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

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

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