Abstract
The general method to find irreducible representations of GL2(Fq) and GL3(Fq) is using induction from Borel subgroup B or parabolic subgroup P. But there exists some irreducible characters which are not easy to find. We have to compose such characters with not obvious way. Deligne-Lusztig give us better method to find irreducible characters of GL2(Fq) and GL3(Fq) by virtual characters RGT\theta for a pair of a maximal torus T and a character \theta of T. In this thesis we work out all Deligne-Lusztig virtual characters of GL2(Fq) and GL3(Fq).