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