  We define and study an analogue of Category O in the context of Kazhdan and
Laumon's gluing construction for perverse sheaves on the basic affine space. We
explicitly describe the simple objects in this category, and we show its
linearized Grothendieck group is isomorphic to a natural submodule of Lusztig's
periodic Hecke module. We then provide a categorification of these results by
showing that the Kazhdan-Laumon Category O is equivalent to a full subcategory
of a suitably-defined category of perverse sheaves on the semi-infinite flag
variety.
