Theorem 1.4.1 in arxiv:0810.2076 answers some of your questions for generic semisimple irreducible representations. Emmanuel has hitherto unpublished results where he does answer your question for generic irreducible representations in terms of intersection cohomology of certain quiver varieties. We do not know about other results on the representation ring of $GL_n({\mathbb F}_q)$.
Tamas Hausel
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