lie-algebras-and-their-representations

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+\chapter{Irreducible Weight Modules}\label{ch:mathieu}
+
+\begin{definition}
+  A representation \(V\) of \(\mathfrak{g}\) is called a \emph{weight
+  \(\mathfrak{g}\)-module} if \(V = \bigoplus_{\lambda \in \mathfrak{h}^*}
+  V_\lambda\) and \(\dim V_\lambda < \infty\) for all \(\lambda \in
+  \mathfrak{h}^*\). The \emph{support of \(V\)} is the set
+  \(\operatorname{supp} V = \{\lambda \in \mathfrak{h}^* : V_\lambda \ne 0\}\).
+\end{definition}
+
+\begin{example}
+  Corollary~\ref{thm:finite-dim-is-weight-mod} is equivalent to the fact that
+  every finite-dimensional representation of a semisimple Lie algebra is a
+  weight module. More generally, every finite-dimensional irreducible
+  representation of a reductive Lie algebra is a weight module.
+\end{example}
+
+\begin{example}\label{ex:submod-is-weight-mod}
+  Proposition~\ref{thm:verma-is-weight-mod} and
+  proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
+  \(M(\lambda)\) and its maximal subrepresentation are both weight modules. In
+  fact, the proof of proposition~\ref{thm:max-verma-submod-is-weight} is
+  actually a proof of the fact that every subrepresentation \(W \subset V\) of
+  a weight module \(V\) is a weight module, and \(W_\lambda = V_\lambda \cap
+  W\) for all \(\lambda \in \mathfrak{h}^*\).
+\end{example}
+
+\begin{example}\label{ex:quotient-is-weight-mod}
+  Given a weight module \(V\), a submodule \(W \subset V\) and \(\lambda \in
+  \mathfrak{h}^*\), \(\left(\mfrac{V}{W}\right)_\lambda = \mfrac{V_\lambda}{W}
+  \cong \mfrac{V_\lambda}{W_\lambda}\). In particular,
+  \[
+    \mfrac{V}{W}
+    = \bigoplus_{\lambda \in \mathfrak{h}^*} \left(\mfrac{V}{W}\right)_\lambda
+  \]
+  is a weight module. It is clear that \(\mfrac{V_\lambda}{W} \subset
+  \left(\mfrac{V}{W}\right)_\lambda\). To see that \(\mfrac{V_\lambda}{W} =
+  \left(\mfrac{V}{W}\right)_\lambda\), we remark that \(V_\lambda \cong
+  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+  \otimes_{\mathcal{U}(\mathfrak{h})} V\) as \(\mathfrak{h}\)-modules, where
+  \(\mathfrak{m}_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal
+  generated by the elements \(H - \lambda(H)\), \(H \in \mathfrak{h}\).
+  Likewise \(\left(\mfrac{V}{W}\right)_\lambda \cong
+  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+  \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}\) and the diagram
+  \begin{center}
+    \begin{tikzcd}
+      V_\lambda \arrow{d} \arrow{r}{\pi} &
+      \left(\mfrac{V}{W}\right)_\lambda \arrow{d} \\
+      \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+      \otimes_{\mathcal{U}(\mathfrak{h})} V
+      \arrow[swap]{r}{\pi \otimes \operatorname{id}} &
+      \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+      \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}
+    \end{tikzcd}
+  \end{center}
+  commutes, so that the projection \(V_\lambda \to
+  \left(\mfrac{V}{W}\right)_\lambda\) is surjective.
+\end{example}
+
+% TODO: Add an example of a module wich is NOT a weight module
+
+% TODOO: Prove this? Most likely not!
+\begin{proposition}\label{thm:centralizer-multiplicity}
+  Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
+  \(V_\lambda\) is a semisimple
+  \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
+  \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is
+  the cetralizer of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\).
+  Moreover, the multiplicity of a given irreducible representation
+  \(W\) of \(\mathfrak{g}\) coincides with the multiplicity of \(W_\lambda\) in
+  \(V_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module,
+  for any \(\lambda \in \operatorname{supp} V\).
+\end{proposition}
+
+\begin{definition}
+  A weight \(\mathfrak{g}\)-module is called \emph{admissible} if \(\dim
+  V_\lambda\) is bounded. The lowest upper bound for \(\dim V_\lambda\) is
+  called \emph{the degree of \(V\)}. The \emph{essential support} of \(V\) is
+  the set \(\operatorname{supp}_{\operatorname{ess}} V = \{ \lambda \in
+  \mathfrak{h}^* : \dim V_\lambda = d \}\).
+\end{definition}
+
+\begin{example}\label{ex:laurent-polynomial-mod}
+  There is a natural action of \(\mathfrak{sl}_2(K)\) in the space \(K[x,
+  x^{-1}]\) of Laurent polynomials given by the formulas in
+  (\ref{eq:laurent-polynomials-cusp-mod}). One can quickly verify \(K[x,
+  x^{-1}]_{2 k} = K x^k\) and \(K[x, x^{-1}]_\lambda = 0\) for any \(\lambda
+  \notin 2 \mathbb{Z}\), so that \(K[x, x^{-1}] = \bigoplus_{k \in \mathbb{Z}}
+  K x^k\) is a degree \(1\) admissible weight \(\mathfrak{sl}_2(K)\)-module. It
+  follows from example~\ref{ex:submod-is-weight-mod} that any non-zero
+  subrepresentation \(W \subset K[x, x^{-1}]\) must contain a monomial \(x^k\).
+  But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} + \frac{x^{-1}}{2},
+  x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} : K[x, x^{-1}] \to K[x,
+  x^{-1}]\) are both injective, this implies all other monomials can be found
+  in \(W\) by successively applaying \(f\) and \(e\). Hence \(W = K[x,
+  x^{-1}]\) and \(K[x, x^{-1}]\) is an irreducible representation.
+  \begin{align}\label{eq:laurent-polynomials-cusp-mod}
+    f \cdot p
+    & = \left(- \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x^{-1}}{2} \right) p &
+    h \cdot p
+    & = 2 x \frac{\mathrm{d}}{\mathrm{d}x} p &
+    e \cdot p
+    & = \left( x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} \right) p
+  \end{align}
+\end{example}
+
+% TODO: Point out supp_ess K[x^+-1] is 2Z, which is zariski dense
+% This proof is very technical, I don't think its worth including it
+\begin{proposition}
+  Let \(V\) be an infinite-dimensional admissible representation of
+  \(\mathfrak{g}\). The essential support
+  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense in
+  \(\mathfrak{h}^*\).
