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Consider a semi-simple simply-connected Lie algebra and the corresponding big quantum group with divided powers (defined by Lusztig) where the quantizing parameter is an even order root of unity. Consider in addition the subalgebra in the big quantum group called the "small" quantum group. In this talk we establish the following relation between the categories of representations of the two algebras: we show that the category of modules over the small quantum group is naturally equivalent to the category of modules over the big quantum group, which have a so called Hecke eigen-property with respect to representations lifted by means of the quantum Frobenius map from the category of representations of the Langlands dual Lie algebra. This description allows to express the regular linkage class in the category of modules over the small quantum group in terms of perverse sheaves on the enhanced Affine Flag variety with a Hecke eigen-property.
