# Part 5: Representation Theory (Monday, November 14)

## $\S 13$ Abstract theory of integral weights

:::{.definition title="Integral weights and the root lattice"}
Let $\EE \contains \Phi \contains \Delta$ with Weyl group $W$. 
An element $\lambda\in \EE$ is an **integral weight** if $\inp \lambda \beta = (\lambda, \beta\dual)\in \ZZ$ for all $\beta \in \Phi$, where $\beta\dual \da {2\beta \over (\beta, \beta)}$.
We write the set of all weights as $\Lambda$, and write $\Lambda_r \da \ZZ \Phi$ for the **root lattice**.
:::

:::{.remark}
Recall $\Delta\dual \da \ts{\alpha\dual \st \alpha\in \Delta}$ is a base for $\Phi\dual = \ts{ \beta\dual \st \beta \in \Phi }$, and so 
\[
\lambda \in \Lambda\iff (\lambda, \alpha\dual) = \inp \lambda \alpha \in \ZZ\forall \alpha \in \Delta
.\]
:::

:::{.definition title="Dominant weights"}
A weight \( \lambda\in \Lambda \) is **dominant** iff $\inp \lambda \alpha \geq 0$ for all \( \alpha\in \Delta \), and we denote the set of all such dominant weights $\Lambda^+$.
The weight $\lambda$ is **strongly dominant** if $\inp \lambda \alpha > 0$ for all \( \alpha \in \Delta\).
Writing $\Delta = \tsl \alpha 1 \ell$, let \( \ts{ \lambda_i }_{1\leq i \leq \ell} \)  be the dual basis for $\EE$ relative to $\inp\wait\wait$, so $\inp{ \lambda_i}{\alpha_i} = \delta_{ij}$.
The $\lambda_i$ are referred to as the **fundamental dominant weights**, written $\lambda_i = \omega_i = \varpi_i$.
:::

:::{.remark}
If \(\lambda \in \Lambda \) then one can write \( \lambda = \sum_{i=1}^\ell m_i \lambda_i \) where $m_i \da \inp{ \lambda}{\alpha_i}$, so $\Lambda$ is a $\ZZ\dash$lattice with lattice basis $\ts{\lambda_i}_{1\leq i\leq \ell}$ containing the root lattice as a sublattice, so in fact $\Lambda_r = \ZZ \Delta$.
Writing the Cartan matrix as $A = (\inp{ \alpha_i}{\alpha_j})$ we have $\alpha_i = \sum_{j=1}^\ell \inp{\alpha_i}{\alpha_j} \lambda_j$ coming from the $i$th row of $A$.
So this matrix expresses how to write simple roots in terms of fundamental dominant roots, and inverting it allows writing the fundamental roots in terms of simple roots.
:::

:::{.fact}
The entires of $A\inv$ are all nonnegative rational numbers, so each fundamental dominant root is a nonnegative rational linear combination of simple roots.
:::

:::{.example title="?"}
For $A_3$ one has $A = \mattt 2 {-1} 0 {-1} 2 {-1} 0 {-1} 2$, so
\[
\alpha_1 &= 2 \lambda_1 - \lambda_2 \\
\alpha_2 &= - \lambda_1 + 2 \lambda_2 - \lambda_3 \\
\alpha_3 &= - \lambda_2 + 2 \lambda_3
.\]
:::

:::{.definition title="Fundamental group"}
The quotient \( \Lambda/ \Lambda_r \) is called the **fundamental group** of $\Phi$, and the index $f\da [\Lambda: \Lambda_r]$ is called its **index of connection**.
:::

:::{.remark}
The index is generally small: 

- $A_\ell$ has $f=\ell + 1$
- $f=1$ is obtained from $F_4, G_2, E_8$,
- $f=2$ is obtained from types $B, C, E_7$,
- $f=3$: $E_6$,
- $f=4$: type $D$.
:::

## $\S 13.2$ Dominant weights

:::{.remark}
Note that
\[
s_i \lambda_j = \lambda_j - \inp{ \lambda_j}{\alpha_i} \alpha_i = \lambda_j - \delta_{ij} \alpha_i
,\]
so $\Lambda$ is invariant under $W$.
In fact, any sublattice of \( \Lambda \) containing \( \Lambda_r \) is $W\dash$invariant.

