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PreprintCC0arXiv

Tempered Symplectic Distinction For Quaternionic GLn(D): A Conjecture of PRASAD

  • Mahendra Kumar Verma
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Abstract

Let F be a non-Archimedean local field of characteristic zero, let D/F be the quaternion division algebra, and put Gn = GLn(D) and Hn = Spn(D). We prove Prasad's revised conjecture on tempered representations with symplectic period. If ρ is a unitary Hr-distinguished supercuspidal representation and δm(ρ) denotes the generalized Steinberg representation associated to ρ, set B0(ρ) = ρ, Bm(ρ) = δm+1(ρ) × δm(ρ), m ≥ 1. We prove that an irreducible tempered representation π of Gn is Hn-distinguished if and only if π is isomorphic to the product over i of Bmi(ρi), where each ρi is a unitary distinguished supercuspidal representation.

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Cite

BibTeXRIS
BibTeX
@misc{verma2026tempered,
  title = {{Tempered Symplectic Distinction For Quaternionic $GL_n(D)$: A Conjecture of PRASAD}},
  author = {Mahendra Kumar Verma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05364},
  eprint = {2610.05364},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05364},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Tempered Symplectic Distinction For Quaternionic GL_n(D): A Conjecture of PRASAD
AU  - Mahendra Kumar Verma
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05364
UR  - https://arxiv.org/abs/2610.05364
AB  - Let F be a non-Archimedean local field of characteristic zero, let D/F be the quaternion division algebra, and put G_n = GL_n(D) and H_n = Sp_n(D). We prove Prasad's revised conjecture on tempered representations with symplectic period. If ρ is a unitary H_r-distinguished supercuspidal representation and δ_m(ρ) denotes the generalized Steinberg representation associated to ρ, set B_0(ρ) = ρ, B_m(ρ) = δ_m+1(ρ) × δ_m(ρ), m ≥ 1. We prove that an irreducible tempered representation π of G_n is H_n-distinguished if and only if π is isomorphic to the product over i of B_m_i(ρ_i), where each ρ_i is a unitary distinguished supercuspidal representation.
ER  -