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Bessel functions and local converse conjecture of Jacquet 6 1 0 2 Jingsong Chai v o N 9 2 ] T Abstract N . Inthispaper,weprovethelocalconverseconjectureofJacquetoverp-adicfields h forGL usingBesselfunctions. t n a m Keywords.Besselfunctions, Howevectors, localconverse conjecture ofJacquet [ 2 1 Introduction v 0 5 4 Let F be a p-adic field, and ψ be a nontrivial additive character of F with conductor 5 precisely O, the ring of integers of F. Let π be an irreducible admissible generic repre- 0 1. sentationofGLn(F).Ifρis an irreducibleadmissiblegenericrepresentation ofGLr(F), 0 onecan attach an importantinvariantlocal gammafactor γ(s,π ×ρ,ψ) viathetheory of 6 1 localRankin-SelbergintegralsbyJacquet,Piatetski-ShapiroandShalika([JPSS83]).This : v invariantcan alsobedefined by Langlands-Shahidimethod([Sh84]). i X These invariants can be used to determine the representation π up to isomorphism.In r a [He], Henniart proved that, for irreducible admissible generic representations π and π 1 2 of GL (F), if the family of invariants γ(s,π × ρ,ψ) = γ(s,π × ρ,ψ), for all r with n 1 2 1 ≤ r ≤ n− 1, and for all irreducible admissible generic representations ρ of GL (F), r then π ∼= π . This is then strengthened by J.Chen ([Ch06]) and Cogdell and Piatetski- 1 2 Shapiro ([CPS99]) by decreasing r from n − 1 to n − 2. Such type of results together withglobalversionfirst appeared in [JLa]forGL(2) and [JPSS79]forGL(3). A general conjecture, whichisdueto Jacquetcan beformulatedas follows. Conjecture 1 (Jacquet). Assumen ≥ 2. Letπ and π beirreduciblegenericsmooth 1 2 representations of GL (F). Suppose for any integer r, with 1 ≤ r ≤ [n], and any irre- n 2 duciblegenericsmoothrepresentationρofG , wehave r Jingsong Chai: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China;e-mail:[email protected] MathematicsSubjectClassification(2010):Primary11F70;Secondary22E50 1 2 JingsongChai γ(s,π ×ρ,ψ) = γ(s,π ×ρ,ψ), 1 2 thenπ and π are isomorphic. 1 2 In the present paper, we will prove Conjecture 1 using Bessel functions. The author wasrecentlyinformedthatthisconjecturehasalsobeenprovedbyH.JacquetandBaiying Liuindependentlyusingadifferent method,see[JL]. By the work of Dihua Jiang, Chufeng Nien and Shaun Stevens in section 2.4, [JNS], this conjecture has been reduced to the following conjecture when both π ,π are unita- 1 2 rizableirreduciblesupercuspidalrepresentations. Conjecture 2.Assumen ≥ 2.Letπ andπ beirreducibleunitarizableandsupercus- 1 2 pidal smooth representations of GL (F). Suppose for any integer r, with 1 ≤ r ≤ [n], n 2 and anyirreduciblegeneric smoothrepresentationρofG , wehave r γ(s,π ×ρ,ψ) = γ(s,π ×ρ,ψ), 1 2 thenπ and π are isomorphic. 1 2 It is this conjecture that we will prove in this paper. The main result can be stated as follows. Theorem 1.1. Conjecture 2 istrue,and sois Conjecture1. AcrucialingredientintheproofisBesselfunction,whichhasitsowninterests.Given an irreducible admissible generic representation π of GL (F), one can attach a Bessel n function j to π. Such functions were first defined over p-adic fields by D.Soudry in [S] π forGL (F),andthenweregeneralizedtoGL (F)byE.M.Baruchin[B05],toothersplit 2 n groups by E.Lapid and Zhengyu Mao in [LM13], respectively. A general philosophy is thatthelocalgammafactorsγ(s,π×ρ,ψ)areintimatelyrelatedtotheBesselfunctionsof π,ρ.Inmanycases,weknowthatlocalgammafactorscanbeexpressedascertainMellin