Local integrability of Bessel functions on split groups

TL;DR

This paper proves the local integrability of Bessel functions on all connected split reductive groups over p-adic fields, extending previous results known only for specific cases like GL(2) and GL(3).

Contribution

It generalizes the known local integrability of Bessel functions from specific groups to all connected split reductive groups over p-adic fields.

Findings

Bessel functions are locally integrable on all connected split reductive groups.

Bessel distributions can be expressed as integrals against these Bessel functions.

Extension of previous results from GL(2) and GL(3) to broader classes of groups.

Abstract

In this paper, we prove that the Bessel functions are locally integrable for all connected split reductive linear algebraic groups over a p-adic field $F$ and the Bessel distributions are given by integrals against these Bessel functions, which are previously known only for $GL(2),GL(3)$ proved by Baruch.