Another Realization of the Category of Modules over the Small Quantum Group Sergey Arkhipov and Dennis Gaitsgory

0.1. Let g be a semi-simple Lie algebra. Given a root of unity (cf. Sect. 1.2), one can consider two remarkable Hopf algebras, Ul and ul, called the big and the small quantum group, respectively. Let Ul -mod and ul -mod denote the corresponding categories of modules. It is explained in [14] and [1] that the former is an analog in characteristic 0 of the category of algebraic representations of the corresponding group G over a field of positive characteristic, and the latter is an analog of the category of representations of its first Frobenius kernel. It is a fact of crucial importance, that although Ul is introduced as an algebra defined by an explicit set of generators and relations, the category Ul -mod (or, rather, its regular block, cf. Sect. 5.1) can be described in purely geometric terms, as perverse sheaves on the (enhanced) affine flag variety F̃l, cf. Sect. 6.5. This is obtained by combining the Kazhdan-Lusztig equivalence between quantum groups and affine algebras and the Kashiwara-Tanisaki localization of modules over the affine algebra on F̃l. This paper is a first step in the project of finding a geometric realization of the category ul -mod. We should say right away that one such realization already exists, and is a subject of [4]. We would like to investigate other directions. We were motivated by a set of conjectures proposed by B. Feigin, E. Frenkel and G. Lusztig, which, on the one hand, tie the category ul -mod to the (sill hypothetical) category of perverse sheaves on the semi-infinite flag variety (cf. [5], [6]), and on the other hand, relating the latter to the category of modules over the affine algebra at the critical level. Since we already know the geometric interpretation for modules over the big quantum group, it is a natural idea to first express ul -mod entirely in terms of Ul -mod. This is exactly what we do in this paper.