Symplectic period for a representation of GL_n(D)

2023/11/20 by Hariom Sharma, Sharma, Hariom, Mahendra K. Verma +1
Mathematics · #11F70 #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2311.11674

openalex publication_date 2023/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D be a quaternion division algebra over a non-archimedean local field K of characteristic zero, and let Sp_n(D) be the unique non-split inner form of the symplectic group Sp_2n(K). This paper classifies the irreducible admissible representations of GL_n(D) with a symplectic period for n = 3 and 4, i.e., those irreducible admissible representations (π, V) of GL_n(D) which have a linear functional l on V such that l(π(h)v) = l(v) for all v ∈ V and h ∈ Sp_n(D). Our results also contain all unitary representations having a symplectic period, as stated in Prasad's conjecture.

Related

• On Symplectic Periods for Inner forms of \rm GL_n
• Cuspidal representations of quaternionic GL_n(D) with symplectic periods
• On representations of \rm GL_2n(F) with a symplectic period
• Symplectic model for ladder and unitary representations
• Symplectic models for Unitary groups