ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS OVER p-adic LOCAL FIELDS

Transcription

1 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS OVER p-adic LOCAL FIELDS DIHUA JIANG AND CHUFENG NIEN Abstract. We discuss two issues related to the local Langlands conjecture over p-adic local fields. One is about the uniqueness of the local Langlands reciprocity map, which is a consequence of the local converse theorem; and the other is to establish the local Langlands conjecture for quasi-split classical groups based on the result for the tempered case, which is obtained by J. Arthur for symplectic and orthogonal groups and by C.-P. Mok for unitary groups. 1. Introduction Let F be a p-adic local field of characteristic zero. The local Langlands conjecture for GL n (F ), which is proved by Harris and Taylor ([HT01]) and by Henniart ([H00]), is a correspondence between admissible n-dimensional complex representations of the Weil-Deligne group of F and admissible complex representations of GL n (F ), with certain constraints. It is regarded as the non-abelian extension of the classical Artin reciprocity law in the local abelian classfield theory. For a brief review of the progress towards the complete proof of the local Langlands conjecture for GL n (F ), we refer the readers to [HT01]. In this paper, we pursue two issues related to the local Langlands conjecture. One is about the uniqueness of the local Langlands reciprocity map, which follows from the local converse theorem for irreducible admissible generic representations of F -quasi-split reductive algebraic groups. In this aspect, we will discuss how to refine the local converse theorem, and in particular, discuss the recent progress towards the conjecture of H. Jacquet. Date: October, Mathematics Subject Classification. Primary 11F70, 22E50; Secondary 11F85, 22E55. Key words and phrases. Local Gamma Factors, Local Langlands Conjecture, Local Converse Theorem, Admissible Representations. The work of the firsted named author is supported in part by NSF grants DMS and DMS ; and the work of the second named author is supported by NSC grants M

2 2 DIHUA JIANG AND CHUFENG NIEN Another is about the local Langlands conjecture for classical groups. In order to extend the local Langlands conjecture from GL n (F ) to other classical groups, one may take an approach using the local Langlands functorial transfer from classical groups to the general linear group, via the standard embedding of the corresponding dual groups. We will discuss the problems involved in this approach. In Section 2, we review briefly the local Langlands conjecture for GL n (F ), which is a Theorem of Harris and Taylor ([HT01]) and of Henniart ([H00]). To discuss the issue on the uniqueness of the local Langlands reciprocity map, we continue with Section 3 on the local converse problem, where we explain the formulation of the local converse problem and our recent progress, joint with S. Stevens, on the Jacquet conjecture. The implication of the any positive answer to the local converse problem towards the uniqueness of the local Langlands reciprocity map is also discussed there. In Section 4, we discuss the second issue, i.e. the local Langlands conjecture for classical groups. First, we recall the tempered case from the work of Arthur ([Ar13]) and of Mok ([Mk13]), and then show that the general case follows from the tempered case by the Langlands classification theory. Section 5 discuss more detailed issues on how to use the local Langlands functorial transfer from classical groups to the general linear groups. In addition to the establishment of the local Langlands functorial transfer, one has to characterize the image of the transfer and show the subjectivity of the local reciprocity map. We explain all the issues for SO 2n+1, which should be applicable to all other classical groups. Finally, we would like to thank the Mathematics Division of the National Center for Theoretical Sciences, Taiwan, for funding support which makes parts of the collaborating work of the authors over the years possible. 2. The Local Langlands Conjecture Let F be the algebraic closure of the local field F and Γ F = Gal(F /F ) be the absolute Galois group. Let F ur be the maximal unramified extension of F and denote by Γ ur F = Gal(F ur /F ) the corresponding Galois group. Consider the exact sequence 1 I F Γ F Γ ur F 1, with I F the inertia group. The Galois group Γ ur F = Ẑ is generated by the Frobenius element Frob. The Weil group W F is defined through the following exact sequence 1 I F W F Z =< Frob > 1

