# Hodge theory of classifying stacks

## TL;DR

This paper computes the Hodge and de Rham cohomology of classifying stacks for reductive groups over various fields, revealing new connections with representation theory and comparing algebraic and topological cohomology.

## Contribution

It provides explicit calculations of cohomology for classifying stacks in algebraic geometry, extending understanding to fields of small characteristic and linking to representation theory.

## Key findings

- Cohomology calculations for classifying stacks over multiple fields.
- New results connecting algebraic cohomology with representation theory.
- Comparison between algebraic and topological classifying space cohomology.

## Abstract

We compute the Hodge and de Rham cohomology of the classifying space BG (defined as etale cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology.