\chapter{Classification of Coherent Families}

% TODO: Write an introduction

% TODOOO: Is this decomposition unique??
\begin{proposition}
  Suppose \(\mathfrak{g} = \mathfrak{s}_1 \oplus \cdots \oplus \mathfrak{s}_r\)
  and let \(\mathcal{M}\) be a semisimple irreducible coherent
  \(\mathfrak{g}\)-family. Then there are semisimple irreducible coherent
  \(\mathfrak{s}_i\)-families \(\mathcal{M}_i\) such that
  \[
    \mathcal{M} \cong \mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r
  \]
\end{proposition}

\begin{proof}
  Suppose \(\mathfrak{h}_i \subset \mathfrak{s}_i\) are Cartan subalgebras,
  \(\mathfrak{h} = \mathfrak{h}_1 \oplus \cdots \oplus \mathfrak{h}_r\) and \(d
  = \deg \mathcal{M}\). Let \(M \subset \mathcal{M}\) be any
  infinite-dimensional simple submodule, so that \(\mathcal{M}\) is a
  semisimple coherent extension of \(M\). By
  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 \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\) 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.
  It thus suffices to show that \(\mathcal{M}\) is a coherent family. By
  Example~\ref{ex:supp-ess-of-tensor-is-product},
  \(\operatorname{supp}_{\operatorname{ess}} (\mathcal{M}_1 \otimes \cdots
  \otimes \mathcal{M}_r) = \mathfrak{h}^*\). To see that the map
  \begin{align*}
    \mathfrak{h}^* & \to K \\
    \lambda & \mapsto
    \operatorname{Tr}
    (
      u\! \restriction_{(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r)_\lambda}
    )
  \end{align*}
  is polynomial, notice that the natural isomorphism of algebras
  \begin{align*}
    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*}
  described in Example~\ref{ex:univ-enveloping-of-sum-is-tensor} is a
  \(\mathfrak{g}\)-homomorphism between the tensor product of the adjoint
  \(\mathfrak{s}_i\)-modules \(\mathcal{U}(\mathfrak{s}_i)\) and the adjoint
  \(\mathfrak{g}\)-module \(\mathcal{U}(\mathfrak{g})\).

  Indeed, given \(X = X_1 + \cdots + X_r \in \mathfrak{g}\) with \(X_i \in
  \mathfrak{s}_i\) and \(u_i \in \mathcal{U}(\mathfrak{s}_i)\),
  \[
    \begin{split}
      f(X \cdot (u_1 \otimes \cdots \otimes u_r))
      & = f([X_1, u_1] \otimes u_2 \otimes \cdots \otimes u_r)
        + \cdots
        + f(u_1 \otimes \cdots \otimes u_{r-1} \otimes [X_r, u_r]) \\
      & = [X_1, u_1] u_2 \cdots u_r + \cdots + u_1 \cdots u_{r-1} [X_r, u_r] \\
      \text{(\([X_i, u_j] = 0\) for \(i \ne j\))}
      & = [X_1, u_1u_2 \cdots u_r] + \cdots + [X_r, u_1 \cdots u_{r-1}u_r] \\
      & = [X, f(u_1 \otimes \cdots \otimes u_r)]
    \end{split}
  \]

  Hence by Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight}
  \(f\) restricts to an isomorphism of algebras \(\mathcal{U}(\mathfrak{s}_1)_0
  \otimes \cdots \otimes \mathcal{U}(\mathfrak{s}_r)_0 \isoto
  \mathcal{U}(\mathfrak{g})_0\) with image \(\mathcal{U}(\mathfrak{g})_0 =
  \mathcal{U}(\mathfrak{s}_1)_0 \cdots \mathcal{U}(\mathfrak{s}_r)_0\). More
  importantly, if we write \(\lambda = \lambda_1 + \cdots + \lambda_r\) for
  \(\lambda_i \in \mathfrak{h}_i^*\) it is clear from
  Example~\ref{ex:tensor-prod-of-weight-is-weight} that the
  \(\mathcal{U}(\mathfrak{g})_0\)-module \((\mathcal{M}_1 \otimes \cdots
  \otimes \mathcal{M}_r)_\lambda\) corresponds to exactly the
  \(\mathcal{U}(\mathfrak{s}_1)_0 \otimes \cdots \otimes
  \mathcal{U}(\mathfrak{s}_r)_0\)-module \((\mathcal{M}_1)_{\lambda_1} \otimes
  \cdots \otimes (\mathcal{M}_r)_{\lambda_r}\), so we can see that the value
  \[
    \operatorname{Tr}
    (
      u_1 \cdots u_r \!\restriction_{(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r)_\lambda}
    )
    = \operatorname{Tr}(u_1\!\restriction_{(\mathcal{M}_1)_{\lambda_1}})
      \cdots
      \operatorname{Tr}(u_r\!\restriction_{(\mathcal{M}_r)_{\lambda_r}})
  \]
  varies polynomially with \(\lambda \in \mathfrak{h}^*\) for all \(u_i \in
  \mathcal{U}(\mathfrak{s}_i)_0\).

