A Fourier transform for unipotent representations of p-adic groups

In recent work, Aubert-Ciubotaru-Romano gave a conjecture involving an involution on a certain elliptic space of unipotent representations of a p-adic group: they defined this involution and conjectured a relation between it and Lusztig's nonabelian Fourier transform for unipotent representations of finite groups of Lie type. The goal of the project is to prove new cases of the conjecture, starting with exceptional groups.