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](https://en.wikipedia.org/wiki/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](http://www.math.rwth-aachen.de/%7EFrank.Luebeck/chev/DegMult/index.html) 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](https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/numbers-of-conjugacy-classes-in-some-finite-classical-groups/98990E5FA1506E488ED8FEC336E46320) and [Wall - On the conjugacy classes in the unitary, symplectic, and orthogonal groups](https://doi.org/10.1017/S1446788700027622) among others.



Number of conjugacy classes of finite reductive groups
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.


