lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -268,6 +268,12 @@
 
 \section{Existance of Coherent Extensions}
 
+% TODO: Comment on the intuition behind the proof: we can get vectors in a
+% given eigenspace by translating by the F's and E's, but neither of those are
+% injective in general, so the translation could take nonzero vectors to zero.
+% If the F's were invertible this problem wouldn't exist, so we might as well
+% invert them by force!
+
 % TODO: Add the results on Ore's localization
 
 % TODO: Define what a set commuting roots is
@@ -281,43 +287,33 @@
 \begin{corollary}
   Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
   \((F_\beta : \beta \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
-  the multiplicative subset generated by \(F_\beta\), \(\beta \in \Sigma\).
-  The \(K\)-algebra \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
-  \Sigma)}\) is well defined and the localization map
+  the multiplicative subset \((F_\beta)_{\beta \in \Sigma}\) generated by the
+  \(F_\beta\)'s.
+  The \(K\)-algebra \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta
+  \in \Sigma}^{-1} \mathcal{U}(\mathfrak{g})\) is well defined. Moreover, if we
+  denote by \(\Sigma^{-1} V\) the localization of \(V\) by \((F_\beta)_{\beta
+  \in \Sigma}\), the localization map
   \begin{align*}
-    V &
-    \to \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
-    \otimes V \\
-    u & \mapsto 1 \otimes u
+    V & \to     \Sigma^{-1} V \\
+    v & \mapsto 1 \otimes v
   \end{align*}
   is injective.
 \end{corollary}
 
+% TODO: Fix V and Sigma beforehand
 \begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
-  Let \(V\) be an irreducible infinite-dimensional admissible
-  \(\mathfrak{g}\)-module. Then \(V\) is contained in a
-  \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\)-module \(W\),
-  whose restriction to the scalars in \(\mathcal{U}(\mathfrak{g})\) is a weight
-  module of degree \(d\) such that \(\operatorname{supp} W = Q +
-  \operatorname{supp} V\) and \(\dim W_\lambda = d\) for all \(\lambda \in
-  \operatorname{supp} W\).
+  The the restriction of the localization \(\Sigma^{-1} V\) is a weight
+  \(\mathfrak{g}\)-module of degree \(d\) with \(\operatorname{supp}
+  \Sigma^{-1} V = Q + \operatorname{supp} V\) and \(\dim \Sigma^{-1} V_\lambda
+  = d\) for all \(\lambda \in \operatorname{supp} \Sigma^{-1} V\).
 \end{proposition}
 
 \begin{proof}
-  Take \(W = \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
-  \otimes_{\mathcal{U}(\mathfrak{g})} V\). Since each \(F_\beta\) acts
-  injectively in \(V\), the localization map
-  \begin{align*}
-    V & \to     W           \\
-    v & \mapsto 1 \otimes v
-  \end{align*}
-  is injective. In particular, we may regard \(V\) as a
-  \(\mathfrak{g}\)-submodule of \(W\).
-
   Fix some \(\beta \in \Sigma\). We begin by show that \(F_\beta\) and
   \(F_\beta^{-1}\) map the weight space \(V_\lambda\) to the weight spaces
-  \(W_{\lambda - \beta}\) and \(W_{\lambda + \beta}\) respectively. Indeed,
-  given \(v \in V_\lambda\) and \(H \in \mathfrak{h}\) we have
+  \(\Sigma^{-1} V_{\lambda - \beta}\) and \(\Sigma^{-1} V_{\lambda + \beta}\)
+  respectively. Indeed, given \(v \in V_\lambda\) and \(H \in \mathfrak{h}\) we
+  have
   \[
     H F_\beta v
     = ([H, F_\beta] + F_\beta H)v
@@ -344,99 +340,90 @@
 
