On the sharpness of the bound for the Local Converse Theorem of p-adic GLprime

Moshe Adrian, Baiying Liu, Shaun Stevens, Geo Kam-Fai Tam

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We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group GL(N) over a nonarchimedean local field, based on distinguishability by twisted gamma factors. In the case that N is prime and the residual characteristic is greater than or equal to N/2, we prove that, for any natural number i at most N/2, there are pairs of cuspidal irreducible representations whose logarithmic distance in this ultrametric is precisely i. This implies that, under the same conditions on N, the bound N/2 in the Local Converse Theorem for GL(N) is sharp.
Original languageEnglish
Pages (from-to)6-17
Number of pages12
JournalProceedings of the American Mathematical Society, Series B
Publication statusPublished - 20 Feb 2018

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