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A Note on the Spectral Transfer Morphisms for Affine Hecke Algebras

E. Opdam introduced the tool of spectral transfer morphism (STM) of affine Hecke algebras to study the formal degrees of unipotent discrete series representations. He established a uniqueness property of STM for the affine Hecke algebras associated of unipotent discrete series representations. Based on this result, Opdam gave an explanation for Lusztig's arithmetic/geometric correspondence (in Lusztig's classification of unipotent representations of $p$-adic adjoint simple groups) in terms of harmonic analysis, and partitioned the unipotent discrete series representations into $L$-packets based on the Lusztig-Langlands parameters. The present paper provides some omitted details for the argument of the uniqueness property of STM. In the last section, we prove that three finite morphisms of algebraic tori are spectral transfer morphisms, and hence complete the proof of the uniqueness property.