Stability of certain elements in the Bernstein center

Abstract

Let G be a split connected reductive algebraic group over a non-archimedean local field F of characteristic zero. The stable center conjecture asserts that the subspace of stable elements in the Bernstein center of G(F) is a sub-algebra, and this subring is conjecturally isomorphic to the regular functions on the moduli of semi-simple Langlands parameters. Bezrukavnikov-Kazhdan-Varshavsky suggested an approach towards the stable center conjecture by explicitly constructing elements and proving their stability directly without assuming LLC, and proved the stability of the depth-r projectors. Following the idea, the first author and Tsao-Hsien Chen produced a family of elements in the center which are conjecturally stable using Deligne-Lusztig parameters attached to irreducible representations. This paper proves the stability of the projectors corresponding to strongly regular Deligne-Lusztig parameters in depth-zero, and for all the positive depth elements with some restrictions on the residue characteristic.

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BibTeX RIS

BibTeX

@misc{bhattacharya2026stability,
  title = {{Stability of certain elements in the Bernstein center}},
  author = {Sarbartha Bhattacharya and Shenghao Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33327},
  eprint = {2610.33327},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33327},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Stability of certain elements in the Bernstein center
AU  - Sarbartha Bhattacharya
AU  - Shenghao Li
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.33327
UR  - https://arxiv.org/abs/2610.33327
AB  - Let G be a split connected reductive algebraic group over a non-archimedean local field F of characteristic zero. The stable center conjecture asserts that the subspace of stable elements in the Bernstein center of G(F) is a sub-algebra, and this subring is conjecturally isomorphic to the regular functions on the moduli of semi-simple Langlands parameters. Bezrukavnikov-Kazhdan-Varshavsky suggested an approach towards the stable center conjecture by explicitly constructing elements and proving their stability directly without assuming LLC, and proved the stability of the depth-r projectors. Following the idea, the first author and Tsao-Hsien Chen produced a family of elements in the center which are conjecturally stable using Deligne-Lusztig parameters attached to irreducible representations. This paper proves the stability of the projectors corresponding to strongly regular Deligne-Lusztig parameters in depth-zero, and for all the positive depth elements with some restrictions on the residue characteristic.
ER  -

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