diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -430,13 +430,13 @@ Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) in
\(\varphi_\lambda K[x, x^{-1}]\) is given by
\begin{align*}
p & \overset{f}{\mapsto}
- \left(
+ \left(
- \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1 - \lambda}{2} x^{-1}
\right) p &
p & \overset{h}{\mapsto}
\left( 2 x \frac{\mathrm{d}}{\mathrm{d}x} + \lambda \right) p &
p & \overset{e}{\mapsto}
- \left(
+ \left(
x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1 + \lambda}{2} x
\right) p,
\end{align*}
@@ -444,21 +444,20 @@ so we can see \((\varphi_\lambda K[x, x^{-1}])_{2 k + \frac{\lambda}{2}} = K
x^k\) for all \(k \in \mathbb{Z}\) and \((\varphi_\lambda K[x, x^{-1}])_\mu =
0\) for all other \(\mu \in \mathfrak{h}^*\).
-Hence \(\varphi_\lambda K[x, x^{-1}]\)
-is a degree \(1\) admissible \(\mathfrak{sl}_2(K)\)-module with
-\(\operatorname{supp} \varphi_\lambda K[x, x^{-1}] = \frac{\lambda}{2} + 2
-\mathbb{Z}\). One can also quickly check that if \(\lambda \notin 1 + 2
-\mathbb{Z}\) then \(e\) and \(f\) act injectively in \(\varphi_\lambda K[x,
-x^{-1}]\), so that \(\varphi_\lambda K[x, x^{-1}]\) is irreducible. In
-particular, if \(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with \(\lambda \notin
-\mu + 2 \mathbb{Z}\) then \(\varphi_\lambda K[x, x^{-1}]\) and \(\varphi_\mu
-K[x, x^{-1}]\) are non-isomorphic irreducible cuspidal \(\mathfrak{sl}_2(K)\),
-since their supports differ.
-These cuspidal representations can be ``glued toghether'' in a \emph{monstrous
-concoction} by summing over \(\lambda \in K\), as in
+Hence \(\varphi_\lambda K[x, x^{-1}]\) is a degree \(1\) admissible
+\(\mathfrak{sl}_2(K)\)-module with \(\operatorname{supp} \varphi_\lambda K[x,
+x^{-1}] = \frac{\lambda}{2} + 2 \mathbb{Z}\). One can also quickly check that
+if \(\lambda \notin 1 + 2 \mathbb{Z}\) then \(e\) and \(f\) act injectively in
+\(\varphi_\lambda K[x, x^{-1}]\), so that \(\varphi_\lambda K[x, x^{-1}]\) is
+irreducible. In particular, if \(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with
+\(\lambda \notin \mu + 2 \mathbb{Z}\) then \(\varphi_\lambda K[x, x^{-1}]\) and
+\(\varphi_\mu K[x, x^{-1}]\) are non-isomorphic irreducible cuspidal
+\(\mathfrak{sl}_2(K)\), since their supports differ. These cuspidal
+representations can be ``glued toghether'' in a \emph{monstrous concoction} by
+summing over \(\lambda \in K\), as in
\[
\mathcal{M}
- = \bigoplus_{\lambda + 2 \mathbb{Z} \in \mfrac{K}{2 \mathbb{Z}}}
+ = \bigoplus_{\lambda + 2 \mathbb{Z} \in \mfrac{K}{2 \mathbb{Z}}}
\varphi_\lambda K[x, x^{-1}],
\]
@@ -521,10 +520,9 @@ named \emph{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 \(\lambda \in
- \mathfrak{h}^*\).
+ A coherent family \(\mathcal{M}\) is called \emph{simple} if
+ \(\mathcal{M}_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module
+ for some \(\lambda \in \mathfrak{h}^*\).
\end{definition}
\begin{definition}
@@ -769,7 +767,7 @@ named \emph{coherent families}.
\end{proof}
\begin{theorem}[Mathieu]
- Let \(\mathcal{M}\) be an irreducible coherent family of degree \(d\) and
+ Let \(\mathcal{M}\) be a simple coherent family of degree \(d\) and
\(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent.
\begin{enumerate}
\item \(\mathcal{M}[\lambda]\) is irreducible.
@@ -796,7 +794,7 @@ named \emph{coherent families}.
is a cuspidal representation, and its degree is bounded by \(d\). We claim
\(\mathcal{M}[\lambda] = V\).
- Since \(\mathcal{M}\) is irreducible and
+ Since \(\mathcal{M}\) is simple 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
@@ -1116,7 +1114,7 @@ named \emph{coherent families}.
\(\operatorname{Ext}(V)\) 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.
+ a simple coherent family.
\end{theorem}
\begin{proof}
@@ -1124,11 +1122,11 @@ named \emph{coherent families}.
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
+ To see that \(\operatorname{Ext}(V)\) is simple, 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