Langlands–Shahidi method

In mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Selberg products for cuspidal automorphic representations of general linear groups. The method develops the theory of the local coefficient, which links to the global theory via Eisenstein series. The resulting L-functions satisfy a number of analytic properties, including an important functional equation.

1 The local coefficient
2 Local factors and functional equation
3 Examples of automorphic L-functions
4 Analytic properties of L-functions
5 Applications to functoriality and to representation theory of -adic groups
6 References

The local coefficient

The setting is in the generality of a connected quasi-split reductive group $G$ , together with a Levi subgroup $M$ , defined over a local field $F$ . For example, if $G = G_l$ is a classical group of rank $l$ , its maximal Levi subgroups are of the form ${\rm GL}_m\times {\rm GL}_n$ , $l=m%2Bn$ . F. Shahidi develops the theory of the local coefficient for irreducible generic representations of $M(F)$ ^[1]. The local coefficient is defined by means of the uniqueness property of Whittaker models paired with the theory of intertwining operators for representations obtained by parabolic induction from generic representations.

The global intertwining operator appearing in the functional equation of Langlands' theory of Eisenstein series^[2] can be decomposed as a product of local intertwining operators. When $M$ is a maximal Levi subgroup, local coefficients arise from Fourier coefficients of appropriately chosen Eisenstein series and satisfy a crude functional equation involving a product of partial L-functions.

Local factors and functional equation

An induction step refines the crude functional equation of a globally generic cuspidal automorphic representation $\pi = \otimes' \pi_v$ to individual functional equations of partial $L$ -functions and $\gamma$ -factors^[3]:

$L^S(s,\pi,r_i) = \prod_{v \in S} \gamma_i(s,\pi_v,\psi_v) L^S(1-s,\tilde{\pi},r_i).$

The details are technical: $s \in \mathbf{C}, S$ is a finite set of places (of the underlying global field) with $\pi_v$ unramified for $v \not\in S$ , and $r = \oplus r_i$ is the adjoint action of $M$ on the complex Lie algebra of a specific subgroup of the Langlands dual group of $G$ . When $G$ is the special linear group ${\rm SL}_2$ , and $M = T$ is the maximal torus of diagonal matrices, $\gamma$ -factors ( $\pi = \otimes' \pi_v$ is a character of idèle classes in this situation) are the local factors of Tate's thesis.

The $\gamma$ -factors are uniquely characterized by their role in the functional equation and a list of local properties, including multiplicativity with respect to parabolic induction. They satisfy a relationship involving Artin L-functions and Artin root numbers when $v$ gives an archimedean local field or when $v$ is non-archimedean and $\pi_v$ is a constituent of an unramified principal series representation of $M(F)$ . Local $L$ -functions and root numbers ε $(s,\pi_v,r_{i,v},\psi_v)$ are then defined at every place, including $v \in S$ , by means of Langlands classification for $p$ -adic groups. The functional equation takes the form

$L(s,\pi,r_i) = \epsilon(s,\pi,r_i) L(1-s,\tilde{\pi},r_i),\text{ where } L(s,\pi,r_i)\text{ and }\varepsilon(s,\pi,r_i)$

are the completed global $L$ -function and root number.

Examples of automorphic L-functions

$L(s,\pi_1 \times \pi_2)$ , the Rankin–Selberg $L$ -function of cuspidal automorphic representations $\pi_1$ of ${\rm GL}_m$ and $\pi_2$ of ${\rm GL}_n$ .

$L(s,\tau \times \pi)$ , where $\tau$ is a cuspidal automorphic representation of ${\rm GL}_m$ and $\pi$ is a globally generic cuspidal automorphic representation of a classical group $G$ .

$L(s,\tau,r)$ , with $\tau$ as before and $r$ a symmetric square, an exterior square, or an Asai representation of the dual group of ${\rm GL}_n$ .

A full list of Langlands–Shahidi L-functions^[4] depends on the quasi-split group $G$ and maximal Levi subgroup $M$ . More specifically, the decomposition of the adjoint action $r = \oplus r_i$ can be classified using Dynkin diagrams.

Analytic properties of L-functions

Global $L$ -functions are said to be nice^[5] if they satisfy:

$L(s,\pi,r), \ L(s,\tilde{\pi}, r) \$ extend to entire functions of the complex variable $s$ .
$L(s,\pi,r), \ L(s,\tilde{\pi},r) \$ are bounded in vertical strips.
(Functional Equation) $L(s,\pi,r) = \epsilon(s,\pi,r) L(1-s,\tilde{\pi},r)$ .

