Motivic homotopy theory of the classifying stack of finite groups of Lie type

Let $G$ be a reductive group over ${\mathbb{F}}_p$ with associated finite group of Lie type $G^F$. Let $T$ be a maximal torus contained inside a Borel $B$ of $G$. We relate the (rational) Tate motives of $\text{B}{G}^F$ with the $T$-equivariant Tate motives of the flag variety $G\mathrm{/}B$. On the way, we show that for a reductive group $G$ over a field $k$, with maximal Torus $T$ and absolute Weyl group $W$, acting on a smooth finite type $k$-scheme $X$, we have an isomorphism $A_G^n(X,m{)}_{\mathbb{Q}}\cong A_T^n(X,m{)}_{\mathbb{Q}}^W$ extending the classical result of Edidin-Graham to higher equivariant Chow groups in the non-split case. We also extend our main result to reductive group schemes over a regular base that admit maximal tori. Further, we apply our methods to more general quotient stacks. In this way, we are able to compute the motive of the stack of $G$-zips introduced by Pink-Wedhorn-Ziegler for reductive groups over fields of positive characteristic.