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Braid group actions are ubiquitous in algebra, geometry and combinatorics. I will discuss one such occurrence in the context of highest weight representations of finite dimensional Lie algebras. I will explain how this braid group action provides a unifying explanation for several phenomena observed at the level of representation theory and combinatorics. To keep things concrete I will emphasize the case of sl_2 and sl_3.

Finally, I will discuss what all of this has to do with the title of the talk. Namely, I will try to explain how this braid group action factors through an action of the Hecke algebra, the tie-in with Kazhdan-Lusztig theory and will talk about ongoing efforts to develop a manageable `higher representation theory' for the Hecke algebra.