Epsilon factors of representations of finite general linear groups

TL;DR

This paper introduces epsilon factors for irreducible representations of finite general linear groups, showing their multiplicativity and expressing them as products of Gauss sums, linking various gamma factors.

Contribution

It defines epsilon factors using Macdonald's correspondence and relates them to Rankin-Selberg and exterior square gamma factors, establishing new connections and equalities.

Findings

Epsilon factors satisfy multiplicativity.

Epsilon factors can be expressed as products of Gauss sums.

Exterior square epsilon factors coincide in a special case.

Abstract

We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums. The exterior square epsilon factors relate the Jacquet-Shalika exterior square gamma factors and the Langlands-Shahidi exterior square gamma factors for level zero supercuspidal representations. We prove that these exterior square factors coincide in a special case.