Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = \mathrm{Frac}(W)$ its fraction field. I'll abuse notation by also writing $\boldsymbol{G}$ for the corresponding (unramified) reductive group over $K$.

When $\boldsymbol{G}$ has connected center, Lusztig proves the following:

**Theorem.** There is a canonical bijection between (isomorphism classes of) irreducible $\overline{K}$-representations of $G$ and special conjugacy classes in $\widehat{\boldsymbol{G}}(\overline{K})$ stable under $g \mapsto g^q$.

Here $\widehat{\boldsymbol{G}}$ is the dual group of $\boldsymbol{G}$. The condition of being special is a condition of Lusztig relating to special representations of Weyl groups through the Springer correspondence.

On the other hand, one could write a Langlands-type statement as follows: irreducible $\overline{K}$-representations of $G$ should correspond to $\widehat{\boldsymbol{G}}(\overline{K})$-conjugacy classes of $L$-parameters for $G$.

One possible definition of an $L$-parameter is that of a homomorphism over $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ of the form
$$\varphi \colon \langle \mathrm{Frob}_q \rangle \times \mathrm{SL}_2(\overline{K}) \to {}^L \boldsymbol{G}(\overline{K}), $$
where ${}^L \boldsymbol{G} = \widehat{\boldsymbol{G}} \rtimes \mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ is the Langlands dual group of $\boldsymbol{G}$, and where we require $\mathrm{Frob}_q$ to have semisimple image in $\widehat{\boldsymbol{G}}(\overline{K})$, and the restriction of $\varphi$ to $\mathrm{SL}_2(\overline{K})$ to be algebraic.

Of course, in this situation, the data of an $L$-parameter $\varphi$ is equivalent to that of a particular kind of $\widehat{\boldsymbol{G}}(\overline{K})$-conjugacy class in ${}^L \boldsymbol{G}(\overline{K})$. However, Lusztig's conditions on the conjugacy classes do not appear anywhere. For instance, when $\boldsymbol{G}$ is split, an $L$-parameter is simply a $\widehat{\boldsymbol{G}}(\overline{K})$-conjugacy class in $\widehat{\boldsymbol{G}}(\overline{K})$.

Instead, we might want to add an extra condition which would pin down the image of $\mathrm{Frob}_q$ to lie in $\widehat{\boldsymbol{G}}(K)$, in order to parallel the following result, pertaining to the semisimple part of the correspondence:

**Theorem.** There is a canonical bijection between (isomorphism classes of) irreducible semisimple $\overline{K}$-representations of $G$ and semisimple conjugacy classes in $\widehat{\boldsymbol{G}}(\mathbb{F}_q)$.

However, such a modification would not account for the unipotent part, nor would it account for the condition of being special.

It seems, then, that this notion of $L$-parameter is the wrong one. What is the fix?