Supercuspidal characters of SL ℓ over a p -adic field, ℓ a prime

Abstract

Let F be a p -adic field of characteristic 0. For odd primes ℓ with certain tameness restrictions, techniques due to Assem allow us to write supercuspidal characters of SL ℓ ( F ) in terms of a family of distributions defined by Kazhdan. Following an argument of Harish-Chandra, we derive an integral formula for these distributions. Results of DeBacker and Kazhdan allow us to evaluate this integral formula. This in turn gives supercuspidal character formulas for SL ℓ ( F ), explicit up to the constants appearing in the GL ℓ ( F ) local character expansion. Our result also describes the explicit local Langlands correspondence for those positive-depth supercuspidal representations of GL ℓ ( F ) which do not restrict irreducibly to SL ℓ ( F ).