diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -449,7 +449,7 @@ basic}. In fact, all we need to know is\dots
\begin{theorem}\label{thm:ext-1-classify-short-seqs}
Given \(\mathfrak{g}\)-modules \(W\) and \(U\), there is a one-to-one
- correspondence between elements of \(\operatorname{Ext}^1(W, U)\) and
+ correspondence between elements of \(\operatorname{Ext}^1(U, W)\) and
isomorphism classes of short exact sequences
\begin{center}
\begin{tikzcd}
@@ -457,7 +457,7 @@ basic}. In fact, all we need to know is\dots
\end{tikzcd}
\end{center}
- In particular, \(\operatorname{Ext}^1(W, U) = 0\) if, and only if every short
+ In particular, \(\operatorname{Ext}^1(U, W) = 0\) if, and only if every short
exact sequence of \(\mathfrak{g}\)-modules with \(W\) and \(U\) in the
extremes splits.
\end{theorem}
@@ -685,7 +685,7 @@ establish\dots
exact sequence of the form
\begin{equation}\label{eq:exact-seq-h1-vanishes}
\begin{tikzcd}
- 0 \arrow{r} & K \arrow{r} & W \arrow{r}{\pi} & V \arrow{r} & 0
+ 0 \arrow{r} & V \arrow{r} & W \arrow{r}{\pi} & K \arrow{r} & 0
\end{tikzcd}
\end{equation}
splits.
@@ -718,37 +718,54 @@ establish\dots
(\ref{eq:trivial-extrems-exact-seq}).
Now suppose that \(V\) is non-trivial, so that \(C_V\) acts on \(V\) as
- \(\lambda \operatorname{Id}\) for some \(\lambda \ne 0\). Given an eigenvalue
- \(\mu \in K\) of the action of \(C_V\) on \(W\), denote by \(W^\mu\) its
- associated generalized eigenspace. We claim \(W^0\) is the image of the
- inclusion \(K \to W\). Since \(C_V\) acts as zero in \(K\), this image is
- clearly contained in \(W^0\). On the other hand, if \(w \in W\) is such that
- \(C_V^n w = 0\) then
+ \(\lambda \operatorname{Id}\) for some \(\lambda \ne 0\). Denote by \(W^\mu\)
+ the generalized eigenspace of \(C_V\!\restriction_W : W \to W\) associated
+ with \(\mu \in K\). If we identify \(V\) with its image under the inclusion
+ \(V \to W\), it is clear that \(V \subset W^\lambda\). The exactness of
+ (\ref{eq:exact-seq-h1-vanishes}) then implies \(\dim W = \dim V + 1\), so
+ that either \(W^\lambda = V\) or \(W^\lambda = W\). But if \(W^\lambda = W\)
+ then there is some nonzero \(w \in W^\lambda\) with \(w \notin V = \ker
+ \pi\) such that
\[
- \lambda^n \pi(w)
- = C_V^n \pi(w)
- = \pi(C_V^n w)
- = 0,
+ 0
+ = (C_V - \lambda)^n w
+ = \sum_{k = 0}^n (-1)^k \binom{n}{k} \lambda^k C_V^{n - k} w
\]
- so that \(w \in \ker \pi\) -- because \(\lambda^n \ne 0\). The exactness of
- (\ref{eq:exact-seq-h1-vanishes}) then implies the desired conclusion.
+ for some \(n \ge 1\) -- given that \(C_V \in \mathcal{U}(\mathfrak{g})\) is
+ central.
- We furthermore claim that the only eigenvalues of \(C_V\) in \(W\) are \(0\)
- and \(\lambda\). Indeed, if \(\mu \ne 0\) is eigenvalue and \(w\) is an
- associated eigenvector, then
+ In particular,
\[
- \mu \pi(w) = \pi(C_V w) = C_V \pi(w) = \lambda \pi(w)
+ (- \lambda)^n \pi(w)
+ = \sum_{k = 0}^{n - 1} (-1)^k \binom{n}{k} \lambda^k \pi(C_V^{n - k} w)
+ = \sum_{k = 0}^{n - 1} (-1)^k \binom{n}{k} \lambda^k
+ \underbrace{C_V^{n - k} \pi(w)}_{= \; 0}
+ = 0,
\]
+ wich is a contradiction in light of the fact that neither \((-\lambda)^n\)
+ nor \(\pi(w)\) are nil. Hence \(V = W^\lambda\) and there must be some other
+ eigenvalue \(\mu\) of \(C_V\!\restriction_W\). For any such \(\mu\) and any
+ \(w \in W^\mu\),
+ \[
+ \mu \pi(w)
+ = \pi(\mu w)
+ = \pi(C_V w)
+ = C_V \pi(w)
+ = 0
+ \]
+ implies \(\mu = 0\), so that the eigenvalues of the action of \(C_V\) in
+ \(W\) are precisely \(\lambda\) and \(0\).
- Since \(w \notin W^0\), \(\pi(w) \ne 0\) and therefore \(\mu = \lambda\).
- Hence \(W = W^0 \oplus W^\lambda\) as vector space. The fact that \(C_V\) is
- central implies \((C_V - \lambda \operatorname{Id})^n X v = X (C_V - \lambda
- \operatorname{Id})^n v\) for all \(v \in V\), \(X \in \mathfrak{g}\) and \(n
- > 0\). In particular, \(W^\lambda\) is stable under the action of
- \(\mathfrak{g}\) -- i.e. \(W^\lambda\) is a subrepresentation. Since \(W^0\)
- is precisely the kernel of \(\pi\), we have an isomorphism of representations
- \(W^\lambda \cong \sfrac{W}{W^0} \isoto V\), which induces a splitting \(W
- \cong K \oplus V\).
+ Now notice that \(W^0\) is in fact a subrepresentation of \(W\). Indeed,
+ given \(w \in W^0\) and \(X \in \mathfrak{g}\), it follows from the fact that
+ \(C_V\) is central that
+ \[
+ C_V^n X w = X C_V^n w = X \cdot 0 = 0
+ \]
+ for some \(n\). Hence \(W = V \oplus W^0\) as representations. The
+ homomorphism \(\pi\) thus induces an isomorphism \(W^0 \cong \mfrac{W}{V}
+ \isoto K\), which translates to a splitting of
+ (\ref{eq:exact-seq-h1-vanishes}).
Finally, we consider the case where \(V\) is not irreducible. Suppose
\(H^1(\mathfrak{g}, W) = 0\) for all \(\mathfrak{g}\)-modules with \(\dim W <