3 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS 3 where Z is equipped with the discrete topology, which yields the topology on the Weil group W F. The Weil-Deligne group is defined to be the product W F SL 2 (C) (a popular version). A Frobenius semisimple representation ρ of the Weil-Deligne group W F SL 2 (C) is a representation satisfying the following two conditions: (1) ρ WF is continuous and ρ(frob) is semisimple; (2) ρ SL2 (C) is algebraic. Define Φ(GL n ) to be the set of equivalence classes of n-dimensional Frobenius semisimple representations of W F SL 2 (C). Let (V, π) be a representation of GL n (F ), where V is a C-vector space with dim C V. A C-representation V π of GL n (F ) is called smooth if v V π, the stabilizer of v in GL n (F ) Stab GLn(F )(v) := {g GL n (F ) π(g)(v) = v} is an open subgroup of GL n (F ). A smooth representation V π of GL n (F ) is called admissible if for any open compact subgroup K of GL n (F ), the subspace K-invariants V K π := {v V π π(k)(v) = v, k K} is of finite-dimension. Let Π(GL n ) be the set of equivalence classes of irreducible admissible representations of GL n (F ). The local Langlands Conjecture for GL n (F ) is proved by Harris and Taylor ([HT01]) and by Henniart ([H00] and can be stated as follows. Theorem 2.1 (Harris-Taylor and Henniart). For each integer n 1, there is a unique bijection rec F,n : Π(GL n ) Φ(GL n ) such that for π Π(GL n ), π Π(GL n ) and χ Π(GL 1 ), the following hold: (1) the Artin reciprocity law holds rec F,1 (χ) = χ Art 1 F, where Art F is the Artin map from F to WF ab; (2) the local factors are preserved and L(s, π π ) = L(s, rec F,n (π) rec F,n (π )) ɛ(s, π π, ψ F ) = ɛ(s, rec F,n (π) rec F,n (π ), ψ F ) for a given nontrivial additive character ψ F of F ; (3) the twisting by character is preserved rec F,n (π (χ det)) = rec F,n (π) rec F,1 (χ);

4 4 DIHUA JIANG AND CHUFENG NIEN (4) the central character condition holds det rec F,n (π) = rec F,n (ω π ), where ω π is the central character of π; and (5) the duality holds rec F,n (π ) = rec F,n (π), where denotes the contragredient. The uniqueness of the local Langlands reciprocity rec F,n follows from the local converse theorem for irreducible admissible representations of GL n (F ), which will be discussed with detail in the next section. 3. The Local Converse Problem Recall that the local L-factor L(s, π π ) and the local ɛ-factor ɛ(s, π π, ψ F ) in Theorem 2.1 were defined by Jacquet, Piatetski- Shapiro, and Shalika in [JPSS83]. The corresponding local gamma factor of the Rankin-Selberg type is defined by γ(s, π π, ψ F ) = ɛ(s, π π, ψ F ) L(1 s, π π ). L(s, π π ) On the Galois side, for φ, φ Φ(GL n ), J. Tate, following Deligne and Langlands, defined in [T79] the local gamma factors of Artin type by γ(s, φ φ, ψ F ) = ɛ(s, φ φ, ψ F ) L(1 s, φ φ ). L(s, φ φ ) Theorem 2.1 also implies that the following identity holds: γ(s, π π, ψ F ) = γ(s, rec F,n (π) rec F,n (π ), ψ F ) for π Π(GL n ) and π Π(GL n ). It is clear that for π Π(GL n ), the local gamma factors for pair, γ(s, π τ, ψ F ), with τ Π(GL m ) and with all m = 1, 2,, are invariants attached to π. The question is: Is this family of invariants enough to determine π, up to equivalence? It is easy to show that if m n, the family γ(s, π τ, ψ F ) with τ Π(GL m ) determines π completely. Hence it remains to ask the following question: Local Converse Problem: Find n 0 < n such that the family γ(s, π τ, ψ F ) with τ Π(GL m ) for m = 1, 2,, n 0 determine π completely. We remark that any positive answer to the Local Converse Problem implies the uniqueness of the collection of the reciprocity maps in the local Langlands conjecture for GL n, i.e. in Theorem 2.1.

