Spectral correspondences for affine Hecke algebras

We introduce the notion of spectral transfer morphisrns between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hec,ke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations. (C) 2015 Elsevier Inc. All rights reserved.