# Wednesday January 15th

## Review

:::{.definition title="Weyl Vector"}
The Weyl vector is given by 
$$
\rho = \bar \omega_1 + \cdots + \bar \omega_\ell = \frac 1 2 \sum_{\beta \in \Phi^+} \beta \in \Lambda^+
.$$
:::
:::{.proposition title="Properties of the Weyl vector"}
envlist
- If $\alpha \in \Delta$ then $(\rho, \alpha\dual) = 1$
- $s_\alpha(\rho) = \rho - \alpha$.
:::

:::{.fact}
Let $\lambda \in \Lambda^+$; a few facts:

1. The size of $\theset{\mu\in \Lambda^+ \suchthat \mu \leq \lambda}$ (with the partial order from last time) is finite.
2. $w\lambda < \lambda$ for all $w\in W$.
:::

:::{.definition title="Weyl Chamber"}
The **Weyl chamber** for a fixed root in $E$ a Euclidean space is 
\[
C = \theset{\lambda \in E \suchthat (\lambda, \alpha) > 0 ~ \forall \alpha\in\Delta}
\]

:::

:::{.remark}
Note that the hyperplane splits $E$ into connected components, we mark this component as distinguished.

- A connected component of the union of hyperplanes is orthogonal to roots.
- They're in bijection with $\Delta$.
- They're permuted simply transitively by $W$.

We also let $\bar C$ denote the **fundamental domain**.

:::


## Weight Representations

:::{.definition title="Weights, Weight Spaces, and Multiplicities"}
For $\lambda \in \lieh\dual$, we let 
\[  
M_\lambda \da
\theset{v\in M \suchthat h\cdot v = \lambda(h) v ~\forall h\in\lieh}
.\]
denote a **weight space** of $M$ if $M_\lambda \neq 0$.
In this case, $\lambda$ is a **weight** of $M$.
The dimension of $M_\lambda$ is the **multiplicity** of $\lambda$ in $M$, and we define the set of weights as 
\[  
\Pi(M) \da
\theset{\lambda \in \lieh\dual \suchthat M_\lambda \neq 0}
.\]
:::

:::{.example title="?"}
If $M = \lieg$ under the adjoint action, then $\Pi(M) = \Phi \union \theset{0}$.
:::

:::{.remark}
The weight vectors for distinct weights are linearly independent.
Thus there is a $\lieg\dash$submodule given by $\sum_\lambda M_\lambda$, which is in fact a direct sum.
It may not be the case that $\sum_\lambda M_\lambda = M$, and can in fact be zero, although this is an odd situation.
[^zeroweight]
In our case, we'll have a *weight module* $M = \bigoplus_\lambda M_\lambda$, where $\lieh\actson M$ semisimply.

[^zeroweight]: See Humphreys \#1, \#20.2, p. 110.

:::

## Finite Dimensional Modules

Recall **Weyl's complete reducibility theorem**, which implies that any finite dimensional $\lieg\dash$module is a weight module.
In fact, $\lien, \lien^- \actson M$ nilpotently.

:::{.fact}
\envlist

- $\Pi(M) \subset \Lambda$ is a subset of the integral lattice.
- $\Pi(M)$ is $W\dash$invariant.
- $\dim M_\lambda = \dim M_{w\lambda}$ for any $\lambda \in \Pi(M), w\in W$.
:::


## Simple Finite Dimensional $\liesl(2, \CC)\dash$modules

Fix the standard basis $\theset{x, h, y}$ of $\liesl(2, \CC)$ with 

\[  
[h x] &= 2x \\
[h y] &= -2y \\
[x y] &= h
.\]

Since $\dim \lieh = 1$, there is a bijection 
\[  
\lieh\dual &\leftrightarrow \CC \\
\Lambda &\leftrightarrow \ZZ \\
\Lambda_r &\leftrightarrow 2\ZZ  \\
\alpha &\to 2\\
\rho &\to 1
.\]

There is a correspondence between weights and simple modules:

\[
\correspond{\text{Isomorphism classes of simple modules}} &\iff \Lambda^+ = \theset{0,1,2,3,\cdots} \\
L(\lambda) &\iff \lambda
.\]

Moreover, $L(\lambda)$ has a 1-dimensional weight spaces with weights $\lambda, \lambda - 2, \cdots, -\lambda$
and thus $\dim L(\lambda) = \lambda + 1$.