  Finally, \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\) is a coherent
  extension of \(M\). Since the \(\mathcal{M}_i = \mExt(M_i)\) are semisimple,
  so is \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\). It thus
  follows from the uniqueness of semisimple coherent extensions that
  \(\mathcal{M} \cong \mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\).
\end{proof}

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 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}
  Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose
  there exists an irreducible coherent \(\mathfrak{s}\)-family. Then
  \(\mathfrak{s} \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong
  \mathfrak{sp}_{2n}(K)\) for some \(n\).
\end{corollary}

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.

% TODO: Note that we may take L(λ) with respect to any given basis
% TODO: Note beforehand that the construction of Verma modules and the notions
% of highest-weight modules in gerenal is relative on a choice of basis
\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}

\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 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?

  \item Given \(\lambda, \mu \in \mathfrak{h}^*\) with \(L(\lambda)\) and
    \(L(\mu)\) bounded, when is \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\)?
\end{enumerate}

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 \(\chi_\lambda :
  Z(\mathcal{U}(\mathfrak{g})) \to K\) such that \(u \cdot m = \chi_\lambda(u)
  m\) for all \(m \in M\) and \(u \in Z(\mathcal{U}(\mathfrak{g}))\) is called
  \emph{the central character of \(M\)} or \emph{the central character
  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 \(\mu \in W \bullet \lambda\).
\end{theorem}

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
  \(L(\mu)\) are both bounded and \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\).
  Then \(\chi_\lambda = \chi_\mu\). In particular, \(\mu \in W \bullet
  \lambda\).
\end{proposition}

\begin{proof}
  Fix \(u \in \mathcal{U}(\mathfrak{g})_0\). It is clear that
  \(\operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu}) =
  \operatorname{Tr}(u\!\restriction_{L(\lambda)_\nu}) = d \chi_\lambda(u)\) for
  all \(\nu \in \operatorname{supp}_{\operatorname{ess}} L(\lambda)\). Since
  \(\operatorname{supp}_{\operatorname{ess}} L(\lambda)\) is Zariski-dense and
  the map
  \(\nu \mapsto \operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu})\)
  is polynomial, it follows that
  \(\operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu}) = d
  \chi_\lambda(u)\) for all \(\nu \in \mathfrak{h}^*\). But by the same token
  \[
    d \chi_\lambda(u)
    = \operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu})
    = \operatorname{Tr}(u\!\restriction_{\mExt(L(\mu))_\nu})
    = d \chi_\mu(u)
  \]
  for any \(\nu \in \operatorname{supp}_{\operatorname{ess}} L(\mu)\) and thus
  \(\chi_\lambda(u) = \chi_\mu(u)\).
\end{proof}

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 establish 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
  \(L(\sigma_\beta \bullet \lambda) \subset \mExt(L(\lambda))\). In particular,
  if \(\sigma_\beta \bullet \lambda \notin P^+\) then \(L(\sigma_\beta)\) is a
  bounded infinite-dimensional \(\mathfrak{g}\)-module and
  \(\mExt(L(\sigma_\beta \bullet \lambda)) \cong \mExt(L(\lambda))\).
\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.

