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On a basepoint issue in the tempered dual of p-adic SL(n)

  • Marie Dautheville
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Abstract

Let F be a non-archimedean local field of characteristic zero. Let M be a Levi subgroup of G = SL(n,F). Consider the action on the discrete series of M of the group of unitary unramified characters of M (where a character acts by twisting). Given an orbit 𝒪 for that action, let WO be the global stabilizer of 𝒪 in the Weyl group of M. We construct families of orbits 𝒪 for which the action of WO has no fixed point. More precisely, let n ≥ 8 be an integer that is not prime and not equal to $9$. We attach families of orbits without fixed points to any divisor m ≥ 2 of n such that m | q−1 where q is the order of the residue field of F. The Levi subgroup M ⊂ G for the corresponding families has m blocks of size k≥ 2 and $2$ blocks of size m. The core of the paper is the construction of superculpidal representations of GL(m,F) and GL(k,F) satisfying some conditions that we require to construct the families of orbits 𝒪 that we expects. In addition, we study the fixed point problem for the dimensions n that we excluded above, i.e., prime or small dimensions. We prove that there is always a fixed point under the action of WO in those cases.

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Cite

BibTeXRIS
BibTeX
@misc{dautheville2026basepoint,
  title = {{On a basepoint issue in the tempered dual of $p$-adic $SL(n)$}},
  author = {Marie Dautheville},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05055},
  eprint = {2609.05055},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05055},
}
RIS
TY  - UNPB
TI  - On a basepoint issue in the tempered dual of p-adic SL(n)
AU  - Marie Dautheville
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05055
UR  - https://arxiv.org/abs/2609.05055
AB  - Let F be a non-archimedean local field of characteristic zero. Let M be a Levi subgroup of G = SL(n,F). Consider the action on the discrete series of M of the group of unitary unramified characters of M (where a character acts by twisting). Given an orbit 𝒪 for that action, let W_O be the global stabilizer of 𝒪 in the Weyl group of M. We construct families of orbits 𝒪 for which the action of W_O has no fixed point. More precisely, let n ≥ 8 be an integer that is not prime and not equal to $9$. We attach families of orbits without fixed points to any divisor m ≥ 2 of n such that m | q−1 where q is the order of the residue field of F. The Levi subgroup M ⊂ G for the corresponding families has m blocks of size k≥ 2 and $2$ blocks of size m. The core of the paper is the construction of superculpidal representations of GL(m,F) and GL(k,F) satisfying some conditions that we require to construct the families of orbits 𝒪 that we expects. In addition, we study the fixed point problem for the dimensions n that we excluded above, i.e., prime or small dimensions. We prove that there is always a fixed point under the action of W_O in those cases.
ER  -