Reduced C*-Algebra of the p-Adic Group GL( n) II

Abstract

The reduced C *-algebra of the p-adic group GL( n) admits a Bernstein decomposition. We give a minimal refinement of this decomposition, and provide structure theorems for the reduced Iwahori–Hecke C *-algebra and the reduced spherical C *-algebra. This leads to a very explicit description of the tempered dual of GL( n) in terms of Bernstein parameters and extended quotients. We also prove that Plancherel measure (on the tempered dual of a reductive p-adic group) is rotation-invariant.