Fourier transforms of semisimple orbital integrals on the Lie algebra of SL₂

Loren Spice

Vol. 254 (2011), No. 2, 407–448

DOI: 10.2140/pjm.2011.254.407

Abstract

The Harish-Chandra–Howe local character expansion expresses the characters of reductive, -adic groups in terms of Fourier transforms of nilpotent orbital integrals on their Lie algebras, and Murnaghan–Kirillov theory expresses many characters of reductive, -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 , the nilpotent orbital integrals have already been computed. We compute Fourier transforms of semisimple orbital integrals using a variant of Huntsinger’s integral formula and the theory of -adic special functions.

Keywords

p-adic, orbital integral, special function

Mathematical Subject Classification 2010

Primary: 22E35, 22E50

Milestones

Received: 7 October 2010

Accepted: 11 April 2011

Published: 27 February 2012