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On the Jacquet Conjecture on the local converse problem for $p$-adic $\mathrm {GL}_N$

Authors: Moshe Adrian, Baiying Liu, Shaun Stevens and Peng Xu
Journal: Represent. Theory 20 (2016), 1-13
MSC (2010): Primary 11S70, 22E50; Secondary 11F85, 22E55.
Published electronically: January 27, 2016
MathSciNet review: 3452696
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Abstract: Based on previous results of Jiang, Nien and the third-named author, we prove that any two minimax unitarizable supercuspidals of $p$-adic $\mathrm {GL}_N$ that have the same depth and central character admit a special pair of Whittaker functions. As a corollary of our result, we prove Jacquet’s conjecture on the local converse problem for $\mathrm {GL}_N$, when $N$ is prime.

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Additional Information

Moshe Adrian
Affiliation: Department of Mathematics, Queens College, Queens, New York 11367-1597

Baiying Liu
Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
MR Author ID: 953254

Shaun Stevens
Affiliation: School of Mathematics, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom
MR Author ID: 678092

Peng Xu
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
MR Author ID: 1099916

Keywords: Local converse problem, special pairs of Whittaker functions
Received by editor(s): March 4, 2015
Received by editor(s) in revised form: October 8, 2015
Published electronically: January 27, 2016
Additional Notes: The second author was supported in part by NSF Grant DMS-1302122, and in part by a postdoc research fund from Department of Mathematics, University of Utah
The third and fourth authors were supported by the Engineering and Physical Sciences Research Council (grant EP/H00534X/1)
Article copyright: © Copyright 2016 American Mathematical Society