# Demazure Character Formula (Monday, October 25)

> See Anderson 1985

:::{.remark}
Today: $A(T) = \ZZ[X(T)] \cong \K_T(\pt)$, where we write characters multiplicatively as $e^{\lambda}$.
For $\pi: G/B\to G/P$ for $P$ a simple parabolic corresponding to $s\in S$, we can push-pull to get an endomorphism of $\K_G(G/B)$, using that this morphism is both flat and proper.
The goal is to compute $\pi^* \pi_*[G\mix{B} \CC_{\lambda}]$, and the major tool in $\K\dash$theory is induction.
Write $G/B = G\mix{P} P/B = G\mix{B} \pt$ and $P = LU$, then there is a diagram

% https://q.uiver.app/?q=WzAsMTAsWzAsMSwiXFxLX0coRy9CKSJdLFswLDMsIlxcS19HKEcvUCkiXSxbMiwxLCJcXEtfTChQL0IpIl0sWzIsMywiXFxLX0woUC9QKSJdLFs0LDEsIlxcS19UKFxccHQpIl0sWzQsMywiXFxLX1QoXFxwdClee1dfU30gXFxzdWJzZXRlcSBcXEtfVChcXHB0KSJdLFs1LDEsImVee1xcbGFtYmRhfSJdLFs1LDMsIj8iXSxbMCwwLCJbR1xcbWl4e0J9IFxcQ0Nfe1xcbGFtYmRhfV0iXSxbMiwwLCJbUFxcbWl4e0J9IFxcQ0Nfe1xcbGFtYmRhfV0iXSxbMCwyXSxbMiw0XSxbMSwzXSxbMyw1XSxbNCw1XSxbMiwzXSxbMCwxLCJcXHBpXyoiLDFdLFs2LDcsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=
\begin{tikzcd}
	{[G\mix{B} \CC_{\lambda}]} && {[P\mix{B} \CC_{\lambda}]} \\
	{\K_G(G/B)} && {\K_L(P/B)} && {\K_T(\pt)} & {e^{\lambda}} \\
	\\
	{\K_G(G/P)} && {\K_L(P/P)} && {\K_T(\pt)^{W_S} \subseteq \K_T(\pt)} & {?}
	\arrow[from=2-1, to=2-3]
	\arrow[from=2-3, to=2-5]
	\arrow[from=4-1, to=4-3]
	\arrow[from=4-3, to=4-5]
	\arrow[from=2-5, to=4-5]
	\arrow[from=2-3, to=4-3]
	\arrow["{\pi_*}"{description}, from=2-1, to=4-1]
	\arrow[maps to, from=2-6, to=4-6]
\end{tikzcd}

:::

:::{.fact}

$\K_G(\pt) = \K_T(\pt)^W$.

:::

:::{.remark}
Writing $W = X^T \subseteq X = G/B$, we can use something due Bill Graham.
It's a fact that $i^* \K_T(X) \to \K_T(X^T)$ is injective, and Bill shows $i^*$ is an isomorphism after inverting certain elements.

:::

:::{.corollary title="Chris-Ginzburg, 5.11.3"}
The composite $i^*i_* \K_T(X^T) \to \K_T(X^T)$ is multiplication by $\lambda_T$, so here $\lambda_{-1}$.
Moreover 
\[
\lambda_T = \sum (-1)^i \Lambda^i N\dual \in \K_T(X_T)
\]
where $N\dual$ is the conormal.
:::

:::{.example title="?"}
For $X = \PP^1$ and $W = \ts{1, s}$, we have

- $\T_{B/B}(G/B) = \lieg/\lieb = \CC_{-\alpha}$,
- $\T_{sB/B}(G/B) = \lieg/s\lieb = \CC_{\alpha}$,
- $N_1\dual = \CC_{\alpha}$,
- $N_s\dual = \CC_{-\alpha}$.


:::

:::{.proposition title="?"}
A formula due to Bill, there is an element:
\[
\K_T(X) \ni \alpha = \sum_{w\in X^T} (i_w)_* \qty{ (i_w)^* \alpha \over \lambda_{-1} (N\dual_w)}
.\]
:::

:::{.remark}
Write $\pi:G/B\to G/P$ and its restriction $P/B\to P/P$.
Pullbacks are easy enough to compute, and we have formulas

- $(i_1)^* [P\mix{B} \CC_{\lambda}] = [B\mix{B} \CC_{\lambda}]$,
- $(i_s)^*[P\mix{B} \CC_{\lambda}] = [sB \mix{B} \CC_{ \lambda}]$.