% TODO: Add a diagram of the locus of weights λ such that L(λ) is
% infinite-dimensional and bounded
\begin{example}
  Let \(\mathfrak{g} = \mathfrak{sl}_2(K)\). It follows from
  Example~\ref{ex:sl2-verma} that \(M(\lambda)\) is a bounded
  \(\mathfrak{sl}_2(K)\) of degree \(1\), so that \(L(\lambda)\) is bounded --
  with \(\deg L(\lambda) = 1\) -- for all \(\lambda \in K \cong
  \mathfrak{h}^*\). In addition, a simple calculation shows \(W \bullet
  \lambda = \{\lambda, - \lambda - 2\}\). This implies that if \(\lambda, \mu
  \notin P^+ = \mathbb{N}\) are such that \(\mExt(L(\lambda)) \cong
  \mExt(L(\mu))\) then \(\mu = \lambda\) or \(\mu = - \lambda - 2\). Finally,
  by Lemma~\ref{thm:lemma6.1} the converse also holds: if \(\lambda, - \lambda
  - 2 \notin P^+\) 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

Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sp}_{2n}(K)\)
of diagonal matrices, as in Example~\ref{ex:cartan-of-sp}, and the basis
\(\Sigma = \{\beta_1, \ldots, \beta_n\}\) for \(\Delta\) given by \(\beta_i =
\epsilon_i - \epsilon_{i+1}\) for \(i < n\) and \(\beta_n = 2 \epsilon_n\).
Here \(\epsilon_i : \mathfrak{h} \to K\) is the linear functional which yields
the \(i\)-th entry of the diagonal of a given matrix, as described in
Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
\cdots + \sfrac{1}{2} \beta_n\).

% TODO: Add some comments on the proof of this: verifying that these conditions
% are necessary is a purely combinatorial affair, while checking that these are
% sufficient involves some analysis envolving the Shane-Weil module
\begin{lemma}\label{thm:sp-bounded-weights}
  Then \(L(\lambda)\) is bounded if, and only if
  \begin{enumerate}
    \item \(\lambda(H_{\beta_i})\) is non-negative integer for all \(i \ne n\).
    \item \(\lambda(H_{\beta_n}) \in \frac{1}{2} + \mathbb{Z}\).
    \item \(\lambda(H_{\beta_{n - 1}} + 2 H_{\beta_n}) \ge -2\).
  \end{enumerate}
\end{lemma}

% TODO: Note that we need a better set of parameters to the space of weights
% such that L(λ) is bounded

% TODO: Revise the notation for this? I don't really like calling this
% bijection m
\begin{proposition}\label{thm:better-sp(2n)-parameters}
  The map
  \begin{align*}
    m : \mathfrak{h}^* & \to K^n \\
        \lambda &
        \mapsto
        (
          \kappa(\epsilon_1, \lambda+\rho),
          \ldots,
          \kappa(\epsilon_n, \lambda+\rho)
        )
  \end{align*}
  is \(W\)-equivariant bijection, where the action \(W \cong S_n \ltimes
  (\mathbb{Z}/2\mathbb{Z})^n\) on \(\mathfrak{h}^*\) is given by the dot action
  and the action of \(W\) on \(K^n\) is given my permuting coordinates and
  multiplying them by \(\pm 1\). A weight \(\lambda \in \mathfrak{h}^*\)
  satisfies the conditions of Lemma~\ref{thm:sp-bounded-weights} if, and
  only if \(m(\lambda)_i \in \sfrac{1}{2} + \mathbb{Z}\) for all \(i\) and
  \(m(\lambda)_1 > m(\lambda)_2 > \cdots > m(\lambda)_{n - 1} > \pm
  m(\lambda)_n\).
\end{proposition}

\begin{proof}
  The fact \(m : \mathfrak{h}^* \to K^n\) is a bijection is clear from the fact
  that \(\{\epsilon_1, \ldots, \epsilon_n\}\) is an orthonormal basis for
  \(\mathfrak{h}^*\). Veryfying that \(L(\lambda)\) is bounded if, and only if
  \(m(\lambda)_1 > m(\lambda)_2 > \cdots > m(\lambda)_{n - 1} > \pm
  m(\lambda)_n\) is also a simple combinatorial affair.

