# Definitions

- Indecomposable: doesn't decompose as $A \oplus B$. Weaker than irreducible.
- Irreducible: simple, i.e. no nontrivial proper submodules. Implies indecomposable.
- Completely reducible: Direct sum of irreducibles.
- Solvable: Derived series terminates.
- Borel: maximal solvable subalgebra.
- Radical: Largest solvable ideal.
- Semisimple: Direct sum of simple modules.
  - Acts in a diagonalizable way.
- Antidominant weight: $\inner{\lambda + \rho}{\alpha\dual} \not\in\ZZ^{>0}$, equivalently $M(\lambda) = L(\lambda)$.
- Dominant weight: $\inner{\lambda + \rho}{\alpha\dual} \not\in \ZZ^{< 0}$.
- Regular weight: $\lambda$ is regular iff the isotropy/stabilizer group $\stab_W(\lambda) \definedas \theset{w\in W\suchthat w\lambda = w}= 1$, equivalently $\abs{W\lambda} = \abs{W}$ so $\inner{\lambda + \rho}{\alpha\dual} \neq 0$ for all $\alpha\in \Phi$.
- Singular weight: Not regular.
- Linked: $\mu \sim \lambda \iff \mu \in W\cdot \lambda$, the orbit of $\lambda$ under $W$, a.k.a. the linkage class of $\lambda$.
- Socle: Direct sum of all simple submodules.
- Radical: Intersection of all maximal submodules, smallest submodule such that quotient is semisimple.
- Head: $M / \mathrm{rad}(M)$.


# List of Notation

- $M(\lambda)$: Verma Modules

- $L(\lambda)$: Unique simple *quotient* of $M(\lambda)$.

- $N(\lambda)$ the maximal *submodule* of $M(\lambda)$

- The root system $$\Phi = \ts{\alpha \in \lieh\dual \suchthat [hx] = \alpha(h)x ~\forall h\in \lieh}$$ containing roots $\alpha$
  - Abstractly: spans a Euclidean space, $\lambda \alpha \in \phi \implies \lambda = \pm 1$, and closed under reflections about orthogonal hyperplanes.
- $\Phi^+$ the corresponding positive system (choose a hyperplane not containing any root), $\Phi \definedas \Phi^+ \disjoint \Phi^-$.
- $$s_\alpha(\wait) \definedas(\wait) - 2\inner{\wait}{\alpha} \frac{\alpha}{\norm{\alpha}^2}$$ the corresponding reflection about the hyperplane $H_\alpha$
- $\lieg_\alpha \definedas \theset{x\in \lieg \suchthat [hx] = \alpha(h)x ~\forall h\in \lieh}$ the corresponding root space
- The triangular decomposition $$\lieg = \bigoplus_{\alpha\in \Phi^+} \lieg_{\alpha} \oplus \lieh \oplus \bigoplus_{\alpha \in \Phi^-} \lieg_{-\alpha} \definedas \lien^{-} \oplus \lieh \oplus \lien^{+}$$
- $\Delta$ the corresponding simple system of size $\ell$, i.e $\alpha = \sum_{\delta_k \in\Delta} c_\delta \delta_k$ with $c_\delta \in \ZZ^{\geq 0}$. 
- $\Lambda = \theset{\lambda \in E \suchthat \inner{\lambda}{\alpha\dual} \in \ZZ ~\forall \alpha\in\Phi }$ the integral weight lattice
- $\Lambda^+ = \ZZ^+\Omega$ the dominant integral weights
  - $\Omega \definedas \theset{\bar \omega_1, \cdots, \bar \omega_\ell}$ the fundamental weights
- $[A: B]$ the composition factor multiplicity of $B$ in a composition series for $A$.
- $(A: B)$ the composition factor multiplicity of $B$ in a *standard filtration* for $A$.
- $\Phi_{[\lambda]} = \theset{\alpha\in \Phi \suchthat \inner{\lambda}{\alpha\dual} \in \ZZ}$ the integral root system of $\lambda$
- $\Delta_{[\lambda]}$ the corresponding simple system
- $W_{[\lambda]}$ the integral Weyl group of $\lambda$
- $\mu \uparrow \lambda$: strong linkage of weights
- $\OO_{\chi_\lambda}$: the block corresponding to $\lambda$.
- $\ch M \definedas \sum_{\lambda \in \lieh\dual} \qty{\dim M_\lambda} e^{\lambda}$ the formal character.