   % TODO: Remark beforehand that any element of the localization of V may be
   % written as an element of v tensored by an element of the form 1/s
-  From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(W_{\lambda \pm
-  \beta}\) follows our first conclusion: since \(V\) is a weight module and
-  every element of \(W\) has the form \(s^{-1} v = s^{-1} \otimes v\) for \(s
-  \in (F_\beta : \beta \in \Sigma)\) and \(v \in V\), we can see that \(W =
-  \bigoplus_\lambda W_\lambda\). Furtheremore, since the action of each
-  \(F_\beta\) in \(W\) is bijective and \(\Sigma\) is a basis of \(Q\) we
-  obtain \(\operatorname{supp} W = Q + \operatorname{supp} V\). 
-
-  % TODOOOOOO: Change the notation for the subspace U
+  From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(\Sigma^{-1}
+  V_{\lambda \pm \beta}\) follows our first conclusion: since \(V\) is a weight
+  module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
+  s^{-1} \otimes v\) for \(s \in (F_\beta)_{\beta \in \Sigma}\) and \(v \in
+  V\), we can see that \(\Sigma^{-1} V = \bigoplus_\lambda \Sigma^{-1}
+  V_\lambda\). Furtheremore, since the action of each \(F_\beta\) in
+  \(\Sigma^{-1} V\) is bijective and \(\Sigma\) is a basis of \(Q\) we obtain
+  \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\). 
+
   Again, because of the bijectivity of the \(F_\beta\)'s, to see that \(\dim
-  W_\lambda = d\) for all \(\lambda \in \mathfrak{h}^*\) it suffices to show
-  that \(\dim W_\lambda = d\) for some \(\lambda \in \operatorname{supp} W\).
-  We may take \(\lambda \operatorname{supp} V\) with \(\dim V_\lambda = d\).
-  For any finite-dimensional subspace \(U \subset W_\lambda\) we can find \(s
-  \in (F_\beta)_{\beta \in \Sigma}\) such that \(s U \subset V\). If \(s =
-  F_{\beta_{i_1}} \cdots F_{\beta_{i_n}}\), it is clear \(s U \subset
-  V_{\lambda - \beta_{i_1} - \cdots - \beta_{i_n}}\), so \(\dim U \le d\) --
-  \(s\) is injective. This holds for all finite-dimensional \(U \subset
-  W_\lambda\), so \(\dim W_\lambda \le d\). It then follows from the fact that
-  \(V_\lambda \subset W_\lambda\) that \(\dim W_\lambda = d\).
+  \Sigma^{-1} V_\lambda = d\) for all \(\lambda \in \operatorname{supp}
+  \Sigma^{-1} V\) it suffices to show that \(\dim \Sigma^{-1} V_\lambda = d\)
+  for some \(\lambda \in \operatorname{supp} \Sigma^{-1} V\). We may take
+  \(\lambda \in \operatorname{supp} V\) with \(\dim V_\lambda = d\). For any
+  finite-dimensional subspace \(W \subset \Sigma^{-1} V_\lambda\) we can find
+  \(s \in (F_\beta)_{\beta \in \Sigma}\) such that \(s W \subset V\). If \(s =
+  F_{\beta_{i_1}} \cdots F_{\beta_{i_n}}\), it is clear \(s W \subset
+  V_{\lambda - \beta_{i_1} - \cdots - \beta_{i_n}}\), so \(\dim W \le d\) --
+  \(s\) is injective. This holds for all finite-dimensional \(W \subset
+  \Sigma^{-1} V_\lambda\), so \(\dim \Sigma^{-1} V_\lambda \le d\). It then
+  follows from the fact that \(V_\lambda \subset \Sigma^{-1} V_\lambda\) that
+  \(V_\lambda = \Sigma^{-1} V_\lambda\) and therefore \(\dim
+  \Sigma^{-1}_\lambda = d\).
 \end{proof}
 
 % TODO: Remark that any module over the localization is a g-module if we
 % restrict it via the localization map, wich is injective in this case
 \begin{proposition}\label{thm:nice-automorphisms-exist}
-  There is a family of automorphisms \(\{\theta_\lambda :
-  \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)} \to
-  \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\}_{\lambda \in
-  \mathfrak{h}^*}\) such that
+  There is a family of automorphisms \(\{\theta_\lambda : \Sigma^{-1}
+  \mathcal{U}(\mathfrak{g}) \to \Sigma^{-1}
+  \mathcal{U}(\mathfrak{g})\}_{\lambda \in \mathfrak{h}^*}\) such that
   \begin{enumerate}
-    \item \(\theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(u) =
-      F_{\beta_1}^{k_1} \cdots F_{\beta_n}^{k_n} u F_{\beta_1}^{- k_n}
-      \cdots F_{\beta_n}^{- k_1}\) for all \(u \in
-      \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\) and \(k_1,
+    \item \(\theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r) = F_{\beta_1}^{k_1}
+      \cdots F_{\beta_n}^{k_n} r F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{-
+      k_1}\) for all \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and \(k_1,
       \ldots, k_n \in \mathbb{Z}\).
 
