Some Remarks on Discrete Series Characters for Reductive p-adic Groups

Abstract

This chapter discusses the discrete series characters for reductive p-adic groups. The discrete series, which is well understood for real groups, has presented one of the most vexing problems in the representation theory of p-adic groups. There has been some progress in the construction of discrete series ([Hi], [KL], [Kub]), but only a few results exist on the specific nature of their characters. It has been noted that, before Bernstein proved the admissibility of irreducible unitary representations, Harish-Chandra gave a simple proof of the admissibility of discrete series representations. The chapter describes the partition of the discrete series of G into supercuspidal representations and generalized special representations. It has been conjectured by many people that supercuspidal representations can be induced irreducibly from open, compact mod center subgroups of G. For the generalized special representations, there are essentially no general techniques available for obtaining explicit character values except for the Steinberg type representations.