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Theorem 1.4.1 in arxiv:0810.2076 answers some of your questions for generic semisimple irreducible representations. Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties. We did not know about other results on the representation ring of $GL_n({\mathbb F}_q)$. EDIT: but see Victor's answer for related results of Lusztig.

Theorem 1.4.1 in arxiv:0810.2076 answers some of your questions for generic semisimple irreducible representations. Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties. We did not know about other results on the representation ring of $GL_n({\mathbb F}_q)$. EDIT: but see Victor's answer for related results of Lusztig. EDIT 2 (added 16/03/11) Letellier's paper is now available at: http://arxiv.org/abs/1103.2759

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 Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties.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

Theorem 1.4.1 in arxiv:0810.2076 answers some of your questions for generic semisimple irreducible representations. Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties. We did not know about other results on the representation ring of $GL_n({\mathbb F}_q)$. EDIT: but see Victor's answer for related results of Lusztig.

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 Emmanuel has hitherto unpublished results where he does answer your question for generic irreducible representations in terms of intersection cohomology of certain quiver varieties.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 $GL_n({\mathbb F}_q)$.

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 Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties.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