Milnor-Witt motivic cohomology and linear algebraic groups

Abstract

This article presents two key computations in MW-motivic cohomology. Firstly, we compute the MW-motivic cohomology of the symplectic groups $S{p}_{2n}$ for any $n\in \mathbb{N}$ using the $Sp$-orientation and the associated Borel classes. Secondly, following the classical computations and using the analogue in ${\mathbb{A}}^1$-homotopy of the Leray spectral sequence, we compute the $\hat{I}\cdot$-inverted MW-motivic cohomology of general Stiefel varieties, obtaining in particular the computation of the $\hat{I}\cdot$-inverted MW-motivic cohomology of the general linear groups $G{L}_n$ and the special linear groups $S{L}_n$ for any $n\in \mathbb{N}$. Finally, we determine the multiplicative structures of these total cohomology groups.