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 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 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 and M^c) are isomorphic.