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

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 $\pi_2$.