# Tuesday, December 06

## Weyl Dimension Formula

:::{.remark}
Last time:

- $q \da \sum_{ \alpha\in \Phi^+} (e_{\alpha\over 2} - e_{-{\alpha\over 2}}) = \sum_{w\in W} (-1)^{\ell(w)} e_{w \rho}$.
- The WCF: $q\convolve \ch_{ \lambda} = \sum_{w\in W} (-1)^{\ell(w)} e_{w ( \lambda+ \rho)}$
- An alternative writing of the WCF: 
\[
\ch_ \lambda= { \sum_{w\in W} (-1)^{\ell(w)} e_{w (\lambda+ \rho)} \over \sum_{w\in W} (-1)^{\ell(w)} e_{w \rho}}
,\]
where the denominator is denoted the *Weyl denominator*.
:::

:::{.corollary title="Weyl dimension formula"}
\[
\dim L( \lambda) = { \prod_{ \sigma\in \Phi^+} \inp{ \lambda+ \rho}{ \alpha} \over \prod_{ \alpha\in \Phi^+} \inp \rho \alpha } 
,\]
which is a quotient of two integers.
:::

:::{.exercise title="?"}
Show that $W$ always has an equal number of even and odd elements, so $\sum_{w\in W} (-1)^{\ell(w)} = 0$.
:::

:::{.proof title="?"}
Note $\ch_{ \lambda} = \sum _{\mu} \dim L( \lambda)_ \mu e_ \mu\in \ZZ[ \Lambda]$, and $\dim L( \lambda) = \sum_{ \mu\in \Lambda} m_ \lambda( \mu)$.
Viewing $\ch_ \lambda: \Lambda\to \ZZ$ as a restriction of a function $H\dual \to \CC$, $\dim L( \lambda)$ is the sum of all values of $\ch_ \lambda$.
Work in the $\CC\dash$subalgebra $\mcx_0$ of $\mcx$ generated by the characteristic functions $S \da \ts{e_ \mu\st \mu\in \Lambda}$; this equals the span of $S$ since $e_{ \mu} \convolve e_{\nu} = e_{ \mu+ \nu}$.
We have a map
\[
v: \mcx_0 &\to \CC \\
f &\mapsto \sum_{ \mu\in \Lambda} f( \mu)
,\]
which makes sense since $\size \supp f < \infty$.
This function sums the values we're after, so the goal is to compute $v( \ch_ \lambda)$.
By the exercise, attempting to apply this directly to the numerator and denominator yields $0/0$, and we get around this by using a variant of L'Hopital's rule.
Define $\del_ \alpha (e_ \mu) = ( \mu, \alpha)e_ \mu$, extended linearly to $\mcx_0$.
In the basis $S$ this operator is diagonal, and this is a derivation relative to convolution:
\[
\del_ \alpha \qty{ e_ \mu \convolve e_{ \nu} } 
&= \del_ \alpha( e_{ \mu+ \nu} ) \\
&= ( \mu+ \nu, \alpha)e_{ \mu+ \nu} \\
&= \qty{ (\mu, \alpha) e_ \mu} \convolve e_ \nu + e_ \mu \convolve \qty{ (\nu, \alpha) e_ \nu} \\
&= (\del_ \alpha e_ \mu) \convolve e_ \nu + e_ \mu \convolve (\del _{\alpha} e _{\nu})
.\]
Moreover they commute, i.e. $\del_ \alpha \del_ \beta = \del _{\beta} \del _{\beta}$.
Set $\del \da \prod_{ \alpha\in \Phi^+} \del_ \alpha$ where the product here is composition, and view $\del$ as an $m$th order differential operator.
Write $\omega( \lambda+ \rho)\da \sum_{w\in W} (-1)^{\ell(w)} e_{w (\lambda+ \rho)}$ for \( \lambda\in \Lambda^+ \), so $q = \omega ( \rho)$.
Rewriting the WCF we have 
\[
\omega( \rho) \convolve \ch_ \lambda = \omega( \lambda+ \rho) \qquad \star_1
,\]
and
\[
\prod_{ \alpha\in \Phi^+}\qty{e_{ \alpha\over 2} - e_{- {\alpha\over 2}}} \convolve \ch_ \lambda = \omega( \lambda+ \rho)
.\]
We now try to apply $\del$ to both sides, followed by $v$.
Note that if any two factors of $\del$ hit the same factor on the LHS, then noting that $v(e_{\alpha\over 2} - e_{-{ \alpha\over 2}}) = 0$, such terms will vanish.
So the total result will be zero unless all of the factors of $\del$ are applied to the $q$ factor in the LHS.