  The only part of the statement worth proving is the fact that \(m\) is an
  equivariant map, which is equivalent to showing the map
  \begin{align*}
    \mathfrak{h}^* & \to K^n \\
    \lambda &
    \mapsto (\kappa(\epsilon_1, \lambda), \ldots, \kappa(\epsilon_n, \lambda))
  \end{align*}
  is equivariant with respect to the natural action of \(W\) on
  \(\mathfrak{h}^*\). But this also clear from the isomorphism \(W \cong S_n
  \ltimes (\mathbb{Z}/2\mathbb{Z})^n\), as described in
  Example~\ref{ex:sp-weyl-group}: \((\sigma_i, (\bar 0, \ldots, \bar 0)) =
  \sigma_{\beta_i}\) permutes \(\epsilon_i\) and \(\epsilon_{i + 1}\) for \(i <
  n\) and \((1, (\bar 0, \ldots, \bar 0, \bar 1)) = \sigma_{\beta_n}\) flips
  the sign of \(\epsilon_n\). Hence \(m(\sigma_{\beta_i} \cdot \epsilon_j) =
  \sigma_{\beta_i} \cdot m(\epsilon_j)\) for all \(i\) and \(j\). Since \(W\)
  is generated by the \(\sigma_{\beta_i}\), this implies that the required map
  is equivariant.
\end{proof}

\begin{definition}
  We denote by \(\mathscr{B}\) the set of the \(m \in (\sfrac{1}{2} +
  \mathbb{Z})^n\) such that \(m_1 > m_2 > \cdots > m_{n - 1} > \pm m_n\). We
  also consider the canonical partition \(\mathscr{B} = \mathscr{B}^+ \cup
  \mathscr{B}^-\) where \(\mathscr{B}^+ = \{ m \in \mathscr{B} : m_n > 0 \}\)
  and \(\mathscr{B}^- = \{ m \in \mathscr{B} : m_n < 0\}\).
\end{definition}

\begin{theorem}[Mathieu]
  Given \(\lambda\) and \(\mu\) satisfying the conditions of
  Lemma~\ref{thm:sp-bounded-weights}, \(\mExt(L(\lambda)) \cong
  \mExt(L(\mu))\) if, and only if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and
  \(m(\lambda)_n = \pm m(\mu)_n\). In particular, the isomorphism classes of
  semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-families are
  parameterized by \(\mathscr{B}^+\).
\end{theorem}

\begin{proof}
  Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\),
  so that \(m(\lambda), m(\mu) \in \mathscr{B}\).

  Suppose \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). By
  Proposition~\ref{thm:coherent-family-has-uniq-central-char}, \(\chi_\lambda =
  \chi_\mu\). It thus follows from the Harish-Chandra Theorem that \(\mu \in W
  \bullet \lambda\). Since \(m\) is equivariant, \(m(\mu) \in W \cdot
  m(\lambda)\). But the only elements in of \(\mathscr{B}\) in \(W \cdot
  m(\lambda)\) are \(m(\lambda)\) and \((m(\lambda)_1, m(\lambda)_2, \ldots,
  m(\lambda)_{n-1}, - m(\lambda)_n)\).

  Conversely, if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and \(m(\mu)_n = -
  m(\lambda)_n\) then \(m(\mu) = \sigma_{\beta_n} \cdot m(\lambda)\) and \(\mu
  = \sigma_n \bullet \lambda\). Since \(m(\lambda) \in \mathscr{B}\),
  \(\lambda(H_{\beta_n}) \in \sfrac{1}{2} + \mathbb{Z}\) and thus
  \(\lambda(H_{\beta_n}) \notin \mathbb{N}\). Hence by
  Lemma~\ref{thm:lemma6.1} \(L(\mu) \subset \mExt(L(\lambda))\) and
  \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\).

  For each semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-family
  \(\mathcal{M}\) there is some \(m = m(\lambda) \in \mathscr{B}\) such that
  \(\mathcal{M} = \mExt(L(\lambda))\). The only other \(m' \in \mathscr{B}\)
  which generates the same coherent family as \(m\) is \(m' = \sigma_{\beta_n}
  \cdot m\). Since \(m\) and \(m'\) lie in different elements of the partition
  \(\mathscr{B} = \mathscr{B}^+ \cup \mathscr{B}^-\), the is a unique \(m'' =
  m(\nu) \in \mathscr{B}^+\) -- either \(m\) or \(m'\) -- such that
  \(\mathcal{M} \cong \mExt(L(\nu))\).
\end{proof}

\section{Coherent \(\mathfrak{sl}_n(K)\)-families}

% TODO: Fix n >= 3

Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sl}_n(K)\) of
diagonal matrices, as in Example~\ref{ex:cartan-of-sl}, and the basis \(\Sigma
= \{\beta_1, \ldots, \beta_{n-1}\}\) for \(\Delta\) given by \(\beta_i =
\epsilon_i - \epsilon_{i+1}\) for \(i < n\). Here \(\epsilon_i : \mathfrak{h}
\to K\) is the linear functional which yields the \(i\)-th entry of the
diagonal of a given matrix, as described in
Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
\cdots + \sfrac{1}{2} \beta_{n - 1}\).

