Title: A Note on the Spectral Transfer Morphisms for Affine Hecke Algebras,
Abstract: 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.