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

TL;DR

This paper constructs an explicit module isomorphism between the tensor product of cohomologies of partial flag varieties and the full flag variety, revealing new bases and analyzing group actions.

Contribution

It introduces a new explicit morphism realizing the isomorphism and explores its applications to module bases and group representations.

Findings

The module isomorphism is explicitly constructed as the product in $H^*_T(G/B)$.

New module bases for $H^*_T(G/B)$ are described, differing from traditional Schubert bases.

The Springer representation's character is shown to be a multiple of a restricted $W_P$-representation.

Abstract

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 $H^*_T(P/B)$.