5 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS 5 In fact, this can be verified as follows. If there are two reciprocity s σ n and σ n from Φ(GL n ) to Π(GL n ), then by the compatibility of local gamma factors, one has γ(s, σ n (φ) τ, ψ F ) = γ(s, σ n(φ) τ, ψ F ) for all τ Π(GL m ) with m = 1, 2,, n 1. Hence σ n (φ) = σ n(φ) holds for φ Φ(GL n ) and n > 1. The progress towards the best answer to the Local Converse Problem is briefly recalled here. In 1970, Jacquet and Langlands ([JL70]) show that for n = 2 and n 0 = 1, the Local Converse Problem has a positive answer. Then in 1979, Jacquet, Piatetski-Shapiro and Shalika ([JPSS79]) show that for n = 3 and n 0 = 1, the Local Converse Problem has a positive answer. After fourteen years, G. Henniart in [H93] proves that for general n > 1 with n 0 = n 1, the Local Converse Problem has a positive answer. This general result is refined by J.-P. Chen in his 1996 Ph.D. thesis at Yale University and show that for general n > 1, it is enough to take n 0 = n 2. This paper is published in 2006 ([Ch06]). On the other hand, J. Cogdell and Piatetski-Shapiro use the global method to prove in [CPS99] that the same result holds. In the early 1980s, Jacquet made his conjecture on the possible answer to the Local Converse Problem. Conjecture 3.1 (Jacquet). The Local Converse Problem should have a positive answer for general n > 1 with n 0 = [ n ], that is, for any 2 π 1, π 2 Π(GL m ), if the identity for the local gamma factors for pairs γ(s, π 1 τ, ψ F ) = γ(s, π 2 τ, ψ F ) holds for all τ Π(GL m ) with m = 1, 2,, [ n], then π 2 1 must be isomorphic to π 2. It is clear that the progress to the Local Converse Problem accumulated over the years is still far away from the expectation of the Jacquet conjecture for general n > 4. Recently, we started to investigate the finite field analogy of the Jacquet conjecture. The second named author is able to give a complete proof over finite fields of the Jacquet conjecture, based on the thesis work of E.-A. Roditty under the supervision of David Soudry at Tel Aviv University. The main idea and the new ingredient in her proof is to find special Bessel function with extra symmetry in the space of Whittaker models of irreducible cuspidal representations of GL n over finite fields. This appears in [N12]. This idea has been carried over to general p-adic local fields. To be more precise, we recall from [JNS13] the basic notions and facts.

6 6 DIHUA JIANG AND CHUFENG NIEN Let U n be the unipotent radical of the standard Borel subgroup B n of GL n, which consists of all upper-triangular matrices. Denote by P n the mirabolic subgroup of GL n (F ), consisting of matrices with last row equal to (0,..., 0, 1). We also fix a standard non degenerate character ψ n of U n, so that all Whittaker functions are implicitly ψ n - Whittaker functions. Let π be an irreducible unitarizable supercuspidal representation of GL n (F ). Take K to be an open subgroup of GL n (F ), which is compact-mod-center. A Whittaker function W π for π is called K-special if it satisfies: W πi (g 1 ) = W πi (g) for all g K, and SuppW π U n K, where denotes complex conjugation. Further, let π 1 and π 2 be irreducible unitarizable supercuspidal representations of GL n (F ) with the same central character and let W π1 and W π2 be Whittaker functions for π 1 and π 2, respectively. We call (W π1, W π2 ) a special pair (of Whittaker functions) for (π 1, π 2 ) if there exists a compact-mod-centre open subgroup K of GL n (F ) such that W π1 and W π2 are both K-special and W π1 (p) = W π2 (p), for all p P n. In [JNS13], the Jacquet conjecture to the Local Converse Problem was first proved under the assumption that for any pair of irreducible unitarizable supercuspidal representations (π 1, π 2 ) of GL n (F ), a special pair of Whittaker functions exists for (π 1, π 2 ), by using a refinement of the argument in [Ch96] and [Ch06]. In other words, we prove in [JNS13] the following theorem. Theorem 3.2. Let π 1 and π 2 be irreducible unitarizable supercuspidal representations of GL n (F ). Assume that a special pair (W π1, W π2 ) exists for (π 1, π 2 ). If the local gamma factors for pairs, γ(s, π 1 τ, ψ) and γ(s, π 2 τ, ψ), are equal as functions in the complex variable s, for all irreducible supercuspidal representations τ of GL r (F ) with r = 1,..., [ n 2 ], then W π 1 = W π2 and π 1 and π 2 are equivalent as representations of GL n (F ). It remains critical to prove the existence of special pairs for irreducible unitarizable supercuspidal representations of GL n (F ). By using the construction of supercuspidal representations in terms of maximal simple types of Bushnell and Kutzko ([BK93]) and the explicit construction of Bessel functions of supercuspidal representations due to Paskunas and Stevens ([PS08]), we are able to handle this issue under certain constraints, which can be explained as follows. Given an