:::{.example title="?"}
\envlist
- $L(0) = \CC$, the trivial representation,
- $L(1) = \CC^2$, the natural representation where $\liesl(2, \CC)$ acts by matrix multiplication,
- $L(2) = \lieg$, the adjoint representation.
:::

Choose a basis $\theset{v_1, \cdots, v_\lambda}$ for $L(\lambda)$ so that $\CC v_0 = M_{\lambda}$, $\CC v_1 = M_{\lambda - 2}$, $\cdots \CC v_{\lambda} M_{-\lambda}$.
Then 

- $h\cdot v_i = (\lambda - 2i) v_i$
- $x \cdot v_i = (\lambda - i + 1) v_{i-1}$, where $v_{-1} \definedas 0$
- $y \cdot v_i = (i + 1)v_{i+1}$ where $v_{\lambda + 1} \definedas 0$.

We then say $L(\lambda)$ is a **highest weight module**, since it is generated by a highest weight vector $\lambda$.
Then $W = \theset{1, s_\alpha}$, where $s_\alpha$ is reflection through 0 by the identification $\alpha = 2$.

![Weyl group reflection in $\liesl_2(\CC)$](figures/2020-01-15-09:38.png){width=250px}


# Chapter 1: Category $\mathcal O$ Basics

The category of $U(\lieg)\dash$modules is too big.
Motivated by work of Verma in 60s, started by Bernstein-Gelfand-Gelfand in the 1970s.
Used to solve the Kazhdan-Lusztig conjecture.

## Axioms and Consequences

:::{.definition title="Category $\OO$"}
$\mathcal O$ is the full subcategory of $U(\lieg)$ modules consisting of $M$ such that 

1. $M$ is finitely generated as a $U(\lieg)\dash$module.
2. $M$ is $\lieh\dash$semisimple, i.e. $M$ is a weight module 
\[
M = \bigoplus_{\lambda \in \lieh\dual} M_\lambda
\]
3. $M$ is locally $n\dash$finite, i.e. 
\[  
\dim_\CC U(\lien) v < \infty \qquad \forall v\in M
.\]

:::

:::{.example title="?"}
If $\dim M < \infty$, then $M$ is $\lieh\dash$semisimple, and axioms 1, 3 are obvious.
:::

:::{.lemma title="?"}
Let $M \in \OO$, then

4. $\dim M_\mu < \infty$ for all $\mu \in \lieh\dual$.
5. There exist $\lambda_1, \cdots \lambda_r \in \lieh\dual$ such that 
\[
\Pi(M) \subset \Union_{i=1}^\lambda (\lambda_i - \ZZ^+ \Phi^+)
\]

![Forest structure of weights](figures/2020-01-15-09:50.png){width=350px}

:::

:::{.proof title="?"}
By axiom 2, every $v\in M$ is a finite sum of weight vectors in $M$.
We can thus assume that our finite generating set consists of weight vectors.
We can then reduce to the case where $M$ is generated by a single weight vector $v$. 
So consider $U(\lieg) v$.
By the PBW theorem, there is a triangular decomposition 
\[
U(\lieg) = U(\lien^-) U(\lieh) U(\lien)
\]


By axiom 3, $U(\lien) \cdot v$ is finite dimensional, so there are finitely many weights of finite multiplicity in the image.
Then $U(\lieh)$ acts by scalar multiplication, and $U(\lien^-)$ produces the "cones" that result in the tree structure:

![Cones under tree structure of weights](figures/2020-01-15-09:57.png){width=350px}

A weight of the form $\mu = \lambda_i - \sum n_i \alpha_i$ can arise from $y_{n_1}^{n_1} \cdots$.

\todo[inline]{Missing end of lecture.}

:::