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.

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 GL(m) × G_n, where G_n is a classical group of rank n and of the same type as G_l, 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 ${\displaystyle \pi =\otimes '\pi _{v}}$ to individual functional equations of partial L-functions and γ-factors:

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