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

Abstract

We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group  GL N {\operatorname {GL}}_{N} over a non-archimedean local field, based on distinguishability by twisted gamma factors. In the case that  N N is prime and the residual characteristic is greater than or equal to  ⌊ N 2 ⌋ \left \lfloor \frac {N}{2}\right \rfloor , we prove that, for any natural number  i ≤ ⌊ N 2 ⌋ 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 -i . This implies that, under the same conditions on  N N , the bound  ⌊ N 2 ⌋ \left \lfloor \frac {N}{2}\right \rfloor in the Local Converse Theorem for  GL N \operatorname {GL}_N is sharp.