diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -103,7 +103,7 @@
\begin{theorem}[Fernando]
Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to
\(L_{\mathfrak{p}}(V)\) for some parabolic subalgebra \(\mathfrak{p} \subset
- \mathfrak{g}\) and some irreducible cuspital \(\mathfrak{p}\)-module \(V\).
+ \mathfrak{g}\) and some irreducible cuspidal \(\mathfrak{p}\)-module \(V\).
Furthermore, if \(\mathfrak{p}_1, \mathfrak{p}_2 \subset \mathfrak{g}\) are
parabolic and \(V_i\) is an irreducible cuspidal \(\mathfrak{p}_i\)-module
then \(L_{\mathfrak{p}_1}(V_1) \cong L_{\mathfrak{p}_2}(V_2)\) if, and only
@@ -206,7 +206,7 @@
We know from examples~\ref{ex:submod-is-weight-mod} and
\ref{ex:quotient-is-weight-mod} that each quotient \(\mfrac{\mathcal{M}_{i j
+ 1}}{\mathcal{M}_{i j}}\) is a weight module. Hence
- \(\mathcal{M}^{\operatorname{ss}}\) is a weight module. Furtheremore, given
+ \(\mathcal{M}^{\operatorname{ss}}\) is a weight module. Furthermore, given
\(\mu \in \lambda_k + Q\)
\[
\mathcal{M}_\mu^{\operatorname{ss}}
@@ -250,7 +250,7 @@
\begin{theorem}[Mathieu]
Let \(V\) be an infinite-dimensional admissible irreducible
\(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
- cohorent extension \(\operatorname{Ext}(V)\) of \(V\). The central characters
+ coherent extension \(\operatorname{Ext}(V)\) of \(V\). The central characters
of the irreducible submodules of \(\operatorname{Ext}(V)\) are all the same.
Furthermore, if \(\mathcal{M}\) is any coherent extension of \(V\), then
\(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
@@ -269,7 +269,7 @@
Fix some coherent extension \(\mathcal{M}\) of \(V\), so that \(V\) is a
subquotient of \(\mathcal{M}\). More precisely, since \(V\) is irreducible it
is a subquotient of \(\mathcal{M}[\lambda]\) -- its support is contained in
- \(\lambda + Q\). Furtheremore, once again it follows from the irreducibility
+ \(\lambda + Q\). Furthermore, once again it follows from the irreducibility
of \(V\) that it can be realized as the quotient of consecutive terms of a
composition series \(0 = \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots
\subset \mathcal{M}_n = \mathcal{M}[\lambda]\). But