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

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

2010-08-09

arxiv.org/abs/1008.1592v2

Mathematics

Representation Theory

35 pages; v2: updated introduction to refer to work of Langlands

Scientific paper

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.

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