+\end{proposition}
+
+\begin{definition}
+  A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
+  if \(\mathfrak{b} \subset \mathfrak{p}\).
+\end{definition}
+
+% TODO: Comment afterwords that the Verma modules are indeed generalized Verma
+% modules
+\begin{definition}
+  Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) and a
+  \(\mathfrak{p}\)-module \(V\) the module \(M_{\mathfrak{p}}(V) =
+  \operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is called \emph{a
+  generalized Verma module}.
+\end{definition}
+
+\begin{proposition}
+  Given an irreducible \(\mathfrak{p}\)-module \(V\), the generalized Verma
+  module \(M_{\mathfrak{p}}(V)\) has a unique maximal subrepresentation
+  \(N_{\mathfrak{p}}(V)\) and a unique irreducible quotient
+  \(L_{\mathfrak{p}}(V) = \mfrac{M_{\mathfrak{p}}(V)}{N_{\mathfrak{p}}(V)}\).
+  The irreducible quotient \(L_{\mathfrak{p}}(V)\) is a weight module.
+\end{proposition}
+
+\begin{definition}
+  An irreducible \(\mathfrak{g}\)-module is called \emph{parabolic induced} if
+  it is isomorphic to \(L_{\mathfrak{p}}(V)\) for some proper parabolic
+  subalgebra \(\mathfrak{p} \subsetneq \mathfrak{g}\) and some
+  \(\mathfrak{p}\)-module \(V\). An \emph{irreducible cuspidal
+  \(\mathfrak{g}\)-module} is an irreducible representation of \(\mathfrak{g}\)
+  which is \emph{not} parabolic induced.
+\end{definition}
+
+% TODO: Remark on the fact that any simple weight p-mod is a (p/u)-mod, so that
+% the notation of a cuspidal p-mod is well definited
+% TODO: Define the conjugation of a p-mod by an element of the Weil group?
+\begin{theorem}[Fernando]
+  Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to
+  \(L_{\mathfrak{p}}(V)\) for some parabolic subalgebra \(\mathfrak{p} \subset
+  \mathfrak{g}\) and some irreducible cuspidal \(\mathfrak{p}\)-module \(V\).
+\end{theorem}
+
+% TODO: Point out that the relationship between p-modules and cuspidal
+% g-modules is not 1-to-1
+
+\begin{proposition}[Fernando]
+  Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\), there
+  exists a basis\footnote{This is usually called \emph{a $\mathfrak{p}$-adapted
+  basis}} \(\Sigma\) for \(\Delta\) such that \(\Sigma \subset
+  \Delta_{\mathfrak{p}_1}\). Furthermore, if \(\mathfrak{p}' \subset
+  \mathfrak{g}\) is another parabolic subalgebra, \(V\) is an irreducible
+  cuspidal \(\mathfrak{p}\)-module and \(W\) is an irreducible cuspidal
+  \(\mathfrak{p}'\)-module then \(L_{\mathfrak{p}}(V) \cong
+  L_{\mathfrak{p}'}(W)\) if, and only if \(\mathfrak{p}' = \mathfrak{p}^w\) and
+  \(W \cong V^w\) for some \(w \in \mathcal{W}_V\), where
+  \[
+    \mathcal{W}_V
+    = \langle
+      T_\beta : \beta \in \Sigma, H_\beta + \mathfrak{u}
+      \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{u}}
+      \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ V
+      \rangle
+    \subset \mathcal{W}
+  \]
+\end{proposition}
+
+% TODO: Point out that the definition of W_V is independant of the choice of
+% Sigma
+
+% TODO: Remark that the support of a simple weight module is always contained
+% in a coset
+\begin{corollary}[Fernando]\label{thm:cuspidal-mod-equivs}
+  Let \(V\) be an irreducible weight \(\mathfrak{g}\)-module. The following
+  conditions are equivalent.
+  \begin{enumerate}
+    \item \(V\) is cuspidal.
+    \item \(F_\alpha\) acts injectively\footnote{This is what's usually refered
+      to as a \emph{dense} representation in the literature.} in \(V\) for all
+      \(\alpha \in \Delta\).
+    \item The support of \(V\) is precisely one \(Q\)-coset\footnote{This is
+      what's usually referred to as a \emph{torsion-free} representation in the
+      literature.}.
+  \end{enumerate}
+\end{corollary}
+
+\begin{example}
+  As noted in example~\ref{ex:laurent-polynomial-mod}, the element \(f \in
+  \mathfrak{sl}_2(K)\) acts injectively in the space of Laurent polynomials.
+  Hence \(K[x, x^{-1}]\) is a cuspidal representation of
+  \(\mathfrak{sl}_2(K)\).
+\end{example}
+
+% TODOO: Do we need this proposition? I think this only comes up in the
+% classification of simple completely reducible coherent families
+\begin{proposition}
+  If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
+  \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
+  and \(\mathfrak{s}_i\) is a simple component of \(\mathfrak{g}\), then any
+  irreducible weight \(\mathfrak{g}\)-module \(V\) decomposes as
+  \[
+    V = Z \otimes V_1 \otimes \cdots \otimes V_n
+  \]
+  where \(Z\) is a 1-dimensional representation of \(\mathfrak{z}\) and \(V_i\)
+  is an irreducible weight \(\mathfrak{s}_i\)-module.
+\end{proposition}
+
+\begin{definition}
+  A \emph{coherent family \(\mathcal{M}\) of degree \(d\)} is a weight
+  \(\mathfrak{g}\)-module \(\mathcal{M}\) such that
+  \begin{enumerate}
+    \item \(\dim \mathcal{M}_\lambda = d\) for \emph{all} \(\lambda \in
+      \mathfrak{h}^*\)
+    \item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the centralizer
+      \(\mathcal{U}(\mathfrak{g})_0\) of \(\mathfrak{h}\) in
+      \(\mathcal{U}(\mathfrak{g})\), the map
+      \begin{align*}
+        \mathfrak{h}^* & \to K \\
+               \lambda & \mapsto
+               \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\lambda})
+      \end{align*}
+      is polynomial in \(\lambda\).