transform of Bessel functions, see for example [CPS98, Sh02, S]. This expression is the starting point of proving stability of local gamma factors, which is crucial in applying converse theorem to Langlands functoriality problems. For the case at hand, such Mellin transform is also expected but has not yet been proved according to the author’s knowl- edge.However,itisstillpossibletoderiveanequalityofBesselfunctionsfromequalities oflocalgammafactorsvialocalRankin-Selbergintegrals.Thentheaboveconjecturefol- lows as the Bessel function j determines the representation π up to isomorphismby the π weak kernel formula(Theorem 4.2in [Chai15]). Thisisthebasicideaofourproof. There are much progress made towards this conjecture in recent years. In particular, ChufengNienin[N]provedananalogueofthisconjectureinfinitefieldcase.DihuaJiang, Chufeng Nien and Shaun Stevens in [JNS] formulated an approach using constructions of supercuspidal representations to attack this conjecture in p-adic case, and proved it in BesselfunctionsandlocalconverseconjectureofJacquet 3 many cases, including the cases when π ,π are supercuspidal representations of depth 1 2 zero. Later on based on this approach, Moshe Adrian, Baiying Liu, Shaun Stenvens and Peng Xu in [ALSX] proved the conjecture for GL (F) when n is prime. There are also n some work on similar problems for other groups. See [B95, B97, JS, Zh1, Zh2] for ex- amples.Foramorecomprehensivesurveyonlocalconverseproblemsandrelatedresults, seerelevantsectionsin[Jiang,JN]. An important ingredient of the approach suggested in [JNS] is to reduce the conjec- turetoshowtheexistenceofcertainWhittakerfunctions(calledspecialpairofWhittaker functions) for a pair of unitarizable supercuspidal representations π ,π of GL (F). In 1 2 n [JNS] and [ALSX], such Whittaker functions were found in many cases using the con- structionsofsupercuspidalrepresentations. We will explain our proof in more details, and the above ingredient is also important. To prove Conjecture 2, as explained above it suffices to show that, under the as- sumptions of the conjecture, the unitarizable supercuspidal representations π ,π have 1 2 the same Bessel functions. By Proposition 5.3 in [Chai15], it is reduced to show that the normalized Howevectors (see Definition 3.1 below), which are certain partial Bessel functions in the Whittaker models providing nice approximations to Bessel functions, satisfy W1 (aω ) = W2 (aω ) for any diagonal matrix a, where ω is the longest Weyl vm n vm n n element. For this purpose, when n = 2r + 1 is odd, we consider the following Rankin- Selberg integrals on GL (F)×GL (F) (for the unexplained notations, see section 2 2r+1 r fordetails) g γ(s,πi∗×ρ,ψ−1)Z Z Wωin.vmx Ir (cid:18)ω2r 1(cid:19)αr+1aW′(g)|det(g)|s−r+21dxdg Nr\Gr M(r×r) 1 f    = ωρ(−1)r−1Z Wiωn.vm(cid:18)(cid:18)g Ir+1(cid:19)ωn,r(cid:18)ω2r 1(cid:19)αr+1a−1(cid:19)W′(g)|det(g)|1−s−r+21dg, f f f whereρis anygenericirreduciblesmoothrepresentationofGL (F). r By Lemma2.3 below,itsuffices toshow g g ω ω W1 x I  2r αr+1a = W2 x I  2r αr+1a ωn.vm r (cid:18) 1(cid:19) ωn.vm r (cid:18) 1(cid:19) 1 1 f    f    oncertainopendensesubsetofthedomainintheintegrals.Inspiredbytheworkof[JNS], wewillfirstinsection3showthat,thenormalizedHowevectorssatisfycertainproperties similar to special pair of Whittaker functions in a slightly weak form on certain Bruhat cells. Then combining with the work of Jeff Chen in [Ch06], these properties will imply the above identities, which finishes the proof in the odd case. The even case can then be deduced fromtheoddcase. 