Gamma Factors of Distinguished Representations of GL_n(C)

Let $(π,V)$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$, let $π'$ be an irreducible, admissible, $GL_m(\mathbb{R})$-distinguished representation of $GL_m(\mathbb{C})$, and let $ψ$ 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 $γ(\frac{1}{2},π\times π'; ψ) = 1$. The result follows as a simple consequence from the characterisation of $GL_n(\mathbb{R})$-distinguished representations in terms of their Langlands data.

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