### 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) |
---|---|

Pages (from-to) | 929-962 |

Number of pages | 34 |

Journal | Commentarii Mathematici Helvetici |

Volume | 87 |

Issue number | 4 |

DOIs | |

State | Published - Nov 1 2012 |

### Fingerprint

### 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

}

*Commentarii Mathematici Helvetici*, vol. 87, no. 4, pp. 929-962. https://doi.org/10.4171/CMH/274

**Spherical pairs over close local fields.** / Aizenbud, Avraham; Avni, Nir; Gourevitch, Dmitry.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Spherical pairs over close local fields

AU - Aizenbud, Avraham

AU - Avni, Nir

AU - Gourevitch, Dmitry

PY - 2012/11/1

Y1 - 2012/11/1

N2 - 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 nCk.(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.

AB - 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 nCk.(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.

KW - Gelfand pair

KW - Multiplicity

KW - Reductive group

KW - Uniqueness of linear periods

UR - http://www.scopus.com/inward/record.url?scp=84867967287&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867967287&partnerID=8YFLogxK

U2 - 10.4171/CMH/274

DO - 10.4171/CMH/274

M3 - Article

VL - 87

SP - 929

EP - 962

JO - Commentarii Mathematici Helvetici

JF - Commentarii Mathematici Helvetici

SN - 0010-2571

IS - 4

ER -