# Toward the Demazure Character Formula (Friday, October 22)

> References: Chris-Ginzburg 

:::{.remark}
Recall that we discussed proper pushforward and flat pullback.
:::

:::{.remark title="on induction"}
For $H\leq G\in \Alg\Grp$ linear groups and $X\in \gspaces{H}$, it is a fact that $G\mix{H} X \in \gspaces{G}$.
There is a functor inducing an equivalence of categories:
\[
\ind_H^G: \Coh^H(X) \iso \Coh^G(X)
,\]
yielding an isomorphism of groups $K_i^H(X) \to K_i^G(G\mix{H} X)$.
Induction can be constructed by quotienting the projection map:

\begin{tikzcd}
	\textcolor{rgb,255:red,92;green,92;blue,214}{\pr\inv(\mcf)} &&& \textcolor{rgb,255:red,92;green,92;blue,214}{\mcf \in \Coh^H(X)} \\
	& {G\times X} && X \\
	\\
	\textcolor{rgb,255:red,92;green,92;blue,214}{\Ind_H^G(\mcf)\in\Coh^H(G\mix{H} X)} & {G\mix{H} X}
	\arrow["\pr", from=2-2, to=2-4]
	\arrow["{\wait/H}"', from=2-2, to=4-2]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-4, to=1-1]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=4-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMSwxLCJHXFx0aW1lcyBYIl0sWzMsMSwiWCJdLFsxLDMsIkdcXG1peHtIfSBYIl0sWzMsMCwiXFxtY2YgXFxpbiBcXENvaF5IKFgpIixbMjQwLDYwLDYwLDFdXSxbMCwwLCJcXHByXFxpbnYoXFxtY2YpIixbMjQwLDYwLDYwLDFdXSxbMCwzLCJcXEluZF9IXkcoXFxtY2YpXFxpblxcQ29oXkgoR1xcbWl4e0h9IFgpIixbMjQwLDYwLDYwLDFdXSxbMCwxLCJcXHByIl0sWzAsMiwiXFx3YWl0L0giLDJdLFszLDQsIiIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs0LDUsIiIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)


:::

:::{.remark}
There is also a restriction functor inducing $\Res: K_i^G(X) \to K_i^H(X)$:

\begin{tikzcd}
	X && {G\mix{H} X} \\
	\\
	{H/H} && {G/H}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYIl0sWzIsMCwiR1xcbWl4e0h9IFgiXSxbMCwyLCJIL0giXSxbMiwyLCJHL0giXSxbMCwxXSxbMCwyXSxbMiwzXSxbMSwzXV0=)

Any linear $G\in \Alg\Grp$ can be written as $G\cong R\semidirect U$ where $R$ is reductive and $U$ is unipotent.
:::

:::{.proposition title="?"}
For any $X\in \gspaces{G}$,
\[
K^G(X) \cong K^R(X)
.\]
:::

:::{.slogan}
Only the reductive groups matter for equivariant \(\K\dash\)theory.
:::

:::{.proof title="?"}
Define the morphism $K^G(X)\to K^R(X)$ by forgetting the action away from the subgroup $R \leq X$:
\[
G\mix{R} X &\mapsvia{\phi} G/R \times X \\
[g, x] &\mapsto (gR/R, gx)
.\]
This induces
\[
\K^R(X) \mapsvia{\Ind_R^G} \K^G(G\mix{R} X) \isovia{\K\phi } \K^G(G/R \times X) \iso \K^G(X)
,\]
using that $G/R$ is affine.

> More generally, for $E\to X\in \VectBundle^G$, the fibers are contractible and thus $\K^G(E) \cong \K^G(X)$.
> See the Thom isomorphism, referenced in Borbo-Brylinksi-MacPherson.

:::

:::{.remark}
Let $\pi: G/B \to G/P$ where $P$ corresponds to the simple reflection $s$, so $P$ is the smallest parabolic not equal to the Borel.
Then

- Any map between projective varieties is proper, so $\pi$ is proper and the fibers are copies of $\PP^1$, i.e. $\pi\inv(gP/P) = gP/B \cong \PP^1$.
- $\pi$ is smooth in the sense of Hartshorne, i.e. so smooth fibers that are "the same".
  
