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European Mathematical Society Publishing House
2016-09-19 17:04:52
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
87
2012
4
Spherical pairs over close local fields
Avraham
Aizenbud
The Weizmann Institute of Science, REHOVOT, ISRAEL
Nir
Avni
Harvard University, CAMBRIDGE, UNITED STATES
Dmitry
Gourevitch
Institute for Advanced Study, PRINCETON, UNITED STATES
Multiplicity, Gelfand pair, reductive group, uniqueness of linear periods
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.
Group theory and generalizations
Algebraic geometry
General
929
962
10.4171/CMH/274
http://www.ems-ph.org/doi/10.4171/CMH/274