diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -475,7 +475,7 @@ On a perhaps less derogatory note, \(\mathcal{M}\) also deserves to be called
\emph{a family}. This is because \(\mathcal{M}\) consists of lots of smaller
cuspidal representations which fit together inside of it in a \emph{coherent}
fashion. Mathieu's engineous breaktrough was the realization that
-\(\mathcal{M}\) is a particular example of a more general pattern, which ha
+\(\mathcal{M}\) is a particular example of a more general pattern, which he
named \emph{coherent families}.
\begin{definition}
@@ -517,7 +517,32 @@ named \emph{coherent families}.
\(x^k\) we can see \((\mathcal{M}(\sfrac{1}{2}))[0] \cong K[x, x^{-1}]\).
\end{example}
-% TODO: Move this somewhere else
+Our hope is that given an irreducible cuspidal representation \(V\), we can
+somehow find \(V\) inside of a coherent \(\mathfrak{g}\)-family, such as in the
+case of \(K[x, x^{-1}]\) and \(\mathcal{M}\) from
+example~\ref{ex:sl-laurent-family}. This leads us to the following definition.
+
+\begin{definition}
+ Given an admissible representation \(V\) of \(\mathfrak{g}\) of degree \(d\),
+ a coherent extension \(\mathcal{M}\) of \(V\) is a coherent family
+ \(\mathcal{M}\) of degree \(d\) that contains \(V\) as a subquotient.
+\end{definition}
+
+Our goal is now showing that every admissible representation has a coherent
+extension. The idea then is to classify coherent extensions, and classify which
+submodules of a given coherent extension are actually irreducible cuspidal
+representations. If every admissible \(\mathfrak{g}\)-module fits inside a
+coherent extension, this would lead to classification of all irreducible
+cuspidal representations, which we now know is the key for the solution of out
+classification problem. However, there are some complications to this scheme.
+
+Leaving aside the question of existence for a second, we should point out that
+coherent families turn out to be rather complicated on their own. In fact they
+are too complicated to classify in general. Idealy, we would like to find
+\emph{nice} coherent extensions -- ones we can actually classify. For instance,
+we may search for \emph{simple} coherent extensions, which are defined as
+follows.
+
\begin{definition}
A coherent family \(\mathcal{M}\) is called \emph{simple} if it contains no
proper coherent subfamilies -- i.e. \(\mathcal{M}\) is a simple object in the
@@ -534,86 +559,86 @@ named \emph{coherent families}.
\mathfrak{h}^*\).
\end{definition}
-\begin{definition}
- Given an admissible representation \(V\) of \(\mathfrak{g}\) of degree \(d\),
- a coherent extension \(\mathcal{M}\) of \(V\) is a coherent family
- \(\mathcal{M}\) of degree \(d\) that contains \(V\) as a subquotient.
-\end{definition}
+Another natural cadidate for the role of ``nice extensions'' are the completely
+reducible coherent families -- i.e. families wich are completely reducible as
+\(\mathfrak{g}\)-modules. These turn out to be very easy to produce. Namely,
+there is a construction, known as \emph{the semisimplification\footnote{Recall
+that a ``semisimple'' is a synonim for ``completely reducible'' in the context
+of modules.} of a coherent family}, which takes a coherent extension of \(V\)
+to a completely reducible coherent extension of \(V\).
+% TODO: Note somewhere that M[mu] is a submodule
% Mathieu's proof of this is somewhat profane, I don't think it's worth
% including it in here
-% TODO: Define the notation for M[mu] somewhere else
-% TODO: Note somewhere that M[mu] is a submodule
\begin{lemma}
Given a coherent family \(\mathcal{M}\) and \(\lambda \in \mathfrak{h}^*\),
\(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
\end{lemma}
-% TODO: From this we may conclude that any admissible submodule is a submodule
-% of the semisimplification of any of its coherent extensions
+% TODO: Point out this construction is NOT functorial, since it depends on the
+% choice of composition series
\begin{corollary}
Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a
unique completely reducible coherent family
\(\mathcal{M}^{\operatorname{ss}}\) of degree \(d\) such that the composition
series of \(\mathcal{M}^{\operatorname{ss}}[\lambda]\) is the same as that of
\(\mathcal{M}[\lambda]\) for all \(\lambda \in \mathfrak{h}^*\), called
- \emph{the semisimplification\footnote{Recall that a ``semisimple'' is a
- synonim for ``completely reducible'' in the context of modules.} of
- \(\mathcal{M}\)}.
-
- Namely, if \(\{\lambda_i\}_i\) is a set of representatives of the
- \(Q\)-cosets of \(\mathfrak{h}^*\) and \(0 = \mathcal{M}_{i 0} \subset
- \mathcal{M}_{i 1} \subset \cdots \subset \mathcal{M}_{i n_i} =
- \mathcal{M}[\lambda_i]\) is a composition series,
+ \emph{the semisimplification of \(\mathcal{M}\)}.
