These notes are from the Database of Automorphic L-functions at http://www.math.rutgers.edu/∼sdmiller/l-functions . They were written up by Stephen Miller, and are based on discussions with Freydoon Shahidi, Purdue University. They are meant to serve as an introduction to the Langlands-Shahidi method of studying L-functions through the Fourier coefficients of Eisenstein series.

To analyze the standard intertwining operators for induced representations of a reductive group, one is naturally led to study certain integrals over the corresponding unipotent radical N . A Levi component M acts on N by the adjoint action, and so we may decompose the integral according to the Ad(M)-orbits. We treat unitary and classical cases in which M is the product G×H of two groups relate...

We sharpen some estimates of Rankin on power sums of Hecke eigenvalues, by using Kim & Shahidi’s recent results on higher order symmetric powers. As an application, we improve Kohnen, Lau & Shparlinski’s lower bound for the number of Hecke eigenvalues of same signs.

‎We illustrate a method of computing Langlands-Shahidi local coefficients via the theory of types and covers‎. ‎The purpose of this paper is to illustrate a method of computing the Langlands-Shahidi local coefficients using the theory of types and covers.

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 a...

The purpose of this note is to announce an extension of the descent method of Ginzburg, Rallis, and Soudry to the setting of essentially self dual representations. This extension of the descent construction provides a complement to recent work of Asgari and Shahidi [A-S] on the generic transfer for general Spin groups as well as to the work of Asgari and Raghuram [A-R] on cuspidality of the ext...

Let π ⊗ σ be a supercuspidal representation of GL(2n)× SO(2n) over a p-adic field with π selfdual, where SO(2n) stands for a quasisplit even special orthogonal group. In order to study its normalized parabolic induction to SO(6n), Goldberg and Shahidi defined a pairing R between the matrix coefficients of π and σ which controls the residue of the standard intertwining operator. The elliptic par...