Gamma Factors of Distinguished Representations of GL_n(C)

Abstract

Let $(\pi ,V)$ be a $G{L}_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $G{L}_n(\mathbb{C})$, let ${\pi }^{\prime }$ be an irreducible, admissible, $G{L}_m(\mathbb{R})$-distinguished representation of $G{L}_m(\mathbb{C})$, 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 $\gamma (\frac{1}{2},\pi \times {\pi }^{\prime };\psi )=1$. The result follows as a simple consequence from the characterisation of $G{L}_n(\mathbb{R})$-distinguished representations in terms of their Langlands data.

28 pages, version 2, added converse theorem

References in corpus (1)

• (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F

Cited by in corpus (2)

• On Whittaker--Fourier coefficients of automorphic forms on unitary groups: reduction to a local identity
• Representations of reductive groups distinguished by symmetric subgroups