7 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS 7 irreducible supercuspidal representation π of GL n (F ), one of the invariants associated to it, by Bushnell and Henniart in [BH96], is its endo-class Θ(π). We prove: Proposition 3.3. Let π 1, π 2 be irreducible unitarizable supercuspidal representations of GL n (F ) with the same endo-class. Then there is a special pair (W π1, W π2 ) for (π 1, π 2 ). Theorem 3.2 with Proposition 3.3 implies, for example, that the Jacquet conjecture holds for two level zero irreducible unitarizable supercuspidal representations π 1, π 2 of GL n (F ). In fact, this is a special case of a more general result proved in [JNS13], as follows. Attached to an irreducible supercuspidal representation π of GL n (F ), via its endo-class Θ(π), is an invariant which we call its degree deg(π). The degree is an integer dividing n: for example, deg(π) = 1 if and only if π is a twist of a level zero representation. By using results on the conductor of pairs of supercuspidal representations from [BHK98, BH03], we obtain the following corollary. Corollary 3.4. Let π 1 and π 2 be irreducible unitarizable supercuspidal representations of GL n (F ) and suppose that deg(π 1 ) < n. If the twisted local gamma factors γ(s, π 1 τ, ψ) and γ(s, π 2 τ, ψ) are equal as functions in the complex variable s, for all irreducible supercuspidal representations τ of GL r (F ) with r = 1,..., [ n 2 ], then π 1 and π 2 are equivalent as representations of GL n (F ). Hence it remains to show the existence of special pairs for irreducible unitarizable supercuspidal representations π 1 and π 2 of GL n (F ) with the condition that deg(π 1 ) = deg(π 2 ) = n. In this aspect, Moshe Adrian and Baiying Liu has recently checked that special pairs for irreducible unitarizable supercuspidal representations π 1 and π 2 of GL n (F ) exists when both π 1 and π 2 are of epipelagic type, which is a case of deg(π 1 ) = deg(π 2 ) = n. 4. The Local Langlands Conjecture for Classical Groups For simplicity, we may assume that the classical groups discussed here are F -split, and hence they are either SO 2n+1, Sp 2n, or SO 2n, which are denoted by G n. The Langlands dual group is L G n := G n(c) Γ F, where G n(c) is the complex dual group given by the following diagram. An F -split of G n is given by a pair (B, T ), where B is an F -Borel subgroup and T is the maximal F -torus contained in B. This pair (B, T ) determines the root data (X, ; X, ) for G n. The dual of (X, ; X, ) is (X, ; X, ), which determines the complex dual

8 8 DIHUA JIANG AND CHUFENG NIEN group G n(c) of G n as displayed in the following diagram: (4.1) G n (X, ; X, ) G n(c) (X, ; X, ) For examples, one has that GL n(c) = GL n (C), SO 2n+1 (C) = Sp 2n (C), Sp 2n(C) = SO 2n+1 (C), and SO 2n(C) = SO 2n (C). We refer to [Ar13], [Mk13], [JZ13] and [GGP12] for more explicit description of the Langlands dual groups of F -quasisplit groups. The local Langlands parameters for G n (F ) can be described as follows. Let ι : G n(c) GL n (C) be the natural embedding, where GL n = GL 2n if G n is an orthogonal group; and GL n = GL 2n+1 if G n is Sp 2n. The set Φ(G n ) of local Langlands parameters for G n (F ) consists of all local Langlands parameters φ Φ(GL n ) which factor through G n(c): (4.2) W F SL 2 (C) GL n (C) G n(c) The Local Langlands Conjecture asserts that there is a reciprocity map (4.3) rec F,Gn : Π(G n ) Φ(G n ) with finite fibres and with certain constraints, where Π(G n ) is the set of equivalence classes of irreducible admissible representations of G n (F ). The main problems are (1) How to define the reciprocity rec F,Gn? and (2) How to construct the fibres, i.e. local L-packets? 4.1. The local Langlands conjecture for classical groups: tempered case. Let Π temp (G n ) be the subset of Π(G n ) consisting of all irreducible admissible representations π whose matrix coefficients are L 2+ɛ on G n (F ) modulo the center Z Gn. Let Φ bdd (G n ) be the subset of Φ(G n ) consisting of all local Langlands parameters φ such that the image φ(w F SU 2 ) in G (C) is bounded modulo the center, here SL 2 (C) is replaced by SU 2. Conjecture 4.1 (Local Langlands Conjecture (Tempered Case)). The local Langlands reciprocity map is given by rec t F,G n : Π temp (G n ) Φ bdd (G n )

