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.
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