Cohomology of algebraic groups with coefficients in twisted representations

TL;DR

This survey explores the cohomology of reductive algebraic groups with twisted coefficients, emphasizing advances from strict polynomial functor theory and introducing new proofs for untwisting theorems.

Contribution

It provides a comprehensive overview of cohomology with twisted coefficients and offers a novel proof connecting universal classes to untwisting theorems in polynomial functor theory.

Findings

Advances in cohomology theory for algebraic groups with twisted coefficients.

Connection between universal classes and untwisting theorems.

New proof of untwisting theorems using cohomological methods.

Abstract

This article is a survey on the cohomology of a reductive algebraic group with coefficients in twisted representations. A large part of the paper is devoted to the advances obtained by the theory of strict polynomial functors initiated by Friedlander and Suslin in the late nineties. The last section explains that the existence of certain `universal classes' used to prove cohomological finite generation is equivalent to some recent `untwisting theorems' in the theory of strict polynomial functors. We actually provide thereby a new proof of these theorems.