Spherical Pairs Over Close Local Fields
arXiv:0910.3199 · doi:10.4171/CMH/274
Abstract
Extending results of Kazhdan 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. One of the Ingredients of the proof is a condition for finite generation of some modules over the Hecke algebra. We apply our results to show that the pair (GL_{n+1},GL_n) is a strong Gelfand pair for all local fields, and that the pair (GL_{n+k},GL_n x GL_k) is a Gelfand pair for all local fields of odd characteristic.
v2. Section 2 changed. v3. A change in the definition of uniform spherical pair, which also caused some changes in the formulations of the theorems. To appear in Commentarii Mathematici Helvetici
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