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

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.