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Hiraga, Ichino and Ikeda have conjectured an explicit expression for thePlancherel density of the group of points of a reductive group defined over alocal field $F$, in terms of local Langlands parameters. In these lectures weshall present a proof of these conjectures for Lusztig's class ofrepresentations of unipotent reduction if $F$ is $p$-adic and $G$ is of adjointtype and splits over an unramified extension of $F$. This is based on theauthor's paper [Spectral transfer morphisms for unipotent affine Heckealgebras, Selecta Math. (N.S.) 22 (2016), no. 4, 2143--2207].  More generally for $G$ connected reductive (still assumed to be split over anunramified extension of $F$), we shall show that the requirement ofcompatibility with the conjectures of Hiraga, Ichino and Ikeda essentiallydetermines the Langlands parameterisation for tempered representations ofunipotent reduction. We shall show that there exist parameterisations for whichthe conjectures of Hiraga, Ichino and Ikeda hold up to rational constantfactors. The main technical tool is that of spectral transfer maps betweennormalised affine Hecke algebras used in op. cit.

Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda