lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -1,5 +1,7 @@
 \chapter{Classification of Coherent Families}
 
+% TODO: Write an introduction
+
 % TODOOO: Is this decomposition unique??
 \begin{proposition}
   Suppose \(\mathfrak{g} = \mathfrak{s}_1 \oplus \cdots \oplus \mathfrak{s}_r\)
@@ -20,8 +22,8 @@
   Example~\ref{thm:simple-weight-mod-is-tensor-prod}, there exists (unique)
   simple weight \(\mathfrak{s}_i\)-modules \(M_i\) such that \(M \cong M_1
   \otimes \cdots \otimes M_r\). Take \(\mathcal{M}_i = \mExt(M_i)\). We will
-  show that \(\mathcal{M}_1 \otimes \cdots \mathcal{M}_r\) is a coherent
-  extension of \(M\).
+  show that \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\) is a
+  coherent extension of \(M\).
 
   It is clear that \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\) is a
   degree \(d\) bounded \(\mathfrak{g}\)-module containing \(M\) as a submodule.
@@ -39,7 +41,8 @@
   \end{align*}
   is polynomial, notice that the natural isomorphism of algebras
   \begin{align*}
-    f : \mathcal{U}(\mathfrak{s}_1) \otimes \cdots \mathcal{U}(\mathfrak{s}_1)
+    f : \mathcal{U}(\mathfrak{s}_1) \otimes
+    \cdots \otimes \mathcal{U}(\mathfrak{s}_1)
     & \isoto \mathcal{U}(\mathfrak{g}) \\
     u_1 \otimes \cdots \otimes u_r & \mapsto u_1 \cdots u_r
   \end{align*}
@@ -95,15 +98,16 @@
   \(\mathcal{M} \cong \mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\).
 \end{proof}
 
-% TODO: Rework this
-In addition, it turns out that very few simple Lie algebras admit cuspidal
-modules at all. Specifically\dots
+This last result allows us to concentrate on focus exclusive on classifying
+coherent \(\mathfrak{s}\)-families for the simple Lie algebras
+\(\mathfrak{s}\). In addition, it turns out that very few simple algebras admit
+irreducible coherent families at all. Namely\dots
 
 \begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
   Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose
-  there exists a cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s} \cong
-  \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\) for
-  some \(n\).
+  there exists a infinite-dimensional cuspidal \(\mathfrak{s}\)-module. Then
+  \(\mathfrak{s} \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong
+  \mathfrak{sp}_{2 n}(K)\) for some \(n\).
 \end{proposition}
 
 \begin{corollary}
@@ -113,33 +117,50 @@ modules at all. Specifically\dots
   \mathfrak{sp}_{2n}(K)\) for some \(n\).
 \end{corollary}
 
-% TODO: Remove this: we will only focus on the combinatorial classification
-% TODO: Simply notice that a more explicit "geometric" description of the
-% cohorent families exists
-Hence it suffices to classify the irreducible semisimple coherent families of
-\(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\). These can be described
-either algebraically, using combinatorial invariants -- which Mathieu does in
-sections 7, 8 and 9 of his paper -- or geometrically, using algebraic varieties
-and differential forms -- which is done in sections 11 and 12. While rather
-complicated on its own, the geometric construction is more concrete than its
-combinatorial counterpart.
-
-% TODO: Add some notes on the proof of this?
-% I really don't think its worth proving this
-\begin{proposition}
+The problem of classifying the semisimple irreducible coherent
+\(\mathfrak{g}\)-families for some arbitrary semisimple \(\mathfrak{g}\) can
+thus be reduced to a proof by exaustion: it suffices to classify coherent
+\(\mathfrak{sl}_n(K)\)-families and coherent
+\(\mathfrak{sp}_{2n}(K)\)-families. We will follow this path by analysing each
+case -- \(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2n}(K)\) -- separately,
+classifying coherent families in terms of combinatorial invariants -- as does
+Mathieu in \cite[sec.~8,sec.~9]{mathieu}. Alternatively, Mathieu also provides
+a more explicit ``geometric'' construction of the coherent families for both
+\(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2n}\) in sections 11 and 12 of his
+paper.
+
+Before we proceed to the individual case analysis, however, we would like
+discuss some further reductions to our general problem, the first of which is a
+crutial refinement to Proposition~\ref{thm:coherent-families-are-all-ext} due
+to Mathieu.
+
+\begin{proposition}\label{coh-family-is-ext-l-lambda}
   Let \(\mathcal{M}\) be a semisimple irreducible coherent
   \(\mathfrak{g}\)-family. Then there exists some \(\lambda \in
   \mathfrak{h}^*\) such that \(L(\lambda)\) is bounded and \(\mathcal{M} \cong
   \mExt(L(\lambda))\).
 \end{proposition}
 
