On symplectic periods for inner forms of GLn

Mahendra Kumar Verma

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In this paper we study the question of determining when an irreducible admissible representation of GLn(D) admits a symplectic model, that is when such a representation has a linear functional invariant under Spn(D), where D is a quaternion division algebra over a non-archimedean local field k and Spn(D) is the unique non-split inner form of the symplectic group Sp2n(k). We show that if a representation has a symplectic model it is necessarily unique. For GL2(D) we completely classify those representations which have a symplectic model. Globally, we show that if a discrete automorphic representation of GLn(DA) has a non-zero period for Spn(DA), then its Jacquet-Langlands lift also has a non-zero sym-plectic period. A somewhat striking difference between distinction question for GL2n(k), and GLn(D)(with respect to Sp2n(k) and Spn(D) resp.) is that there are supercuspidal representations of GLn(D) which are distinguished by Spn(D). The paper ends by formulating a general question classifying all unitary distinguished representations of GLn(D), and proving a part of the local conjectures through a global conjecture.

Original languageEnglish
Pages (from-to)309-334
Number of pages26
JournalMathematical Research Letters
Issue number1
StatePublished - 1 Jan 2018

ASJC Scopus subject areas

  • General Mathematics


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