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Distinguished standard modules for GL2m(ℂ)/GLm(ℍ)
Abstract
We characterize the standard modules of GL2m(C) that are distinguished by GLm(HH). Let δ1,…, δ2m be characters of ℂ×. Assume that S = δ1 × ⋯ × δ2m is a standard module of GL2m(ℂ). For each i, define δi* (z) = δi(z̅)−1 for $ z \in \mathbb{C}^\times$. In particular, we conclude that a standard module for GL2m(ℂ) is distinguished by GLm(ℍ) if and only if there exists an involution p∈ S2m without fixed points such that δp(i)=δi* for every i. We first verify the hypotheses of the multiplicity estimate theorem of Suzuki and Tamori in \cite{ST}. The orbit calculation of Matringe, Offen, and Yang in \cite{MOYglobal} then gives the necessary condition and a dimension bound. Then local intertwining periods prove sufficiency.
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BibTeX
@misc{hou2026distinguished,
title = {{Distinguished standard modules for $\mathrmGL_2m(\mathbbC)/\mathrmGL_m(\mathbbH)$}},
author = {Alan Xuelun Hou and Tudor Popescu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10483},
eprint = {2609.10483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10483},
}
RIS
TY - UNPB
TI - Distinguished standard modules for GL_2m(ℂ)/GL_m(ℍ)
AU - Alan Xuelun Hou
AU - Tudor Popescu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10483
UR - https://arxiv.org/abs/2609.10483
AB - We characterize the standard modules of GL_2m(C) that are distinguished by GL_m(HH). Let δ_1,…, δ_2m be characters of ℂ^×. Assume that S = δ_1 × ⋯ × δ_2m is a standard module of GL_2m(ℂ). For each i, define δ_i^* (z) = δ_i(z̅)^−1 for $ z \in \mathbb{C}^\times$. In particular, we conclude that a standard module for GL_2m(ℂ) is distinguished by GL_m(ℍ) if and only if there exists an involution p∈ S_2m without fixed points such that δ_p(i)=δ_i^* for every i. We first verify the hypotheses of the multiplicity estimate theorem of Suzuki and Tamori in \cite{ST}. The orbit calculation of Matringe, Offen, and Yang in \cite{MOYglobal} then gives the necessary condition and a dimension bound. Then local intertwining periods prove sufficiency.
ER -