9
$\begingroup$

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.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ 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 $s_i$. $\endgroup$
    – Victor Petrov
    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. $\endgroup$
    – Marc Besson
    Apr 8, 2019 at 22:57
  • 1
    $\begingroup$ 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. $\endgroup$
    – Victor Petrov
    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. $\endgroup$
    – Marc Besson
    Aug 7, 2021 at 6:38

1 Answer 1

Reset to default
5
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.