Let G be a split reductive group over Fp with associated finite group of Lie type GF. Let T be a split maximal torus contained inside a Borel B of G. We relate the (rational) Tate motives of BGF with the T-equivariant Tate motives of the flag variety G/B. We explain how this could lead to representation theoretic applications, using recent advances in motivic geometric representation theory of the flag variety. On the way, we show that for a split reductive group G over a field k acting on a smooth k-scheme X, we get an isomorphism AnG(X, m)Q ? AnT(X, m)QW extending the classical result of Edidin-Graham to higher equivariant Chow groups.