9 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS 9 and the fibre: rec 1 F,G n (φ) is called the local tempered L-packet attached to φ, and denoted by Π Gn (φ), which is parameterized by the dual of the component group of the centralizer of φ in G (C), and hence it is nonempty. Moreover, Π temp (G n ) is a disjoint union of Π Gn (φ), for φ Φ bdd (G n ). This conjecture has been proved by Arthur in [Ar13] for quasi-split orthogonal groups and symplectic groups; and by Mok in [Mk13] for quasi-split unitary groups. We sketch here how to establish the reciprocity map rec t F,G n. One first needs to establish the local Langlands functorial transfer l : Π temp (G n ) Π temp (GL n ) through the global method, i.e. by comparing the stable trace formula on G n to the stable, twisted trace formula on GL n and using the Fundamental Lemma proved by B.-C. Ngô and the local transfer established by J.-L. Waldspurger. Then by the local Langlands conjecture for GL n with tempered representations rec F,GL : Π temp (GL n ) Φ bdd (GL n ) where φ Φ bdd (GL n ) is of form: φ = (ρ 1, b 1 ) (ρ 2, b 2 ) (ρ r, b r ) with ρ i irreducible with bounded image in GL ai (C) modulo the center and b i denoting the b i -dimensional algebraic representation of SL 2 (C), and n = r i=1 a ib i. It follows that for any irreducible admissible tempered representation π Π temp (G n ), there exists a local Langlands parameter φ Φ bdd (GL n ) which corresponds to π. One has to show that this local Langlands parameter φ actually belongs to the set Φ bdd (GL n ). This is done in [Ar13] and in [Mk13]. Next, one has to show that the local Langlands reciprocity map, established above, rec t F,G n : Π temp (G n ) Φ bdd (G n ) is actually surjective. Let Π gen (G n ) be the subset of Π(G n ) consisting of all irreducible admissible representations which are generic, i.e. have nonzero Whittaker model; and let Π gen temp(g n ) be the intersection of Π gen (G n ) and Π temp (G n ). By the local descent method and the classification of irreducible generic representations of G n (F ), it is proved in [JS03] and [JS04] (see also [JNQ10]), in [Liu11], and in [JtL13] that the map rec gen F,G n : Π gen temp(g n ) Φ bdd (G n )

10 10 DIHUA JIANG AND CHUFENG NIEN is surjective for all split classical groups G n, and the local factors are preserved: L(s, π π ) := L(s, rec F,Gn (π) rec F,Gn (π )); γ(s, π π, ψ F ) := γ(s, rec F,Gn (π) rec F,Gn (π ), ψ F ). We remark that this should also be true for quasi-split classical groups. By the Shahidi Conjecture, which is a consequence of the local descent method ([JS12]), for φ Φ bdd (G n ), the subset Π gen G n (φ) of the local tempered L-packet Π Gn (φ)is not empty. By Arthur s definition of the local factors of the local tempered L-packets ([Ar13] and [Mk13]), these two reciprocity maps are the same and hence the local reciprocity map rec t F,G n : Π temp (G n ) Φ bdd (G n ) is surjective. This establishes the reciprocity map in the local Langlands conjecture for split classical groups G n. Note that the surjectivity of the reciprocity map for the tempered case is part of the theorem in [Ar13] and [Mk13]. Of course, it is impossible at least for the moment to define the tempered local L-packets in general without using the trace formula method The local Langlands conjecture for split classical groups: general case. We are going to deduce the general case of the local Langlands conjecture for split classical groups from the tempered case via the Langlands classification theory. The Langlands classification of irreducible admissible representations of p-adic reductive algebraic groups in terms of essentially tempered representations, for any π Π(G n ), there exists a pair (P, σ), where P = MN is a parabolic subgroup of G n and σ is essentially tempered on M(F ), s.t. Ind Gn(F ) P (F ) (σ) π gives the unique irreducible quotient, up to Weyl group action. More precisely, we write a Levi subgroup of G n, and write M = GL n1 GL nt G n0, σ = σ 1 det s 1 σ t det st σ 0 with σ i for i = 1, 2,, t being all tempered and s i being all real, so that s 1 s t. The local Langlands transfer from G n to GL n is

