
Journal of Lie Theory 30 (2020), No. 4, 939964 Copyright Heldermann Verlag 2020 Singularities of Intertwining Operators and Decompositions of Principal Series Representations Taeuk Nam Dept. of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada taeuk.nam@alumni.ubc.ca Avner Segal Dept. of Mathematics, Bar Ilan University, Ramat Gan 5290002, Israel segalavner@gmail.com Lior Silberman Dept. of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada lior@math.ubc.ca [Abstractpdf] \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 semisimple 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 simplyconnected simple groups of rank at least $2$. Keywords: Representation theory, Lie groups, padic groups, principle series, intertwining operators. MSC: 22E50, 47G10, 22E46. [ Fulltextpdf (231 KB)] for subscribers only. 