lie-algebras-and-their-representations

diff --git a/preamble.tex b/preamble.tex
@@ -174,6 +174,9 @@
 % Macro for Mathieu's coherent extension
 \newcommand{\mExt}{\mathcal{E\!x\!t}}
 
+% Notation for twisted modules
+\newcommand{\twisted}[2]{{}^{#2}\!{#1}}
+
 % Isomorphism arrow
 \newcommand{\isoto}{\xlongrightarrow{\sim}}
 

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -288,16 +288,17 @@ is well understood. Namely, Fernando himself established\dots
   \Delta_{\mathfrak{p}} \subset \Delta\), where \(\Delta_{\mathfrak{p}}\)
   denotes the set of roots of \(\mathfrak{p}\). Furthermore, if \(\mathfrak{p}'
   \subset \mathfrak{g}\) is another parabolic subalgebra, \(M\) is a simple
-  cuspidal \(\mathfrak{p}\)-module and \(N\) is a simple
-  cuspidal \(\mathfrak{p}'\)-module then \(L_{\mathfrak{p}}(M) \cong
+  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 \(M \cong N^\sigma\) 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 $N^\sigma$ 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
+  \twisted{\mathfrak{p}}{\sigma}\) and \(M \cong \twisted{N}{\sigma}\) for
+  some\footnote{Here $\twisted{\mathfrak{p}}{\sigma}$ 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
+  $\twisted{N}{\sigma}$ 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
   \[
     W_M
     = \langle
@@ -380,15 +381,16 @@ 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 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 \(\varphi_\lambda(P)\).
+and consider the twisted module \(\twisted{K[x, x^{-1}]}{\varphi_\lambda} =
+K[x, x^{-1}]\), where some operator \(P \in \operatorname{Diff}(K[x, x^{-1}])\)
+acts as \(\varphi_\lambda(P)\).
 
 By composing the action map \(\operatorname{Diff}(K[x, x^{-1}]) \to
-\operatorname{End}(K[x, x^{-1}]^{\varphi_\lambda})\) with the homomorphism of
-algebras \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x,
-x^{-1}])\) we can give \(K[x, x^{-1}]^{\varphi_\lambda}\) the structure of an
-\(\mathfrak{sl}_2(K)\)-module. Diagrammatically, we have
+\operatorname{End}(\twisted{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 \(\twisted{K[x,
+x^{-1}]}{\varphi_\lambda}\) the structure of an \(\mathfrak{sl}_2(K)\)-module.
+Diagrammatically, we have
 \begin{center}
   \begin{tikzcd}
     \mathcal{U}(\mathfrak{sl}_2(K))   \rar                  &
@@ -402,7 +404,7 @@ 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
-\(K[x, x^{-1}]^{\varphi_\lambda}\) is given by
+\(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) is given by
 \begin{align*}
   p & \overset{e}{\mapsto}
   \left(
@@ -415,25 +417,26 @@ 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 \((K[x, x^{-1}]^{\varphi_\lambda})_{2 k + \frac{\lambda}{2}} = K
-x^k\) for all \(k \in \mathbb{Z}\) and \((K[x, x^{-1}]^{\varphi_\lambda})_\mu =
-0\) for all other \(\mu \in \mathfrak{h}^*\).
+so we can see \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}_{2 k +
+\frac{\lambda}{2}} = K x^k\) for all \(k \in \mathbb{Z}\) and \(\twisted{K[x,
+x^{-1}]}{\varphi_\lambda}_\mu = 0\) for all other \(\mu \in \mathfrak{h}^*\).
 
-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
+Hence \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) is a degree \(1\) admissible
+\(\mathfrak{sl}_2(K)\)-module with \(\operatorname{supp} \twisted{K[x,
+x^{-1}]}{\varphi_\lambda} = \frac{\lambda}{2} + 2 \mathbb{Z}\). 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
+act injectively in \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\), so that
+\(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) is simple. In particular, if
+\(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with \(\lambda \notin \mu + 2
+\mathbb{Z}\) then \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) and
+\(\twisted{K[x, x^{-1}]}{\varphi_\mu}\) 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
 \[
   \mathcal{M}
   = \bigoplus_{\lambda + 2 \mathbb{Z} \in \mfrac{K}{2 \mathbb{Z}}}
-    K[x, x^{-1}]^{\varphi_\lambda},
+    \twisted{K[x, x^{-1}]}{\varphi_\lambda},
 \]
 
 To a distracted spectator, \(\mathcal{M}\) may look like just another,
@@ -481,8 +484,8 @@ families}.
 
