diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -186,8 +186,10 @@
% comes from the relationship between highest weight modules and coherent
% families
+% TODOO: I think this is equivalent to M being a simple object in the
+% category of coherent families. This is perhaps a more intuitive formulation
\begin{definition}
- A coherent family \(\mathcal{M}\) called \emph{irreducible} if
+ A coherent family \(\mathcal{M}\) is called \emph{irreducible} if
\(\mathcal{M}_\lambda\) is a simple
\(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
\mathfrak{h}^*\).
@@ -277,8 +279,7 @@
= d
\]
- Likewise, given \(u \in \mathcal{U}(\mathfrak{g})_0\) the
- number
+ Likewise, given \(u \in \mathcal{U}(\mathfrak{g})_0\) the value
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}})
= \sum_j
@@ -382,8 +383,8 @@
\operatorname{End}(\mathcal{M}_\lambda)) =
\operatorname{rank}(\operatorname{End}(\mathcal{M}_\lambda)^* \to
\mathcal{U}(\mathfrak{g})_0^*)\), imply that \(\operatorname{rank} B_\lambda
- = d^2\). Hence \(U\) is precisely the set of \(\lambda\) such that
- \(B_\lambda\) has maximal rank \(d^2\). We now show that \(U\) is
+ = d^2\). This goes to show that \(U\) is precisely the set of \(\lambda\)
+ such that \(B_\lambda\) has maximal rank \(d^2\). We now show that \(U\) is
Zariski-open. First, notice that
\[
U =
@@ -517,8 +518,8 @@
\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 \(\operatorname{ad}(s)^n r = [s, [s, \cdots
- [s, r]]\cdots] = 0\) for sufficiently large \(n\). Then \(S\) satisfies Ore's
+ that for each \(r \in R\) there exists \(n > 0\) such that \(\operatorname{ad}(s)^n r = [s, [s, \cdots
+ [s, r]]\cdots] = 0\). Then \(S\) satisfies Ore's
localization condition.
\end{lemma}
@@ -548,6 +549,7 @@
% localization map
% TODO: Point out that each element of the localization has the form s^-1 r
+% TODO: Point out that Sigma depends on V!
\begin{lemma}\label{thm:nice-basis-for-inversion}
Let \(V\) be an irreducible infinite-dimensional admissible
\(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
@@ -574,11 +576,11 @@
\end{proposition}
\begin{proof}
- 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
- \(\Sigma^{-1} V_{\lambda - \beta}\) and \(\Sigma^{-1} V_{\lambda + \beta}\)
- respectively. Indeed, given \(v \in V_\lambda\) and \(H \in \mathfrak{h}\) we
- have
+ Fix some \(\beta \in \Sigma\). We begin by showing that \(F_\beta\) and
+ \(F_\beta^{-1}\) map the weight space \(\Sigma^{-1} V_\lambda\) to the weight
+ spaces \(\Sigma^{-1} V_{\lambda - \beta}\) and \(\Sigma^{-1} V_{\lambda +
+ \beta}\) respectively. Indeed, given \(v \in V_\lambda\) and \(H \in
+ \mathfrak{h}\) we have
\[
H F_\beta v
= ([H, F_\beta] + F_\beta H)v
@@ -600,7 +602,7 @@
H F_\beta^{-1} v
= ([H, F_\beta^{-1}] + F_\beta^{-1} H) v
= F_\beta^{-1} (\beta(H) + H) v
- = F_\beta^{-1} (\lambda + \beta)(H) \cdot v
+ = (\lambda + \beta)(H) \cdot F_\beta^{-1} v
\]
From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(\Sigma^{-1}
@@ -620,12 +622,11 @@
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 \(\dim
- \Sigma^{-1}_\lambda = d\).
+ V_{\lambda - \beta_{i_1} - \cdots - \beta_{i_n}}\), so \(\dim W = \dim sW \le
+ d\). 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 \(\dim \Sigma^{-1} V_\lambda = d\).
\end{proof}
\begin{proposition}\label{thm:nice-automorphisms-exist}
@@ -688,8 +689,8 @@
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
+ 1)}{k!}\) can be uniquely extended to polynomial functions in \(x \in K\), we
+ may in general define
\[
\theta_\lambda(r)
= \sum_{i_1, \ldots, i_n \ge 0}
@@ -707,7 +708,7 @@
\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
+ \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
@@ -717,7 +718,7 @@
identicaly zero.
Finally, let \(M\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
- whose restriction is weight module. If \(m \in M\) then
+ whose restriction is a weight module. If \(m \in M\) then
\[
m \in (\theta_\lambda M)_{\mu + \lambda}
\iff \theta_\lambda(H) m = (\mu + \lambda)(H) \cdot m