:::

:::{.lemma title="A"}
Each integral weight is $W\dash$conjugate to exactly one dominant weight. 
If $\lambda$ is dominant, then $w \lambda\leq \lambda$ for all $w\in W$, and if $\lambda$ is strongly dominant then $w \lambda= \lambda\iff w=1$.
:::

:::{.proof title="Sketch"}
Most of this follows from Theorem 10.3, exercise 10.14, lemma 10.3B, and corollary 10.2C, along with induction on $\ell(w)$.
We'll omit the details.
:::

:::{.remark}
The ordering $\leq$ on $\Lambda$ is not well-behaved with respect to dominant weights, i.e. one can have $\mu \Lambda$ with $\mu\in \Lambda^+$ dominant but $\lambda\not\in \Lambda^+$ not dominant.
:::

:::{.example title="?"}
Let $\Phi$ be indecomposable of type $A_1$ with two roots $\alpha, \beta$, then $0\in \Lambda^+$ is dominant, but $0 < \alpha \in \Delta$ is not dominant: $(\alpha,\beta) < 0 \implies \ip\alpha \beta< 0$.
:::

:::{.lemma title="?"}
Let \( \lambda \in \Lambda^+ \) be dominant, then the number of dominant $\mu \in \Lambda^+$ with $\mu\leq \lambda$ is finite.
:::

:::{.proof title="?"}
Let \( \lambda, \mu\in \Lambda^+ \) and write \( \lambda - \mu \) as a nonnegative integer linear combination of simple roots. 
Note
\[
0 \leq (\lambda+ \mu, \lambda- \mu ) = (\lambda, \lambda ) - (\mu, \mu) = \norm{ \lambda}^2 - \norm{\mu}^2
,\]
so $\mu$ lies in the compact set of vectors whose length is $\norm{\lambda}$ and also in the discrete set \( \Lambda^+ \).
The intersection of a compact set and a discrete set is always finite.
:::

## $\S 13.3$ The weight $\rho$

:::{.definition title="$\rho$"}
\[
\rho \da {1\over 2} \sum_{\alpha\in \Phi^+} \alpha
.\]
:::

:::{.remark}
This section shows \( \rho = \sum_{i = l}^\ell \lambda_i \) and \( \norm{\lambda+ \rho}^2 \geq\norm{w \lambda+ \rho}^2 \) when $\lambda$ is the unique dominant weight in the orbit $W \lambda$.
:::

## $\S 13.4$: Saturated sets of weights

:::{.remark}
This section will be used later to analyze the set of weights in a finite-dimensional module for semisimple Lie algebra over $\CC$.
:::

## $\S 20$: Weights and maximal vectors.

:::{.remark}
Let $L$ be finite-dimensional semisimple over $\CC$ containing $H$ its toral subalgebra.
This corresponds to $\Phi \contains \Delta$ with Weyl group $W$ and \( \Phi \subseteq \EE = \RR \Phi \).
:::

## $\S 20.1$

:::{.definition title="Weight spaces and weights for $L\dash$modules"}
Let $V$ be a finite-dimensional $L\dash$module.
By corollary 6.4, $H\actson V$ semisimply (diagonally) and we can simultaneously diagonalize to get a decomposition 
\[
V= \bigoplus _{\lambda\in H\dual} V_{\lambda}, \qquad V_{ \lambda} \da \ts{v\in V \st h.v = \lambda(h) v\,\,\forall h\in H}
.\]
If $V_{\lambda}\neq 0$ then $\lambda$ is a **weight**.
:::

:::{.example title="?"}
If $\phi = \ad$ and $V=L$, then $L = H \oplus \oplus _{\alpha\in \Phi} L_{\alpha}$ where $H = L_0$.
:::

:::{.warnings}
If $\dim V = \infty$, $V_{ \lambda}$ still makes sense but $V$ may no longer decompose as a direct sum of its weight spaces.
E.g. take $V = \mcu(L)$ and the left regular representation given by left-multiplication in the algebra $\mcu(L) \actson \mcu(L)$.
This restricts to $L = L_0 \actson \mcu(L)$, the *regular action* of $L$ on $\mcu(L)$.
Note that there are no eigenvectors, since taking a PBW basis one can write $\prod h_i^{n_i} \cdot \prod_{\alpha\in \Phi} x_{\alpha}^{n_ \alpha}$, which strictly increases monomial degrees and thus there are no eigenspaces.
So $V_ \lambda= 0$ for all $\lambda$, i.e. there are no weight spaces at all.
:::