lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -1,27 +1,36 @@
 \chapter{Classification of Coherent Families}
 
-% This is a very important theorem, but since we won't classify the coherent
-% extensions in here we don't need it, and there is no other motivation behind
-% it. Including this would also require me to explain what central characters
-% are, which is a bit of a pain
-%\begin{proposition}[Mathieu]
-%  The central characters of the irreducible submodules of
-%  \(\operatorname{Ext}(M)\) are all the same.
-%\end{proposition}
-
-First and foremost, notice that because of
-Example~\ref{thm:simple-weight-mod-is-tensor-prod} the problem of classifying
-the simple weight \(\mathfrak{g}\)-modules can be reduced to that of
-classifying the simple weight modules of its simple components. In addition, it
-turns out that very few simple Lie algebras admit cuspidal modules at all.
-Specifically\dots
+% TODOO: Is this decomposition unique??
+% TODO: Prove this
+\begin{proposition}
+  Suppose \(\mathfrak{g} = \mathfrak{s}_1 \oplus \cdots \oplus \mathfrak{s}_r\)
+  and let \(\mathcal{M}\) be a semisimple irreducible coherent
+  \(\mathfrak{g}\)-family. Then there are semisimple irreducible coherent
+  \(\mathfrak{s}_i\)-families \(\mathcal{M}_i\) such that
+  \[
+    \mathcal{M} \cong \mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r
+  \]
+\end{proposition}
+
+% TODO: Rework this
+In addition, it turns out that very few simple Lie algebras admit cuspidal
+modules at all. Specifically\dots
 
+% TODO: Add sp(2n) to the list of simple Lie algebras!
 \begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
-  Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra. Suppose
+  Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose
   there exists a cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s} \cong
-  \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\).
+  \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\) for
+  some \(n\).
 \end{proposition}
 
+\begin{corollary}
+  Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose
+  there exists an irreducible coherent \(\mathfrak{s}\)-family. Then
+  \(\mathfrak{s} \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong
+  \mathfrak{sp}_{2n}(K)\) for some \(n\).
+\end{corollary}
+
 % TODO: Remove this: we will only focus on the combinatorial classification
 % TODO: Simply notice that a more explicit "geometric" description of the
 % cohorent families exists
@@ -33,6 +42,332 @@ and differential forms -- which is done in sections 11 and 12. While rather
 complicated on its own, the geometric construction is more concrete than its
 combinatorial counterpart.
 
+% TODO: Add some notes on the proof of this?
+% I really don't think its worth proving this
+\begin{proposition}
+  Let \(\mathcal{M}\) be a semisimple irreducible coherent
+  \(\mathfrak{g}\)-family. Then there exists some \(\lambda \in
+  \mathfrak{h}^*\) such that \(L(\lambda)\) is bounded and \(\mathcal{M} \cong
+  \mExt(L(\lambda))\).
+\end{proposition}
+
+% TODO: Finish this remark
+\begin{note}
+  I once had the opportunity to ask Olivier Mathieu himself how he first came
+  across the notation of coherent families and what was the intuition behind
+  it.
+\end{note}
+
+\begin{enumerate}
+  \item When is \(L(\lambda)\) bounded?
+
+  \item Given \(\lambda, \mu \in \mathfrak{h}^*\) with \(L(\lambda)\) and
+    \(L(\mu)\) bounded, when is \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\)?
+\end{enumerate}
+
+% TODO: Explain beforehand why central characters exist and are unique
+\begin{definition}
+  Given a highest weight \(\mathfrak{g}\)-module \(M\) of highest weight
+  \(\lambda\), the unique algebra homomorphism \(\chi_\lambda :
+  Z(\mathcal{U}(\mathfrak{g})) \to K\) such that \(u \cdot m = \chi_\lambda(u)
+  m\) for all \(m \in M\) and \(u \in Z(\mathcal{U}(\mathfrak{g}))\) is called
+  \emph{the central character of \(M\)} or \emph{the central character
+  associated with the weight \(\lambda\)}.
+\end{definition}
+
+% TODO: Define the dot-action beforehand
+\begin{theorem}[Harish-Chandra]
+  Given \(\lambda, \mu \in \mathfrak{h}^*\), \(\chi_\lambda = \chi_\mu\) if,
+  and only if \(\mu \in W \bullet \lambda\). All algebra homomorphism
+  \(Z(\mathcal{U}(\mathfrak{g})) \to K\) have the form \(\chi_\lambda\) for
+  some \(\lambda\).
