Number of conjugacy classes of finite reductive groups Let $G$ be a connected reductive group over $\mathbb{Z}$.  Let $c_{G(\mathbb{F}_q)}$ be the number of conjugacy classes of $G(\mathbb{F}_q)$.
Question: Is it true that $c_{G(\mathbb{F}_q)}$ is a quasi-polynomial in $q$? I.e. is it true that there exists an integer $N$ (depending only on $G$) such that for every $i\in \{0,1,\dotsc,N-1\}$ there exists a polynomial $P_i$ such that $c_{G(\mathbb{F}_q)}=P_i(q)$ for all $q\equiv i \bmod N$?
If the answer is yes, then a type-independent explanation would be desirable. I'm mainly interested in the case $i=1$ but the general question seems worthwhile.
Frank Lübeck's tables at Character Degrees and their Multiplicities for some Groups of Lie Type of Rank < 9 illustrate that the question has a positive answer for (simply connected form of) exceptional groups. For instance, for $G_2$, $E_6$ and $E_7$, the site states that $N=6$ and gives the explicit polynomials. We also have a lot of positive evidence for classical groups coming from explicit computations in Macdonald - Numbers of conjugacy classes in some finite classical groups and Wall - On the conjugacy classes in the unitary, symplectic, and orthogonal groups among others.
 A: Assume that $G$ is adjoint split over $\mathbb F_q$. Let $G^*$ be the (Langlands) dual group; it is also split over $\mathbb F_q$. For each semisimple $s \in G^*( \mathbb F_q)$ let $N_s$ be the number of unipotent representations
of the centralizer $Z_{G^*}(s)(\mathbb F_q)$. The number of conjugacy classes in $G(\mathbb F_q)$ is equal to the number of irreducible characters of $G(\mathbb F_q)$ and this equals (by my 1984 classification) the sum $\sum_s N_s$ where $s$ runs over the $G^*(\mathbb F_q)$-conjugacy classes of semisimple elements in $G^*(\mathbb F_q)$. Since the $N_s$ are very well behaved (much better behaved that number of unipotent classes) this reduces the problem to a problem of counting semisimple classes in $G^*(\mathbb F_q)$ with centralizer of fixed type. From this the desired result follows.
The main point is that the number of unipotent classes in $G(\mathbb F_q)$ behaves differently in small characteristic while the number of uniptent representations of a reductive group over $\mathbb F_q$ is independent of characteristic. This makes it easier to count irreducible representations than conjugacy classes.
