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We give an explicit (new) morphism of modules between $H^*_T(G/P) \otimesH^*_T(P/B)$ and $H^*_T(G/B)$ and prove (the known result) that the two modulesare isomorphic. Our map identifies submodules of the cohomology of the flagvariety that are isomorphic to each of $H^*_T(G/P)$ and $H^*_T(P/B)$. With thisidentification, the map is simply the product within the ring $H^*_T(G/B)$. Weuse this map in two ways. First we describe module bases for $H^*_T(G/B)$ thatare different from traditional Schubert classes and from each other. Second weanalyze a $W$-representation on $H^*_T(G/B)$ via restriction to subgroups$W_P$. In particular we show that the character of the Springer representationon $H^*_T(G/B)$ is a multiple of the restricted representation of $W_P$ on$H^*_T(P/B)$.

A module isomorphism between $H^*_T(G/P)\otimes H^*_T(P/B)$ and $H^*_T(G/B)$

A module isomorphism between $H^_T(G/P)\otimes H^_T(P/B)$ and $H^*_T(G/B)$