Gross–Prasad periods for reducible representations

We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F product extensions total degree 3. For irreducible representations, theorem Prasad shows that the space such has dimension at most 1, non-zero when certain epsilon-factor condition holds. give an extension this result to class reducible (of Whittaker type), extending results Harris--Scholl A split algebra x F.