On Representations of $\mathrm{GL}_n(\mathrm{D})$ admitting a generalized linear period

Abstract: Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, and let $\mathrm{G}<em>n=\mathrm{GL}_n(\mathrm{D})$. Let $$\mathrm{H}</em>{1,n-1}={\mathrm{diag}(g_1,g_2)\in\mathrm{G}<em>1,\ g_2\in\mathrm{G}</em>{n-1}}$$ and, for $s \in \mathbb{R}$, define $$χ<em>s(\mathrm{diag}(g_1,g_2))=ν(g_1)<sup>{2s}ν(g_2)<sup>{-2s}$$. We classify, for $n=3,4$, the irreducible smooth representations of $\mathrm{G}_n$ admitting a generalized linear period with respect to $(\mathrm{H}</em>{1,n-1},χ<em>s)$. Motivated by these results, we conjecture a complete classification for all $n&gt;2$. Assuming this conjecture, we characterize such representations in terms of Langlands parameters: an irreducible smooth representation $π$ of $\mathrm{G}_n$ admits such a period if and only if $\mathfrak{L}(π)$ contains a Weil-Deligne subrepresentation isomorphic to $\mathfrak{L}(ν<sup>{-2s})$, the $(2n-4)$-dimensional parameter of $ν<sup>{-2s}$ on $\mathrm{G}</em>{n-2}$, and the four-dimensional quotient is the Langlands parameter of either the trivial representation of $\mathrm{G}<em>2$, with $s=\pm\frac{n-2}{2}$, or an irreducible infinite-dimensional $\mathrm{H}</em>{1,1}$-distinguished representation of $\mathrm{G}<em>2$. We also verify the Lapid-Prasad conjecture in this setting: the $L$-packet of an irreducible representation of $\mathrm{G}_n$ admitting a linear period with respect to $\mathrm{H}</em>{1,n-1}$ is invariant under $ρ\mapsto\widetildeρ<sup>θ$.