Eilenberg-Moore spectral sequence and Hodge cohomology of classifying stacks

Let $G$ be a smooth connected reductive group over a field $k$ and $\Gamma$ be a central subgroup of $G$. We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of $B(G/\Gamma)$. As an application, building upon work of Toda and using Totaro's inequality, we show that for all $m\geq 0$ the Hodge and de Rham cohomology algebras of the classifying stacks $B\mathrm{PGL}_{4m+2}$ and $B\mathrm{PSO}_{4m+2}$ 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 $H^{>0}(\mathrm{GL}_{4m+2}, \operatorname{Sym}^j(\mathfrak{pgl}_{4m+2}^\vee))$ and $H^{>0}(\mathrm{SO}_{4m+2}, \operatorname{Sym}^j(\mathfrak{pso}_{4m+2}^\vee))$ over $\mathbb{F}_2$.