+\end{theorem}
+
+% TODO: Note we will prove that central characters are also invariants of
+% coherent families
+% TODO: Prove this
+\begin{proposition}[Mathieu]
+  Suppose \(\lambda, \mu \in \mathfrak{h}^*\) are such that \(L(\lambda)\) and
+  \(L(\mu)\) are both bounded and \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\).
+  Then \(\chi_\lambda = \chi_\mu\).
+\end{proposition}
+
+\section{Coherent \(\mathfrak{sp}_{2n}(K)\)-families}
+
+% TODO: Fix n >= 2: sp_2 = sl_2
+
+% TODOO: Note this beforehand
+% TODOO: Note beforehand that the Weyl group of sp(2n) is S_n ⋉ (ℤ/2)^n. Write
+% down the isomorphism explicitly in terms of the basis Σ
+% TODOO: Perhaps its best to keep this information in here?
+% TODO: Change the notation for this? We use the notation α_i instead of ϵ_i in
+% the previous chapters
+We can find an orthonormal basis \(\{\epsilon_1, \ldots, \epsilon_n\}\) for
+\(\mathfrak{h}^*\) such that \(\Delta = \{\pm \epsilon_i \pm \epsilon_j\}_{i
+\ne j} \cup \{2 \epsilon_i\}_i\). Take the basis \(\Sigma = \{ \beta_1, \cdots,
+\beta_n \}\) for \(\Delta\) given by \(\beta_n = 2 \epsilon_n\) and \(\beta_i =
+\epsilon_i - \epsilon_{i + 1}\) for \(i < n\).
+
+% TODO: Prove this? This is the core of the classification for sp(2n), but it
+% is profoundly technical
+\begin{lemma}\label{thm:sp-bounded-weights}
+  Then \(L(\lambda)\) is bounded if, and only if
+  \begin{enumerate}
+    \item \(\lambda(H_{\beta_i})\) is non-negative integer for all \(i \ne n\).
+    \item \(\lambda(H_{\beta_n}) \in \frac{1}{2} + \mathbb{Z}\).
+    \item \(\lambda(H_{\beta_{n - 1}} + 2 H_{\beta_n}) \ge -2\).
+  \end{enumerate}
+\end{lemma}
+
+% TODO: Note that we need a better set of parameters to the space of weights
+% such that L(λ) is bounded
+
+% TODO: Prove this. The only part worth proving is the fact this is
+% W-equivariant, and this is clear from the isomorphism W ≅ S_n ⋉ (ℤ/2)^n
+% TODO: Revise the notation for this? I don't really like calling this
+% bijection m
+\begin{proposition}
+  The map
+  \begin{align*}
+    m : \mathfrak{h}^* & \to K^n \\
+        \lambda &
+        \mapsto
+        (
+          \kappa(\epsilon_1, \lambda+\rho),
+          \cdots,
+          \kappa(\epsilon_n, \lambda+\rho)
+        )
+  \end{align*}
+  is \(W\)-equivariant bijection, where the action \(W \cong S_n \ltimes
+  (\mathbb{Z}/2\mathbb{Z})^n\) on \(\mathfrak{h}^*\) is given by the dot-action
+  and the action of \(W\) on \(K^n\) is given my permuting coordinates and
+  multiplying them by \(\pm 1\). A weight \(\lambda \in \mathfrak{h}^*\)
+  satisfies the conditions of Lemma~\ref{thm:sp-bounded-weights} if, and
+  only if \(m(\lambda)_i \in \sfrac{1}{2} + \mathbb{Z}\) for all \(i\) and
+  \(m(\lambda)_1 > m(\lambda)_2 > \cdots > m(\lambda)_{n - 1} > \pm
+  m(\lambda)_n\).