+
+ Namely, if \(\lambda \in \mathfrak{h}^*\) and \(0 = \mathcal{M}_{\lambda 0}
+ \subset \mathcal{M}_{\lambda 1} \subset \cdots \subset \mathcal{M}_{\lambda
+ n_\lambda} = \mathcal{M}[\lambda]\) is a composition series,
\[
\mathcal{M}^{\operatorname{ss}}
- \cong \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+ \cong \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}}
+ \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}
\]
\end{corollary}
\begin{proof}
The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear:
since \(\mathcal{M}^{\operatorname{ss}}\) is completely reducible, so is
- \(\mathcal{M}^{\operatorname{ss}}[\lambda_i]\). Hence
+ \(\mathcal{M}^{\operatorname{ss}}[\lambda]\). Hence
\[
- \mathcal{M}^{\operatorname{ss}}[\lambda_i]
- \cong \bigoplus_j \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+ \mathcal{M}^{\operatorname{ss}}[\lambda]
+ \cong
+ \bigoplus_i \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}
\]
As for the existence of the semisimplification, it suffices to show
\[
\mathcal{M}^{\operatorname{ss}}
- = \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+ = \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}}
+ \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}
\]
is indeed a completely reducible coherent family of degree \(d\).
We know from examples~\ref{ex:submod-is-weight-mod} and
- \ref{ex:quotient-is-weight-mod} that each quotient \(\mfrac{\mathcal{M}_{i j
- + 1}}{\mathcal{M}_{i j}}\) is a weight module. Hence
- \(\mathcal{M}^{\operatorname{ss}}\) is a weight module. Furthermore, given
- \(\mu \in \lambda_k + Q\)
+ \ref{ex:quotient-is-weight-mod} that each quotient
+ \(\mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}\) is a weight
+ module. Hence \(\mathcal{M}^{\operatorname{ss}}\) is a weight module.
+ Furthermore, given \(\mu \in \mathfrak{h}^*\)
\[
\mathcal{M}_\mu^{\operatorname{ss}}
- = \bigoplus_{i j}
+ = \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}}
\left(
- \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
+ \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}
\right)_\mu
- = \bigoplus_j
+ = \bigoplus_i
\left(
- \mfrac{\mathcal{M}_{k j + 1}}{\mathcal{M}_{k j}}
+ \mfrac{\mathcal{M}_{\mu i + 1}}{\mathcal{M}_{\mu i}}
\right)_\mu
- \cong \bigoplus_j
- \mfrac{(\mathcal{M}_{k j + 1})_\mu}
- {(\mathcal{M}_{k j})_\mu}
+ \cong \bigoplus_i
+ \mfrac{(\mathcal{M}_{\mu i + 1})_\mu}
+ {(\mathcal{M}_{\mu i})_\mu}
\]
In particular,
\[
\dim \mathcal{M}_\mu^{\operatorname{ss}}
- = \sum_j
- \dim (\mathcal{M}_{k j + 1})_\mu
- - \dim (\mathcal{M}_{k j})_\mu
- = \dim \mathcal{M}[\lambda_k]_\mu
+ = \sum_i
+ \dim (\mathcal{M}_{\mu i + 1})_\mu - \dim (\mathcal{M}_{\mu i})_\mu
+ = \dim \mathcal{M}[\mu]_\mu
= \dim \mathcal{M}_\mu
= d
\]
@@ -621,20 +646,24 @@ named \emph{coherent families}.
Likewise, given \(u \in \mathcal{U}(\mathfrak{g})_0\) the value
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}})
- = \sum_j
- \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j + 1})_\mu})
- - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j})_\mu})
- = \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\lambda_k]_\mu})
+ = \sum_i
+ \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{\mu i + 1})_\mu})
+ - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{\mu i})_\mu})
+ = \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\mu]_\mu})
= \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
\]
is polynomial in \(\mu \in \mathfrak{h}^*\).
\end{proof}
-\begin{corollary}\label{thm:admissible-is-submod-of-extension}
+As promised, if \(\mathcal{M}\) is a coherent extension of \(V\) then so is
+\(\mathcal{M}^{\operatorname{ss}}\).
+
+\begin{proposition}
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}
+ \(\mathcal{M}\) be a coherent extension of \(V\). Then
+ \(\mathcal{M}^{\operatorname{ss}}\) is a coherent extension of \(V\) and
+ \(V\) is in fact a subrepresentation of \(\mathcal{M}^{\operatorname{ss}}\).
+\end{proposition}
\begin{proof}
Since \(V\) is irreducible, its support is contained in a single \(Q\)-coset.
@@ -649,12 +678,70 @@ named \emph{coherent families}.
\to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
\cong \mathcal{M}^{\operatorname{ss}}[\lambda]
\]
+\end{proof}
+
+Given the uniqueness of the semisimplification, the semisimplification of any
+completely reducible coherent extension \(\mathcal{M}\) is \(\mathcal{M}\)
+itself and therefore\dots
+
+\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}
+
+This last results provide a partial answer to the question of existence of nice
+coherent extensions. A complementary question now is: wich submodules of a nice
+coherent family are cuspidal representations?
