Character bounds for finite groups of Lie type

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $\lvert \chi(g) \rvert \leqslant c \chi (1)^{\alpha g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular, we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.

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The first author was partially supported by the NSF grants DMS-1102434 and DMS-1601953. The second and third authors acknowledge the support of EPSRC grant EP/H018891/1. The third author acknowledges the support of ERC advanced grant 247034, ISF grants 1117/13 and 686/17, BSF grant 2016072 and the Vinik chair of mathematics which he holds. The fourth author was partially supported by the NSF grants DMS-1839351 and DMS-1840702, the Simons Foundation Fellowship 305247, the EPSRC, and the Mathematisches Forschungsinstitut Oberwolfach.

Received 2 November 2017

Received revised 7 August 2018

Accepted 10 August 2018

Published 6 November 2018