Bessel Functions and Kloosterman Integrals on GL(n)

TL;DR

This paper proves the local integrability of Bessel functions on GL(n) over p-adic fields by leveraging relations with Kloosterman integrals, Shalika germs, and generalized Kloosterman sum estimates.

Contribution

It establishes the local integrability of Bessel functions on GL(n) (p-adic case) using advanced relations and estimation techniques from prior works.

Findings

Proved local integrability of Bessel functions for GL(n) over p-adic fields.

Connected Bessel functions with Kloosterman integrals and Shalika germs.

Utilized estimates of generalized Kloosterman sums for the proof.

Abstract

This paper will focus on the proof of local integrability of Bessel functions for GL(n) (p-adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [Ba03] [Ba04] [Ba05], the theory of the (relative) Shalika germs established by H. Jacquet and Y. Ye in [JY96] [JY99] and G. Stevens' approach [Ste87] on estimating certain GL(n) generalized Kloosterman sums.