-    \item For each \(u \in \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
-      \Sigma)}\) the map
+    \item For each \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) the map
       \begin{align*}
-        \mathfrak{h}^* &
-        \to \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)} \\
-        \lambda & \mapsto \theta_\lambda(u)
+        \mathfrak{h}^* & \to     \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
+               \lambda & \mapsto \theta_\lambda(r)
       \end{align*}
       is polynomial.
 
-    \item If \(W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
-      \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
-      \Sigma)} \otimes V\), \(\lambda, \mu \in \mathfrak{h}^*\) and
-      \(\theta_\lambda W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\beta :
-      \beta \in \Sigma)}\)-module \(W\) twisted by the automorphism
-      \(\theta_\lambda\) then \(W_\mu = (\theta_\lambda W)_{\mu +
-      \lambda}\).
+    \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
+      \(\Sigma^{-1} V_\mu = (\theta_\lambda \Sigma^{-1} V)_{\mu + \lambda}\).
   \end{enumerate}
 \end{proposition}
 
 \begin{proposition}[Mathieu]
-  Let \(V\) be an infinite-dimensional admissible irreducible
-  \(\mathfrak{g}\)-module of degree \(d\). There exists a coherent extension
-  \(\mathcal{M}\) of \(V\) with degree \(d\).
+  There exists a coherent extension \(\mathcal{M}\) of \(V\) with degree \(d\).
 \end{proposition}
 
 \begin{proof}
   Let \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
-  \(\mathfrak{h}^*\) with \(0 \in \Lambda\), \(W =
-  \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
-  \otimes_{\mathcal{U}(\mathfrak{g})} V\) be as in
-  proposition~\ref{thm:irr-admissible-is-contained-in-nice-mod} and take
+  \(\mathfrak{h}^*\) with \(0 \in \Lambda\) and take
   \[
     \mathcal{M}
-    = \bigoplus_{\lambda \in \Lambda} \theta_\lambda W
+    = \bigoplus_{\lambda \in \Lambda} \theta_\lambda \Sigma^{-1} V
   \]
 
-  On the one hand, \(V\) lies in \(W = \theta_0 W\) -- notice that
-  \(\theta_0\) is just the identity operator -- and therefore \(V \subset
-  \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
-  \theta_\lambda W_\mu = \dim W_{\mu - \lambda} = d\) for all \(\mu \in
-  \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given \(u \in
-  C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) and \(\mu \in \lambda + Q\),
+  On the one hand, \(V\) lies in \(\Sigma^{-1} V = \theta_0 \Sigma^{-1} V\) --
+  notice that \(\theta_0\) is just the identity operator -- and therefore \(V
+  \subset \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
+  \theta_\lambda \Sigma^{-1} V_\mu = \dim \Sigma^{-1} V_{\mu - \lambda} = d\)
+  for all \(\mu \in \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given
+  \(u \in C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) and \(\mu \in \lambda +
+  Q\),
   \[
     \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
-    = \operatorname{Tr}(\theta_\lambda(u)\!\restriction_{W_{\mu - \lambda}})
+    = \operatorname{Tr}
+      (\theta_\lambda(u)\!\restriction_{\Sigma^{-1} V_{\mu - \lambda}})
   \]
   is polynomial in \(\mu\) because of the second item of
   proposition~\ref{thm:nice-automorphisms-exist}.
 \end{proof}
 
 \begin{theorem}[Mathieu]
-  Let \(V\) be an infinite-dimensional admissible irreducible
-  \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
-  coherent extension \(\operatorname{Ext}(V)\) of \(V\). More precisely, if
-  \(\mathcal{M}\) is any coherent extension of \(V\), then
-  \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\). Furthermore,
-  \(V\) is itself contained in \(\operatorname{Ext}(V)\) and
+  There exists a unique semisimple coherent extension \(\operatorname{Ext}(V)\)
+  of \(V\). More precisely, if \(\mathcal{M}\) is any coherent extension of
+  \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
+  Furthermore, \(V\) is itself contained in \(\operatorname{Ext}(V)\) and
   \(\operatorname{Ext}(V)\) is irreducible as a coherent family.
 \end{theorem}