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Do the following two filtrations of the affine Grassmannian agree?

This statement isn't true; as you've currently defined everything, your $X_i$ are infinite dimensional (in fact they are unions of connected components.) You want instead to take the $X_i$ unions of $ …
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7 votes

Why should affine lie algebras and quantum groups have equivalent representation theories?

(Written on my phone - apologies for any typos.) A few comments: a) First, as to the source of the braided monoidal structure on the Kazhdan-Lusztig category. The category of integrable affine Lie …
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