Nonabelian Fourier Kernels on ${\mathrm{S}\mathrm{L}}_2$ and ${\mathrm{G}\mathrm{L}}_2$

For $G={\mathrm{S}\mathrm{L}}_2$ or ${\mathrm{G}\mathrm{L}}_2$, 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.