Uniform character bounds for finite classical groups

Abstract

For every finite quasisimple group of Lie type $G$, every irreducible character $\chi $ of $G$, and every element $g$ of $G$, we give an exponential upper bound for the character ratio $|\chi (g)|/\chi (1)$ with exponent linear in $\log_{|G|} |g^G|$, or, equivalently, in the ratio of the support of $g$ to the rank of $G$. We give several applications, including a proof of Thompson’s conjecture for all sufficiently large simple symplectic groups, orthogonal groups in characteristic $2$, and some other infinite families of orthogonal and unitary groups.