Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Fixed a major error in the proof of complete reducibility
Fixed an error Kostya pointed out regarding the Ext functors
1 file changed, 45 insertions, 28 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/complete-reducibility.tex | 73 | 45 | 28 |
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 <