4 JingsongChai WefinallyremarkthatNienusedBesselfunctionsinher’sproofoffinitefieldanalogue ([N]),andinthispaperwegivethefirstBesselfunctionproofofConjecture1overp-adic fields. Thepaperisorganizedasfollows.Insection2,werecallsomebackgroundsandprepa- rationsonRankin-Selberg integralsandBesselfunctions.WethenstudyHowevectorsin detailin section3.In thelastsection,weprovethetheorem. 2 Preparations Use G to denote GL (F), and embed G into G on the left upper corner. Let N n n n−1 n n besubgroupoftheuppertriangularunipotentmatrices. A thesubgroupofdiagonalma- n trices. Use P to denote the mirabolic subgroup consisting of matrices with the last row n (0,...,0,1).We extendtheadditivecharacter ψ toN , stilldenotedas ψ, bysetting n n−1 ψ(u) = ψ( u ) u = (u ) ∈ N . i,i+1 ij n Xi=1 If π is an irreducible admissiblegeneric representation of G , use W(π,ψ) to denote n theWhittakermodelofπ withrespect toψ. IfW ∈ W(π,ψ),define W(g) := W(ω ·tg−1) n f 1 whereωn =  ... , thenthespaceoffunctions 1    {W(g) : W ∈ W(π,ψ)} f is theWhittakermodelof π∗ withrespect toψ−1, where π∗ denotes thecontragredientof π. Supposeπ and π′ are irreducible admissiblegeneric representations ofG and G re- n r spectively,withassociated WhittakermodelsW(π,ψ) and W(π′,ψ−1). Forourpurpose, we will assume r < n. For any W ∈ W(π,ψ), W′ ∈ W(π′,ψ−1), s ∈ C a complex number,and anyintegerj withn−r −1 ≥ j ≥ 0, setk = n−r −1−j, and let g 0 0 I(s,W,W′,j) = W x I 0 W′(g)|detg|s−(n−r)/2dxdg, j Z Z Nr\Gr M(j×r) 0 0 I k+1   whereM(j ×r)denotesthespaceofmatrices ofsizej ×r. Wehavethefollowingbasicresultinthetheoryoflocal Rankin-Selberg integrals. BesselfunctionsandlocalconverseconjectureofJacquet 5 Theorem 2.1. ([JPSS83])Foranypairofgenericirreducibleadmissiblerepresentations π andπ′ on G andG , wehave: n r (1). TheintegralsI(s,W,W′,j) convergeabsolutelyforRe(s) large; (2). The integrals I(s,W,W′,j) span a fractional ideal in C[qs,q−s] with a unique generatorL(s,π×π′)suchthatL(s,π×π′)hastheformP(q−s)−1 forsomepolynomial P ∈ C[x] withP(0) = 1; (3). There exists a meromorphic function γ(s,π × π′,ψ), independent of the choices ofW, W′, suchthat I(1−s,π∗(ωn,r)W,W′,k) = ωπ′(−1)n−1γ(s,π ×π′,ψ)I(s,W,W′,j), f f I where n−r −1 ≥ j ≥ 0, k = n−r−1−j, ωn,r = (cid:18) r ω (cid:19)andωπ′ isthecentral n−r characterof π′. Remark. Themeromorphicfunctionγ(s,π×π′,ψ)iscalled thelocalγ-factor ofπ and π′. Now let π be a generic irreducible unitarizable representation of G . Consider the n spaceoffunctions W := {W(g) : W(g) ∈ W(π,ψ)} where′¯′ denotes thecomplexconjugate. Then with the right translation by G , W is an irreducible representation of G , with n n W(ug) = ψ−1(u)W(g)foru ∈ N ,g ∈ G .ThusW istheWhittakermodelwithrespect n n toψ−1 ofsomegenericirreducibleunitarizablerepresentation τ ofG . n Sinceπ isunitarizable,wehaveaG -invariantinnerproduct(W ,W )onsomecom- n 1 2 plex Hilbert space. Then for W ∈ W, view it as a smooth linear functional on W(π,ψ) viatheaboveform l : W (g) → (W ,W). W v v Since (∗) is G -invariant, this gives an isomorphism between the representation τ on W n and π∗. Werecord it as aproposition. Proposition2.2. IftherepresentationπisgenericirreducibleunitarizablewithWhittaker modelW(π,ψ),thentherepresentation(τ,W)isaWhittakermodelofπ∗ withrespectto ψ−1. Wealsoneed thefollowinglemma,whichis Corollary2.1 in[Ch06]. 