Abstract
We prove vanishing of \(\mathfrak {z}\)-eigen distributions on a split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup. This is a generalization of a result by Shalika, that concerned the group case. Shalika’s result was crucial in the proof of his multiplicity one theorem. We view our result as a step in the study of multiplicities of quasi-regular representations on spherical varieties. As an application we prove non-vanishing of spherical Bessel functions.
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Acknowledgments
We thank Erez Lapid for pointing out to us the application to non-vanishing of spherical Bessel functions. We also thank Vladimir Hinich, Omer Offen and Eitan Sayag for useful discussions. D.G. was partially supported by ISF Grant 756/12, ERC grant 291612, and a Minerva foundation grant. A.A. was partially supported by NSF Grant DMS-1100943 and ISF Grant 687/13.
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Aizenbud, A., Gourevitch, D. Vanishing of certain equivariant distributions on spherical spaces. Math. Z. 279, 745–751 (2015). https://doi.org/10.1007/s00209-014-1391-6
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DOI: https://doi.org/10.1007/s00209-014-1391-6