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}