We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnicket al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019,arXiv:1904.08070] by different methods.

中文翻译：

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字符级别和字符范围

我们为有限、一般或特殊、线性和酉群的复杂不可约特征开发了特征级别的概念。我们根据 Lusztig 标签和程度来描述一个字符的级别。然后，我们证明了具有不太大中心化器的元素的字符值的明确上限，并推导出与这些共轭类相对应的随机游走的覆盖数和混合时间的上限。我们还根据某些对偶对来表征字符的级别，并证明字符值的显式指数字符边界，前提是该级别不太大。还提供了几个进一步的应用程序。其他有限经典群的相关结果在续集 [Guralnick等。“有限经典群的字符级别和字符界限”，预印本，2019，arXiv:1904.08070] 通过不同的方法。