lie-algebras-and-their-representations
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
- Commit
- e9b03cf14827bf128b52d5ca3aee477dba4d8532
- Parent
- 2e316a8a4b72d2f668afff9eaa0118021a522060
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Moved the proof that an admissible module is contained in any of its semisimple coherent extensions to a separate theorem
Diffstat
1 file changed, 40 insertions, 30 deletions
diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -266,6 +266,31 @@
is polynomial in \(\mu \in \mathfrak{h}^*\).
\end{proof}
+\begin{corollary}\label{thm:admissible-is-submod-of-extension}
+ Let \(V\) be an irreducible admissible \(\mathfrak{g}\)-module and
+ \(\mathcal{M}\) be a completely reducible coherent extension of \(V\). Then
+ \(V\) is contained in \(\mathcal{M}\).
+\end{corollary}
+
+\begin{proof}
+ Since \(V\) is irreducible, its support is contained in a single \(Q\)-coset.
+ This implies that \(V\) is a subquotient of \(\mathcal{M}[\lambda]\) for any
+ \(\lambda \in \operatorname{supp} V\). If we fix some composition series \(0
+ = \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
+ \mathcal{M}[\lambda]\) of \(\mathcal{M}[\lambda]\) with \(V \cong
+ \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}\), there is a natural inclusion
+ \[
+ V
+ \isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+ \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
+ \cong \mathcal{M}^{\operatorname{ss}}[\lambda]
+ \]
+
+ It then follows from the uniqueness of the semisimplification of
+ \(\mathcal{M}\) that \(\mathcal{M} \cong \mathcal{M}^{\operatorname{ss}}\),
+ so we have an inclusion \(V \to \mathcal{M}\).
+\end{proof}
+
\begin{theorem}[Mathieu]
Let \(\mathcal{M}\) be an irreducible coherent family and \(\lambda \in
\mathfrak{h}^*\). The following conditions are equivalent.
@@ -486,42 +511,27 @@
There exists a unique completely reducible coherent extension
\(\operatorname{Ext}(V)\) of \(V\). More precisely, if \(\mathcal{M}\) is any
coherent extension of \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong
- \operatorname{Ext}(V)\). Furthermore, \(V\) is itself contained in
- \(\operatorname{Ext}(V)\) and \(\operatorname{Ext}(V)\) is irreducible as a
- coherent family.
+ \operatorname{Ext}(V)\). Furthermore, \(\operatorname{Ext}(V)\) is
+ irreducible as a coherent family.
\end{theorem}
-% TODOOO: Prove the uniqueness
+% TODOOO: Prove the uniqueness!
\begin{proof}
- The existence part should be clear from the previous discussion: let
- \(\mathcal{M}\) be a coherent extension of \(V\) and take
- \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of \(V\).
-
- % TODOOOOOO: Extract this to a general result: any admissible module is
- % contained in the semisimplification of any of its coherent extensions
- We claim \(V\) is contained in \(\operatorname{Ext}(V)\). Indeed, if we fix
- some composition series \(0 = \mathcal{M}_0 \subset \mathcal{M}_1 \subset
- \cdots \subset \mathcal{M}_n = \mathcal{M}[\lambda]\) of
- \(\mathcal{M}[\lambda]\) with \(V \cong \mfrac{\mathcal{M}_{i +
- 1}}{\mathcal{M}_i}\), \(\lambda \in \operatorname{supp} V\), there is a
- natural inclusion
- \[
- V
- \isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
- \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
- = \mathcal{M}^{\operatorname{ss}}[\lambda]
- \subset \operatorname{Ext}(V)
- \]
+ The existence part should be clear from the previous discussion: it suffices
+ to fix some coherent extension \(\mathcal{M}\) of \(V\) and take
+ \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
% TODOOO: Prove that the weight spaces of any simple g-module are all simple
% C(h)-modules
- Since the degree of \(V\) is the same as the degree of
- \(\operatorname{Ext}(V)\), some of its weight spaces have maximal dimension
- inside of \(\operatorname{Ext}(V)\). In particular, it follows from the
- irreducibility of \(V\) that \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is
- a simple \(C_{\mathcal{U}(\mathfrak{h})}(\mathfrak{h})\)-module for some
- \(\lambda \in \operatorname{supp} V\), so that \(\operatorname{Ext}(V)\) is
- irreducible as a coherent family.
+ To see that \(\operatorname{Ext}(V)\) is irreducible as a coherent family,
+ recall from corollary~\ref{thm:admissible-is-submod-of-extension} that \(V\)
+ is a subrepresentation of \(\operatorname{Ext}(V)\). Since the degree of
+ \(V\) is the same as the degree of \(\operatorname{Ext}(V)\), some of its
+ weight spaces have maximal dimension inside of \(\operatorname{Ext}(V)\). In
+ particular, it follows from the irreducibility of \(V\) that
+ \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
+ \(C_{\mathcal{U}(\mathfrak{h})}(\mathfrak{h})\)-module for some \(\lambda \in
+ \operatorname{supp} V\).
\end{proof}
\begin{proposition}[Mathieu]