Another realization of the category of modules over the small quantum group

AbstractLet g be a semi-simple simply-connected Lie algebra and let Uℓ be the corresponding quantum group with divided powers, where ℓ is an even order root of unity. Let in addition uℓ⊂Uℓ be the corresponding “small” quantum group. In this paper we establish the following relation between the categories of representations of Uℓ and uℓ. We show that the category of uℓ-modules is naturally equivalent to the category of Uℓ-modules, which have a Hecke eigen-property with respect to representations lifted by means of the quantum Frobenius map Uℓ→U(ǧ), where ǧ is the Langlands dual Lie algebra. This description allows to express the regular linkage class in the category uℓ-mod in terms of perverse sheaves on the affine flag variety with a Hecke eigen-property. Moreover, it can serve as a basis to the program to understand the connection between the category uℓ-mod and the category of representations of the corresponding affine algebra at the critical level