Relation between unipotent cuspidal representations and cuspidal local systems

Question

This could well be a question for reading suggestion. Hope it's not too bad and thanks a lot.

So the question is as in the title. What are the relations between the notion of unipotent cuspidal representations of $G(\mathbb{F}_q)$, and that of cuspidal local systems as in generalized Springer theory, if any? For example, I notice that for classical groups of type $\mathbf{C}$, there exists a cuspidal local system for $\mathbf{C}_n$ iff there exists a unipotent cuspidal representations for $\mathbf{C}_{2n}$. Same for type $\mathbf{D}$. (For type $\mathbf{B}$, while there exist cuspidal local systems for $\mathrm{SO}_{n^2}$ and $\mathrm{SO}_{(n+1)^2}$, there exists a unipotent cuspidal representations for $SO_{n^2+(n+1)^2}$.)

This doesn't look like coincidence to me. Though by the time of the 1977 paper of Lusztig on representations of finite classical groups, he hasn't invented generalized Springer theory yet...

Answer 1

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.

Answer 2

I am mostly likely only adding to the question here and not giving an answer. First, Are the unipotent cuspidal representations in Lusztig's 1977 work related to cuspidal charachter sheaves with non-trivial restriction to the Unipotent Variety ? If the answer to this is yes, then this paper of A.M Aubert may be useful : "Some properties of Charachter Sheaves" .

(I am posting this as an answer so that I could provide the link - This is more of a comment really).