 \begin{example}\label{ex:sl-laurent-family}
   The module \(\mathcal{M} = \bigoplus_{\lambda + 2 \mathbb{Z} \in
-  \mfrac{K}{2 \mathbb{Z}}} K[x, x^{-1}]^{\varphi_\lambda}\) is a degree \(1\)
-  coherent \(\mathfrak{sl}_2(K)\)-family.
+  \mfrac{K}{2 \mathbb{Z}}} \twisted{K[x, x^{-1}]}{\varphi_\lambda}\) is a
+  degree \(1\) coherent \(\mathfrak{sl}_2(K)\)-family.
 \end{example}
 
 \begin{example}
@@ -1175,12 +1178,12 @@ 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 \((\Sigma^{-1} M^{\theta_\beta})_\lambda = \Sigma^{-1} M_{\lambda +
-\beta}\). 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
-have to twist by automorphisms that shift its support by \(\lambda \in
-\mathfrak{h}^*\) lying \emph{outside} of \(Q\).
+see that \(\twisted{(\Sigma^{-1} M)}{\theta_\beta}_\lambda = \Sigma^{-1}
+M_{\lambda + \beta}\). 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 have to twist by automorphisms that shift its support
+by \(\lambda \in \mathfrak{h}^*\) lying \emph{outside} of \(Q\).
 
 The situation is much less obvious in this case. Nevertheless, it turns out we
 can extend the family \(\{\theta_\beta\}_{\beta \in \Sigma}\) to a family of
@@ -1207,11 +1210,11 @@ 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
-      \(N^{\theta_\lambda}\) is the \(\Sigma^{-1}
+      \(\twisted{N}{\theta_\lambda}\) is the \(\Sigma^{-1}
       \mathcal{U}(\mathfrak{g})\)-module \(N\) twisted by the automorphism
-      \(\theta_\lambda\) then \(N_\mu = (N^{\theta_\lambda})_{\mu + \lambda}\).
-      In particular, \(\operatorname{supp} N^{\theta_\lambda} = \lambda +
-      \operatorname{supp} N\).
+      \(\theta_\lambda\) then \(N_\mu = \twisted{N}{\theta_\lambda}_{\mu +
+      \lambda}\). In particular, \(\operatorname{supp}
+      \twisted{N}{\theta_\lambda} = \lambda + \operatorname{supp} N\).
   \end{enumerate}
 \end{proposition}
 
@@ -1278,7 +1281,7 @@ 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
   \[
-    n \in (N^{\theta_\lambda})_{\mu + \lambda}
+    n \in \twisted{N}{\theta_\lambda}_{\mu + \lambda}
     \iff \theta_\lambda(H) \cdot n = (\mu + \lambda)(H) n
     \, \forall H \in \mathfrak{h}
   \]
@@ -1296,14 +1299,14 @@ Explicitly\dots
   and hence
   \[
     \begin{split}
-      n \in (N^{\theta_\lambda})_{\mu + \lambda}
+      n \in \twisted{N}{\theta_\lambda}_{\mu + \lambda}
       & \iff (\lambda(H) + H) \cdot n = (\mu + \lambda)(H) n
         \; \forall H \in \mathfrak{h} \\
       & \iff H \cdot n = \mu(H) n \; \forall H \in \mathfrak{h} \\
       & \iff n \in N_\mu
     \end{split},
   \]
-  so that \((N^{\theta_\lambda})_{\mu + \lambda} = N_\mu\).
+  so that \(\twisted{N}{\theta_\lambda}_{\mu + \lambda} = N_\mu\).
 \end{proof}
 
 It should now be obvious\dots
@@ -1315,20 +1318,20 @@ 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
-  $\Sigma^{-1} M^{\theta_\lambda} \isoto \Sigma^{-1} M^{\theta_\mu}$ 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.}
+  $\twisted{(\Sigma^{-1} M)}{\theta_\lambda} \isoto \twisted{(\Sigma^{-1}
+  M)}{\theta_\mu}$ 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.}
   \[
     \mathcal{M}
     = \bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}
-      \Sigma^{-1} M^{\theta_\lambda}
+      \twisted{(\Sigma^{-1} M)}{\theta_\lambda}
   \]
 
-  It is clear \(M\) lies in \(\Sigma^{-1} M = \Sigma^{-1} M^{\theta_0}\) and
-  therefore \(M \subset \mathcal{M}\). On the other hand, \(\dim
-  \mathcal{M}_\mu = \dim (\Sigma^{-1} M^{\theta_\lambda})_\mu = \dim
-  \Sigma^{-1} M_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\) --
+  It is clear \(M\) lies in \(\Sigma^{-1} M = \twisted{(\Sigma^{-1}
+  M)}{\theta_0}\) and therefore \(M \subset \mathcal{M}\). On the other hand,
+  \(\dim \mathcal{M}_\mu = \dim \twisted{(\Sigma^{-1} M)}{\theta_\lambda}_\mu =
+  \dim \Sigma^{-1} M_{\mu - \lambda} = d\) 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 \(\mu \in
   \lambda + Q\),