# Friday February 14th

Recall that we have a decomposition $$\OO = \bigoplus_{\chi \in \hat\mcz(\lieg)} \OO_\chi$$ into infinitesimal blocks, where $\OO_0 \definedas \OO_{\chi_0}$ is the principal block.
Since $0\in\lieh\dual$, we can associate $\chi_0, M_0, L(0) = \CC$ the trivial module for $\lieg$.

## 1.14 -- 1.15: Formal Characters

Some background from finite dimensional representation theory of a finite group $G$ over $\CC$.
The hope is to find matrices for each element of $G$, but this isn't basis invariant.
Instead, we take traces of these matrices, which is less data and basis-independent
This is referred to as the *character* of the representation, and in nice situations, the characters determine the irreducible representations.

For a semisimple lie algebra $\lieg$ and a finite dimensional representation $M$, it's enough to keep track of weight multiplicities when $\lieg$ is the lie algebra associated to a compact lie group $G$.
From this data, the characters can be recovered.
So the data of all pairs $(\dim M_\lambda, \lambda \in \lieh\dual)$ suffices.
To track this information, we introduce a *formal character*.

*Remark:*
If $G$ is a group and $k$ is a commutative ring, $kG$ is the group ring of $G$.
This has the following properties:

- $\sum a_i g_i + \sum b_i g_i = \sum(a_i + b_i) g_i$
- $\qty{ \sum a_i g_i  } \qty{ \sum b_i g_i  } = \sum_{i, j} a_i b_j g_i g_j$

Let $\ZZ \Lambda$ be the integral group ring of the lattice.
Since $\Lambda$ is an abelian group, and the additive notation would be more difficult.
So we write $\Lambda$ multiplicatively and introduce $e(\lambda)$ for $\lambda \in \lieh\dual$, where $e(\lambda) e(\mu) = e(\lambda + \mu)$.
For $M$ a finite dimensional $\lieg\dash$module, the formal character of $M$ is given by

\begin{align*}
\ch M = \sum_{\lambda\in \Lambda} \qty{ M(\lambda)  } e(\lambda) \quad\in \ZZ\Lambda
.\end{align*}

This satisfies

- $\ch(M\oplus N) = \ch(M) + \ch(N)$
- $\ch(M\tensor N) = \ch(M)\ch(N)$
- For $\ch(M) = \sum a_\mu e(\mu)$ and $\ch(N) = \sum b_\nu e(\nu)$, we have $$\ch(M) \ch(N) = \sum_{\lambda} \qty{ \sum_{\mu + \nu = \lambda} a_\mu b_\nu } e(\lambda)$$

By Weyl's complete reducibility theorem, any semisimple module decomposes into a sum of simple modules.
Thus it suffices to determine that characters of simple modules $L(\lambda)$ for $\lambda \in \Lambda^+$, corresponding to dominant integral weights.
Then we can reconstruct $\ch(M)$ from $\ch L(\lambda)$ for $M\in \OO$.

Specifying the weight spaces dimensions is equivalent to a function $\ch_M: \lieh\dual \to \ZZ^+$ where $\ch_M(\lambda) = \dim M_\lambda$.
The analogy of $e(\lambda)$ in this setting is the characteristic function $e_\lambda$ where $e_\lambda(\mu) = \delta_{\lambda \mu}$ for $\mu \in \lieh\dual$.
We can thus write the function

\begin{align*}
\ch_M = \sum_{\lambda \in \lieh\dual} \qty{ \dim M_\lambda } e_\lambda
.\end{align*}

When $\dim M < \infty$, $\ch_M$ has finite support, although we generally don't have this in $\OO$.
In this setting, multiplication of formal characters corresponds to convolution of functions, i.e. $$(f\ast g)(\lambda) = \sum_{\mu + \nu = \lambda} f(\mu) g(\nu).$$
Define 
$$
\mcx = \theset{f: \lieh\dual \to \ZZ \suchthat \supp(f) \subset \union_{i\leq n} \qty{ \lambda_i - \ZZ^+ \Phi^+  }  ~~\text{ for some } \lambda_1, \cdots, \lambda_n \in \lieh\dual }
$$

> Idea: this is a "cone" below some weights.

