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We prove that a certain full subcategory of equivariant perverse sheaves on the affine Grassmannian of PGL(2, R) is equivalent to representations of Lusztig's quantum SL(2) at a primitive fourth root of unity. Nadler's functor of nearby cycles from the complex affine Grassmannian corresponds to quantum Frobenius.
Macerato et al. (Thu,) studied this question.
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