### Abstract

Extending results of [Kaz86] to the relative case, we relate harmonic analysis over some spherical spaces G.(F)/H.(F), where F is a field of positive characteristic, to harmonic analysis over the spherical spaces G.(E)=H.(E), where E is a suitably chosen field of characteristic 0. We apply our results to show that the pair .(GLnC1.F ); GL _{n}.(F)) is a strong Gelfand pair for all local fields of arbitrary characteristic, and that the pair .GL _{n}Ck.(F); GL _{n}.(F)̃GLk.(F)) is a Gelfand pair for local fields of any characteristic different from 2. We also give a criterion for finite generation of the space of K-invariant compactly supported functions on G.E/=H.E/ as a module over the Hecke algebra.

Original language | English (US) |
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Pages (from-to) | 929-962 |

Number of pages | 34 |

Journal | Commentarii Mathematici Helvetici |

Volume | 87 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 1 2012 |

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### Keywords

- Gelfand pair
- Multiplicity
- Reductive group
- Uniqueness of linear periods

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Commentarii Mathematici Helvetici*,

*87*(4), 929-962. https://doi.org/10.4171/CMH/274