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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...

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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.

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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).

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  • $\begingroup$ Thanks! I don't know as well though... $\endgroup$ – Cheng-Chiang Tsai Apr 15 '16 at 1:21

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