The characters of the finite general linear groups

Introduction. In this paper we show how to calculate the irreducible characters of the group GL(n, q) of all nonsingular matrices of degree n with coefficients in the finite field of q elements. These characters have been given for n = 2 by H. Jordan [8], Schur [10], and others, and for n =3 and n =4 by Steinberg [12], who has also [13] done important work in the general case. We are concerned here with characters, that is, characters of representations by matrices with complex coefficients. Let Xi, * * *, XA be the distinct absolutely irreducible ordinary characters of a group 5 of order g. By a of 6 (often called a generalised character or difference character) we mean a class-function 4 on 5 of the form