lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -343,8 +343,8 @@
 \begin{lemma}
   Let \(S \subset R\) be a multiplicative subset generated by locally
   \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\) such
-  that for each \(r \in R\) there \([s, [s, \cdots [s, r]]\cdots] = 0\) for
-  sufficiently many applications of the commutator. Then \(S\) satisfies Ore's
+  that for each \(r \in R\) there \(\operatorname{ad}(s)^n r = [s, [s, \cdots
+  [s, r]]\cdots] = 0\) for sufficiently large \(n\). Then \(S\) satisfies Ore's
   localization condition.
 \end{lemma}
 
@@ -470,15 +470,110 @@
       \end{align*}
       is polynomial.
 
-    \item If \(\lambda, \mu \in \mathfrak{h}^*\) and \(\theta_\lambda
-      \Sigma^{-1} V\) 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, \(\operatorname{supp} \theta_\lambda \Sigma^{-1} V =
-      \lambda + Q + \operatorname{supp} V\).
+    \item If \(\lambda, \mu \in \mathfrak{h}^*\), \(M\) is a \(\Sigma^{-1}
+      \mathcal{U}(\mathfrak{g})\)-module whose restriction to
+      \(\mathcal{U}(\mathfrak{g})\) is a weight \(\mathfrak{g}\)-module and
+      \(\theta_\lambda M\) is the \(\Sigma^{-1}
+      \mathcal{U}(\mathfrak{g})\)-module \(M\) twisted by the automorphism
+      \(\theta_\lambda\) then \(M_\mu = (\theta_\lambda M)_{\mu + \lambda}\).
+      In particular, \(\operatorname{supp} \theta_\lambda M = \lambda +
+      \operatorname{supp} M\).
   \end{enumerate}
 \end{proposition}
 
+\begin{proof}
+  Since the elements \(F_\beta\), \(\beta \in \Sigma\) commute with one
+  another, the endomorphisms
+  \begin{align*}
+    \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}
+    : \Sigma^{-1} \mathcal{U}(\mathfrak{g}) &
+    \to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
+    r & \mapsto
+    F_{\beta_1}^{k_1} \cdots F_{\beta_n}^{k_n}
+    r
+    F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{- k_1}
+  \end{align*}
+  are well defined for all \(k_1, \ldots, k_n \in \mathbb{Z}\).
+
+  Fix some \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\). For any \(s \in
+  (F_\beta)_{\beta \in \Sigma}\) and \(k > 0\) we have \(s^k r = \binom{k}{0}
+  \operatorname{ad}(s)^0 r s^{k - 0} + \cdots + \binom{k}{k}
+  \operatorname{ad}(s)^k r s^{k - k}\). Now if we take \(m\) such
+  \(\operatorname{ad}(F_\beta)^{m + 1} r = 0\) for all \(\beta \in \Sigma\) we
+  find
+  \[
+    \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r)
+    = \sum_{i_1, \ldots, i_n = 1, \ldots m}
+    \binom{k_1}{i_1} \cdots \binom{k_n}{i_n}
+    \operatorname{ad}(F_{\beta_1})^{i_1} \cdots
+    \operatorname{ad}(F_{\beta_n})^{i_n}
+    r
+    F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
+  \]
+  for all \(k_1, \ldots, k_n \in \NN\).
+
+  Since the binomial coeffients \(\binom{x}{k} = \frac{x (x -1) \cdots (x - k +
+  1)}{k!}\) can be uniquely extended to polynomial functions in \(x\), we may
+  in general define
+  \[
+    \theta_\lambda(r)
+    = \sum_{i_1, \ldots, i_n \ge 0}
+    \binom{\lambda_1}{i_1} \cdots \binom{\lambda_n}{i_n}
+    \operatorname{ad}(F_{\beta_1})^{i_1} \cdots
+    \operatorname{ad}(F_{\beta_n})^{i_n}
+    r
+    F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
+  \]
+  for \(\lambda_1, \ldots, \lambda_n \in K\), \(\lambda = \lambda_1 \beta_1 +
+  \cdots + \lambda_n \beta_n \in \mathfrak{h}^*\)
+
+  It is clear that the \(\theta_\lambda\) are endmorphisms. To see that the
+  \(\theta_\lambda\) are indeed automorphisms, notice \(\theta_{- k_1 \beta_1 -
+  \cdots - k_n \beta_n} = \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}^{-1}\).
+  The uniqueness of the polynomial extensions then implies \(\theta_{- \lambda}
+  = \theta_\lambda^{-1}\) in general: given \(r \in \Sigma^{-1}
+  \mathcal{U}(\mathfrak{g})\) the map
+  \begin{align*}
+    \mathfrak{h}^* & \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})        \\
+           \lambda & \mapsto \theta_\lambda(\theta_{-\lambda}(r)) - r
+  \end{align*}
+  is a polynomial extension of the zero map \(\ZZ \beta_1 \oplus \cdots \oplus
+  \ZZ \beta_n \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is therefore
+  identicaly zero.
+
+  Finally, let \(M\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
+  whose restriction is weight module. If \(m \in M\) then
+  \[
+    m \in (\theta_\lambda M)_{\mu + \lambda}
+    \iff \theta_\lambda(H) m = (\mu + \lambda)(H) \cdot m
+    \, \forall H \in \mathfrak{h}
+  \]
+
+  But
+  \[
+    \begin{split}
+      \theta_\beta(H)
+      & = F_\beta H F_\beta^{-1} \\
+      & = ([F_\beta, H] + H F_\beta) F_\beta^{-1} \\
+      & = (\beta(H) + H) F_\beta F_\beta^{-1} \\
+      & = \beta(H) + H
+    \end{split}
+  \]
+  for all \(H \in \mathfrak{h}\) and \(\beta \in \Sigma\). In general,
+  \(\theta_\lambda(H) = \lambda(H) + H\) for all \(\lambda \in \mathfrak{h}^*\)
+  and hence
+  \[
+    \begin{split}
+      m \in (\theta_\lambda M)_{\mu + \lambda}
+      & \iff (\lambda(H) + H) m = (\mu + \lambda)(H) \cdot m
+        \; \forall H \in \mathfrak{h} \\
+      & \iff H m = \mu(H) \cdot m \; \forall H \in \mathfrak{h} \\
+      & \iff m \in M_\mu
+    \end{split},
+  \]
+  so that \((\theta_\lambda M)_{\mu + \lambda} = M_\mu\).
+\end{proof}
+
 \begin{proposition}[Mathieu]
   There exists a coherent extension \(\mathcal{M}\) of \(V\).
 \end{proposition}