# Wednesday, October 13


:::{.remark}
Goal: show Schubert varieties are normal.
:::

:::{.theorem title="8.2.2"}
Let $v\leq w\in W$, $\lambda \in D_\ZZ \intersect \lieh\dual_{\ZZ, Y}$ where we take the extension $P_Y \mapsvia{\lambda} \CC\units$ to the parabolic.
Then part (b) of the theorem states that $X_W^Y$ is normal.
:::

:::{.proof title="?"}
Let $w\in W_Y'$ such that $w'$ is a minimal length representative in $w W_Y$.
Write $\pi(w') = w$ for the element obtained by multiplying the elements in the word $w'$, and choose a word \( \mathcal{w} \in \mathcal{W}   \) such that \( \pi \mathcal{w}' = w'  \).
Then $m_{\mcw}^Y: Z_{\mcw'} \to X_{w'}^Y$ is surjective and birational, and so the following induced hom is an isomorphism
\[
(m_{\mcw}^Y)^*: H^0(X_W^Y, \mcl_w^Y( \lambda)) \iso H^0(Z_{w'}, \mcl_{w'} ( \lambda)) 
.\]
Taking any $\lambda^0 \in D^0_Y$ and applying A.32 (a deep AG fact) to the ample line bundle $\mcl = \mcl_W^Y(\lambda_0)$, we get the following important formula:

\[
(m_{w'}^Y)_* \OO_{Z_{w'}} = \OO_{X_W^Y}
.\]
This is what Kumar spends most of the time showing, and is essentially equivalent to the following:



:::{.fact title="Zariski's Main Theorem"}
Let $f:X\to Y$ be birational and proper such that $X$ is normal.
Then $Y$ is normal iff $f_* \OO_X = \OO_Y$, which implies that the fibers are connected.
This is proved in Hartshorne.
:::

> Vogan: there are more statements in representation theory that say "**if** normal" than there are that say "**then** normal".


Recall that the normalization $\tilde Y \mapsvia{\nu} Y$ satisfies a universal property with respect to maps from normal varieties.
Using functoriality, we have
\[
f_* \OO_X 
&= (\nu \circ \tilde f)_* \OO_X \\
&= \nu_* (\tilde f_* \OO_X) \\
&= \nu_* \OO_{\tilde Y} &&\text{Zariski's Main Theorem, forward direction}\\
&= \OO_Y &&\text{by assumption}
.\]
Use that $\tilde f$ is birational and proper, where properness can be shown by exhibiting it as the pullback of a proper morphism.
Using that $Y$ is normal iff every open affine $U \subseteq Y$ is normal, we have 
\[
\OO_Y(U) = (\nu_* \OO_{\tilde Y})(U) = \OO_{\tilde Y}(\nu\inv(U))
.\]
:::

## Borel-Weil Homomorphism

:::{.remark}
For any $V\in \mods{\CC}$ with $\dim_\CC \leq \infty$, define a morphism
\[
\beta_V: V\dual &\to H^0( \PP V, \mcl_V\dual ) \\
f &\mapsto (\delta \mapsto (\delta, \ro{f}{\delta }))
,\]
where taking the dual of the tautological amounts to, for each line $\delta \in \PP V$, quotienting by the annihilator to get $V\dual/ \delta^\perp$.
Note that there is a projection $\pi: \mcl_V\dual \to \PP V$.

Take $\lambda \in D_\ZZ$ and define a morphism of $\mods{G}$
\[
\beta = \beta(\lambda): L^{\max}( \lambda)\dual \to H^0(\mcX, \mcl( \lambda))
,\]
where $\mcX$ denotes that this works in the Kac-Moody setting.
Note that $\mcg$ acts naturally on $\mcl^Y(\lambda)$ and thus on $H^p(\mcx^Y, \mcl^Y( \lambda))$, and recall $G\mix{P_Y}\CC_{- \lambda} \to G/P_Y = X^Y$.
Then $X_w \subseteq X$ and $\beta_w( \lambda): L^{\max}( \lambda)\dual \to H^0(X_w, \mcl_w( \lambda))$
:::

:::{.remark}
How does this relate to representation theory?
Let $V$ be an irreducible integrable $\lieg\dash$module with highest weight $\lambda$, then every $w\in W$ induces $V_w$, and $U(\lieb)\dash$submodule generated by extremely weight vectors $w_{w \lambda}$.
Then $\beta$ acts by pushing weights "up", and so e.g. if one has weights \( \lambda, w_1 \lambda, w_2 \lambda, \cdots \) one can consider the **Demazure submodule** generated by any given $w_i \lambda$.
Often we set $V = L^{\max}(\lambda)$, and so 
\[
(L^{\max} (\lambda))_w = L_w^{\max}( \lambda)
.\]
:::

:::{.remark}
Going back to part (a) of the theorem, we have isomorphisms:

\[
\bar{\beta}_w^Y: L_w^{\max}( \lambda)\dual &\iso H^0( X_w^Y; \mcl_w^Y(\lambda)) \\
\alpha\dual: H^0( X_w^Y; \mcl_w^Y( \lambda)) &\iso H^0(X_w, \mcl_w( \lambda))
.\]
We have the following geometric picture:

\begin{tikzcd}
	{G_w \da \bar{BwB}} && G \\
	\\
	{X_w} && {G/B} \\
	\\
	{X_w^Y} && {G/P}
	\arrow[from=1-1, to=3-1]
	\arrow[from=3-1, to=5-1]
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-3, to=5-3]
	\arrow["\subseteq", hook, from=5-1, to=5-3]
	\arrow["\subseteq", hook, from=3-1, to=3-3]
	\arrow["\subseteq", hook, from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJHX3cgXFxkYSBcXGJhcntCd0J9Il0sWzIsMCwiRyJdLFsyLDIsIkcvQiJdLFsyLDQsIkcvUCJdLFswLDIsIlhfdyJdLFswLDQsIlhfd15ZIl0sWzAsNF0sWzQsNV0sWzEsMl0sWzIsM10sWzUsMywiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzQsMiwiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzAsMSwiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV1d)

The connection between representation theory and geometry is th following:
\[
H^0(Z_\infty; \mcl_\infty ( \lambda ) ) \iso L^{\max}( \lambda)\dual
.\]
:::

:::{.remark}
These statements are easy to remember and use but hard to prove!
So we'll move on and look at the Demazure character formula.
:::