Relation between unipotent cuspidal representations and cuspidal local systems

Have you looked at Lusztig's 1995 IMRN paper "Classification of Unipotent Representations of Simple p-adic Groups"? He discusses this relationship in general, and it is indeed not a coincidence. To the question in Aswin's answer: see Sections 6.6. and 6.7. of the paper.