diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -272,10 +272,10 @@ to introduce some basic tools which will come in handy later on, known as\dots
\section{Invariant Bilinear Forms}
\begin{definition}\index{invariant bilinear form}
- A symmetric bilinear \(B : \mathfrak{g} \times \mathfrak{g} \to K\) is called
- \emph{\(\mathfrak{g}\)-invariant} if the operator \(\operatorname{ad}(X) :
- \mathfrak{g} \to \mathfrak{g}\) is antisymmetric with respect to \(B\) for
- all \(X \in \mathfrak{g}\).
+ A symmetric bilinear form \(B : \mathfrak{g} \times \mathfrak{g} \to K\) is
+ called \emph{\(\mathfrak{g}\)-invariant} if the operator
+ \(\operatorname{ad}(X) : \mathfrak{g} \to \mathfrak{g}\) is antisymmetric
+ with respect to \(B\) for all \(X \in \mathfrak{g}\).
\[
B(\operatorname{ad}(X) Y, Z) + B(Y, \operatorname{ad}(X) Z) = 0
\]
@@ -625,18 +625,18 @@ a \(\mathfrak{g}\)-module}.
i.e. the unique basis for \(\mathfrak{g}\) satisfying \(\kappa_M(X_i, X^j) =
\delta_{i j}\). We call
\[
- C_M = X_1 X^1 + \cdots + X_r X^r \in \mathcal{U}(\mathfrak{g})
+ \Omega_M = X_1 X^1 + \cdots + X_r X^r \in \mathcal{U}(\mathfrak{g})
\]
the \emph{Casimir element of \(M\)}.
\end{definition}
\begin{lemma}
- The definition of \(C_M\) is independent of the choice of basis
+ The definition of \(\Omega_M\) is independent of the choice of basis
\(\{X_i\}_i\).
\end{lemma}
\begin{proof}
- Whatever basis \(\{X_i\}_i\) we choose, the image of \(C_M\) under the
+ Whatever basis \(\{X_i\}_i\) we choose, the image of \(\Omega_M\) under the
canonical isomorphism \(\mathfrak{g} \otimes \mathfrak{g} \isoto \mathfrak{g}
\otimes \mathfrak{g}^* \isoto \operatorname{End}(\mathfrak{g})\) is the
identity operator\footnote{Here the isomorphism $\mathfrak{g} \otimes
@@ -646,16 +646,16 @@ a \(\mathfrak{g}\)-module}.
\end{proof}
\begin{proposition}
- The Casimir element \(C_M \in \mathcal{U}(\mathfrak{g})\) is central, so that
- \(C_M\!\restriction_N : N \to N\) is a \(\mathfrak{g}\)-homomorphism for any
- \(\mathfrak{g}\)-module \(N\). Furthermore, \(C_M\) acts on \(M\) as a
- nonzero scalar operator whenever \(M\) is a non-trivial finite-dimensional
- simple \(\mathfrak{g}\)-module.
+ The Casimir element \(\Omega_M \in \mathcal{U}(\mathfrak{g})\) is central, so
+ that \(\Omega_M\!\restriction_N : N \to N\) is a
+ \(\mathfrak{g}\)-homomorphism for any \(\mathfrak{g}\)-module \(N\).
+ Furthermore, \(\Omega_M\) acts on \(M\) as a nonzero scalar operator whenever
+ \(M\) is a non-trivial finite-dimensional simple \(\mathfrak{g}\)-module.
\end{proposition}
\begin{proof}
- To see that \(C_M\) is central fix a basis \(\{X_i\}_i\) for \(\mathfrak{g}\)
- and denote by \(\{X^i\}_i\) its dual basis as in
+ To see that \(\Omega_M\) is central fix a basis \(\{X_i\}_i\) for
+ \(\mathfrak{g}\) and denote by \(\{X^i\}_i\) its dual basis as in
Definition~\ref{def:casimir-element}. Let \(X \in \mathfrak{g}\) and denote
by \(\lambda_{i j}, \mu_{i j} \in K\) the coefficients of \(X_j\) and \(X^j\)
in \([X, X_i]\) and \([X, X^i]\), respectively.
@@ -672,29 +672,30 @@ a \(\mathfrak{g}\)-module}.
