Let $X=G/B$ , where $G=GL_n(\mathbb{C}^n)$ and $B$ be the upper triangular matrices. I am curoius about the structure of $H^*_T(G/B)$ which I consider as a $H_T^*(pt)$-module. If we just consider ordinary cohmology, we know that the Schubert classes $[X_w]$ form a $\mathbb{Z}$-linear basis , their product is a positive combination of the others and it is described by Schubert polynomial. In Knutson and Tao's paper, they provided a nice puzzle to calculate the equivariant cohomology of $G/P$, the grassmanian variety.

Do we have something similar happening in $H_T^*(G/B)$? To be more precise, do we have the following properites?

- $H_T^*(G/B)$ is graded module over $H_T^*(pt)$.
- We have a natural basis $[\tilde{X_w}]$ for this $H_T^*$-module.
- The forgetful functor from $H_T^*(-)$ to $H^*(-)$ maps $[\tilde{X_w}]$ to $[X_w]$.

If the above listed properties are true, is it possible to create a simlar puzzle as in Knutson and Tao's paper? What are the progress in this area?