On the Jacquet Conjecture on the local converse problem for 𝑝-adic 𝐺𝐿_{𝑁}

Abstract

Based on previous results of Jiang, Nien and the third-named author, we prove that any two minimax unitarizable supercuspidals of  p p -adic  G L N \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  G L N \mathrm {GL}_N , when  N N is prime.