The research of this paper is based on the Ph. D. thesis of the second author at Columbia University (1989). The basic technique is a relative version of the Deligne-Kazhdan “simple trace formula”.
Let \(E\) be a quadratic separable extension of a global field \(F\), \(G\) the \(F\)-group \(GL(2)\), and \(G'\) the \(F\)-group obtained from \(G\) by restricting scalars from \(E\) to \(F\). Let \(D\) be an inner form of \(G\) and denote by \(D(F)\) the multiplicative group of a quaternion division algebra central over \(F\). Consider a representation \(\pi\) of \(G'\) over the adele ring \(A\) of \(F\). \(\pi\) is said to be \(G(A)\)-distinguished (resp. \(D(A)\)-distinguished) if there exists a form in the space of \(\pi\) such that its integral over \(G(A)\) (resp. \(D(A))\), modulo the rational points and the center, is nonzero. One can also define \(G(F_v)\)- distinguished (resp. \(D(F_v)\)-distinguished) representations of \(G'\) over a local field \(F_v\). Let \(V\) be a finite set (with even cardinality) of places of \(F\) where \(D\) ramifies.
The authors prove that an irredicuble, automorphic discrete-series representation \(\pi\) of \(G'(A)\) is \(D(A)\)-distinguished if and only if it is \(G(A)\)-distinguished and its components \(\pi_v\) at \(v\) in \(V\) are not of the form \(I(\mu_1, \mu_2)\) induced from trivial characters \(\mu_1\), \(\mu_2\) on \(F^x_v\). A more general result is proved which implies the main theorem of H. Jacquet and K. Lai, [Compos. Math. 54, 243-310 (1985; Zbl 0587.12006)]. Local distinguished representations are also studied in the paper.