Hence
\[
\begin{split}
- [X, C_M]
+ [X, \Omega_M]
& = \sum_i [X, X_i X^i] \\
& = \sum_i [X, X_i] X^i + \sum_i X_i [X, X^i] \\
& = \sum_{i j} \lambda_{i j} X_j X^i + \sum_{i j} \mu_{i j} X_i X^j \\
& = 0
\end{split},
\]
- and \(C_M\) is central. This implies that \(C_M\!\restriction_N : N \to N\)
- is a \(\mathfrak{g}\)-homomorphism for all \(\mathfrak{g}\)-modules \(N\):
- its action commutes with the action of any other element of \(\mathfrak{g}\).
+ and \(\Omega_M\) is central. This implies that \(\Omega_M\!\restriction_N : N
+ \to N\) is a \(\mathfrak{g}\)-homomorphism for all \(\mathfrak{g}\)-modules
+ \(N\): its action commutes with the action of any other element of
+ \(\mathfrak{g}\).
In particular, it follows from Schur's Lemma that if \(M\) is
- finite-dimensional and simple then \(C_M\) acts on \(M\) as a scalar
+ finite-dimensional and simple then \(\Omega_M\) acts on \(M\) as a scalar
operator. To see that this scalar is nonzero we compute
\[
- \operatorname{Tr}(C_M\!\restriction_M)
+ \operatorname{Tr}(\Omega_M\!\restriction_M)
= \operatorname{Tr}(X_1\!\restriction_M X^1\!\restriction_M)
+ \cdots
+ \operatorname{Tr}(X_r\!\restriction_M X^r\!\restriction_M)
= \dim \mathfrak{g},
\]
- so that \(C_M\!\restriction_M = \lambda \operatorname{Id}\) for \(\lambda =
- \frac{\dim \mathfrak{g}}{\dim M} \ne 0\).
+ so that \(\Omega_M\!\restriction_M = \lambda \operatorname{Id}\) for
+ \(\lambda = \frac{\dim \mathfrak{g}}{\dim M} \ne 0\).
\end{proof}
As promised, the Casimir element of a \(\mathfrak{g}\)-module can be used to
@@ -732,60 +733,59 @@ establish\dots
% because the action of every element of g is strictly upper triangular -- and
% semisimple -- because it is a quotient of g, which is semisimple. We thus
% have rho(g) = 0, so that W is trivial
- Since \(\dim N = 2\), the simple component \(\mathcal{U}(\mathfrak{g})
- \cdot n\) of \(n\) in \(N\) is either \(K n\) or \(N\) itself. But this
- component cannot be \(N\), since the image of \(f\) is a
- \(1\)-dimensional \(\mathfrak{g}\)-module -- i.e. a proper nonzero submodule.
- Hence \(K n\) is invariant under the action of \(\mathfrak{g}\). In
- particular, \(X \cdot n = 0\) for all \(X \in \mathfrak{g}\). Since \(n\) lies
- outside the image of \(f\), \(g(w) \ne 0\) -- which is
- to say, \(n \notin \ker g = \operatorname{im} f\). This implies the map \(K
- \to N\) that takes \(1\) to \(\sfrac{n}{g(n)}\) is a splitting of
- (\ref{eq:trivial-extrems-exact-seq}).
-
- Now suppose that \(M\) is non-trivial, so that \(C_M\) acts on \(M\) as
- \(\lambda\) for some \(\lambda \ne 0\). Denote by \(N^\mu\)
- the generalized eigenspace of \(C_M\!\restriction_N : N \to N\) associated
- with \(\mu \in K\). If we identify \(M\) with \(f(M)\),
- it is clear that \(M \subset N^\lambda\). The exactness of
- (\ref{eq:exact-seq-h1-vanishes}) implies \(\dim N = \dim M + 1\), so
- that either \(N^\lambda = M\) or \(N^\lambda = N\). But if \(N^\lambda = N\)
- then there is some nonzero \(n \in N^\lambda\) with \(n \notin M = \ker g\)
- such that
+ Since \(\dim N = 2\), the simple component \(\mathcal{U}(\mathfrak{g}) \cdot
+ n\) of \(n\) in \(N\) is either \(K n\) or \(N\) itself. But this component
+ cannot be \(N\), since the image of \(f\) is a \(1\)-dimensional
+ \(\mathfrak{g}\)-module -- i.e. a proper nonzero submodule. Hence \(K n\) is
+ invariant under the action of \(\mathfrak{g}\). In particular, \(X \cdot n =
+ 0\) for all \(X \in \mathfrak{g}\). Since \(n\) lies outside the image of
+ \(f\), \(g(n) \ne 0\) -- which is to say, \(n \notin \ker g =
+ \operatorname{im} f\). This implies the map \(K \to N\) that takes \(1\) to
+ \(\sfrac{n}{g(n)}\) is a splitting of (\ref{eq:trivial-extrems-exact-seq}).
