Eilenberg-Moore spectral sequence and Hodge cohomology of classifying stacks

Let G G be a smooth connected reductive group over a field k k and Γ \Gamma be a central subgroup of G G . We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of B ( G / Γ ) B(G/\Gamma ) . As an application, building upon work of Toda and using Totaro’s inequality, we show that for all m ≥ 0 m\geq 0 the Hodge and de Rham cohomology algebras of the classifying stacks B P G L 4 m + 2 B\mathrm {PGL}_{4m+2} and B P S O 4 m + 2 B\mathrm {PSO}_{4m+2} over F 2 \mathbb {F}_2 are isomorphic to the singular F 2 \mathbb {F}_2 -cohomology of the classifying space of the corresponding Lie group. From this we obtain a full description of H > 0 ( G L 4 m + 2 , Sym j ⁡ ( p g l 4 m + 2 ∨ ) ) H^{>0}(\mathrm {GL}_{4m+2}, \operatorname {Sym}^j(\mathfrak {pgl}_{4m+2}^\vee )) and H > 0 ( S O 4 m + 2 , Sym j ⁡ ( p s o 4 m + 2 ∨ ) ) H^{>0}(\mathrm {SO}_{4m+2}, \operatorname {Sym}^j(\mathfrak {pso}_{4m+2}^\vee )) over F 2 \mathbb F_2 .