# Geometric Interpretations of Nil-Hecke Ring and Affine Hecke Algebra

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.

• 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 $s_i$. – Victor Petrov Apr 8 '19 at 18:54
• 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. – Marc Besson Apr 8 '19 at 22:57
• 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. – Victor Petrov Apr 9 '19 at 7:36