A module isomorphism betweenH_T^*(G/P) ⊗ H_T^*(P/B) andH_T^*(G/B)

Abstract

We give an explicit (new) morphism of modules between HT∗(G∕P)⊗HT∗(P∕B) and HT∗(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 HT∗(G∕P) and HT∗(P∕B). With this identification, the map is simply the product within the ring HT∗(G∕B). We use this map in two ways. First, we describe module bases for HT∗(G∕B) that are different from traditional Schubert classes and from each other. Second, we analyze a W-representation on HT∗(G∕B) via restriction to subgroups WP. In particular, we show that the character of the Springer representation on HT∗(G∕B) is a multiple of the restricted representation of WP on HT∗(P∕B).