+
+\begin{theorem}[Mathieu]
+ Let \(\mathcal{M}\) be a simple coherent family of degree \(d\) and
+ \(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent.
+ \begin{enumerate}
+ \item \(\mathcal{M}[\lambda]\) is irreducible.
+ \item \(F_\alpha\!\restriction_{\mathcal{M}[\lambda]}\) is injective for
+ all \(\alpha \in \Delta\).
+ \item \(\mathcal{M}[\lambda]\) is cuspidal.
+ \end{enumerate}
+\end{theorem}
+
+\begin{proof}
+ The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
+ from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
+ corollary that \strong{(iii)} implies \strong{(ii)}. All it's left is to show
+ \strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
+ corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
+ $\mathcal{M}[\lambda]$ may not be irreducible for some $\lambda$ satisfying
+ \strong{(ii)}. We will show this is never the case.}.
- 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 \(V \to \mathcal{M}\).
+ Suppose \(F_\alpha\) acts injectively in the subrepresentation
+ \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since
+ \(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains
+ an infinite-dimensiona irreducible \(\mathfrak{g}\)-submodule \(V\).
+ Moreover, again by corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(V\)
+ is a cuspidal representation, and its degree is bounded by \(d\). We want to
+ show \(\mathcal{M}[\lambda] = V\).
+
+ We claim the set \(U = \{\mu \in \mathfrak{h}^* : \mathcal{M}_\mu \ \text{is
+ a simple $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is Zariski-open. If we
+ suppose this is the case for a moment or two, it follows from the fact that
+ \(\mathcal{M}\) is simple and \(\operatorname{supp}_{\operatorname{ess}} V\)
+ is Zariski-dense that \(U \cap \operatorname{supp}_{\operatorname{ess}} V\)
+ is non-empty. In other words, there is some \(\mu \in \mathfrak{h}^*\) such
+ that \(\mathcal{M}_\mu\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module
+ and \(\dim V_\mu = \deg V\).
+
+ % Here we use thm:centralizer-multiplicity
+ In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any
+ irreducible \(\mathfrak{g}\)-module \(W\), the multiplicity of \(W\) in
+ \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
+ \(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module, which is, of
+ course, \(1\) if \(W \cong V\) and \(0\) otherwise. Hence
+ \(\mathcal{M}[\lambda] = V\) and \(\mathcal{M}[\lambda]\) is cuspidal.
\end{proof}
+To finish the proof, we now show\dots
+
\begin{lemma}
Let \(\mathcal{M}\) be a coherent family. The set \(U = \{\lambda \in
\mathfrak{h}^* : \mathcal{M}_\lambda \ \text{is a simple
@@ -775,51 +862,6 @@ named \emph{coherent families}.
the union of Zariski-open subsets and is therefore open. We are done.
\end{proof}
-\begin{theorem}[Mathieu]
- Let \(\mathcal{M}\) be a simple coherent family of degree \(d\) and
- \(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent.
- \begin{enumerate}
- \item \(\mathcal{M}[\lambda]\) is irreducible.
- \item \(F_\alpha\!\restriction_{\mathcal{M}[\lambda]}\) is injective for
- all \(\alpha \in \Delta\).
- \item \(\mathcal{M}[\lambda]\) is cuspidal.
- \end{enumerate}
-\end{theorem}
-
-\begin{proof}
- The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
- from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
- corollary that \strong{(iii)} implies \strong{(ii)}. All it's left is to show
- \strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
- corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
- $\mathcal{M}[\lambda]$ may not be irreducible for some $\lambda$ satisfying
- \strong{(ii)}. We will show this is never the case.}.
-
- Suppose \(F_\alpha\) acts injectively in the subrepresentation
- \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since
- \(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains
- an infinite-dimensiona irreducible \(\mathfrak{g}\)-submodule \(V\).
- Moreover, again by corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(V\)
- is a cuspidal representation, and its degree is bounded by \(d\). We claim
- \(\mathcal{M}[\lambda] = V\).
-
- Since \(\mathcal{M}\) is simple and
- \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense, \(U = \{\mu
- \in \mathfrak{h}^* : \mathcal{M}_\mu \ \text{is a simple
- $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is a non-empty open set, and \(U
- \cap \operatorname{supp}_{\operatorname{ess}} V\) is non-empty. In other
- words, there is some \(\mu \in \mathfrak{h}^*\) such that \(\mathcal{M}_\mu\)
- is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module and \(\dim V_\mu = \deg
- V\).
-
- In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any
- irreducible \(\mathfrak{g}\)-module \(W\), the multiplicity of \(W\) in
- \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
- \(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module, which is, of
- course, \(1\) if \(W \cong V\) and \(0\) otherwise. Hence
- \(\mathcal{M}[\lambda] = V\) and \(\mathcal{M}[\lambda]\) is cuspidal.
-\end{proof}
-
\section{Localizations \& the Existance of Coherent Extensions}
% TODO: Comment on the intuition behind the proof: we can get vectors in a