Singularities of Intertwining Operators and Decompositions of Principal Series Representations

In this paper, we show that, under certain assumptions, a parabolic induction $Ind_B^G\lambda$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: $$Ind_B^G\lambda = \left(Ind_P^G St_M\otimes \chi_0\right) \oplus \left(Ind_P^G \mathbf{1}_M\otimes \chi_0\right),$$ where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\mathbf{1}_M$ is the trivial representation of $M$, $St_M$ is the Steinberg representation of $M$ and $\chi_0$ is a certain character of $M$. We construct examples of this phenomenon for all simply-connected simple groups of rank at least $2$.