+\end{proposition}
+
+% TODO: Prove this
+% Given the Harish-Chandra theorem and the previous proposition, all its left
+% is to show that of m(λ)_n = - m(μ)_n then indeed Ext(L(λ)) = Ext(L(μ)). I
+% think this is cavered in Lemma 6.1 of Mathieu
+\begin{theorem}[Mathieu]
+  Given \(\lambda\) and \(\mu\) satisfying the conditions of
+  Lemma~\ref{thm:sp-bounded-weights}, \(\mExt(L(\lambda)) \cong
+  \mExt(L(\mu))\) if, and only if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and
+  \(m(\lambda)_n = \pm m(\mu)_n\). In particular, the isomorphism classes of
+  semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-families are
+  parameterized by the \(n\)-tuples \(m \in (\sfrac{1}{2} + \mathbb{Z})^n\)
+  with \(m_1 > m_2 > \cdots > m_n > 0\).
+\end{theorem}
+
+% TODOO: Change the notation in here to work with sl(n) (n >= 3) instead of
+% sl(n + 1) (n >= 2)
+\section{Coherent \(\mathfrak{sl}_{n + 1}(K)\)-families}
+
+% TODOO: Add notes about this basis beforehand
+Take the standard Cartan subalgebra \(\mathfrak{h} = \{ X \in \mathfrak{sl}_{n
++ 1}(K) : X \ \text{is diagonal}\}\) of \(\mathfrak{sl}_{n + 1}(K)\) as in
+Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\alpha_i,
+\ldots, \alpha_{n + 1} \in \mathfrak{h}^*\) such that \(\alpha_i(H)\) is the
+\(i\)-th entry of the diagonal of \(H\). Let \(\Sigma = \{ \beta_1, \ldots,
+\beta_n \}\) be the basis for \(\Delta\) given by \(\beta_i = \alpha_i -
+\alpha_{i + 1}\).
+
+% TODO: Add some comments on the proof of this: while the proof that these
+% conditions are necessary is a purely combinatorial affair, the proof of the
+% fact that conditions (ii) and (iii) imply L(λ) is bounded requires some
+% results on the connected components of of the graph ℘ (which we will only
+% state later down the line)
+% TODO: Change the notation for A(λ)? I hate giving objets genereic names, but
+% there really isn't any reasonable name for it
+\begin{lemma}\label{thm:sl-bounded-weights}
+  Let \(\lambda \notin P^+\) and \(A(\lambda) = \{ i : \lambda(H_{\beta_i})\
+  \text{is not a non-negative integer}\}\). Then \(L(\lambda)\) is bounded if,
+  and only if one of the following assertions holds.
+  \begin{enumerate}
+    \item \(A(\lambda) = \{1\}\) or \(A(\lambda) = \{n\}\).
+    \item \(A(\lambda) = \{i\}\) for some \(1 < i < n\) and \((\lambda +
+      \rho)(H_{\beta_{i - 1}} + H_{\beta_i})\) or \((\lambda +
+      \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive integer.
+    \item \(A(\lambda) = \{i, i + 1\}\) for some \(1 \le i < n\) and
+      \((\lambda + \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive
+      integer.
+  \end{enumerate}
+\end{lemma}
+
+% TODO: Change the notation: these should be called "sl_n+1-sequences", not
+% "sl_n+1(K)-sequences"
+\begin{definition}
+  A \emph{\(\mathfrak{sl}_{n + 1}(K)\)-sequence} \(m\) is a \(n + 1\)-tuple \(m
+  \in K^{n + 1}\) such that \(m_1 + \cdots + m_{n + 1} = 0\).
+\end{definition}
+
+% TODO: Revise the notation for this? I don't really like calling this
+% bijection m
+% TODO: Note that this prove is similar to the previous one, and that the
+% equivariance of the map follows from the nature of the isomorphism W ≅ S_n+1
+% TODOO: Describe this isomorphism beforehand
+\begin{proposition}
+  The map
+  \begin{align*}
+    m : \mathfrak{h}^* & \to K^{n + 1} \\
+        \lambda &
+        \mapsto
+        (
+          \kappa(\alpha_1, \lambda + \rho),
+          \cdots,
+          \kappa(\alpha_{n + 1}, \lambda + \rho)
+        )
+  \end{align*}
+  is \(W\)-equivariant bijection onto the space of all \(\mathfrak{sl}_{n +
+  1}(K)\)-sequences, where the action \(W \cong S_{n + 1}\) on
+  \(\mathfrak{h}^*\) is given by the dot-action and the action of \(W\) on
+  \(K^n\) is given my permuting coordinates. A weight \(\lambda \in
+  \mathfrak{h}^*\) satisfies the conditions of
+  Lemma~\ref{thm:sl-bounded-weights} if, and only if 
+  the diferences between all but one consecutive coordinates of \(m(\lambda)\)
+  are positive integers -- i.e. there is some unique \(i \le n\) such that
+  \(m(\lambda)_i - m(\lambda)_{i + 1}\) is not a positive integer.