11 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS 11 given by the following diagram: Π temp (M) Π temp (M ) Π(G) Π(GL ) where M is the Levi subgroup GL n corresponding to the Levi subgroup M of G n, which is given by GL n1 GL nt GL n 0 GL nt GL n1. Take τ 0 = l(σ 0 ) to be the local Langlands transfer of σ 0 from G n0 to GL n 0 and define σ := σ 1 det s 1 σ r det st τ 0 σ t det st σ 1 det s 1. The local Langlands functorial transfer l(π) of π is given by Ind GL n (F ) P (F ) (σ ) l(π) which is the unique irreducible quotient. The conditions on local factors are satisfied: for π Π(G n ) and τ Π(GL m ), define L(s, π τ) := L(s, l(π) τ); γ(s, π τ, ψ F ) := γ(s, l(π) τ, ψ F ). The right hand side is defined by Jacquet, Piatetski-Shapiro and Shalika (1983). It is clear that rec F,Gn = rec F,n l : Π(G n ) Φ(G n ) is a local Langlands reciprocity map in general. The surjectivity of the local Langlands reciprocity map has been checked for split classical groups through the work ([JS03]. [JS04], [Liu11], and [JtL13]). This is also expected to be true for quasi-split classical groups. For general reductive algebraic groups over F, we refer to a recent work [ABPS13]. 5. Related Problems We explain here some interesting issues related to the establishment of the reciprocity map in the local Langlands conjecture for split classical groups G n as sketched in the previous section. We treat the case when G n = SO 2n+1 and make comments on the other classical groups, including the quasi-split unitary groups. As explained before, one general strategy to establish the local Langlands conjecture for G n (F ) is to go with the local Langlands functorial transfer from Π(G n ) to Π(GL n ). For G n = SO 2n+1, the complex dual group is G n(c) = Sp 2n (C). In this case, we have that n = 2n. For

12 12 DIHUA JIANG AND CHUFENG NIEN a local Langlands parameter φ Φ(SO 2n+1 ), one has the following structure: Sp 2n (C) GL 2n (C) (5.1) W F SL 2 (C) The local Langlands functorial transfer asserts that the following diagram is commutative: Π(SO 2n+1 ) Π(GL 2n ) (5.2) Φ(SO 2n+1 ) Φ(GL 2n ) In order to carry out this strategy with details, one has to deal with the following problems: (1) Local Langlands Functorial Transfer: l : Π(SO 2n+1 ) Π(GL 2n ); (2) Characterization of Image: l(π(so 2n+1 )) Π(GL 2n ); (3) Surjectivity of the Reciprocity Maps: l(π(so 2n+1 )) = rec 1 F,GL 2n (ι(φ(so 2n+1 ))); (4) Uniqueness of the Reciprocity Maps: rec F,SO2n+1 : Π(SO 2n+1 ) Φ(SO 2n+1 ). In the process, one has to check that the local reciprocity map is compatible with the local factors and goes with other constraints. We will discuss them with some details in the following subsections. Note that the local Langlands functorial transfer is already discussed before Characterization of the image. We consider the local Langlands functorial transfer: l : Π(SO 2n+1 ) Π(GL 2n ) and ask for the characterization of the image of this transfer. When an irreducible admissible representation τ Π(GL 2n ) is supercuspidal, we have the following characterization which is proved in our paper joint with Y.-J. Qin based on previous work of many others. We refer to [JNQ08] for detailed discussion on the background of this theorem and

13 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS 13 what was proved in [JNQ08]. However, when τ is not supercuspidal, much more work needs to be done. Theorem 5.1 ([JNQ08]). Let τ be an irreducible supercuspidal representation of GL 2n (F ). Then the following are equivalent: (1) τ has a nonzero Shalika model. (2) The local exterior square L-factor L(s, τ, 2 ) has a pole at s = 0. (3) The square γ-factor γ(s, τ, 2, ψ F ) has a pole at s = 1. (4) τ is a local Langlands functorial transfer from SO 2n+1 (F ). If one of the above holds for τ, then τ is self-dual. Remark 5.2. By the classification theory for GL 2n (F ), it is enough to describe the image of l explicitly. In principle, this can be established for other classical groups. It remains in progress to extend the above characterization to more general representations in Π(GL 2n ), in order to study other relevant problems Surjectivity of the reciprocity map. In the case of G n = SO 2n+1, one has to show that for any local Langlands parameter φ Φ(GL 2n ), which is of symplectic type, i.e. the image φ(w F SL 2 (C) Sp 2n (C), in other words, φ Φ(SO 2n+1 ), one has to find an irreducible admissible representation π Π(SO 2n+1 ), which has the local Langlands parameter φ. It is a theorem of the first named author and David Soudry in [JS04] which is stated as below. Theorem 5.3 ([JS04]). For any local parameter φ Φ(SO 2n+1 ), there exists (can be explicitly constructed) a member π Π(SO 2n+1 ) such that L(s, π τ) := L(s, rec 1 F,GL (φ) τ); γ(s, π τ, ψ F ) := γ(s, rec 1 F,GL (φ) τ, ψ F ). Hence the reciprocity map is surjective. rec F,SO2n+1 : Π(SO 2n+1 ) Φ(SO 2n+1 ). We remark that Baiying Liu in [Liu11] extended the above theorem to Sp 2n, based on the local descent theory of the first named author and D. Soudry, which is published in [JS12]. Liu with Chris Jantzen in [JtL13] extended the above theorem to split SO 2n. The result should also be true for quasi-split unitary groups, but it still remains to be checked.

