lie-algebras-and-their-representations

diff --git a/TODO.md b/TODO.md
@@ -1,5 +1,4 @@
 # TODO
 
-* Change the notation for twisted modules
 * Change the notation for boxtimes
 * Move proposition 5.39 to the start of the notes?

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -277,10 +277,10 @@ is well understood. Namely, Fernando himself established\dots
   cuspidal \(\mathfrak{p}\)-module and \(N\) is a simple
   cuspidal \(\mathfrak{p}'\)-module then \(L_{\mathfrak{p}}(M) \cong
   L_{\mathfrak{p}'}(N)\) if, and only if \(\mathfrak{p}' =
-  \mathfrak{p}^\sigma\) and \(M \cong \sigma N\) for some\footnote{Here
+  \mathfrak{p}^\sigma\) and \(M \cong N^\sigma\) for some\footnote{Here
   $\mathfrak{p}^\sigma$ denotes the image of $\mathfrak{p}$ under the
   automorphism of $\sigma : \mathfrak{g} \to \mathfrak{g}$ given by the
-  canonical action of $W$ on $\mathfrak{g}$ and $\sigma N$ is the
+  canonical action of $W$ on $\mathfrak{g}$ and $N^\sigma$ is the
   $\mathfrak{p}$-module given by composing the map $\mathfrak{p}' \to
   \mathfrak{gl}(N)$ with the restriction $\sigma\!\restriction_{\mathfrak{p}} :
   \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in W_M\), where
@@ -289,7 +289,7 @@ is well understood. Namely, Fernando himself established\dots
     = \langle
       \sigma_\beta : \beta \in \Sigma, H_\beta + \mathfrak{nil}(\mathfrak{p})
       \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}
-      \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ M
+      \ \text{and}\ H_\beta\ \text{acts on \(M\) as a positive integer}
       \rangle
     \subset W
   \]
@@ -366,14 +366,14 @@ automorphisms of \(\operatorname{Diff}(K[x, x^{-1}])\). For example, given
   \frac{\mathrm{d}}{\mathrm{d} x} & \mapsto \frac{\mathrm{d}}{\mathrm{d} x} +
   \frac{\lambda}{2} x^{-1}
 \end{align*}
-and consider the module \(\varphi_\lambda K[x, x^{-1}] = K[x, x^{-1}]\) where
-some operator \(P \in \operatorname{Diff}(K[x, x^{-1}])\) acts as
-\(\varphi_\lambda(P)\).
+and consider the twisted module \(K[x, x^{-1}]^{\varphi_\lambda} = K[x,
+x^{-1}]\), where some operator \(P \in \operatorname{Diff}(K[x, x^{-1}])\) acts
+as \(\varphi_\lambda(P)\).
 
 By composing the action map \(\operatorname{Diff}(K[x, x^{-1}]) \to
-\operatorname{End}(\varphi_\lambda K[x, x^{-1}])\) with the homomorphism of
+\operatorname{End}(K[x, x^{-1}]^{\varphi_\lambda})\) with the homomorphism of
 algebras \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x,
-x^{-1}])\) we can give \(\varphi_\lambda K[x, x^{-1}]\) the structure of an
+x^{-1}])\) we can give \(K[x, x^{-1}]^{\varphi_\lambda}\) the structure of an
 \(\mathfrak{sl}_2(K)\)-module. Diagrammatically, we have
 \begin{center}
   \begin{tikzcd}
@@ -388,7 +388,7 @@ x^{-1}])\) and \(\operatorname{Diff}(K[x, x^{1}]) \to \operatorname{End}(K[x,
 x^{-1}])\) are the ones from the previous diagram.
 
 Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) on
-\(\varphi_\lambda K[x, x^{-1}]\) is given by
+\(K[x, x^{-1}]^{\varphi_\lambda}\) is given by
 \begin{align*}
   p & \overset{e}{\mapsto}
   \left(
@@ -401,25 +401,25 @@ Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) on
   p & \overset{h}{\mapsto}
   \left( 2 x \frac{\mathrm{d}}{\mathrm{d}x} + \lambda \right) p,
 \end{align*}
-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 =
+so we can see \((K[x, x^{-1}]^{\varphi_\lambda})_{2 k + \frac{\lambda}{2}} = K
+x^k\) for all \(k \in \mathbb{Z}\) and \((K[x, x^{-1}]^{\varphi_\lambda})_\mu =
 0\) 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
-simple. 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 simple cuspidal
-\(\mathfrak{sl}_2(K)\)-modules, since their supports differ. These cuspidal
-modules can be ``glued together'' in a \emph{monstrous concoction} by
-summing over \(\lambda \in K\), as in
+Hence \(K[x, x^{-1}]^{\varphi_\lambda}\) is a degree \(1\) admissible
+\(\mathfrak{sl}_2(K)\)-module with \(\operatorname{supp} K[x,
+x^{-1}]^{\varphi_\lambda} = \frac{\lambda}{2} + 2 \mathbb{Z}\). One can also
+quickly check that if \(\lambda \notin 1 + 2 \mathbb{Z}\) then \(e\) and \(f\)
+act injectively in \(K[x, x^{-1}]^{\varphi_\lambda}\), so that \(K[x,
+x^{-1}]^{\varphi_\lambda}\) is simple. In particular, if \(\lambda, \mu \notin
+1 + 2 \mathbb{Z}\) with \(\lambda \notin \mu + 2 \mathbb{Z}\) then \(K[x,
+x^{-1}]^{\varphi_\lambda}\) and \(K[x, x^{-1}]^{\varphi_\mu}\) are
+non-isomorphic simple cuspidal \(\mathfrak{sl}_2(K)\)-modules, since their
+supports differ. These cuspidal modules can be ``glued together'' in a
+\emph{monstrous concoction} by summing over \(\lambda \in K\), as in
 \[
   \mathcal{M}
   = \bigoplus_{\lambda + 2 \mathbb{Z} \in \mfrac{K}{2 \mathbb{Z}}}
-    \varphi_\lambda K[x, x^{-1}],
+    K[x, x^{-1}]^{\varphi_\lambda},
 \]
 
 To a distracted spectator, \(\mathcal{M}\) may look like just another,
@@ -467,7 +467,7 @@ families}.
 
 \begin{example}\label{ex:sl-laurent-family}
   The module \(\mathcal{M} = \bigoplus_{\lambda + 2 \mathbb{Z} \in
-  \mfrac{K}{2 \mathbb{Z}}} \varphi_\lambda K[x, x^{-1}]\) is a degree \(1\)
+  \mfrac{K}{2 \mathbb{Z}}} K[x, x^{-1}]^{\varphi_\lambda}\) is a degree \(1\)
   coherent \(\mathfrak{sl}_2(K)\)-family.
 \end{example}
 
