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 $Sp_{2n}$ 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 $\eta$-inverted MW-motivic cohomology of general Stiefel varieties, obtaining in particular the computation of the $\eta$-inverted MW-motivic cohomology of the general linear groups $GL_n$ and the special linear groups $SL_n$ for any $n\in\mathbb{N}$. Finally, we determine the multiplicative structures of these total cohomology groups.