14 14 DIHUA JIANG AND CHUFENG NIEN 5.3. Uniqueness of the reciprocity map. In general, one has to show that the local Langlands reciprocity map established above rec F,SO2n+1 : Π(SO 2n+1 ) Φ(SO 2n+1 ). is in fact unique. From the deduction of the general case of the local Langlands conjecture to the tempered case of the local Langlands conjecture, it is enough to prove the uniqueness of the reciprocity map for rec t F,SO 2n+1 : Π temp (SO 2n+1 ) Φ bdd (SO 2n+1 ). Now by the Arthur classification of the tempered spectrum of SO 2n+1 (F ) in [Ar13] and by the local descent theory of the first named author and D. Soudry in [JS03] and [JS12], the Shahidi Conjecture holds. This means that for any local Langlands parameter φ Φ bdd (SO 2n+1 ), the local tempered L-packet Π SO2n+1 (φ) has the property that the subset (φ) of generic members in the packet is nonempty. Hence it is enough to show the uniqueness of the reciprocity map Π gen G rec gen F,SO 2n+1 with the property: : Π gen temp(so 2n+1 ) Φ bdd (SO 2n+1 ) γ(s, π τ, ψ F ) = γ(s, rec gen F,SO 2n+1 (π) rec F,GL (τ), ψ F ) for all τ Π(GL m /F ) and for all m 1. As in the case of GL n, the uniqueness follows from the a positive solution to the Local Converse Problem for GL n. Hence the uniqueness of the local Langlands reciprocity map follows from a positive solution to the Local Converse Problem for G n, which will be discussed in the next section. 6. Local Converse Problems in General In [J06], the first named author formulated the local converse problem for general reductive groups over p-adic local fields. We will specialize the general formulation in [J06] to the split classical groups and make the problem more precise. As remarked in the previous sections, we consider the local converse problem for irreducible generic representations of G n (F ). To recall the notion of Whittaker models, we fix an F -Borel subgroup B = T U with a maximal split torus T, which defines the root system Φ(G n, T ) with the positive roots Φ + determined by U and gives the set of the simple roots. Take a basis vector X α in the one-dimensional F -root space of α, which gives the following isomorphism (6.1) U/[U, U] = α F X α

15 ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS 15 as abelian groups. Any character ψ of U(F ) factors through the quotient U(F )/[U(F ), U(F )], By (6.1), a character ψ of U(F ) is called generic if ψ is nontrivial at each of the simple root α. By the Pontryagin duality, such characters of U(F ) is parameterized by n-tuples a = (a 1,, a n ) (F ) n. An irreducible admissible representation (π, V π ) of G n (F ) is called generic or ψ-generic if the following space Hom U(F ) (V π, ψ) = Hom Gn(F )(V π, Ind Gn(F ) U(F ) (ψ)) is nonzero. Any nonzero functional l ψ Hom U(F ) (V π, ψ) yields a G n (F )-equivariant homomorphism v V π W ψ v (g) = l ψ (π(g)(v)). The subspace {Wv ψ (g) v V π } is called the ψ-whittaker model associated to π. It is of multiplicity one in the space Ind Gn(F ) U(F ) (ψ) because of the uniqueness of local Whittaker models ([Shl74]). For t T (F ), we have that t ψ(u) = ψ(t 1 ut), via the adjoint action. It follows that a π Π(G n ) is ψ-generic if and only if π is t ψ-generic for any t T (F ). In other words, the genericity of π depends on the T (F )-orbit of the generic characters. An easy exercise leads to Proposition 6.1 (([K02])). The set of T (F )-orbits the generic characters of U(F ) is in one-to-one correspondence with H 1 (Γ F, Z Gn ), where Γ F is the absolute Galois group of F and Z Gn is the center of G n. For a π Π gen (G n ), we define (6.2) F(π) = {ψ π is ψ-generic}, which is clearly T (F )-stable. Hence the genericity of π is completely described by the T (F )-orbits F(π)/T (F ). Moreover, we recall from [J06] the following refinement of the Shahidi conjecture. Conjecture 6.2 ([J06]). In a generic local L-packet Π(φ), for any generic members π 1, π 2 Π(φ), the sets F(π 1 ) and F(π 2 ) are disjoint and the union of T (F )-orbits of generic characters over the subset Π gen (φ) of Π(φ) consisting of all generic members in Π(φ), i.e. π Π g (φ) F(π)/T (F ) is in one-to-one correspondence with H 1 (Γ F, Z G ). Based on Conjecture 6.2, we recall from [J06] a general conjecture on the local converse theorem.

16 16 DIHUA JIANG AND CHUFENG NIEN Conjecture 6.3 (Local Converse Theorem). For π 1, π 2 Π gen (G n ), assume that (1) the intersection of F(π 1 ) and F(π 2 ) is not empty, and (2) the twisted local γ-factors are equal, i.e. Then π 1 = π2. γ(s, π 1 τ, ψ) = γ(s, π 2 τ, ψ) holds for all supercuspidal τ Π(GL l ) with l = 1, 2,, n. In [JS03], the first named author with D. Soudry proved a weak version of Conjecture 6.3 for G n = SO 2n+1. Theorem 6.4 ([JS03]). For π, π Π gen (SO 2n+1 ), assume that the twisted local gamma factors γ(π τ, s, ψ) and γ(π τ, s, ψ) are the same, i.e. γ(π τ, s, ψ) = γ(π τ, s, ψ) for all supercuspidal τ Π(GL l ) with l = 1, 2,, 2n 1. Then the representations π and π are equivalent. Note that for π, π Π gen (SO 2n+1 ), it is easy to check that F(π) = F(π ). We refer to [J06] for more detailed discussion on relevant topics. [Ar13] [ABPS13] [BH96] [BH03] [BHK98] [BK93] [Ch96] References Arthur, J. The endoscopic classification of representations: orthogonal and symplectic groups. Colloquium Publication Vol. 61, 2013, American Mathematical Society. Aubert, Anne-Marie; Baum, Paul; Plymen, Roger; Solleveld, Maarten On the local Langlands correspondence for non-tempered representations. arxiv: Bushnell, Colin J. and Henniart, Guy M. Local tame lifting for GL(N). I. Simple characters. Inst. Hautes Études Sci. Publ. Math. 83 (1996), Bushnell, Colin J. and Henniart, Guy M. Local tame lifting for GL(n). IV. Simple characters and base change, Proc. London Math. Soc. (3) 87 (2003), no. 2, Bushnell, Colin J., Henniart, Guy M. and Kutzko, Philip C. Local Rankin-Selberg convolutions for GL n : explicit conductor formula. J. Amer. Math. Soc. 11 (1998), no. 3, Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of GL(N) via compact open subgroups. Annals of Mathematics Studies, 129. Princeton University Press, Princeton, NJ, (1993). Chen, Jiang-Ping Jeff Local Factors, Central Characters, and Representations of the General Linear Group over Non-Archimedean Local Fields. Thesis, Yale University, May (1996).

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18 18 DIHUA JIANG AND CHUFENG NIEN [JZ13] Jiang, Dihua; Zhang, Lei A product of tensor product L-functions for classical groups of hermitian type. submitted (2013) [K02] Kuo, W. Principal nilpotent orbits and reducible principal series. Represent. Theory 6 (2002), [L70] Langlands, R. P. Problems in the theory of automorphic forms. Lectures in modern analysis and applications, III, pp Lecture Notes in Math., Vol. 170, Springer, Berlin, [Liu11] Liu, Baiying Genericity of representations of p-adic Sp 2n and local Langlands parameters. Canad. J. Math. 63(2011), [Mk13] Mok, Chung Pang Endoscopic classification of representations of quasi-split unitary groups. To appear in Memoirs of the American Mathematical Society. [N12] Nien, Chufeng A proof of finite field analogue of Jacquet s conjecture. Accepted by American Journal of Mathematics, [PS08] Paskunas, Vytautas and Stevens, Shaun On the realization of maximal simple types and epsilon factors of pairs. Amer. J. Math. 130 (2008), no. 5, [Shl74] Shalika, J. The multiplicity one theorem for GL n. Ann. of Math. (2) 100 (1974), [T79] Tate, J. Number theoretic background. Proc. Sympos. Pure Math., 33, Automorphic forms, representations and L-functions Part 2, pp. 3 26, Amer. Math. Soc., [Z80] Zelevinsky, A. Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA address: dhjiang@math.umn.edu Department of Mathematics, National Cheng Kung University and National Center for Theoretical Sciences(South), Tainan 701, Taiwan address: nienpig@mail.ncku.edu.tw