Exceptional poles of archimedean Rankin-Selberg L-functions for principal series representations of GL(n,R)

TL;DR

This paper establishes the equivalence of two types of exceptional poles for Rankin-Selberg L-functions of principal series representations of GL(n,R) and expresses these L-functions via exceptional factors related to derivatives.

Contribution

It proves the coincidence of exceptional pole types for principal series representations and relates L-functions to exceptional factors from derivatives.

Findings

Exceptional poles of type 1 and 2 coincide for principal series representations.

L(s,π₁×π₂) can be expressed using exceptional L-factors from derivatives.

The results apply to irreducible principal series in general position.

Abstract

We prove that for any pair of irreducible principal series representations $(\pi_1,\pi_2)$ of $\operatorname{GL}_n(\mathbb{R})$ in general position, the notions of exceptional pole of type 1 and type 2 coincide. Using this identification, we express the Rankin--Selberg $L$-function $L(s,\pi_1\times\pi_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of the derivatives of $\pi_1$ and $\pi_2$.