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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.

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    $\begingroup$ Historical remark: "Bernstein-Gelfand-Gelfand approach" is due to Bott-Samelson, and "Ginzburg approach" is due to Kostant-Kumar. A simple observation to relate these two: if you make pushforward $K(G/B)\to K(G/P_i)$ and then pullback to $K(G/B)$, the resulting operator is precisely Demazure operator associated to $s_i$. $\endgroup$ Apr 8, 2019 at 18:54
  • $\begingroup$ This is helpful from the nomenclature aspect, but this still only uses one of the $\mathbb{P}^1$ bundles, with no mention of the other. In particular in "$T$-equivariant $K$-theory of generalized flag varieties" by Kostant and Kumar, the $G/B \rightarrow G/P_i$ fibration is the only one I can find. $\endgroup$ Apr 8, 2019 at 22:57
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    $\begingroup$ My guess then that the ${\mathbb P}^1$-bundle over $G/B$ is just the pullback of $G/B\to G/P_{i}$ along itself. $\endgroup$ Apr 9, 2019 at 7:36
  • $\begingroup$ I am confused about referring both to Borel subgroups and minimal parabolic subgroups. Is $P_i$ meant to be a maximal proper parabolic subgroup? A minimal non-Borel parabolic subgroup? $\endgroup$
    – LSpice
    Aug 6, 2021 at 1:57
  • $\begingroup$ Dear LSpice, you are quite right in that here the $P_i$ are not minimal parabolic but rather the smallest standard parabolic subgroups such that $Lie(P_i)$ contains the root space spanned by $e_{-\alpha_i}$; this was an error in my terminology. $\endgroup$ Aug 7, 2021 at 6:38

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The subvariety $\overline{Y}_{s_i}\subset G/B \times G/B$ is the fiber product $G/B\times_{G/P_i}G/B$. The set $\overline{Y}_{s_i}$ is the saturation for the diagonal $G$-action of $\{B/B\}\times P_i/B$, by definition, and of course, that also lies in the fiber product; since they are smooth irreducible varieties of the same dimension, inclusion shows they are equal.

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