Bessel Functions on GL(n), II -- The case $n=4$

Abstract

The purpose of this article is to verify the conjectures of the previous paper in the particular case of $GL(4)$. We accomplish this in general, but observe two failures of the conjectures: First, that the Strong Interchange of Integrals conjecture is perhaps false for a single Weyl element $w_{2,2}$, though we prove the Weak Interchange of Integrals still holds. Second, again for a single Weyl element $w_{2,1,1}$ and its conjugate $w_{1,1,2}$, it appears that the space of solutions to the Bessel differential equations may not be spanned by the Frobenius series solutions. We discuss what refinements, namely to the Asymptotics Theorem, would be necessary to uniquely identify the Bessel functions for such Weyl elements, and prove them in the exceptional cases for $GL(4)$.