6 JingsongChai Lemma 2.3. Let H bea complexsmoothfunctiononG satisfying r H(ug) = ψ(u)H(g) foranyg ∈ G , u ∈ N . r r If for any irreducible generic smooth representation ρ of G , and for any Whittaker r function W ∈ W(ρ,ψ−1), the function defined by the following integral vanishes for Re(s) large H(g)W(g)|det(g)|s−kdg = 0, Z Nr\Gr where k issomefixed constant,thenH ≡ 0. Remark. We don’t need to require the function H to be in the Whittaker model of some generic representation as in [Ch06]. The proof there works for general Whittaker functions. Next we introduce the Bessel functions briefly. For more details see [B05, Chai15]. Let (π,V) be a generic irreducible representation of G . Take W ∈ W(π,ψ). Consider n v theintegralforg ∈ N A ω N n n n n W (gu)ψ−1(u)du. v Z Nn Thisintegralstabilizesalong anycompactopen filtrationofN , and themap n ∗ v → W (gu)ψ−1(u)du, v Z Nn ∗ where denotes the stabilized integral, defines a Whittaker functional on V. By the uniquenRessofWhittakerfunctional,thereexistsascalarj (g)suchthat π,ψ ∗ W (gu)ψ−1(u)du = j (g)W (I). v π,ψ v Z Nn Definition 2.4. The assignment g → j (g) = j defines a function on N A ω N , π π,ψ n n n n whichis calledtheBessel functionofπ attached toω . n We extend j to G by putting j (g) = 0 if g ∈/ N A ω N , and still use j to π n π n n n n π denote it and call it the Bessel function of π. It is easy to check that j is locally con- π stant on N A ω N (See Theorem 1.7 and remarks above it in [B05]) and j (u gu ) = n n n n π 1 2 ψ(u )ψ(u )j (g)foranyu ,u ∈ N ,g ∈ G . 1 2 π 1 2 n n A property of Bessel function which is important to us is the following weak kernel formula,seeTheorem 4.2in [Chai15]. BesselfunctionsandlocalconverseconjectureofJacquet 7 Theorem 2.5. (Weak Kernel Formula) For any bω , b = diag(b ,...,b ) ∈ A , and any n 1 n n W ∈ W, wehave W(bω ) = n −1 a a 1 1   x a    x a  21 2 21 2 . . jπbωn ..  W  ..  Z        x ··· x a   x ··· x a    n−1,1 n−1,n−2 n−1    n−1,1 n−1,n−2 n−1    1   1       |a |−(n−1)da |a |−(n−2)dx da ···|a |−1dx ···dx da , 1 1 2 21 2 n−1 n−1,1 n−1,n−2 n−1 wheretherightsideisaniteratedintegral,a isintegratedoverF× ⊂ F fori = 1,...,n− i 1, x is integrated over F for all relevant i,j, and all measures are additive self-dual ij Haarmeasures onF. 3 Howe vectors Inthissection,wewillintroduceandstudyHowevectors.Following[B05],forapositive integerm, let Km = I +M (pm), herep is themaximalideal ofO and O is theringof n n n integersofF.Use̟ to denotean uniformizerofF. Let 1  ̟2  ̟4 d = .    ..   .     ̟2n−2   Put J = dmKmd−m, N = N ∩J , N¯ = N¯ ∩J , B¯ = B¯ ∩J n,m n n,m n n,m n,m n n,m n,m n n,m and A = A ∩J , then n,m n m J = N¯ A N = B¯ N = N B¯ . n,m n,m n,m n,m n,m n,m n,m n,m For j ∈ J , write j = ¯b n with respect to the above decomposition, as in [B05], n,m j j defineacharacter ψ on J by m n,m ψ (j) = ψ(n ). m j Remark. WewillwriteJ forJ , and ψ forψ when thereis noconfusion. m n,m m In thissection weassumeπ isagenericirreducibleunitarizablerepresentationofG . n 8 JingsongChai Definition3.1. W ∈ W(π,ψ)iscalledaHowevector ofπ oflevelmwithrespecttoψ m if W(gj) = ψ (j)W(g) (3.1) m forallg ∈ G , j ∈ J = J . n m n,m Remark. Itfollowsthatthelevelmmustbegreaterthanorequaltotheconductorofthe central character ω ofπ iftheHowevectorexists. π ForeachW ∈ W(π,ψ),letM beapositiveconstantsuchthatR(KM)W = W where n R denotestheactionofrightmultiplication.Foranym > 3M, put W (g) = W(gu)ψ−1(u)du, m Z Nn,m thenby Lemma7.1 