Cohomology of algebraic groups, Lie algebras, and finite groups of Lie type

TL;DR

This paper explores the interconnected cohomology theories of reductive algebraic groups, their Lie algebras, Frobenius kernels, and finite Chevalley groups, highlighting key developments and open questions over the past two decades.

Contribution

It provides an overview of recent advances in the cohomology of algebraic groups and related structures, emphasizing their interrelations and open problems.

Findings

Identification of key cohomological properties of algebraic groups and finite groups of Lie type

Connections established between the cohomology of Lie algebras and algebraic groups

Open questions highlighted for future research in the field

Abstract

Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The representation theories of these structures are highly interconnected. This expository article will focus specifically on the cohomology theories of these structures and the relationships between them with the aim of highlighting a few key developments over the past 20 years and related open questions.