The mixed Tate property of reductive groups

Abstract: This thesis is concerned with the mixed Tate property of reductive algebraic groups $G$, which in particular guarantees a Chow Kunneth property for the classifying space $BG$. Toward this goal, we first refine the construction of the compactly supported motive of a quotient stack. In the first section, we construct the compactly supported motive $M^c(X)$ of an algebraic space $X$ and demonstrate that it satisfies expected properties, following closely Voevodsky's work in the case of schemes. In the second section, we construct a functorial version of Totaro's definition of the compactly supported motive $M^c([X/G])$ for any quotient stack $[X/G]$ where $X$ is an algebraic space and $G$ is an affine group scheme acting on it. A consequence of functoriality is a localization triangle for these motives. In the third section, we study the mixed Tate property for the classical groups as well as the exceptional group $G_2$. For these groups, we demonstrate that all split forms satisfy the mixed Tate property, while exhibiting non-split forms that do not. Finally, we prove that for any affine group scheme $G$ and normal split unipotent subgroup $J$ of $G$, the motives $M^c(BG)$ and $M^c(B(G/J))$ are isomorphic.