Gamma factors of distinguished representations of $\operatorname{GL}_n(\mathbb{C})$

Alexander Kemarsky

Vol. 278 (2015), No. 1, 137–172

DOI: 10.2140/pjm.2015.278.137

Abstract

Let $\left(\pi ,V\right)$ be a ${GL}_n\left(ℝ\right)$-distinguished, irreducible, admissible representation of ${GL}_n\left(ℂ\right)$, let ${\pi }^{\prime }$ be an irreducible, admissible, ${GL}_m\left(ℝ\right)$-distinguished representation of ${GL}_m\left(ℂ\right)$, and let $\psi$ be a nontrivial character of $ℂ$ which is trivial on $ℝ$. We prove that the Rankin–Selberg gamma factor at $s=1∕2$ is $\gamma \left(1∕2,\pi \times {\pi }^{\prime };\psi \right)=1$. The result follows as a simple consequence from the characterization of ${GL}_n\left(ℝ\right)$-distinguished representations in terms of their Langlands data.

Keywords

Rankin–Selberg gamma factor, distinguished representations, Langlands classification, Whittaker model

Mathematical Subject Classification 2010

Primary: 22E30

Secondary: 20G05, 11F70

Milestones

Received: 12 October 2014

Revised: 12 March 2015

Accepted: 9 April 2015

Published: 30 September 2015