Hodge theory of classifying stacks

Abstract

We compute the Hodge and the de Rham cohomology of the classifying space $\mathit{BG}$ (defined as étale cohomology on the algebraic stack $\mathit{BG}$) 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.