Commentarii Mathematici Helvetici

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Volume 87, Issue 4, 2012, pp. 929–962
DOI: 10.4171/CMH/274

Published online: 2012-10-10

Spherical pairs over close local fields

Avraham Aizenbud[1], Nir Avni[2] and Dmitry Gourevitch[3]

(1) The Weizmann Institute of Science, Rehovot, Israel
(2) Harvard University, Cambridge, USA
(3) Institute for Advanced Study, Princeton, USA

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

Keywords: Multiplicity, Gelfand pair, reductive group, uniqueness of linear periods

Aizenbud Avraham, Avni Nir, Gourevitch Dmitry: Spherical pairs over close local fields. Comment. Math. Helv. 87 (2012), 929-962. doi: 10.4171/CMH/274