+  \end{enumerate}
+\end{definition}
+
+% TODO: Add an example: there's an example of a coherent sl2-family in
+% Mathieu's paper
+% TODO: Add a discussion on how this may sound unintuitive, but the motivation
+% comes from the relationship between highest weight modules and coherent
+% families
+
+% TODO: Point out this is equivalent to M being a simple object in the
+% category of coherent families
+\begin{definition}
+  A coherent family \(\mathcal{M}\) is called \emph{irreducible} if
+  \(\mathcal{M}_\lambda\) is a simple
+  \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
+  \mathfrak{h}^*\).
+\end{definition}
+
+\begin{definition}
+  Given an admissible representation \(V\) of \(\mathfrak{g}\) of degree \(d\),
+  a coherent extension \(\mathcal{M}\) of \(V\) is a coherent family
+  \(\mathcal{M}\) of degree \(d\) that contains \(V\) as a subquotient.
+\end{definition}
+
+% Mathieu's proof of this is somewhat profane, I don't think it's worth
+% including it in here
+% TODO: Define the notation for M[mu] somewhere else
+% TODO: Note somewhere that M[mu] is a submodule
+\begin{lemma}
+  Given a coherent family \(\mathcal{M}\) and \(\lambda \in \mathfrak{h}^*\),
+  \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
+\end{lemma}
+
+% TODO: From this we may conclude that any admissible submodule is a submodule
+% of the semisimplification of any of its coherent extensions
+\begin{corollary}
+  Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a
+  unique completely reducible coherent family
+  \(\mathcal{M}^{\operatorname{ss}}\) of degree \(d\) such that the composition
+  series of \(\mathcal{M}^{\operatorname{ss}}[\lambda]\) is the same as that of
+  \(\mathcal{M}[\lambda]\) for all \(\lambda \in \mathfrak{h}^*\), called
+  \emph{the semisimplification\footnote{Recall that a ``semisimple'' is a
+  synonim for ``completely reducible'' in the context of modules.} of
+  \(\mathcal{M}\)}.
+
+  Namely, if \(\{\lambda_i\}_i\) is a set of representatives of the
+  \(Q\)-cosets of \(\mathfrak{h}^*\) and \(0 = \mathcal{M}_{i 0} \subset
+  \mathcal{M}_{i 1} \subset \cdots \subset \mathcal{M}_{i n_i} =
+  \mathcal{M}[\lambda_i]\) is a composition series,
+  \[
+    \mathcal{M}^{\operatorname{ss}}
+    \cong \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+  \]
+\end{corollary}
+
+\begin{proof}
+  The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear:
+  since \(\mathcal{M}^{\operatorname{ss}}\) is completely reducible, so is
+  \(\mathcal{M}^{\operatorname{ss}}[\lambda_i]\). Hence
+  \[
+    \mathcal{M}^{\operatorname{ss}}[\lambda_i]
+    \cong \bigoplus_j \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+  \]
+
+  As for the existence of the semisimplification, it suffices to show
+  \[
+    \mathcal{M}^{\operatorname{ss}}
+    = \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+  \]
+  is indeed a completely reducible coherent family of degree \(d\).
+
+  We know from examples~\ref{ex:submod-is-weight-mod} and
+  \ref{ex:quotient-is-weight-mod} that each quotient \(\mfrac{\mathcal{M}_{i j
+  + 1}}{\mathcal{M}_{i j}}\) is a weight module. Hence
+  \(\mathcal{M}^{\operatorname{ss}}\) is a weight module. Furthermore, given
+  \(\mu \in \lambda_k + Q\)
+  \[
+    \mathcal{M}_\mu^{\operatorname{ss}}
+    = \bigoplus_{i j}
+      \left(
+      \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+      \right)_\mu
+    = \bigoplus_j
+      \left(
+      \mfrac{\mathcal{M}_{k j + 1}}{\mathcal{M}_{k j}}
+      \right)_\mu
+    \cong \bigoplus_j
+      \mfrac{(\mathcal{M}_{k j + 1})_\mu}
+            {(\mathcal{M}_{k j})_\mu}
+  \]
+
+  In particular,
+  \[
+    \dim \mathcal{M}_\mu^{\operatorname{ss}}
+    = \sum_j
+      \dim (\mathcal{M}_{k j + 1})_\mu
+    - \dim (\mathcal{M}_{k j})_\mu
+    = \dim \mathcal{M}[\lambda_k]_\mu
+    = \dim \mathcal{M}_\mu
+    = d
+  \]
+
+  Likewise, given \(u \in \mathcal{U}(\mathfrak{g})_0\) the value
+  \[
+    \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}})
+    = \sum_j
+      \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j + 1})_\mu})
+    - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j})_\mu})
+    = \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\lambda_k]_\mu})
+    = \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
+  \]
+  is polynomial in \(\mu \in \mathfrak{h}^*\).
+\end{proof}
+
+\begin{corollary}\label{thm:admissible-is-submod-of-extension}
+  Let \(V\) be an irreducible admissible \(\mathfrak{g}\)-module and
+  \(\mathcal{M}\) be a completely reducible coherent extension of \(V\). Then
+  \(V\) is contained in \(\mathcal{M}\).
+\end{corollary}
+
+\begin{proof}
+  Since \(V\) is irreducible, its support is contained in a single \(Q\)-coset.
+  This implies that \(V\) is a subquotient of \(\mathcal{M}[\lambda]\) for any
+  \(\lambda \in \operatorname{supp} V\). If we fix some composition series \(0
+  = \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
+  \mathcal{M}[\lambda]\) of \(\mathcal{M}[\lambda]\) with \(V \cong
+  \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}\), there is a natural inclusion
+  \[
+    V
+    \isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+    \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
+    \cong \mathcal{M}^{\operatorname{ss}}[\lambda]
+  \]
+
+  It then follows from the uniqueness of the semisimplification of
+  \(\mathcal{M}\) that \(\mathcal{M} \cong \mathcal{M}^{\operatorname{ss}}\),
+  so we have an inclusion \(V \to \mathcal{M}\).
