Title : A Geometric Proof of a Modular Schur-Weyl Theorem
Abstract: Modular representation theory is the study of representation theory over fields of positive characteristic. Generally, modular representation theory is more complicated than its characteristic zero counterpart. Recently, a number of theorems have appeared giving geometric descriptions of categories of modular representations. A version of the geometric Satake theorem due to Mirkovic-Vilonen implies that the modular representation theory of the general linear group is encoded in a category of geometric objects called perverse sheaves on an affine Grassmannian. On the other hand, a version of Springer theory due to Juteau relates the modular representation theory of the symmetric group with the geometry of the nilpotent cone in gl_n. This talk will explain how these two pieces fit together to give a geometric explanation for connections between the modular representation theory of the general linear group and that of the symmetric group.
Title : Lusztig's conjectures on modular representations
Abstract: This is the first of a series of related talks where certain mechanisms will be applied to representation theory, Langlands program, algebraic geometry and hopefully to knot invariants and quantum field theory.
The representation theoretic aspect is a proof of Lusztig's conjectures which describe numerical structure of modular representation theory (with Bezrukavnikov).
The two key geometric ideas: (i) a construction of Azumaya algebras in positive characteristic as a tool for math/physics, (ii) action of the affine braid groups on coherent sheaves on cotangent bundles of flag varieties.
Title : Lusztig's conjectures on modular representations (part II)
Title : Nilpotent orbits in characteristic 2 and Springer correspondence
Abstract: Let k and F_q be an algebraically closed and a finite field of characteristic 2 respectively. Let G be an adjoint (resp. simply connected) algebraic group of type B,C or D over k, g the Lie algebra of G and g^* the dual vector space of g. We construct the Springer correspondence for g (resp. g^*) following Lusztig's method. The correspondence is a bijective map from the set A_g (resp. A_g^*) to the set of irreducible characters of the Weyl group of G, where A_g (resp. A_g^*) is the set of all pairs (c,F) with c a nilpotent G-orbit in g (resp. g^*) and F an irreducible G-equivariant local system on c (up to isomorphism). In particular, we obtain classifications of nilpotent orbits in orthogonal Lie algebras over F_q and in the duals of classical Lie algebras over k and F_q. Finally, we describe the explicit correspondence using similar combinatorics that appears in the description of generalized Springer correspondence (defined by Lusztig) for classical groups in the case of characteristic not equal 2 and unipotent case in characteristic 2.
Title : Faces of polytopes and Koszul algebras
Abstract: Given a simple Lie algebra g and a finite-dimensional simple g-module V, we study the category G of graded finite-dimensional modules of the corresponding semidirect product Lie algebra. This framework includes the truncated current Lie algebras as well as those associated to folding of complex simple Lie algebras. Given a face of the polytope formed by the weights of V, we introduce a partial order on the simple objects in G. For certain finite subsets of the affine weight lattice, we produce Koszul algebras of finite global dimension equal to the number of weights of V which are on the face. This is joint work with Vyjayanthi Chari and Tim Ridenour.