Eilenberg-Moore spectral sequence and Hodge cohomology of classifying stacks

Let $G$ be a smooth connected reductive group over a field $k$ and $\mathrm{\Gamma }$ be a central subgroup of $G$. We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of $B(G\mathrm{/}\mathrm{\Gamma })$. As an application, building upon work of Toda and using Totaro’s inequality, we show that for all $m\ge 0$ the Hodge and de Rham cohomology algebras of the classifying stacks $B{\mathrm{P}\mathrm{G}\mathrm{L}}_{4m+2}$ and $B{\mathrm{P}\mathrm{S}\mathrm{O}}_{4m+2}$ over ${\mathbb{F}}_2$ are isomorphic to the singular ${\mathbb{F}}_2$-cohomology of the classifying space of the corresponding Lie group. From this we obtain a full description of $H^{>0}({\mathrm{G}\mathrm{L}}_{4m+2},{\mathrm{Sym}}^j({\mathfrak{p}\mathfrak{g}\mathfrak{l}}_{4m+2}^{\vee }))$ and $H^{>0}({\mathrm{S}\mathrm{O}}_{4m+2},{\mathrm{Sym}}^j({\mathfrak{p}\mathfrak{s}\mathfrak{o}}_{4m+2}^{\vee }))$ over ${\mathbb{F}}_2$.