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. 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. 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 = Gl is a classical group of rank l, its maximal Levi subgroups are of the form GL(m) × Gn, where Gn is a classical group of rank n and of the same type as Gl, l = m + n. F. Shahidi develops the theory of the local coefficient for irreducible generic representations of M(F). 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 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. An induction step refines the crude functional equation of a globally generic cuspidal automorphic representation π = ⊗ ′ π v {displaystyle pi =otimes 'pi _{v}} to individual functional equations of partial L-functions and γ-factors: The details are technical: s a complex variable, S a finite set of places (of the underlying global field) with π v {displaystyle pi _{v}} unramified for v outside of S, and r = ⊕ r i {displaystyle 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 SL(2), and M = T is the maximal torus of diagonal matrices, then π is a Größencharakter and the corresponding γ-factors are the local factors of Tate's thesis. The γ-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 π v {displaystyle pi _{v}} is a constituent of an unramified principal series representation of M(F). Local L-functions and root numbers ε ( s , π v , r i , v , ψ v ) {displaystyle (s,pi _{v},r_{i,v},psi _{v})} are then defined at every place, including v ∈ S {displaystyle vin S} , by means of Langlands classification for p-adic groups. The functional equation takes the form where L ( s , π , r i ) {displaystyle L(s,pi ,r_{i})} and ϵ ( s , π , r i ) {displaystyle epsilon (s,pi ,r_{i})} are the completed global L-function and root number. A full list of Langlands–Shahidi L-functions depends on the quasi-split group G and maximal Levi subgroup M. More specifically, the decomposition of the adjoint action r = ⊕ r i {displaystyle r=oplus r_{i}} can be classified using Dynkin diagrams. A first study of automorphic L-functions via the theory of Eisenstein Series can be found in Langlands' Euler Products, under the assumption that the automorphic representations are everywhere unramified. What the Langlands–Shahidi method provides is the definition of L-functions and root numbers with no other condition on the representation of M other than requiring the existence of a Whittaker model. Global L-functions are said to be nice if they satisfy: