Basic sets in defining characteristic for general linear groups of small rank

Let qq be a power of some prime number pp. Let G be a connected reductive group defined over the field with qq elements and let FF be the corresponding Frobenius map. In this note, we give methods to find relations between the restrictions on semisimple elements of the irreducible characters of GF. As illustration, we explicitly determine a pp-basic set for GL2(q), GL3(q) and GL4(q).