Character levels and character bounds for finite classical groups (Q6139246)

Let \(G\) be a finite group and \(\chi \in \mathrm{Irr}(G)\) an irreducible character. For all \(g \in G\) we have the trivial bound \(|\chi(g)|\leq \chi(1)\), but stronger bounds typically holds. Indeed, the centralizer bound \(|\chi(g)| \leq|C_{G}(g)|^{1/2}\), which follows from Schur's lemma, is often much better than \(\chi(1)\). In particular, good bounds are most easily obtained for elements with \(|C_{G}(g)| \ll |G|\) and characters with \(\chi(1)\) not too much smaller than \(|G|^{1/2}\). The main results of the paper under review develop a level theory and establish strong character bounds for finite classical groups, in the case that the centralizer of the element has small order compared to \(|G|\) in a logarithmic sense.