Langlands-Shahidi Method Research Articles

The Langlands-Shahidi method for pairs via types and covers

We compute the local coefficient attached to a pair ( π 1 , π 2 ) (\pi _1,\pi _2) of supercuspidal (complex) representations of the general linear group using the theory of types and covers à la Bushnell-Kutzko. In the process, we obtain another proof of a well-known formula of Shahidi for the corresponding Plancherel constant. The approach taken here can be adapted to other situations of arithmetic interest within the context of the Langlands-Shahidi method, particularly to that of a Siegel Levi subgroup inside a classical group.

Langlands-Shahidi $L$-functions for $GSpin$ groups and the generic Arthur packet conjecture

We prove that $L$-functions from Langlands-Shahidi method in the case of $GSpin$ groups over a non-Archimedean local field of characteristic zero are Artin $L$-functions through the local Langlands correspondence. It has an application on the proof of a weak version of the generic Arthur packet conjecture. Furthermore, we study and describe a local $L$-packet that contains a generic member in the case of $GSpin$ groups. Using this description of a local $L$-packet, we strengthen a weak version of the generic Arthur packet conjecture in those cases (local version of the generalized Ramanujan conjecture).

Local Langlands Correpondence for the twisted exterior and symmetric square epsilon-factors of GL(N)

In this paper, we prove the equality of the local arithmetic and analytic epsilon- and L-factors attached to the twisted exterior and symmetric square representations of GL(N). We will construct the twisted symmetric square local analytic gamma- and L-factor of GL(N) by applying Langlands-Shahidi method to odd GSpin groups. Then we reduce the problem to the stablity of local coefficients, and eventually prove the analytic stabitliy in this case by some analysis on the asymptotic behavior of certain partial Bessel functions.

On the generic local Langlands correspondence for 𝐺𝑆𝑝𝑖𝑛 groups

In the case of split G S p i n GSpin groups, we prove an equality of L L -functions between automorphic local L L -functions defined by the Langlands-Shahidi method and local Artin L L -functions. Our method of proof is based on previous results of the first author which allow us to reduce the problem to supercuspidal representations of Levi subgroups of G S p i n GSpin , by constructing Langlands parameters for general generic irreducible admissible representations of G S p i n GSpin from the one for generic irreducible supercuspidal representations of its Levi subgroups.

On automorphic L-functions in positive characteristic

We give a proof of the existence of Asai, exterior square, and symmetric square local $L$-functions, $\gamma$-factors and root numbers in characteristic $p$, including the case of $p = 2$. Our study is made possible by developing the Langlands-Shahidi method over a global function field in the case of a Siegel Levi subgroup of a split classical group or a quasi-split unitary group. The resulting automorphic $L$-functions are shown to satisfy a rationality property and a functional equation. A uniqueness result of the author and G. Henniart allows us to show that the definitions provided in this article are in accordance with the local Langlands conjecture for ${\rm GL}_n$. Furthermore, in order to be self contained, we include a treatise of $L$-functions arising from maximal Levi subgroups of general linear groups.

Strongly positive representations of $$GSpin_{2n+1}$$ G S p i n 2 n + 1 and the Jacquet module method

We explicitly construct the structure of Jacquet modules of parabolically induced representations of $$GSpin_{2n+1}$$ over a $$p$$ -adic field $$F$$ of any characteristic. Using this construction of the Jacquet module, we obtain a classification of strongly positive representations of $$GSpin_{2n+1}$$ over $$F$$ and describe the general discrete series representations of $$GSpin_{2n+1}$$ over $$F$$ , assuming the half-integer conjecture. One application of this paper is the proof of the equality of $$L$$ -functions from the Langlands–Shahidi method and Artin $$L$$ -functions through the local Langlands correspondence (Kim in Langlands–Shahidi $$L$$ -functions for $$GSpin$$ groups and the generic Arthur packet conjecture, preprint).