# Useful Facts

- $\lambda$ dominant integral $\implies w\lambda \leq \lambda$ for all $W$.
- $M(\lambda)$ is simple $\iff \lambda$ is antidominant.
- The dot action is given by $w\cdot \lambda = w(\lambda + \rho) - \rho$.
- For any filtration $0 \injects M^n \injects M^{n-1} \injects \cdots \injects M^1 \injects M^0 = M$, we have $$\ch M = \sum_{i=1}^n \ch \qty{M^i/M^{i-1}},$$
  i.e. the character of $M$ is the sum of the characters of its composition factors (with multiplicity).
- $\hd(M(\lambda)) = L(\lambda)$
- $\rad(M(\lambda)) = N(\lambda)$
- $\soc(M(\lambda)) =_? M(w_0 \cdot \lambda) = L(\mu)$ for $\mu$ the unique antidominant highest weight in the block determined by $\lambda$ (?)
- $\soc(M(w \cdot \lambda)) = L(w_0 \cdot \lambda)$.
- $$[M(\lambda) : L(\mu)] \geq 1 \iff \mu \uparrow \lambda \qtext{(strong linkage)}$$

# SL2 Theory

Definition
The group and the algebra:

\begin{align*}
\liesl(n, \CC)     &= \theset{M \in \gl(n, \CC) \suchthat \det(M) = 1} \\
\liesl(n, \CC)  &= \theset{M \in \gl(n, \CC) \suchthat \tr(M) = 0}
.\end{align*}


- The usual representation on $\CC^2$: $h$ has eigenvalues $\pm 1$, yields $L(1)$.
- The adjoint representation on $\CC^3$: $\ad h = \mathrm{diag}(2, 0, -2)$ with eigenvalues $0, \pm 2$, yields $L(2)$.

Generated by
\begin{align*}
x =
\begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}
,\quad
h =
\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
,\quad
y =
\begin{bmatrix}
0 & 0 \\
1 & 0
\end{bmatrix}
\end{align*}

with relations

\begin{align*}
[hx] &= 2x \\
[hy] &= -2y \\
[xy] &= h \\
.\end{align*}


Some identifications:
\begin{align*}
\Phi &= A_1 \\
\dim \lieh &= 1\\
\Lambda &\cong \ZZ \\
\Lambda_r & \cong \ZZ/2\ZZ \\
\Lambda^+ &= \theset{0, 1, 2, 3, \cdots} \\
W &= \theset{1, s_0} \quad \lambda \overset{s_0}\iff -\lambda \\
\chi_\lambda = \chi_\mu &\iff \mu = \lambda, -\lambda-2 \qtext{(linked)}\\
\Pi(M(\lambda)) &= \theset{\lambda, \lambda-2, \cdots} \\
\rho &= 1 \\
\alpha &= 2 \\
s_\alpha \cdot \lambda &= - \lambda - 2
.\end{align*}

For $\lambda$ dominant integral
\begin{align*}
N(\lambda) &\cong L(-\lambda - 2) \\
\dim L(\lambda) &= \lambda + 1 \\
\Pi(L(\lambda)) &= \theset{\lambda, \lambda-2, \cdots, -\lambda} \\
\dim \qty{L(\lambda)}_\mu &= 1 \quad\quad\forall \mu = \lambda-2i 
.\end{align*}


- Simple modules are parameterized by dominant integral weights:
  $$M(\lambda) \text{ is simple } \iff \lambda \not\in\ZZ^{\geq 0} = \Lambda^+ \iff \dim L(\lambda) = \infty$$

![Image](figures/2020-03-16-13:59.png)\


Finite-dimensional irreducible representations (i.e. simple modules) of $\liesl(2, \CC)$ are in bijection with dominant integral weights $n\in \Lambda$, i.e. $n\in \ZZ^{\geq 0}$, are denoted $M(n)$, and each admits a basis $\theset{\vector v_i\suchthat 0\leq i \leq n}$ where
\begin{align*}
h \cdot v_{i} &= (n-2 i) v_{i}\\
x \cdot v_{i} &= (n-i+1) v_{i-1}\\
y \cdot v_{i} &= (i+1)v_{i+1}
,\end{align*}
setting $v_{-1} = v_{n + 1}=0$ and letting $v_0$ be the unique vector in $L(n)$ annihilated by $x$.


- $\mathrm{rad}~M(\lambda) = N(\lambda)$
- $\mathrm{hd}~M(\lambda) = L(\lambda)$.
- $M(\lambda)$ for $\lambda > 0$ not integral is simple, however $-\lambda-2\not\in W\cdot \lambda$.
- $\lambda \geq 0 \implies \ch L(\lambda) = \ch M(\lambda) - \ch M(s_\alpha \cdot \lambda)$ where $s_\alpha \cdot \lambda = -\lambda - 2$.
- For $\lambda \geq 0$, $\dim L(\lambda) = \lambda + 1$ and so $$\ch L(\lambda) = e^\lambda + e^{\lambda-2} + \cdots + e^{-\lambda} = {e^{\lambda + 1} - e^{\lambda - 1} \over e^1 - e^{-1}}.$$
- For $\lambda \neq \rho\in \ZZ$, the composition factors of $M(\lambda)$ are $M(\lambda), L(-\lambda - 2)$.
- There is an exact sequence

\begin{center}
\begin{tikzcd}
0 \ar[r] \ar[equal]{d} & N(\lambda) \ar[r]\ar[equal]{d}  & M(\lambda) \ar[r]\ar[equal]{d}  & L(\lambda) \ar[r]\ar[equal]{d}  & 0\ar[equal]{d}  \\
0 \ar[r] & L(-\lambda-2) \ar[r] & M(\lambda) \ar[r] & L(\lambda) \ar[r] & 0
\end{tikzcd}
\end{center}

Characters:
\begin{align*}
\ch M(\lambda) &= \ch L(\lambda) + \ch L(s_\alpha \cdot \lambda) \\
\ch M(s_\alpha \cdot \lambda) &= \ch L(s_\alpha \cdot \lambda)
.\end{align*}

We can think of this pictorially as the 'head' on top of the socle:

\begin{align*}
M(\lambda) = \frac{L(\lambda)}{L(s_\alpha \cdot \lambda)}
.\end{align*}

We can invert the formula to get equation (2), which corresponds to inverting this matrix:

\begin{align*}
\ch L(\lambda) &= \ch M(\lambda) - \ch M(s_\alpha \cdot \lambda) \\
\ch L(s_\alpha \cdot \lambda) &= \ch M(s_\alpha \cdot \lambda)
.\end{align*}

If $\lambda \not\in\Lambda^+$, then $\ch L(\lambda) = \ch M(\lambda)$ and $b_{\lambda, 1} = 1, b_{\lambda, s_\alpha} = 0$ are again independent of $\lambda \in \lieh\dual \setminus \Lambda^+$.

# SL3

$\liesl(3, \CC)$ has root system $A_2$:


![](figures/image_2020-05-01-16-37-30.png)\


\begin{align*}
\Phi &= \theset{\pm \alpha, \pm \beta, \pm\gamma \definedas \alpha + \beta} \\
\Delta &= \theset{\alpha, \beta} \\
\Phi^+ &= \theset{\alpha, \beta, \gamma} \\
W &= \theset{1, s_\alpha, s_\beta, s_\alpha s_\beta, s_\beta s_\alpha, w_0 = s_\alpha s_\beta s_\alpha = s_\beta s_\alpha s_\beta}
.\end{align*}

For $\lambda$ regular, integral, and antidominant:

- $M(\lambda) = L(\lambda)$
- No other $M(w\cdot \lambda)$ is simple
- $\soc(M(w\cdot \lambda)) = L(\lambda)$.
-  $[M(w\cdot \lambda) : L(\lambda)] = [M(w\cdot \lambda) : L(w\cdot \lambda)] = 1$ for all $w$.
- $\ch L(s_\alpha \cdot \lambda) = \ch M(s_\alpha \cdot \lambda) - \ch M(\lambda)$.
- $\ch M(s_\alpha \cdot \lambda) = \ch L(s_\alpha \cdot \lambda) + \ch L(\lambda)$.
- The Jantzen filtration when $w \in \theset{s_{\alpha\beta}, s_{\beta\alpha}, w_0}$ is given by
\begin{align*}
M(w\cdot \lambda)^0 &= M(w\cdot \lambda) \\
M(w\cdot \lambda)^1 &= ? \\
M(w\cdot \lambda)^2 &= L(\lambda) \\
M(w\cdot \lambda)^{\geq 3} &= 0
.\end{align*}