+\end{proposition}
+
+% TODO: Change the notation for 𝓟 ?
+% TODO: Explain the intuition behind defining the arrows like so: the point is
+% that if there is an arrow m(λ) → m(μ) then μ = σ_i ∙ λ for some i, which
+% implies L(μ) is contained in 𝓔𝔁𝓽(L(λ)) - so that L(μ) is also bounded and
+% 𝓔𝔁𝓽(L(λ)) ≅ 𝓔𝔁𝓽(L(μ))
+\begin{definition}
+  Denote by \(\mathcal{P}\) the set of \(\mathfrak{sl}_{n + 1}(K)\)-sequences
+  \(m\) such that \(m_i - m_{i + 1}\) is a positive integer for all but a
+  single \(i \le n\). Given \(m, m' \in \mathcal{P}\), say there is an arrow
+  \(m \to m'\) if the unique \(i\) such that \(m_i - m_{i + 1}\) is not a
+  positive integer is such that \(m' = \sigma_i \cdot m\).
+\end{definition}
+
+It should then be obvious that\dots
+
+\begin{proposition}
+  Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that
+  \(m(\lambda) \in \mathcal{P}\) -- and suppose that \(\mu \in \mathfrak{h}^*\)
+  is such that \(m(\mu) \in \mathcal{P}\) and there is an arrow \(m(\lambda)
+  \to m(\mu)\). Then \(L(\mu)\) is also bounded and \(\mExt(L(\mu)) \cong
+  \mExt(L(\lambda))\).
+\end{proposition}
+
+\begin{definition}
+  Let \(\mathcal{P}^+ = \{m \in \mathcal{P} : m_1 - m_2 \ \text{is not a
+  positive integer}\}\) and \(\mathcal{P}^- = \{m \in \mathcal{P} : m_n - m_{n
+  + 1} \ \text{is not a positive integer}\}\).
+\end{definition}
+
+\begin{definition}
+  A \(\mathfrak{sl}_{n + 1}(K)\)-sequence \(m\) is called \emph{integral} if
+  \(m_i - m_j \in \mathbb{Z}\) for all \(i\) and \(j\).
+\end{definition}
+
+% TODOO: Discuss the notion of a regular weight beforehand
+\begin{definition}
+  A \(\mathfrak{sl}_{n + 1}(K)\)-sequence \(m\) is called \emph{regular} if
+  \(m_i \ne m_j\) for all \(i \ne j\). A sequence is called \emph{singular} if
+  it is not regular.
+\end{definition}
+
+% TODO: Add notes on what are the sets W . m ∩ 𝓟  : the connected component of
+% a given element is contained in its orbit, but a given orbit may contain
+% multiple connected components. When m is regular and integral then its orbit
+% is the union of n connected components, but otherwise its orbit is precisely
+% its connected component (see Lemma 8.3)
+% TODO: Perhaps this could be incorporated into the proof of the following
+% theorem? Perhaps it's best to create another lemma for this
+% TODOO: Define the notation for σ_i beforehand
+\begin{proposition}
+  The connected component of some \(m \in \mathcal{P}\) is given by the
+  following.
+  \begin{enumerate}
+    \item If \(m\) is regular and integral then there exists\footnote{Notice
+      that in this case $m' \notin \mathscr{P}$, however.} a unique \(m' \in W
+      \cdot m\) such that \(m_1' > m_2' > \cdots > m_{n + 1}'\), in which case
+      the connected component of \(m\) is given by
+      \[
+        \begin{tikzcd}[cramped, sep=small]
+          \sigma_1 \sigma_2 \cdots \sigma_i \cdot m'           \rar &
+          \sigma_2 \cdots \sigma_i \cdot m'                    \rar &
+          \cdots                                               \rar &
+          \sigma_{i-1} \sigma_i \cdot m'
+            \ar[rounded corners,
+                to path={ -- ([xshift=4ex]\tikztostart.east)
+                          |- (X.center) \tikztonodes
+                          -| ([xshift=-4ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dlll}[at end]{}      \\
+          \sigma_i \cdot m'                                         &
+          \sigma_{i+1} \sigma_i \cdot m'                       \lar &
+          \cdots                                               \lar &
+          \sigma_n \cdots \sigma_i \cdot m'                    \lar &
+        \end{tikzcd}
+      \]
+      for some unique \(i\), with \(\sigma_1 \cdots \sigma_i \cdot m' \in
+      \mathcal{P}^+\) and \(\sigma_n \cdots \sigma_i \cdot m' \in
+      \mathcal{P}^-\).