Complementary results on the Rankin–Selberg gamma factors of classical groups

Let G be an orthogonal or symplectic group, defined over a local field, or the metaplectic group. We study the γ-factor for a pair of irreducible generic representations of G×GLn, defined using the Rankin–Selberg method. In the metaplectic case we use Shimura type integrals. We prove that the γ-factor satisfies a list of fundamental properties, stated by Shahidi, which define it uniquely. In particular, we show full multiplicativity for symplectic and metaplectic groups. It is important for applications to relate this γ-factor to the one arising from the Langlands–Shahidi method. As a corollary of our results, these factors coincide. This is a refinement of previous works on orthogonal groups, showing such an equality up to certain normalization factors.

Uniqueness of Rankin–Selberg products

In the present paper, we show the equality of the γ -factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands–Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local–global principle for GL n which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of γ -factors is made via a uniqueness result for Rankin–Selberg γ -factors.

Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method

Let \(\overline {S{p_{2n}}({\rm{ F }})} \) be the metaplectic double cover of F where F is a local field of characteristic 0. We use the Uniqueness of Whittaker model to define a metaplectic analog to Shahidi local coefficients and we use these coefficients to define gamma factors. We show that these gamma factors are multiplicative and satisfy the crude global functional equation. Then, we compute these factors in various cases and obtain explicit formulas for Plancherel measures. These computations are then used to prove some irreducibility theorems for parabolic induction on the metaplectic group over p-adic fields. In particular, we show that all principal series representations induced from unitary characters are irreducible. We also prove that parabolic induction from unitary supercuspidal representation of the Siegel parabolic sub group is irreducible if and only if a certain parabolic induction on F is irreducible.

Local-to-global extensions for GL n in non-zero characteristic: a characterization of γ F ( s , π, Sym 2 , ψ) and γ F ( s , π, ∧ 2 , ψ)

Let $F$ be a locally compact field of positive characteristic. In his thesis, the second author used the Langlands-Shahidi method to define $\gamma$-factors for the symmetric square and exterior square representations of the dual group of ${\rm GL}_n(F)$. We prove here that such $\gamma$-factors are characterized by local properties --- including a multiplicativity property with respect to parabolic induction --- and their role in global functional equations for $L$-functions. For this, we prove that a given cuspidal representation of ${\rm GL}_n(F)$ is the component at some place of a global automorphic representation, the ramification of which is controlled at other places. In fact, we use an equivalent result for $\ell$-adic Galois representations, due to O.~Gabber and N.~Katz, and transport it via the global Langlands correspondence proved by L.~Lafforgue.

Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Langlands Functoriality Conjecture

Functoriality conjecture is one of the central and influential subjects of the present day mathematics. Functoriality is the profound lifting problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository article, I describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which are powerful tools for the establishment of functoriality of some important cases, and survey the interesting results related to functoriality conjecture.

Functoriality for the Classical Groups over Function Fields

Langlands' functoriality for generic representations from the split classical groups to an appropriate GL N is established. The functorial lift or transfer to GL N is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands-Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands-Shahidi method has only been developed over number fields. To overcome many difficulties, stability of γ-factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.

Square Integrable Representations and the Standard Module Conjecture for General Spin Groups

Abstract.In this paper we study square integrable representations and L -functions for quasisplit general spin groups over a p-adic field. In the first part, the holomorphy of L -functions in a half plane is proved by using a variant formof Casselman's square integrability criterion and the Langlands–Shahidi method. The remaining part focuses on the proof of the standard module conjecture. We generalize Muić's idea via the Langlands–Shahidimethod towards a proof of the conjecture. It is used in the work of M. Asgari and F. Shahidi on generic transfer for general spin groups.

On the residual spectrum of Hermitian quaternionic inner form of SO_8

In this paper we decompose the residual spectrum sup- ported in the minimal parabolic subgroup of an inner form of the split group SO8. The approach is the Langlands spectral theory. However, since the group is non-quasi-split, it is out of the scope of the Langlands-Shahidi method and the new technique for the normalization of standard intertwin- ing operators is developed. The decomposition shows interesting parts of the residual spectrum not appearing in the case of quasi-split groups.

