  We define an involution on the space of elliptic unipotent Langlands
parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split
adjoint exceptional, the composition of this involution with the hyperspecial
parahoric restriction map agrees with Lusztig's nonabelian Fourier transform
for unipotent representations of the finite reductive quotient. This is
inspired by recent works of Lusztig on the almost unipotent characters of
$p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier
transform of odd orthogonal groups.
