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

TL;DR

This paper computes Fourier transforms of semisimple orbital integrals on the Lie algebra of SL2, building on known nilpotent integral results and applying p-adic special functions to advance harmonic analysis in p-adic groups.

Contribution

It introduces a method to evaluate semisimple orbital integrals for SL2 using Huntsinger's formula and p-adic special functions, extending previous nilpotent integral computations.

Findings

Explicit formulas for semisimple orbital integrals obtained.

Enhanced understanding of Fourier transforms in p-adic harmonic analysis.

Method applicable to other reductive p-adic groups?

Abstract

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 $\operatorname{SL}_2$, 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.