The residual spectrum of an inner form of 𝑆𝑝₈ supported in the minimal parabolic subgroup

The part of the residual spectrum of an inner form of the split group S p 8 Sp_8 supported in the minimal parabolic subgroup is decomposed. Since the considered inner form is not quasi–split, the normalization of the standard intertwining operators, required for the calculation of the poles of the Eisenstein series, is out of the reach of the Langlands–Shahidi method. Hence, a normalization technique, based on the transfer of the Plancherel measure between the split group and its inner form, is applied. The obtained decomposition reveals certain features of the residual spectrum of the inner form which do not appear for the split group.

Langlands–Shahidi method and poles of automorphic L-functions III: Exceptional groups

In this paper, we apply Langlands–Shahidi method to exceptional groups, with the assumption that the cuspidal representations have one spherical tempered component. A basic idea is to use the fact that the local components of residual automorphic representations are unitary representations, and use the classification of the unitary dual. We prove non-unitarity of certain spherical representations of exceptional groups. We need to divide into five different cases, and in two cases we can prove that the completed L-functions are holomorphic except possibly at 0, 1/2, 1 under some local assumptions.

On the fundamental automorphic L-functions of SO(2n + 1)

The fundamental automorphic L-functions of SO2n+1 are by definition the Langlands automorphic L-functions attached to irreducible cuspidal automorphic representations σ of SO2n+1(A) and the fundamental complex representations ρ1, ρ2, . . . , ρn of the complex dual group Sp2n(C) of SO2n+1,whereA is the ring of adeles of the number field k. These Lfunctions are denoted by L(s, σ, ρj)which is given by a Euler product of all local L-factors. The precise definition will be given in Section 2. It is known by a theorem of Langlands that the L-functions L(s, σ, ρj) converge absolutely for the real part of s large [6]. The Langlands conjecture asserts that L(s, σ, ρj) should have meromorphic continuation to the whole complex plane C, satisfy a functional equation relating the value at s to the value at 1−s, and have finitely many poles on the real line. When j = 1, L(s, σ, ρ1) is the standard L-function. In this case, the Langlands conjecture has been verified through the doubling method of Gelbart, Piatetski-Shapiro, and Rallis [11] and through the Langlands-Shahidi method [12]. For j ≥ 2, it is clear that the Langlands conjecture for L(s, σ, ρj) is beyond reach via the Langlands-Shahidi method or via any currently known integral representation of the Rankin-Selberg-type (except for certain cases of n ≤ 4 and j = 2 [7]).

On Local L-Functions and Normalized Intertwining Operators

AbstractIn this paper we make explicit all L-functions in the Langlands–Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local L-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for Re(s) ≥ 1/2 in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrumand determining poles of automorphic L-functions.

Infinite Dimensional Groups and Automorphic L-Functions

In this note we will speculate on the possibility of establishing higher symmetric power transfers for cusp forms on GL2(AF), where AF is the ring of adeles of a number field F, by means of extending the Langlands–Shahidi method to infinite dimensional groups (Section 6). It appears that a straightforward generalization [14, 15] of the theory of Eisenstein series from the finite dimensional theory [39, 40] may not suffice. A conceptually different generalization would be necessary if one wants to establish further transfers from forms on GLm(AF) × GLn(AF) to GLmn(AF), from which functoriality for higher symmetric powers could be obtained. We note that to obtain the fifth symmetric power transfer of a cusp form π on GL2(AF), it would be enough to prove that π × Sym 4 π can be transferred to an automorphic representation π ⊠ Sym 4 π of GL10(AF), from which the existence of Sym 5 π follows by an appeal to the classification of automorphic forms on GLN(AF). This was successfully accomplished for the transfer of forms on GL2(AF)×GL3(AF) to GL6(AF) in [34]. No higher transfers of forms on products of general linear groups is available at present. We refer to [44] for GL2 × GL2. We invite the reader to read Section 6 carefully to appreciate different aspects of this observation and evidence for it as well as its positive consequences towards Garland’s work [14, 15] on establishing meromorphic continuation of Eisenstein series for loop groups. We motivate the paper by discussing some very important conjectures in number theory such as those of Ramanujan–Petersson and Selberg on cuspidal representations of GL2(AF), as well as generic forms on groups whose connected L–groups have a classical derived group. We concentrate on the former and refer the reader to [56] for a detailed expository treatment of latter. We conclude