The answer may depend on what you mean by "classified", but the best results so far seem to be those of Brundan-Kleshchev and Brown-Goodwin.    See for example the preprint <a href="http://front.math.ucdavis.edu/1009.3869">here</a> and its references.   (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)