Theory of Fundamental Bessel Functions of High Rank

Abstract

In this article, we shall study fundamental Bessel functions for ${\mathrm{GL}}_n(\mathbb{F})$ arising from the Voronoï summation formula for any rank $n$ and field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, with focus on developing their analytic and asymptotic theory. The main implements and subjects of our study of fundamental Bessel functions are their formal integral representations and Bessel differential equations. We shall prove the asymptotic formulae for fundamental Bessel functions and explicit connection formulae for the Bessel differential equations.

126 pages. Remark 17.6 added for the $G{L}_2$ kernel formula. This is the combination of two papers "Theory of Fundamental Bessel Functions of High Rank - I: Fundamental Bessel Functions" [arXiv:1408.5652 ] and "II: Hankel Transforms and Fundamental Bessel Kernels" [arXiv:1411.6710]

Cited by in corpus (2)

• Cancellation in the Additive Twists of Fourier Coefficients for ${\mathrm{GL}}_2$ and ${\mathrm{GL}}_3$ over Number Fields
• On the Kuznetsov Trace Formula for ${\mathrm{PGL}}_2(\mathbb{C})$