A module isomorphism between $H_T^{\ast }(G/P)\otimes H_T^{\ast }(P/B)$ and $H_T^{\ast }(G/B)$

Abstract

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