Quaternion distinguished generic representations of ${\mathrm{GL}}_{2n}$

Abstract

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, ${\mathrm{GL}}_m(D)$ can be regarded as a subgroup of ${\mathrm{GL}}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic ${\mathrm{GL}}_m(D)$-distinguished representations of ${\mathrm{GL}}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representation in the image of the unstable base change lift from the unitary group ${\mathrm{U}}_{2m}$ to be ${\mathrm{GL}}_m(D)$-distinguished.