Langlands–Shahidi $L$ -functions satisfy the functional equation. Progress towards boundedness in vertical strips was made by S. S. Gelbart and F. Shahidi^[6]. And, after incorporating twists by highly ramified characters, Langlands–Shahidi $L$ -functions do become entire^[7].

Another result is the non-vanishing of $L$ -functions. For Rankin–Selberg products of general linear groups it states that $L(1%2Bit,\pi_1 \times \pi_2)$ is non-zero for every real number t.

Applications to functoriality and to representation theory of $p$ -adic groups

Functoriality for the classical groups: A cuspidal globally generic automorphic representation of a classical group admits a Langlands functorial lift to an automorphic representation of ${\rm GL}_N$ ^[8], where $N$ depends on the classical group. Then, the Ramanujan bounds of W. Luo, Z. Rudnick and P. Sarnak^[9] for ${\rm GL}_N$ over number fields yield non-trivial bounds for the generalized Ramanujan conjecture of the classical groups.

Symmetric powers for ${\rm GL}_2$ : Proofs of functoriality for the symmetric cube and for the symmetric fourth^[10] powers of cuspidal automorphic representations of ${\rm GL}_2$ were made possible by the Langlands–Shahidi method. Progress towards higher Symmetric powers leads to the best possible bounds towards the Ramanujan–Peterson conjecture of automorphic cusp forms of ${\rm GL}_2$ .

Representations of $p$ -adic groups: Applications involving Harish-Chandra $\mu$ functions (from the Plancherel formula) and to complementary series of $p$ -adic reductive groups are possible. For example, ${\rm GL}_n$ appears as the Siegel Levi subgroup of a classical group G. If $\pi$ is a smooth irreducible ramified supercuspidal representation of ${\rm GL}_n(F)$ over a field $F$ of $p$ -adic numbers, and $I(\pi) = I(0,\pi)$ is irreducible, then:

$I(s,\pi)$ is irreducible and in the complementary series for $0<s<1$ ;
$I(1,\pi)$ is reducible and has a unique generic non-supercuspidal discrete series subrepresentation;
$I(s,\pi)$ is irreducible and never in the complementary series for $s>1$ .

Here, $I(s,\pi)$ is obtained by unitary parabolic induction from

$\pi \otimes |\det|^s$ if $G ={\rm SO}_{2n}$ , ${\rm Sp}_{2n}$ , or ${\rm U}(n%2B1,n)$ ;
$\pi \otimes |\det|^{s/2}$ if $G={\rm SO}_{2n%2B1}$ , or ${\rm U}(n,n)$ .

References

^ F. Shahidi, On certain $L$ -functions, American Journal of Mathematics 103 (1981), 297–355.
^ R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., Vol. 544, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
^ F. Shahidi, A proof of Langlands conjecture on Plancherel measures; Complementary series for $p$ -adic groups, Annals of Mathematics 132 (1990), 273–330.
^ F. Shahidi, Eisenstein Series and Automorphic $L$ -functions, Colloquium Publications, Vol. 58, American Mathematical Society, Providence, Rhode Island, 2010.
^ J. W. Cogdell and I. I. Piatetski–Shapiro, Converse theorems for ${\rm GL}_n$ , Publications Mathématiques de l'IHÉS 79 (1994), 157–214.
^ S. Gelbart and F. Shahidi, Boundedness of automorphic $L$ -functions in vertical strips, Journal of the American Mathematical Society, 14 (2001), 79–107.
^ H. H. Kim and F. Shahidi, Functorial products for ${\rm GL}_2 \times {\rm GL}_3$ and the symmetric cube for ${\rm GL}_2$ , Annals of Mathematics 155 (2002), 837–893.
^ J. W. Cogdell, H. H. Kim, I. I. Piatetski–Shapiro, and F. Shahidi, Functoriality for the classical groups, Publications Mathématiques de l'IHÉS 99 (2004), 163–233
^ W. Luo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for ${\rm GL}(n)$ , Proceedings of Symposia in Pure Mathematics 66, part 2 (1999), 301–310.
^ H. H. KimFunctoriality for the exterior square of ${\rm GL}_4$ and the symmetric fourth of ${\rm GL}_2$ , Journal of the American Mathematical Society 16 (2002), 131–183.