Reducibility of certain induced representations of \(E_6\) and \(E_7\) (Q2377431)

The author studies the reducibility of the induced representations of certain parabolic subgroups of the split simply-connected groups of types \(E_{6}\) and \(E_{7}\) over a non-Archimedean field of characteristic zero. Necessary and sufficient conditions under which certain representations of \(E_{6}\) and \(E_{7}\) that are parabolically induced from unitary supercuspidal ones reduce are given by explicit calculation of the Weyl group elements and poles of the first two Langlands-Shahidi \(L\)-functions at \(s=0\). Further, related complementary series are also described in detail.