Discrete series representations of quaternionic GL 𝑛 (𝐷) with symplectic periods

For a non-Archimedean locally compact field 𝐹 of odd residue characteristic and characteristic 0, we prove a conjecture of Prasad predicting that, for an integer n ⩾ 1 n 1 and a non-split quaternionic 𝐹-algebra 𝐷, a discrete series representation of GL n ⁡ ( D ) GLₙ(D) has a symplectic period if and only if it is cuspidal and its Jacquet–Langlands transfer to GL 2 ⁢ n ⁡ ( F ) GL₂ₙ(F) is non-cuspidal.