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