Journal of Lie Theory 30 (2020), No. 4, 939--964
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
[Abstract-pdf]
\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 $2$.
Keywords: Representation theory, Lie groups, p-adic groups, principle series, intertwining operators.
MSC: 22E50, 47G10, 22E46.
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