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We determine the unitary dual of the geometric graded Hecke algebras with{unequal} parameters which appear in Lusztig's classification of unipotentrepresentations for {exceptional} $p$-adic groups. The largest such algebra isof type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to thissetting, this is equivalent to the identification of the corresponding unitaryunipotent representations with real central character of the $p$-adic groups.In order for this correspondence to be applicable here, we show (followingLusztig's geometric classification, and Barbasch and Moy's original argument)that the set of tempered modules with real central character for a geometricgraded Hecke algebra is linearly independent when restricted to the Weyl group.

On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters