Multiplicity one for a linear spherical pair and its unitary analogue
Abstract
We prove multiplicity one for a linear spherical pair associated with GL4(F)×GL2(F) and its unitary similitude analogue associated with GU(2,2)(F)×GU(1,1)(F). Both results hold for all irreducible admissible representations over non-Archimedean local fields of characteristic zero. Generalized Harish-Chandra descent reduces the proofs to the analysis of closed orbits and their normal representations. The unitary proof combines reductions to the linear case with a rational classification of closed orbits and a separate analysis of the remaining normal representations. We also establish multiplicity one for unitary Shalika periods of GU(2,2)(F) with arbitrary irreducible admissible coefficients. A theta-period identity relates these periods to the unitary spherical pair. By varying the splitting character, we prove that every irreducible admissible representation of GU(2,2)(F) occurs as a quotient of a suitable big theta lift, allowing the multiplicity-one bound to pass to the Shalika periods. The linear argument also recovers Rallis's multiplicity-one theorem for twisted Jacquet modules of GL4(F).
Related
- Multiplicity One for Unitary Ginzburg--Rallis Models over Local Fields
- Multiplicity One for Five Strongly Tempered Spherical Varieties
- Componentwise linearity, Fröberg's analogue, and classification of linear support-two monomial ideals
- Multiplicity and solution construction in linear elliptic problems with nonlocal nonlinearities
- Hilbert Series in Four Variables and Linear Factors of High Multiplicity
BibTeX
@misc{ge2026multiplicity,
title = {{Multiplicity one for a linear spherical pair and its unitary analogue}},
author = {Yuxuan Ge},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02670},
eprint = {2611.02670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02670},
}
RIS
TY - UNPB
TI - Multiplicity one for a linear spherical pair and its unitary analogue
AU - Yuxuan Ge
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02670
UR - https://arxiv.org/abs/2611.02670
AB - We prove multiplicity one for a linear spherical pair associated with GL_4(F)×GL_2(F) and its unitary similitude analogue associated with GU(2,2)(F)×GU(1,1)(F). Both results hold for all irreducible admissible representations over non-Archimedean local fields of characteristic zero. Generalized Harish-Chandra descent reduces the proofs to the analysis of closed orbits and their normal representations. The unitary proof combines reductions to the linear case with a rational classification of closed orbits and a separate analysis of the remaining normal representations. We also establish multiplicity one for unitary Shalika periods of GU(2,2)(F) with arbitrary irreducible admissible coefficients. A theta-period identity relates these periods to the unitary spherical pair. By varying the splitting character, we prove that every irreducible admissible representation of GU(2,2)(F) occurs as a quotient of a suitable big theta lift, allowing the multiplicity-one bound to pass to the Shalika periods. The linear argument also recovers Rallis's multiplicity-one theorem for twisted Jacquet modules of GL_4(F).
ER -