For any semisimple Lie algebra, the action on categories generated by a simple module with generic central character is semi-simple, simple transitive, and described by weight multiplicities.
Translated simple modules for Lie algebras and simple supermodules for Lie superalgebras
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Combinatorics of monoidal actions in Lie-algebraic context
For any semisimple Lie algebra, the action on categories generated by a simple module with generic central character is semi-simple, simple transitive, and described by weight multiplicities.