On the sharpness of the bound for the Local Converse Theorem of 𝑝-adic 𝐺𝐿_{𝑝𝑟𝑖𝑚𝑒}

Adrian, Moshe; Liu, Baiying; Stevens, Shaun; Tam, Geo

doi:10.1090/bproc/32

On the sharpness of the bound for the Local Converse Theorem of $p$-adic ${\operatorname {GL}}_{prime}$
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by Moshe Adrian, Baiying Liu, Shaun Stevens and Geo Kam-Fai Tam HTML | PDF

Proc. Amer. Math. Soc. Ser. B 5 (2018), 6-17

Abstract:

We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group ${\operatorname {GL}}_{N}$ over a non-archimedean 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 $\left \lfloor \frac {N}{2}\right \rfloor$, we prove that, for any natural number $i\le \left \lfloor \frac {N}{2}\right \rfloor$, 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 $\left \lfloor \frac {N}{2}\right \rfloor$ in the Local Converse Theorem for $\operatorname {GL}_N$ is sharp.

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