A natural enlargement of the BGG Category O for a semisimple Lie algebra is the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. I will present a new geometric realization of this category in terms of gluing sheaves on the flag variety; this realization is Koszul dual to a well-studied gluing construction of Kazhdan and Laumon, (which I will introduce in the first half of the talk as a basic combinatorial construction generalizing the BGG Category O). We will explain its relationship to a proposed Koszul duality relating the small quantum group and the semi-infinite flag variety to the geometry of affine Springer fibers.