So apply $v\circ \del$ to $\star_1$ to get
\[
v( \del \omega( \rho)) v( \ch_ \lambda) = v( \del \omega( \lambda+ \rho))
.\]
We can compute
\[
v( \del \omega ( \rho)) 
= v\qty{ \del \sum_{w\in W} (-1)^{\ell(w)} e_{ w \rho} }
= \sum_{w\in W} (-1)^{\ell(w)} v(\del( e_{ w \rho} ) )
.\]
We have
\[
v ( \del( e_{ w \rho})) 
&= v\qty{ \qty{ \prod_{\alpha\in \Phi^+} \del_\alpha } e_{ w \rho} } \\
&= v\qty{\prod_{ \alpha\in \Phi^+} (w \rho, \alpha) e_{ w \rho} } \\
&= \prod_{ \alpha\in \Phi^+} ( w \rho, \alpha) \\
&= \prod_{ \alpha\in \Phi^+} ( \rho, w\inv \alpha) \qquad \star_2
.\]
Note that $w\inv \cdot \Phi^+$ is a permutation of $\Phi^+$, just potentially with some signs changed -- in fact, exactly $n(w\inv)$, the number of positive roots sent to negative roots, and $n(w\inv) = \ell(w\inv)$.
Thus the above is equal to
\[
(-1)^{\ell(w)} \prod_{ \alpha\in \Phi^+}( \rho, \alpha)
.\]
Continuing $\star_2$, we have
\[
v( \del( e_{ w \rho})) 
&= \sum_{w\in W} (-1)^{\ell(w)} (-1)^{\ell(w)} \prod_{ \alpha\in \Phi^+}( \rho, \alpha) \\
&= \size W \prod_{ \alpha\in \Phi^+}( \rho, \alpha)
,\]
which is the LHS.
Similarly for the RHS,
\[
v(\del( \lambda+ \rho)) = \size W \prod_{ \alpha\in \Phi^+} (\lambda+ \rho, \alpha)
.\]

Taking the quotient yields
\[
\dim L(\lambda) = 
{\size W \prod_{ \alpha\in \Phi^+}( \rho, \alpha)
\over 
\size W \prod_{ \alpha\in \Phi^+} (\lambda+ \rho, \alpha) } = 
{\prod_{ \alpha\in \Phi^+}( \rho, \alpha)
\over 
\prod_{ \alpha\in \Phi^+} (\lambda+ \rho, \alpha) }
.\]
Multiplying the numerator and denominator by $\prod_{ \alpha\in \Phi^+} {2\over (\alpha, \alpha)}$ yields
\[
\prod_{ \alpha\in \Phi^+} \inp{\rho}{\alpha}
\over 
\prod_{ \alpha\in \Phi^+} \inp{ \lambda+ \rho} {\alpha} 
.\]
:::

:::{.remark}
If $\alpha\in \Phi^+$, using that $\alpha\dual$ is a basis of $\Phi\dual$, one can write \( \alpha\dual = \sum_{i=1}^\ell c_i^{ \alpha} \alpha_i\dual \) for some $c_{i}^{\alpha} \in \ZZ_{\geq 0}$ and $\lambda = \sum_{i=1}^\ell m_i \lambda_i$ for $m_i \in \ZZ_{ \geq 0}$, using that $( \rho, \alpha_i\dual) = \inp{ \rho}{ \alpha_i} = 1$ one can rewrite the dimension formula in terms of the integers $c_i^{\alpha}$ and $m_i$.
:::

## New Directions

:::{.remark}
Where you could go after studying semisimple finite-dimensional Lie algebras over $\CC$:

- Infinite-dimensional representations of such algebras, e.g. the Verma modules $M( \lambda)$.
  One has a SES $K( \lambda) \injects M( \lambda) \surjects L( \lambda)$, which doesn't split since $M( \lambda)$ is indecomposable.

- Category $\OO$, expressing characters of simples in terms of characters of Vermas.

- Parabolic versions of Verma modules: we've looked at modules induced from $B = H + N$, but one could look at parabolics $P = U + \sum L_i$.

- Coxeter groups, i.e. groups generated by reflections, including Weyl groups.
  These can be infinite, which ones are finite?

- Quantize Coxeter groups to get Hecke algebras, which are algebras over $\CC[q, q\inv]$.
  See Humphreys.

- Representations of Lie groups over $\RR$, semisimple algebraic groups, representations of finite groups of Lie type (see the classification of finite simple groups, e.g. algebraic groups over finite fields).

- Characteristic $p$ representation theory, which is much more difficult.

- Infinite-dimensional Lie algebras over $\CC$, e.g. affine/Kac-Moody algebras using the Serre relations on generalized Cartan matrices.
  See also current algebras, loop algebras.

- Quantum groups (quantized enveloping algebras), closely tied to modular representation theory.
:::