Twisted Jacquet modules and induction from Speh representations
Abstract
Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field F of characteristic zero. Let qF be the cardinality of its residue field. Let Js be the twisted Jacquet module of the representation induced from Δ(τ,m+i)|det|s. It carries an action of G× H, where G and H are split symplectic or special orthogonal groups. Let σ be an admissible representation of G of finite length. We prove that (Js⊗σ)G is induced from Δ(τ,i)|det|s and a fixed conjugate of σ, outside a finite set of values of qF−s depending on τ and σ. No genericity assumption on σ is needed. For irreducible σ, this determines all irreducible quotients of Js of the form σ∨boxtimesπ. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal σ unless G is the split group SO2. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when τ is cuspidal on GLn and n is larger than the rank of G.
Related
- Siegel-Weil type identities for general linear groups related to parabolic induction from two Speh representations
- One-periodic category of twisted complexes, Cohen-Macaulay modules and differential modules
- Crossed modularity of twisted VOA modules
- G-crossed braided tensor categories of twisted modules
- Twisted Deligne modules and abelian varieties over finite fields
BibTeX
@misc{hou2026twisted,
title = {{Twisted Jacquet modules and induction from Speh representations}},
author = {Alan Xuelun Hou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12221},
eprint = {2610.12221},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12221},
}
RIS
TY - UNPB
TI - Twisted Jacquet modules and induction from Speh representations
AU - Alan Xuelun Hou
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12221
UR - https://arxiv.org/abs/2610.12221
AB - Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field F of characteristic zero. Let q_F be the cardinality of its residue field. Let J_s be the twisted Jacquet module of the representation induced from Δ(τ,m+i)|det|^s. It carries an action of G× H, where G and H are split symplectic or special orthogonal groups. Let σ be an admissible representation of G of finite length. We prove that (J_s⊗σ)_G is induced from Δ(τ,i)|det|^s and a fixed conjugate of σ, outside a finite set of values of q_F^−s depending on τ and σ. No genericity assumption on σ is needed. For irreducible σ, this determines all irreducible quotients of J_s of the form σ^∨boxtimesπ. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal σ unless G is the split group SO_2. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when τ is cuspidal on GL_n and n is larger than the rank of G.
ER -