Distinguished representations, base change, and reducibility for unitary groups

We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square-integrable representation of GLn(E). As a consequence, we characterize reducibility of certain induced representations of U(n,n), and the image of the base change map from U(n) to GLn(E) in terms of GLn(F)-distinguishedness. We also determine precisely when the Steinberg representation of GLn(E) is distinguished with respect to a character of F^*.