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 here and its references. (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)
Jim Humphreys
- 52.9k
- 4
- 120
- 240