On representations of GL_n distinguished by GL₁ × GL_n−1 over a quaternion division algebra

Let D be a quaternion division algebra over a non-Archimedean local field F of characteristic zero, and let G_n=GL_n(D). Let H_1,n−1 denote the subgroup of G_n consisting of block-diagonal matrices of the form diag(g₁,g₂), where g₁∈G₁ and g₂∈G_n−1. In this article, we classify all irreducible smooth H_1,n−1-distinguished representations of G_n for n=3 and 4. Furthermore, we conjecture that an irreducible smooth representation π of G_n, for n>2, is H_1,n−1-distinguished if and only if either π is a trivial representation or it is parabolically induced from the trivial representation of G_n−2 and an infinite-dimensional irreducible H_1,1-distinguished representation of G₂.