I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.

In Langlands' program, Satake correspondence gives a correspondence between unramified representation of a reductive group $G$ over a local field and conjugacy classes in the Langlands dual group ${}^{L}G$ whose projection to $\hat{G}$ is semisimple and projection to $W_K$ is Frob.

In page 11 of this article, there is similar but different correspondence. It gives a bijection between irreducible representations of $\mathrm{GL}(2, \mathbb{F}_{q})$ and conjugacy classes of $\mathrm{GL}(2, \mathbb{F}_{q})$. Also, the type of the conjugacy class (Jordan form) determines the type of representation (principal series, special, cuspidal, 1-dimensional).

Is there a general theory for such correspondence over finite field? Can we generalize this to arbitrary reductive groups over finite field? If it is, what is the correspondence? In the article, author said that the correspondence is kind of *ad hoc*, which is not canonical at all. However, if we fix a generator of $\mathbb{F}_{q}^{\times}$, than I think it may possible to find some canonical way to do it.

I'm trying to verify this for other cases by using GAP. This is true for $\mathrm{GL}(2, \mathbb{F}_{q})$ as in the note, and it seems also true for $\mathrm{GL}(3, \mathbb{F}_{5})$. I computed dimension of irreducible representations, size of conjugacy classes, and number of each stuff (number of irreducible representations of given dimension or number of conjugacy classes of given size). GAP give the result that dimensions of irreducible representations are $$ [ 1, 30, 31, 96, 124, 125, 155 ] $$ and size of conjugacy classes are $$ [ 1, 744, 775, 12000, 14880, 15500, 18600 ] $$ And the correspondence (at least as a set) exists since both of them has $$ [ 4, 4, 12, 40, 4, 40, 12 ] $$ many different things.

`Finite groups of Lie type: Conjugacy classes and complex characters`

is an excellent introduction textbook that covers much ground. Digne and Michel's`Representations of Finite Groups of Lie Type`

is also highly recommended. There are also many related question on MO, such as this one: mathoverflow.net/questions/127691/reconciling-lusztigs $\endgroup$ – Dror Speiser Mar 23 at 19:47