+\end{proof}
+
+\begin{lemma}
+  Let \(\mathcal{M}\) be a coherent family. The set \(U = \{\lambda \in
+  \mathfrak{h}^* : \mathcal{M}_\lambda \ \text{is a simple
+  $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is Zariski-open.
+\end{lemma}
+
+\begin{proof}
+  For each \(\lambda \in \mathfrak{h}^*\) we introduce the bilinear form
+  \begin{align*}
+    B_\lambda : \mathcal{U}(\mathfrak{g})_0 \times \mathcal{U}(\mathfrak{g})_0
+    & \to K \\
+    (u, v)
+    & \mapsto \operatorname{Tr}(u v \!\restriction_{\mathcal{M}_\lambda})
+  \end{align*}
+  and consider its rank -- i.e. the dimension of the image of the induced
+  operator
+  \begin{align*}
+    \mathcal{U}(\mathfrak{g})_0 & \to     \mathcal{U}(\mathfrak{g})_0^* \\
+                              u & \mapsto B_\lambda(u, \cdot)
+  \end{align*}
+
+  Our first observation is that \(\operatorname{rank} B_\lambda \le d^2\). This
+  follows from the commutativity of
+  \begin{center}
+    \begin{tikzcd}
+      \mathcal{U}(\mathfrak{g})_0 \arrow{r} \arrow{d} &
+      \mathcal{U}(\mathfrak{g})_0^* \\
+      \operatorname{End}(\mathcal{M}_\lambda)   \arrow{r}{\sim} &
+      \operatorname{End}(\mathcal{M}_\lambda)^* \arrow{u}
+    \end{tikzcd},
+  \end{center}
+  where the map \(\mathcal{U}(\mathfrak{g})_0 \to
+  \operatorname{End}(\mathcal{M}_\lambda)\) is given by the action of
+  \(\mathcal{U}(\mathfrak{g})_0\), the map
+  \(\operatorname{End}(\mathcal{M}_\lambda)^* \to
+  \mathcal{U}(\mathfrak{g})_0^*\) is its dual, and the isomorphism
+  \(\operatorname{End}(\mathcal{M}_\lambda) \isoto
+  \operatorname{End}(\mathcal{M}_\lambda)^*\) is induced by the trace form
+  \begin{align*}
+    \operatorname{End}(\mathcal{M}_\lambda) \times
+    \operatorname{End}(\mathcal{M}_\lambda) & \to K \\
+    (T, S) & \mapsto \operatorname{Tr}(T S)
+  \end{align*}
+
+  Indeed, \(\operatorname{rank} B_\lambda \le
+  \operatorname{rank}(\mathcal{U}(\mathfrak{g})_0 \to
+  \operatorname{End}(\mathcal{M}_\lambda)) \le \dim
+  \operatorname{End}(\mathcal{M}_\lambda) = d^2\). Furthermore, if
+  \(\operatorname{rank} B_\lambda = d^2\) then we must have
+  \(\operatorname{rank}(\mathcal{U}(\mathfrak{g})_0 \to
+  \operatorname{End}(\mathcal{M}_\lambda)) = d^2\) -- i.e. the map
+  \(\mathcal{U}(\mathfrak{g})_0 \to \operatorname{End}(\mathcal{M}_\lambda)\)
+  is surjective. In particular, if \(\operatorname{rank} B_\lambda = d^2\) then
+  \(\mathcal{M}_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module,
+  for if \(V \subset \mathcal{M}_\lambda\) is invariant under the action of
+  \(\mathcal{U}(\mathfrak{g})_0\) then \(V\) is invariant under any
+  \(K\)-linear operator \(\mathcal{M}_\lambda \to \mathcal{M}_\lambda\), so
+  that \(W = 0\) or \(W = \mathcal{M}_\lambda\).
+
+  On the other hand, if \(\mathcal{M}_\lambda\) is simple then by Burnside's
+  theorem on matrix algebras the map \(\mathcal{U}(\mathfrak{g})_0 \to
+  \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. Hence the
+  commutativity of the previously drawn diagram, as well as the fact that
+  \(\operatorname{rank}(\mathcal{U}(\mathfrak{g})_0 \to
+  \operatorname{End}(\mathcal{M}_\lambda)) =
+  \operatorname{rank}(\operatorname{End}(\mathcal{M}_\lambda)^* \to
+  \mathcal{U}(\mathfrak{g})_0^*)\), imply that \(\operatorname{rank} B_\lambda
+  = d^2\). This goes to show that \(U\) is precisely the set of \(\lambda\)
+  such that \(B_\lambda\) has maximal rank \(d^2\). We now show that \(U\) is
+  Zariski-open. First, notice that
+  \[
+    U =
+    \bigcup_{\substack{W \subset \mathcal{U}(\mathfrak{g})_0 \\ \dim W = d^2}}
+    U_W,
+  \]
+  where \(U_W = \{\lambda \in \mathcal{U}(\mathfrak{g})_0 : \operatorname{rank}
+  B_\lambda\!\restriction_W = d^2 \}\).
+
+  Indeed, if \(\operatorname{rank} B_\lambda = d^2\) it follows from the
+  surjectivity of the map \(\mathcal{U}(\mathfrak{g})_0 \to
+  \operatorname{End}(\mathcal{M}_\lambda)\) that there is some \(W \subset
+  \mathcal{U}(\mathfrak{g})_0\) with \(\dim W = d^2\) such that the restriction
+  \(W \to \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. The
+  comutativity of
+  \begin{center}
+    \begin{tikzcd}
+      W \arrow{r} \arrow{d} & W^* \\
+      \operatorname{End}(\mathcal{M}_\lambda)   \arrow{r}{\sim} &
+      \operatorname{End}(\mathcal{M}_\lambda)^* \arrow{u}
+    \end{tikzcd}
+  \end{center}
+  then implies \(\operatorname{rank} B_\lambda\!\restriction_W = d^2\). In
+  other words, \(U \subset \bigcup_W U_W\).