Consequently, $\pi$ is flat, and $G/B \cong G\mix{P} P/B \to G/P$ with $G/B\to G/P$ flat.
We can push forward along proper maps and pull back along flat maps, so here we can do both.
So define a map
\[
D: \K^G(G/B) &\to \K^G(G/B) \\
[\mcf] &\mapsto \pi^*\pi_*[\mcf]
.\]
Note that this factors as $\K^G(G/B) \mapsvia{\pi_*} \K^G(G/P) \mapsvia{\pi_*} \K^G(G/B)$.
The question is now what $\pi^* \pi_* [\mcf]$ actually is.
:::

:::{.slogan}
The idea: we can recover representations as $\K^G(\pt)$, which is hard, so we apply these $D$ operators to larger parabolics to get to a point one step at a time.
:::

:::{.remark}
We have $A(T) = \ZZ[X(T)] \cong \K^T(\pt)$ for $A(T)$ representations of the torus.
On notation: write $\lambda \in X(T)$ as $e^{\lambda} \in A(T)$.
Note that $\K^P(P/B) \iso \K^P(P\mix{B} \pt) \iso \K^B(\pt)$ and $A(T) \iso \K^T(\pt)$, so writing $B = T\semidirect U$, there is an isomorphism
\[
A(T) &\iso \K^P(P/B) \\
C^{\lambda} &\mapsto [P\mix{B} \CC_{ \lambda}] \mapsto [G\mix{B} \CC_{\lambda}]
,\]
which is a composition $\Ind_P^B \circ \Ind_T^P$.
One can regard this as a line bundle on $\CP^1$ via the projection $P\mix{B} \CC_{\lambda} \to P/B \iso \PP^1$.
:::

:::{.remark}
A trick: recovering $\K^G$ from $\K^T$ and the Weyl group action on it.
This is why we reduce to $\K^T$ so often!
Write $\K^G(\pt) = R(G)$ on one hand and $A(T)^W$ on the other (taking Weyl group invariants), define a map $[V] \mapsto \sum_{\lambda \in X(T)} n_\lambda e^{\lambda}$.
Now assemble some maps:

\begin{tikzcd}
	{\K^L(\pt)} && {\K^P(\pt)} && {\K^G(G/P)} \\
	\\
	{A(T)} && {\K^P(P/B)} && {\K^G(G/B)}
	\arrow["\cong", from=1-1, to=1-3]
	\arrow["\cong", from=1-3, to=1-5]
	\arrow["{\pi^*}", from=3-3, to=1-3]
	\arrow["{D_s}", from=3-1, to=1-1]
	\arrow["{\pi_*}"', from=3-5, to=1-5]
	\arrow["\cong", from=3-1, to=3-3]
	\arrow["\cong", from=3-3, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJcXEteTChcXHB0KSJdLFsyLDAsIlxcS15QKFxccHQpIl0sWzQsMCwiXFxLXkcoRy9QKSJdLFswLDIsIkEoVCkiXSxbMiwyLCJcXEteUChQL0IpIl0sWzQsMiwiXFxLXkcoRy9CKSJdLFswLDEsIlxcY29uZyJdLFsxLDIsIlxcY29uZyJdLFs0LDEsIlxccGleKiJdLFszLDAsIkRfcyJdLFs1LDIsIlxccGlfKiIsMl0sWzMsNCwiXFxjb25nIl0sWzQsNSwiXFxjb25nIl1d)

What is $D_s(e^{\lambda})$?
By defining of pushforward along proper morphisms, we can write
Using these identifications, write 
\[
\pi_*[G\mix{B} \CC_{\lambda}] 
&= \pi_* [P\mix{B} \CC_{\lambda}] \\
&= \sum_i (-1)^i [\RR^i \pi_* (P\mix{B} \CC_{ \lambda}) ] \\
&= [H^0(P/B, e^{\lambda})] - [H^1(P/B, e^{ \lambda})] \\
&= [H^0(\PP^1, e^{\lambda})] - [H^1(\PP^1, e^{ \lambda}) ]
.\]
Recall that for $\OO(k)$, we have a pairing $-1, 0 \iff -2, 1 \iff -3, \cdots$ and $\ip{ \lambda}{ \alpha\dual} = k$.
:::

:::{.remark}
Next time: the Demazure character formula.
:::