# More Broad Overview (Wednesday, August 25)

:::{.remark}
We'll assume background in affine varieties, but not necessarily sheaves.
Today's material: see Springer.
:::

:::{.definition title="Ringed Spaces"}
Let $X\in \Top$, then a **ringed space** is the data of $X$ and for all $U\in \Open(X)$ an assignment $\OO(U) \in \Alg_{\CC}$ a $\CC\dash$algebra of complex functions satisfying *restriction* and *extension*, also known as a sheaf of $\CC\dash$valued functions.
A **morphism** of ringed spaces $\xi:X\to Y$ is a continuous function such that for all $W\in \Open(Y)$, one can form the pullback
\[
\xi_W^*f: \xi\inv(W) \mapsvia{\xi} W \mapsvia{f} \CC
,\]
and we require that there is a well-defined induced map $\xi_W^*: \OO_Y(W) \to \OO_X(\xi\inv(W))$.
:::

:::{.example title="?"}
For $X$ an affine variety, the sheaf $\OO_X$ of regular functions satisfies this property.
Note that $\OO$ can be an arbitrary sheaf though, not necessarily just regular functions.
:::

:::{.definition title="Prevariety"}
A **prevariety** $X$ is a quasicompact space $X$ such that every $x\in X$ admits a neighborhood $U \subseteq X$ such that $(U, \Res(\OO_X, U))$ is isomorphic to an affine variety.
A prevariety is a **variety** if it is additionally separated, so $\Delta_X \subseteq X^{\times 2}$ is closed.
:::

:::{.remark}
Last time we said that $\mcl(\lambda)$ is an *algebraic variety*, so it satisfies the above definitions.
:::

:::{.remark}
From now on $G$ will be a connected reductive group.
$\pi: G\to \mcl(\lambda)$ will always be the map from the group to the flag variety.
:::

:::{.remark}
Let $X \in \Alg\Var\slice{\CC}$ and $H\in \Alg\Grp$ be linear where $H\actson X$.
Then $X/H$ is a quotient in $\Top$, by just taking the quotient topology.
Let $\rho: X\to X/H$ be the projection, then define the ring of functions as
\[
\OO_{X/H}(U) \da \ts{f\in \Hom(U, \CC) \st \Res(f \circ \rho, \rho\inv(U) ) \in \OO_X(\rho\inv(U))}
.\]
In this way $\OO_{X/H}(U)$ can be identified with $H\dash$invariant functions $\OO_X(\rho\inv(U))^H$.
This makes $X/H$ a ringed space, which is often (but not necessarily) an algebraic variety.
:::

:::{.example title="?"}
This is not always an algebraic variety, e.g. taking $\CC\units \actson \CC$ by multiplication.
This yields two orbits (0 and everything else) and isn't a variety.
:::

:::{.remark}
If $\pi: G\to G/H$ has local sections, then $(G\times X)/H \in \Alg\Var$ using $(g, x)h \da (gh, h\inv x)$.
Note that this is a fiber bundle for the Zariski topology, and doesn't have local sections (contrasting the analytic topology).
:::

:::{.claim}
The map $\pi: G\to G/B$ has local sections (but no global sections).
:::

:::{.remark}
Side note: we have the Bruhat decomposition $G = \disjoint_{w\in W} BwB$ as a partition into double cosets, quotienting by an action of $B\times B$.
The theorem is that these are parameterized by the Weyl group.
:::

:::{.remark}
Let $B = TU$ where $T$ is a torus and $U$ unipotent (so upper triangular, ones along the diagonal) and set $U^-$ to be the *opposite unipotent radical* (e.g. lower triangular, ones along diagonal).
Define a map
\[
\phi: U^i \times B &\to G \\
(\bar u, b) &\mapsto \bar{u}b\inv 
.\]
Then $\im(\phi) = U^- B$, and $\phi$ is injective since $U^- \intersect B = \ts{e}$.
The argument on matrices holds more generally: $B$ are the upper triangular matrices and $U^-$ has ones on the diagonal, so these intersect only at the identity.
$\phi$ is an open embedding: one can show that the derivative is surjective:
\[
d\phi(1, 1): \lieu^- \times \lieb &\to \liey \\
(x, y) &\mapsto x-y
.\]
Rewriting the target as $\lieu^- \oplus \lieh \oplus \lieu^+$ where $\lieb = \lieh \oplus \lieu^+$, one can find preimages of any element.

Define a local section: $\sigma: U\to G$ where $U \subseteq G/B$.
Use the composite $U^- \times B\to G \to G/B \supseteq U^-$ to view $U^-$ as a subset of the flag variety.
An explicit formula for section is the following: 
\[
\sigma(\bar u) \da (\bar u, 1) \in U^- \times B \subseteq G
.\]
Although this only constructs a section for one open set, translating by elements of $g$ yields an open cover, and everything is equivariant.
:::

:::{.remark}
Using this, $(G\times X)/B$ is always an algebraic variety, since $G\to G/B$ always has local sections.
For other groups, $X$ quasiprojective will also make the quotient algebraic, but the proof is more difficult.
However it still involves constructing local sections.
It turns out that $G\fiberprod{B}X\to G/B$ is a locally trivial fiber bundle.
:::

:::{.remark}
A note on notation: $(G\times X)B$ is sometimes written $G\fiberprod{B} X$ (as above), but this is *not* a fiber product.
In this notation, $\mcl(\lambda) = G\fiberprod{B} \CC_{- \lambda}$.
Note that this is a line bundle on $G/B$, so we can take sections.
:::

:::{.theorem title="Borel-Weil"}
\envlist

1. There is a correspondence
  \[
  H^0(G/B; \mcl(\lambda)) \cong \ts{f:G\to \CC \st f(g) = bf(gb) } && G \in \CC[G]
  .\]

  A section $\sigma: G/B \to G\fiberprod{B} \CC_{- \lambda}$ gets sent to $\sigma(gB/B) = [g, f(g)]$.
  Use that the quotient acts like a tensor over $B$, so 
  \[
  gB/B = gbB/B = [gb, f(gb)] = [g, b f(gb)]
  .\]

2. $H^0(G/B; \mcl(\lambda)) = L_{\lambda}\dual$ for $\lambda$ a dominant character in $D_\ZZ$, where $L_\lambda$ is the irreducible finite dimensional representation of $G$ with highest weight $\lambda$.
  Note that in the finite case, we have $L_\lambda\dual = L_{w_0 \lambda}$, but in the Kac-Moody case one doesn't have $w_0$.
:::

:::{.example title="?"}
For $\lambda = 0 \in X(T)$ a character, we get 
\[
\ts{f:G\to \CC \st f(g) = f(gb)} = \CC[G/B] = \OO_{G/B}(G/B)=\CC
.\]
:::

:::{.remark}
Chapter 1 of Kumar, Cartan matrices.
:::