sub-quotient of i G Pσ.By the Langlands quotient theorem,there is a standard parabolic subgroup P1=M1U1of G,an irreducible tempered representationτofM1andν∈(a∗M1)+,such thatπis the unique irreducible quotient of i G P1τν.As any representation in the supercuspidal support ofτνmust lie in the super-cuspidal support ofπ,any such representation must be conjugated toσ.So,after conjugation by an element of G,we can assume that M⊆M1and thatτνis asub-representation of i M1P∩M1σ.We will actually show that P1=G,which means thatπis tempered and by2.1 in fact square-integrable.Following1.3,τmust beψ-generic and it is enough to show that Cψ(·,τ,w)has a pole inν,if P1=G.For this we will use the assumption that i G Pσhas an irreducible sub-quotient which is square-integrable.By the main result of[H]this implies thatµhas a pole of order equal to rk ss G−rk ss M inσ.Remark thatµM1can have at most a pole of order rk ss M1−rk ss M inσ.The order of the pole ofµinτνis equal to the one ofµ/µM1inσ.It follows that the order of this pole must be>0,if P1=G.Asµ(τ,·)=Cψ(·,τ,w)Cψ(w(·),wτ,w−1),it follows that either Cψ(·,τ,w)or Cψ(w(·),wτ,w−1)must have a pole inν.As ν∈(a∗M)+,it follows from1.5that Cψ(w(·),wτ,w−1)cannot have a pole inν.So Cψ(·,τ,w)does.This gives us the desired contradiction.Now assume that i G Pσonly has a tempered sub-quotientτ.Then there is a standard parabolic subgroup P1=M1U1of G and a discrete series representationπ1of M1,such thatτis a sub-representation of i G P1π1.As the supercuspidal supportofπ1is contained in the G-conjugacy class ofσ,it follows that there is a standard Levi subgroup M′⊇M,such that i M′P∩M′σhas a discrete series sub-quotient.By what we have just shown,there exists a uniqueψ-generic subquotientπ′of i M′P∩M′σ,which lies in the discrete series.As i G Pσand i G P′π′have each one a unique irreducibleψ-generic sub-quotient and any sub-quotient of i G P′π′is a sub-quotient of i G Pσ,these irreducibleψ-generic sub-quotients must be equal and therefore tempered.22.3Theorem:Let G be a group that satisfies property(TL).Let P=MU bea F-standard Levi subgroup of G,τan irreducible tempered generic representation of M andν∈a∗+M.STANDARD MODULE CONJECTURE7 Then the Langlands quotient J(τ,ν)is generic,if and only if i G Pτνis irreducible.Proof:As i G Pτνalways has a generic sub-quotient,one direction is trivial.So, assume i G Pτνis reducible.We will show thatπ=J(ν,τ)is notψ-generic for anyψ.We can consider(and will)νπ:=νas an element of a∗T.We denote by<the partial order on a∗T explained in([BW],Chapter XI,2.1)(for our purpose it is not important to write it explicitly).Letπ′be an irreducible sub-quotient of i G Pτν,which is not isomorphic toπ. Let P′=M′U′be a F-standard parabolic subgroup,τ′an irreducible temperedrepresentation of M′andν′∈a∗+M′,such thatπ′=J(ν′,τ′).Letνπ′:=ν′.Then[BW,XI,Lemma2.13](2.1)νπ′<νπ.Choose an F-standard parabolic subgroup P1=M1U1,M1⊆M,with an ir-reducibleψ–generic supercuspidal representationσof M1,such thatτis a sub-quotient of i M P1∩M σ.Thenσνlies as well in the supercuspidal support ofπas in the supercuspidal support ofπ′.It lies also in the G-conjugacy class of the supercuspidal support of τ′ν′andτν.Letπψbe the uniqueψ-generic irreducible sub-quotient of i G P1σν.By2.2,the uniqueψ-generic irreducible sub-quotientτ′′of i M′P1∩M′σis tempered.Theinduced representation i G P′τ′′ν′admits a uniqueψ-generic irreducible sub-quotient,which is equal to the uniqueψ-generic sub-quotient of i G P1σν.Letπ′′=J(ν′,τ′′)bethe Langlands quotient of i G P′τ′′ν′.Since(2.1)impliesνπ′′=νπ′<νπ,πcannot be a sub-quotient of i G P′τ′′ν′by[BW,XI,Lemma2.13].Therefore,πis notψ-generic.2References[BW] A.Borel and N.Wallach,Continuous Cohomology,dsicrete subgroups and representa-tions of reductive groups,Princeton University Press,Princeton,1980.[CSh]W.Casselman and F.Shahidi,On irreducibility of standard modules for generic repre-sentations,Ann.Sci.´Ecole Norm.Sup.31(1998),561–589.[H1]V.Heiermann,D´e composition spectrale d’un groupe r´e ductif p-adique,J.Inst.Math.Jussieu3(2004),327–395.[H2]V.Heiermann,Orbites unipotents et pˆo les d’ordre maximal de la fonctionµde Harish-Chandra,to appear in Cand.J.Math..[K]H.Kim,On Local L-Functions and Normalized Intertwining Operators,Canad.J.Math.57(2005),535–597.8V.HEIERMANN AND G.MUI´C[M]G.Mui´c,Some results on square integrable representations;Irreducibility of standard representations,Intern.Math.Research Notices41(1998),705–726..[M1]G.Mui´c,A proof of Casselman-Shahidi’s Conjecture for quasi-split classical groups, Canad.Math.Bull.43(2000),90–99.[R] F.Rodier,Whittaker models for admissible representations,Proc.Sympos.Pure Math.AMS26(1973),425–430.[Sh1] F.Shahidi,On certain L-functions,Amer.J.Math.103(1981),297–356.[Sh2] F.Shahidi,A proof of Langlands’conjecture on Plancherel measures;complementary series for p-adic groups,Ann.Math.132(1990),273–330.[Si] A.Silberger,Discrete Series and classification of p-adic groups I,Amer.J.Mathematics 103(1981),1241–1321.V.H.:Institut f¨u r Mathematik,Humboldt-Universit¨a t zu Berlin,Unter den Lin-den6,10099Berlin,AllemagneE-mail address:heierman@math.hu-berlin.deG.M.:Department of Mathematics,University of Zagreb,Bijenicka30,10000 Zagreb,CroatiaE-mail address:gmuic@math.hr。