The Murnaghan-Nakayama rule for S_n is a combinatorial rule to compute the irreducible characters of the symmetric group. Is there a q-analogue of this rule for GL(n,q) to compute the irreducible characters? For example, exhibiting that the value of the unipotent characters of GL(n,q) on a unipotent class is given by the cocharge Kostka-Foulkes polynomials, and showing other special cases.
I presume you're talking about characters over C. In which case Green's classic paper "The characters of the finite general linear groups" is the only thing I know on this subject. He defines a whole load of polynomials (that have since come to be known as Green polynomials), out of which he's able to extract a list of the irreducible characters.
Green's paper is the mathematical ancestor of the Deligne-Lusztig theory of irreducible characters.