Another Realization of the Category of Modules Over the Small Quantum Group

Abstract

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