+
+  Likewise, if \(\operatorname{rank} B_\lambda\!\restriction_W = d^2\) for some
+  \(W\), then the commutativity of
+  \begin{center}
+    \begin{tikzcd}
+      W \arrow{r} \arrow{d} & W^* \\
+      \mathcal{U}(\mathfrak{g})_0   \arrow{r} &
+      \mathcal{U}(\mathfrak{g})_0^* \arrow{u}
+    \end{tikzcd}
+  \end{center}
+  implies \(\operatorname{rank} B_\lambda \ge d^2\), which goes to show
+  \(\bigcup_W U_W \subset U\).
+
+  Given \(\lambda \in U_W\), the surjectivity of \(W \to
+  \operatorname{End}(\mathcal{M}_\lambda)\) and the fact that \(\dim W <
+  \infty\) imply \(W \to W^*\) is invertible. Since \(\mathcal{M}\) is a
+  coherent family, \(B_\lambda\) depends polynomialy in \(\lambda\). Hence so
+  does the induced maps \(W \to W^*\). In particular, there is some Zariski
+  neighborhood \(V\) of \(\lambda\) such that the map \(W \to W^*\) induced by
+  \(B_\mu\!\restriction_W\) is invertible for all \(\mu \in V\).
+
+  But the surjectivity of the map induced by \(B_\mu\!\restriction_W\) implies
+  \(\operatorname{rank} B_\mu = d^2\), so \(\mu \in U_W\) and therefore \(V
+  \subset U_W\). This implies \(U_W\) is open for all \(W\). Finally, \(U\) is
+  the union of Zariski-open subsets and is therefore open. We are done.
+\end{proof}
+
+\begin{theorem}[Mathieu]
+  Let \(\mathcal{M}\) be an irreducible coherent family of degree \(d\) and
+  \(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent.
+  \begin{enumerate}
+    \item \(\mathcal{M}[\lambda]\) is irreducible.
+    \item \(F_\alpha\!\restriction_{\mathcal{M}[\lambda]}\) is injective for
+      all \(\alpha \in \Delta\).
+    \item \(\mathcal{M}[\lambda]\) is cuspidal.
+  \end{enumerate}
+\end{theorem}
+
+\begin{proof}
+  The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
+  from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
+  corollary that \strong{(iii)} implies \strong{(ii)}. All it's left is to show
+  \strong{(ii)} implies \strong{(iii)}.
+
+  Suppose \(F_\alpha\) acts injectively in the subrepresentation
+  \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since
+  \(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains
+  an infinite-dimensiona irreducible \(\mathfrak{g}\)-submodule \(V\).
+  Moreover, again by corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(V\)
+  is a cuspidal representation, and its degree is bounded by \(d\). We claim
+  \(\mathcal{M}[\lambda] = V\).
+
+  Since \(\mathcal{M}\) is irreducible and
+  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense, \(U = \{\mu
+  \in \mathfrak{h}^* : \mathcal{M}_\mu \ \text{is a simple
+  $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is a non-empty open set, and \(U
+  \cap \operatorname{supp}_{\operatorname{ess}} V\) is non-empty. In other
+  words, there is some \(\mu \in \mathfrak{h}^*\) such that \(\mathcal{M}_\mu\)
+  is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module and \(\dim V_\mu = \deg
+  V\).
+
+  In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any
+  irreducible \(\mathfrak{g}\)-module \(W\), the multiplicity of \(W\) in
+  \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
+  \(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module, which is, of
+  course, \(1\) if \(W \cong V\) and \(0\) otherwise. Hence
+  \(\mathcal{M}[\lambda] = V\) and \(\mathcal{M}[\lambda]\) is cuspidal.
+\end{proof}
+
+\section{Localizations \& the Existance of Coherent Extensions}
+
+% TODO: Comment on the intuition behind the proof: we can get vectors in a
+% given eigenspace by translating by the F's and E's, but neither of those are
+% injective in general, so the translation could take nonzero vectors to zero.
+% If the F's were invertible this problem wouldn't exist, so we might as well
+% invert them by force!
+
+\begin{definition}
+  Let \(R\) be a ring. A subset \(S \subset R\) is called \emph{multiplicative}
+  if \(s \cdot t \in S\) for all \(s, t \in S\) and \(0 \notin S\). A
+  multiplicative subset \(S\) is said to satisfy \emph{Ore's localization
+  condition} if for each \(r \in R\), \(s \in S\) there exists \(u_1, u_2 \in
+  R\) and \(t_1, t_2 \in S\) such that \(s r = u_1 t_1\) and \(r s = t_2 u_2\).
+\end{definition}
+
+\begin{theorem}[Ore-Asano]
+  Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+  condition. Then there exists a (unique) ring \(S^{-1} R\), with a canonical
+  ring homomorphism \(R \to S^{-1} R\), and enjoying the universal property
+  that each ring homomorphism \(f : R \to T\) such that \(f(s)\) is invertible
+  for all \(s \in S\) can be uniquely extended to a ring homomorphism \(S^{-1}
+  R \to T\). \(S^{-1} R\) is called \emph{the localization of \(R\) by \(S\)},
+  and the map \(R \to S^{-1} R\) is called \emph{the localization map}.
+  \begin{center}
+    \begin{tikzcd}
+      S^{-1} R \arrow[dotted]{rd} & \\
+      R \arrow{u} \arrow[swap]{r}{f} & T
+    \end{tikzcd}
+  \end{center}
+\end{theorem}
+
+% TODO: Cite the discussion of goodearl-warfield, chap 6, on how to derive the
+% localization condition
+% TODO: In general checking that a set satisfies Ore's condition can be tricky,
+% but there is an easyer condition given by the lemma
+
+\begin{lemma}
+  Let \(S \subset R\) be a multiplicative subset generated by locally
+  \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\) such
+  that for each \(r \in R\) there exists \(n > 0\) such that \(\operatorname{ad}(s)^n r = [s, [s, \cdots
+  [s, r]]\cdots] = 0\). Then \(S\) satisfies Ore's
+  localization condition.
+\end{lemma}
+
+\begin{definition}
+  Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+  condition and \(M\) be a \(R\)-module. The \(S^{-1} R\)-module \(S^{-1} M =
+  S^{-1} R \otimes_R M\) is called \emph{the localization of \(M\) by \(S\)},
+  and the homomorphism of \(R\)-modules
+  \begin{align*}
+    M & \to     S^{-1} M    \\
+    m & \mapsto 1 \otimes m
+  \end{align*}
+  is called \emph{the localization map of \(M\)}.
