Note on restriction maps of Chow rings to Weyl group invariants

Abstract

Let $G$ be an algebraic group over $\mathbf{C}$ corresponding a compact simply connected Lie group. When $H^{*}(G)$ has $p$-torsion, we see $ρ^{*}_{CH} : CH^{*}(BG) → CH^{*}(BT)^{W_{G}(T)}$ is always not surjective. We also study the algebraic cobordism version $ρ^{*}_{Ω}$. In particular when $G = Spin(7)$ and $p = 2$, we see each Griffiths element in $CH^{*}(BG)$ is detected by an element in $Ω^{*}(BT)$.