lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -216,6 +216,8 @@ the following definition.
 \end{definition}
 
 % TODO: Define nilpotent algebras beforehand
+% TODO: Define the nilradical in the introduction and state that the quotient
+% of an alebra by its nilradical is reductive
 % TODO: State the universal property of quotients in the introduction
 Parabolic subalgebras thus give us a process for constructing weight
 \(\mathfrak{g}\)-modules from representations of smaller (parabolic)
@@ -290,7 +292,7 @@ We should point out that the relationship between irreducible weight
 cuspidal \(\mathfrak{p}\)-module -- is not one-to-one. Nevertheless, this
 relationship is well understood. Namely, Fernando himself established\dots
 
-% TODO: Define the conjugation of a p-mod by an element of the Weil group
+% TODOO: Define the conjugation of a p-mod by an element of the Weil group
 % beforehand
 \begin{proposition}[Fernando]
   Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\), there
@@ -343,7 +345,8 @@ characterizations of cuspidal modules.
 \end{example}
 
 %% TODOO: Do we need this proposition? I think this only comes up in the
-%% classification of simple completely reducible coherent families
+%% classification of simple completely reducible coherent families. This could
+%% stated in the end when we discuss the classification
 %\begin{proposition}
 %  If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
 %  \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
@@ -373,7 +376,8 @@ a cuspidal representations we have encoutered so far: the
 \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\) of Laurent polynomials -- i.e.
 example~\ref{ex:laurent-polynomial-mod}.
 
-% TODO: Add a definition of the ring of differential operators somewhere?
+% TODO: Add a reference to Coutinho or something on the definition of the ring
+% of differential operators of a given algebra
 Our first observation is that \(\mathfrak{sl}_2(K)\) acts in \(K[x, x^{-1}]\)
 via differential operators. In other words, the action map
 \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{End}(K[x, x^{-1}])\)
@@ -422,7 +426,7 @@ 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.
 
-% TODO: Point out that the twisting automorphism does not fact to an
+% 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
 
@@ -575,7 +579,7 @@ to a completely reducible coherent extension of \(V\).
   \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
 \end{lemma}
 
-% TODO: Point out this construction is NOT functorial, since it depends on the
+% TODOO: Point out this construction is NOT functorial, since it depends on the
 % choice of composition series
 \begin{corollary}
   Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a
@@ -1267,8 +1271,8 @@ It should now be obvious\dots
   There exists a coherent extension \(\mathcal{M}\) of \(V\).
 \end{proposition}
 
-% TODO: Point out that here we have to fix representatives, so this
-% construction is, once again, not functorial
+% TODOO: Point out that here we have to fix representatives of the cosets, so
+% this construction is, once again, not functorial
 \begin{proof}
   Take
   \[
@@ -1359,7 +1363,6 @@ Lo and behold\dots
   \(\operatorname{Ext}(V)\) is unique.
 \end{proof}
 
-% TODO: Remove this
 % 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