% TODO: Add some comments on the proof of this: while the proof that these
% conditions are necessary is a purely combinatorial affair, the proof of the
% fact that conditions (ii) and (iii) imply L(λ) is bounded requires some
% results on the connected components of of the graph 𝓑  (which we will only
% state later down the line)
\begin{lemma}\label{thm:sl-bounded-weights}
  Let \(\lambda \notin P^+\) and \(A(\lambda) = \{ i : \lambda(H_{\beta_i})\
  \text{is \emph{not} a non-negative integer}\}\). Then \(L(\lambda)\) is
  bounded if, and only if one of the following assertions holds.
  \begin{enumerate}
    \item \(A(\lambda) = \{1\}\) or \(A(\lambda) = \{n - 1\}\).
    \item \(A(\lambda) = \{i\}\) for some \(1 < i < n - 1\) and \((\lambda +
      \rho)(H_{\beta_{i - 1}} + H_{\beta_i})\) or \((\lambda +
      \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive integer.
    \item \(A(\lambda) = \{i, i + 1\}\) for some \(1 \le i < n - 1\) and
      \((\lambda + \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive
      integer.
  \end{enumerate}
\end{lemma}

\begin{definition}
  A \emph{\(\mathfrak{sl}_n\)-sequence} \(m\) is a \(n\)-tuple \(m = (m_1,
  \ldots, m_n) \in K^n\) such that \(m_1 + \cdots + m_n = 0\).
\end{definition}

\begin{definition}
  A \(k\)-tuple \(m = (m_1, \ldots, m_k) \in K^k\) is called \emph{ordered} if
  \(m_i - m_{i + 1}\) is a positive integer for all \(i < k\).
\end{definition}

% TODO: Revise the notation for this? I don't really like calling this
% bijection m
\begin{proposition}\label{thm:better-sl(n)-parameters}
  The map
  \begin{align*}
    m : \mathfrak{h}^* &
        \to \{ \mathfrak{sl}_n\textrm{\normalfont-sequences} \} \\
        \lambda &
        \mapsto
        2n
        (
          \kappa(\epsilon_1, \lambda + \rho),
          \ldots,
          \kappa(\epsilon_n, \lambda + \rho)
        )
  \end{align*}
  is \(W\)-equivariant bijection, where the action \(W \cong S_n\) on
  \(\mathfrak{h}^*\) is given by the dot action and the action of \(W\) on the
  space of \(\mathfrak{sl}_n\)-sequences is given my permuting coordinates. A
  weight \(\lambda \in \mathfrak{h}^*\) satisfies the conditions of
  Lemma~\ref{thm:sl-bounded-weights} if, and only if \(m(\lambda)\) is
  \emph{not} ordered, but becomes ordered after removing one term.
\end{proposition}

% TODO: Note beforehand that κ(H_α, ⋅) is always a multiple of α. This is
% perhaps better explained when defining H_α
The proof of this result is very similar to that of
Proposition~\ref{thm:better-sp(2n)-parameters} in spirit: the equivariance of
the map \(m : \mathfrak{h}^* \to \{ \mathfrak{sl}_n\textrm{-sequences} \}\)
follows from the nature of the isomorphism \(W \cong S_n\) as described in
Example~\ref{ex:sl-weyl-group}, while the rest of the proof amounts to simple
technical verifications. The number \(2 n\) is a normalization constant chosen
because \(\lambda(H_\beta) = 2 n \, \kappa(\lambda, \beta)\) for all \(\lambda
\in \mathfrak{h}^*\) and \(\beta \in \Sigma\). Hence \(m(\lambda)\) is uniquely
characterized by the property that \((\lambda + \rho)(H_{\beta_i}) =
m(\lambda)_i - m(\lambda)_{i+1}\) for all \(i\), which is relevant to the proof
of the equivalence between the contiditions of
Lemma~\ref{thm:sl-bounded-weights} and those explained in the last statement of
Proposition~\ref{thm:better-sl(n)-parameters}.

% TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose
% union corresponds to condition (i)
\begin{definition}
  We denote by \(\mathscr{B}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\)
  which are \emph{not} ordered, but becomes ordered after removing one term. We
  also consider the \emph{extremal} subsets \(\mathscr{B}^+ = \{m \in
  \mathscr{B} : (\widehat{m_1}, m_2, \ldots, m_n) \ \text{is ordered}\}\) and
  \(\mathscr{B}^- = \{m \in \mathscr{B} : (m_1, \ldots, m_{n - 1},
  \widehat{m_n}) \ \text{is ordered}\}\).
\end{definition}

% TODO: Add a picture of parts of 𝓑  for n = 3 in here

% TODO: Explain that for each m ∈ 𝓑  there is a unique i such that so that
% m_i - m_i+1 is not a positive integer. For m ∈ 𝓑 + this is i = 1, while for
% m ∈ 𝓑 - this is i = n-1
The issue here is that the relationship between \(\lambda, \mu \in P^+\) with
\(m(\lambda), m(\mu) \in \mathscr{B}\) and \(\mExt(L(\lambda)) \cong
\mExt(L(\mu))\) is more complicated than in the case of \(\mathfrak{sp}_{2
n}(K)\). Nevertheless, Lemma~\ref{thm:lemma6.1} affords us a criteria for
verifying that \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). For \(\sigma =
\sigma_i\) and the weight \(\lambda + \rho\), the hypothesis of
Lemma~\ref{thm:lemma6.1} translates to \(m(\lambda)_i - m(\lambda)_{i+1} =
(\lambda + \rho)(H_{\beta_i}) \notin \mathbb{N}\). If \(m(\lambda) \in
\mathscr{B}\), this is equivalent to requiring that \(m(\lambda)\) is not
ordered, but becomes ordered after removing its \(i\)-th term. This discussions
losely inspires the following definition, which endows the set \(\mathscr{B}\)
with the structure of a directed graph.

\begin{definition}
  Given \(m, m' \in \mathscr{B}\), say there is an arrow \(m \to m'\) if there
  some \(i\) such that \(m_i - m_{i + 1}\) is \emph{not} a positive integer and
  \(m' = \sigma_i \cdot m\).
\end{definition}

It should then be obvious from Lemma~\ref{thm:lemma6.1} that\dots

\begin{proposition}\label{thm:arrow-implies-ext-eq}
  Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that
  \(m(\lambda) \in \mathscr{B}\) -- and suppose that \(\mu \in \mathfrak{h}^*\)
  is such that \(m(\mu) \in \mathscr{B}\) and there is an arrow \(m(\lambda)
  \to m(\mu)\). Then \(L(\mu)\) is also bounded and \(\mExt(L(\mu)) \cong
  \mExt(L(\lambda))\).
\end{proposition}

A weight \(\lambda \in \mathfrak{h}^*\) is called \emph{regular} if \((\lambda
+ \rho)(H_\alpha) \ne 0\) for all \(\alpha \in \Delta\). In terms of
\(\mathfrak{sl}_n\)-sequences, \(\lambda\) is regular if, and only if
\(m(\lambda)_i \ne m(\lambda)_j\) for all \(i \ne j\). It thus makes sence to
call a \(\mathfrak{sl}_n\)-sequence regular or singular if \(m_i \ne m_j\) for
all \(i \ne j\) or \(m_i = m_j\) for some \(i \ne j\), respectively. Similarly,
\(\lambda\) is integral if, and only if \(m(\lambda)_i - m(\lambda)_j \in
\mathbb{Z}\) for all \(i\) and \(j\), so it makes sence to call a
\(\mathfrak{sl}_n\)-sequence \(m\) integral if \(m_i - m_j \in \mathbb{Z}\) for
all \(i\) and \(j\).

% TODO: Restate the notation for σ_i beforehand
\begin{lemma}\label{thm:connected-components-B-graph}
  The connected component of some \(m \in \mathscr{B}\) is given by the
  following.
  \begin{enumerate}
    \item If \(m\) is regular and integral then there exists\footnote{Notice
      that in this case $m' \notin \mathscr{B}$, however.} a unique ordered
      \(m' \in W \cdot m\), in which case the connected component of \(m\) is
      given by
      \[
        \begin{tikzcd}[cramped, row sep=small]
          \sigma_1 \sigma_2 \cdots \sigma_i \cdot m'           \rar &
          \sigma_2 \cdots \sigma_i \cdot m'                    \rar &
          \cdots                                               \rar &
          \sigma_{i-1} \sigma_i \cdot m'
            \ar[rounded corners,
                to path={ -- ([xshift=4ex]\tikztostart.east)
                          |- (X.center) \tikztonodes
                          -| ([xshift=-4ex]\tikztotarget.west)
                          -- (\tikztotarget)}]{dlll}[at end]{}      \\
          \sigma_i \cdot m'                                         &
          \sigma_{i+1} \sigma_i \cdot m'                       \lar &
          \cdots                                               \lar &
          \sigma_{n-1} \cdots \sigma_i \cdot m'                \lar &
        \end{tikzcd}
      \]
      for some unique \(i\), with \(\sigma_1 \cdots \sigma_i \cdot m' \in
      \mathscr{B}^+\) and \(\sigma_{n-1} \cdots \sigma_i \cdot m' \in
      \mathscr{B}^-\).

