Bessel functions and local converse conjecture of Jacquet

TL;DR

This paper proves a kernel formula for Bessel functions associated with supercuspidal representations of p-adic GL(n), establishes their equivalence via different definitions, and applies these results to prove Jacquet's local converse conjecture.

Contribution

It introduces a kernel formula for Bessel functions, confirms their equivalence through different definitions, and proves Jacquet's local converse conjecture.

Findings

Kernel formula for Bessel functions established

Equivalence of Bessel functions via distribution and Whittaker models proven

Proof of Jacquet's local converse conjecture provided

Abstract

In this paper, we prove a kernel formula of Bessel functions attached to irreducible smooth supercuspidal representations of p-adic $GL(n)$. We also show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell. As an application we give a proof of the local converse conjecture of Jacquet.