The main motivation for considering such matters is the role of affine Hecke algebras in the harmonic analysis of reductive p-adic groups. The most general point of view in this context is provided by the theory of types (see [6]). This theory seeks to describe a given block in the Bernstein decomposition of the category of smooth representations of a p-adic reductive group G via Morita equivalence as the representation category of the Hecke algebra of an associated "type". In many cases this is known, and in many impor-tant cases it was shown that the emerging Hecke algebras associated to types are isomor-phic to affine Hecke algebras in the above sense (see e.g. [15], [24], [20]). These Morita equivalences respect the harmonic analysis: The spectral measure of the Hilbert algebra of the affine Hecke algebra h arising as the Hecke algebra of a type of G can be transferred (up to a known positive factor) by the Morita equivalence to the Plancherel measure of G restricted to the corresponding Bernstein block [7]. In this way the affine Hecke algebra may be considered as a tool to disclose parts of the Plancherel measure of a reductive p-adic group, a point of view that was advocated by several authors (e.g. [28], [29], [14]).
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