Wednesday, September 23, 2020
Time  Items 

All day 

4:00pm 
09/23/2020  4:15pm Abstract: Representation theory seeks to understand ways in which a given algebraic object (a group, an associative algebra, a Lie algebra etc.) can be represented via linear operators on a vector space over a field. What the representations are going to look like very much depends on the field in question, and, in particular, on its characteristic. Many important questions are settled in characteristic 0, for example, when we work over the complex numbers. But in the case of positive characteristic fields, which the word “modular” refers to, even basic questions are wide open. In my talk I will concentrate on one of the most important algebraic objects, semisimple Lie algebras, and explain what we know about about their irreducible (=no subrepresentations) modular representations. I will start with the case of sl_2 explaining the results of Rudakov and Shafarevich from 1967 describing the irreducible representations. Then I will talk about recent work on the general case including my paper with Bezrukavnikov from 2020, where we get the most explicit description of irreducible representations available to date. Our primary tool is relating the modular representations of semisimple Lie algebras to the (affine) Hecke category, the fundamental object in modern Representation theory. Location: 