For $[P\mix{B} \in \CC_{\lambda}]$, we can compute
\[
\pi_*[ P\mix{B} \CC_{ \lambda}] &=
\pi_* \sum_{1, s} (i_w)_* \qty{ (i_w)^* [P\mix{B} \CC_{ \lambda}] \over \lambda_{-1} (N_w\dual) } \\
&=
\pi_* \qty{
(i_1)_* \qty{ [B\mix{B} \CC_{ \lambda}] \over 1 - e^{\alpha}}
+
(i_s)_* \qty{ [sB\mix{B} \CC_{ \lambda}] \over 1 - e^{-\alpha}}
}\\
&=
\pi_* \qty{
(i_1)_* \qty{ e^{ \lambda} \over 1- e^{ \lambda}}
+
(i_s)_* \qty{ e^{ s\lambda} \over 1- e^{ \lambda}}
}\\
&= 
\qty{ e^{ \lambda} \over 1- e^{ \lambda}}
+
\qty{ e^{ s\lambda} \over 1- e^{ \lambda}}
\in A(T)
.\]

:::

:::{.proposition title="?"}
\[
\pi_* [P\mix{B} \CC_{\lambda}] = {e^{\lambda} - s^{s \lambda + \alpha} \over 1 - e^{ \alpha}}
.\]

:::

:::{.proof title="?"}
Let $q = e^{\alpha}$ and $k \da \inner{\lambda}{ \alpha\dual}$.
Note that 
\[
s \lambda- \lambda = \lambda- \ip{\lambda}{ \alpha\dual} - \lambda = -\ip{\lambda}{\alpha\dual} \alpha
.\]
Then
\[
(e^{ \lambda} - e^{ s \lambda+ \alpha})( 1- e^{ \alpha}) 
= e^{\lambda}(1 - e^{ - \alpha}) + e^{s \lambda}(1 - e^{ \alpha})
,\]
and we can write the RHS as 
\[
e^{ \lambda}\qty{ 1 - e^{ s \lambda- \lambda+ \alpha} \over 1 - e^{- \alpha }} 
=
e^{ \lambda}\qty{ 1 - q^{1-k} \over 1 - q} 
= e^{\lambda} c(q)
\]
where
\[
c(q) 
&= 
\begin{cases}
1 + q + \cdots + q^{-k} & k\leq 0 \\
0 & k=0 \\
-\qty{q^{1-k} + q^{2-k} + \cdots + q\inv }& k\geq 1.
\end{cases}\\ \\
&=
\begin{cases}
e^{ \lambda} + e^{ \lambda+ \alpha} + \cdots + e^{s \lambda}& k\leq 0 \\
0 & k=0 \\
-\qty{ e^{s \lambda+ \alpha} + e^{s \lambda+ 2 \alpha} + \cdots + e^{s \lambda + (k-1) \alpha} }& k\geq 1.
\end{cases}
.\]

:::

:::{.remark}
By Kumar,
\[
D_s(e^{ \lambda}) \da {e ^{ \lambda} - e^{s \lambda- \alpha} \over 1 - e^{- \alpha}}
,\]
where $e^{ \lambda}$ corresponds to $\mcl( \lambda)$.
:::

:::{.theorem title="8.?"}
For any $w\in W$, not necessarily reduced, and finite dimensional $M$ of $B$, 

1. There is an Euler characteristic formula
\[
\chi(Z_w, \mcl_w(M)) = \bar{D}_w(\bar{\character M})
,\]
where $\chi$ is given by $\sum (-1)^p \character\qty{H^p(Z_w, \mcl_w(M))} \in A(T)$.

2. $\chi(X_w, \mcl_w( \lambda)) = \bar{D}_w(e^{\lambda})$.

Then if $\lambda \in D_\ZZ$,

3. $\character H^0(X_w, \mcl_w( \lambda)) = \bar{D}_w (e ^{\lambda})$
4. $\character L_w^{\max}( \lambda) = D_w(e^{ \lambda})$.

:::