Kazhdan-Laumon categories, semi-infinite flags, and the algebra of braids and ties

We study D. Kazhdan and G. Laumon's 1988 gluing construction for perverse sheaves on the basic affine space G/U and explore unexpected connections to other interesting objects in representation theory. We first define an analogue of Category O in the context of Kazhdan-Laumon categories and explicitly classify its simple objects, and then use this combinatorial data to discuss its connections to Braverman-Kazhdan's Schwartz space on G/U and perverse sheaves on the semi-infinite flag variety. Finally, we study the action of the braid group appearing in the definition of Kazhdan-Laumon categories and give a categorification of the "algebra of braids and ties" occuring in the context of knot theory.