On the Jacquet Conjecture on the Local Converse Problem for p-adic GL_n

TL;DR

This paper advances the proof of Jacquet's conjecture on the local converse problem for p-adic GL_n by establishing a key property of supercuspidal representations, particularly for prime N.

Contribution

It proves that minimax unitarizable supercuspidals with the same depth and central character admit special Whittaker functions, reducing the conjecture's proof to a more manageable case.

Findings

Established existence of special Whittaker functions for certain supercuspidals.

Reduced the proof of Jacquet's conjecture to prime N cases.

Proved Jacquet's conjecture for all prime N.

Abstract

Based on previous results of Jiang, Nien and the third author, we prove that any two minimax unitarizable supercuspidals of GL_N that have the same depth and central character admit a special pair of Whittaker functions. This result gives a new reduction towards a final proof of Jacquet's conjecture on the local converse problem for GL_N. As a corollary of our result, we prove Jacquet's conjecture for GL_N, when N is prime.