Theta distinguished representations, inflation and the symmetric square L-function

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A theta distinguished representation is a quotient of a tensor of exceptional representations, where “exceptional” is in the sense of Kazhdan and Patterson. We study relations between theta distinguished representations of GL n and GSpin 2 n + 1. In the case of GSpin 2 n + 1 (or SO 2 n + 1) exceptional (or small) representations were constructed by Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation τ is distinguished if and only if the representation I (τ) induced to GSpin 2 n + 1 is distinguished, and the multiplicity of both quotients is at most one. If τ is supercuspidal and distinguished, then so is the Langlands quotient of I (τ). As a corollary, we characterize supercuspidal distinguished representations, in terms of the pole of the symmetric square L-function at s= 0.

Original languageEnglish
Pages (from-to)909-936
Number of pages28
JournalMathematische Zeitschrift
Issue number3-4
StatePublished - 1 Aug 2016
Externally publishedYes


  • Distinguished representations
  • Exceptional representations
  • Symmetric square

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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