This makes $\mcx$ into a $\ZZ\dash$module with a well-defined convolution, thus $\mcx$ is a commutative ring where

- $e_\lambda \in \mcx$ for all $\lambda$
- $e_0 = 1$
- $e_\lambda \ast e_\mu = e_{\lambda + \mu}$.

If $M\in \OO$, then $\ch_M \in \mcx$ by axiom O5 (local finiteness).

*Example:*
$\ch L(\lambda) = e(\lambda) + \sum_{\mu < \lambda} m_{\lambda \mu} e(\mu)$, where $m_{\lambda \mu} = \dim L(\lambda)_{\mu} \in \ZZ^\pm$.

Definition (Principal Block)
: 	Let $\mcx_0$ be the additive subgroup of $\mcx$ generated by all $\ch M$ for $M \in \OO$.

Proposition (Additivity of Characters, Correspondence with K(O) )
:   \hfill 
    
    a. If $0 \to M' \to M \to M'' \to 0$ is a SES in $\OO$, then $\ch M = \ch M' + \ch M''$.

    b. There is a 1-to-1 correspondence

    \begin{align*}
    \mcx_0 &\iff K(\OO) \\
    \ch M &\iff [M]
    ,\end{align*}

    where $K$ is the Grothendieck group.

    c. If $M\in \OO$ and $\dim L < \infty$, then $\ch(L\tensor M) = \ch L \ast \ch M$.

*Remark:*
(a) implies that $\ch M$ is the sum of the formal characters of its composition factors with multiplicities.
Thus 
$$
\ch M = \sum_{L \text{ simple }} [M:L] ~\ch L
.$$

Proof (of a)
: 	Use the fact that $\dim M_\lambda = \dim M_\lambda' + \dim M_\lambda''$

Proof (of b)
: 	Check that the obvious maps are well-defined and mutually inverse.

Proof (of c)
: 	Because of the module structure we've put on the tensor product $(L \tensor M)_\lambda = \sum_{\mu + \nu = \lambda} L_\mu \tensor M_\nu$.

*Remark:*
The natural action of $W$ on $\Lambda$ or on $\lieh\dual$ extends to $\ZZ \Lambda$ and $\mcx$ if we define

\begin{align*}
w\cdot e(\lambda) \definedas e(w\lambda) \quad w\in W,~~\lambda \in \Lambda \text{ or } \lieh\dual
.\end{align*}

If $\lambda \in \Lambda^+$, then $w( \ch L(\lambda) ) = \ch L(\lambda)$ since $\dim L(\lambda)_\mu = \dim L(\lambda)_{w\mu}$.
Thus the characters of simple finite-dimensional modules are $W\dash$invariant.

## 1.16: Formal Characters of Verma Modules

1: We have a similar formula

\begin{align*}
\ch M(\lambda) = \ch L(\lambda) + \sum_{\mu < \lambda} a_{\lambda \mu} \ch L(\mu) \\
\quad \text{ with } a_{\lambda \mu} \in \ZZ^+ 
\text{ and } a_{\lambda \mu} = [M(\lambda): L(\mu)]
.\end{align*}

This all happens in a single block of $\OO$, which has finitely many simple and Verma modules.  
In fact, the sum will be over $\theset{ \mu \in W\cdot \lambda \suchthat \mu < \lambda}$.
But computing $L(\mu)$ is difficult in general.

Since the set of weights $W\cdot \lambda$ is finite, we can totally order it in a way that's compatible with the partial order on $\lieh\dual$ (so $\leq$ in the partial order implies $\leq$ in the total order).
So if we order the weights $\mu_i$ indexing the Verma modules in columns and indexing the simple modules in the rows, this is an upper triangular matrix with 1s on the diagonal.
This can inverted since it's unipotent, with the inverse of same upper triangular form.

2: We can write 
$$
\ch L(\lambda) 
= \ch M(\lambda) + 
\sum_{\mu < \lambda,~\mu \in W\cdot \lambda} b_{\lambda \mu} \ch M(\mu) \quad b_{\lambda \mu} \in \ZZ
$$
This expresses the character in terms of Verma modules, which are easier to compute.

Next time: formulas for the characters