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

Let $g$ be a semi-simple simply-connected Lie algebra and let $U_\ell$ be the corresponding quantum group with divided powers, where $\ell$ is an even order root of unity. Let in addition $u_\ell\subset U_\ell$ be the corresponding "small" quantum group. In this paper we establish the following relation between the categories of representations of $U_\ell$ and $u_\ell$: We show that the category of $u_\ell$-modules is naturally equivalent to the category of $U_\ell$-modules, which have a {\it Hecke eigen-property} with respect to representations lifted by means of the quantum Frobenius map $U_\ell\ti U(\check g)$, where $g$ is the Langlands dual Lie algebra. This description allows to express the regular linkage class in the category $u_\ell$-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 $u_\ell$-mod and the category of representations of the corresponding affine algebra at the critical level.