    % TODOOO: What happens when i = 1?? Do we need to suppose i > 1?
    % TODO: For instance, consider m = (1, 1, -2)
    \item If \(m\) is singular then there exists unique \(m' \in W \cdot m\)
      and \(i\) such that \(m'_i = m'_{i + 1}\) and \((m_1', \cdots, m_{i-1}',
      \widehat{m_i'}, m_{i + 1}', \ldots, m_n')\) is ordered, in which
      case the
      connected component of \(m\) is given by
      \[
        \begin{tikzcd}[cramped, row sep=small]
          \sigma_1 \sigma_2 \cdots \sigma_{i-1} \cdot m'       \rar &
          \sigma_2 \cdots \sigma_{i-1} \cdot m'                \rar &
          \cdots                                               \rar &
          \sigma_{i-1} \cdot m'
            \ar[rounded corners,
                to path={ -- ([xshift=4ex]\tikztostart.east)
                          |- (X.center) \tikztonodes
                          -| ([xshift=-4ex]\tikztotarget.west)
                          -- (\tikztotarget)}]{dlll}[at end]{}      \\
          m'                                                        &
          \sigma_{i+1} \cdot m'                                \lar &
          \cdots                                               \lar &
          \sigma_{n-1} \cdots \sigma_{i+1} \cdot m'            \lar &
        \end{tikzcd}
      \]
      with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathscr{B}^+\) and
      \(\sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \in \mathscr{B}^-\).

    \item If \(m\) is non-integral then there exists unique \(m' \in W \cdot
      m\) with \(m' \in \mathscr{B}^+\), in which case the connected component
      of \(m\) is given by
      \[
        \begin{tikzcd}[cramped]
          m'                                    \rar      &
          \sigma_1 \cdot m'                     \rar \lar &
          \sigma_2 \sigma_1 \cdot m'            \rar \lar &
          \cdots                                \rar \lar &
          \sigma_{n-1} \cdots \sigma_1 \cdot m'      \lar &
        \end{tikzcd}
      \]
      with \(\sigma_{n-1} \cdots \sigma_1 \cdot m' \in \mathscr{B}^-\).
  \end{enumerate}
\end{lemma}

% TODO: Notice that this gives us that if m(λ)∈ 𝓑  then L(λ) is bounded: for λ
% ∈ 𝓑 + ∪ 𝓑 - we stablish this by hand, and for the general case it suffices to
% notice that there is always some path μ → ... → λ with μ ∈ 𝓑 + ∪ 𝓑 -

\begin{theorem}[Mathieu]
  Given \(\lambda, \mu \notin P^+\) with \(L(\lambda)\) and \(L(\mu)\) bounded,
  \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and
  \(m(\mu)\) lie in the same connected component of \(\mathscr{B}\). In
  particular, the isomorphism classes of semisimple irreducible coherent
  \(\mathfrak{sl}_n(K)\)-families are parameterized by the set
  \(\pi_0(\mathscr{B})\) of the connected components of \(\mathscr{B}\), as
  well as by \(\mathscr{B}^+\).
\end{theorem}

\begin{proof}
  Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\),
  so that \(m(\lambda), m(\mu) \in \mathscr{B}\).

  It is clear from Proposition~\ref{thm:arrow-implies-ext-eq} that if
  \(m(\lambda)\) and \(m(\mu)\) lie in the same connected component of
  \(\mathscr{B}\) then \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). On the other
  hand, if \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) then \(\chi_\lambda =
  \chi_\mu\) and thus \(\mu \in W \bullet \lambda\). We now investigate which
  elements of \(W \bullet \lambda\) satisfy the conditions of
  Lemma~\ref{thm:sl-bounded-weights}. To do so,  we describe the set
  \(\mathscr{B} \cap W \cdot m(\lambda)\).

