lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -326,9 +326,8 @@ characterizations of cuspidal modules.
   \(\mathfrak{sl}_2(K)\).
 \end{example}
 
-%% TODOO: Do we need this proposition? I think this only comes up in the
-%% classification of simple completely reducible coherent families. This could
-%% stated in the end when we discuss the classification
+%% TODO: Move this to the section where we discuss the classification of
+%% 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 \(\mathfrak{g}\)
@@ -408,10 +407,6 @@ where the maps \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x,
 x^{-1}])\) and \(\operatorname{Diff}(K[x, x^{1}]) \to \operatorname{End}(K[x,
 x^{-1}])\) are the ones from the previous diagram.
 
-% TODOO: Point out that the twisting automorphism does not fact to an
-% automorphism of the universal enveloping algebra of sl2, but it factor
-% trought an automorphism of the localization of this algebra by f
-
 Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) in
 \(\varphi_\lambda K[x, x^{-1}]\) is given by
 \begin{align*}
@@ -1125,7 +1120,33 @@ if twist \(\Sigma^{-1} V\) by an automorphism which shifts its support by some
 \(\lambda \in \mathfrak{h}^*\), we can construct a coherent family by summing
 this modules over \(\lambda\) as in example~\ref{ex:sl-laurent-family}.
 
-For \(\lambda = \beta \in \Sigma\) the map
+% TODO: Are you sure these maps factor trought automorphisms of the
+% localization?
+For \(K[x, x^{-1}]\) this was achieved by twisting the
+\(\operatorname{Diff}(K[x, x^{-1}])\)-module \(K[x, x^{-1}]\) by the
+automorphisms \(\varphi_\lambda : \operatorname{Diff}(K[x, x^{-1}]) \to
+\operatorname{Diff}(K[x, x^{-1}])\) and restricting the results to
+\(\mathcal{U}(\mathfrak{sl}_2(K))\) via the map
+\(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x, x^{-1}])\), but
+we could just as well twist \(K[x, x^{-1}]\) by automorphisms of
+\(\mathcal{U}(\mathfrak{sl}_2(K))_f\) -- where
+\(\mathcal{U}(\mathfrak{sl}_2(K))_f\) denotes the localization of
+\(\mathcal{U}(\mathfrak{sl}_2(K))\) by the multiplicative subset generated by
+\(f\). In fact, \(\varphi_\lambda\) factors trought an automorphism
+\(\theta_\lambda :\mathcal{U}(\mathfrak{sl}_2(K))_f \to
+\mathcal{U}(\mathfrak{sl}_2(K))_f\).
+\begin{center}
+  \begin{tikzcd}
+    \mathcal{U}(\mathfrak{sl}_2(K))_f \rar \dar[dotted, swap]{\theta_\lambda} &
+    \operatorname{Diff}(K[x, x^{-1}]) \dar{\varphi_\lambda} \\
+    \mathcal{U}(\mathfrak{sl}_2(K))_f \rar &
+    \operatorname{Diff}(K[x, x^{-1}])
+  \end{tikzcd}
+\end{center}
+
+In general, we may twist the \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
+\(\Sigma^{-1} V\) by automorphisms of \(\Sigma^{-1}
+\mathcal{U}(\mathfrak{g})\). For \(\lambda = \beta \in \Sigma\) the map
 \begin{align*}
   \theta_\beta : \Sigma^{-1} \mathcal{U}(\mathfrak{g}) & \to
                  \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\