Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods

Nadir Matringe; Vincent Sécherre; Shaun Stevens; Miyu Suzuki

Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods

Representation Theory 2026-01-28 v2 Number Theory

Authors: Nadir Matringe , Vincent Sécherre , Shaun Stevens , Miyu Suzuki

Abstract

For a non-Archimedean locally compact field $F$ of odd residue characteristic and characteristic $0$ , we prove a conjecture of D. Prasad predicting that, for an integer $n \geq 1$ and a non-split quaternionic $F$ -algebra $D$ , a discrete series representation of ${\rm GL}_n(D)$ has a symplectic period if and only if it is cuspidal and its Jacquet--Langlands transfer to ${\rm GL}_{2n}(F)$ is non-cuspidal.

Keywords

discrete series representation symplectic period quaternionic algebra cuspidal representation Jacquet-Langlands transfer

Cite

@article{arxiv.2503.08955,
  title  = {Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods},
  author = {Nadir Matringe and Vincent Sécherre and Shaun Stevens and Miyu Suzuki},
  journal= {arXiv preprint arXiv:2503.08955},
  year   = {2026}
}

Comments

This new version results from the merging of the first version arXiv:2503.08955v1 together with arXiv:2507.16364v2

Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods

Abstract

Keywords

Cite

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