# Equivariant \(\K\dash\)theory of $G/P$ (Monday, November 01)

:::{.remark}
We'll stick to the finite-type case for today.
Setup: let $G\in \Alg\Grp\slice \CC$ be connected, semisimple, simply connected, with $T\leq G$ a maximal torus.
Goal: describe $\K_T(G/P)$.

:::

:::{.remark}
Note that 
\[
\K_G(G/B) \iso \K_G(G\mix{B} \pt) \iso \K_B(\pt) \iso \K_T(\pt)
,\]
which we sometimes write as $\K_T$ or $A(T)$, the representation ring of $T$.
General pattern: for $\K_G(\wait)$, look at $\K_T(\wait)^W$ instead, using that $\K_G(\pt) = \K_T(\pt)^W = A(T)^W$.
Writing $P = LU\contains T$ for $P$ a parabolic and $L$ a Levi, we have
\[
\K_G(G/P) \iso \K_P(\pt) \iso \K_L(\pt) \iso \K_T(\pt)^{W_Y} \iso A(T)^{W_Y}
.\]

Thus there is a chain of isomorphisms:
\[
\K_T(G/P) 
&\iso \K_B(G/P) && \text{doesn't see unipotent radical} \\
&\iso \K_G(G\mix{B} G/P) && \text{induction}\\
&\iso \K_G(G/B \mix{B} G/P) && \text{trivialization for algebraic fiber bundles} \\
&\iso \K_G(G/B) \tensor_{\K_G(\pt)} \K_G(G/P) && \text{Kunneth} \\
&\iso A(T) \tensor_{A(T)^W} A(T)^{W_Y}
.\]
Note that $A(T) = \ZZ[X(T)] = \ZZ\cartpower{\ell}$ for some $\ell$.
:::

:::{.remark}
This formula may hold in more generality, but we're sticking with what's in the literature for now.
:::

:::{.remark}
Phrasing this in terms of equivariant line bundles:
starting with \( \lambda\in X(T) \), we write it as \( e ^{\lambda} \in A(T) \), and we have two morphisms $A(T) \to \K_T(G/B)$:

1. $F_1: e^{ \lambda} \to G/B \times \CC_{\lambda} \in \K_T(G/B)$.
2. $F_2: e^{\lambda} \to G\mix{B} \CC_{ \lambda}\in \K_T(G/B)$.

Note that the latter can be projected onto $G/B$.
If $e^{\lambda}\in R(G) = A(T)^W$, then $G\mix{B} \CC_{\lambda} \cong G/B \times \CC_{\lambda}$ since the $B\dash$action extends to a $G\dash$action.
So these assemble to a map
\[
F_1 \tensor F_2: A(T) \tensor_{A(T)^W} A(T) \to \K_T(G/B)
.\]
The claim is that this is equivalent to the isomorphism from above.
:::

## Equivariant Cohomology

> Perhaps don't try to learn this from Kumar as a first pass! 
> See [Anderson-Fulton](https://people.math.osu.edu/anderson.2804/eilenberg/) for a good treatment.
> For fiber bundles, see *Husemoller*.
> For algebraic topology, see May's "Concise Course..", chapter 18.

:::{.slogan}
Studying the equivariant geometry of a space $X$ is the same as studying fiber bundles with fiber $X$.
:::

:::{.remark}
Recalling some notions of axiomatic cellular cohomology: fix $M\in \Ab\Grp$ and consider pairs $(X, A)\in \Top$.
Then there exist functors $H^k(X, A; M): \ho\Top\cartpower{2} \to \Ab\Grp$ with natural transformations $\delta: H^k(A; M) \to H^{k+1}(X, A; M)$, where $H^k(A; M)\da H^k(A, \emptyset; M)$.
These satisfy and are characterized by a set of 5 axioms, which we'll omit.
Note that these constructions will work for any space we run into in this setting.
:::

:::{.exercise title="?"}
If $B \subseteq A \subseteq X$, show that there is a LES

\begin{tikzcd}
	{H^{k+1}(X, A; M)} && \cdots \\
	\\
	{H^k(X, A; M)} && {H^k(X, B; M)} && {H^k(A, B; M)}
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=1-1]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwyLCJIXmsoWCwgQTsgTSkiXSxbMiwyLCJIXmsoWCwgQjsgTSkiXSxbNCwyLCJIXmsoQSwgQjsgTSkiXSxbMCwwLCJIXntrKzF9KFgsIEE7IE0pIl0sWzIsMCwiXFxjZG90cyJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdXQ==)

:::

:::{.definition title="Equivariant cohomology"}
Let $G\in \Lie\Grp$ with $G\actson X\in \Top$ acting on the left.[^subcats]
Write $E\in \Top$ for any contractible space with a free right $G\dash$action, then define the **$G\dash$equivariant cohomology** of $X$ as
\[
H_G^k(X) \da H^k(E \mix{G} X)
.\]


[^subcats]: 
Note that $\Alg\Grp\slice \CC \leq \Lie\Grp$!

:::

:::{.fact}
Some facts:

- $X \homotopic E\times X$.
- $H_G^k(X)$ does not depend on the homotopy representative of $E$
- $\B G \da E/G$ is the **classifying space** of $G$.
- If $\xi:X\to Y$ equivariant with respect to $\phi:G\to H$, there is a map $EG\mix{G} X\to EH\mix{H} Y$ which induces $\xi^*: H_H^*(Y) \to H_G^*(X)$.
  In particular, $X\to \pt$ always exists, which is why $H^*(\B G)$ plays a large role.
- If $G\contains H$ and $EG$ is given, then $EH = EG$.

:::

:::{.example title="?"}
\[
H_G^*(\pt) \iso H^*(E\mix{G} \pt) \iso H^*(E/G) \iso H^*(\B G)
.\]
:::

:::{.example title="?"}
Examples of $\BG$:

- For $G = \CC^n$, we have $\EG = G = \CC^n$ and $H_G^*(\pt) = H^*(\pt)= M$.
- For $G = \CC\units$, $E = \CC^\infty\smz$ which is a contractible Ind-variety, and $\BG = \EG/G = \PP^\infty\slice \CC$.

:::