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

Nadir Matringe; Vincent Sécherre; Shaun Stevens; Miyu Suzuki

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arXiv · 2503.08955

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

Nadir Matringe·Vincent Sécherre·Shaun Stevens·Miyu Suzuki

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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.

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Nadir Matringe, Vincent Sécherre, Shaun Stevens, Miyu Suzuki. 2025-03-11. Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods. https://arxiv.org/abs/2503.08955

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