lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -740,7 +740,7 @@ over the ring of \(G\)-invariant differential operators -- i.e.
 \(G\).
 
 Proposition~\ref{thm:geometric-realization-of-uni-env} is in fact only the
-beginning of a profound connection between the theory of \(D\)-modules and and
+beginning of a profound connection between the theory of \(D\)-modules and
 \emph{representation theory}, the latter of which we now explore in the
 following section.
 

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -926,8 +926,8 @@ are really interested in is\dots
 
 We should point out that proposition~\ref{thm:verma-is-finite-dim} fails for
 non-dominant \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
-\(M(\lambda)\), one can show show that if \(\lambda \in P\) is not dominant
-then \(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if
+\(M(\lambda)\), one can show that if \(\lambda \in P\) is not dominant then
+\(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if
 \(\mathfrak{g} = \mathfrak{sl}_2(K)\) and \(\lambda : h \mapsto -2\) then the
 action of \(\mathfrak{g}\) on \(M(\lambda)\) is given by
 \begin{center}

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -821,7 +821,7 @@ eigenvalue \(\lambda([E_{1 2}, E_{2 1}]) - 2k\) of the action of \(h\) on
 \(\bigoplus_{k \in \mathbb{N}} V_{\lambda + k (\alpha_1 - \alpha_2)}\) is
 \(V_{\lambda + k (\alpha_1 - \alpha_2)}\), the weights of \(V\) appearing the
 string \(\lambda, \lambda + (\alpha_1 - \alpha_2), \ldots, \lambda + k
-(\alpha_1 - \alpha_2), \ldots\) must be symmetric with respect to the the line
+(\alpha_1 - \alpha_2), \ldots\) must be symmetric with respect to the line
 \(B(\alpha_1 - \alpha_2, \alpha) =  0\). The picture is thus
 \begin{center}
   \begin{tikzpicture}