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Jordan induction for GLn(š¯’Ŗ)

  • Zhe Chen
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Abstract

Let š¯’Ŗ be a complete discrete valuation ring with a finite residue field. We introduce a simple framework of constructing smooth representations of GLn(š¯’Ŗ) modulo the knowledge of nilpotent orbits, called Jordan induction, which enjoys several remarkable properties: It yields only irreducible representations, it yields all the even level irreducible representations, and up to conjugation it yields non-isomorphic representations. In the even level case, this gives an explicit realisation of Hill's analogue of Lusztig's Jordan decomposition, as well as a vast generalisation of GĆ©rardin's construction for GLn(š¯’Ŗ). As a simple application, we construct an explicit section to the orbit map. We also propose a conjecture linking Jordan induction and Lusztig induction, aiming at generalising the algebraisation theorem obtained in recent joint works with Stasinski.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026jordan,
  title = {{Jordan induction for $\mathrmGL_n(\mathcalO)$}},
  author = {Zhe Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35393},
  eprint = {2609.35393},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35393},
}
RIS
TY  - UNPB
TI  - Jordan induction for GL_n(š¯’Ŗ)
AU  - Zhe Chen
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.35393
UR  - https://arxiv.org/abs/2609.35393
AB  - Let š¯’Ŗ be a complete discrete valuation ring with a finite residue field. We introduce a simple framework of constructing smooth representations of GL_n(š¯’Ŗ) modulo the knowledge of nilpotent orbits, called Jordan induction, which enjoys several remarkable properties: It yields only irreducible representations, it yields all the even level irreducible representations, and up to conjugation it yields non-isomorphic representations. In the even level case, this gives an explicit realisation of Hill's analogue of Lusztig's Jordan decomposition, as well as a vast generalisation of GĆ©rardin's construction for GL_n(š¯’Ŗ). As a simple application, we construct an explicit section to the orbit map. We also propose a conjecture linking Jordan induction and Lusztig induction, aiming at generalising the algebraisation theorem obtained in recent joint works with Stasinski.
ER  -