Pablo Boixeda Alvarez (Northeastern): Weight modules and Kazhdan-Laumon category

Abstract:     In work of Mathieu and Fernando, the modules of the enveloping algebra with finite-dimensional weight spaces are understood. These conditions can be translated into a support condition for the associated graded or some singular support condition for some sheaves on G/B. This singular support is given by taking the union of W copies of the conditions for category O.
    In work of Kazhdan and Laumon, they construct a category by glueing W copies of the category of perverse sheaves on G/U. This category was studied by Bezrukavnikov, Polishchuk and Morton-Ferguson. In particular, some subcategory known as Kazhdan-Laumon category O was related to the representation theory of the small quantum group u_q.
    In joint work with Morton-Ferguson, we relate the Kazhdan-Laumon category O to some subcategory of weight modules. This connection should explain the relation to the representation theory of u_q.  In this talk I will present Kazhdan-Laumon’s category O construction and describe the relation with weight modules. Time permitting, I will discuss the connection to representation theory of the small quantum group u_q via the joint work with Bezrukavnikov, McBreen and Yun and the geometry of affine Springer fibers.