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Peer-reviewedCC BYJournal für die reine und angewandte Mathematik (Crelles Journal)

Local Langlands in families: The banal case

  • Jean-François Dat1
  • David Helm2
  • Robert Kurinczuk3
  • Gilbert Moss4
4 affiliations
  1. 1IMJ-PRG , Sorbonne Université , Université Paris Cité, CNRS, 4 Place Jussieu, 75005 , Paris , France
  2. 2Department of Mathematics , Imperial College , London , SW7 2AZ , United Kingdom
  3. 3School of Mathematics and Statistics , University of Sheffield , Sheffield , S3 7RH , United Kingdom
  4. 4Department of Mathematics & Statistics , University of Maine , 5752 Neville Hall , Orono , ME 04469 , USA
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Abstract

We state a conjecture, local Langlands in families , connecting the centre of the category of smooth representations on Z ⁢ [ q − 1 ] \mathbb{Z}[\sqrt{q}^{-1}] -modules of a quasi-split 𝑝-adic group G \mathrm{G} (where 𝑞 is the cardinality of the residue field of the underlying local field), the ring of global functions on the stack of Langlands parameters for G \mathrm{G} over Z ⁢ [ q − 1 ] \mathbb{Z}[\sqrt{q}^{-1}] , and the endomorphisms of a Gelfand–Graev representation for G \mathrm{G} . For a class of classical 𝑝-adic groups (symplectic, unitary, or odd special orthogonal groups), we prove this conjecture after inverting an integer depending only on G \mathrm{G} . Along the way, we show that the local Langlands correspondence for classical 𝑝-adic groups (1) preserves integrality of ℓ-adic representations, (2) satisfies an “extended” (generic) packet conjecture, (3) is compatible with parabolic induction up to semisimplification (generalizing a result of Moussaoui), hence induces a semisimple local Langlands correspondence, and (4) the semisimple correspondence is compatible with automorphisms of ℂ fixing q \sqrt{q} .

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Cite

BibTeXRIS
BibTeX
@article{dat2026local,
  title = {{Local Langlands in families: The banal case}},
  author = {Jean-François Dat and David Helm and Robert Kurinczuk and Gilbert Moss},
  journal = {Journal für die reine und angewandte Mathematik (Crelles Journal)},
  year = {2026},
  month = may,
  doi = {10.1515/crelle-2026-0034},
  url = {https://doi.org/10.1515/crelle-2026-0034},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Local Langlands in families: The banal case
AU  - Jean-François Dat
AU  - David Helm
AU  - Robert Kurinczuk
AU  - Gilbert Moss
PY  - 2026
DA  - 2026/05/20
JO  - Journal für die reine und angewandte Mathematik (Crelles Journal)
DO  - 10.1515/crelle-2026-0034
UR  - https://doi.org/10.1515/crelle-2026-0034
AB  - We state a conjecture, local Langlands in families , connecting the centre of the category of smooth representations on Z ⁢ [ q − 1 ] \mathbb{Z}[\sqrt{q}^{-1}] -modules of a quasi-split 𝑝-adic group G \mathrm{G} (where 𝑞 is the cardinality of the residue field of the underlying local field), the ring of global functions on the stack of Langlands parameters for G \mathrm{G} over Z ⁢ [ q − 1 ] \mathbb{Z}[\sqrt{q}^{-1}] , and the endomorphisms of a Gelfand–Graev representation for G \mathrm{G} . For a class of classical 𝑝-adic groups (symplectic, unitary, or odd special orthogonal groups), we prove this conjecture after inverting an integer depending only on G \mathrm{G} . Along the way, we show that the local Langlands correspondence for classical 𝑝-adic groups (1) preserves integrality of ℓ-adic representations, (2) satisfies an “extended” (generic) packet conjecture, (3) is compatible with parabolic induction up to semisimplification (generalizing a result of Moussaoui), hence induces a semisimple local Langlands correspondence, and (4) the semisimple correspondence is compatible with automorphisms of ℂ fixing q \sqrt{q} .
SN  - 0075-4102
SN  - 1435-5345
ER  -