Gamma Factors of Distinguished Representations of GL_n(C)

TL;DR

This paper proves that the Rankin-Selberg gamma factor at s=1/2 equals 1 for certain distinguished representations of complex general linear groups, based on their Langlands parameters.

Contribution

It establishes a simple proof that gamma factors equal 1 for $GL_n(R)$-distinguished representations of $GL_n(C)$, linking gamma factors to Langlands data.

Findings

Gamma factor at s=1/2 is 1 for distinguished representations.

Characterization of distinguished representations via Langlands data.

Simplifies understanding of gamma factors in this context.

Abstract

Let $(\pi,V)$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$, let $\pi'$ be an irreducible, admissible, $GL_m(\mathbb{R})$-distinguished representation of $GL_m(\mathbb{C})$, and let $\psi$ be a non-trival character of $\mathbb{C}$ which is trivial on $\mathbb{R}$. We prove that Rankin-Selberg gamma factor at $s=\frac{1}{2}$ is $\gamma(\frac{1}{2},\pi \times \pi'; \psi) = 1$. The result follows as a simple consequence from the characterisation of $GL_n(\mathbb{R})$-distinguished representations in terms of their Langlands data.