## Abstract

The notion of a period of a cusp form on GL(2,D(double-struck)), with respect to the diagonal subgroup D(double-struck A)^{×} × D(double-struck A)^{×}, is defined. Here D is a simple algebra over a global field F with a ring A of adeles. For D^{×} =GL(1), the period is the value at 1/2 of the L-function of the cusp form on GL(2, double-struck). A cuspidal representation is called cyclic if it contains a cusp form with a non zero period. It is investigated whether the notion of cyclicity is preserved under the Deligne-Kazhdan correspondence, relating cuspidal representations on the group and its split form, where D is a matrix algebra. A local analogue is studied too, using the global technique. The method is based on a new bi-period summation formula. Local multiplicity one statements for spherical distributions, and non-vanishing properties of bi-characters, known only in a few cases, play a key role.

Original language | English |
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Pages (from-to) | 91-111 |

Number of pages | 21 |

Journal | Mathematische Nachrichten |

Volume | 183 |

DOIs | |

State | Published - 1997 |

Externally published | Yes |

## Keywords

- Bi-period summation formula
- Cuspidal representations
- Deligne-Kazhdan correspondence
- Periods
- Simple algebras
- Spherical distributions