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what happens to y?
can you lift this algebra to the whole Hecke algebras, not just the centers? thus deforming the malvenuto-reutenauer hopf alg?
Darij, In thinking about your question, I think it is healthy to think of the finite Hecke algebra as a special cyclotomic Hecke algebra and then go across Brundan-Kleshchev so that everything is happening in the KLR algebra world. I think the observation that type A KLR is deformed Malvenuto-Reutenaruer is very healthy and has not been sufficiently studied.