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}