Langlands correspondence is a conjectural correspondence between automorphic forms over a reductive group and Galois representation. This is widely open subject in number theory and there are some known results for special cases (special groups over special fields). By the way, I think that representation theory of finite groups is much easier than general Lie groups because it is finite - we can actually count things and we don't need to concern about convergence issues. So I believe we can formulate Langlands correspondence over a finite field and even easily prove it, but I can't find any reference for this. (I strongly believe that it would be not that hard to prove, but I don't know what is a correct way to formulate it.) Am I right? Thanks in advance.
FYI, I read chapter 4.1 of Bump, which introduces the representation theory of $\mathrm{GL}(\mathbb{F}_{p})$ which can be applied to $p$-adic general linear groups with some modification. But I don't know how to attach Galois representations to these representation of $\mathrm{GL}(\mathbb{F}_{p})$.