+
+ Now suppose that \(M\) is non-trivial, so that \(\Omega_M\) acts on \(M\) as
+ \(\lambda\) for some \(\lambda \ne 0\). Denote by \(N^\mu\) the generalized
+ eigenspace of \(\Omega_M\!\restriction_N : N \to N\) associated with \(\mu
+ \in K\). If we identify \(M\) with \(f(M)\), it is clear that \(M \subset
+ N^\lambda\). The exactness of (\ref{eq:exact-seq-h1-vanishes}) implies \(\dim
+ N = \dim M + 1\), so that either \(N^\lambda = M\) or \(N^\lambda = N\). But
+ if \(N^\lambda = N\) then there is some nonzero \(n \in N^\lambda\) with \(n
+ \notin M = \ker g\) such that
\[
0
- = (C_M - \lambda)^r \cdot n
- = \sum_{k = 0}^r (-1)^k \binom{r}{k} \lambda^k C_M^{r - k} \cdot n
+ = (\Omega_M - \lambda)^r \cdot n
+ = \sum_{k = 0}^r (-1)^k \binom{r}{k} \lambda^k \Omega_M^{r - k} \cdot n
\]
for some \(r \ge 1\).
In particular,
\[
(- \lambda)^{r - 1} g(n)
- = \sum_{k = 0}^{r - 1} (-1)^k \binom{r}{k} \lambda^k g(C_M^{r - k} \cdot n)
= \sum_{k = 0}^{r - 1} (-1)^k \binom{r}{k} \lambda^k
- \underbrace{C_M^{r - k} \cdot g(n)}_{= \; 0}
+ g(\Omega_M^{r - k} \cdot n)
+ = \sum_{k = 0}^{r - 1} (-1)^k \binom{r}{k} \lambda^k
+ \underbrace{\Omega_M^{r - k} \cdot g(n)}_{= \; 0}
= 0,
\]
- which is a contradiction -- given that neither \((-\lambda)^{r - 1}\)
- nor \(g(n)\) are nil. Hence \(M = N^\lambda\) and there must be some other
- eigenvalue \(\mu\) of \(C_M\!\restriction_N\). For any such \(\mu\) and any
- eigenvector \(n \in N_\mu\),
+ which is a contradiction -- given that neither \((-\lambda)^{r - 1}\) nor
+ \(g(n)\) are nil. Hence \(M = N^\lambda\) and there must be some other
+ eigenvalue \(\mu\) of \(\Omega_M\!\restriction_N\). For any such \(\mu\) and
+ any eigenvector \(n \in N_\mu\),
\[
\mu g(n)
= g(\mu n)
- = g(C_M \cdot n)
- = C_M \cdot g(n)
+ = g(\Omega_M \cdot n)
+ = \Omega_M \cdot g(n)
= 0
\]
- implies \(\mu = 0\), so that the eigenvalues of the action of \(C_M\) on
+ implies \(\mu = 0\), so that the eigenvalues of the action of \(\Omega_M\) on
\(N\) are precisely \(\lambda\) and \(0\).
Now notice that \(N^0\) is in fact a submodule of \(N\). Indeed,
given \(n \in N^0\) and \(X \in \mathfrak{g}\), it follows from the fact that
- \(C_M\) is central that
+ \(\Omega_M\) is central that
\[
- C_M^r \cdot (X \cdot n) = X \cdot (C_V^r \cdot n) = X \cdot 0 = 0
+ \Omega_M^r \cdot (X \cdot n) = X \cdot (\Omega_M^r \cdot n) = X \cdot 0 = 0
\]
for some \(r\). Hence \(N = M \oplus N^0\) as \(\mathfrak{g}\)-modules. The
homomorphism \(g\) thus induces an isomorphism \(N^0 \cong \mfrac{N}{M}
@@ -888,8 +888,7 @@ We are now finally ready to prove\dots
Now notice \(\operatorname{Hom}(L, L')^{\mathfrak{g}} =
\operatorname{Hom}_{\mathfrak{g}}(L, L')\) for all \(\mathfrak{g}\)-modules
- \(L'\). Indeed, given
- a \(K\)-linear map \(f : L \to L'\)
+ \(L'\). Indeed, given a \(K\)-linear map \(f : L \to L'\)
\[
\begin{split}
f \in \operatorname{Hom}(L, L')^{\mathfrak{g}}