Quaternion distinguished generic representations of GL2n

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, GLm(D) can be regarded as a subgroup of GL2m(E). Using the method of Matringe, we classify irreducible generic GLm(D)-distinguished representations of GL2m(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 U2m to be GLm(D)-distinguished.