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Multiplicity One for Unitary Ginzburg--Rallis Models over Local Fields

  • Yuxuan Ge
  • Sagnik Mukherjee
  • Junyi Zhai
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Abstract

We prove multiplicity at most one for the unitary similitude Ginzburg--Rallis model over local fields of characteristic zero, for both the quasi-split and non-quasi-split groups. The theorem holds for all irreducible smooth representations over non-Archimedean fields and all irreducible Casselman--Wallach representations over the real field. We first describe the decomposition into parabolic double cosets and compute the required stabilizers. Over a non-Archimedean field we use character obstructions, localization, and transpose symmetry. Over the real field we use a Casimir identity and explicit normal vectors to prove the corresponding statement for distributions with prescribed infinitesimal character. The Gelfand--Kazhdan criterion and a dualizing involution then give the multiplicity bound.

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Cite

BibTeXRIS
BibTeX
@misc{ge2026multiplicity,
  title = {{Multiplicity One for Unitary Ginzburg--Rallis Models over Local Fields}},
  author = {Yuxuan Ge and Sagnik Mukherjee and Junyi Zhai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36294},
  eprint = {2610.36294},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36294},
}
RIS
TY  - UNPB
TI  - Multiplicity One for Unitary Ginzburg--Rallis Models over Local Fields
AU  - Yuxuan Ge
AU  - Sagnik Mukherjee
AU  - Junyi Zhai
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36294
UR  - https://arxiv.org/abs/2610.36294
AB  - We prove multiplicity at most one for the unitary similitude Ginzburg--Rallis model over local fields of characteristic zero, for both the quasi-split and non-quasi-split groups. The theorem holds for all irreducible smooth representations over non-Archimedean fields and all irreducible Casselman--Wallach representations over the real field. We first describe the decomposition into parabolic double cosets and compute the required stabilizers. Over a non-Archimedean field we use character obstructions, localization, and transpose symmetry. Over the real field we use a Casimir identity and explicit normal vectors to prove the corresponding statement for distributions with prescribed infinitesimal character. The Gelfand--Kazhdan criterion and a dualizing involution then give the multiplicity bound.
ER  -