# Wednesday November 6th

Last time: 
We considered the finite dimensional representation theory of $\lieg$ a semisimple Lie algebra over $\CC$.
We showed Weyl's complete reducibility theorem: any finite dimensional $\lieg$ module is semisimple and $\lieg = \bigoplus \mathfrak{s}_i$, a sum of simple modules.

Therefore, it suffices to understand the *characters* for simple modules, i.e. what are the dimensions of the weight spaces?

We can answer this question for $\lieg = \liesl(2, \CC)$: we have $L(\lambda) = \mathrm{span}_\CC \theset{v_i}_{i=1}^\lambda$ where
$$
\dim L(\lambda)_\mu = \begin{cases} 1 & \mu \in \theset{\lambda, \lambda-2, \cdots, -\lambda} \\ 0 & otherwise \end{cases}
$$

For an arbitrary $\lieg$, what is $L(\lambda)$? We'll describe this using Weyl's character theorem, the Verma module (which is an infinite-dimensional highest weight module), and the PBW theorem of the universal enveloping algebra.

In general, the representation of $\lieg$ is complicated, so we restrict ourselves to a subcategory $BGG$ category $\mathcal O$, which contains the simple and Verma modules.
Here, the irreducible character problem is solved if we know that the *multiplicity* of simple modules in any Verma module. 
The multiplicity is the number of simple modules occurring in a filtration, and the Kazhdan-Lusztig conjecture says that this multiplicity should be the evaluation of a certain $KL$ polynomial at 1.
This was first proved using perverse sheaves and $D\dash$modules in the 1980s (geometric), and then with purely algebraic proof is due to Williamson around 2013.
This was obtained using something called the Soergel bimodule.
This is all over $\CC$, and there are some generalizations that work for characteristic $p$. 
It was thought that the original polynomial would work here, but it turns out that there is another one called the $p-KL$ polynomial.

> These come from the KLR algebra, where there is a change of basis that induces a change of basis on the Hecke algebra, where the $KL$ polynomial takes that standard basis to the $KL$ basis.

Recall that $M$ is a weight module if $M = \oplus_\lambda M_\lambda$, where $M_\lambda \coloneqq \theset{m\in M \mid h.m = \lambda(h)m ~\forall h\in\lieg}$.

*Non-example:*
Take $\lieg = \liesl(2, \CC) = \generators{e, h, f}$ and $M = U(\lieg)/ I$ where $I = U(\lieg) (1-e) \normal U(\lieg)$ is a left ideal.
Then $M$ has basis $\theset{f^a h^b + I \mid a, b \in \ZZ_{\geq 0}}$.
The claim is that $h + I$ is not in any weight space.
If so, we would have $h \actson (h+I) = h^2 + I$, which is not a multiple of $h +I$, i.e. it's not in $\CC(h+I)$.
So it is not a weight module.

## Section 20.2: Highest Weight Modules

**Definition:**
A *maximal vector* $v^+ \in M$ is a nonzero vector such that $\eta v^+ = 0$, i.e. $\lieg_\alpha v^+$ for all $\alpha \in \Phi^+$.

**Definition:**
A $\lieg\dash$Module $M$ is a *highest weight module* of weight $\lambda$ if $M = U(\lieg)v^+$ for some maximal vector $v^+$.

*Example:*
Consider $\liesl(2, \CC)$ and $L(\lambda)$, we have the following situation:

\begin{center}
\begin{tikzcd} 2 \cdot \mathbf{0} \\ 
2v_0 					\arrow[u, "e"', bend right] \arrow[d, "f"', bend right]          	\\ 
2v_1 					\arrow[u, "e"', bend right] \arrow[d, "f"', bend right]          	\\ 
2v_2 					\arrow[u, "e"', bend right] \arrow[d, "f"', bend right]          	\\ 
{\vdots}			\arrow[d, "f"', bend right] \arrow[u, "e"', bend right] \\ 
2v_{-\lambda}	\arrow[u, "e"', bend right] \arrow[d, "f"', bend right] 	\\ 
2 						\arrow[u, "e"', bend right]
\end{tikzcd}
\end{center}

Then a similar picture holds for $M(\lambda)$, thus $v_0$ is a maximal vector, and $L(\lambda), M(\lambda)$ are weight modules.

**Theorem:**
Let $M$ by a highest weight module of weight $\lambda$ with maximal vector $v^+$. 
Fix an ordering $\Phi^+ = \theset{\beta_1, \beta_2, \cdots, \beta_m}$ where $m = \abs \Phi^+$.
Pick a nonzero $e_i \in \lieg_{\beta_i}$, then there exists a nonzero $f_i \in \lieg_{-\beta_i}$ such that $[e_i, f_i] = h_i$ (a Cartan element) for all $i$.

a. We can write a basis for the highest weight module, $M = \mathrm{span}_\CC\theset{ \prod_{i=1}^m f_i^{r_i} v^+  \mid r_I \in \ZZ_{\geq 0} }$, 
b. $\mathrm{Wt}(M) \subseteq \theset{\mu \in \lieh^* \mid \mu \leq \lambda}$.
c. $\dim M_\lambda = 1$ and $\dim M_\mu < \infty$ for all $\mu \in \lieh^*$.
d. Submodules of $M$ are weight modules.
e. If $M$ has a *unique* submodule, then $M$ has a unique simple quotient and $M$ is indecomposable.
f. Every non-zero homomorphic image of $M$ is a highest weight module of weight $\lambda$.

*Proof of (a):*
$M = U(\lieg)v^+$, which is in $\sum \CC f \ldots h \ldots e \ldots v^+$ where $e\ldots v^+ = 0$, which is in $\sum \CC f \ldots v^+$.