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 ${\mathrm{Sp}}_{2n}$ for any $n\in N$ using the Sp-orientation and the associated Borel classes.

Secondly, following the classical computations and using the analogue in $A^1$-homotopy of the Leray spectral sequence, we compute the η-inverted MW-motivic cohomology of general Stiefel varieties, obtaining in particular the computation of the η-inverted MW-motivic cohomology of the general linear groups ${\mathrm{GL}}_n$ and the special linear groups ${\mathrm{SL}}_n$ for any $n\in N$.

Finally, we determine the multiplicative structures of these total cohomology groups.