Prasad–Takloo-Bighash conjecture for distinguished representations of inner forms of general linear groups

Let $F$ be a non-Archimedean local field of characteristic not $2$, let $E/F$ be a quadratic extension, and let $D$ be a central division algebra over $F$ of dimension $d^2$. Set $G=\operatorname{GL}_n(D)$, assume that $E$ embeds in $\operatorname{Mat}_n(D)$, and let $H=C_G(E^\times)$. Write $\eta_{E/F}$ for the quadratic character of $F^\times$ associated with $E/F$. An admissible representation is $H$-distinguished if $\operatorname{Hom}_H(\pi,1)\neq 0$.

Prasad–Takloo-Bighash conjecture. Let $\pi$ be an irreducible admissible representation of $G$ with trivial central character. If $\pi$ is $H$-distinguished, then:

1. The Langlands parameter of $\pi$ takes values in

$$\operatorname{Sp}_{nd}(\mathbb{C}).$$

2. Its root number satisfies

$$\varepsilon(\pi)\varepsilon(\pi\otimes\eta_{E/F})=(-1)^n\eta_{E/F}(-1)^{nd/2}.$$

Conversely, if $\pi$ is essentially square integrable and satisfies these two conditions, then $\pi$ is $H$-distinguished.

The conjecture describes distinguished representations through both their Langlands parameters and root numbers. In the source, the original formulation is noted to assume that $\pi$ corresponds to a generic representation of $\operatorname{GL}_{nd}(F)$; the paper reduces the classification of standard modules with nonzero linear periods to the essentially square-integrable case.