diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -158,7 +158,7 @@
% TODO: Add an example: there's an example of a coherent sl2-family in
% Mathieu's paper
-% TODOO: Add a discussion on how this may sound unintuitive, but the motivation
+% TODO: Add a discussion on how this may sound unintuitive, but the motivation
% comes from the relationship between highest weight modules and coherent
% families
@@ -170,9 +170,9 @@
\end{definition}
\begin{definition}
- Given a representation \(V\) of \(\mathfrak{g}\), a coherent extension
- \(\mathcal{M}\) of \(V\) is a coherent family \(\mathcal{M}\) that contains
- \(V\) as a subquotient.
+ 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
@@ -283,7 +283,6 @@
s^{-1} = t_2^{-1} u_2\).
\end{definition}
-% TODOOO: Does R need to be Noetherian?
\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
@@ -440,11 +439,13 @@
\Sigma^{-1} V\) is the \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
\(\Sigma^{-1} V\) twisted by the automorphism \(\theta_\lambda\) then
\(\Sigma^{-1} V_\mu = (\theta_\lambda \Sigma^{-1} V)_{\mu + \lambda}\).
+ In particular, \(\operatorname{supp} \theta_\lambda \Sigma^{-1} V =
+ \lambda + Q + \operatorname{supp} V\).
\end{enumerate}
\end{proposition}
\begin{proposition}[Mathieu]
- There exists a coherent extension \(\mathcal{M}\) of \(V\) with degree \(d\).
+ There exists a coherent extension \(\mathcal{M}\) of \(V\).
\end{proposition}
\begin{proof}
@@ -480,12 +481,10 @@
\end{theorem}
% TODOOO: Prove the uniqueness
-% TODOO: Does every coherent extension of V have the same degree as V?
\begin{proof}
The existence part should be clear from the previous discussion: let
- \(\mathcal{M}\) be a coherent extension of \(V\) with \(\deg \mathcal{M} =
- d\) and take \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of
- \(V\).
+ \(\mathcal{M}\) be a coherent extension of \(V\) and take
+ \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of \(V\).
We claim \(V\) is contained in \(\operatorname{Ext}(V)\). Indeed, if we fix
some composition series \(0 =