Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a locally compact division algebra

In this paper we study the representations $\operatorname{Ind}(G,P,\pi)$ of the group $G=GL(n,D)$, where $D$ is a locally compact nondiscrete division algebra, that are induced by irreducible representations $\pi$ of an arbitrary parabolic subgroup $P\subset G$. If $D$ is totally disconnected, $\pi$ is assumed to be either supercuspidal (in the sense of Harish-Chandra; this is the same as absolutely cuspidal in the sense of Jacquet), or one-dimensional; we also allow combinations of these cases of a specific sort.
We give a construction of intertwining operators in this class of representations generalizing the construction of Schiffmann, Knapp and Stein. Using these intertwining operators, we prove that for the “principal series” representation $\operatorname{Ind}(G,P,\pi)$ to be contained in the “complementary series” the necessary formal condition of symmetry on $(P,\pi)$ turns out to also be sufficient. If $\pi$ is one-dimensional we estimate the width of the “critical interval”. Under certain conditions this estimate is best possible.
Bibliography: 28 titles.