Suppose $G$ is a classical matrix group over a finite field.If $C$ is any conjugacy class of $G$ , then is $|C|$ is a polynomial inorder $q$.
If $C$ is a conjugacy class in $G$ , is $|C|$ a polynomial in $q$?
This question is encouraged fromsupported by the fact that whenever I have calculated all conjugacy classes and itstheir sizes for very small groups, for (for example, $GL_{2}(\mathbb{F}_{q})$, $GL_{3}(\mathbb{F}_{q})$, $SL_{2}(\mathbb{F}_{q})$, $SL_{3}(\mathbb{F}_{q})$, it does) the sizes of conjugacy classes turn out to be polynomialspolynomial in $q$.
So, is it true indoes this property hold for all finite classical group, or at least even in the case of $GL_n$ or $SL_n$?
Thanks!