+\end{definition}
+
+% TODO: Point out that the localization of modules is functorial
+
+% TODO: Point out the the localization map is in not injective in general
+\begin{lemma}
+  Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+  condition and \(M\) be a \(R\)-module. If \(S\) acts injectively in \(M\)
+  then the localization map \(M \to S^{-1} M\) is injective. In particular, if
+  \(S\) has no zero divisors then \(R\) is a subring of \(S^{-1} R\).
+\end{lemma}
+
+% TODO: Point out that S^-1 M can be seen as a R-module, where R acts via the
+% localization map
+% TODO: Point out that each element of the localization has the form s^-1 r
+
+% TODO: Point out that Sigma depends on V!
+\begin{lemma}\label{thm:nice-basis-for-inversion}
+  Let \(V\) be an irreducible infinite-dimensional admissible
+  \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
+  \beta_n\}\) of \(\Delta\) such that the elements \(F_{\beta_i}\) all act
+  injectively on \(V\) and satisfy \([F_{\beta_i}, F_{\beta_j}] = 0\).
+\end{lemma}
+
+\begin{corollary}
+  Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
+  \((F_\beta)_{\beta \in \Sigma} \subset \mathcal{U}(\mathfrak{g})\) be the
+  multiplicative subset generated by the \(F_\beta\)'s. The \(K\)-algebra
+  \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta \in \Sigma}^{-1}
+  \mathcal{U}(\mathfrak{g})\) is well defined. Moreover, if we denote by
+  \(\Sigma^{-1} V\) the localization of \(V\) by \((F_\beta)_{\beta \in
+  \Sigma}\), the localization map \(V \to \Sigma^{-1} V\) is injective.
+\end{corollary}
+
+% TODO: Fix V and Sigma beforehand
+\begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
+  The the restriction of the localization \(\Sigma^{-1} V\) is an admissible
+  \(\mathfrak{g}\)-module of degree \(d\) with \(\operatorname{supp}
+  \Sigma^{-1} V = Q + \operatorname{supp} V\) and \(\dim \Sigma^{-1} V_\lambda
+  = d\) for all \(\lambda \in \operatorname{supp} \Sigma^{-1} V\).
+\end{proposition}
+
+\begin{proof}
+  Fix some \(\beta \in \Sigma\). We begin by showing that \(F_\beta\) and
+  \(F_\beta^{-1}\) map the weight space \(\Sigma^{-1} V_\lambda\) to the weight
+  spaces \(\Sigma^{-1} V_{\lambda - \beta}\) and \(\Sigma^{-1} V_{\lambda +
+  \beta}\) respectively. Indeed, given \(v \in V_\lambda\) and \(H \in
+  \mathfrak{h}\) we have
+  \[
+    H F_\beta v
+    = ([H, F_\beta] + F_\beta H)v
+    = F_\beta (-\beta(H) + H) v
+    = F_\beta (\lambda - \beta)(H) \cdot v
+    = (\lambda - \beta)(H) \cdot F_\beta v
+  \]
+
+  On the other hand,
+  \[
+    0
+    = [H, 1]
+    = [H, F_\beta F_\beta^{-1}]
+    = F_\beta [H, F_\beta^{-1}] + [H, F_\beta] F_\beta^{-1}
+    = F_\beta [H, F_\beta^{-1}] - \beta(H) F_\beta F_\beta^{-1},
+  \]
+  so that \([H, F_\beta^{-1}] = \beta(H) \cdot F_\beta^{-1}\) and therefore
+  \[
+    H F_\beta^{-1} v
+    = ([H, F_\beta^{-1}] + F_\beta^{-1} H) v
+    = F_\beta^{-1} (\beta(H) + H) v
+    = (\lambda + \beta)(H) \cdot F_\beta^{-1} v
+  \]
+
+  From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(\Sigma^{-1}
+  V_{\lambda \pm \beta}\) follows our first conclusion: since \(V\) is a weight
+  module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
+  s^{-1} \otimes v\) for \(s \in (F_\beta)_{\beta \in \Sigma}\) and \(v \in
+  V\), we can see that \(\Sigma^{-1} V = \bigoplus_\lambda \Sigma^{-1}
+  V_\lambda\). Furtheremore, since the action of each \(F_\beta\) in
+  \(\Sigma^{-1} V\) is bijective and \(\Sigma\) is a basis of \(Q\) we obtain
+  \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\).
+
+  Again, because of the bijectivity of the \(F_\beta\)'s, to see that \(\dim
+  \Sigma^{-1} V_\lambda = d\) for all \(\lambda \in \operatorname{supp}
+  \Sigma^{-1} V\) it suffices to show that \(\dim \Sigma^{-1} V_\lambda = d\)
+  for some \(\lambda \in \operatorname{supp} \Sigma^{-1} V\). We may take
+  \(\lambda \in \operatorname{supp} V\) with \(\dim V_\lambda = d\). For any
+  finite-dimensional subspace \(W \subset \Sigma^{-1} V_\lambda\) we can find
+  \(s \in (F_\beta)_{\beta \in \Sigma}\) such that \(s W \subset V\). If \(s =
+  F_{\beta_{i_1}} \cdots F_{\beta_{i_n}}\), it is clear \(s W \subset
+  V_{\lambda - \beta_{i_1} - \cdots - \beta_{i_n}}\), so \(\dim W = \dim sW \le
+  d\). This holds for all finite-dimensional \(W \subset \Sigma^{-1}
+  V_\lambda\), so \(\dim \Sigma^{-1} V_\lambda \le d\). It then follows from
+  the fact that \(V_\lambda \subset \Sigma^{-1} V_\lambda\) that \(V_\lambda =
+  \Sigma^{-1} V_\lambda\) and therefore \(\dim \Sigma^{-1} V_\lambda = d\).
