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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

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Degenerate principal series for even-orthogonal groups
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by Dubravka Ban and Chris Jantzen PDF
Represent. Theory 7 (2003), 440-480 Request permission

Abstract:

Let $F$ be a $p$-adic field of characteristic 0 and $G=O(2n,F)$ (resp. $SO(2n,F)$). A maximal parabolic subgroup of $G$ has the form $P=MU$, with Levi factor $M \cong GL(k,F) \times O(2(n-k),F)$ (resp. $M \cong GL(k,F) \times SO(2(n-k),F)$). We consider a one-dimensional representation of $M$ of the form $\chi \circ det_k \otimes triv_{(n-k)}$, with $\chi$ a one-dimensional representation of $F^{\times }$; this may be extended trivially to get a representation of $P$. We consider representations of the form $\mbox {Ind}_P^G(\chi \circ det_k \otimes triv_{(n-k)}) \otimes 1$. (Our results also work when $G=O(2n,F)$ and the inducing representation is $(\chi \circ det_k \otimes det_{(n-k)}) \otimes 1$, using $det_{(n-k)}$ to denote the nontrivial character of $O(2(n-k),F)$.) More generally, we allow Zelevinsky segment representations for the inducing representations. In this paper, we study the reducibility of such representations. We determine the reducibility points, give Langlands data and Jacquet modules for each of the irreducible composition factors, and describe how they are arranged into composition series. For $O(2n,F)$, we use Jacquet module methods to obtain our results; the results for $SO(2n,F)$ are obtained via an analysis of restrictions to $SO(2n,F)$.
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Additional Information
  • Dubravka Ban
  • Affiliation: Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
  • MR Author ID: 658785
  • Email: dban@math.siu.edu
  • Chris Jantzen
  • Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
  • MR Author ID: 316181
  • Email: jantzenc@mail.ecu.edu
  • Received by editor(s): May 7, 2002
  • Received by editor(s) in revised form: September 22, 2003
  • Published electronically: October 9, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Represent. Theory 7 (2003), 440-480
  • MSC (2000): Primary 22E50
  • DOI: https://doi.org/10.1090/S1088-4165-03-00166-3
  • MathSciNet review: 2017065