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

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).