Nonabelian Fourier Kernels on $\mathrm{SL}_2$ and $\mathrm{GL}_2$

TL;DR

This paper derives explicit formulas for nonabelian Fourier kernels, orbital Hankel transforms, and stable orbital integrals on $ ext{SL}_2$ and $ ext{GL}_2$, advancing understanding of harmonic analysis on these groups.

Contribution

It provides explicit formulas for nonabelian Fourier kernels and orbital Hankel transforms on $ ext{SL}_2$ and $ ext{GL}_2$, confirming conjectures and extending applicability to various local fields.

Findings

Explicit formulas for nonabelian Fourier kernels on $ ext{SL}_2$ and $ ext{GL}_2$

Explicit formulas for the orbital Hankel transform on these groups

Formulas for stable orbital integrals of the basic function

Abstract

For $G=\mathrm{SL}_2$ or $\mathrm{GL}_2$, we present explicit formulas for the nonabelian Fourier kernels on $G$, as conjectured by A. Braverman and D. Kazhdan. Additionally, we furnish explicit formulas for the orbital Hankel transform on $G$, a topic investigated by the second author, and provide an explicit formula for the stable orbital integral of the basic function. These results are applicable to local fields with residual characteristics other than two.