# Generalized Flag Varieties, 7.1 (Wednesday, September 15)

:::{.remark}
Most of the things we'll look at will be motivated by the finite-type case, but the statements still go through more generally.
The setup: $A$ a GCM $\leadsto$ root datum $(\lieh, \pi, \pi\dual) \leadsto \lieg$ a Kac-Moody Lie algebra $\leadsto (W, S)$ a Coxeter group $\leadsto T \subseteq B$ a maximal torus, where $T = \Hom_\ZZ(\lieh_\ZZ, \CC\units)$ and $B$ plays the role of the Borel, $\leadsto \mcg$ a Kac-Moody group.
Here $\lieh_\ZZ$ is the integer span of coroots, using that $\lieh \subseteq \pi\dual$.
Note that since $\mcg$ arises from a Tits system, so even though we haven't described it set-theoretically yet, we know many nice properties it has by previous propositions.
:::

:::{.fact}
For $G\in \Alg\Grp$ arbitrary and $H\leq G$, the quotient space $G/H$ is a variety (See Springer's book for a proof).
Write $G/H = (X, a)$ where $a= H/H$ is a distinguished point.
Quotients have a universal property: for any pair $(Y, b)$ of pointed $G\dash$spaces whose isotropy (stabilizer) group contains $H$, there exists a unique equivariant pointed morphism $\phi: G/H\to Y$ such that $\phi(a) = b$.
:::

:::{.remark}
Today: we defined a flag variety to be any projective homogeneous space, and today we'll see that $G/B$ is a *projective* variety.
In fact, we'll show that $\mcg/P_Y$ is a projective *ind-variety*, where $P_Y$ is the standard parabolic coming from the Tits system.
:::

:::{.definition title="Ind-varieties"}
An **Ind-variety** is a set with a countable filtration $X_0 \subseteq X_1 \subseteq \cdots$ such that

- $X = \colim_n X_n = \Union_n X_n$,
- Each $X_n \embeds X_{n+1}$ is a closed embedding of finite-dimensional varieties.

$X$ will be projective/affine iff its filtered pieces are projective/affine.
:::

:::{.remark}
Note that we don't require a stratification here, but there will be a stratification on the flag varieties we'll use, which induces a filtration.
:::

:::{.example title="?"}
Infinite affine space $\AA^\infty\slice k$ can be written as
\[
\AA^\infty\slice k = \ts{ (a_1,a_2,\cdots) \st a_i \in k, \text{ finitely many } a_i\neq 0}
.\]
The filtration is given by
\[
\AA^1 &\injects \AA^\infty \\
x &\mapsto (x,0,0,\cdots) \\ \\
\AA^2 &\injects \AA^\infty \\
(x, y) &\mapsto (x,y,0,\cdots) \\
\vdots & 
\]

:::

:::{.example title="?"}
For $V\in \mods{\CC}$ with $\dim_\CC V = \infty$, we have $V\cong \AA^\infty\slice\CC$ as Ind-varieties.
:::

:::{.example title="?"}
For any $V\in \mods{\CC}$, the space $\PP(V) \da \Gr_1(V)$ (the space of lines in $V$) is a projective Ind-variety.
:::

:::{.remark}
For any integrable highest weight $\lieg\dash$module $V = V(\lambda)$ for $\lambda \in D_\ZZ$ an integral dominant weight, this will yield a $\mcg\dash$module.
Here for $\lieg$ semisimple, it integrates to the simply connected $\mcg$.
:::

:::{.definition title="?"}
For any $v_\lambda \neq 0\in V$, define
\[
\bar\iota_v: \mcg &\to \PP(V) \\
g &\mapsto [gv_\lambda]
,\]
:::

:::{.definition title="?"}
For any $Y \subseteq \ts{1,\cdots,\ell}$, define $D_Y^0$ the **$Y\dash$regular weights** by
\[
D_Y^0 \da \ts{ \lambda\in D_\ZZ \st \inner{ \lambda}{ \alpha_i} = 0 \iff i \in Y}
.\]
This partitions the integral dominant chamber:

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:::

:::{.lemma title="?"}
For \( \lambda \in D_Y^0 \) the map $\bar \iota_v$ factors through $\mcg/P_Y$ to give an injection
\[
\iota_v: G/P_Y \injects \PP(V)
.\]
So any Kac-Moody maps into an Ind-variety.
:::

:::{.remark}
We'll show that $\im \iota_v \subseteq \PP(V)$ is closed, i.e. that its intersection with any finite filtered piece is closed.
The variety structure will be induced from this embedding.
:::

:::{.proof title="?"}
We have a distinguished point $[v_\lambda] \in \PP(V)$, so $\Stab_G([v_ \lambda]) \contains P_Y$.
Showing this amounts to showing that for all $s\in Y$, $\bar s\in G$ fixes $[v _{\lambda} ]$, but this follows from the definition of \( v_\lambda \).
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:::{.remark}
A great class of varieties: Bott-Samelson-Demazure-Hansen varieties, which capture the geometry of words in Coxeter groups.
We'll have $w\in W, \bar w\in N$, and we'll define $\lieW\ni w$ as words:
\[
\lieW \da \ts{w = (s_{i_1}, \cdots, s_{i_n} ) \st n\geq 0}
,\]
which is a poset under deleting symbols.
For any $w\in \lieW$, define $Z_w \da \prod_{k\leq n} P_{i_k} / B\cartpower{n}$, where the action of the Borel is the *right mixed space action*:
\[
(p_1, \cdots, p_n)(b_1, \cdots, b_n) = (p_1 b_1, b_1\inv p_2 b_2, b_2\inv p_3 b_3,\cdots, b_{n-1}\inv p_n b_n)
.\]
:::

:::{.example title="?"}
Take $G = \GL_3(\CC)$, so $S = (s_1, s_2)$ and $w= (s_{i_1}, s_{i_2})$, then
\[
Z_w = (P_{i_1} \times P_{i_2})/B\cartpower{2} = P_{i_1} \mix{B} P_{i_2}/B = P_{i_1}\fiberprod{B} P_{i_2}/B \to P_{i_1}/M \cong \PP^1
,\]
so these are all bundles over $\PP^1$ with fibers $\PP^1$, and are in fact Hirzebruch surfaces.
:::

:::{.fact}
\envlist

1. $Z_w$ is an irreducible smooth variety with a $P_{i_1}\dash$action.

2. $Z_w\to P_{i_1}/B$ is locally trivial with fiber $Z_{w'}$ where $w'$ is obtained from $w$ by deleting the first reflection, so $s' = (s_{i_2}, \cdots, s_{i_n})$.

3. $Z_w \mapsvia{\psi} Z_{w_1}$ where $w_1 \da w[n-1] \da (s_1, \cdots, s_{i_n - 1})$ where $[p_1,\cdots, p_n] \mapsvia{\psi} [p_1,\cdots, p_{n-1}]$.
  This admits a section $[p_1,\cdots, p_{n-1}] \mapsvia{\sigma} [p_1,\cdots, p_{n-1}, 1]$.

4. $Z_w$ is a projective variety.

:::

:::{.remark}
Why projective: it's a fiber bundle of compact varieties, thus compact and complete.
A bit more goes into fully showing projectivity.
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:::{.definition title="?"}
Define a map
\[
m_w: Z_w &\to \mcg/B \\
[p_1, \cdots, p_n] &\mapsto p_1\cdots p_n B
,\]
then $\im m_w = \Union_{v\leq w} BvB/B \subseteq \mcg/B$.
:::

:::{.remark}
This is where the projective variety structure comes from, and we'll discuss when the image hits Schubert varieties.
:::