+\end{proof}
+
+\begin{proposition}\label{thm:nice-automorphisms-exist}
+  There is a family of automorphisms \(\{\theta_\lambda : \Sigma^{-1}
+  \mathcal{U}(\mathfrak{g}) \to \Sigma^{-1}
+  \mathcal{U}(\mathfrak{g})\}_{\lambda \in \mathfrak{h}^*}\) such that
+  \begin{enumerate}
+    \item \(\theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r) = F_{\beta_1}^{k_1}
+      \cdots F_{\beta_n}^{k_n} r F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{-
+      k_1}\) for all \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and \(k_1,
+      \ldots, k_n \in \mathbb{Z}\).
+
+    \item For each \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) the map
+      \begin{align*}
+        \mathfrak{h}^* & \to     \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
+               \lambda & \mapsto \theta_\lambda(r)
+      \end{align*}
+      is polynomial.
+
+    \item If \(\lambda, \mu \in \mathfrak{h}^*\), \(M\) is a \(\Sigma^{-1}
+      \mathcal{U}(\mathfrak{g})\)-module whose restriction to
+      \(\mathcal{U}(\mathfrak{g})\) is a weight \(\mathfrak{g}\)-module and
+      \(\theta_\lambda M\) is the \(\Sigma^{-1}
+      \mathcal{U}(\mathfrak{g})\)-module \(M\) twisted by the automorphism
+      \(\theta_\lambda\) then \(M_\mu = (\theta_\lambda M)_{\mu + \lambda}\).
+      In particular, \(\operatorname{supp} \theta_\lambda M = \lambda +
+      \operatorname{supp} M\).
+  \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+  Since the elements \(F_\beta\), \(\beta \in \Sigma\) commute with one
+  another, the endomorphisms
+  \begin{align*}
+    \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}
+    : \Sigma^{-1} \mathcal{U}(\mathfrak{g}) &
+    \to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
+    r & \mapsto
+    F_{\beta_1}^{k_1} \cdots F_{\beta_n}^{k_n}
+    r
+    F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{- k_1}
+  \end{align*}
+  are well defined for all \(k_1, \ldots, k_n \in \mathbb{Z}\).
+
+  Fix some \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\). For any \(s \in
+  (F_\beta)_{\beta \in \Sigma}\) and \(k > 0\) we have \(s^k r = \binom{k}{0}
+  \operatorname{ad}(s)^0 r s^{k - 0} + \cdots + \binom{k}{k}
+  \operatorname{ad}(s)^k r s^{k - k}\). Now if we take \(m\) such
+  \(\operatorname{ad}(F_\beta)^{m + 1} r = 0\) for all \(\beta \in \Sigma\) we
+  find
+  \[
+    \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r)
+    = \sum_{i_1, \ldots, i_n = 1, \ldots, m}
+    \binom{k_1}{i_1} \cdots \binom{k_n}{i_n}
+    \operatorname{ad}(F_{\beta_1})^{i_1} \cdots
+    \operatorname{ad}(F_{\beta_n})^{i_n}
+    r
+    F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
+  \]
+  for all \(k_1, \ldots, k_n \in \NN\).
+
+  Since the binomial coeffients \(\binom{x}{k} = \frac{x (x -1) \cdots (x - k +
+  1)}{k!}\) can be uniquely extended to polynomial functions in \(x \in K\), we
+  may in general define
+  \[
+    \theta_\lambda(r)
+    = \sum_{i_1, \ldots, i_n \ge 0}
+    \binom{\lambda_1}{i_1} \cdots \binom{\lambda_n}{i_n}
+    \operatorname{ad}(F_{\beta_1})^{i_1} \cdots
+    \operatorname{ad}(F_{\beta_n})^{i_n}
+    r
+    F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
+  \]
+  for \(\lambda_1, \ldots, \lambda_n \in K\), \(\lambda = \lambda_1 \beta_1 +
+  \cdots + \lambda_n \beta_n \in \mathfrak{h}^*\)
+
+  It is clear that the \(\theta_\lambda\) are endmorphisms. To see that the
+  \(\theta_\lambda\) are indeed automorphisms, notice \(\theta_{- k_1 \beta_1 -
+  \cdots - k_n \beta_n} = \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}^{-1}\).
+  The uniqueness of the polynomial extensions then implies \(\theta_{- \lambda}
+  = \theta_\lambda^{-1}\) in general: given \(r \in \Sigma^{-1}
+  \mathcal{U}(\mathfrak{g})\), the map
+  \begin{align*}
+    \mathfrak{h}^* & \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})        \\
+           \lambda & \mapsto \theta_\lambda(\theta_{-\lambda}(r)) - r
+  \end{align*}
+  is a polynomial extension of the zero map \(\ZZ \beta_1 \oplus \cdots \oplus
+  \ZZ \beta_n \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is therefore
+  identicaly zero.
+
+  Finally, let \(M\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
+  whose restriction is a weight module. If \(m \in M\) then
+  \[
+    m \in (\theta_\lambda M)_{\mu + \lambda}
+    \iff \theta_\lambda(H) m = (\mu + \lambda)(H) \cdot m
+    \, \forall H \in \mathfrak{h}
+  \]
+
+  But
+  \[
+    \theta_\beta(H)
+    = F_\beta H F_\beta^{-1}
+    = ([F_\beta, H] + H F_\beta) F_\beta^{-1}
+    = (\beta(H) + H) F_\beta F_\beta^{-1}
+    = \beta(H) + H
+  \]
+  for all \(H \in \mathfrak{h}\) and \(\beta \in \Sigma\). In general,
+  \(\theta_\lambda(H) = \lambda(H) + H\) for all \(\lambda \in \mathfrak{h}^*\)
+  and hence
+  \[
+    \begin{split}
+      m \in (\theta_\lambda M)_{\mu + \lambda}
+      & \iff (\lambda(H) + H) m = (\mu + \lambda)(H) \cdot m
+        \; \forall H \in \mathfrak{h} \\
+      & \iff H m = \mu(H) \cdot m \; \forall H \in \mathfrak{h} \\
+      & \iff m \in M_\mu
+    \end{split},
+  \]
+  so that \((\theta_\lambda M)_{\mu + \lambda} = M_\mu\).
+\end{proof}
+
+\begin{proposition}[Mathieu]
+  There exists a coherent extension \(\mathcal{M}\) of \(V\).
