I am interested in two related constructions which give us either the cohomology or the $T \times \mathbb{C}^*$-equivariant $K$-theory of flag varieties.

Let $G$ be a semisimple, simply connected algebraic group, with $T \subset B \subset G$ a chosen maximal torus and Borel subgroup. In order to gain geometric information about the flag variety $G/B$, we make use of a collection of $\mathbb{P}^1$ bundles. To be a little more explicit:

In what I would (perhaps erroneously) call the Bernstein-Gelfand-Gelfand approach, to find the cohomology of $G/B$, we would use minimal parabolic subgroups $P_i$ and maps $G/B \rightarrow G/P_i$. This gives $G/B$ the structure of a $\mathbb{P}^1$-bundle over $G/P_i$. In this situation we can use the Leray-Serre spectral sequence to obtain $H^*(G/B)$ in terms of $H^*(\mathbb{P}^1)$ and $H^*(G/P_i)$, and in particular we can get our hands on classes $[\overline{X}_{s_i}]$ where $X_{s_i}$ is the Schubert cell associated to a simple reflection $s_i$.

In the construction of $K^{T \times \mathbb{C}^*}(G/B)$, following Ginzburg (and possibly originally due to Kazhdan and Lusztig?), we construct a different $\mathbb{P}^1$-bundle. Namely, the $G$-diagonal orbits in $G/B \times G/B$ are parametrized by $w \in W$. Let $Y_{s_i}$ denote the orbit associated to $(B/B, s_iB/B)$, and $\overline{Y_{s_i}}$ its orbit closure. Then via projection onto the first factor, $\pi_1 :\overline{Y_{s_i}} \rightarrow G/B$ is a $\mathbb{P}^1$-bundle. Ginzburg goes on to construct all sorts of sheaves in this setup, and constructs the affine Hecke algebra geometrically.

The two constructions should be related in the following way: "The $T_{s_i}$ action on $K^{T \times \mathbb{C}^*}(T^*G/B)$ is given by $e^{\lambda} \mapsto \frac{e^{\lambda}-e^{s_{i} (\lambda)}}{e^{\alpha_i}-1}-q \frac{e^{\lambda}-e^{s_{i}(\lambda)+\alpha_i}}{e^{\alpha_i}-1}.$ The fraction on the left is the Demazure operator associated to $s_i$, which is used to find the $K$-theory of $G/B$" (here I am paraphrasing from Chriss-Ginzburg Thm 7.2.16).

I know I am mixing cohomology and all sorts of $K$ theory here, but it seems that there should be a more accessible topological relationship. My question is: on the most simple (purely topological) level, how do these two distinct $\mathbb{P}^1$-bundles give us similar information about the cohomology (or $K$-theory) of $G/B$?

I feel like I've simultaneously included too many details and left out too many details, and I'd be happy to edit for clarification.

andminimal parabolic subgroups. Is $P_i$ meant to be amaximalproper parabolic subgroup? A minimal non-Borel parabolic subgroup? $\endgroup$