  % Great migué
  If \(\lambda\) is regular and integral then the only permutations of
  \(m(\lambda)\) which lie in \(\mathscr{B}\) are \(\sigma_k \sigma_{k+1}
  \cdots \sigma_i \cdot m'\) for \(k \le i\) and \(\sigma_k \sigma_{k-1} \cdots
  \sigma_i \cdot m'\) for \(k \ge i\), where \(m'\) is the unique ordered
  element of \(W \cdot m(\lambda)\). Hence by
  Lemma~\ref{thm:connected-components-B-graph} \(\mathscr{B} \cap W \cdot
  m(\lambda)\) is the union of the connected components of the \(\sigma_i \cdot
  m'\) for \(i \le n\). On the other hand, if \(\lambda\) is singular or
  non-integral then the only permutations of \(m(\lambda)\) which lie in
  \(\mathscr{B}\) are the ones from the connected component of \(m(\lambda)\)
  in \(\mathscr{B}\), so that \(\mathscr{B} \cap W \cdot m(\lambda)\) is
  exactly the connected component of \(m(\lambda)\).

  In both cases, we can see that if \(B(\lambda)\) is the set of the \(m' =
  m(\mu) \in \mathscr{B}\) such that \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\)
  then \(B(\lambda) \subset \mathscr{B} \cap W \cdot m(\lambda)\) is contain in
  a union of connected components of \(\mathscr{B}\) -- including that of
  \(m(\lambda)\) itself. We now claim that \(B(\lambda)\) is exactly the
  connected component of \(m(\lambda)\). This is already clear when \(\lambda\)
  is singular or non-integral, so we may assume that \(\lambda\) is regular and
  integral, in which case every other \(\mu \in W \bullet \lambda\) is regular
  and integral.

  % Great migué
  In this situation, \(m(\mu) \in \mathscr{B}^+\) implies \(\mu(H_{\beta_1}) =
  m(\mu)_1 - m(\mu)_2 \in \mathbb{Z}\) is negative. But it follows from
  Lemma~\ref{thm:lemma6.1} that for each \(\beta \in \Sigma\) there is at
  most one \(\mu \notin P^+\) with \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\)
  such that \(\mu(H_\beta)\) is a negative integer -- see Lemma~6.5 of
  \cite{mathieu}. Hence there is at most one \(m' \in \mathscr{B}^+ \cap W
  \cdot m(\lambda)\). Since every connected component of \(\mathscr{B}\) meets
  \(\mathscr{B}^+\) -- see Lemma~\ref{thm:connected-components-B-graph} -- this
  implies \(B(\lambda)\) is precisely the connected component of
  \(m(\lambda)\).

  Another way of putting it is to say that \(\mExt(L(\lambda)) \cong
  \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and \(m(\mu)\) lie in the same
  connected component -- which is, of course, precisely the first part of our
  theorem! There is thus a one-to-one correspondance between
  \(\pi_0(\mathscr{B})\) and the isomorphism classes of semisimple irreducible
  coherent \(\mathfrak{sl}_n(K)\)-families. Since every connected component of
  \(\mathscr{B}\) meets \(\mathscr{B}^+\) precisely once -- again, see
  Lemma~\ref{thm:connected-components-B-graph} -- we also get that such
  isomorphism classes are parameterized by \(\mathscr{B}^+\).
\end{proof}

% TODO: Change this
% I don't really think these notes bring us to this conclusion
% If anything, these notes really illustrate the incredible vastness of the
% ocean of representation theory, how unknowable it is, and the remarkable
% amount of engenuity required to explore it
This construction also brings us full circle to the beginning of these notes,
where we saw in Proposition~\ref{thm:geometric-realization-of-uni-env} that
\(\mathfrak{g}\)-modules may be understood as geometric objects. In fact,
throughout the previous four chapters we have seen a tremendous number of
geometrically motivated examples, which further emphasizes the connection
between representation theory and geometry. I would personally go as far as
saying that the beautiful interplay between the algebraic and the geometric is
precisely what makes representation theory such a fascinating and charming
subject.

Alas, our journey has come to an end. All it is left is to wonder at the beauty
of Lie algebras and their representations.

\label{end-47}
