Discrete series representations of quaternionic GLn(D) with symplectic periods

Key Points

• Discrete series representations have symplectic periods if and only if they are cuspidal, demonstrating a clear link with their structure.
• Key evidence shows that for representations of GLn(D), the presence of symplectic periods directly relates to their cusp behavior.
• With observational analysis on quaternionic algebras, we determine how discrete series representations can be fully characterized.
• The findings highlight the importance of Jacquet–Langlands transfers, particularly in understanding the non-cuspidal nature of these representations.