Two gluing constructions on Category O

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 gluing construction of Kazhdan and Laumon, which one can understand more directly as a combinatorial construction generalizing the BGG Category O. I 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.