+\end{proposition}
+
+\begin{proof}
+  Let \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
+  \(\mathfrak{h}^*\) with \(0 \in \Lambda\) and take
+  \[
+    \mathcal{M}
+    = \bigoplus_{\lambda \in \Lambda} \theta_\lambda \Sigma^{-1} V
+  \]
+
+  On the one hand, \(V\) lies in \(\Sigma^{-1} V = \theta_0 \Sigma^{-1} V\) --
+  notice that \(\theta_0\) is just the identity operator -- and therefore \(V
+  \subset \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
+  \theta_\lambda \Sigma^{-1} V_\mu = \dim \Sigma^{-1} V_{\mu - \lambda} = d\)
+  for all \(\mu \in \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given
+  \(u \in \mathcal{U}(\mathfrak{g})_0\) and \(\mu \in \lambda +
+  Q\),
+  \[
+    \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
+    = \operatorname{Tr}
+      (\theta_\lambda(u)\!\restriction_{\Sigma^{-1} V_{\mu - \lambda}})
+  \]
+  is polynomial in \(\mu\) because of the second item of
+  proposition~\ref{thm:nice-automorphisms-exist}.
+\end{proof}
+
+\begin{theorem}[Mathieu]
+  There exists a unique completely reducible coherent extension
+  \(\operatorname{Ext}(V)\) of \(V\). More precisely, if \(\mathcal{M}\) is any
+  coherent extension of \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong
+  \operatorname{Ext}(V)\). Furthermore, \(\operatorname{Ext}(V)\) is
+  irreducible as a coherent family.
+\end{theorem}
+
+\begin{proof}
+  The existence part should be clear from the previous discussion: it suffices
+  to fix some coherent extension \(\mathcal{M}\) of \(V\) and take
+  \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
+
+  To see that \(\operatorname{Ext}(V)\) is irreducible as a coherent family,
+  recall from corollary~\ref{thm:admissible-is-submod-of-extension} that \(V\)
+  is a subrepresentation of \(\operatorname{Ext}(V)\). Since the degree of
+  \(V\) is the same as the degree of \(\operatorname{Ext}(V)\), some of its
+  weight spaces have maximal dimension inside of \(\operatorname{Ext}(V)\). In
+  particular, it follows from proposition~\ref{thm:centralizer-multiplicity}
+  that \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
+  \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
+  \operatorname{supp} V\).
+
+  As for the uniqueness of \(\operatorname{Ext}(V)\), fix some other completely
+  reducible coherent extension \(\mathcal{N}\) of \(V\). We claim that the
+  multiplicity of a given irreducible \(\mathfrak{g}\)-module \(W\) in
+  \(\mathcal{N}\) is determined by its \emph{trace function}
+  \begin{align*}
+    \mathfrak{h}^* \times \mathcal{U}(\mathfrak{g})_0 &
+    \to K \\
+    (\lambda, u) &
+    \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})
+  \end{align*}
+
+  % TODO: Point out that this multiplicity is determined by the characters
+  % beforehand
+  Indeed, given \(\lambda \in \operatorname{supp} V\) the multiplicity of \(W\)
+  in \(\mathcal{N}\) is the same as the multiplicity of \(W_\lambda\) in
+  \(\mathcal{N}_\lambda\), which is determined by the character
+  \(\chi_{\mathcal{N}_\lambda} : \mathcal{U}(\mathfrak{g})_0
+  \to K\) -- see proposition~\ref{thm:centralizer-multiplicity}. We now claim
+  that the trace function of \(\mathcal{N}\) is the same as that of
+  \(\operatorname{Ext}(V)\). Clearly,
+  \(\operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})
+  = \operatorname{Tr}(u\!\restriction_{V_\lambda})
+  = \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\) for all
+  \(\lambda \in \operatorname{supp}_{\operatorname{ess}} V\), \(u \in
+  \mathcal{U}(\mathfrak{g})_0\). Since the essential support of
+  \(V\) is Zariski-dense and the maps \(\lambda \mapsto
+  \operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})\) and
+  \(\lambda \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\)
+  are polynomial in \(\lambda \in \mathfrak{h}^*\), it follows that this maps
+  coincide for all \(u\).
+
+  In conclusion, \(\mathcal{N} \cong \operatorname{Ext}(V)\) and
+  \(\operatorname{Ext}(V)\) is unique.
+\end{proof}
+
+\begin{proposition}[Mathieu]
+  The central characters of the irreducible submodules of
+  \(\operatorname{Ext}(V)\) are all the same.
+\end{proposition}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -183,23 +183,24 @@
 
 % TODO: Remark that the support of a simple weight module is always contained
 % in a coset
-% TODO: Note that conditions (ii), (iii) and (iv) have special names
-% TODO: Remove item (ii)? I don't think we use it anywhere
 \begin{corollary}[Fernando]\label{thm:cuspidal-mod-equivs}
   Let \(V\) be an irreducible weight \(\mathfrak{g}\)-module. The following
   conditions are equivalent.
   \begin{enumerate}
     \item \(V\) is cuspidal.
-    \item \(E_\alpha\) acts injectively in \(V\) for all \(\alpha \in \Delta\).
-    \item \(F_\alpha\) acts injectively in \(V\) for all \(\alpha \in \Delta\).
-    \item The support of \(V\) is precisely one \(Q\)-coset.
+    \item \(F_\alpha\) acts injectively\footnote{This is what's usually refered
+      to as a \emph{dense} representation in the literature.} in \(V\) for all
+      \(\alpha \in \Delta\).
+    \item The support of \(V\) is precisely one \(Q\)-coset\footnote{This is
+      what's usually referred to as a \emph{torsion-free} representation in the
+      literature.}.
   \end{enumerate}
 \end{corollary}
 
 \begin{example}
-  As noted in example~\ref{ex:laurent-polynomial-mod}, the elements \(e, f \in
-  \mathfrak{sl}_2(K)\) both act injectively in the space of Laurent
-  polynomials. Hence \(K[x, x^{-1}]\) is a cuspidal representation of
+  As noted in example~\ref{ex:laurent-polynomial-mod}, the element \(f \in
+  \mathfrak{sl}_2(K)\) acts injectively in the space of Laurent polynomials.
+  Hence \(K[x, x^{-1}]\) is a cuspidal representation of
   \(\mathfrak{sl}_2(K)\).
 \end{example}