Fourier transforms of orbital integrals on the Lie algebra of ${\mathrm{SL}}_2$

The Harish-Chandra–Howe local character expansion expresses the characters of reductive, $p$-adic groups in terms of Fourier transforms of nilpotent orbital integrals on their Lie algebras, and Murnaghan–Kirillov theory expresses many characters of reductive, $p$-adic groups in terms of Fourier transforms of semisimple orbital integrals (also on their Lie algebras). In many cases, the evaluation of these Fourier transforms seems intractable; but, for ${\mathrm{SL}}_2$, the nilpotent orbital integrals have already been computed. In this paper, we use a variant of Huntsinger’s integral formula, and the theory of $p$-adic special functions, to compute semisimple orbital integrals.