The value of the global intertwining operators on spherical vectors

TL;DR

This paper computes the meromorphic functions from global intertwining operators acting on spherical vectors in automorphic forms, revealing well-behaved pole ratios, using Langlands-Shahidi theory for unramified groups over global fields.

Contribution

It provides an explicit computation of the meromorphic functions associated with global standard intertwining operators on spherical vectors, demonstrating their pole behavior in the context of unramified reductive groups.

Findings

Explicit formulas for meromorphic functions of intertwining operators

Analysis of pole ratios in the positive Weyl chamber

Application of Langlands-Shahidi theory to unramified groups

Abstract

Let F be a global field, G an unramified quasi-split reductive group over F and chi an everywhere unramified automorphic character of a maximal maximally split torus of G. Using Langlands-Shahidi theory, we compute the meromorphic function defined by the action of a global standard intertwining operator associated to chi on a spherical vector and show that the ratio of its poles in the positive Weyl chamber is well behaved.