+
+    \item If \(m\) is singular then there exists unique \(m' \in W \cdot m\)
+      and \(i\) such that \(m_1' > m_2' > \cdots > m_i' = m_{i + 1}' > \cdots >
+      m_{n + 1}'\), in which case the connected component of \(m\) is given by
+      \[
+        \begin{tikzcd}[cramped, sep=small]
+          \sigma_1 \sigma_2 \cdots \sigma_{i-1} \cdot m'       \rar &
+          \sigma_2 \cdots \sigma_{i-1} \cdot m'                \rar &
+          \cdots                                               \rar &
+          \sigma_{i-1} \cdot m'
+            \ar[rounded corners,
+                to path={ -- ([xshift=4ex]\tikztostart.east)
+                          |- (X.center) \tikztonodes
+                          -| ([xshift=-4ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dlll}[at end]{}      \\
+          m'                                                        &
+          \sigma_{i+1} \cdot m'                                \lar &
+          \cdots                                               \lar &
+          \sigma_n \cdots \sigma_{i+1} \cdot m'                \lar &
+        \end{tikzcd}
+      \]
+      with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathcal{P}^+\) and
+      \(\sigma_n \cdots \sigma_{i+1} \cdot m' \in \mathcal{P}^-\).
+
+    \item If \(m\) is non-integral then there exists unique \(m' \in W \cdot
+      m\) such that \(m_2' > m_3' > \cdots > m_{n + 1}'\), in which case the
+      connected component of \(m\) is given by
+      \[
+        \begin{tikzcd}[cramped]
+          m'                                \rar      &
+          \sigma_1 \cdot m'                 \rar \lar &
+          \sigma_2 \sigma_1 \cdot m'        \rar \lar &
+          \cdots                            \rar \lar &
+          \sigma_n \cdots \sigma_1 \cdot m'      \lar &
+        \end{tikzcd}
+      \]
+      with \(m' \in \mathcal{P}^+\) and \(\sigma_n \cdots \sigma_1 \cdot m' \in
+      \mathcal{P}^-\).
+  \end{enumerate}
+\end{proposition}
+
+% TODO: Add pictures of parts of the graph 𝓟 ?
+
+% TODO: Notice that this gives us that if m(λ)∈ 𝓟  then L(λ) is bounded: for λ
+% ∈ 𝓟 + ∪ 𝓟 - we stablish this by hand, and for the general case it suffices to
+% notice that there is always some path μ → ... → λ with μ ∈ 𝓟 + ∪ 𝓟 -
+% TODO: Perhaps this could be incorporated into the discussion of the lemma
+% that characterizes the weights of sl(n + 1) whose L is bounded
+
+% TODO: Prove this
+\begin{theorem}[Mathieu]
+  Given \(\lambda, \mu \in P^+\) with \(L(\lambda)\) and \(L(\mu)\) bounded,
+  \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and
+  \(m(\mu)\) lie in the same connected component of \(\mathcal{P}\). In
+  particular, the isomorphism classes of semisimple irreducible coherent
+  \(\mathfrak{sl}_{n + 1}(K)\)-families are parameterized by the set
+  \(\pi_0(\mathcal{P})\) of the connected components of \(\mathcal{P}\).
+\end{theorem}
+
 % TODO: Change this
 % I don't really think these notes bring us to this conclusion
 % If anything, these notes really illustrate the incredible vastness of the

diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex
@@ -103,7 +103,7 @@ We have already seen some concrete examples. Namely\dots
   self-normalizing.
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:cartan-of-sl}
   Let \(\mathfrak{h}\) be as in Example~\ref{ex:cartan-of-gl}. Then the
   subalgebra \(\mathfrak{h} \cap \mathfrak{sl}_n(K)\) of traceless diagonal
   matrices is a Cartan subalgebra of \(\mathfrak{sl}_n(K)\).