-% TODO: Finish this remark
 \begin{note}
   I once had the opportunity to ask Olivier Mathieu himself how he first came
-  across the notation of coherent families and what was the intuition behind
-  it.
+  across the notation of coherent families and what was his intuition behind
+  it. Unfortunately, his responce was that ``he did not remember.'' However,
+  Mathieu was able to tell me that ``the \emph{trick} is that I managed to show
+  that they all come from simple highest-weight modules, which were already
+  well understood.'' I personally find it likely that Mathieu first considered
+  the idea of twisting \(L(\lambda)\) -- for \(\lambda\) with \(L(\lambda)\)
+  bounded -- by a suitable automorphism \(\theta_\mu : \Sigma^{-1}
+  \mathcal{U}(\mathfrak{g}) \isoto \Sigma^{-1} \mathcal{U}(\mathfrak{g})\), as
+  in the proof of Proposition~\ref{thm:coh-ext-exists}, and only after decided
+  to agregate this data in a coherent family by summing over the \(Q\)-cosets
+  \(\mu + Q\), \(\mu \in \mathfrak{h}^*\).
 \end{note}
 
+In case the significance of Proposition~\ref{coh-family-is-ext-l-lambda} is
+unclear, the point is that it allows is to reduce the problem of classifying
+the coherent \(\mathfrak{g}\)-families to that of aswering the following two
+questions:
+
 \begin{enumerate}
   \item When is \(L(\lambda)\) bounded?
 
@@ -147,8 +168,33 @@ combinatorial counterpart.
     \(L(\mu)\) bounded, when is \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\)?
 \end{enumerate}
 
-% TODO: Explain beforehand why central characters exist and are unique
-% TODO: Cite the discussion on central characters of [humphreys-cat-o] here
+These are the questions which we will attempt to answer for \(\mathfrak{g} =
+\mathfrak{sl}_n(K)\) and \(\mathfrak{g} = \mathfrak{sp}_{2n}(K)\). We begin by
+providing a partial answer to the second answer by introducing an invariant of
+coherent families, known as its \emph{central character}.
+
+To describe this invariant, we consider the Verma module \(M(\lambda) =
+\mathcal{U}(\mathfrak{g}) \cdot m^+\). Given \(\mu \in \mathfrak{h}^*\) and \(m
+\in M(\lambda)_\mu\), it is clear that \(u \cdot m \in M(\lambda)_\mu\) for all
+central \(u \in \mathcal{U}(\mathfrak{g})\). In particular, \(u \cdot m^+ \in
+M(\lambda)_\lambda = K m^+\) is a scalar multiple of \(m^+\) for all \(u \in
+Z(\mathcal{U}(\mathfrak{g}))\), say \(\chi_\lambda(u) m^+\) for some
+\(\chi_\lambda(u) \in K\). More generally, if we take any \(m = v \cdot m^+ \in
+M(\lambda)\) we can see that
+\[
+  u \cdot m
+  = v \cdot (u \cdot m^+)
+  = \chi_\lambda(u) \, v \cdot m^+
+  = \chi_\lambda(u) m
+\]
+
+Since every highest-weight module is a quotient of a Verma module, it follows
+that \(u \in Z(\mathcal{U}(\mathfrak{g}))\) acts on a highest-weight module
+\(M\) of highest-weight \(\lambda\) via multiplication by \(\chi_\lambda(u)\).
+In addition, it is clear that the function \(\chi_\lambda :
+Z(\mathcal{U}(\mathfrak{g})) \to K\) must be an algebra homomorphism. This
+leads us to the following definition.
+
 \begin{definition}
   Given a highest weight \(\mathfrak{g}\)-module \(M\) of highest weight
   \(\lambda\), the unique algebra homomorphism \(\chi_\lambda :
@@ -158,16 +204,20 @@ combinatorial counterpart.
   associated with the weight \(\lambda\)}.
 \end{definition}
 
