We give an explicit (new) morphism of modules between H*(T)(G/P) circle times 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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