## Abstract

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 language | English |
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Pages (from-to) | 909-936 |

Number of pages | 28 |

Journal | Mathematische Zeitschrift |

Volume | 283 |

Issue number | 3-4 |

State | Published - 1 Aug 2016 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2016, Springer-Verlag Berlin Heidelberg.

## Keywords

- Distinguished representations
- Exceptional representations
- Symmetric square