Singularities of Intertwining Operators and Decompositions of Principal Series Representations

Abstract

In this paper, we show that, under certain assumptions, a parabolic induction $In{d}_B^G\lambda$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \[ Ind_B^Gλ= \left(Ind_P^G St_M\otimes χ_0\right) \oplus \left(Ind_P^G \mathbf{1}_M\otimes χ_0\right), \] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $1_M$ is the trivial representation of $M$, $S{t}_M$ 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 $2$.