@@ -1161,7 +1161,7 @@ In general, we may twist the \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
                  u & \mapsto F_\beta u F_\beta^{-1}
 \end{align*}
 is a natural candidate for such a twisting automorphism. Indeed, we will soon
-see that \((\theta_\beta \Sigma^{-1} M)_\lambda = \Sigma^{-1} M_{\lambda +
+see that \((\Sigma^{-1} M^{\theta_\beta})_\lambda = \Sigma^{-1} M_{\lambda +
 \beta}\). However, this is hardly useful to us, since \(\beta \in Q\) and
 therefore \(\beta + \operatorname{supp} \Sigma^{-1} M = \operatorname{supp}
 \Sigma^{-1} M\). If we want to expand the support of \(\Sigma^{-1} M\) we will
@@ -1193,10 +1193,10 @@ Explicitly\dots
     \item If \(\lambda, \mu \in \mathfrak{h}^*\), \(N\) is a \(\Sigma^{-1}
       \mathcal{U}(\mathfrak{g})\)-module whose restriction to
       \(\mathcal{U}(\mathfrak{g})\) is a weight \(\mathfrak{g}\)-module and
-      \(\theta_\lambda N\) is the \(\Sigma^{-1}
+      \(N^{\theta_\lambda}\) is the \(\Sigma^{-1}
       \mathcal{U}(\mathfrak{g})\)-module \(N\) twisted by the automorphism
-      \(\theta_\lambda\) then \(N_\mu = (\theta_\lambda N)_{\mu + \lambda}\).
-      In particular, \(\operatorname{supp} \theta_\lambda N = \lambda +
+      \(\theta_\lambda\) then \(N_\mu = (N^{\theta_\lambda})_{\mu + \lambda}\).
+      In particular, \(\operatorname{supp} N^{\theta_\lambda} = \lambda +
       \operatorname{supp} N\).
   \end{enumerate}
 \end{proposition}
@@ -1264,7 +1264,7 @@ Explicitly\dots
   Finally, let \(N\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
   whose restriction is a weight module. If \(n \in N\) then
   \[
-    n \in (\theta_\lambda N)_{\mu + \lambda}
+    n \in (N^{\theta_\lambda})_{\mu + \lambda}
     \iff \theta_\lambda(H) \cdot n = (\mu + \lambda)(H) n
     \, \forall H \in \mathfrak{h}
   \]
@@ -1282,14 +1282,14 @@ Explicitly\dots
   and hence
   \[
     \begin{split}
-      n \in (\theta_\lambda N)_{\mu + \lambda}
+      n \in (N^{\theta_\lambda})_{\mu + \lambda}
       & \iff (\lambda(H) + H) \cdot n = (\mu + \lambda)(H) n
         \; \forall H \in \mathfrak{h} \\
       & \iff H \cdot n = \mu(H) n \; \forall H \in \mathfrak{h} \\
       & \iff n \in N_\mu
     \end{split},
   \]
-  so that \((\theta_\lambda N)_{\mu + \lambda} = N_\mu\).
+  so that \((N^{\theta_\lambda})_{\mu + \lambda} = N_\mu\).
 \end{proof}
 
 It should now be obvious\dots
@@ -1301,22 +1301,23 @@ It should now be obvious\dots
 \begin{proof}
   Take\footnote{Here we fix some $\lambda_\xi \in \xi$ for each $Q$-coset $\xi
   \in \mfrac{\mathfrak{h}^*}{Q}$. While there is a natural isomorphism
-  $\theta_\lambda \Sigma^{-1} M \isoto \theta_\mu \Sigma^{-1} M$ for each $\mu
-  \in \lambda + Q$, they are not the same \(\mathfrak{g}\)-modules strictly
-  speaking. This is yet another obstruction to the functoriality of our
-  constructions.}
+  $\Sigma^{-1} M^{\theta_\lambda} \isoto \Sigma^{-1} M^{\theta_\mu}$ for each
+  $\mu \in \lambda + Q$, they are not the same \(\mathfrak{g}\)-modules
+  strictly speaking. This is yet another obstruction to the functoriality of
+  our constructions.}
   \[
     \mathcal{M}
     = \bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}
-      \theta_\lambda \Sigma^{-1} M
+      \Sigma^{-1} M^{\theta_\lambda}
   \]
 
-  It is clear \(M\) lies in \(\Sigma^{-1} M = \theta_0 \Sigma^{-1} M\) and
+  It is clear \(M\) lies in \(\Sigma^{-1} M = \Sigma^{-1} M^{\theta_0}\) and
   therefore \(M \subset \mathcal{M}\). On the other hand, \(\dim
-  \mathcal{M}_\mu = \dim \theta_\lambda \Sigma^{-1} M_\mu = \dim \Sigma^{-1}
-  M_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\) -- \(\lambda\)
-  standing for some fixed representative of its \(Q\)-coset. Furthermore, given
-  \(u \in \mathcal{U}(\mathfrak{g})_0\) and \(\mu \in \lambda + Q\),
+  \mathcal{M}_\mu = \dim (\Sigma^{-1} M^{\theta_\lambda})_\mu = \dim
+  \Sigma^{-1} M_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\) --
+  \(\lambda\) standing for some fixed representative of its \(Q\)-coset.
+  Furthermore, given \(u \in \mathcal{U}(\mathfrak{g})_0\) and \(\mu \in
+  \lambda + Q\),
   \[
     \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
     = \operatorname{Tr}