  This paper contributes to the study of the fibers of the commutator map on
special linear groups in characteristic zero. Specifically, we show that the
fibers over non-central elements all have the same dimension. Also we explain
that the fibers over central elements can be of larger dimension and compute
how large. We use the character tables of finite general linear groups
constructed by J.A. Green to count solutions to the commutator equation
$[x,y]=g$ over finite fields and use algebraic geometry to go from
characteristic $p$ to characteristic $0$. To deal with fibers over central
elements, we compute the orbits of the conjugation action of $\mathrm{GL}_n$ on
these fibers.