in[B05], wehave W (gj) = ψ (j)W (g),∀j ∈ J , ∀g ∈ G . m m m m n This gives the existence of Howe vectors when m is large enough. The followinglemma establishesits uniquenessin Kirillovmodel,see Theorem5.2 in[Chai15]fortheproof. Lemma 3.2. AssumeW ∈ W(π,ψ) satisfying(3.1).Let h ∈ G ,if n−1 h W 6= 0, (cid:18) 1(cid:19) thenh ∈ N B¯ . Moreover n−1 n−1,m h W = ψ(u)W(I) (cid:18) 1(cid:19) ifh = u¯b,with u ∈ N ,¯b ∈ B¯ . n−1 n−1,m Remark. Thus, given π, if m is large enough, there exists a unique vector W ∈ vm W(π,ψ) satisfying (3.1) and W (I) = 1. We will call this vector as the normalized vm Howevectoroflevelmwithrespect to ψ . m Remark. Bytheconstructionsabove,Howevectorsexistiftheirlevelsmaresufficiently large.SowhenwetalkaboutHowevectors,weimplicitlymeanthelevelsarelargeenough sothat thesevectorsexist. Proposition3.3. IfW isthenormalizedHowevectorofπoflevelmwithrespecttoψ , vm m then W is the normalized Howe vector of level m with respect to ψ−1 for (π∗,W), ωn.vm m and f f W (g) = W (ω tg−1ω ) ∀g ∈ G . vm vm n n n BesselfunctionsandlocalconverseconjectureofJacquet 9 Proof. Consider the Whittaker function W . For j ∈ J , one can check ω tj−1ω ∈ ωn.vm m n n J , and m f ψ (ω tj−1ω ) = ψ−1(j). m n n m Thusforanyg ∈ G ,j ∈ J , n m Wωn.vm(gj) = Wtj−1ωn.vm(g) = Wψm−1(j)ωn.vm(g) = ψm−1(j)Wωn.vm(g). f f f f ThisshowsthatW isthenormalizedHowevectoroflevelmwithrespecttoψ−1 for ωn.vm m (π∗,W). f Ofn the other hand, by Proposition 2.2, W is a Whittaker model of π∗ with respect to ψ−1. TakeW ∈ W, foranyg ∈ G ,j ∈ J , vm n m W (gj) = W (g)ψ (j) = ψ−1(j)W (g), vm vm m m vm which implies that W is the normalized Howe vector of level m with respect to ψ−1. vm m Thusby theuniquenessofHowevectors,wehave W (g) = W (g) ∀g ∈ G ······(2), ωn.vm vm n f whichprovestheproposition. u 0 u ω a′u 0 a r r r Proposition 3.4. Assume n = 2r + 1. Let g = 0 1 0 , with (cid:18)bω 0(cid:19) r 0 0 I r   a = diag(a ,...,a ) ∈ A , a′ = diag(a′,...,a′) ∈ A , b = diag(b ,...,b ) ∈ A , 1 r+1 r+1 1 r r 1 r r u ∈ N ,u = (u ) ∈ N ,. W is the normalized Howe vector. If W (g) 6= 0, then r r ij r vm vm I 0 ω a′u a r r i ∈ 1+pm for i = 1,2,...,r,0 1 0  ∈ J ,and m a i+1 0 0 I r   u 0 0 0 a r W (g) = W  0 1 0. vm vm (cid:18)bω 0(cid:19) r 0 0 I r    I 0 0 r Proof. Takej = 0 1 x  ∈ J withx = (x ,...,x ).Asπ(j ).v = ψ (j )v , 1 1 m 1 11 1r 1 m m 1 m 0 0 I r   wefind 10 JingsongChai ψ(x )W (g) = W (gj ) 11 vm vm 1 u 0 u ω a′u 0 a r r r = W  0 1 0 j  vm (cid:18)bω 0(cid:19) 1 r 0 0 I r     u 0 u ω a′u 0 a r r r = W  j 0 1 0  vm (cid:18)bω 0(cid:19) 1 r 0 0 I r    1 x′ 0 u 0 u ω a′u 1 0 a r r r = W 0 I 0 0 1 0  vm r (cid:18)bω 0(cid:19) 0 0 I r 0 0 I r r     = ψ(a a−1x )W (g), 1 2 11 vm wherex′ = (a a−1x ,...,a a−1 x ). 1 1 2 11 1 r+1 1r Since W (g) 6= 0, we have ψ(x (1 − a a−1)) = 1. As x ∈ p−m is arbitrary, we vm 11 1 2 11 get 1−a a−1 ∈ pm, which meansa a−1 ∈ 1+pm. 1 2 1 2 I 0 0 r Letj′ = y 1 0 ∈ J withy = (y ,...,y ).Then wehave 1 1 m 1 11 1r 0 0 I r   W (g) = W (gj′) vm vm 1 u 0 u ω a′u 0 a r r r = W  0 1 0 j′ vm (cid:18)bω 0(cid:19) 1 r 0 0 I r     I 0 0 u 0 u ω a′u 0 a r r r r = W  y u−1 1 −y ω a′u0 1 0  vm (cid:18)bω 0(cid:19) 1 r 1 r r 0 0 I 0 0 I r r     1 y′ ∗ u 0 u ω a′u 1 0 a r r r = W 0 I 0 0 1 0  vm r (cid:18)bω 0(cid:19) 0 0 I r 0 0 I r r     = ψ(a a−1y a′)W (g), 1 2 1r 1 vm wherey′ = (a a−1y a′,...). 1 1 2 1r 1 Since W (g) 6= 0, we have ψ(a a−1y a′) = 1. As a a−1 ∈ 1+pm, and y ∈ p3m vm 1 2 1r 1 1 2 1r isarbitrary, wefind a′ ∈ p−3m. 1