Journal of Lie Theory

\newcommand{\Ind}{\textrm{Ind}\,} \newcommand\IBG{\Ind_B^G} \newcommand\IBGl{\IBG\lambda} \newcommand\IPG{\Ind_P^{G}} \newcommand{\St}{\textrm{St}\,} \newcommand\tr{\operatorname{\mathbf{1}}} We show that, under certain assumptions, a parabolic induction $\IBGl$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \[ \IBGl = \big( \IPG\, \St_M\otimes \chi_0 \big) \oplus \big( \IPG \tr_M\otimes \chi_0 \big), \] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\tr_M$ 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 $2$.