+Since a simple highest-weight \(\mathfrak{g}\)-module is uniquelly determined
+by is highest-weight, it is clear that central characters are invariants of
+simple highest-weight modules. We should point out that these are far from
+perfect invariants, however. Namelly\dots
+
 % TODO: Cite the definition of the dot action
 \begin{theorem}[Harish-Chandra]
   Given \(\lambda, \mu \in \mathfrak{h}^*\), \(\chi_\lambda = \chi_\mu\) if,
-  and only if \(\mu \in W \bullet \lambda\). All algebra homomorphism
-  \(Z(\mathcal{U}(\mathfrak{g})) \to K\) have the form \(\chi_\lambda\) for
-  some \(\lambda\).
+  and only if \(\mu \in W \bullet \lambda\).
 \end{theorem}
 
-% TODO: Note we will prove that central characters are also invariants of
-% coherent families
+This and much more can be found in \cite[ch.~1]{humphreys-cat-o}. What is
+interesting about all this to us is that, as it turns out, central character
+are also invariants of coherent families. More specifically\dots
 
 \begin{proposition}\label{thm:coherent-family-has-uniq-central-char}
   Suppose \(\lambda, \mu \in \mathfrak{h}^*\) are such that \(L(\lambda)\) and
@@ -197,9 +247,14 @@ combinatorial counterpart.
   \(\chi_\lambda(u) = \chi_\mu(u)\).
 \end{proof}
 
-% TODO: Remark that the probability of σ_β ∙ λ ∈ P+ is slight: there precisely
-% one element in the orbit of λ which is dominant integral, so the odds are
-% 1/|W ∙ λ|
+Central characters may thus be used to distinguished between two semisimple
+irreducible coherent families. Unfortunately for us, as in the case of simple
+highest-weight modules, central characters are not perfect invariants of
+coherent families: there are non-isomorphic semisimple irreducible coherent
+families which share a common central character. Nevertheless, Mathieu was able
+to at least provide a somewhat \emph{precarious} version of the converse of
+Proposition~\ref{thm:coherent-family-has-uniq-central-char}. Namelly\dots
+
 \begin{lemma}\label{thm:lemma6.1}
   Let \(\beta \in \Sigma\) and \(\lambda \notin P^+\) be such that.
   \(L(\lambda)\) is bounded and \(\lambda(H_\beta) \notin \mathbb{N}\). Then
@@ -209,6 +264,17 @@ combinatorial counterpart.
   \(\mExt(L(\sigma_\beta \bullet \lambda)) \cong \mExt(L(\lambda))\).
 \end{lemma}
 
+\begin{note}
+  We should point out that, while it may very well be that \(\sigma_\beta
+  \bullet \lambda \in P^+\), there is generally only a slight chance of such an
+  event happening. Indeed, given \(\lambda \in \mathfrak{h}^*\), its orbit \(W
+  \bullet \lambda\) meets \(P^+\) precisely once, so that the probability of
+  \(\sigma_\beta \bullet \lambda \in P^+\) for some random \(\lambda \in
+  \mathfrak{h}^*\) is only \(\sfrac{1}{|W \bullet \lambda|}\). With the odds
+  stacked in our favor, we will later be able to exploit the second part of
+  Lemma~\ref{thm:lemma6.1} without much difficulty!
+\end{note}
+
 While technical in nature, this lemma already allows us to classify all
 semisimple irreducible coherent \(\mathfrak{sl}_2(K)\)-families.
 
@@ -227,6 +293,8 @@ semisimple irreducible coherent \(\mathfrak{sl}_2(K)\)-families.
   - 2 \notin P^+\) then \(\mExt(L(\lambda)) \cong \mExt(L(- \lambda - 2))\).
 \end{example}
 
+% TODO: Add a transition here
+
 \section{Coherent \(\mathfrak{sp}_{2n}(K)\)-families}
 
 % TODO: Fix n >= 2

diff --git a/sections/simple-weight.tex b/sections/simple-weight.tex
@@ -1471,7 +1471,7 @@ Explicitly\dots
 
 It should now be obvious\dots
 
-\begin{proposition}[Mathieu]
+\begin{proposition}[Mathieu]\label{thm:coh-ext-exists}
   There exists a coherent extension \(\mathcal{M}\) of \(M\).
 \end{proposition}
 
@@ -1571,7 +1571,7 @@ Lo and behold\dots
 A sort of ``reciprocal'' of Theorem~\ref{thm:mathieu-ext-exists-unique} also
 holds. Namely\dots
 
-\begin{proposition}
+\begin{proposition}\label{thm:coherent-families-are-all-ext}
   Let \(\mathcal{M}\) be a semisimple irreducible coherent family and \(M
   \subset \mathcal{M}\) be an infinite-dimensional simple submodule. Then
   \(\mathcal{M} \cong \mExt(M)\). In particular, all semisimple coherent