In mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups

Reductive group

In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...

 over a number field

Algebraic number field

In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...

. This includes Rankin–Selberg

Rankin–Selberg method

In mathematics, the Rankin–Selberg method, introduced by and , also known as the theory of integral representations of L-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic L-functions. Some authors reserve the term for a...

 products for cuspidal automorphic representations of general linear groups

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

. The method develops the theory of the local coefficient, which links to the global theory via Eisenstein series

Eisenstein series

Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...

. The resulting L-functions satisfy a number of analytic properties, including an important functional equation.

The local coefficient

The setting is in the generality of a connected quasi-split reductive group , together with a Levi subgroup , defined over a local field

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

 . For example, if is a classical group

Classical group

In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...

 of rank , its maximal Levi subgroups are of the form , . F. Shahidi

Freydoon Shahidi

Freydoon Shahidi is an Iranian mathematician who is currently a Distinguished Professor of Mathematics at Purdue University in the U.S. He is known for a method of automorphic L-functions which is now known as the Langlands–Shahidi method. He is a member of the American Academy of Arts and...

 develops the theory of the local coefficient for irreducible generic representations of . The local coefficient is defined by means of the uniqueness property of Whittaker models

Whittaker model

In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space of functions on the group. It is named after E. T...

 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

Robert Langlands

Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...

' theory of Eisenstein series can be decomposed as a product of local intertwining operators. When 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 to individual functional equations of partial -functions and -factors:


The details are technical: is a finite set of places (of the underlying global field) with unramified for , and is the adjoint action of on the complex Lie algebra of a specific subgroup of the Langlands dual group

Dual group

In mathematics, the dual group may be:* The Pontryagin dual of a locally compact abelian group* The Langlands dual of a reductive algebraic group* The Deligne-Lusztig dual of a reductive group over a finite field....

 of . When is the special linear group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

 , and is the maximal torus of diagonal matrices, -factors ( is a character of idèle classes in this situation) are the local factors of Tate's thesis

John Tate

John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...

.

The -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

Artin L-function

In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...

 and Artin root numbers when gives an archimedean local field or when is non-archimedean and is a constituent of an unramified principal series representation of . Local -functions and root numbers ε are then defined at every place, including , by means of Langlands classification for -adic groups. The functional equation takes the form

are the completed global -function and root number.

Examples of automorphic L-functions

• , the Rankin–Selberg -function of cuspidal automorphic representations of and of .

• , where is a cuspidal automorphic representation of and is a globally generic cuspidal automorphic representation of a classical group .

• , with as before and a symmetric square, an exterior square, or an Asai representation of the dual group of .

A full list of Langlands–Shahidi L-functions depends on the quasi-split group and maximal Levi subgroup . More specifically, the decomposition of the adjoint action can be classified using Dynkin diagrams.

Analytic properties of L-functions

Global -functions are said to be nice if they satisfy:

1. extend to entire functions of the complex variable .
2. are bounded in vertical strips.
3. (Functional Equation) .

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

Another result is the non-vanishing of -functions. For Rankin–Selberg products of general linear groups it states that is non-zero for every real number t.

Applications to functoriality and to representation theory of -adic groups

• Functoriality for the classical groups: A cuspidal globally generic automorphic representation of a classical group admits a Langlands functorial
  Langlands program
  The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
  
   lift to an automorphic representation of , where depends on the classical group. Then, the Ramanujan bounds of W. Luo, Z. Rudnick and P. Sarnak for over number fields yield non-trivial bounds for the generalized Ramanujan conjecture of the classical groups.

• Symmetric powers for : Proofs of functoriality for the symmetric cube and for the symmetric fourth powers of cuspidal automorphic representations of 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 .

• Representations of -adic groups: Applications involving Harish-Chandra
  Harish-Chandra
  Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...
  
  functions (from the Plancherel formula) and to complementary series of -adic reductive groups are possible. For example, appears as the Siegel Levi subgroup of a classical group G. If is a smooth irreducible ramified supercuspidal representation of over a field of -adic numbers, and is irreducible, then:

1. is irreducible and in the complementary series for ;
2. is reducible and has a unique generic non-supercuspidal discrete series subrepresentation;
3. is irreducible and never in the complementary series for .

Here, is obtained by unitary parabolic induction from

• if , , or ;
• if , or .