JournalscmhVol. 87, No. 4pp. 929–962

Spherical pairs over close local fields

  • Avraham Aizenbud

    The Weizmann Institute of Science, Rehovot, Israel
  • Nir Avni

    Harvard University, Cambridge, USA
  • Dmitry Gourevitch

    Institute for Advanced Study, Princeton, USA
Spherical pairs over close local fields cover
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Abstract

Extending results of [Kaz86] to the relative case, we relate harmonic analysis over some spherical spaces G(F)/H(F)G(F)/H(F), where FF is a field of positive characteristic, to harmonic analysis over the spherical spaces G(E)/H(E)G(E)/H(E), where EE is a suitably chosen field of characteristic 0.

We apply our results to show that the pair (GLn+1(F),GLn(F))(\mathrm{GL}_{n+1}(F),\mathrm{GL}_n(F)) is a strong Gelfand pair for all local fields of arbitrary characteristic, and that the pair (GLn+k(F),GLn(F)×GLk(F))(\mathrm{GL}_{n+k}(F),\mathrm{GL}_n(F)\times\mathrm{GL}_k(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 KK-invariant compactly supported functions on G(E)/H(E)G(E)/H(E) as a module over the Hecke algebra.

Cite this article

Avraham Aizenbud, Nir Avni, Dmitry Gourevitch, Spherical pairs over close local fields. Comment. Math. Helv. 87 (2012), no. 4, pp. 929–962

DOI 10.4171/CMH/274