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Parabolically induced representations of GLn(Ol)

  • Prem Dagar
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Abstract

Let π’ͺ be the ring of integers of a non-Archimedean local field with uniformizer Ο–, and let π’ͺβ„“=π’ͺ/Ο–β„“π’ͺ for β„“β‰₯1. In this article, we study representations of GLn(π’ͺβ„“) parabolically induced from characters of the Borel subgroup Bn(π’ͺβ„“). For n=3, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary n. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple (GLn(π’ͺβ„“),Bn(π’ͺβ„“),Ο‡) is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for GL3(π’ͺβ„“) when two relative conductors attain their maximal values. Using the decomposition theory for GL2(π’ͺβ„“), we describe the irreducible constituents and determine their dimensions.

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Cite

BibTeXRIS
BibTeX
@misc{dagar2026parabolically,
  title = {{Parabolically induced representations of $GL_n(O_l)$}},
  author = {Prem Dagar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15762},
  eprint = {2610.15762},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15762},
}
RIS
TY  - UNPB
TI  - Parabolically induced representations of GL_n(O_l)
AU  - Prem Dagar
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15762
UR  - https://arxiv.org/abs/2610.15762
AB  - Let π’ͺ be the ring of integers of a non-Archimedean local field with uniformizer Ο–, and let π’ͺ_β„“=π’ͺ/Ο–^β„“π’ͺ for β„“β‰₯1. In this article, we study representations of GL_n(π’ͺ_β„“) parabolically induced from characters of the Borel subgroup B_n(π’ͺ_β„“). For n=3, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary n. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple (GL_n(π’ͺ_β„“),B_n(π’ͺ_β„“),Ο‡) is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for GL_3(π’ͺ_β„“) when two relative conductors attain their maximal values. Using the decomposition theory for GL_2(π’ͺ_β„“), we describe the irreducible constituents and determine their dimensions.
ER  -