Pseudo-spherical representations and types for covers of split tori

Abstract

We introduce a notion of pseudo-spherical representations for any Brylinski-Deligne (BD) cover of a split torus over a p-adic field. In particular, we do not assume that the cover is tame. We enumerate the pseudo-spherical representations and compute their dimension. When the torus is the maximal split torus of a connected split reductive group G, we enumerate a subset of distinguished pseudo-spherical representations, under mild assumptions on the BD datum. We expect these distinguished representations to provide the torus input for a Borel-Casselman Theorem for BD covers of G. We have verified this expectation for certain non-tame double covers in a separate work. While investigating the pseudo-spherical representations, we also prove that the restriction of irreducible genuine representations of covering tori to their maximal compact subgroup is multiplicity-free. From this we see that the irreducible genuine representations of this maximal compact subgroup are types in the sense of Bushnell-Kutzko. We conclude by describing the Hecke algebras attached to these types.

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BibTeX RIS

BibTeX

@misc{karasiewicz2026pseudospherical,
  title = {{Pseudo-spherical representations and types for covers of split tori}},
  author = {Edmund Karasiewicz and Shuichiro Takeda},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29548},
  eprint = {2608.29548},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29548},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Pseudo-spherical representations and types for covers of split tori
AU  - Edmund Karasiewicz
AU  - Shuichiro Takeda
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29548
UR  - https://arxiv.org/abs/2608.29548
AB  - We introduce a notion of pseudo-spherical representations for any Brylinski-Deligne (BD) cover of a split torus over a p-adic field. In particular, we do not assume that the cover is tame. We enumerate the pseudo-spherical representations and compute their dimension. When the torus is the maximal split torus of a connected split reductive group G, we enumerate a subset of distinguished pseudo-spherical representations, under mild assumptions on the BD datum. We expect these distinguished representations to provide the torus input for a Borel-Casselman Theorem for BD covers of G. We have verified this expectation for certain non-tame double covers in a separate work. While investigating the pseudo-spherical representations, we also prove that the restriction of irreducible genuine representations of covering tori to their maximal compact subgroup is multiplicity-free. From this we see that the irreducible genuine representations of this maximal compact subgroup are types in the sense of Bushnell-Kutzko. We conclude by describing the Hecke algebras attached to these types.
ER  -

Abstract

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